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Design, Development and Control of Mobile

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1. e 4 3 External Sensing Architecture o eee ee eee vi 66 66 67 67 69 73 75 77 TT 79 86 96 5 Control 110 SL Introductio enges tee A te a a ala aa 110 5 2 Ground Based Control eos wee ear oe te Mo are ee A 111 Sa Sphere Based Controls 44 46 8 XS Aw 46S OO t 116 5 3 1 Power Optimal Control Fixed Final Time Unconstrained Input 117 5 3 2 Power Optimal Control Free Final Time Constrained Input 120 5 3 3 LQR Overview a 66 ost oe ed Bk ee bo 124 534 Stable LOR S srt un Ser see Sw Se ds A ed ac 126 5 3 9 A E a 130 6 Estimation 137 6 1 a A IS Be Bhs 137 6 2 Continuous Discrete Extended Kalman Filter 138 6 3 Application of Continuous Discrete EKF 140 6 4 Sampling Time Analysis for C D EKF 148 7 Implementation 153 Tal Jitrod ctio nerden pod era a A He EGD AE Se Ra 153 7 2 Implementation Challenges aoa aa a 154 7 3 Ground Based Implementation 158 7 4 Balancing Implementation as a ah oo al ach a 169 8 Conclusions and Future Work 176 Bibliography 181 A Supplementary Concepts 184 Alo Tntrod ctio as 3 g aie ye he oe Be Se ba Be Le IAI E 184 vil nd A he how ol ee Sow ok Ba ed ee Sg 184 A 3 Translational Motion 2d hoe Sk ewe Ge Le BE 186 Assi Actuation e sci hae sae ee AE Jk RS Se Oe ae t 188 A 5 Control Superposition 12 ebrios Qe atrae 190 Pr A tice oO sence a Re Oe Yee ie Wt AOS He IOS GK eS 192
2. LQR Ground Track With P Yaw 20 pi Retrograde f 1 f f f fi f f f f f 05 0 4 0 3 0 2 01 0 01 02 03 04 05 x position m c Circle trajectory tracking with propor d Circle trajectory tracking with propor tional yaw control for 107 retrograde norm tional yaw control for 207 retrograde norm error 0 7973 error 1 3933 Figure 7 8 Circle reference trajectory tracking while using proportional yaw control for various retrogrades 167 Jonathan Missel Ground Based Implementation To analyze how prescribing both trajectories affected performance experiments were run on the standardized test trajectory with an increasing number of retrograde yaw rotations Figure 7 8 demonstrates how the norm error increased with the number of retrograde rotations As mentioned earlier this is because control is held constant between sampling times As the body rotates directional translation skews and a straight trajectory in the body frame becomes a curved path in the inertial frame The greater this curvature becomes the further out the track will move radially for retrograde rotation If the yaw trajectory were in the other direction the track plot would drift inward as the number of rotations increased The two conservative trajectories did not yield much more error than the yaw fixed case but in Figure 7 8 c and 7 8 d where the rotations were much faster there was appreciable error The plots of yaw measurement and tra
3. 0 01 T T T T T T T T _ 0 0051 02 T D OF amp 0 005 E La fi 1 i 1 i 1 281 4 6 8 10 12 14 16 18 20 Time 0 01 T T T T T T T _ 0 005 2 9 0 amp 0 005 E i 1 1 1 oui 4 6 8 10 12 14 16 18 20 Time Figure 6 4 C D EKF angle state errors LQR Track with EKF Rate Error 27 Sphere Rotation 0 2 T T T T T T T T o T O ate Tadians sec o o N o gt o o N T o a T gt ate fadians sec o N o o Time Figure 6 5 C D EKF rate state errors 147 Jonathan Missel Sampling Time Analysis for C D EKF that the estimated response has effectively represented the true response and filtered the noise For further comparison the errors between the true and estimated states were plotted with their respective 3d bounds Figures 6 4 and 6 5 show that the errors were small and well contained by the bounds therefore the filter was well tuned and succeeded in improving upon the measurement data 6 4 Sampling Time Analysis for C D EKF In some applications if the measurement sampling time is well known the filter can be run multiple times between measurements The objective of this is to try and improve the accuracy of the estimate without requiring more measurements When executed the update portion of the filter is not used unless a measurement is available at that time step In the absence of a measurement the propagation from t
4. 0 015 ang rate control 0 1 E 3 4 5 6 7 8 9 10 Figure 5 11 Response with stabilizing LQR controller wrt vertical equilibrium but requires about 4 5 seconds to do so To mitigate the undesired characteristics the A matrix was augmented This effectively includes an additional component to the cost function that continues to penalize the cost with time Augmentation for this system assumed the following form Ag SA 5 21 where A 005 and is the identity matrix This augmented A matrix was applied in the LQR procedure to generate new gains Conservative values were necessary for the augmentation because it is used to manipulate the state space model which is al ready a linearized representation of the full equations of motion Small augmentation prevented extensive departure from the real system and diminished validity of the controller The Q matrix weightings required retuning for the augmented controller 128 Jonathan Missel Sphere Based Control 6 radians 6 dot rad s 0 03 o a iS o Time radians 2 o Q o Time 1 o o a o dot rad s S N Time State response with augmented stabilizing LQR controller Vertigo Response 0 2 4 6 8 10 Time Figure 5 12 0 02 0 015 A 0 01F Magnitude wrt Vertical radian units Figure 5 13 Response with augmented sta
5. 144 Jonathan Missel Application of Continuous Discrete EKF States ratian units States ratian units LQR Track with EKF True State Response 2x Sphere Rotation 8 T T T T T T T T T 6 4 2 0 2 4 theta 6 phi L theta rate phi rate 8 1 1 1 L 1 T 0 2 4 6 8 10 12 14 16 18 20 Time Figure 6 2 C D EKF true state response LQR Track with EKF Estimated State Response 2x Sphere Rotation 8 T T T T T T T T T 6 4 2 0 2 4 theta 6 phi L theta rate phi rate 8 1 1 1 L T 0 2 4 6 8 10 12 14 16 18 20 Time Figure 6 3 C D EKF estimated state response 145 Jonathan Missel Application of Continuous Discrete EKF the cycle is repeated in the next time step This filter was constructed and applied in simulation to ensure it was properly incorporated Not all uncertainties are readily determined in the continuous discrete nonlinear problem so tuning of the Q and R matrices was required to improve the estimator results after it was applied This was a sensitive procedure for the extended Kalman filter because factors such as linearization errors are not easily quantified so discerning their influence took time A baseline for this tuning was formed with knowledge of the noise in the process and measurement from there refinement was done by trial and error In the implemented simulati
6. Program from 0x01fdf000 0x01fff000 at Ox60fe0000 RedBoot gt fis create b 0x80000 1 0x180000 e 0x80000 zImage Erase from 0x60340000 0x604c0000 Program from 0x00080000 0x00200000 at 0x60340000 Erase from 0x60fe0000 0x61000000 Program from 0x01fdf000 Ox01fff000 at Ox60fe0000 Now we ll load the optfs image into flash The command to load the optfs image differs slightly depending on which Qwerk revision you have The Qwerk revision is printed on the board between the analog inputs terminal and the top of the enclosure It should read either Qwerk 1 1A c 2006 or Qwerk 1 2A c 2006 Determine which version you have and then use the appropriate command below to load the optfs image into flash If you have a revision 1 1 qwerk use this command don t worry if the numbers in the output differ RedBoot gt fis create b 0x1000000 1 0x300000 optfs Erase from 0x603c0000 0x606cO0000 ee Program from 0x01000000 0x01300000 at Ox603C0000 ee Erase from 0x607e0000 0x60800000 Program from 0x01fdf000 Ox01fff000 at Ox607e0000 RedBoot gt If you have a revision 1 2 qwerk use this command don t worry if the numbers in the output differ RedBoot gt fis create b 0x1000000 1 0xb00000 optfs Erase from 0x604c0000 0x60fcO000 0 0 eee 234 Jonathan Missel Changing Qwerk s Embedded Code Program from 0x01000000 0x01b00000
7. Response with augmented stabilizing LQR controller wrt vertical State response with tracking LQR controller heavily weighted rates Response with tracking LQR controller wrt vertical heavily weighted Tales iras a a eet a GY che St E ee eet State response with tracking LQR controller heavily weighted angles Response with tracking LQR controller wrt vertical heavily weighted PINS IC SS tl See i hit di o AE RN E Ale xi 95 113 115 116 5 18 5 19 5 20 5 21 6 1 6 2 6 3 6 4 6 5 6 6 6 7 6 8 6 9 6 10 Gl 152 7 3 7 4 State response with augmented tracking LQR controller heavily weighted a IN 135 Response with augmented tracking LQR controller wrt vertical heavily weighted angles eur sa gora aa A 135 State response with augmented tracking LQR controller realistic weight EN EEEE A EE TA 136 Response with augmented tracking LQR controller wrt vertical real ISO WEIB RCNE i p gua ws Se el nite o Sk Soe Ser de Beas sae 136 Planar measurement model for sphere based Vertigo 141 C D EKF true state response e 20004 145 C D EKF estimated state response 04 145 C D EKF angle state errors os hsb o AS 147 C D EKF rate state errors Lea 2 eo oe 147 C D EKF true state response fm 25Hz fs 100Hz 149 C D EKF estimated state response fm 25Hz fs 100Hz 149 C D EKF angle state errors fm 25Hz f 100Hz 150 C D EKF rate stat
8. 227 Jonathan Missel Changing Qwerk s Embedded Code G 38400 W 1 H 57600 X 2 I 115200 Q 8 N 1 J 230400 R 7 E 1 Choice or lt Enter gt to exit Enter H to select 57600 baud then Q to select 8 N 1 As you do so you should see the line at the top which displays the current settings change When you re done type ENTER to close the Comm Parameters menu You should now see that the Bps Par Bits value has changed A Serial Device dev ttyS0 B Lockfile Location var lock C Callin Program D Callout Program E Bps Par Bits 57600 8N1 F Hardware Flow Control Yes G Software Flow Control No Change which setting Finally type F to turn off Hardware Flow Control The settings should now look like this 228 Jonathan Missel Changing Qwerk s Embedded Code A Serial Device dev ttyS0 B Lockfile Location var lock C Callin Program D Callout Program E Bps Par Bits 57600 8N1 F Hardware Flow Control No G Software Flow Control No Change which setting Type ENTER to exit this menu Now that we re back at the configuration menu use the arrow keys to highlight the Save setup as dfl option and type ENTER to select it Now use the arrow keys to highlight the Exit option and type ENTER to select it Finally quit minicom by typing CRTL A x Minicom is now configured to
9. Also the framework of implementing LQR meant that measurements had to be related to the mathematical states of the system This used a geometric measurement model that relied on how accurately the Vicon object was defined to represent the physical system Errors from this were apparent because the performance of the controller quickly worsened as balance was attempted further from the origin where the uncertainties were magnified Based on the success of the estimation simulations the continuous discrete ex tended Kalman filter seemed well suited to accompany LQR control and solve the problems with noise modeling error and measurement error However three main issues prevented this from being successful The first was with initialization Man ually placing Vertigo close enough to the filters initial conditions was difficult and often caused immediate failure Also modifying the filter to account for larger ini tialization errors diminished its effectiveness The second problem was that Vicon 174 Jonathan Missel Balancing Implementation did not always define the object well enough This had similar repercussions to the initialization error The third and most influential problem was the inconsistent loop sampling time as signals navigated the tortuous communication architecture EKF re quires knowledge of the sampling time for tuning and to propagate a priori estimates for the next time step When sampling time varies propagations ar
10. Figure 5 20 State response with augmented tracking LQR controller realistic weight ing Augmented LQR Track Response 2pi rotation k 1 o for T o o T e ES o N Magnitude wrt Vertical radian units o H 0 4 0 6 ang 0 87 rate control lt A T 1 1 J 0 2 4 6 8 10 12 14 16 18 20 Time Figure 5 21 Response with augmented tracking LQR controller wrt vertical realistic weighting 136 Chapter 6 Estimation 6 1 Introduction As an unstable system Vertigo requires very responsive and accurate control to op erate successfully In developing feedback control techniques it was assumed that perfectly accurate measurements were available for all states and times however this is not realistic when considering implementation on the actual system Once all functions of Qwerk are made available information on all states can be measured directly Until then the Vicon system had to be used for external measurement and none of the states are measured directly Even if full state sensing were available all instruments are rated with a level of accuracy which means their measurements are inherently erroneous The focus of this chapter is to establish a method for extracting reliable state information that can be fed back into the control loop Here the continuous discrete extended Kalman filter was developed and simulated for Vertigo One of the dominant cha
11. Technology Video capture Up to 640 x 480 Still image capture Up to 1 3 megapixel with software enhancement e Built in microphone with RightSound Technology Frame rate 30 frames per second USB 2 0 certified 2 4 4 Power Over the course of this project Vertigo has been powered by a few different methods Manufacturer defects made finding a suitable power supply that lasted challenging The criteria for selecting a power supply were based on size cost and the demands of the equipment For wireless autonomy Vertigo needed to be battery powered and lithium polymer LiPo batteries have the most attractive capacity size and cost combination for this application Based on the specifications of Qwerk and the other electronic components it was determined that the power supply should provide roughly 12 VDC to the system The amperage required varies greatly depending on 37 Jonathan Missel Structure the usage but an estimate of about 2 6 A hr was determined from the component specifications With this knowledge packs of 2 Zippy R 4150 mAh 11 1 V three cell LiPo batteries were chosen Wired in parallel with a Y connector a pairs of these power bricks should last for more than 3 hours of constant operation Two of these brick bundles were purchased so that a fully charged pair would always be on hand They take about 1 5 hours each to recharge so even if Vertigo is constantly in use the other pack will have time to rech
12. To Out 3 Amplitude To Out 1 To Out 2 To Out 4 To Out 3 To Out 2 To Out 1 To Out 4 Linear Simulation Resutts 200 200 200 200 500 500 500 500 0 0 2 0 4 0 6 0 5 1 1 2 14 Time sec Figure 5 5 Gramian based control u max 55 Linear Simulation Results 200 200 200 200 500 500 500 500 0 Time sec Figure 5 6 Gramian based control u max 45 121 Jonathan Missel Sphere Based Control Amplitude Amplitude To Out 3 To Out 2 To Out 1 To Out 4 To Out 1 To Out 2 Linear Simulation Resutts 100 100 100 100 500 500 500 500 Time sec Figure 5 7 Gramian based control u max 20 Linear Simulation Results 1000 0 5 1 15 2 25 3 Time sec 0 Figure 5 8 Gramian based control u max 15 122 Jonathan Missel Sphere Based Control u_max vs tf_min y 31 831x 260 86x 815 59x 1171 1x 677 66 60 50 40 kj 30 a 20 10 0 1 1 2 1 4 1 6 1 8 2 22 2 4 2 6 tf_min sec Figure 5 9 Plot of successful u max convergences and associated final times In Figure 5 8 when Uma is reduced to 15 the minimum time is tf 2 5143 but the states do not even approach their specified location In fact it was found that for any values below Umar 15 the code would not converge due to singularity errors Even for Umar 15 many attempts and mani
13. and be appropriately compact To satisfy this exacting list Faulhaber 2232V0050 gear head DC motors with embedded quadrature encoders were eventu ally called upon after extensively researching the market Motor specifications 22 e Motors Nominal voltage 6 V Terminal resistance lt 8 Q Max output lower 11 W Max efficiency 86 56 Jonathan Missel Design Iteration Max speed 8 000 rpm Max current 035 A Stall torque gt 60 mNm Permissible torque 10 mNm Speed constant 1 200 rpm V Back EMF constant 8 mV rpm Mechanical time constant 6 ms Rotor inertia 3 8 gem Max angular acceleration 120 10 rad s e Planetary Gearhead Gear ratio 20 1 All metal geartrain and housing Backlash lt 1 Preloaded bearings e Encoders Liners per revolution 512 Supply voltage 5 5 VDC Max frequency 160 kHz Code disc inertia 09 gem These outstanding motor packages offered plenty of torque and speed for this ap plication and were only half the volume of the 1 0 generation motor encoder assembly 57 Jonathan Missel Design Iteration Additionally their stainless steel planetary gears output the driveshaft concentrically with the motor casing At the opposite end the embedded encoders were secured up against the motor casing They were completely enclosed and conveniently fitted with a ribbon that not onl
14. controllability and observability matrices insight was gained as to how states influ enced their condition This also guided the hardware assembly configuration to help achieve desired performance Future work on mathematical modeling should focus on further improving accuracy in matching the physical system as this is a suspect for implementation difficulty Deriving the equations with various methods and as sumptions should give an idea of how to better represent the dynamics Also system identification techniques could be employed to extract parameters from the physical response These efforts would almost certainly enhance implementation success and the validity of simulations 177 Jonathan Missel Conclusions and Future Work Establishing the communication architecture was a laborious process requiring the use of five different coding languages Eventually an external sensing loop was closed where Vicon measurements were sent to Simulink and then passed on to MATLAB to be conditioned for feedback in to a control loop The output of this loop was sent to Qwerk through a Java based wireless connection Qwerk received this control signal as input to its own PID control loop that tracked a velocity trajectory Output from this embedded controller was sent to the motors and their back EMF measurements were fed back to the PID loop as they carried out the control In the future this network will be greatly simplified because a new versio
15. it must be rotated by a torque T Fl If the control input is a force F at the base then the length l is the lever arm for the torque When the force is applied both linear and rotational motion will ensue at the center of gravity A longer lever arm means less control force is required to impart the same torque Additionally smaller F will result in smaller linear motion so the pendulum will sway less when balancing This is easily seen when balancing a baseball bat or broom it is much less challenging when the heavier end is at the top Similarly the rotational inertia has an effect on the system If the mass is more concentrated the rotational inertia is reduced and less input is required to rotate the pendulum 197 Appendix B Powering Vertigo and Qwerk Through the evolution of Vertigo and as its demands changed several powering methods were used Currently only two remain in use e LiPo battery packs e Tethered power supply LiPo Battery Pack Zippy R 4150 mAh 11 1 V 3slp X2 This method is best for mobile powering There are two identical battery packs allowing a charged pack to be on hand at all times Both packs bundle two 4150 mAh 11 1 V bricks that are charged separately see charging instructions below They are connected in parallel so they deliver their original voltage for twice as long The batteries were fitted with Deans connectors that enable quick secure and high per formance connection to the Y
16. matrices The condition number of a matrix is the largest singular value divided by the smallest singular value The singular values of a matrix A mxn Where m n 80 Jonathan Missel Analysis Observability Rank phi radians a theta radians Figure 3 4 Rank of observability matrix while varying 0 and Q are the positive roots of the eigenvalues of 44 A Matrix A is non singular if and only if iff all of its singular values are greater than zero It can be shown that if C can be put in to Control Canonical Form CCF it is completely controllable in order to do so it must be nonsingular because its inverse must be taken This is also true of O for the Observer Canonical Form OCF Therefore the condition number provides a sensitivity bound for the solution of linear equations of the matrix By definition the condition number is always greater than or equal to one so calculating the inverse of the condition number gives an easy to visualize scaled measure between one and zero If the inverse of the condition number is close to one the matrix is said to be well conditioned This means that the inverse can be computed with high accuracy A smaller inverse condition number means that the matrix is nearing singularity and computation of its inverse or the solution to linear systems of equations is prone to large numerical errors If the inverse of the condition number is exactly zero that corresponds to a condition
17. sphere of infinite radius Being able to drive the platform using two completely different methods is very attractive to the overall goal of experimental diversity and adaptability This method does however have disadvantages As with concept A this needs passive contact with the sphere to help support the robot because the two actuators cannot do this alone These two actuators each control their respective 24 Jonathan Missel Actuation Method degree of freedom and do not allow for direct control in yaw which is not pertinent to omni directionality but would improve the mobility of the system The remaining two concepts solve the problem of including yaw along with omni directional actuation Keeping to the theme of one actuator per degree of freedom Figure 2 2 c shows how concept C proposes to completely decouple control in all three degrees of freedom by using three orthogonal actuators Here roll is controlled by an omni wheel on the equator pitch is controlled at the top of the sphere and yaw is also controlled at the equator exactly opposite to the one that controls roll Two variations exist for this configuration the yaw omni wheel can be placed 90 degrees from where it is either direction along the equator of the sphere These configurations put an actuator at each apex of a spherical quadrant and orient them orthogonally in Euclidian space The omni wheel orientation is critical to this approach because each location depen
18. tion error was fed into an LQR controller which determined the appropriate control torque to be applied by the motors Before this control signal could be sent to Vertigo it had to be conditioned First it was converted into a representative angular veloc ity for these motors this was calculated from a calibration chart bundled with the 103 Jonathan Missel External Sensing Architecture motor specifications 23 Next the input was transformed into the body frame so its orientation matched the Vicon object which in turn approximated that of the actual body Finally the control was converted into representative scalar values recognized by Qwerk Yaw input did not necessitate transformation into the body frame because its measurements were made in Euler angles In sphere based control the same was true for all associated angle measurements transformation was not required only scaling for input to Qwerk In the above schematic scaling the yaw input was jointly accomplished by the proportional controller gain The conditioned translational in puts were then superimposed with the yaw input a convenience permitted by the unique manner in which Vertigo is actuated See Appendix A for more on control superposition After the control input had been converted into a form that Qwerk was able to implement in the body fixed frame it had to be delivered Forming the link between Qwerk and an external computer turned out to be one of the more chal
19. 50 5 10 15 20 o 5 10 15 20 Time Time 100 100 ae 50 50 Ka Do Do oO oO E E o z 3 o En 50 505 5 10 15 20 100 5 10 15 20 Time Time Figure 5 16 State response with tracking LQR controller heavily weighted angles LOR Track Response 2pi rotation oc e o e N gt D o Magnitude wrt Vertical radian units 0 2 4 6 8 10 12 14 16 18 20 Figure 5 17 Response with tracking LQR controller wrt vertical heavily weighted angles 133 Jonathan Missel Sphere Based Control trols If thought of as a standard transient step response less weighting on the rates improves the rise time but increases chatter and control effort Determining the proper tuning is case specific and finicky This is where the augmented controller shines because it is time dependant and helps blend the advantages of both extremes Low weighting can initially be given to the velocities to improve the rise time and as time progresses the augmentation penalizes the rate errors more heavily to smooth out the response To demonstrate this characteristic the system was augmented with A 1 and simulated with the same aggressive tuning as the previous simulation Fig ures 5 18 and 5 19 show the response of the augmented LQR controller with heavily weighted angle By comparing this to the two un augmented responses it is clear that the augmented response is far superior It marries the smoothness accuracy and e
20. Al Sphere Properties ada a Om dw asd ce What A Gee 195 A 8 Weight Distribution soe este eee Be eh eek oe ed ae 196 Powering Vertigo and Qwerk 198 Connection to Vertigo 201 Changing Qwerk s Embedded Code 214 Maintenance 237 vill List of Figures 1 1 1 2 1 3 2 1 2 2 2 3 2 4 2 5 2 6 2 2 8 2 9 3 1 a Stationary manipulator b Statically stable vehicles c Omni directional statically stable vehicle d Differential drive balancing ve hicle e Bipedal walker E SA ARA 2 Statically and dynamically stable vehicles encountering terrain 5 Anatomical planes uo ec oe e oe a al 8 Fixed biaxial inverted pendulum concept 20 Front 45 deg and top view of omni directional sphere actuation con cepts a Concept A inverse mouse b Concept B omni wheel Eu clidian perpendicularity c Concept C omni wheel Euclidian orthog onality d Concept D omni wheel spherical perpendicularity 23 Vertigo 1 0 assembly photograph 40 Vertigo 1 0 motor encoder wheel exploded subassembly CAD 44 Vertigo 1 0 motor encoder wheel compressed subassembly photograph 45 Vertigo design generation dius rear ee ee A eee Ee ee 50 CNC mill machining Delrin legs for Vertigo 2 1 55 Modifying omni wheels for Vertigo 2 1 use photographs 59 Vertigo 2 1 motor encoder wheel exploded subassembly CAD 61 Planar sphere based Vertigo model o a
21. Faulhaber Static continuous callibration tables micromotor Jan 2008 T T R Kit Qwerkbot software api quick reference pp 1 6 2006 K Ogata Modern Control Engineering Prentice Hall Inc Addison Wesley 2007 C T Chen Linear System Theory and Design Oxford University Press 1999 C T Chen Linear System Theory and Design Transfer Function State Space and Algebraic Methods Oxford University Press 1993 R E Kalman A new approach to linear filtering and prediction problems Transactions of the ASME Journal of Basic Engineering vol 82 no Series D pp 35 45 1960 J L Crassidis and J L Junkins OPTIMAL ESTIMATION OF DYNAMIC SYSTEMS CRC Press 2004 183 Appendix A Supplementary Concepts A 1 Introduction The objective of this appendix is to provide cogent explanations for some of this platform s non intuitive aspects Many of the items addressed can be attributed to Vertigo s nonlinearity and unique design Mathematical backing is provided accord ingly throughout this thesis however this appendix focuses on clearly conveying the ideas in plain English The topics of directionality translational motion actuation control superposition pivot drift sphere properties and weight distribution are com monly misinterpreted by those first being introduced to this system They are also critical to the proper understanding of how it works and therefore warrant discussion A 2 Directi
22. a biaxial inverted pendulum is what makes it unique Vertigo was designed to be simplistic With only four actively moving parts all of which are identical the chance of mechanical failure is minimized therefore its simplicity translates in to robustness With a height of only 17 inches and weight of 5 6 lbs it is easily transported which is convenient for its role as a mobile experi mental platform The structural assembly was carefully designed with four legs that support two interchangeable mounting plates to allow reconfigurability and fortify its defenses against failure from impact The combined influence of these traits granted Vertigo with longevity and the ability to contribute experiments that pertain to vastly different systems One of Vertigo s applications is modeling under actuated systems these are sys tems that do not have direct control of all feasible degrees of freedom Fully exploring the dynamics of these systems is currently an active area of research its mission is to discover and carry out maneuvers that control more degrees of freedom then are directly actuated Requiring fewer actuators permits significant reduction in weight cost and energy needed to perform tasks In addition when fully actuated systems experience failure these techniques may save the mission by using the remaining ac tuators to compensate for the loss Laboratories in Japan are experimenting with passive joints to help improve the dexterity o
23. a system near its equilibrium but subject to large perturbations the inherent nonlinearities will decrease the validity of the simplification In this analysis the observability O and controllability C matrices are calculated respectively as O C CA CA 3 24 C B AB A B 3 25 where n is the number of states The system is observable if the observability matrix has full column rank and controllable if the controllability matrix has full row rank All four states will be varied over a range of values and this calculation carried out for the relinearized system at each point The rank criterion for O and C only gives insight as to whether the system is or is not observable or controllable it tells nothing of a measure or degree of observability and controllability To dig deeper into this the inverse of the condition number and determinants of these matrices are examined Many factors contributed to the structural development of this platform Balanc ing machines are fundamentally dependant on the location of their center of gravity length of the inverted pendulum so understanding the nature of this dependence was important to the design process Furthermore Vertigo was intentionally created to be reconfigurable which allows its center of mass to be altered so it was necessary to determine how the physical changes should be expected to affect the performance making experiment design more predictable Two main components a
24. actuator orientation and axial symmetry means that the full state space equation with x and y position and velocity is decoupled and identically redundant for both directions Therefore control and implementation can be carried out for a planar model and applied to both planes independently The continuous equations of motion for the ground based robot were put in standard state space form q Aq Bu with the output taking the form y Cq Du With the state vector defined as q rt and control as u T the model can be expressed as he ya 3 20 z y 1 o o 7 3 21 Where r m and T are the radius of the wheels mass of the robot and imposed torque of an individual motor respectively This model makes the assumptions that all components are rigid bodies dynamic affects can be neglected and opposing 75 Jonathan Missel Mathematical Model N 4 N 4 N 4 Figure 3 2 Ground based free body diagram frictions at the wheels are negligible As a result of the dynamic simplifications all points of contact with the ground are assumed to carry equal portions of the total normal force N In reality this is not true for two main reasons all four wheels will not always be in contact with the ground due to uneven terrain and the discontinuity of the wheels and as the body accelerates the leading wheel experiences less normal force than the trailing wheel discussed further in the Implementation Challenges section Ne
25. all to generate very accurate measurements For this application RE201 Kit Incremental encoders were chosen Quadrature encoder specifications 18 1 inch code disc Maximum line count of 4096 2 data channels in quadrature Standard 90 index pulse CMOS ASIC design 500 kHz frequency response 5 pin headers In addition to these sensors Qwerk supports electromotive force EMF feedback which is a measure of energy per unit charge that the motor provides to the circuit This is a function of rotational velocity for the motor so EMF feedback is a sen sorless way of measuring the motor rates This method is not as precise as encoder measurements but it can be combined with low weighting to improve overall accu racy Qwerk also supports video feed via webcam To verify which specific devices are compatible the Terk website was consulted Terk is an organization through Carnegie Mellon that provides support and advisement for the Qwerk unit and was 36 Jonathan Missel Components referred to throughout this project 19 From their list of webcams the Logitech Communicate STX was selected for its performance Not only is the inclusion of this webcam a means of video surveillance but with proper image processing it is high enough quality to back out attitude data Further development of this could even lead to real time obstacle avoidance Webcam specifications 20 High quality VGA sensor with RightLight
26. and sphere radius The determinant of the controllability matrix mesh is shown in Figure 3 20 It reveals that the determinant is not close to zero and its highly nonlinear manner of leveling off suggests that it will not approach zero near this range Therefore it is safe to assume these results are valid for reasonable sphere properties The inertial influences were also isolated for the sphere design parameters the inverse condition for controllability with fixed inertia is shown in Figure 3 21 The radius of the sphere appears in both body and sphere rotational inertia calculations and the influence from holding them constant tends to cancel out so no significant 95 Jonathan Missel Conclusions Inverse of Controllability Condition E D 4 rb m Sphere Mass kg Figure 3 21 Inverse condition number of controllability matrix while varying sphere mass and sphere radius with fixed rotational inertias change is observed However when the sphere mass is varied it is clear that the rota tional inertia of the sphere was the major contributor to the improved controllability condition because this parameter does not appear in the body inertia calculations 3 4 Conclusions This chapter derived the equations of motion for Vertigo and conducted an extensive analysis to determine their controllability and observability characteristics It began with the derivation of nonlinear equations of motion for sphere based Ve
27. angle deviated to the point where the front edge of the board was on the ground roll angle was held constant despite the efforts of y control Therefore Vertigo moved in the y direction to try to account for the small angle of the uneven ground By the time x control brought pitch back to balance y had veered far enough to deflect the roll angle to rest on the adjacent edge This was not a problem when starting from a balanced point and the objective of establishing convention was met therefore further development was not necessary At this point an informative collection of controllers had been created to prepare for balancing on the sphere Although it surpassed the objectives of this project early stages of sphere based implementation were attempted Building off the findings of previous controllers and experiments made it easier to understand the origins and severity of problems encountered in this Proportional control was first attempted with the sphere With Vertigo s angles as Vicon feedback this controller applied input in the correct direction for balance but was insufficient Gains that were too low did not apply enough control to recover from any appreciable disturbance but gains that were large enough to bring Vertigo back to the unstable equilibrium would overshoot to an unrecoverable angle However this controller was beneficial because 172 Jonathan Missel Balancing Implementation PID board pitch 0 01 ao D m
28. assumptions Examples of these include the inertias friction coefficients and the relation between signal input and actual torque that is applied by the mo tors To minimize the impact of these errors filters can be applied that account for uncertainty or even predict unknown parameters 3 2 2 Model Derivation Viewed by many as the most important step in control and simulation modeling is charged with balancing accuracy and practicality Following the Ballbot team s pro cedure Lagrangian mechanics are used here to develop Vertigo s nonlinear equations of motion This method starts with the identification of kinetic energy K K Kep and potential energy V V Vz where the subscript b represents the sphere and B represents the body of the robot From the energies the Lagrangian can be constructed and the Euler Lagrange equations of motion calculated as follows L qq K V 3 1 d Where T is the input torque for the motors and q is the state vector 69 Jonathan Missel Mathematical Model Figure 3 1 Planar sphere based Vertigo model Figure 3 1 depicts the state and parameter convention that will be used throughout this thesis for the sphere based configuration It is worth noting here that the angle of the body is measured with respect to the angle of the sphere 0 This is done for mathematical convenience however it makes some of the results and simulations less intuitive to interoperate For this reaso
29. beyond one rotation there was no distinction between 27 and 47 Many commonly used rotation transformation methods including MATLAB s mod command did not fully address this issue This is because they conditioned the measurement back to zero when it crossed 27 and the error appeared to be 27 Methods were also attempted that conditioned the trajectory by resetting it to 0 at 27 rather than the other way around These did not work either because the error became 2r and the robot quickly completed a full rotation in the opposite direction to pick up the trajectory again Eventually a measurement conditioning method was developed that allowed any number of rotations This bit of coding is convoluted as it works with three incrementing counters the mod function and a time delay however it effectively shifts the measurements to a continuous scale Including yaw trajectories was not without drawback Its problem was similar to the error from undesired yaw disturbances when controlled form the inertial frame That is as control was applied the robot translated and rotated together thus curv 157 Jonathan Missel Ground Based Implementation ing the translations For slow trajectories this effect was negligible because the sampling rate updated yaw measurements reasonably fast As yaw rate increased so did the error State Feedback LQR is a full state feedback method however Vicon does not measure rates Therefore half of the
30. commonly have four wheels where two steer similar to a car or have differential drives like a wheelchair In this way they only have two degrees of freedom because they can move forward backward and turn but cannot move side to side Accordingly the mobility of these systems leaves something to be desired especially in close quarters The acceleration of these platforms is also limited when their center of gravity is above the axles of their wheels This is due to a likelihood of falling over Similarly ex treme terrain can cause them to tip or have wheels lift off the ground Mitigating these issues requires a low center of gravity and large base making them even more cumbersome Omni directional vehicles similar to the one in Figure 1 1 c have made advances by solving some of the directionality limitations These robots can move in any planar direction Their rotation does not affect their direction of motion so a complex path can be followed without having to spend time and power rotating Further explanation of directionality can be found in Appendix A Traditionally these are also statically stable and therefore inherit the risk of becoming dynamically unstable Balancing dynamically stable differential drive robots also called self balancing Jonathan Missel Motivation robots have begun to take advantage of dynamic stability These are variations of inverted pendulums as shown in Figure 1 1 d and require unremitting cont
31. decoupled This is true for all forms For exam ple if the actuators all have equal projected angular spacing with respect to each other then control is not applied in perpendicular directions Even if two actuators are positioned perpendicularly the control of the third motor would still be coupled Additionally yaw would be coupled with directional control because the perpendic ular actuators would not apply control through the center of the sphere causing a 26 Jonathan Missel Actuation Method resultant torque With four actuators only yaw is coupled both directional control components are completely decoupled allowing for the previously mentioned planar development The major advantage of this over concept C which also included yaw control is that any sized sphere can be used here and it can be placed directly on the ground while preserving all three planar degrees of freedom Concept of of Sym Ctrl Self Omni Ground Any Actu DOF Sym Support Wheel Based Sphere ators A 2 2 No Yes No No No No B 2 2 No Yes No Yes Yes Yes C 3 3 No No No Yes No No D 4 3 Yes Yes Yes Yes Yes Yes Table 2 1 Features of drive concepts Table 2 1 shows a comparison of significant features for the four drive concepts The gravity of these statistics is subjective and should be established to make an in formed choice To maximize versatility a system that can accept any
32. dynamic response and its small size 14 Jonathan Missel System Description and robust design make it practical for a single person to safely operate with minimal risk of failure As an experimental platform applications of Vertigo are not limited to modeling other systems it promises original contributions as well particularly to the field of rover locomotion As previously discussed a biaxial inverted pendulum has excep tional performance and agility because it takes advantage of dynamic motion rather than fighting it With fully developed control Vertigo can exploit omni directional motion with independent yaw scale inclines and terrain in any direction and can ac tively adapt to unexpected loadings These qualities make it ideal for accomplishing the objectives of many roving scenarios Its fluid motion also has enormous potential to bridge the gap for human and robotic coexistence Building this cooperation is a driving motivation behind Carnegie Mellon s Ballbot project Smooth robotic inter action with humans mandates that machines have high centers of gravity and small footprints however this limits the maneuverability of statically stable robots due to their likelihood of tipping Developing dexterous robotics may help the interface be tween lifeless machines and skeptic humans In doing so the importance of humanlike motion and dimensions is more than just psychological cohabited environments ex pect both parties to o
33. ground based states were not measured for feedback and none were for sphere based mode Derivatives of the position had to be taken for rates however implementation was a discrete process so numerical derivatives were taken as the difference in position from the current and previous time step divided by the sampling time between the two In addition controllers had to be implemented in the time domain This was challenging because many continuous and discrete methods for controller design are not carried out in the time domain accordingly either extra measures had to be taken with their application or they were deemed too disparate to be applicable 7 3 Ground Based Implementation Implementation began with the ground based mode because it was a safer and more attainable starting point than sphere based control The first code to successfully make the link between Vicon and MATLAB tested the position and attitude of the Vicon object Using the sim command MATLAB extracted the Vicon information through a manufacturer provided Simulink plant that was able to record measure ments This program was useful throughout the project for checking the coordinates of the robot and analyzing how well the object was defined by Vicon Next a program was built to complete the full communication loop by sending 158 Jonathan Missel Ground Based Implementation commands to the robot and using the Vicon information as feedback This controller t
34. ground without the sphere and the sphere has to satisfy very exacting dimensional tolerances This is a result of the method used to actuate the sphere which is similar to that of the ball and encoder assembly in an old computer mouse but inverted Two cylindrical rollers are used to drive the sphere in the x and y directions but in order to prevent conflicting control inputs on the surface of the sphere they must 10 Jonathan Missel System Description be perpendicular and placed on the equator largest cross sectional diameter When one direction is controlled the sphere rotates about the point of contact with the perpendicular roller This method decouples directional control but requires that the sphere be a very specific size to ensure the friction is sufficiently large enough to prevent slip yet small enough avoid impeding motion Since the actuation had to be placed at the equator it was not able to support the weight of the robot on the sphere and passive rollers had to be included in the design for this Though relatively new the mobile inverted pendulum has quickly proven its use fulness From its humble beginnings as a bidirectional bench top experiment to its most recent manifestation as the biaxial omni directional and fully autonomous Ball bot this platform has made rapid and fruitful progress With this thesis providing a solution for its only remaining absence in motion direct control of yaw this platform awaits
35. harness that wires the two bricks in parallel To protect Qwerk and insure that the minimum limits of the battery packs are not breached a 9 0V LiPo battery voltage cut off circuit was fitted between the power supply 198 Jonathan Missel Powering Vertigo and Qwerk and the power input for Qwerk http www rctoys com rc toys and parts DF DIV BATTCUT RC PARTS WIRELESS VIDEO BATTERY ACCESS html Battery Charging For single 4150 mAh 11 1 V LiPo battery pack 1 Plug single must charge both of the packs individually battery into TRITON 2 charger through the Equinox interface 2 On the charger click Battery Type to select LiPo charge and then choose 4150 mAh 11 1 V 3 Click the Equinox interface box to switch to interface mode 4 Press dial on charger and hold to initiate the charging sequence Charging takes roughly an hour and a half per pack Always charge both bricks before use It is a good idea to charge the pack immediately after they run out to ensure that it will be ready when needed NOTE The charger has a safety timer that may expire before full charge is reached It will say why it stopped on the display If it timed out simply proceed as though you were charging it again and it will finish Tethered Power Supply Input AC 100 230 V 0 7 0 4 A 50 60 HZ Output DC 12V 1 6 A This method is best for stationary powering programming diagnostics etc Programming of any type should really be don
36. in stage five is carried out similarly to acceleration except the lean is in the opposite direction and the brakes are applied Stage six acts to terminate the lean This can be carried out in two slightly different ways the first is by applying the brakes perfectly so that Vertigo pivots back to vertical with a velocity of zero and exactly at the desired destination The second is the opposite to lean initiation in stage two the final destination is overshot slightly as the motion is slowed then the recovery acts to resume vertical and find the final destination The second way is more time efficient because the first theoretically takes an infinite amount of time however both are actually different manifestations of the same process Stage seven is the final state of the robot where point B is reached and all translation has ceased A 4 Actuation The method used here for actuating the angles of an inverted pendulum is unique to Vertigo When balancing an inverted pendulum the goal is to control the base in this case the sphere What sets Vertigo apart is its actuator orientation and use of omni directional wheels that allow control to be applied to the same surface in various directions without interference Four spherically orthogonal motors allow almost any 188 Jonathan Missel Actuation size sphere restricted only to a minimum size to be used as a base for Vertigo Taking this to the extreme results in a sphere of infinite
37. it was found that the most accurate results can be obtained near the vertical unstable equilibrium This is ideal because most of Vertigo s motion is confined to this region To further explore the characteristic of Vertigo the controllability and observabil ity analysis that was outlined for the state variations was done while varying physical parameters These results were used to draw conclusions about the design of Vertigo and learn what was to be expected for controls and estimation to come It was found that for the current length between the center of gravity of the body and the center of the sphere Vertigo is at an optimal mass for controllability condition The same analysis was conducted for the design parameters while holding the rotational iner tias constant and the results compared with the previous analysis This comparison helped isolate the influence of the many factors that contribute to the controllabil ity of this system The range of values chosen for the body and sphere parameter 97 Jonathan Missel Conclusions variations were chosen based on physical constraints yielding an inclusive analysis of feasible options 98 Chapter 4 Communication Architecture 4 1 Introduction In order to implement closed loop control methods on Vertigo a network of commu nication had to be established The main objective of this architecture is to send sensory input to a processor for control law to determine the desire
38. m l L L L if l i i i l 0 5 0 4 0 3 0 2 0 1 0 01 02 03 04 05 x position m Figure 7 6 4 circle trajectory tracking laps with proportional yaw control have enough information to track Vertigo Vicon then sets the object s position to the origin and a signature spike will be seen in the track plot that goes to the origin and then back out to Vertigo s location once it is observed again This results in an inaccurately high position error that cause the controller to send large inputs and huge departures occur The frequency of this problem was increased for balancing controllers because changing roll and pitch angles were more prone to hiding markers One final modification was made to the ground based LQR controller that helped to prove the concept of omni directionality with yaw This was done by transform ing the control signal from the inertial frame into the body frame This allowed independent yaw and position trajectories to be specified It also meant that yaw 165 Jonathan Missel Ground Based Implementation LQR ground track with transformed input 0 5 0 4 y position m 5 o o o o rm w o mo 0 3 0 4 0 5 05 04 03 02 01 0 01 02 03 04 05 x position m Figure 7 7 Circle trajectory tracking with transformed control norm error 0 5402 control could be omitted and inertial trajectories could be tracked accurately in the body frame despite yaw disturbances Figure 7 7 shows the sta
39. my Value Instructions After Initial Installation and Setup 1 Open MATLAB 2 Open vertigo qwerk folder in MATLAB 210 Jonathan Missel Connection to Vertigo 3 Enter Ts 0 1 in the command window 4 Run vertigo_nonlinear m 5 Open MyFirstRobot m make sure the directory locations are all correct see code below a The classpath txt file altered above in the initial setup must be returned to its original state see reference a section copied from the rn pdf in the QWERK dir 6 Run MyFirstRobot m GUI should pop up just like other connection methods a Click Connect b Choose to connect directly to a peer c Enter Vertigo s IP address something like 192 168 1 100 d Click Finish then Play and minimize the GUI 7 Now all of the Terk commands can be used in MATLAB just as they were used in JCreator for Java 8 Run vertigo_java_vicon_loop mdl found in the vertigo simulink directory AAA P P A PP AAP VV PV 70 V0 V0 P00 LV POV V0 VOV0 Yo 070 AS setting the path to include required dir javaaddpath C Users Jon Desktop VERTIGO QWERK MyFirst Vertigo terk client MyFirstRobot 211 Jonathan Missel Connection to Vertigo setting the path to terk jar also terk jar is where the motor connection and other commands are located javaaddpath C Users Jon Desktop VERTIGO QWERK MyFirst Vertigo terk client MyFirstRobot terk jar add text to speech fu
40. nature of existing mobile robotic platforms has lead to an impressive understanding of like systems but this convention has done little for the progression of unique or complex systems By following this paradigm designers have imposed significant limitations on their creations The most severe of these constraints confine the maximum acceleration and deceleration ability to withstand unexpected loading and directional control In the past these obstacles have been individually overcome with robots that either balance but only have two degrees of freedom or have omni directionality but are prone to become dynamically unstable Recent attempts to completely liberate machines from such inabilities have led to the expensive and of ten unnecessary complications of bipedal walkers These platforms have proved to be unreliable and well beyond most budgets of those interested in implementing them For simplicity many experimental platforms are designed to be linear though non linear systems dominate the real world Linear control methods are only guaranteed to work on nonlinear systems over small realms limiting the usefulness of these plat Jonathan Missel Motivation forms to test many practical applications To form a genuine knowledge of dynamics and controls there is a need to develop versatile robotics that can model a diverse range of other systems and contribute original experiments without being overly convoluted or costly There ar
41. o Q D pitch radians a o E 0 05 0 06 o E ro i D a o wN co 10 time sec Figure 7 11 Balancing plank pitch with PID control PID board pitch pitch radians D 1 o N an E n mn od co w 10 time sec Figure 7 12 Balancing plank pitch with PID control by tracking Vertigo 173 Jonathan Missel Balancing Implementation it started to demonstrate the specific challenges of balancing on a sphere PID control was then implemented in a more sophisticated attempt at balance This gave better results however it could not sustain balance only keeping Vertigo on the sphere for about two seconds During this time oscillations would diverge to an unrecoverable angle Tweaking the gains to reduce overshoot limited the recovery angle such that it was not able to cope with the small deviations from noisy measure ments and external disturbances Since the damping component of the PID controller was most beneficial for balance and tuning was difficult LQR was attempted next in hopes that its optimal gains would solve the problem For LQR control many techniques were tried for implementation and defining the Vicon object but none were successful LQR was appealing because it included the damping aspect of the PID control and it gave optimal gains for the mathematical model however it is possible that model errors were sufficient enough to invalidate the results for the physical system
42. of maintaining an accurate and up to date CAD model 2 6 Design Iteration Before moving on to the Vertigo 2 1 design iteration a quick recap of the project s evolution to this point is given The discussion is complemented by the images in Figure 2 6 which illustrate some of the major phases in Vertigo s design generation Figure 2 6 a shows the first CAD rendering that depicted the conceptual fundamen tals and a primitive vision for the design It established the idea of using an inverted pendulum and balancing on a sphere it even began to work out the orientation of the actuators for omni directionality This sketch launched the project by helping to explain its main ideas Once the project was accepted the model in Figure 2 6 b was created to help determine what components were needed and how they might be assembled This model established the actuation method and sensing equipment to be used by includ 48 Jonathan Missel Design Iteration ing components that were dimensioned based on the published specifications of the manufacturer At this stage early ideas for efficient design and reconfigurability were also being explored As the components were ordered and the design took shape the continuously updated working model evolved into that of Figure 2 6 c 1 This included all components of Vertigo 1 0 modeled precisely to measurements Most importantly this was the final design for Vertigo 1 0 that was analyzed and app
43. of the wheel and threaded at the end so that a screw and washer could clamp the wheel against the hub stop Though it increased the offset slightly the encoder was mounted between the motor and the omni wheel to protect its delicate exposed wheel For all of its intricacy the final design was very compact as seen from the photograph of the compressed assembly in Figure 2 5 As a final modification flats had to be milled between the rollers of the wheels to allow clearance with the curved surface of the sphere Figure 2 5 Vertigo 1 0 motor encoder wheel compressed subassembly photograph Since the conception of this project much thought and debate was directed toward determining the ideal sphere for the robot to balance on Though it was intended to fit any size sphere a choice had to be made for acquiring the first one The design controllability and observability analysis shed light on how this decision could be made 45 Jonathan Missel Structure to simplify the problem but it was understood that this was an experimental platform so a degree of challenge was welcomed It was concluded that a hollow sphere would minimize rotational inertia and increase agility and that a rubberized finish would mitigate concerns for traction with both the omni wheels and ground Eventually a basketball was settled upon to launch the project because they satisfied all these requirements are inexpensive come in standard sizes and are read
44. output sampling time As a measure of accuracy the 30 values for the states were stored once it had settled for each combination of sampling times The values were taken 5 time steps from the end rather than exactly at the end because this is a cyclic process and breaking that to exit the simulation causes the last 30 values to be invalid Figure 6 10 is the mesh of the 30 values for 0 as the sampling times were varied Analysis of all four states yielded the same trends therefore only the results for 0 were included here as a representative set The previous analysis showed that increased output frequency improved the estimator this is confirmed here though it is not obvious There are several reasons for this the most pronounced being that its impact is outshined by that of the measurement frequency variation Also the maximum and minimum values for both sampling times were chosen to encompass practical values and ones that could all be simulated with the same tuning This fixed tuning was acceptable for all points however if it were possible to specify ideal tunings for all simulations the influence of varied output frequency would have been more apparent Lastly is the affect of diminishing return Running the filter with fs 2fm will improve the results increasing that to fs 4fm will show further improvement however a limit exists to how accurate a filter can be without receiving more frequent measurements Because the range of va
45. ridge can be seen where the matrix is best conditioned Consider again the initial angle recovery scenario Increasing mass will increase the moment with gravity making recovery more difficult However linear inertia also increases so it has greater resistance to motion making it easier to move the sphere underneath it and recover from the initial angle The increased mass is assumed to be at a point so the rotational inertia about the center of mass does not change and the control torque required to rotate it does not change However the rotational inertia with respect to the center of the sphere does increase and the body will fall more slowly The slower falling body with no additional control requirements acts to improve controllability with increasing mass These phenomena work against each other and this ridge represents the point at which the increasing inertias are most dominant As marked by the vertical blue line Vertigo s current configuration is at an optimal mass for its pendulum length It is worth noting that just as the mass variations generate an optimal ridge due to counteracting physical principles so too does the length variation However this ridge corresponds to a length that is within the radius of the sphere and was therefore not a feasible design option and was omitted from these results Figure 3 13 shows how the observability matrix s condition is affected by varying Vertigo s design parameters The range of in
46. to use these functions The javaclasspath function when used with no arguments displays both the static and dynamic segments of the Java class path javaclasspath STATIC JAVA PATH D Sys0 Java util jar D Sys0 Javal widgets jar D Sys0 Java beans jar DYNAMIC JAVA PATH User4 Work1 Javal ClassFiles User4 Work Java mywidgets jar You can read more about this feature in the sections The Java Class Path and Making Java Classes Available to MATLAB in the External Interfaces documentation 213 Appendix D Changing Qwerk s Embedded Code Special thanks to Mark Tjersland for all his help in finally cracking this This document goes through how to change Qwerk s embedded code More specif ically the hardcoded gains for the PIDV controller and the trajectory generation function that the PIDV controller tracks This was necessary to attain the respon siveness demanded by Vertigo It details how to acquire the code manipulate it compile it delete the existing code replace it and reinitialize Qwerk It also lists all the required programs for these steps and how they can be acquired This procedure is loosely based on the outline given in the Qwerk Development Guide QDG which can be found at http terk svn sourceforge net viewvc terk embed trunk share docs QwerkDevelo pment Guide html BuildingtheQwerkSource The relevant excerpts can also be found at the end of these instructions Programs For this proce
47. until necessary It was at this time Carnegie Mellon s Ballbot paper was encountered Ballbot did not work on exactly the same principles as Vertigo 1 0 but it was very similar This pushed the project into an early design iteration to make use of drive method D for actuating the sphere This would give Vertigo direct control in yaw which Ballbot did not have It would also eliminate the possibility of motor offset interfering with the dynamics of the system Since both the motor offset issue and new drive method only pertained to the legs the remainder of Vertigo 1 0 for which the design proved to be very successful was preserved This reduced the time and cost of redesign and still allowed the Vertigo 1 0 legs to be fitted if desired Vertigo 2 0 and 2 1 differ only in that new actuators and motor mounts were fitted to Vertigo 2 1 With every other aspect redundant only the design of Vertigo 2 1 is included here As with the explanation of the original design discussion of the redesign is accompanied by visuals to better illustrate ideas Figures 2 6 d 1 and 2 6 d 2 are respectively the CAD model and photograph of the completed Vertigo 2 1 design Features will again be explained from the top down starting with the first modification at the top of the legs The spines that protrude vertically from the tops of the legs were included to give further protection to the components on top of the upper plate They were specifically sized to shi
48. were written to manage various aspects of motion The response for tracking the standardized trajectory and the associated norm errors are included for compar ison with other controllers and the simulated results The first LQR controller was implemented similarly to the PID controller with the exception that it converted the input from torque Nm to angular rate rad sec and then scaled it to a corresponding Qwerk input This allowed mathematically cal culated gains to be written directly into the code and converted internally Observing the performance of this controller confirmed the need for controlling yaw Therefore another controller was created that expanded LQR to include proportional yaw con trol This was applied using control superposition see Appendix A in moderation to gently hold the angle Figures 7 4 and 7 5 illustrate the importance of yaw control for this system The 161 Jonathan Missel Ground Based Implementation P Ground Track 0 5 0 4 0 3 0 2 0 14 y position m 01 02 03 04 05 04 03 02 0 1 0 01 02 03 04 05 x position m Figure 7 2 Circle trajectory tracking using proportional controller with proportional yaw control norm error 2 3897 PID Ground Track 0 5 F 0 4 0 3 0 2 0 1 y position m f f fi 1 1 1 1 1 1 1 1 05 0 4 0 3 0 2 0 1 0 01 02 03 04 05 x position m Figure 7 3 Circle trajectory tracking using PID controller wi
49. D wheel and the gaps are small because of this Additionally the roller axels are made from stainless steel so rust will not be a concern and their light weight will not appreciably hinder the response time of the motors The motors for Vertigo 1 0 were chosen primarily for their torque because this is the control input in the equations of motion The motors also had to match the 7 2 12 volt range of Qwerk s output Finally Jameco Gear Head DC motors were chosen for their compatibility with Qwerk and because they had one of the highest 32 Jonathan Missel Components torque ratings in its class Motor specifications 16 e Rated voltage 12 VDC e Operating range 6 24 VDC Current at maximum efficiency 105 mA Speed at maximum efficiency 8 rpm e Torque at maximum efficiency 3500 g cm Gear ratio 332 01 00 In Vertigo 1 0 passive contact is required for the drive mechanism the same omni wheels were also used for this To let them spin freely two 10mm bearings were press fit into their I D s Spacers were made to position bearings flush with the outer faces of each wheel The bearings inner races had an I D of 4mm so 4mm partially threaded shoulder cap screws were used as axles and threaded into brass slugs These slugs were made to match the mass and diameter of the motors and the cap screw axles were threaded in with the same offset as the motors driveshaft All this attention to detail served to mainta
50. Design Development and Control of Mobile Biaxial Inverted Pendulum by Jonathan Missel Friday August 71 2009 A thesis submitted to the Faculty of the Graduate School of the State University of New York at Buffalo in partial fulfillment of the requirements for the degree of Master of Science Department of Mechanical and Aerospace Engineering Copyright by Jonathan Missel 2009 ii Acknowledgements First and foremost I wish to thank my family for their unwavering support through out my education I have focused heavily on my education over the past several years and they have all made sacrifices on my behalf My parents Dan and Linda Mis sel have been exemplary models of success both as parents and in their respective careers Their dedication to selflessness has always been an inspiration to search for broader meaning My dad gets special thanks for his many hours spent helping design and manufacture Vertigo and my mom for help editing this thesis Thanks to my sister and her husband Becky and Charlie Fennie for their perpetual encouragement sarcastic candor and comic relief I would also like to extend my gratitude for their engenderment of my favorite quartet of playmates Danny Eli Ainsley and Mara Those kids are a blast and their chaos is confoundingly soothing I want to thank my brother Matt Missel for his support companionship ideas and advice as we grew up and now pursue similar academic fields t
51. However there was a fundamental concern with only having three vertical legs in our orthogonal universe the configuration is limited to being asymmetrical at best Therefore even if the actuators are applying orthogonal control the product of inertia a consequence of omitting the symmetry of four legs would still couple their effect on the system Accordingly using only three legs would not fit the planar and decoupled mathematical model and controller design would be more challenging Another advantage to using three legs is that it would ensure that all control contact be firmly secured to the surface of the sphere increasing the frictional force and reducing the likelihood of slippage In the four legged design this issue was mitigated by holding tight tolerances when manufacturing the uniform length legs and designing the legs to flex slightly This permitted flexing has not yet been problematic but may be in the future if more extreme control induces undesired vibrations If this does occur the addition of simple struts that brace the lower legs would restrict their motion and prevent oscillation Another possible issue is with the battery orientation The battery pack is cur rently composed of two identical bricks that are bundled together to make removal and installation convenient Their diagonal orientation causes a dynamically coupling product of inertia that has not yet been addressed but may need to be in the future 64 Jon
52. ICE_API L slice slice IceE BuiltinSequences ice mv BuiltinSequences h include IceE rm f include IceE FacetMap h FacetMap cpp slice2cppe ice include dir IceE dll export ICE_API L slice slice IceE FacetMap ice terk embed trunk share Writable filesystem successfully created Building the share tree may take a while but should eventually finish with the re port Writable filesystem successfully created The writeable filesystem it created is a file named optfs out and is located in the terk embed trunk cirrus arm linux 1 0 4 edb9302 directory The build also copies over the opt directory from which the optsfs out image was created this is necessary since as we ll see next the build for the cirrus arm linux tree also creates the optfs out image by repacking the opt directory that it finds within its terk embed trunk cirrus arm linux 1 0 4 edb9302 directory Now build the cirrus arm linux tree This builds three filesystem images for the Qwerk cd terk embed trunk cirrus arm linux 1 0 4 make edb9302 make 1 Entering directory home YOUR USERNAME terk embed trunk cirrus arm linux 1 0 4 edb9302 Creating root filesystem 221 Jonathan Missel Changing Qwerk s Embedded Code Creating linux source tree Configuring linux Building linux modules Configuring module init tools Building module init tools Building linux zImage C
53. a ai done booting the kernel Wait a few seconds while the kernel boots You should eventually see a message shown below that prompts you to type Enter to activate the console Do so and you should see Please press Enter to activate this console BusyBox v1 00 2006 06 28 21 45 0000 Built in shell ash Enter help for a list of built in commands Congratulations You successfully installed the firmware on the Qwerk and it s ready to use 236 Appendix E Maintenance This appendix lists the routine maintenance suggestions for Vertigo s up keeping Most of these suggestions and timeframes are based on experience working with the system and assume daily use Set screws on wheel hubs should be tightened weekly when in use 1 8 in hex key loosens the wheels retaining washer then a 05 in hex key fits the set screw Be careful not to strip the hex or threads on these small screws Keep an eye on battery life and how well they are charging Watch for wheel erosion Periodically check that all bolts and screws are securely fastened Periodically check that none of the motors have shifted in their mounts so that all four wheels contact a flat surface evenly To mount the motors evenly make the 237 Jonathan Missel Maintenance front face of the gear head flush with the face of the front motor mount Align them on a flat surface applying pressure to keep both in contact with the surface as you t
54. a strong future giving the robotics dynamics and controls communities new tools for discovery 1 3 System Description This thesis proposes a solution to help solve the problem of limited mobility in robotics and its trade off with cost The idea is that relatively inexpensive dynamically stable omni directional motion can be achieved by developing a platform that balances navigates and rotates on a single spherical wheel This original design has potential for exceptional agility and was developed to promote mass dimensional flexibility for adaptability to wide ranging experimentation As a fresh take on motion it offers new avenues for research and broadens the scope of ground vehicle navigation Vertigo as it is called takes the form of a biaxial inverted pendulum with yaw control The name was chosen for its blend of irony and accuracy The word vertigo comes from 11 Jonathan Missel System Description the Latin word vertigin meaning dizziness and can be traced further back to the word verto meaning I turn Today it is used as a medical term describing a form of dizziness that includes the sensation of spinning or swaying often resulting in loss of balance A symptom of balancing disorders seemed perfectly ironic for naming a robot whose party piece is its ability to balance Additionally the meaning I turn form which the word was originally derived is quite fitting because Vertigo s ability to rotate as
55. aa a a a e 70 3 2 3 3 3 4 3 9 3 6 3 7 3 8 3 9 3 10 3 11 3 12 3 13 3 14 3 15 3 16 3 17 3 18 3 19 Ground based free body diagram 0 2004 76 Rank of controllability matrix while varying 0 and 0 80 Rank of observability matrix while varying 0 and 0 81 Inverse condition number of controllability matrix while varying 0 and DAL he Bete A BAe Ets Se he CS BR GE DOA Ee EA 82 Inverse condition number of observability matrix while varying 0 and 83 Determinant of controllability matrix while varying 0 and 84 Rank of controllability matrix while varying and 85 Rank of observability matrix while varying and 85 Rank of controllability matrix while varying O ando 88 Rank of observability matrix while varying and 88 Inverse condition number of controllability matrix while varying Ver tigo mass and Ge a AA be ee oe ee A 89 Inverse condition number of observability matrix while varying Vertigo massand to Lo ute dy hea San Oia with ais Sal Boks be ate eB eee Gi r 89 Determinant of controllability matrix while varying Vertigo mass and 91 Inverse condition number of controllability matrix while varying Ver tigo mass and with fixed rotational inertias 91 Rank of controllability matrix while varying sphere mass and sphere PAUSE ARA A tee hae EAS SAR et 93 Rank of observability matrix while v
56. ag would further dis tance the system from the trajectory 5 3 Sphere Based Control Figure 5 3 Planar sphere based Vertigo model In sphere based control the two primary functions are balance and translation In 116 Jonathan Missel Sphere Based Control this section several optimal controllers are designed to accomplish these tasks with various criteria for optimization The techniques that were successfully executed here are for linear systems so the linearized equations of motion that were derived earlier were used for development This means that their accuracy is only valid near the unstable equilibrium which was the point the equations were linearized about The controllers developed with the linear equations were simulated on the nonlinear equa tions of motion to ensure their applicability on the real system The sphere based planar model shown in Figure 5 3 is used here and yaw control is dismissed for the time being A common condition that many of these optimal control methods require is that the system must be controllable and observable This was proven earlier in the state controllability and observability analysis for all physically practical state combinations 5 3 1 Power Optimal Control Fired Final Time Uncon strained Input The first control method considered is Gramian based power optimal control with fixed final time and unconstrained input It is meant for linear systems and its development begin
57. alance be 92 Jonathan Missel Analysis Controllability Rank 0 5 Sphere Mass kg 0 4 at rb m Figure 3 16 Rank of controllability matrix while varying sphere mass and sphere radius Observability Rank 0 5 Sphere Mass kg 0 4 af rb m Figure 3 17 Rank of observability matrix while varying sphere mass and sphere radius 93 Jonathan Missel Analysis Inverse of Controllability Condition cee SEEN A 0 025 0 02 0 015 0 01 TON OS OS 0 7 Sphere Mass kg 0 4 i rb m Figure 3 18 Inverse condition number of controllability matrix while varying sphere mass and sphere radius Inverse of Observability Condition A Cas HA OS SOS Seta eet Roe Sphere Mass kg 0 4 f rb rn Figure 3 19 Inverse condition number of observability matrix while varying sphere mass and sphere radius 94 Jonathan Missel Analysis tween these two and this is why the plot is contoured thought it is mild Figure 3 19 shows that the observability matrix becomes better conditioned with increasing mass and radius As with the body parameters the observability condition does not ex perience any major changes over this considered range however the contour with increasing mass is more dramatic then that of controllability Determinant of Controllability Matrix 0 5 1 Sphere Mass kg 0 4 rb m Figure 3 20 Determinant of controllability matrix while varying sphere mass
58. ame defined in the initial calibration and setup of Vicon and measuring distances with respect to the inertial origin However 142 Jonathan Missel Application of Continuous Discrete EKF the accuracy of these measurements depends on how well the object is defined with respect to the body it is trying to track and more specifically the orientation of its actuators Vicon observes all of the markers selected to compose an object and chooses a point to center the object The exact location of this object with respect to the body is unknown Similarly the attitude of the object is defined to correspond with the inertial frame and is based on the location of the markers at the time it is created However if the body is not perfectly aligned with the inertial frame this object will not accurately represent the body s attitude Uneven ground and the discontinuous wheels make it extremely difficult to address this misalignment Together with errors in the radius of the sphere every term in the measurement model has uncertainty This means there will be significant errors in the measurement data that will have to be accounted for as best it can by the noise parameter vz in the Kalman framework To commence the cyclic filtering procedure initial values of the components to be calculated must be defined For Vertigo the states and error covariances are initialized as sit 0 0 0 o 6 6 Po E X to to 6 7 where th
59. amplifier with MP3 PCM and WAV audio support USB 2 0 host ports for connecting standard USB PC peripherals 10 100BT Ethernet port This computer was chosen over its alternatives for its dense package of features and compact size Qwerk met all desired specifications and has additional features 30 Jonathan Missel Components that allow for future expansion and experimentation It is able to accept a significant voltage range thus keeping many powering options open and its wireless networking interface is useful for stand alone operation that will support autonomy 2 4 2 Actuation Much of the theoretical requirements for actuation were already covered by determin ing the drive method however selecting and building hardware to carry out these requirements has its own set of challenges Choosing the omni wheels alone had a great deal of tradeoffs Due to the curvature of the wheel gaps exist between the rollers With existing wheels a continuous control surface is achieved by having two rows of rollers as seen in Figure 1 1 c in the Motivation section These rows are staggered such that the gaps of one row align with the rollers of the other There fore on a flat surface these wheels will rotate smoothly However this application is controlling a sphere and having two rows of rollers is not ideal because building a system that makes contact with both rows restricts it to a single sphere size voiding one of the majo
60. and pitch while a combined effort of all control yaw This configuration also solves one of the shared problems of the other three that has not yet been mentioned In every other method control was designed to act through the axis of yaw rotation of the entire system this passes through the sphere s point of contact with the ground and the center of mass of the robot If the actuators are slightly misaligned or the center of gravity is not at the geometric center of the robot from a top view or if the center of gravity shifts during operation control will apply an additional undesired torque that acts to spin the robot and couple its motion By jointly applying control from pairs of motors input can be allocated to account for this if necessary All the previous concepts used one actuator per degree of freedom and are fittingly classified as equal actuated since four actuators are used to control three degrees of freedom concept D is considered over actuated However it is possible to use only three actuators and eliminate the excess With three all of the omni wheels axes of rotation would have to be oriented radially outward from the top view and spherical orthogonality would no longer be required The only constraint on the projected an gle between the motors is that the sum of any two must be greater than 180 degrees so this variation in itself can take on many forms The sacrifice of only using three is that no control components are
61. and the point of linearization is reestablished at each time interval with the current conditions This method is used when applying the filter to Vertigo s nonlinear equations of motion and measurement model to improve and complete data acquisition for its feedback control loop Vertigo relies on external sensing for dynamic measurements however this data contains errors and does not provide direct measurements of the states There are errors in the model relating these measurements to the states Furthermore the mathematical model has errors based on measurement uncertainties and simplifying assumptions examples of these include the inertias friction coefficients and the relation between signal input and actual toque applied by the motors The extended Kalman filter is a well suited tool for estimating Vertigo s states and attenuating the measurement noise More specifically this application employs the continuous discrete EKF because Vertigo is a continuous system but the measurements are discrete The full mathematical development of this filter is detailed with its simulated implementation in the next section 139 Jonathan Missel Application of Continuous Discrete EKF 6 3 Application of Continuous Discrete EKF This section covers the development of the continuous discrete extended Kalman filter and its application to the Vertigo system The general method for this filter is well documented in many sources for consistency the
62. andate a new platform all together To ensure that the inverted pendulum dynamics were incorporated it was necessary that the base only have one point of contact with the ground This base also needed to move in any direction and hold as much symmetry as possible to maintain consistency in its dynamics From these criteria a sphere was chosen to act as a single wheeled base that a pendulum would balance on 2 2 3 Design Challenges As with every new idea there are many hurdles to overcome in the preliminary stages of design With balancing on a sphere as the updated objective the dominant struggle was developing a method for precisely controlling the sphere in two directions while it simultaneously serves as a base for supporting the robot For control in any directions multiple actuators must be interfacing the same surface while oriented in different directions Another challenge in the design of this robot was maintaining the highest pos sible degree of symmetry to preserve the foreseen simplifications it would permit in the modeling and controls development A symmetrical system should respond the same in all directions yielding consistently predictable motion To retain symmetry all components structure electronics actuators etc must be designed chosen and incorporated with precise intention 21 Jonathan Missel Actuation Method 2 3 Actuation Method Pursuing such a unique drive made the method of actuation the most
63. andmade so that the camera could be prescribed to any desired fixed pan and tilt angle This joint has 005 in of interference for the ball and socket which makes it stiff enough to maintain the desired orientation through small bumps but gives it the flexibility to deflect in major collisions to discourage breakage of the mount or camera Preventing deflection from small impacts was important for reducing uncertainty if the camera is to be used 39 Jonathan Missel Structure Figure 2 3 Vertigo 1 0 assembly photograph for attitude sensing The bottom of the mount was made to be flat giving a secure contact surface for fastening with double sided tape It was also center drilled to allow concentric mounting on a motor for actively controlling the pan The metallic cylinder under the camera mount in Figure 3 is a small motor used to control pan it has since been removed Continuing down the assembly the yellow anodized aluminum cube is the IMU that was specified earlier At each of its corners a bumper was attached as a safety measure These bumpers are marketed for child safety and are intended to be fitted to the corners of furniture They work perfectly for this application and their holes lined up with the screws in the top panel of the IMU allowing them to be included without 40 Jonathan Missel Structure modification The electrical connection on the front face is composed of a scavenged DB 15 connector from a deskto
64. ange of inputs and outputs and be able to communicate with other computers Microcontrollers are capable of executing these tasks but tend to be specialized and are not easy to implement without experience There are a handful small computers designed for general robotics use and after careful consideration Qwerk was chosen Qwerk is a compact board that was developed at Carnegie Mellon s Robotic Institute and is produced by Charmed Labs It is a powerful little computer with impressive specifi cations Qwerk specifications 14 e Qwerk Overview Powerful robotics solution for university and hobbyist markets High performance CPU with an excellent I O feature set for robotics and mechinatronics applications e Low cost e Hardware 29 Jonathan Missel Components 200 MHz ARM9 RISC processor with MMU and hardware floating point unit 32 Mbytes SDRAM 8 Mbytes flash memory Linux 2 6 installed WiFi wireless networking support WebCam video input support 4 Amp switching power supply 90 efficient 7 to 30 Volt input range Rugged aluminum enclosure 5 1 x 5 8 x 1 3 11 8 oz e I O 4 closed loop 2 0 Amp motor controllers supports quadrature encoder and back EMF sensorless position feedback as well as current sensing 16 RC servo controllers 16 programmable digital I Os 12 bit analog inputs 2 Rs 232 ports I C ports Built in audio
65. arge A drawback of these batteries is that they can sustain damage if they drop be low a minimum voltage To address this a voltage cutoff circuit was installed to terminate power when the battery provides less than 9 VDC Another drawback to the battery is that they come in long brick shapes which unless mounted vertically do not promote symmetry This would impose a product of inertia that couples the directional dynamics and causes departure from the planar model To assuage this the two batteries could be mounted perpendicular to each other forming an X This would improve the validity of the planar model by canceling the product of inertia Additionally a computer power supply was hacked to provide a second option for powering Vertigo It plugs into a standard outlet and provides 12 VDC at 1 6 A to Qwerk This tether makes it more suitable for use when Vertigo is stationary as it is for programming and diagnostics Further details on powering options can be found in Appendix B 2 5 Structure With the majority of the components now specified design of the structure could proceed This section details the various aspects of the assembly design for Vertigo 38 Jonathan Missel Structure 1 0 with focus on the structural skeleton For the purpose of discussion Figure 2 3 shows a photograph of the assembled prototype It will be referred to as the design is explained starting from the top at the webcam and finishing with the
66. arying sphere mass and sphere f dI S BA AA A ee ee A 93 Inverse condition number of controllability matrix while varying sphere mass and sphere radius o oo a 94 Inverse condition number of observability matrix while varying sphere mass and sphere radius Seras a a 94 3 20 3 21 4 1 4 2 5 1 5 2 5 3 5 4 5 9 5 6 5 7 5 8 5 9 5 10 5 11 5 12 5 13 5 14 5 15 5 16 5 17 Determinant of controllability matrix while varying sphere mass and sphere radis Ww pd fo xe a id is Ne Se ele Inverse condition number of controllability matrix while varying sphere mass and sphere radius with fixed rotational inertias Onboard sensing communication architecture Ground based communication schematic 4 Ground based free body diagram 2 004 Simulation results for a circle reference trajectory norm error 0 7821 Planar sphere based Vertigo model 00004 Gramian based control for 27 translation in 1 5 sec 2 2 Gramian based control u max 55 ao e ee oe Gramian based control u max 45 oaoa foe dae Se Gramian based control usmax 20 234 46 a a a Gramian based control u max 15 coo essere a Plot of successful u max convergences and associated final times State response with stabilizing LQR controller 2 2 Response with stabilizing LQR controller wrt vertical State response with augmented stabilizing LQR controller
67. ased control the first balancing controller exploited the safety and predictability of if statements to establish the coordinate error and control signal convention This if control was designed to balance an 8 foot long 2x12 inch plank that pivoted at the center By translating along the top of the plank 169 Jonathan Missel Balancing Implementation Vertigo was able to effectively control the pitch of the board If the pitch error was negative Vertigo would move forward and vice versa This program was a valuable step because it exposed the need to keep all forms of motion in check not just for performance but safety As the robot moved about small disturbances in y and yaw accumulated eventually causing Vertigo to fall off the board This was successfully addressed by adding y and yaw components to the ifcontrol The only thing preventing this controller from balancing the board indefinitely was battery life and Vicon connection lapses These lapses caused Vertigo to quickly run off the board before there was time to catch it More protection was clearly needed for balancing experimentation to safely continue therefore a 6 inch lawnmower tire inner tube was stretched around the middle of Vertigo Partially inflated this served well as a durable bumper for damping crash impacts Controlling board pitch became slightly more intelligent by converting to pro portional control for x y and yaw The gains and magnitudes discovered in t
68. ast the motor s gear head is to change the diameter of the wheels That being said a larger sphere is still easier to balance on The reason for this is apparent when considering the uncontrolled dynamics As Vertigo tips over it is also rotating the sphere A sphere with a larger diameter will travel along the ground farther per degree of rota tion Since this translation is in the same direction as the control needed to balance a larger diameter sphere is naturally easier to stabilize In addition a larger sphere will raise the center of gravity which is beneficial as will be discussed later in this appendix 195 Jonathan Missel Weight Distribution Inertia is a property of matter which states that more massive objects have greater resistance to change in motion Rotational inertia moment of inertia is a similar principle that deals with resistance to change in rotation rate and is a function of mass and geometry To investigate the affects of sphere mass the size can be assumed fixed so both forms of inertia will increase with mass If we consider an extremely massive sphere it will be extremely resistant to change in motion which means Vertigo will be able to balance much more easily It would be analogous to a fixed sphere where Vertigo would simply have to climb to the top and then no more control would be necessary The fatal drawback is that greater inertia requires more control input to accelerate That is when navigatio
69. at Ox604cO000 0 0 eee Erase from 0x60fe0000 0x61000000 Program from 0x01fdf000 Ox01fff000 at Ox60fe0000 RedBoot gt The difference is simply the value for the l argument which specifies the length Revision 1 1 qwerks have 8 megs of flash RAM but revision 1 2 qwerks have 16 megs of flash RAM Using a value of 0xb00000 for the revision 1 2 qwerks enables the use of the extra 8 megs of RAM Testing the Linux Installation Now either enter reset or power cycle the Qwerk so that your changes take effect but don t type CTRL C because we want the Qwerk to boot up into Linux You should something similar to the following RedBoot gt Ethernet eth0 MAC address 00 dc 6c 7d 6b 25 IP 192 168 0 3 255 255 255 0 Gateway 192 168 0 1 Default server 192 168 0 2 DNS server IP 192 168 1 1 RedBoot tm bootstrap and debug environment ROMRAM Non certified release version v2_0 built 13 09 18 Mar 21 2006 Platform Cirrus Logic EDB9302 Board ARM920T Rev A Copyright C 2000 2001 2002 Red Hat Inc RAM 0x00000000 0x02000000 0x00041fa8 0x01fdd000 available FLASH 0x60000000 0x60800000 64 blocks of 0x00020000 bytes each Executing boot script in 1 000 seconds enter C to abort RedBoot gt fis load ramdisk RedBoot gt fis load zImage 235 Jonathan Missel Changing Qwerk s Embedded Code RedBoot gt exec r 0x800000 s 0x600000 Using base address 0x00080000 and length 0x00180000 Uncompressing Li u
70. athan Missel Future Improvements The solution to this is fast and simple Simply mounting them perpendicularly will cancel out their products of inertia and improve the system s dynamics Vertigo s design gives it a high degree of flexibility in its center mass and iner tia However this reconfiguration requires some disassembly which may be too time consuming for a given experiment depending on what is being moved To make the physical properties of Vertigo even more flexible simple appendages with sliding and interchangeable weights could be added They could be designed to mount in sev eral directions and support weights of various magnitudes in different locations This would help experimentally prove the theoretically suggested concept that it is easier to balance with a higher center of gravity It would also help explore the affects that coupling inertias have on the system dynamics Lastly Vertigo is currently initialized in Vicon by being placed on the ground as a reference The controller is then applied and then the body placed on the sphere man ually This should be done quickly and accurately to prevent human interference with the control almost to the point where it is aligned a short distance above the sphere and dropped If hands are trying to guide the body onto the sphere while the con troller is also trying to adjust its position the system oscillates and can even diverge because both inputs are acting in the same di
71. bilizing LQR controller wrt vertical 129 Jonathan Missel Sphere Based Control design In the original LQR tuning equal weighting was assigned to all states how ever the best results for the augmented system were obtained when the angle states had greater weighting to ensure that they were forced to zero Figure 5 12 shows that the balancing LQR controller designed with an augmented A matrix was much more effective at managing the states and increasing performance The steady state bias of the angles was reduced by an order of magnitude and the rates appear to reach zero much faster than they did with the original controller Therefore the augmented controller yields less translational error Figure 5 13 is the response of the body with respect to the vertical inertial axis This clearly shows that the body stabilized in under two seconds which is less than half the original time and required less control to do so 5 3 5 Tracking LOR After balance the second type of motion that must be controlled for Vertigo is trans lation The details of this motion are often conceptually counterintuitive and are discussed in Appendix A With some cleaver manipulation LQR techniques can also be used to develop a tracking controller for translation By definition LQR is a full state feedback method that drives the states to zero therefore if the states are redefined as the difference between the current and desired states the controller wil
72. cific instructions are presented first then the Terk given instructions for installation of programs and running code 1 Open JCreator LE version 4 50 010 2 File open workspace a VERTIGO QWERK MyFirst Vertigo terk client MyFirstRobot b Choose workspace 3 Choose the program you would like to run 4 Compile the project by either hitting F7 or going to Build gt Compile Project in the top menu 5 Run the project by going to Build gt Execute File in the top menu Note that you must execute the file NOT the project 202 Jonathan Missel Connection to Vertigo 6 Running the program will open a graphical interface To connect to a robot click on the Connect button 7 From here follow steps 5 and 6 from the previous section Through TERK Program Launched From Site THESE ARE THE PROVIDED INSTRUCTIONS FOR INSTALLING AND CONNECTING THROUGH JAVA ROBOT CLIENT QUICKSTART Purpose This document describes how to install compile and run the MyFirstRobot pro gram It also briefly describes important other files in this folder Installation Create a directory for the MyFirstRobot program and its associated files Unzip the terk client MyFirstRobot zip file into the directory you just created You will need to have Sun s Java SE JDK 5 0 installed If it s not already installed you can download it at 203 Jonathan Missel Connection to Vertigo http java sun com ja
73. convention and notation from 29 will be used here To apply the filter the model must take on the following form x f x u t G t w t w t N 0 Q t 6 1 Ve h x Vk Vk Y N 0 Ry 6 2 Where x and yz are respectively the state and measurement models f is the system s nonlinear equations of motion with oe D S D as the state vector h x converts the states into measurement space and w t and v are zero mean Gaussian white noise processes The G matrix allocates the process noise and is defined as 140 Jonathan Missel Application of Continuous Discrete EKF 0 1 where the first two rows are zeros because they represent kinematical relationships and it is always true that velocity is the time derivative of position The equations of motion implemented here are those previously derived for Vertigo The measurement model will be based on the Vicon data acquisition process until the onboard sensing is supported To relate the Vicon measurements to the states geometric and kinematic relations are required Figure 6 1 Planar measurement model for sphere based Vertigo 141 Jonathan Missel Application of Continuous Discrete EKF The Vicon motion capture system works by locating the center of spherical retro reflective markers through triangulation with multiple infrared strobe cameras With the accompanying software clusters of reflectors can be made into an object and tracked as such The s
74. cover from the wide conclusion of the upper leg and locate the actuators One of the convenient aspects of having two part legs is that if the desired angle of the actuators changes only the bottom section needs to be redesigned This saves material cost and time In this case the lower legs were designed to position the actuators sufficiently far below Qwerk so as not to limit access to its electrical connections or prevent it from being removed without disassembling the legs but not so far down that the legs were prone to vibration Note that similar to the original legs these were designed to permit deflection for improved contact with the sphere and absorbing impact However this was done in moderation to maintain the rigid body assumption Another concern that was solved by the lower legs was that colliding with walls in ground based mode could damage or dislodge the actuators if they bore the brunt of the impact For this reason the lower legs tuck the actuators under such that the sturdy leg knuckles impact first In practice this was also found to be true when operating with the sphere more often than not the rotation from tipping over had Vertigo impacting the ground horizontally and the knuckles protected all other components from damage by impacting first The angle of the lower leg s bottom face determines the actuator s angle of attack These were designed to orient the 53 Jonathan Missel Design Iteration omni whee
75. cquisition of all major components needed to make the platform functional To determine what components were required the various aspects of an implemented feedback control system were considered and found to be sensing actuation computation and power Sensing gives feedback measure ments to the control loop so that the appropriate input can be determined To carry out the control input actuators are needed to influence the motion of the system A computer is required to send and receive signals in accordance with the control loop and possibly communicate with other computers Finally all of these processes need a power supply to run These components had to be acquired before manufacturing the structure because its design was dependent on their size weight and mounting This section explains how the components were deemed necessary the criteria they had to meet and the specifications of the chosen items 28 Jonathan Missel Components 2 4 1 Onboard Computer Intelligent robots need a brain to support their functions and Vertigo is no excep tion As the central hub of the control loop the computer is charged with taking in sensory signals determining the appropriate control and generating an output signal for actuation To determine the control the computer must either apply its own embedded controller or transmit signals to and from an external computer that does so Therefore the onboard computer should accommodate a broad r
76. critical as pect to its success Controlling one spherical surface in multiple directions is not a straightforward task because multiple actuators are required to be in contact with the sphere while avoiding conflicting control By studying the geometry and motion of a sphere four solutions to this problem were generated Figure 2 shows the front 45 degree and top view for the four concepts The commonality in these methods is their omni directional control but there are many differences in their complexity practicality and performance The first idea concept A came from contemplating the motion of a basketball spinning on a finger If the finger is assumed to be a single point of contact then the sphere can rotate about it and the point will be uninfluenced by the rotation To control the sphere s rotation a second point of contact a roller in this case must be present To avoid interference the roller must be constrained to only apply control in the direction the sphere would naturally rotate about the first point Now if both points of contact are rollers it can be deduced that omni directionality can only be achieved if the axis of both rollers are in a plane that intersects the largest diameter of the sphere Furthermore control will be coupled for all orientations in that plane unless the two rollers are perpendicular Since the sphere must be placed on the ground a third point of uncontrolled contact is introduced To account for t
77. ct is further along and more complex trajectories are attempted the more inclusive dynamics can be exploited Since Ballbot does not have control in yaw Carnegie Mellon s team was also able to derive its equations of motion for a planar model 3 Their model will be followed here because it is very well reasoned with logical assumptions and it closely describes Vertigo s planar motion A vast majority of introductory analysis estimation and control techniques are designed for use on linear systems Nonlinear systems such as Vertigo can take advantage of these methods with local accuracy if the system is linearized about the point of interest Vertigo s nonlinear equations of motion will be linearized with a first order Taylor series expansion approach Ground based equations of motion are then developed for use in that configu ration The model used yields linear equations so linearization is not necessary All three sets of equations proved to be sufficiently accurate for drawing conclusions 68 Jonathan Missel Mathematical Model about the characteristics of Vertigo s motion and were used for the analysis esti mation and control that follows The reconfigurable nature of this platform means that the parameter values may change slightly between the various simulations and implementations but the same model structures are used throughout Mathematical models will have errors based on measurement uncertainties and simplifying
78. cting to improve controllability the matrix becomes more ill conditioned with increasing mass This shows that increasing rotational inertia reduces the required control effort for recovery Sphere Design Analysis In this section controllability and observability are first checked by plotting the rank of these two matrices while the sphere design parameters sphere mass and sphere ra dius are varied Consistent with previous plots the vertical blue line in the following figures represents the current configuration of Vertigo s parameters Figures 3 16 and 3 17 show that the matrices have full rank so it is both control lable and observable for this range of sphere design parameters Figure 3 18 and 3 19 respectively look further and show how the condition of the controllability and ob servability matrices are affected as sphere mass and sphere radius are varied The controllability matrix is shown to become better conditioned with both increasing mass and radius This is because as the radius gets larger the distance from the actuators to the ground increases and it requires less control efforts to generate the torque needed to move the sphere This is similar to increasing As the mass gets larger the sphere has more inertia making it harder to move however the control efforts would then have more of a tendency to rotate Vertigo back to the top of the sphere rather than move the sphere under the body In practice there is a b
79. d course of ac tion The control input must then be delivered to actuators to persuade the system accordingly The response is measured by the sensors and the loop is repeated This generic schematic can take on many forms depending on the application and does so even for the various methods of controlling Vertigo This chapter explains the implemented communication methods for onboard sens ing and external sensing Communication for both 1 0 and 2 1 generations of Vertigo was almost identical so only the 2 1 communication is included here and acceptations for 1 0 are explained as they arise Additionally the sphere and ground based modes of Vertigo 2 1 had relatively similar communication networks differing most dramat ically in their applied control and estimation therefore an instance of ground based control is used to illustrate the method and the variations in control and estimation 99 Jonathan Missel Onboard Sensing Architecture are covered in their respective chapters 4 2 Onboard Sensing Architecture To measure the dynamics of this system Vertigo was fitted with onboard sensors that were intended to transmit measurements through Qwerk for feedback in the control loop These devices included the quadrature encoders Inertial Measurement Unit IMU back EMF and webcam Despite later discovering that Qwerk did not actually support the majority of these sensors as advertized they were not omitted from the system because
80. d package minicom Reading database 86463 files and directories currently installed Unpacking minicom from minicom_2 1 10_i386 deb Setting up minicom 2 1 10 Now run minicom sudo minicom Welcome to minicom 2 1 OPTIONS History Buffer F key Macros Search History Buffer 118n Compiled on Nov 5 2005 15 45 44 Press CTRL A Z for help on special keys We ll now configure minicom to be able to talk to the Qwerk Minicom should be set to use 57600 baud 8 N 1 8 data bits no parity 1 stop bit and no software or 224 Jonathan Missel Changing Qwerk s Embedded Code hardware flow control Type CTRL A Z to bring up the main menu Minicom Command Summary Commands can be called by CTRL A lt key gt Main Functions Other Functions Capture on off L Hangup H eXit and reset X send break F initialize Modem M Quit with no reset Q scroll Back B Select function or press Enter for none Written by Miquel van Smoorenburg 1991 1995 Some additions by Jukka Lahtinen 1997 2000 i18n by Arnaldo Carvalho de Melo 1998 Enter O to configure minicom configuration Filenames and paths 225 Jonathan Missel Changing Qwerk s Embedded Code File transfer protocols Serial port setup Modem and dialing Screen and keyboard Save setup as dfl Save setup as Exit Now use the arrow keys
81. ds on both the permitted slip from the wheels and the point of rotation idea from concept A to decouple control Together these three exhaustively compose the entire set of possible ways to independently control all three degrees of freedom using just three actuated omni wheels If the position or orientation of these actuators is off even slightly control input will conflict and become coupled Accordingly this system is highly dependent on geometry and will only work with a specific size sphere which is one of its flaws Furthermore this concept cannot operate on the ground This is because it relies on the geometry of the sphere to locate points of rotation when decoupling control Additionally in all three configurations of concept C passive contact with the sphere is needed to support the robot In the previous methods symmetry meant that control design was identical for both roll and pitch so a planar model could be used to design a single controller which could be applied to both axes That is not the case here all three degrees of freedom would require their own slightly different controller to account for their varied locations and lack of symmetry 25 Jonathan Missel Actuation Method Concept D shown in Figure 2 2 d also includes yaw and omni directional control This method uses four spherically orthogonal omni wheels and is not dependant on the size of the sphere Opposite pairs of actuators independently control roll
82. dure you will need root access to in Linux so the SENS accounts 214 Jonathan Missel Changing Qwerk s Embedded Code through the school will not work You may need to install a virtual machine on your computer This can be done by installing Sun xVM VirtualBox free online Also ubuntu 6 06 1 desktop i386 is needed More current versions will not compile the code correctly You will also need to install the drivers for the USB Serial adapter Remember the port that the adapter is plugged into when you do this because it will always need to be plugged into the same port when connecting to and altering the embedded Qwerk code Procedure Compiling Firmware Open Sun xVM VirtualBox NOTE Remove all flash drives SD cards and external hard drives Also the blue USD to serial adapter must be plugged in to the port it was in during installation of drivers See figure below for current location on the editing computer Click state It will boot up then ask for username and password Username jwmissel these specifics will change with the user Password password In Linux go to applications accessories terminal In terminal type in cd terk embed trunk share src terkapi Then type gedit Client cpp amp This is case sensitive Executing this command will open up a new window with Qwerk s source code in it You will now need to find the lines with the gains for the motor controller and change them as desired To make this fa
83. e 73 Jonathan Missel Mathematical Model formulation x Ax Bu this can be reduced to taking the Jacobian of the nonlinear equation with respect to the states to obtain the A matrix and with respect to the inputs for the B matrix This process takes on the form A Ox Do Ox _ oy Ou aa A xX xX Xx X u u u u u u u u where the barred values represent the point being linearized about When the system linearization is transformed to the origin all states are zero and the state space becomes 0 0 1 0 0 0 0 0 1 0 A B 3 18 43 83 43 83 0342 0878 43 9 79 31 79 31 0439 1228 61 4 1000 010 0 C D 0 3 19 0 10 C0 0001 This model represents Vertigo with its completed communications architecture where all onboard sensing is assumed fully operational Accordingly is the iden tity matrix because all states are measured The extreme size of the symbolic state space equations made them unreasonable for inclusion here so the above model with numerical values is given Note that these values represent the parameters for a spe 74 Jonathan Missel Mathematical Model cific configuration of Vertigo and is linearized about the vertical equilibrium As the equation parameters and the point of linearization changes so too will these values but the general results that this model yields are still valid 3 2 4 Ground Based Derivation As mentioned before Vertigo s
84. e must be no unobservable modes The above state controllability and observability analysis verifies that this system meets these requirements securing the validity of LQR control for this system Two different objectives were considered for LQR controller design to manage the two types of planar motion that Vertigo is capable of The first was for balance and the second was for point to point translation The resulting controllers were imple mented in a linear simulation to verify their accuracy for theoretically controlling the linear system When it was established that this method was working properly sim ulations were run with the linear control methods on the nonlinear system to mimic the physical system These nonlinear simulations are the focus of this section 125 Jonathan Missel Sphere Based Control 5 3 4 Stabilizing LQR For the balance of Vertigo a linear full state feedback controller was developed using LQR methods This process was executed and simulated in MATLAB Figure 5 10 shows the results of this simulation when applied to the nonlinear system using MAT LAB s ode45 integration command The initial conditions for this and the remainder of the LQR simulations are bounded random values to ensure accuracy for all rea sonable scenarios 0 3 0 1 m w D G Cc S amp D 5 fa la S D e T 3 3 E ko 3 9 Y D_ Figure 5 10 State res
85. e different sizes They suspended the motors such that when assembled the bottoms of the legs made perfect tangential contact with the largest diameter of the motor This formed a very stiff assembly and transferred most of the pressure away from the mounts therefore they did not have to support the weight of the body which allowed them to be smaller and lighter To give the design some flexibility the motors could be adjusted by sliding them further in or out of the clamps This let the posture of Vertigo be very quickly altered to raise or lower the center of gravity and take on a wider or narrower stance 54 Jonathan Missel Design Iteration If desired the motors could even be completely reversed keeping all of Vertigo s control contact very near the top of the sphere Lastly the clamps were designed to concentrate most of their flexing at the bottom This kept the upper half more rigid ensuring a secure and consistent fit to the legs It also gave the assembly more clearance to fit smaller spheres without rubbing on the mounts All leg and motor mount components were again made of Delrin and machined with a CNC mill as seen in Figure 2 7 Figure 2 7 CNC mill machining Delrin legs for Vertigo 2 1 Drive method D utilized four motors rather than two which gave rise to another design decision Purchasing two more of the original motors would be less expensive but their shortcomings would still be present These motors
86. e erroneous and the filter tuning is compromised Despite the shortcomings these attempts were not in vain EKF is extremely powerful and it is likely that continued work will enable it to provide state estimates reliable enough for Vertigo to balance on the sphere 175 Chapter 8 Conclusions and Future Work This thesis introduced an autonomous biaxial robotic inverted pendulum as a new experimental platform for research in controls and dynamics Designed to balance and navigate on a sphere it is theoretically capable of exceptionally agile motion This along with its reconfigurability enables it to model a wide range of other more complex systems Its unique abilities also permit original work in rover locomotion and as an educational tool Covered in this thesis were the design mathematical modeling and analysis communication architecture control estimation and early implementation As an original platform proper hardware design was a major effort The original objective was to create a simplistic and robust testbed with minimal restrictions on its motion The first prototype Vertigo 1 0 balanced on a sphere and had omni directional motion Actuation methods were the cornerstone for enabling its dynam ics A second design iteration Vertigo 2 1 was then developed with new actuation that offered the addition of yaw control to the existing omni directionality and biaxial inverted pendulum dynamics Both of these generations re
87. e errors fm 25Hz fs 100Hz 150 Sampling time mesh Lores ita le OO ee See eS 152 Proportional two dimensional step track 160 Circle trajectory tracking using proportional controller with propor tional yaw control norm error 2 3897 0004 162 Circle trajectory tracking using PID controller with proportional yaw control norm error 1 8976 owe a as deal oe A 162 Circle trajectory tracking without yaw control norm error 0 8614 163 xii 7 5 7 6 T T 7 8 7 9 7 10 7 11 7 12 A l A 2 A 3 A 4 A 5 A 6 D 1 Circle trajectory tracking with proportional yaw control norm error 0 3866 163 4 circle trajectory tracking laps with proportional yaw control 165 Circle trajectory tracking with transformed control norm error 0 5402 166 Circle reference trajectory tracking while using proportional yaw con trol for various retrogrades 2 o ee ee 167 Yaw control for 27 retrograde and 207 retrograde 169 Balancing plank pitch with proportional control 171 Balancing plank pitch with PID control 173 Balancing plank pitch with PID control by tracking Vertigo 173 Directionality a Bi directionality b Bi directionality with yaw c Omni directionality d Omni directionality with yaw 185 Stages of translational motion ERA RA 187 Spherically orthogonal omni directional actuation 189 Ground bas
88. e many desired characteristics for robotic platforms they are mostly rooted in performance requirements and are challenging to satisfy wholly with a sin gle unit Specifications are dependent on objective however autonomy exceptional mobility responsive dynamics and cost effectiveness are among the most commonly sought after traits Figure 1 1 shows a series of robots that give a rough idea of what is available Figure 1 1 a Stationary manipulator b Statically stable vehicles c Omni directional statically stable vehicle d Differential drive balancing vehicle e Bipedal walker Jonathan Missel Motivation Figure 1 1 a shows a stationary robotic arm These are useful in assembly appli cations where their dexterity accuracy and stamina often find them performing tasks that would be too mundane difficult or tedious for humans As with the numerous bench top robotics in existence these are very precise and relatively easy to model but as stationary robots their applications are limited To provide mobility statically stable vehicles like the example in Figure 1 1 b are often the default platform To be statically mechanically stable simply means that on a flat surface the robot will not fall over under the acceleration of gravity when all control is terminated These units are easily understood and well documented which makes their implementation more straightforward however limits circumscribe their mobility They
89. e on this power supply to minimize the risk of undesired loss of power which may potentially damage Qwerk s hardware 199 Jonathan Missel Powering Vertigo and Qwerk or software This supply was lifted off of an old computer The wiring was cleaned up removing all unnecessary leads and it was fitted with a 12 foot power tether that connects directly to Qwerk s power input Also added was a master power switch and green LED that indicates the unit is on 200 Appendix C Connection to Vertigo Method 1 Through TERK program launched from site 1 Go to www terk ri cmu edu software index php a Choose program and launch it 2 Close internet 3 Plug router in 4 Power Vertigo Qwerk a Qwerk will automatically connect i LED 6 will blink when connection is established ii It is normal for LEDs 0 1 2 and or 3 to be blinking or solid A I think LED 3 blinking means it is connected to another comp 5 Connect computer to LAIRS router 201 Jonathan Missel Connection to Vertigo a Wirelessly or cable in i Password lairs b Can check the router status by typing 192 168 1 1 into web browser user name leave this blank password admin i StatusLocal NetworksDHCP Clients Table ii Should see ip addresses of both the computer and Qwerk 6 In the launched software click the Connect button a Connect directly to a peer b Type in ip address of Vertigo and connect Vertigo spe
90. e simulation coding to take a user defined maximum control input and initial guess for final time It then calculated the associated power optimal Gramian based controller and determined the maximum control required to carry this out It defined an error between this and the specified allowable value then updated the guess for final time based on this error and a scaling factors This is effectively a free final time method that finds the minimum power controller to meet the state and input constraints in the minimum time This was carried out for several specified maximum u values and some interesting results were uncovered Figures 5 5 5 8 depict linear simulations of the Gramian based control with decreasing allowable uma values In Figure 5 5 Umar 55 and the minimum time that this maneuver was able to be carried out in while still meeting the u constraint was tf 1 2635 seconds and all of the final states were taken to the appropriate values Figure 5 6 represents the estimated maximum allowable input for Vertigo with Umar 45 and the minimum time that this maneuver was able to be carried out in was tf 1 3749 seconds and all of the final states were taken to the appropriate values In Figure 5 7 Umar 20 the minimum time that this maneuver was able to be carried out in was tf 2 0154 seconds but the final state values do not meet the specified constraints perfectly 120 Jonathan Missel Sphere Based Control Amplitude
91. e taken as inputs in a Simulink or MATLAB control loop The loop calculated the appropriate control to correct the system s motion and then wirelessly sent it back to Qwerk where voltages were prescribed to the motors When the system responded the loop was repeated The intricacies of these lines of communication were rather complex and will be detailed later they were spoken of in generalities here because this method had not yet been implemented leaving no specific details to speak of 101 Jonathan Missel External Sensing Architecture 4 3 External Sensing Architecture All implemented forms of Vertigo s communication architectures enact the same gen eral functions sense compute and actuate They differ mostly in their means of accomplishing these tasks The schematic in Figure 4 2 depicts a specific instance of ground based control as an illustrative example of the methodology To avoid redundancy and excess detail the major aspects of this description can be applied to sphere based control as well Since onboard sensing was not available Vicon an external motion capture system was used for measuring the dynamics Vicon works by locating retro reflective markers through triangulation of multiple infrared strobe cameras The infrared light emitted is a high frequency electromagnetic wave that borders that of visible red light Working with such short wavelengths allows for highly accurate measurement of the position of each mark
92. e zero initial states represent vertical stature that is stationary at the origin and E is the expected value The Kalman gain is calculated with the formula Ky Py He O He amp Py He amp q Ra 6 8 143 Jonathan Missel Application of Continuous Discrete EKF In this equation H x hx x is the Jacobian linearized measurement model Based on the geometry of the model schematic this works out to be 1 1 0 0 poe 6 9 2r l cos 0 9 l cos 0 GH 0 0 When a new measurements is available the Kalman gain can be used to update the state estimate and covariance by RE amp Ke Fe OR 6 10 Pi Es Pe 6 11 If a measurement is not available the estimate remains and is used as the initial condition for propagation forward in time The propagated values are taken as the a priory estimate for the next step just as the initialized values were for the first step The equations to be propagated are x t X t u t t 6 12 P t FR OPE PAF R t t G Q t G7 t 6 13 Where F x t t X t is essentially the A matrix from the linearized equations of motion These covariance and state equations are nonlinear so they must be integrated forward in time Since the covariance and state propagations are coupled the two equations must be integrated simultaneously This was done using the ode45 function in MATLAB by combining both systems Once propagation is carried out
93. ect mode you will now be prompted to enter the IP address of the robot Do so and click Connect then click Finish You should now be connected to the robot If you entered relay mode you will be prompted to enter your TeRK login and password Enter your login click the Login button and then click Next to go to the next page in the connection wizard You will see a list of available robots Highlight the appropriate robot by clicking on it once Then click the Connect and the Finish buttons You should now be connected to the robot Once you are connected to a robot you can begin displaying an image stream from the robot s onboard camera by clicking on Start Video Once started you can pause video at any time by hitting Pause Video and capture images from the video stream by clicking Save Picture To run your program hit the Play button To stop your program hit Stop Several other example files are available and visible in the file view They are Headturner java Photovore java hearingTest java and soundTest java You can compile and run these as you did for MyFirstRobot java Compiling and Running the MyFirstRobot Program from Command Line 205 Jonathan Missel Connection to Vertigo Open a command prompt and set the current directory to the directory you created during installation Type compile no quotes at the command prompt to c
94. ecting them meant acquiring source code for Qwerk deciphering the interactions of hundreds of uncommented C files developing a method for correcting the errors rewriting the source code compiling new firmware deleting Qwerk s existing firmware uploading the new image and reconfiguring the connec tion settings This was an extremely time consuming and sensitive process that had potential to damage Qwerk and required that a virtual machine be installed to gain root access to Linux Even after developing a structured routine this remained a two hour long process when everything worked properly The time and high risk associated with this process made it important that the problems be fixed with minimal iteration Gains that were too high caused extreme actuator saturation and the process became even more risky The Ziegler Nichols tuning method was used to safely improve the PID gains this gave much greater weighting to the P and D gain and almost zero to the I gain 25 Also the embedded trajectory generation files were rewritten to include an acceleration parameter that defined the slope of the angular velocity profile The initial configuration of this code would have had an acceleration parameter of 0 6 this was changed to 200 provoking the original gentle trapezoidal profile into an approximate step function 107 Jonathan Missel External Sensing Architecture For this reason the PID gains were tuned to best track a step funct
95. ed The first method was accomplished by fixing the yaw angle In this situation a proportional controller regulated the angle most often driving it to zero This way Vertigo was only prescribed translational motion Simple proportional control was found to be more than sufficient because yaw requirements were not very demand ing For gain determination it was increased until recovery from a gt step showed 156 Jonathan Missel Implementation Challenges overshoot The gain just prior to overshoot was chosen The second control strategy was similar but assigned a trajectory rather than a fixed angle This most often took a form resembling a retrograde orbit in ground based control Before a yaw trajectory could be applied control input had to be transformed from the inertial to the body frame This was done by multiplying the positions with a transformation matrix that accounted for the yaw angle therefore position control input was made statically independent from yaw at the time of measurement To perform the retrograde reference trajectory a user defined angular yaw frequency was multiplied by time and translation was assigned separately By applying a proportional yaw controller Vertigo was forced to follow the proposed trajectory Implementation of this strategy was difficult because Vicon returned its angle measurements between m and r Adding m to the measurement signal fixed the problem for conservative trajectories but
96. ed trace of translation with rotation 190 Inverted pendulum on cart o oo rta A 193 Uncontrolled pivot drift ases da a 194 Wiring for access to Qwerk aoaaa bed a Gok oe BA es 217 xiii List of Tables 2 1 Features of drive concepts a ee A ifaw ee le abo We ee 2 2 Properties of Delrin y E eras s Blt ok he doe Ik bei Subs A xlv Abstract The limited versatility of existing experimental robotic platforms has left demand for a single practical unit with broad application in dynamics and controls To address this an autonomous biaxial robotic inverted pendulum is proposed that balances and navigates on top of a sphere The spherical base and innovative actuation techniques permit omni directional translation and direct control of yaw With this unprece dented motion Vertigo as it is called is capable of modeling a myriad of other systems and offers great potential for original work To maximize versatility it was designed for reconfigurability can accept spheres of different size and can operate on the ground as a more traditional omni directional vehicle This thesis focuses on the design mathematical modeling analysis communication control estimation and early stages of implementation for Vertigo and its various configurations Methods used in this development have general application making the results and findings broadly relevant XV Chapter 1 Introduction 1 1 Motivation The conforming
97. eld the wireless USB adapter the IMU connector and the 51 Jonathan Missel Design Iteration retro reflective markers used for tracking with Vicon see Chapter 4 Further down the legs larger cutouts were included to make wrapping and securing wires easier Attached to the underside of the top plate is a speaker which makes use of Qwerk s audio compatibility This was initially included for amusement but proved to be very useful in debugging code as different tones were scattered throughout implemented programs to pinpoint errors To fortify protection in the highly chaotic early stages of implementation a pneu matic rubber bumper not shown was stretched around the narrowest span of the legs It consisted of a nine inch lawnmower inner tube inflated to half capacity Rub ber construction made it durable to resist puncture and with its volume primarily occupied by air its mass and inertia did little to affect the system s dynamics The bumper provided an inexpensive and affective buffer that cushioned most components from impact Continuing down the robot the next major design change was the pronounced kinks in the legs These were included for several reasons In Vertigo 1 0 accessing Qwerk s ports was somewhat difficult The USB ports were particularly tight and the plastic casings on the plugs had to be shaved down Increasing the distance between Qwerk and the legs not only freed up all of the electrical connections but
98. empted for sphere based control but acceptable results were not attained These included an analytical power time optimal method a power optimal shooting method for translation and a gradient based method for translation These techniques are highly sensitive to tuning and initial guesses but are very close to converging properly therefore continued work on them will likely be fruitful Success is especially hopeful for the gradient based and shooting methods as they are complementary methods and are both near completion If either one of these methods is worked out its results can be used as initial guesses for the other For estimation the problem was put into the Kalman filter framework and simu lated These results showed its success at providing estimates for unmeasured states while attenuating noisy signals It was then shown that increasing the number of loop iterations between measurements updates improves the estimate further To fully ex plore the impact of sampling times the simulation was run at twenty five different combinations of measurement and output sampling times The magnitudes of the set tled 30 bounds were plotted as a mesh surface which showed more measurements and loop iterations improved accuracy but the affect was much more pronounced for mea surement sampling The EKF is not a true optimal filter because it applies a linear method to a system that is by definition nonlinear however it can still be used with exc
99. end result for this scenario would be a track which resembles a wavy bias error The problem of translational control divergence is a function of the yaw rotational velocity and the sampling time of the system In practice fast sampling times and conservative yaw trajectories help to mitigate these errors For most practical appli cations a yaw trajectory can be determined that allows superposition to be taken advantage of with negligible error If more evasive maneuvers are required empiri cal knowledge of the predicted error can be applied to help compensate This error can also be determined mathematically however constant sampling time must be assumed and this is not the case with Vertigo s current control loop A 6 Pivot Drift A famous benchmark problem in control theory is the inverted pendulum on a cart where the angle and position of the pendulum are controlled with an input force applied to the cart as seen in Figure A 5 In many ways the dynamics of this system are conceptually similar to those of Vertigo both are inverted pendulums and both apply an input force or torque that controls the base However the feature that differentiates these two problems is the sphere that Vertigo balances on Balancing on the sphere has two dominant affects on the dynamics that prevent the control methodology from being used interchangeably between these two problems the first 192 Jonathan Missel Pivot Drift Figure A 5 Inv
100. ent and future states while minimizing the mean of the squared error In this way it is analogous to applying a standard discrete estimator but optimally placing the poles with a minimum variance approach To be implemented the filtered process must be modeled in the frame work of the Kalman filter which assumes a zero mean Gaussian model for both process and mea surement noise This framework further assumes that the model and measurement noise are uncorrelated from each other and in time Once modeled the error covari ances and states must be initialized Based on this knowledge an optimal expression for computing the estimator gain is applied State and covariance estimates are then 138 Jonathan Missel Continuous Discrete Extended Kalman Filter updated using their current values the calculated gain and the measurement model Lastly the states and covariances are propagated forward in time to be used as the a priori estimates for the next time interval The process is repeated for each time step thereafter This explanation is for the discrete linear system and follows the original development presented in 1960 when R E Kalman first published a paper on the topic 28 Further development expanded the filters application to other types of systems The extended Kalman filter EKF is a variation that accommodates nonlinear systems by using their linearized simplifications When linearizing these equations the Jacobian is taken
101. eptional results as demonstrated here More work can be done to use this filter to estimate unknown parameters of the system This is an extremely advantageous feature of the EKF because it adds functionality as a real time system identification technique to update the dynamic model and improve performance For Vertigo the unknown friction coefficients are likely to be the most beneficial parameter to esti mate because they cannot be measured and pertain to slip Work on this has already begun but revealed that defining new states that included an unknown parameter 179 Jonathan Missel Conclusions and Future Work make the system unobservable and the process is not applicable Continued research in this area is likely to uncover a method for circumventing this In accordance with the objectives of this research the majority of future work will be in implementation A large bank of ground based and balancing controllers were successfully implemented to serve as a spring board for continued development of sphere based implementation With the current communication network further work on the implemented EKF and LQR combination is most promising for balanc ing on the sphere However progress should flourish with the addition of improved omni wheels and the updated Qwerk firmware when it is released The onboard sensing loop architecture should solve many of the timing and connection problems currently faced Also as the mathematical model
102. equipment as needed In any given assembly one side of a plate is completely empty and the IMU and Qwerk were left exposed to serve as additional 46 Jonathan Missel Structure mounting surfaces Symmetry was stressed to permit simplifying assumptions in the mathematical modeling This was achieved by designing and purchasing symmetrical components and assembling them such that coronal and sagittal planes had optimally similar inertias The major departure from this was in the motor mount and assembly this offset the wheels but was not severe and was consistent for all four Therefore the angle could be determined and accounted for in the dynamics if deemed necessary Having the privilege of designing and building a system to best fit its equations of motion was unusual but welcomed Lastly manufacturability and assembly was at the forefront of design for every part It is very common that engineers design parts to meet their mechanical objective in an assembly but their prints get sent back from the machine shop because they are infeasible or impractical to manufacture By understanding the processes used in manufacturing the parts cost time and consistency can be improved with intentioned design All of Vertigo 1 0 s structural components were cut out on a CNC mill with a 1 8 in end mill Therefore 2 D designs were used to simplify production and all inner radii were limited to 1 16 in Flat or 2 D components are easier and les
103. er The relative speeds of ground vehicles does not produce a large enough shift for Vicon to register prevent ing the Doppler effect from fully being exploited therefore all rate states had to be calculated externally with numerical time derivatives With Vicon s accompanying software multiple reflectors can be made into an object and position and attitude are tracked as such It is important to recognize that the process by which Vicon defines the object is dependent on the markers however the object s height size attitude and origin do not necessarily reflect that of the true body Therefore the actual center of mass may be slightly misrepresented by the Vicon object s origin Attitude tracking has a similar concern When the object is defined its Euler angles are automatically aligned with the inertial frame of the system However the inertial frame is defined somewhat arbitrarily making orientation of the body for accurate initialization difficult As a result the attitude of the body is often misrepresented by the Vicon object as well These discrepancies 102 Jonathan Missel External Sensing Architecture were made small enough to be assumed negligible in most applications of ground based control by intelligently positioning the markers and with carefully aligned initial conditions However the precision exacted by sphere based control necessitated that estimation techniques be used Unaddressed these erroneous measu
104. erted pendulum on cart is pivot drift and the second is the rotational inertia of the sphere In the pendulum on a cart problem the pivot point of the pendulum is always known to be at the hinge on the cart In Vertigo s case this point is not so easily determined Its center of rotation is a function of the body mass sphere mass iner tia rotational inertia velocity and angular rates When control is being applied this point is shifting wildly and is extremely difficult to determine To help illustrate how complex the pivot drift is Figure A 6 shows Vertigo s response to the most simplistic scenario an uncontrolled fall The figure shows snapshots of Vertigo s central axis at ten degree intervals as it falls uncontrolled Here it is assumed that the robot does not ever fall off the sphere and that only the sphere is constrained by the ground The pivot path can be traced out by the inner arch that is formed by the tangential lines The reason for this path is that Vertigo s center line must pass through the center of the sphere while at the same time the center of the sphere is moving as Vertigo falls over and it rolls along the ground The translation and rotation gen erate a path that is not characteristic of any standard polynomial or trigonometric function This thought experiment is for the most straightforward motion possible 193 Jonathan Missel Pivot Drift Figure A 6 Uncontrolled pivot drift as soon as co
105. es that are to be fed back into the controller and plant The state space representation x Ax Bu y Cx can be rewritten with this input in the following fashion x A Bk x Br 5 18 y Cx 5 19 This form reveals that k can be used to adjust the eigenvalues of A It can also be shown that both the rank and determinant of the controllability matrix do not change with state feedback This means that controllability is invariant under state feedback and the previous result showing that Vertigo is controllable in all cases also 124 Jonathan Missel Sphere Based Control holds for feedback control Establishing that Vertigo is controllable for feedback control verifies that this method can be used for controller design To optimally generate the gains the Linear Quadratic Regulator LQR method was used In LQR for a cost function J where J a Or u Ru dt 5 20 0 the control u Fx minimizes this cost where F R BTP and P is found by solving the Riccati equation From this cost function it is clear that both the states and input are reaching weighted minimums These weightings can be altered to best meet the specifications of a given problem and are often chosen or tuned experimentally when implemented on physical systems The LQR method poses additional conditions that must be met by the system in order be applied these are as follows the linear state space pair A B must be stabilizable and ther
106. ey can be used in sim ulation as proof of concept Additionally their analysis can help detect unforeseen problems that may be addressed early on or maybe avoided all together with design modifications Vertigo s model was derived mathematically because rigid body dynamics can be assumed and data was not available when first modeling the system The equations 67 Jonathan Missel Mathematical Model of motion were developed as Vertigo was being designed and they were analyzed to help make design decisions System identification methods can also be used to verify and improve the model after implementation Having an existing model will aid in that process by focusing efforts appropriately In the derivation that follows a planar model is assumed This is a simplifica tion of the actual system but a reasonable one because Vertigo was intentionally designed to be highly symmetrical so the motion will not only be the same in both the sagittal and coronal body planes but the product of inertia will not couple their motion significantly Using a simplified model helped accelerate the project by re quiring management of simpler equations and turned out to be a beneficial decision In the planar model yaw is not incorporated however this is not debilitating because control superposition can be applied in moderation for trajectories that do not de mand extravagant yaw control see Control Superposition section in Appendix A Once the proje
107. f an inverted pendulum Robotica vol 6 no 3 pp 235 241 1988 Y Ha and S Yuta Tracking control for navigation of self contained mobile inverse pendulum Institute of Information Sciences and Electronics 2000 S Y Ryo NAKAJIMA Takashi TSUBOUCHI and E KOYANAGI A devel opment of a new mechanism of an autonomous unicycle IEEE 1997 S Inc www segway com 2008 K T H Arai and N Shiroma Time scaling control of an underactuated ma nipulator Journal of Robotic Systems vol 15 pp 525 536 Sep 1998 A Vera Human and robotic exploration NASA Ames Research Center Feb 2007 Support Qwerk hardware guide Charmed Labs vol 1 Sep 2006 K Corporation Transwheel powered bidirectional wheels swivel wheels and swivel tables 2008 J Electronics Dc motor specification sheet Nov 1999 Crossbow Imu user s manual modles Imu300cc imu400c imu400cd Docu ment 7480 0008 08 June 2005 Renco Hollow shaft encoders specifications sheet RCH50 2005 T T R Kit www terk ri cmu edu Carnegie Mellon University Robotics 2007 Logitech Webcam c series specifications Communicate Overview 2009 182 21 22 23 24 25 26 27 28 29 P P Inc Technical product data sheets Delrin www plastic products com spec 2009 Faulhaber Product specification tables micromotor Jan 2008
108. f under actuated probes 12 Similarly Vertigo can be controlled to induce yaw rotation without direct control a trajectory 12 Jonathan Missel System Description can be assigned that combines pitch and roll for a resultant rotation in yaw This maneuver is analogous to the way an airplane banks to turn the rudder primary yaw actuator plays little role in carrying out substantial net changes in direction By de sign Vertigo 1 0 is under actuated because it does not have direct control of yaw so simulating this is straightforward Vertigo 2 1 can also be configured or controlled to be under actuated by only employing direct control over any two of the three degrees of freedom With intelligent control any combination of two of the four actuators can be eliminated and the system can still secondarily dictate all three degrees of freedom Understanding how three degrees of freedom can be controlled by actuating only one is somewhat abstract for this system However it can be explained with the aid of a motorcycle stunt called a circle wheelie as an illustrative example In this stunt only the rear wheel of the motorcycle used for propulsion is on the ground while the front wheel normally used for steering is suspended While maintaining this stance the bike and rider carve out circular trajectories by leaning to the side This is an extremely technical stunt because the precise control of three degrees of freedom is comple
109. fficiency of the heavily weighed rate tuning with the high performance of the heavily weighted angle tuning In addition the magnitudes of the control and rates required by the augmented controller are much more realistic than those in the un augmented response however they are still quite demanding To represent a more realistic scenario the same augmented system was more evenly tuned with rate weightings at 75 of the angle weightings Figures 5 20 and 5 21 show these results With realistic tuning this maneuver is executed in about 10 seconds and there are low expectations of the rate and input magnitudes This simulation is thought to be fairly representative of what should be reasonably expected when implemented on the physical system 134 Jonathan Missel Sphere Based Control 2 5 10 15 20 s 5 10 15 20 Time Time 20 30 0 5 10 15 20 0 5 10 15 20 Time Time Figure 5 18 State response with augmented tracking LQR controller heavily weighted angles Augmented LQR Track Response 2pi rotation Magnitude wrt Vertical radian units rate control 0 2 4 i 6 8 10 12 14 16 18 20 Time Figure 5 19 Response with augmented tracking LQR controller wrt vertical heavily weighted angles 135 Jonathan Missel Sphere Based Control 0 5 10 15 20 0 5 10 15 20 Time Time 0 5 10 15 20 0 5 10 15 20 Time Time
110. full rank is four In MATLAB the controllability and observability were calculated while varying all four states and the rank of the matrices at each point were stored in four dimensional matrices Figures 3 3 and 3 4 clearly show that this system has full rank for all values of theta and phi under consideration This range of values was logically chosen based 79 Jonathan Missel Analysis on the physical system s characteristics A range of 1 radian is nearly 60 which tests a comprehensive span of possibilities because by this point Vertigo would have fallen off of the sphere it rests on Controllability Rank a A J E phi radians 1 theta radians Figure 3 3 Rank of controllability matrix while varying 0 and Many of the results in this thesis are presented in terms of 3 0 the angle of Vertigo with respect to vertical because it is more intuitive and combines the four states into just two values However this analysis does not lend itself to such a representation because its results are not unique For given rates multiple combina tions of angle states may give the same 8 but combinations do not necessarily share controllability and observability characteristics The rank values in this analysis only inform of whether or not the system is controllable or observable To further look into what is taking place as these variables change the inverse of the condition number was calculated and plotted for both
111. generating the trajectory the tic toc MATLAB function was used to synchronize the code to real time The trajectory was specified in the same way as the simulation with sine and cosine waves for the two axes Variations of these allowed easy specification of sinusoidal and elliptical trajectories as well The 160 Jonathan Missel Ground Based Implementation proportional gain was chosen empirically the results included in Figure 7 2 represent the most successful gain found With norm error 2 3897 this controller fell short of the norm error 1 criteria but was respectable for such a simplistic form of control To work out how time derivatives would be taken a PID controller was imple mented next Numerical derivatives and integrations were used This required that a delay of one time step be induced to calculate the first time difference The Ziegler Nichols method was used to specify a starting point for the PID gains they were experimentally tuned from there Figure 7 3 shows that increasing the complexity of the controller to PID reduced the norm error by 0 5 compared to the proportional controller LQR can be thought of as a form of PD control which is similar to both P and PID control however LQR s primary advantage is that it is a method for determining the optimal gains so tuning is ideally not necessary Gains from the simulated LQR controller were used here to maintain consistency A progressive series of LQR con trollers
112. glecting opposing frictional forces mainly assumes that the rollers on the wheels do not have any appreciable resistance since they are designed to allow slip If it were deemed significant modeling this friction would depend on the system s dynamics and estimated coefficients making it highly inaccurate During some of the ground based controller development discrete time equations 76 Jonathan Missel Analysis of motion with T 0 04 were used t 1 0 04 lt 0 0240 ja 3 22 T 0 1 x 1 1976 v o o 7 3 23 The sampling time was chosen based on the average sampling time of the implemented closed loop see the Implementation Challenges section for more details The dis cretized equations were obtained with MATLAB s c2d command 3 3 Analysis 3 3 1 Introduction The objective of this section is to determine the conditions for which sphere based Vertigo can be controlled and whether its states can be reliably determined if provided knowledge of its inputs and outputs Investigating these characteristics will help build an understanding of the system and establish what should be expected of the control and estimation processes that it will rely on for balance and navigation To accomplish this an extensive controllability and observability analysis is conducted on the linearized model It is important to note that a linearized model may well capture TT Jonathan Missel Analysis the dynamics of
113. had plenty of torque but this was accomplished by gear reductions which sacrificed speed Vertigo would be able to balance and navigate with them but with severely limited performance In a fruitless attempt to rectify this problem the gearbox was modified to eliminate 59 Jonathan Missel Design Iteration one of the reductions This did increase their speed but not enough During this modification it was found that these high torque motors had gears made of ABS plastic which later failed as a result of their inadequate construction Opening up the gearbox also revealed that they were unnecessarily bulky with cavities of unused space Another drawback was that its method of gear reduction offset the output shaft from the center which made mounting all four in precisely the same way challenging Lastly the original motors required the intricate and fragile encoder assembly to measure its rotation This was unnecessary because it had since been realized that motors and encoders could be purchased as a single unit Faced with a long list of cogent arguments against using the original motors it was concluded that splurging for four completely new motors was well worth the cost New motors for Vertigo 2 1 were carefully selected to eliminate the problems expe rienced with the previous set They had to meet both the torque and speed requests of the controller as well as have embedded quadrature encoders concentric driveshafts metal gearing
114. han Missel Structure plate The aluminum mounting plate screwed into preexisting holes on the face of the motor This would fix the circuit and gave the assembly a solid and consistent stop for securing in the motor mounts A hole in the plate allowed the driveshaft to pass through the encoder assembly The encoder circuit screwed into the thin plate and was positioned such that the wiring harness plugged in along the side of the leg to shield it from impacts Figure 2 4 Vertigo 1 0 motor encoder wheel exploded subassembly CAD The encoder and omni wheels were secured to the driveshaft with an aluminum hub The hub was fitted with a setscrew that tightened on the flat of the shaft The inner half of the hub accepted the encoder wheel which was secured with a setscrew The hub was designed such that when it was seated at the bottom of the shaft the encoder wheel lined up perfectly with its slot The hub was given a stop to consistently locate both wheels from their respective ends and allow clearance between the omni 44 Jonathan Missel Structure wheel and encoder plate The omni wheel also needed to be fastened to the hub but its geometry was not ideal for fitting a setscrew To address this two mechanisms were utilized for securing the wheel For the first the hub radius was made with 002 in of interference with the wheel I D forming a snug press fit Secondly this section was made 005 in shorter than the thickness
115. he ground based proportional control experiments accelerated the process of finding ap propriate gains here The proportional controller easily balanced the board however the response was choppy and certain gain values induced oscillatory overshooting that resonated causing the board to sway This was not an issue for if control because its response was not consistent enough to resonate Figure 7 10 shows the board starting with one end on the ground and plots the response as Vertigo brings it to balance At 3 seconds the board tipped to the point where it almost made contact with the ground It recovered but oscillation persisted To attenuate oscillation PID control was called upon Inclusion of the derivative component was especially beneficial as it helped to damp the response Figure 7 11 shows the significant improvement over proportional control Around 5 seconds into the experiment Vertigo raised the board to the balancing point with very little over 170 Jonathan Missel Balancing Implementation P board pitch 0 02 pitch radians a is 5 y o o o Q o o D 0 ro pars o E o a 0 06 time sec Figure 7 10 Balancing plank pitch with proportional control shoot This plot also demonstrates how much noise is added by the Vicon measure ments For this experiment the plank was fitted with markers and its angle used for feedback to the controller Therefore any variation in the response in the first 4 sec
116. he previous step is used as the current estimate and the filter proceeds as normal The applied code was set up to separately define the measurement and output frequency To investigate how this may affect the estimate a simulation was run using the same realistic measurement frequency of fm 25Hz but with an increased output frequency of fs 100Hz making the filter run four times per update Figures 6 6 through 6 9 show that propagating four times per measurement up date versus only one in the continuous discrete extended Kalman filter also provides improved estimates over the measurement data In fact close comparison between the two simulations shows that propagating multiple times not only improved the esti mate but slightly improved the transient time of the true response as well Improved performance is the goal of incorporating estimation techniques It is important to note that this simulation mandated retuning to converge with the increased output sampling time Modified tuning also affects the performance of the filter so this 148 Jonathan Missel Sampling Time Analysis for C D EKF LQR Track with EKF True State Response 2x Sphere Rotation States ratian units o 2 4 theta 6 phi theta rate phi rate 8 1 1 1 fi T 0 2 4 6 8 10 12 14 16 18 20 Figure 6 6 C D EKF true state response fm 25Hz f 100Hz LQR Track with EKF Estimated State Resp
117. he robot s onboard camera by clicking on Start Video Once started you can pause video at any time by hitting Pause Video and capture images from the video stream by clicking Save Picture To run your program hit the Play button To stop your program hit Stop Important Locations in this Folder javadocs Folder Documentation for every method in the classes included in this project API Reference pdf This file contains descriptions and sample usages of all meth ods used for controlling the Qwerk accessing robot sensor data and using the GUI Read this document or the javadocs before writing any programs MyFirstRobot java Skeleton starter file MyFirstCreate java Skeleton started file for controlling the iRobot Create Examples Example programs which demonstrate some of the available robot control methods All examples are commented and contain a header with a description of the pro gram s operation MyFirstRobot jcp Jcreator project file Double click to open JCreator and the MyFirstRobot project 207 Jonathan Missel Connection to Vertigo MyFirstRobot jew and jcu Other Jcreator files compile bat Batch file for compiling MyFirstRobot java from the command line run bat Batch file for running MyFirstRobot java from the command line clean bat Batch file for removing any files generated by using compile or run bat RobotClient Folder containing classes for accessing and con
118. he way 186 Jonathan Missel Translational Motion a human moves and when explained step by step is rather intuitive Figure A 2 illustrates the following description as Vertigo translates from point A to point B and time passes Vertigo Translation Figure A 2 Stages of translational motion Translational motion of Vertigo or any inverted pendulum can be broken down into seven distinct stages The first stage is simply the initial condition where Vertigo is vertical and stationary at point A Now to move forward the robot must match its acceleration with the angle of lean Accordingly lean must first be initiated in the second stage by shifting the point of contact with the ground backward This stage is frequently confused despite humans subconsciously doing the same thing before 187 Jonathan Missel Actuation walking by rolling weight back onto the heel The third stage is acceleration This stage is essentially a recovery from the initial lean in the previous stage and the greater the angle of lean the greater the acceleration This is why sprinters start a race on all fours as soon as their hands are lifted they are leaning at an extremely large angle and can impart greater force to accelerate than someone starting upright Stage four is cruise Here the robot is moving at a constant velocity and is per fectly vertical assuming no friction or drag Everything is mirrored about stage four so the deceleration
119. his the plane must be parallel with the ground so the equator of the sphere is intersected by the plane Figure 2 2 a shows the geometry of this configuration This is evidently the inverse of how an old computer mouse operates when the ball rolls perpendicular 22 Jonathan Missel Actuation Method rollers on the equator of the ball decompose the rotation and encoders are used to measure the component Therefore when used as an actuation method only two actuators are needed and they independently control a component of the rotation Top 45 deg Side a b c d Figure 2 2 Front 45 deg and top view of omni directional sphere actuation concepts a Concept A inverse mouse b Concept B omni wheel Euclidian perpendicular ity c Concept C omni wheel Euclidian orthogonality d Concept D omni wheel spherical perpendicularity Of the four designs the inverse mouse drive is the easiest to understand but there are several drawbacks Though it only needs two actuators it cannot directly control yaw rotation so it only offers two degrees of freedom The remaining drawbacks are attributable to the mandate of control implementation at the equator Since 23 Jonathan Missel Actuation Method contact at the equator of a sphere does not support the weight of the vehicle above it additional passive points of contact are needed to hold the robot Additionally just as it is with the computer mouse at lea
120. hn 2005 Figure 1 3 Anatomical planes As a robotic platform this system got its start in 1986 when Professor Kazuo Yamafuji from the University of Electro Communications in Tokyo successfully im plemented the first wheeled inverted pendulum 7 It was driven by a two wheeled base with both rotating in unison In this way it could not turn and is conceptually analogues to balancing on a cylinder Lack of emphasis on documentation and pub Jonathan Missel Background lishing his work has brewed debate however this is regarded by many as the first autonomous inverted pendulum In 1988 Professor Yamafuji came up with another very different idea for the two wheeled inverted pendulum 8 An arm at the top of the pendulum was actuated to swing providing a control torque that stabilized the pendulum The two wheeled base was controlled as if it were a statically stable device and could turn like traditional differential drive vehicles Since the wheels did not dictate the pendulum s motion this device was really a fixed inverted pendulum with reaction wheel control for balance that was mounted on a mobile base In 1994 a group in Japan successfully demonstrated the control of the first two wheeled true inverted pendulum balancing robot that was able to turn A differential drive allowed each wheel to be driven independently enabling the base to rotate while its translational position governed the pendulum balance 9 The abilit
121. ighten the mounts 238
122. ill cause inconsistent loop sampling time Many implemented control methods make the assumption of constant sampling time this introduces error The effects of erratic sampling times were most detrimental to implementation of the Kalman filter and will be discussed later in this chapter To best mitigate this problem an average sampling time was used and assumed to be constant There are some cases where the Vicon measurement system momentarily paused and control was held constant This caused tracking controllers to deviate from their desired trajectories and loss of balance in sphere based mode Dynamic Instability Vertigo was intentionally designed with a high center of gravity to improve its controllability and dynamic stability in sphere based mode however as a statically stable ground vehicle this made it prone to dynamic instability This was problematic for large accelerations which caused Vertigo to lean or tip When leaning only one or two wheels contacted the ground and yaw was induced Recovery from this often caused large overshoot sometimes to the point of divergence or erratic behavior de pending on the controller Frequently these departures were caused by the paused Vicon connection In this case dynamic instability was merely the mechanism of failure with poor connection as the cause Embedded Signal Processing 155 Jonathan Missel Implementation Challenges Unfortunately Qwerk was programmed with no mean
123. ily available Once acquired the surface was lightly sanded to reduce the severity of the bumps and groves in the ball In general there were a few critical notions that guided this structural design sim plicity durability reconfigurability symmetry and manufacturability The simplistic and open structure was intended to make assembly and repair straight forward This also gave fewer parts the opportunity to fail and made visually analyzing problems easier Durability was secured by designing a compact and rigid skeleton made of Delrin Safety features like the camera mount IMU bumpers Qwerk mount and en coder assembly also help protect the robot and improve overall hardware robustness The primary service that reconfigurability provided was the ability to easily change the center of mass and inertial properties of the robot without the use of dead weight As an experimental vehicle it was important that the various constituents be flexible to test which setup produced the best results and to make room for additional equip ment For this reason Qwerk was modified to accept the same mounting hole pattern as the IMU Also the two plates were designed to be interchangeable with universal mounting hole patterns on both sides This allows Vertigo 1 0 to be assembled in 12 unique configurations of the major components each with different center of mass and inertia The design was also adaptable in that it had many mounting surfaces to attach new
124. in symmetry throughout the system 2 4 3 Sensing The nature of this system mandates access to very accurate measurements of both the sphere and body dynamics for the feedback control loop There are many types of sensors that measure attitude and acceleration and in general they tend to increase 33 Jonathan Missel Components in accuracy with price Most inexpensive sensors are not accurate enough for this application Some attitude sensors depend on the acceleration of gravity for their measurements which is acceptable for static situations but the dynamics of this system would invalidate their readings An Inertial Measurement Unit IMU is a package of sensors that is specifically designed to measure the rates and acceleration of dynamic systems Attitude is easily backed out from this information so an IMU is well suited for this application The IMU chosen was the Crossbow IMU400CD 100 It is a six axis measurement system designed to measure linear acceleration along three orthogonal axes and rotation rates around three orthogonal axes It uses three accelerometers and three angle rate sensors based on a 3 2 1 Euler angle system to make a complete measurement of the body dynamics Methods such as dead reckon ing can be used with this information to track the position IMU specifications 17 e Performance Update rate gt 100 Hz Start up time lt 1 sec e Angular rate Rate range 100 sec Ra
125. ing control into the body frame are both impos sible The reason behind such difficulties is discretization Yaw information comes from sensors that have a discrete sampling time and transformation takes place in coded loops which do not provide continuous output This naturally brings up the discussion of the discrete time controlled system In the discrete time control system yaw is measured control is calculated and transformed and voltage is applied to the motors at every time step Between steps the voltage to the motors is held constant To demonstrate the problem consider ground based Vertigo under discrete time control with a rather large sampling time of one second Now suppose that translational control for forward motion is determined and yaw control was separately determined to follow a rotational trajectory Now when those two inputs are superimposed and applied as voltages to the motors that signal will remain constant throughout the time step Therefore at the start of 191 Jonathan Missel Pivot Drift the step the state for which control was calculated the input will be appropriate however at the end of the time step one second later the straight line control will have curved as a function of the yaw rotation For the second time step the translation control will try to bring the robot back to the trajectory but again yaw rotation will cause divergence and prevent the desired control from being carried out The
126. ion The detailed process and instructions for changing the embedded PID gains and velocity profile are included in Appendix D Ironing out the kinks of this convoluted communication network meant working with a total of 10 computer programs and five different coding languages With such a tortuous journey maintaining dependable communication was problematic through out implementation However the reward of connecting to an external computer was being able to program all controllers and estimators directly from the familiar MATLAB Simulink environment 108 Jonathan Missel External Sensing Architecture a Yaw Meas Ref Yaw Trajectory u Torque XY Meas Convert to Angular Transform into Convert to scalar GY Velocity Body Frame Qwerk input Ref U i JAVA 3 Wireless Signal Angular Rate value ACTUATION Uvoltage Trajectory Generator Convert to Voltage EMF Feedback Figure 4 2 Ground based communication schematic 109 Chapter 5 Control 5 1 Introduction Vertigo is composed of two major constituents the main body and the sphere The body includes the processor power supply DC motors sensors and structure The sphere is the base which Vertigo was primarily designed to balance and navigate on however it can also be omitted and control implemented in a secondary ground based configuration Ground based mode simply means that the body of the robot is placed directl
127. is applied from the motors is defined by 7 The viscous friction is modeled by defining 199 D q 3 8 Me where ug is the viscous damping coefficient for friction between the sphere and ground and ug is the coefficient for friction between the sphere and body The Euler Lagrange equations can then be expressed with this notation as d 0 Fla la 20 3 9 T The derivative of this gives the equations of motion which are rearranged to take the form M a C q 4 G q D 4 3 10 Where T 2Mprlcos 0 To Mmpgrlcos 0 M y o A Dl dada 3 11 Ta mprplcos 6 q To 72 Jonathan Missel Mathematical Model is the mass and inertia matrix with Ti b 18 m mpg r mgl 3 12 To Ig mg 3 13 The vector of Coriolis and centrifugal forces is defined by i A2 mprilsin 0 d 6 4 0 C q 4 3 14 and the vector of gravitational force components is mpglsin 0 N ie 3 15 mpgtsin 0 Q T By defining u 7 and x a 4 the equations of motion can be expressed in nonlinear state space form as 3 2 3 Linearization Many control estimation and analysis methods are for linear systems so Vertigo s nonlinear equations must first be linearized The linearization used here is based on keeping only the linear terms of a Taylor series expansion about an operating point This is often referred to as Jacobian linearization because for a state spac
128. is blinking it means that it is trying to connect through a relay server This is what was just changed in the previous step Everything should be back to normal and running properly with the new gains and trajectory 219 Jonathan Missel Changing Qwerk s Embedded Code Qwerk Development Guide QDG Only relevant sections are included here This document describes all the steps required to develop software for the Qwerk Blocks with a grey background represent what you ll see in the command prompt Bold text within those blocks is text you enter Building the Qwerk Source Compile the share tree first The share tree contains all the TeRK specific code You can safely ignore all the might be used uninitialized in this function warnings reported during the build cd terk embed trunk share make make C sre make 1 Entering directory home YOUR_USERNAME terk embed trunk share src make C IceE 1 0 0 all make 2 Entering directory home YOUR_USERNAME terk embed trunk share sre IceE 1 0 0 making all in src make 3 Entering directory home YOUR_USERNAME terk embed trunk share src IceE 1 0 0 sre making all in IceE make 4 Entering directory home YOUR_USERNAME terk embed trunk share src IceE 1 0 0 src IceE rm f include IceE BuiltinSequences h BuiltinSequences cpp 220 Jonathan Missel Changing Qwerk s Embedded Code slice2cppe ice include dir IceE dll export
129. is improved implementing theoret ical results will be more straightforward With the accumulated success to this point and expected results of future work the Vertigo platform promises a rich and prolific future 180 Bibliography 1 L H Chang and A C Lee Design of nonlinear controller for bi axial inverted pendulum system JET control Theory Appl vol 1 no 4 pp 979 986 2007 S G Kobilarov M B Near time optimal constrained trajectory planning on outdoor terrain IEEE International Conference on Robotics and Automation ICRA2005 pp 18 22 2005 G A K T B LAUWERS and R L HOLLIS A dynamically stable single wheeled mobile robot with inverse mouse ball drive EEE International Con ference on Robotics and Automation 2006 G A K T B LAUWERS and R L HOLLIS One is enough 12th Intl Symp on Robotics Research 2005 J Dewey and P Byerly Bulletin of the seismological society of america vol 59 no 1 pp 183 227 1969 S N X Y Hui Di Zhang S Hi Rong Liu A neurodynamics based neuron pid controller and its applications to inverted pendulum Third International Conference on Machine Learning and Cybernetics pp 26 29 2004 T Uranaka Scientist claims he made segway predecessor in 86 The Japan Times 2001 181 8 13 14 15 18 19 20 Y K Feng Qing Design and simulation of a control system o
130. it also protected the USB peripherals whose plug ends were exposed in the original design The exact distance the legs extended was determined by the overall assembled geometry of Vertigo Together with the upper spines the top plate and the bottom plate these elbows were spaced so that at no point when Vertigo 2 1 is on its side will the IMU or webcam contact the ground This is not to say that these components will never hit the ground if Vertigo falls off the sphere or tipping over at speed but it is very affective at protecting them from minor falls It is also beneficial for working on Qwerk allowing the body to safely rest on its side Lastly the angle of the elbow was 52 Jonathan Missel Design Iteration determined in accordance with the full leg assembly Excluding the motor mounts the new legs had to be made in two parts for manufacturing reasons The holes that mount the bottom plate had to be drilled and tapped from the underside of the legs This meant that the mill had to have clearance to maneuver and the lower portion of the legs would interfere if made of one solid piece Therefore the legs were designed in two parts and the angle of the kink was assigned for ease of assembly and to form a sturdy perpendicular joint As before all skeletal joints are fastened with the same size screws to make assembly reconfiguration and maintenance easier Making up the bottom half of this joint is the lower leg Its function is to re
131. jectory for the smallest and largest rota tions in Figure 7 9 show that the method devised for conditioning the measurements works for any number of yaw rotations These plots also include the errors that indi cate how far off the trajectory the track is Errors plateau at a small nonzero number because there must always be some error for the proportional controller to produce input signal It is worth noting here that simply looking at how well the track fits the trajectory is not sufficient enough information to make discerned conclusions As mentioned before the time response is also important so the norm error values are an accurate basis for comparison because they take time into account as well as position errors 168 Jonathan Missel Balancing Implementation Yaw Track and Error Versus Time yaw measure yaw trajectory error y position m 0 5 10 15 20 25 x position m a Yaw control during 27 retrograde Yaw Track and Error Versus Time yaw measure yaw trajectory error 30 y position m 40 50 60 70 f 0 5 10 15 20 25 x position m b Yaw control during 207 retrograde Figure 7 9 Yaw control for 27 retrograde and 207 retrograde 7 4 Balancing Implementation Working towards the goal of building a preliminary foundation to expedite future work in balancing on the sphere the next step was to conduct simplified balancing experiments As with ground b
132. k gz load r v b 0x80000 m http zImage fis create b 0x800000 1 0x300000 ramdisk Answer y to the posed question for continuation fis create b 0x80000 1 0x180000 e 0x80000 zImage Answer y to the posed question for continuation Then run the command for version 1 1 Qwerk the first option in the QDG document fis create b 0x1000000 1 0x300000 optfs Answer y to the posed question for continuation 218 Jonathan Missel Changing Qwerk s Embedded Code Type reset in Terminal NOTE Don t hit ctrl C this time because you want it to reboot Wait and press Enter when asked At this point the firmware has been replaced and Qwerk is just like it was out of the package except of course for the embedded code that have been changed Now you have to change its connection settings back to where they were for com munication Change Qwerk s connection settings With everything still plugged in and connected as it was before enter Qwerk s IP address into the web browser Under General Configuration select Direct Connect Then click Save Then under Wireless Configuration check Set the SSID Manually and make the settings as follows Network name SSID LAIRS Data encryption WEP Network key B4A8BFD5F4 Max tune to retry this SSID 60 Max unsuccessful attempts 2 Click Save and Restart Reboot Qwerk After reboot led 6 should be blinking indicating that you are connecting directly If led 7
133. k walled housing that interfered with them contacting the ground Tooling for these wheels as with all of the cylindrical parts and modifications was done primarily on a lathe as seen in Figure 2 8 The objective of this modification was to maximize the exposure of the rollers Modifications from the 1 0 generation had already milled down the material 58 Jonathan Missel Design Iteration in the roller gaps to eliminate interference with the curvature of the sphere The new modifications faced both sides of the wheels and put chamfers at the gaps The upper right corner of Figure 2 8 shows a side by side comparison of the original and modified wheels Since the spokes that support the rollers only act in compression and all stress is transmitted through the axels the material above the spokes and on the walls did not serve much structural purpose In way of torsional strength these wheels were rated for much higher loads than those experienced in this application Therefore slimming down the wheels did not compromise their integrity Figure 2 8 Modifying omni wheels for Vertigo 2 1 use photographs Joining these modified wheels to the new motors was the aluminum hub seen in Figure 2 9 Unlike the original hub this did not share the burden of securing 59 Jonathan Missel Design Iteration and spacing an encoder wheel therefore it was made much smaller and with less complexity which required fewer machine setups to
134. l act to minimize the error In doing so the current states are driven to the final desired states affectively tracking the reference states To implement this a new control input is defined as u r k x xg 5 22 which makes the state space 130 Jonathan Missel Sphere Based Control x A Bk x xf Br 5 23 y C x xf 5 24 As with the stabilizing LQR this technique was implemented and simulated in MATLAB on the nonlinear system equations The simulation takes the system from random initial conditions bounded near zero to xs 27 27 0 0 In other words it is taking Vertigo from near vertical rest to vertical rest at a distance one sphere circumference away Tuning the Q matrix for translation was more difficult than tuning for balance This is because the final angle states were far away from the initial conditions but the rates were not Therefore quickly minimizing the angles required large rates but this fought against the rate minimization If weighting is placed more heavily on the rate states the response will be smooth and predictable but will take longer as less aggressive rate trajectories will be permitted Figures 5 14 and 5 15 show how the LQR tracking response with heavily weighted rates followed a very smooth trajectory but took nearly 20 seconds to reach the final values The conservative nature of this response also means that less control effort was required in only two seconds
135. lenging as pects of this project taking the greatest amount of time Eventually two variations of communication through Java were successfully established These were first dis covered by others working with the Qwerk unit and further matured by the TeRK group 24 The simpler method required some formatting of Qwerk but essentially launched Java based graphical user interfaces GUIs that initiated wireless commu nication through a router and governed Qwerk s various functions These programs were primitive in their facilities but their specialized nature made them excellent for troubleshooting and diagnostic work The second method expanded on the principles of the first but rather than operating from a pre coded GUI it allowed original Java programs to be written and run directly This admitted more fundamental access to Qwerk s faculties but coding was still done in a fairly high level language JCreator was used as the editing program for developing Vertigo s Java code and TeRK pro 104 Jonathan Missel External Sensing Architecture vided a small library of commands that controlled the functions of Qwerk Of these the most applicable to this project were the audio battery voltage and DC motor commands The audio commands come in three flavors In the first a tone was prescribed in Hz and given a duration of time to be played for This was used extensively to assign audible signals throughout implemented code to help de
136. ll be discussed in the remainder of this chapter along with the progression of ground based control and balancing control Finally attempts at sphere based balancing control will be detailed with a discussion of what prevented their success 7 2 Implementation Challenges Uneven Terrain Vertigo has four support contacts in ground based mode so the slightest curva ture in terrain or inconsistency in actuator dimensions will lead to uneven weight distribution amongst the wheels This has several negative effects on performance all of which pertain to friction Friction is needed between the wheels and ground to move the body If a leg is not supporting any weight it has no normal force to generate its maximum frictional force This is the primary cause of slip When this affect is large or prolonged only one pair of motors will be properly applying control inducing undesired yaw If both pairs of motors are slipping time response will suffer from the wasted control When tracking a step slip near the final destination may prevent the robot from achieving the final destination This is because error is small 154 Jonathan Missel Implementation Challenges so the applied control was insufficient to overcome the potential causing slip Inconsistent Loop Sampling Time Vertigo s loop of communication calls upon five different programs two of which require network connections one being wireless A delay in any of these systems w
137. llenges in controlling this vehicle was the infor mation circuit for closed loop feedback control The current sensing method utilizes Vicon an external 8 camera motion capture system for information on the robot s 137 Jonathan Missel Continuous Discrete Extended Kalman Filter position and attitude Despite the celebrated precision of Vicon its measurements do not directly correspond to the states of the system Mathematically transforming the raw measurements to representative state measurements introduces the accumulated uncertainties of the physical system parameters lengths mass etc Vicon measure ments and initialization errors Consequentially half of the states are not measured at all and the others only have flawed indirect measurements available To address this problem the continuous discrete extended Kalman filter EKF was exploited This provided a method for full state estimation and noise filtering however the EKF requires linearization of the nonlinear measurement model and equations of motion This added to the uncertainties and had to be included in the development of the filter to mitigate its affects 6 2 Continuous Discrete Extended Kalman Filter The Kalman filter is a recursive process that efficiently estimates the states of a process from noisy measurements The filter is based on linear dynamical systems that are discretized in the time domain and is very powerful in that it can estimate past pres
138. ller tracked a two dimensional step from 0 0 m to 0 5 0 5 m Plotted with time as the z axis the proportional control is apparent as velocity is greater at the beginning and consistently decreases as the error reduces Plots of the control input showed the inverse of this trend as magnitude diminished with time To kick off yaw control and establish the Vicon defined convention if logic was employed again Once directionality was confirmed proportional control was intro duced It was discovered that the gain and resultant control magnitudes were much 159 Jonathan Missel Ground Based Implementation time sec x f 0 35 0 4 0 45 O05 y position m 0 0 05 0 1 015 0 2 026 03 x position m Figure 7 1 Proportional two dimensional step track more conservative for yaw than translation This was because the error to recover from was limited to 7 and control was cooperatively applied by all four motors One of the goals of implementation was to compare the simulated LQR controller with the physical response This simulation was discussed in the Control chapter and was specified to track a standardized circular trajectory with a 1 m diameter traversed at 0 3 rad sec To begin work on this the proportional controller was applied to both directions and coded with a variable reference point By specifying how the reference should change with time this was effectively modified to track a trajectory rather than steps For
139. llers would make the control surface discontinuous and add noise to the 62 Jonathan Missel Future Improvements measurements The wheels used were chosen to mitigate this problem and were successful to a degree but the problem had not entirely vanished This inspired a handful of new design ideas for omni wheels all of which would perform better than what was currently on the market but one design stood out from the rest The least ambitious design would simply use a larger wheel with smaller rollers to minimize the size of the gaps This is really identical to the existing concept but with more advantageous dimensions The most promising idea directly confronted the need for gaps Currently spaces exist because the spokes that support the roller axels have to fit between the rollers pushing them apart In addition these gaps widen radially outward as the circumference increases but the length of the rollers remains constant Therefore even designs that did not have radially facing spokes and eliminated the inner spacing between the rollers would still have gaps at the control circumference To overcome this two different types of rollers could be used with one resembling the existing rollers and the other being larger with hollows at each end Inside the hollows would seat the spokes and gaps for the smaller rollers The outer surface of both types of rollers would be turned to the same radius as the assembled wheels This configuratio
140. ls near perpendicular with respect to the surface of this sphere however this was not mandatory Since the omni wheels always act through the center of the sphere making contact at an angle is simply analogues to running on different sized wheels Accounting for this in the controller gains or equations of motion can easily be done experimentally or geometrically Designs for adjustable legs were generated for consideration but they were decided against due to their complexity and durability concerns An adjustable joint is not as secure and jarring from impact could easily misalign them This is not only inconvenient but would make fixing all four joints at exactly the same angle difficult Fusing motor mounts into the legs was no longer practical with the redesigned actuation method The new mounts had to secure the driveshafts of the motors in the plane of the legs Logically this meant that the motors should be fixed to the bottom of the legs making Vertigo 2 1 s mounts vastly different than the originals In the final design two clamps secured each motor and were made specifically to fit the motor diameters Therefore if new motors were ever used only the clamps had to be remade Each clamp connected to the leg with only one bolt which permitted them to pivot in the absence of a motor but with a motor their motion was fully constrained These particular motors changed diameter along their length therefore the two clamps had to be mad
141. manufacture With the success of the original fixture to draw upon the wheels were held by a similar two stage fixture press fit diameter and clamped faces The aluminum washers were made thicker to discourage denting and bending on impacts In the new design the hub ends were outward facing so rounded cap screws were used to make it easier to handle and to avoid damaging objects floors or walls it crashed into The hubs were secured with setscrews onto the motor driveshaft This design mandated that it be assembled in a specific order starting with the hubs and setscrews and then placing the wheels and finally the clamping washers and end cap screws Disassembly had to be done in the reverse order Though it was not very complicated or time consuming this requirement was made as a sacrifice to reduce the overall size by burying the setscrew under the wheel when assembled To minimize play the hub had to span as much of the driveshaft as is permitted however the end screw had to be threaded into the other side of the same hole and needed its own space to thread in properly Working out the spacing such that both components could take advantage of the same concentric hole in the hub was accomplished in both 1 0 and 2 1 design generations but was more challenging here because there was less room to work with The primary goal of this design iteration was to implement the new drive mecha nism that controlled all aspects of Ballbot s and Ver
142. mass of Vertigo and length from the 86 Jonathan Missel Analysis center of the sphere to the center of mass of Vertigo are varied Figures 3 10 and 3 11 show that the system is both controllable and observable for all parameter variations because the rank of both matrices is four at each point matching the number of states Now the inverse condition number of these two matrices is considered to evaluate the parameter s affect on how well conditioned they are The contours in Figures 3 12 and 3 13 show the inverse condition number as the parameters are varied The vertical blue lines mark the parameters to which Vertigo is currently configured Figure 3 12 reveals that as the length from the center of the sphere to the center mass of the body increases the controllability matrix monotonically becomes more ill conditioned To help understand this consider the following scenario Vertigo is required to return to a vertical stature from an initial angle of 10 degrees This plot shows that more control effort would be required for the system to recover if the length is greater This makes sense because for a fixed angle the horizontal lever arm creating a torque with gravity grows with this parameter However if this was the only affect of increasing this length the plots would decrease at a constant slope because of the linear relationphip this is clearly not the case The reason is that the rotational inertia also increases as increa
143. mation and control theory Ground based equations of motion are also derived and will be used for control in that configuration Observability and controllability analyses are then conducted for the states and design parameters of the sphere based 66 Jonathan Missel Mathematical Model equations of motion This development should give a quality model that is well un derstood and can reliably be used in the remainder of this thesis 3 2 Mathematical Model 3 2 1 Introduction Having a mathematical model of a system is required for most analysis control and estimation methods In general there are two approaches for obtaining a model system identification and mathematical derivation System identification is a broad topic that is composed of a diverse collection of methods and algorithms in general they are all designed to extract dynamical models from measurement data System identification is most advantageous when applied to complex or flexible systems This is because it does not require knowledge of the system that is to be modeled De riving equations of motion mathematically does require knowledge of the system so it is most readily done for systems with well known dynamics such as rigid bodies Derivation does not require previously collected data so it can be done for theoretical systems or in the preliminary stages of development before data is available Having the equations before the system is even built means that th
144. matrix while varying 0 and To this point the controllability and observability analysis has been devoted to the 0 and state combination as they are varied The same analysis was done for all six combinations of states pairs as the four states were varied Figures 3 8 and 3 9 show that as the two rate states and were varied the controllability and observability are still preserved When this rate analysis was extended to the inverse condition numbers it was found that for both observability and controllability the value was a constant for all rates Accordingly the mesh plot for the inverse condition number of the controlla bility matrix was a flat plane at 1 67 x 107 and the inverse condition number of the observability matrix was a flat plane at 1 17 x 1076 The plot of the determinant of the controllability matrix was also a constant value at 1 6 x 10 Constant values mean that these states do not affect the controllability or observability This was confirmed upon closer inspection of the A matrix where it can be shown that these values cancel each other out and only the angle states can change the A matrix 84 Jonathan Missel Analysis Controllability Rank 0 phi radians 4 theta radians Figure 3 8 Rank of controllability matrix while varying and 0 phi radians 3 theta radians Figure 3 9 Rank of observability matrix while varying and 85 Jonathan Missel Anal
145. n allows the two types of rollers to be placed as close as tolerances allow forming an almost perfectly continuous surface Inclusion of these wheels would greatly decrease the demands on the estimator because measurement noise from the drive mechanism would be nearly eliminated and prescribed control would more accurately be carried out fitting the mathematical model more accurately and improving predictions The drawback to this design is that they are extremely intricate and complex To make just one wheel that had 6 of each type of roller 28 tiny parts would have to be designed and made from scratch that is about twice the number it takes to manufacture the whole of Vertigo 2 1 To make four of these would be quite time consuming but in the future their benefit 63 Jonathan Missel Future Improvements may be deemed worth the effort To this point the 2 1 legs have only solved problems not caused any however this is not to say that problems cannot arise in the future as trajectories increase in complexity and the dynamics of the system are tested to their limit The number of legs is an example of this and the decision to include four rather than three was nontrivial Three legs are all that is needed to support and control the robot on top of the sphere with this actuation method allowing for minimal weight and number of parts It would also make the system equal actuated with three motors to control three degrees of freedom
146. n is desired the system has to work very hard to move the sphere Now a near massless sphere would require very responsive control to balance because there would be little inertial resistance to change However navigation would use comparatively less control Besides pivot drift a massless sphere would cause Vertigo to behave very similarly to the pendulum on a cart because sphere inertias would have no effect It is worth noting here that Vertigo is designed to be a controls experiment so it is important to select a sphere with properties that balance the challenges of control with practical feasibility A 8 Weight Distribution Since inverted pendulums by definition have centers of gravity above their pivot point and are known to be unstable it is a common notion that lowering the center of gravity will make it easier to control To investigate this idea assume the cart in Figure A 5 is fixed making it a simple uncontrolled inverted pendulum The 196 Jonathan Missel Weight Distribution equation of motion for the angle would then be 6 sin where the length of the pendulum is inversely proportional to the acceleration Thus an inverted pendulum with a higher center of gravity will fall more slowly giving the controller more time to respond and making it easier to control These simple equations provide cogent evidence but there is a more intuitive way of understanding this concept To control the inverted pendulum
147. n many of the results include representations that are easier to visualize in addition to the state results The following assumptions are made during this formulation 1 Rigid sphere and body 2 No slip between ball and ground 3 No slip between ball and actuation 70 Jonathan Missel Mathematical Model 4 Inputs are torques applied between roller and ball 5 Sagittal and coronal plane are symmetric and decoupled 6 Neglect friction in actuator rollers 7 Neglect dynamic friction 8 Origin is at the center of the sphere 9 Yaw coupling is negligible The first step in the Lagrangian formulation is to define the kinetic and potential energy for the body and sphere Sphere 1 62 Mp rd Ko 3 3 V 0 3 4 Body Kpg gt ri art 0 05 cos 0 4 4 4 6 4 3 5 Vg mgglcos 0 3 6 Where mp and mg are the mass of the sphere and body respectively J and Ig are the inertias of the sphere and body respectively r is the radius of the sphere and the length from the body s center of mass to the center of the sphere The total energies are considered for the derivation and are calculated as follows Total kinetic 71 Jonathan Missel Mathematical Model energy is K Kp Kp and total potential energy is V V Vg Defining the system configuration vector as q 0 At the Lagrangian formulation is continued with L qq K V 3 7 The input torque that
148. n of Qwerk s firmware that supports the onboard sensing architecture will be released soon Vertigo will no longer be dependent on Vicon measurements a laptop and wireless router will be the only equipment needed This means easy transportation and set up almost anywhere and Vertigo can be made fully autonomous The new firmware also promises to support passing of the embedded PID gains and trajectory parameters This will enable further tuning for improved response and back EMF measurements can be added with low weighting to MATLAB feedback loops Development of control methods spanned a wide range of motion Starting with the ground based configuration a standardized test trajectory was developed to com pare the results of simulated and implemented experiments The majority of the methods simulated in this thesis were optimal control techniques These gave ideal gains for respective specified cost functions and removed some of the guesswork from tuning Several sphere based controllers were developed for balance and translational motion simulation of these yielded exceptional results that confirmed Vertigo s con ceptual principles The most sophisticated and successful was an augmented LQR tracking controller that balanced and translated Vertigo according to a cost function that effectively minimized weightings of states power and time 178 Jonathan Missel Conclusions and Future Work Many other optimal control techniques were att
149. nctionality javaaddpath C lUsersl JonlDesktopl VERTIGO QWERKAMyFirst Vertigo terk client MyFirstRobot TTS Ajavaadpath C Users Jon Desktop VERTIGO QWERK MyFirst Vertigo terk client MyFirstRobot RobotClient applicationTitle java lang String Woof ipaddress java lang String 192 168 1 100 myRobot RobotClient RobotClient applicationTitle ipaddress Ymmy Robot RobotClient SimpleRobotClient trial DAADAA xX Java Interface Adds Dynamic Java Class Path excerpt from rn pdf in the QWERK dir MATLAB loads Java class definitions from files that are on the Java class path The Java class path now consists of two segments the static path and a new segment called the dynamic path The static path is loaded from the file classpath txt at the start of each MATLAB session and cannot be changed without restarting MATLAB This was the only path available in previous versions of MATLAB Thus there was no way to change the Java path without restarting MATLAB The dynamic Java class path can be loaded at any time during a MATLAB session using the javaclasspath function You can define the dynamic path using javaclasspath modify the path 212 Jonathan Missel Connection to Vertigo using javaaddpath and javarmpath and refresh the Java class definitions for all classes on the dynamic path using clear java without restarting MATLAB See the function reference pages for more information on how
150. nd cost weighting The proper mathematical development of LQR tracking is posed as a minimization problem for the performance index J J D Calk ae h PQ Ca h a AR 53 k ko Where q k u k and y k are the state control and output vectors with lengths of 113 Jonathan Missel Ground Based Control 2 1 and 2 respectively Also Q is a 2x2 dimensional positive semidefinite symmetric matrices and R is a 1x1 positive definite symmetric matrix These two matrices are defined to weight the cost function according to the desired performance It is desired that the error e k y k z k be minimized with minimum control effort where z k is reference trajectory After solving the Riccati equation the optimal control input becomes u k L k q k Ly k g k 1 5 4 where Ey k R B P k 1 B 1B 5 5 L k R B P k 1 B B P k DA 5 6 1 A BL k q k BLg k g k 1 Cue glk AHI P k 1 E Ejg k 1 C Q 5 8 P k A P k 1 EP k 1 4 V 5 9 V C OQC 5 10 E BR B 5 11 The above equations show how the matrices Q and R can be viewed as weightings for the state and control minimization In general only the ratio of these weightings is of importance so it is standard procedure to set R as the identity matrix and only manipulate Q Once the optimal gains were determined the command dlsim was used in MATLAB to simulate the response This simulation was done for b
151. ndardized trajectory being traced with transformed input With norm error 0 5402 it outperformed the uncontrolled track however it could not quite match the response of controlled yaw with untransformed input This is because actively controlling yaw increased the control when disturbances were encountered which helped conquer them faster With uncontrolled yaw no additional input was sent to aid the LQR and the re sponse suffered slightly Experiments were run with transformed input and fixed yaw control however they did not improve upon the yaw control significantly 166 Jonathan Missel Ground Based Implementation 04 0 3 0 2 0 1 y position m 0 1 02 03 LOR Ground Track With P Yaw 2 pi Retrograde YN y 05 04 03 02 01 0 01 02 03 04 05 x position m 0 5 0 4 0 3 0 2 0 1 y position m LOR Ground Track With P Yaw 6 pi Retrograde aa alae 05 04 03 02 01 0 01 02 03 04 05 x position m a Circle trajectory tracking with propor b Circle trajectory tracking with propor tional yaw control for 27 retrograde norm tional yaw control for 67 retrograde norm error 0 5486 0 5 0 4 0 3 0 2 0 1 y position m 01 02 03 04 05 LQR Ground Track With P Yaw 10 pi Retrograde EN 05 0 4 0 3 0 2 01 0 01 02 03 04 05 x postion m error 0 5805 0 5 04 03 02 0 1 y position m o 0 1 02 03
152. ned with corrosion resistant stainless steel 8 32 socket head cap screws so it can be completely disassembled 41 Jonathan Missel Structure with only one tool These cap screws are recessed in counterbored holes so they seat flush this maximizes the usable mounting surface area on the plates and maintains the clean lines of the skeleton The bottom plate holds the battery pack which is attached via hook and loop so it can easily be attached removed and relocated On the underside of the bottom plate is Qwerk the onboard computer To further promote reconfigurability mounting holes were made in Qwerk that mirrored those of the IMU Also the holes in the top and bottom plates were completely threaded through this allowed the IMU to be mounted on either side of both plates and Qwerk on the top of the top plate or the bottom of the bottom plate This is true for all structural configurations The larger plate was sized to match the dimensions of Qwerk to help protect it and fit the curves of the legs in the shown configurations To further protect Qwerk it was sent out to be anodized black this reduces pitting from corrosion on its aluminum casing In Vertigo 1 0 the legs not only connect the plates but also serve as motor mounts For simplicity this was all accomplished in one solid piece so only six structural components assemble to form the skeleton four of which are the identical legs The drawback to combining the legs and mo
153. new firmware was being developed by the manufacturer that promised to patch up these connections Therefore their physical properties were included in all dynamic modeling to avoid new models having to be derived once Qwerk is updated This should accelerate the process of incorporating the sensors when they are made functional Of course the sensing equipment was also included in the hardware to match the dynamics of the model this will permit all controllers to be directly applicable after adaptation to fit the new communication methods Working with Qwerk was a long and finicky process therefore none of the control or estimation was hardcoded onboard Vertigo though it was possible Instead the power and familiarity of MATLAB made it the desired program for implementation therefore communication had to be established between Qwerk and MATLAB to pass on sensory and control inputs This in itself was a laborious task and is covered later in this chapter For the onboard sensing architecture Vertigo s brain was rooted in Qwerk and served as a central hub for receiving and transmitting signals As seen in Figure 4 1 it received sensory input from the motor EMF feedback quadrature encoders camera 100 Jonathan Missel Onboard Sensing Architecture COMMUNICATION SCHEMATIC SENSING COMPUTATION ACTUATION Figure 4 1 Onboard sensing communication architecture and IMU and wirelessly transmitted these signals to a PC where they wer
154. ng testbeds that were highly maneuver able but they were overly complex or specialized in their application Some were detached all together from any practical application and served only as experimental amusement Turning to the work of creation it was clear that omni directionality is very common in nature and uncontrolled systems This would be very advantageous for improving mobility however it is notoriously difficult to accomplish properly in robotics Nature also features many variations of the inverted pendulum throughout anatomical motion and its purpose is often improved performance and agility This can be observed when an agile creature like the cheetah turns while at speed Front on the animal carries the tall slender signature of an inverted pendulum As it turns it leans into the corner to cancel the centripetal force from turning By matching the angle of lean with the aggressiveness of the turn the animal experiences no lateral forces and can turn much sharper A statically stable system in these circumstances would either tip over or lose traction with the ground and slip The inverted pendu lum is also attractive because it is famous for being a benchmark problem in control Accordingly an omni directional inverted pendulum was decided on for the desired design To make an omni directional inverted pendulum the bottom of the pendulum would ideally be actuated in both directions independently The first concepts brain st
155. ng three load commands the range of hex addresses displayed may be different RedBoot gt load r v b 0x1000000 m http optfs out Raw file loaded 0x01000000 0x011e099b assumed entry at 0x01000000 232 Jonathan Missel Changing Qwerk s Embedded Code RedBoot gt load r v b 0x800000 m http ramdisk gz Raw file loaded 0x00800000 0x0095c6d4 assumed entry at 0x00800000 RedBoot gt load r v b 0x80000 m http zImage Raw file loaded 0x00080000 0x0019d12b assumed entry at 0x00080000 RedBoot gt Now we ll initialize the flash filesystem Answer yes y when it asks if it s ok initialize The output should look like this don t worry if the numbers in the output differ RedBoot gt fis init f About to initialize format FLASH image system continue y n y Initialize FLASH Image System io Erase from 0x60040000 0x60fc0000 riada ae rede Erase from 0x60fe0000 0x60fe0000 Erase from 0x61000000 0x61000000 Erase from Ox60fe0000 0x61000000 Program from 0x01fdf000 0x01fff000 at Ox60fe0000 RedBoot gt Now we ll load the ramdisk and zlmage images into flash don t worry if the numbers in the output differ RedBoot gt fis create b 0x800000 1 0x300000 ramdisk Erase from 0x60040000 0x60340000 eee Program from 0x00800000 0x00b00000 at Ox60040000 ee Erase from Ox60fe0000 0x61000000 233 Jonathan Missel Changing Qwerk s Embedded Code
156. nto these two categories Omni directionality Figure A 1 c is among the most commonly misused terms because it is considered a holy grail for robotic motion It refers to a vehicles ability to move in any direction on a planar surface This means that the vehicle does not need to be facing in a particular direction to carry out a maneuver but also does not have the ability to intentionally control the direction it is facing Controls in the x and y directions are usually decoupled for these two degree of freedom vehicles This type of motion improves maneuverability and response time because the vehicle does not have to spend time steering to avoid obstacles Omni directionality with yaw Figure A 1 d is the most ambitious form of pla nar motion having all three possible degrees of freedom It means that the vehicle is able to move in any translational direction on a surface and control its yaw angle in dependently The primary advantage of this over omni directionality is that yaw can intentionally be controlled at all times so sensing equipment can always be facing its subject Both ground based and sphere based Vertigo have omni directionality with yaw A 3 Translational Motion The method by which Vertigo is able to navigate from point to point while balancing is often misunderstood The key to this is controlling the point of contact with the ground with respect to the center of mass In this way it is very similar to t
157. ntrol is applied the sphere and body of the robot do not rotate as one and the complexity is magnified 194 Jonathan Missel Sphere Properties A 7 Sphere Properties The properties of the sphere have been established as a critical component of this system this is the second dominant departure from the classic pendulum on a cart problem Exploring exactly how it influences the dynamics is consequently important Beyond features like material and texture which were discussed previously in the Design chapter this section looks at how the physical properties of the sphere such as mass and diameter affect the system If for a moment the mass and inertia of the sphere are somehow assumed con stant and only the diameter is considered it is found that balancing on a larger sphere is easier than balancing on a smaller one This phenomenon is often falsely attributed to a gear reduction in the control where the actuator is the first gear the sphere is a middle gear and the ground is the final gear or pinion This is not true however As with any planar multi gear linkage the only gears that contribute to the overall ratio are the first and last or in this case the wheel and ground The sphere size has no affect on this because as the wheel rotates the sphere that arc length is conveyed exactly to the next gear ground resulting in identical translation across the ground for any size sphere The only way to change Vertigo s gearing p
158. number of infinity and the matrix is singular Here 81 Jonathan Missel Analysis this would translate to not being controllable or observable for the controllability or observability matrices respectively Inverse of Controllability Condition x 10 TEN SRETEN A phi radians theta radians Figure 3 5 Inverse condition number of controllability matrix while varying 0 and From Figures 3 5 and 3 6 it can be seen that the controllability and observability characteristics are definitely affected by the state angles an observation that could not be made from the controllability and observability pots For controllability in Figure 3 5 consider the profile curve for fixed at zero and varying 0 0 is defined with respect to so fixing at zero means that the body and sphere will rotate together Therefore it makes sense that there is a peak about the unstable equilibrium point because it would take less control input to move to another state It is also easy to understand that if the body is initially tipped at an angle more control would be required to work against gravity and resume balance The rotation of the sphere has nothing to do with the gravitational component of the system or controllability except for the fact that the angle of the body is defined with respect to it This is why the general shape of the profile remains the same along curves of fixed 6 and only 82 Jonathan Missel Analysis Inver
159. ogether His mechanical wizardry was of tremendous assistance in designing Vertigo Thanks to all of my friends back in Rochester for their understanding and encouragement as I have spent the majority of my academic career a city away Thanks to several in academia particularly my advisor Dr Puneet Singla for being available and willing to help It was his courage and insight that made gambling on my endeavors end in success It has been a pleasure working with him and I now ill look forward to working under his former advisor as I continue on towards my Ph D at Texas A and M University I would also like to thank John Wadach Dr Tarunraj Singh and Dr David Forliti for their openness and inspiration on both personal and academic matters Thanks to all of my labmates over the years those in Dr Singh s lab Kurt Cavalieri and Jack Lee for weathering my antics with a smile and maintaining an eagerness to assist Special thanks to Shajan Thomas and Mark Tjersland for their computer genius and immense contribution as they volunteered their time and expertise to my research I would like to thank cs for her prolonged belief uplifting humor and studio photography skills Lastly thanks and apologies to anyone I may have inadvertently left out I attribute this accomplishment to a myriad of combined efforts and I thank you all Contents Acknowledgements iii List of Figures xiii List of Tables xiv Abstract XV 1 Introduction 1 lal Moti
160. oller is that it took this simulation about 4 seconds for the transient response to settle This is respectable but reducing this time is desirable if possible because it will further improve overall performance and accuracy Though improvement is possible expecting perfect results is impractical here be cause this is a linear controller design method being implemented on a nonlinear system As the states deviate further from the point of linearization the nonlinear affects of the system have an un modeled influence on the response In this simulation the linearization was taken about a vertical stationary position where all states and control are zero Therefore pushing for greater performance increases the magnitudes of the rates and control input which introduces more linearization error A balance must be reached that provides acceptable accuracy and sufficient performance The convention for defining states in this model make it difficult to visualize the motion of the body with respect to an inertial frame To make this clearer a plot was generated that sums the two angle and rate state pairs As seen in Figure 5 11 this definition represents the response of Vertigo in terms of angles with respect to a vertical reference This clearly shows that the body of the robot returns to 127 Jonathan Missel Sphere Based Control Vertigo Response 0 02 0 015 0 01 0 005 0 005 0 01 Magnitude wrt Vertical radian units
161. ompile the program It will take a few seconds to compile and will print out any syntax errors that may occur If there are no compilation errors type run no quotes to run the program Running the program will open a graphical interface To connect to a robot click on the Connect button A connection dialog will pop up and you will have to select whether you are connecting to the robot via the CMU based relay relay mode or if you will connect over a local network by entering the robot s IP address direct connect For more information about these two modes of operating your qwerk visit http www terk ri cmu edu projects qwerk overview php Make your selection and click Next If you entered direct connect mode you will now be prompted to enter the IP address of the robot Do so and click Connect then click Finish You should now be connected to the robot If you entered relay mode you will be prompted to enter your TeRK login and password Enter your login click the Login button and then click Next to go to the next page in the connection wizard You will see a list of available robots Highlight the appropriate robot 206 Jonathan Missel Connection to Vertigo by clicking on it once Then click the Connect and the Finish buttons You should now be connected to the robot Once you are connected to a robot you can begin displaying an image stream from t
162. on ing devoted to research and development or any other aspect of its engenderment Bipedal walkers are also inundated with single point failures which foster vulnera bility and unreliability ASIMO is 120 lbs of wires actuators sensors circuits and structure failure of any one of these components with few exceptions results in complete malfunction The unreliability and cost of bipedal walkers has pushed their availability out of reach for most researchers causing many facilities such as MIT s Leg lab to shut down The biaxial inverted pendulum was the iconoclastic idea that combined dynamic stability and omni directionality in a frugal and simplistic manner Rather than differential drive self balancing robots the conclusion that should have been drawn from understanding dynamic stability is to balance on a sphere and liberate both axes By simply mirroring the effect of the two wheeled inverted pendulum this idea is not entirely disparate and accordingly seems logical however an ongoing literature study revealed that this has only recently been done by one other group at Carnegie Mellon University with Ballbot in 2006 3 4 The purpose of the Ballbot project is to investigate how to maintain reliable balance in robots to decrease their footprint for improved navigability This is meant to overcome the problem of statically stable platforms having wide bases Ballbot is a commendable project that has yielded impressive results
163. on It re quires thorough knowledge of how mechanical components fit and work together once assembled and comprehension of underlying theory so as not to corrupt the princi ples being tested To simplify and insure the success of this process previous work of others often serves as a foundation to build upon Designing this particular hard ware was especially challenging and critical because much of the projects originality stemmed from its mechanics As a unique platform there were few examples of per tinent work to guide its development therefore extra effort was made to ensure that functionality accuracy and robustness were built into the design for its long lasting success Many factors must be considered when conceiving and producing designs for a brand new robotic system Often these factors have conflicting solutions and de cisions must be made based on priority of tradeoffs This chapter details how these decisions directed the conception design and production of Vertigo Starting with the 17 Jonathan Missel Conception main objective the evolution of the final product is detailed in full Along the way the challenges and logic behind every decision is explained giving both the advan tages and disadvantages of all considered options The techniques and processes for manufacturing are also included as this knowledge is an important tool for efficient design Lastly suggestions for improving or altering future design
164. on a linear quadratic regulator LQR was used for control Together the LQR and Kalman filter solve the famous linear quadratic Gaussian control problem The simulation generated the true data for the response of a translation that equates to a 27 rotation of the sphere the true data is calcu lated in the filter loop and only computes to the next time step This is so it can use the estimated states to calculate the LQR control which mimics how the loop will be implemented on the real system Noise was added to the true data to generate measurements and the filter was applied to them Each time through the filter cal culates its own control using the same LQR method that the true data was generated with For those simulations that run the filter between measurement updates the most frequent measurement was stored for plotting at that step In the first simulation the sampling time was based on the actual average sampling time of the Vicon MATLAB implemented communication loop 0 04 seconds 25 Hz The states and covariance were directly propagated forward to the next measurement so every time step updated the estimate with measurement data Figure 6 2 and 6 3 respectively show the true and measured response to the simulated continuous discrete extended Kalman filter From these results it is clear 146 Jonathan Missel Application of Continuous Discrete EKF LQR Track with EKF Angle Error 2x Sphere Rotation
165. on of the state trajectories and control inputs show that this short time requires very extreme magnitudes and would not be feasible with the implemented system As stated before this method 118 Jonathan Missel Sphere Based Control Linear Simulation Results To Out 1 To Out 2 Amplitude 3 ute o C T To Out 4 Time sec Figure 5 4 Gramian based control for 27 translation in 1 5 sec imposes no limitations on input or trajectory so for situations where final states and time are fixed and all paths and inputs are available the power optimal Gramian based controller is a viable option However for Vertigo and in most other real world applications constraints bind the possibilities for performing a maneuver To accom modate this a technique had to be developed to adjust the final time to fit realistic capabilities 119 Jonathan Missel Sphere Based Control 5 3 2 Power Optimal Control Free Final Time Constrained Input In most real systems actuators are limited in their input capabilities attempting to exceed these limits will lead to saturation and can cause damage to the hardware For the new motors on Vertigo 2 1 and the capabilities of Qwerk according to the man ufacturer specifications a realistic limitation on input u is about 45 Nm To account for this a numerical method was developed that permits imposing limitations on in put It was implemented in th
166. onality The directionality of a system refers to its mobility and is often mislabeled in robotics when describing motion Categories for directionality are loosely defined and many 184 Jonathan Missel Directionality vehicles do not fit nicely into a particular profile This section provides simple def initions and explanations for common types of directionality and establishes the convention to be used throughout this thesis Figure A 1 illustrates the progressively expanding mobility that climaxes at omni directionality with yaw which describes the motion of Vertigo bee CIS ap QD a b c d Figure A 1 Directionality a Bi directionality b Bi directionality with yaw c Omni directionality d Omni directionality with yaw Bi directionality Figure A 1 a simply means that an object is capable of moving forward or backward along a single axis A cart on a track is an example of this where motion is constrained to one degree of freedom Bi directionality with yaw Figure A 1 b is a more common form of motion In this case the vehicle has 185 Jonathan Missel Translational Motion the additional ability to rotate yaw Differential drive vehicles such as tanks and wheelchairs are examples of this They often require only two driven wheels to jointly control their two degrees of freedom and have additional passive wheels or casters to maintain their static stability Nearly all mobile robots fall i
167. onds can be attributed to Vicon noise because the plank was resting on the ground at a fixed angle Though this noise is undesirable tracking the board was a significant improvement over the response when Vertigo was the Vicon object Fig ure 7 12 shows how noisy the response was when Vertigo s angles were tracked by the same PID controller This additional noise is partially due to uneven terrain but the dominant contributor is actuation Large gaps in the omni wheels gave Vertigo a bumpy ride so the measured angle at a given time did not represent net pitch well This noise was so dramatic that performance was affected by it Tracking the board was just a temporary solution to circumvent the true prob 171 Jonathan Missel Balancing Implementation lem Vertigo needed better omni wheels This raised concerns about using Vicon as feedback for balancing on the sphere because tracking a different object would not be an option and the improved wheel design would be extremely expensive and time consuming to manufacture Balance in 2 dimensions began by applying the if controller for the board to both axes The primary objective of this was to confirm the convention of the Vicon roll angle This simple controller was able to balance a flat plate resting on a sphere This was analogous to balancing the plank in 2 dimensions The biggest challenge in this was recovery from large disturbances which is best described with an example If the pitch
168. onse 2x Sphere Rotation N States ratian units o 2 4 theta 6 phi L theta rate phi rate 8 I 0 2 4 6 8 10 12 14 16 18 20 Time Figure 6 7 C D EKF estimated state response fm 25Hz fs 100Hz 149 Jonathan Missel Sampling Time Analysis for C D EKF x10 LQR Track with EKF Angle Error 2x Sphere Rotation 5 T T T T T T T T 6 radians radians o 0 2 4 6 8 10 12 14 16 18 20 Time Figure 6 8 C D EKF angle state errors fm 25Hz fs 100Hz LQR Track with EKF Rate Error 2x Sphere Rotation 0 05 T T T T T T T T T rate Tadians sec Jl L ji i f H L ji l 4 887 2 4 6 8 10 12 14 16 18 20 Time 0 1 T T T T T T T T T F g 0 05 a 4 S 0 g a 0 05 o p 0 1 1 li fi 1 ii 1 fi 1 0 2 4 6 8 10 12 14 16 18 20 Time Figure 6 9 C D EKF rate state errors fm 25Hz fs 100Hz 150 Jonathan Missel Sampling Time Analysis for C D EKF comparison shows that when both filters are properly tuned it is advantageous to have fs gt fm It is clear that the number of propagations influences the filters accuracy but the measurement sampling time also has a significant impact Understanding the nature of this affect is important for maximizing the chances of success with the fully implemented system Therefore a simulation was created that varied both the measurement and
169. opying linux Creating redboot source tree Building redboot Installing redboot Building IceE Building firmware Building r w filesystem make 1 Leaving directory home YOUR_USERNAME terk embed trunk cirrus arm linux 1 0 4 edb9302 This will generate three important files zlmage ramdisk gz and optfs out All of these are in the edb9302 directory under the cirrus arm linux tree Building the cirrus arm linux tree will likely take a while but luckily it s not some thing you ll have to build very often The code in the cirrus arm linux tree rarely changes so you ll most likely build it once to create the zImage and ramdisk gz im ages and then you ll just do builds of the share tree which does change often which creates the optfs out image 222 Jonathan Missel Changing Qwerk s Embedded Code Note that although the cirrus arm linux build generates a new optfs out image all it s really doing is repacking the opt directory which was copied into terk embed trunk cirrus arm linux 1 0 4 edb9302 by the share tree build above It repacks the image by calling the buildoptfs sh script In any case it should be the same as the one created by the share tree build We re now almost ready to download the image files to the Qwerk We ll first need to put the images in the Apache s web root directory so your machine can serve them to the Qwerk To do so copy the three image files to the var
170. ormed for this were fixed platforms that supported the base of a pendulum and 19 Jonathan Missel Conception actuated it in both directions This was often done with perpendicular tracks or rails like those in Figure 2 1 These devices could control the pendulum in any planar direction but their motion was limited to the size of the chassis A larger range of motion would require a larger device which quickly becomes cumbersome and impractical To overcome this restriction a mobile vehicle would need to be used Figure 2 1 Fixed biaxial inverted pendulum concept Omni directional vehicles had been done but they were statically stable with wide footprints and low centers of gravity Inverted pendulum vehicles also existed but they were not omni directional they could only move forward and backward and had to turn when changing directions The idea of placing a pendulum on top of an omni directional vehicle seemed promising at first it would allow both features to be tested as controls experiments This is similar in principal to the fixed platforms 20 Jonathan Missel Conception but would not be limited to the size of the base The major drawback of this con figuration is that it would not fully take advantage of the unstable dynamics of the pendulum Cornering is still mostly dependant on the abilities of the statically stable base with minor influence from the pendulum Properly combining the two features would m
171. ossible to fully control an under actuated system it requires high precision and is more time consuming to execute than having a fully actuated system therefore these methods are generally implemented when actuators fail un expectedly or when weight and cost make including more actuators unreasonable This is often the case with spacecraft In space they are free to move in six degrees of freedom but the cost of building and launching space vehicles makes reducing the number of actuators desirable when possible Also replacing failed actuators is rarely an option therefore these techniques may be exploited to save a mission using only what remains operational Vertigo is a highly maneuverable platform and is capable of modeling more than just under actuated systems Some other examples with related motion to that of Vertigo are the control of in flight refueling systems quad rotor helicopter hover and planar translation and actively controlled rockets These systems are expensive to operate risky to test and require highly trained personnel in controlled settings to ensure safety Additionally their development requires the progressive accomplish ment of many stages that build up to a final result A platform like Vertigo that can mimic the general motion of these systems is ideal for addressing some of the preliminary stages of development in an inexpensive reusable and safe way It can also be reconfigured to best match the simulated
172. oth the continuous using lsim and discrete equations with similar results however one 114 Jonathan Missel Ground Based Control of the focuses of the ground based work was to investigate the response when treated as a discrete system so only those results are included here Figure 5 2 shows the results of the simulated discrete response for the standardized circular trajectory This trajectory was specified by assigning Xref amp x sin w x t and Yref amp x cos w x t where amp is the amplitude and was set to 0 5 m and w is the angular velocity and was set to 0 3 rad sec Discrete LOR Response 0 5 ym 5 35 04 03 02 01 0 01 02 03 04 OS Figure 5 2 Simulation results for a circle reference trajectory norm error 0 7821 As seen in Figure 5 2 the simulation results correspond very well to reference trajectory having a norm error of only 0 7821 This is within the design criteria of norm error lt 1 however it was expected that the physical response of the real sys 115 Jonathan Missel Sphere Based Control tem would have greater error due to the un modeled and unpredictable disturbances This is mostly because the simulation assumes that the yaw angle is perfectly held at zero without the need for additional control input to keep it there In reality this would not be the case because external influences such as slip and uneven ground would accumulate small yaw errors Also slip and response l
173. otors these values had to be defined on a scale and specified as unitless 105 Jonathan Missel External Sensing Architecture To form the link to MATLAB a third communication method had to be devel oped It built on the progress of the first and second methods by dynamically adding a classpath to establish access to the library of Java commands and a generic GUI that provided a means of wirelessly connecting with Qwerk This GUI also supported the video feed applications that worked with Vertigo s webcam Making the last con nection opened the door for the Java based Qwerk commands to be inserted directly into MATLAB code To initiate this union a program was written that dynamically added the Java classpath and opened up the GUI for connection to Qwerk This saved a variable in MATLAB s workspace that gave access to the Java library The nature of this variable did not allow m files that were not in the same directory or functions to call the Java commands Additionally it had unsavable material that could not be saved to a MAT file for reloading in other locations therefore all code had to be implemented accordingly Further explanation and instruction for all three connection methods can be found in Appendix C With these connections made Qwerk was finally able to receive control inputs Figure 4 2 only illustrated the use of the motor commands for Vertigo 2 1 and differed from Vertigo 1 0 s schematic in that all four m
174. otors were controlled rather than just two Qwerk accepted the conditioned control signal as a representative final scalar value for desired motor speed A trapezoidal angular velocity trajectory was generated and an embedded PID controller applied to track the profile The output of the PID was converted to voltage and sent to the motors Back EMF from the motors was measured and converted to a representative velocity to be feed back into the embedded PID controller Qwerk used this embedded control method to regulate the inputs to the motors and protect itself from saturation The downside to this was that the user was unable to directly control the motor inputs Signals were always feed into the trajectory 106 Jonathan Missel External Sensing Architecture generator and then tracked by the PID controller this significantly added to the response time In addition there was no documentation that addressed or even mentioned the problem Therefore when the motors were found to have an extremely slow response taking 20 seconds to accelerate between the positive and negative rotational velocity extremes it took several months to determine that conservative profile trajectories and poorly chosen hardcoded PID gains were the culprits This egregiously lazy response time was about three orders of magnitude too slow for inclusion in this control loop which cycled at about 25 Hz This put it in urgent need of mending As embedded features corr
175. own are shown in black bold type Values that you need to enter which might be different for your network are shown in red bold type You should use the defaults for all other values RedBoot gt fconfig Run script at boot true Boot script load r v b 0x800000 ramdisk gz load r v b 0x80000 zImage exec r 0x800000 s 0x600000 Enter script terminate with empty line gt gt fis load ramdisk gt gt fis load zImage gt gt exec r 0x800000 s 0x600000 gt gt 231 Jonathan Missel Changing Qwerk s Embedded Code Boot script timeout 1000ms resolution 1 Use BOOTP for network configuration false Gateway IP address 192 168 0 1 Local IP address 192 168 0 3 Local IP address mask 255 255 255 0 Default server IP address 192 168 0 2 DNS server IP address 192 168 0 1 Set eth0 network hardware address MAC false GDB connection port 9000 Force console for special debug messages false Network debug at boot time false Update RedBoot non volatile configuration continue y n y Erase from 0x607c0000 0x607c1000 Program from 0x01fde000 0x01fdf000 at 0x607c0000 RedBoot gt Now either enter reset or power cycle the Qwerk so that your changes take effect Either way be ready to type CTRL C to get back in to the RedBoot gt prompt Loading the Linux Images First we ll load three images from the HTTP server running on your machine into the Qwerk s RAM Enter the followi
176. p computer and was ideal for this assembly because its pins exited downward protecting them from being bent A 4 pin connector was then used to plug into the DB 15 outputs and cable into Qwerk Attached to the side of the IMU is the USB extension and Airlink wireless adapter that allows Qwerk to communicate with other computers Further down is a black Delrin plate that the IMU attaches to Delrin is a high grade engineering plastic whose properties are given in Table 2 2 21 In addition to these attractive characteristics it machines well has a clean finish is very durable resists warping resists corrosion and is in the same general price range as alternate materials such as aluminum For these reasons Delrin was used for the entire skeletal structure and motor mounts Properties ASTM Units DELRIN Test Method Density D792 lbs cu in 0 0564 Specific Gravity D792 g cu cm 1 56 Tensile Strength D638 psi 8 500 at 73 F Linear Thermal D696 in in F 4 5 X 10 5 Expansion Coeff Table 2 2 Properties of Delrin The top plate is supported by four identical Delrin legs that link it to the bottom plate The legs have cutouts to reduce excess weight and allow wires to be fed through them The top and bottom holes in the legs that mount to the two plates are coaxial and the hole patterns in both plates are identical so they can be interchanged for reconfigurability All structural components are faste
177. perate in surroundings designed for the human frame Though the setting may be different this is the primary objective of NASA s Human and Robotic Exploration program for designing robots that assist astronauts 13 These machines must be built to work in environments meant for humans without sacri ficing functionality The omni directional motion and dynamic stability of Vertigo means that it is better suited than most platforms to work side by side with humans here on Earth Lastly Vertigo would make an exceptionally intriguing educational tool There are hundreds of platforms and experiments on the market to help teach dynamics 15 Jonathan Missel System Description control and mechatronics but most are limited in focus and fail to inspire students Vertigo tends to invoke curiosity to those first introduced to it and is flexible enough to accommodate work at all levels Simple experiments can be set up to confirm and illustrate concepts taught in the classroom Demonstrations that alter the physical properties and investigate the dynamics in comparison to theory would be an excellent aid for proving concepts at an introductory level and would not be difficult provided a small bank of controllers Additionally its mobility and versatility offer much to be explored in advanced controls and estimation research 16 Chapter 2 Design 2 1 Introduction Designing hardware is a notoriously challenging aspect of experimentati
178. ponse with stabilizing LQR controller It was the primary objective of this controller to balance Vertigo by bring the states from their initial conditions to ones where the sum of the two angle states is zero This signifies a vertical and balanced stature If fully effective all states should be brought to exactly zero as this is their minimum possible value The plots of the rates in Figure 5 10 show that this method was effective at bringing these states back to near zero values accordingly the angle states leveled out as the rates were 126 Jonathan Missel Sphere Based Control brought to zero This controller also attempted to drive the angles back towards zero however they over shot and settled about 2 radians away from zero Offset angle states with zero rate states means that the system settled into a balanced position at a translated location that is away from the origin Purely in terms of balance this is not a problem because the system did not tip over However this error may cause difficulty for implementation on the real system because one of the options for the implemented control of Vertigo is to have two separate controllers acting One of these would be devoted to assuring balance while the other controls translation If the balancing controller is doing its job but causing large translational errors it will create more work for the second controller and will add to the overall error Another issue with this contr
179. pulations of the error scaling factor were required before it would converge The reason is that for values below 15 the controllability Gramian approaches singularity and the equation u t B e 1 9 wet t1 Ax x1 5 16 is not able to be calculated by MATLAB due to its ill condition These results support the basis of the inverse condition number presented in the state controllability and observability analysis Again looking at Figures 5 5 5 8 it is clear that as the 123 Jonathan Missel Sphere Based Control controllability Gramian approaches singularity with smaller allowable Uma values the accuracy of the calculations reduced and the linear simulation was not able to reach the final states This error grew as the condition of the controllability Gramian degraded This also made the code more sensitive to the initial guess for final time so the plot and included equation in Figure 5 9 were created and used to generate close approximations for the initial guesses based on the empirical results 5 3 3 LQR Overview In feedback control the actuating signal is dependent on both the reference signal and the output of the system In state feedback control the plant feedback output contains at least one of the states full state feedback is when all of the states are included as outputs For state feedback input u is defined as follows u r kx 5 17 With this definition x represents the output stat
180. quired only a handful of moving parts and were successful in protecting onboard components from impact 176 Jonathan Missel Conclusions and Future Work However working with them gave rise to ideas for improvement Results from control implementation showed that a need for new omni wheels was the most urgent of these improvements The gaps between the rollers caused chattery motion which affected measurement quality A new design for near seamless omni wheels was generated but cost and time of manufacturing made them impractical to this point In the future it would behoove the project to incorporate these new wheels Also proposed is a design iteration that fits Vertigo with only three legs This would make the Vertigo 2 1 drive equal actuated while preserving all modes of motion and making more consistent contact with the control surface The only drawback is a sacrifice of independent actuation however motion would remain decoupled so control could be decomposed and applied in representative components relatively easily The mathematical modeling of Vertigo proved valuable for its development Its accuracy was confirmed through the response of simulations and agreement of analysis results with theory Analysis of the equations focused on observability and controlla bility These results were useful in confirming the validity of control and estimation methods that would be applied By looking into the inverse condition number of the
181. r and going into Ad vanced system settings b Next click on the Advanced tab then click on the Environmental Vari ables button c Under the System Variables section add a new variable MATLAB_JAVA with the following value C Program Files Java jdk1 6 0_06 jre this was it in my case or a value that corresponds to the version of Java that you are running d Enter version java to ensure that the version is updated restarting Matlab may be required 2 Enter edit classpath txt into the command window and add the directory where your class files are located in a new line C Users Jon Desktop VERTIGO QWERK MyFirst Vertigo terk client MyFirstRobot 209 Jonathan Missel Connection to Vertigo 1 Try constructing a java object Example of constructing a java object in Matlab using two methods Method 1 gt gt j javaObject AddNum 3 2 4 2 gt gt j writeValue 7 4 Method 2 gt gt q AddNum 3 4 1 3 gt gt q writeValue 4 7 This was first tested this with a class file generated from the javac compiler from the command prompt Files generated from an IDE such as JCreator or Eclipse were then successfully tested as well This method has not yet been tested with stuffing the class into a package Sample class public class AddNum private double myValue public AddNum double a double b my Value a b public void writeValue System out println
182. r advantages of this drive Also as the wheel rotates and control application oscillates between the two rows the output is inconsistent Accordingly a single row omni wheel would have to be used for Vertigo Unfortunately the gaps in single row wheels make the affective radius inconsistent and vibration from rolling on a discontinuous surface will add noise to the sensors In selecting single row omni wheels it was important that they have many rollers around the circumference to minimize the size of the gaps between them The rollers should also be made of or coated with material that will provide traction on the var ious surfaces Vertigo will encounter These criteria governed the verdict of choosing 31 Jonathan Missel Components Kornylak Transwheel 2051 10mmX CAT TRAK wheels CAT TRAK is their syn thetic rubber coating that is applied to the rollers for added grip and significantly improved performance Wheels specifications 15 e Standard 2 O D 10mm Plain Bore e Recommended max load Steel bottom 25 lbs 11 3 kg Plywood surface 7 5 lbs 3 4 kg 200 lb test corrugated bottom 5 lbs 2 25 kg e Weight 1 00 oz e 8 rollers The 10mm I D was attractive because it is a common size for hardware which would simplify the process of finding fittings and it was large enough to allow small hardware such as setscrews to fit within its diameter Having eight rollers is dense for a 2 inch O
183. r is the HTTP server i e your machine If the IP addresses are correct then you can skip down to the Loading the Linux Images section Otherwise follow the directions in the Configure the Qwerk For Your Network section to set them for your network Configure the Qwerk For Your Network If the IP Gateway and Default Server IP addresses displayed in minicom when you first power on the Qwerk are incorrect then you won t be able to download the images from your HTTP server The IP address is the address of the Qwerk the Gateway is your router and the Default Server is the HTTP server i e your ma 230 Jonathan Missel Changing Qwerk s Embedded Code chine I usually go ahead and set the DNS Server IP Address too which should in most cases just be the same as the Gateway IP Address Execute the fconfig command to set the IP Gateway and Default Server IP addresses Start by entering true for whether the script should run at boot and then enter the script exactly as shown below Once you ve entered the boot script enter an empty line to terminate the script editing mode Continue on through the other values and enter the correct IP addresses for the Qwerk Gateway and Default Server for your network Accept the default values for all the other options Make sure to en ter yes y when prompted whether to update the RedBoot non volatile configuration In the example shown below values you need to enter exactly as sh
184. racked a one dimensional user defined step function with if statements and constant magnitude control inputs Broken down the logic followed that Vertigo should be moved forward if the difference between desired and measures position was negative and backward if positive This program helped to start piece together the coordinate and actuation sign convention for the loop For programs that introduced new func tions if statements were used rather than proportional control because they allowed exact specification of input preventing motor saturation Building off this a two dimensional proportional controller that tracked a step function in both x and y was constructed By converting the x if control into a pro portional gain that was multiplied by the position error and then mirroring applica tion to the y axis this controller verified that Vertigo has decoupled omni directional motion on the ground It was also used to confirm the inertial axis convention estab lish the safest way to terminate control and experiment with control input and gain values to determine reasonable ranges Running these experiments began to expose some of the flaws in using the Vicon system Nonspecific object definitions meant that the center of the Vicon model did not exactly match that of the actual robot Also concerns about yaw control were seeded here as minor but erratic rotational drifting was observed for large step functions Figure 7 1 shows how this contro
185. radius which is a flat surface this is why the ground based configuration of Vertigo is possible The omni directionality comes from the two inch diameter wheels that are inlaid with rollers around their circumference This configuration allows control in the direction of rotation of the motors and slip perpendicular to this which decouples the directional control Figure A 3 Spherically orthogonal omni directional actuation Here directional control is referring to the motion of the sphere s surface with respect to the robot s body Controlling direction is somewhat specious wording really the motors are applying a torque which acts to rotate the ball This rotation causes Vertigo to lean as mentioned in stage two in the previous section and is how the angles are controlled By calling upon opposing pairs of motors roll and pitch are decoupled and independent To control yaw all four motors rotate in the same direction If the friction between the ground and the ball is sufficient the robot will rotate while the sphere remains stationary 189 Jonathan Missel Control Superposition A 5 Control Superposition Previously it was shown that Vertigo s directional control is completely decoupled through actuation of opposing pairs of motors but its yaw control requires the cooper ation of all four motors For the continuous model the solution to this can be handled by superposition The same principles apply for bo
186. rection causing overshoot Inclusion of retractable statically stabilizing legs may be a solution to avoid this A simple design where servomotors retract the leg extensions as soon as balancing control is applied and quickly extends them again when control is terminated would eliminate the need for placing the body on the sphere or making sure it is off before ending control It would also allow Vicon to initialize the system at its proper balancing height and no bias would have to be included to account for the added height from the sphere The Ballbot team has implemented a similar set of retractable legs with great success 65 Chapter 3 Mathematical Modeling and Analysis 3 1 Introduction The intent of this chapter is to derive the equations of motion for Vertigo and con duct an extensive controllability and observability analysis on these equations A trustworthy mathematical model is invaluable when designing reliable and affective estimators and controllers In development accuracy and convolution must be bal anced using reasonable assumptions to yield dependable equations that avoid excess computational cost Analysis of these equations is also important it helps to verify their accuracy and establish the objective of the control and estimation processes that will be applied later on This chapter derives the nonlinear equations of motion for sphere based Vertigo and then shows how they are linearized for use in linear esti
187. rements plagued the system with implacably whimsical responses that oscillated divergently Once an object was created and tracked its Euler angles and position with re spect to the inertial frame were transmitted to a manufacturer provided Simulink plant for deciphering From there the sim command was used in MATLAB to call the Simulink plant and fetch the measurements for the feedback controller Of all processes executed in the control loop this was the most time consuming casting it as the leading role in defining the measurement sampling time In addition it took an inconsistent amount of time to acquire the data ensuing a variable sampling time Furthermore this communication periodically paused causing the input to be held constant and the controller to fail in sphere based mode The cause of this lapse was never confidently pinpointed but its frequency during waking hours and scarcity throughout the night suggested that it was an interference problem For ground based and the alternate architecture variations MATLAB can be thought of as the central hub for all communication It received and sent signals and was the platform on which controllers and estimators were written and implemented Figure 4 2 shows how the trajectories were generated in MATLAB and how errors were calculated with the measurement information from Simulink For this specific example of control a proportional controller was applied to the yaw error The posi
188. riation was restricted by tuning it was not possible to start this analysis from the baseline point of fm 25Hz 151 Jonathan Missel Sampling Time Analysis for C D EKF Theta 30 Timing Mesh fm Hz Figure 6 10 Sampling time mesh fs 25Hz and expand from there It was also not realistic for the two ranges to overlap because impractical points would exist where fs lt fm Therefore it was necessary that fs gt fm for all points so the more significant initial improvement of running the filter more than once per update could not be captured It is clear from Figure 6 10 that there is a much more influential dependence on the measurement sampling time Lower 30 values tend to signify a more accurate estimate Therefore this mesh verifies the expected results that the estimate will improve as the number of measurements is increased Again the same general results were found with the other three states so they were not included here to avoid redundancy 152 Chapter 7 Implementation 7 1 Introduction Vertigo is an original platform that was built from scratch as such it was necessary to increment its development by introducing controllers from the ground up Taking small steps built a familiarity with the system that made troubleshooting easier as the project progressed to more sophisticated coding The goal here was to develop methods for implementation and apply fundamental programs for testing function ali
189. rience with the material the legs were designed to allow a nonspecific midrange amount of deflection but were kept rigid enough to avoid appreciable departure from the rigid body assumptions in the mathematical modeling In practice this design proved to be successful The motor mounts were designed specifically to clamp tightly down on these par ticular motors Accordingly the diameters of the ballast motor blanks that support the two freely spinning wheels were sized to that of the motors and their length was set to match the motors weight The blanks accepted a 4 mm shoulder screw that functioned as an axel for the bearings that press fit into the wheels A Delrin spacer was made to fit between the bearings and keep them flush with the outside surfaces of the wheels to help them run true The motor encoder wheel assembly was challenging to design The encoder wheels were very fragile and the assembly had to be as compact as possible for minimal compromise of symmetry due to wheel offset For the encoders to function properly the reader circuit had to be fixed with respect to the motor and the encoder wheel must rotate with the omni wheel Both wheels had to be fixed to the motor shaft and the encoder wheel had to line up with the optics slot in the circuit Figure 2 4 shows the exploded CAD model of the final design solution for this assembly The key components to this design are the wheel hub and the encoder circuit mounting 43 Jonat
190. rol to balance because of their instability However this ostensible curse of instability is ac tually advantageous at times Studies 1 have shown that dynamically stable robots are far superior at navigating terrain This is because they are able to shift their weight to keep their center of gravity above their wheels as shown in Figure 1 2 2 The value in this ability is most clearly seen when unexpected loading is applied and the controllers automatically compensate to maintain balance Self balancing robots also have greater potential for agility Controlling where their center of gravity is allows dynamically stable robots to take advantage of their instability and let gravity assist in the task of acceleration This is similar to how jet fighters are intention ally aerodynamically unstable for greater performance In light of these dynamic prospects the two wheeled inverted pendulum configuration has been eagerly em braced as the focus of many research projects worldwide however this is a myopic response to straightforward results Such vehicles do not take full advantage of the very feature that sets them apart their dynamic stability They are self balancing on one axis but maintain dependence with static stability on the other Since they can still tip over in one direction slopes must be dealt with head on and directional control is lost They also have problems with unexpected external loading applied in both directions such as in
191. roved for manufacturing Figure 2 6 c 2 shows a photograph of the physical system after manufacturing Side by side with the CAD rendering it is apparent how accurate the model was and why it could be used with such confidence for designing the structural components 49 Jonathan Missel Design Iteration w Conceptual Sketch Inverted pendulum e Spherical base Motor orientation b Preliminary CAD Model Actuation mechanism Sensing requirement Reconfigurability c 1 Vertigo 1 0 CAD Model Component specification Working drawing c 2 Vertigo 1 0 Photograph e Prototype of Vertigo 1 0 d 1 Vertigo 2 1 CAD Model Yaw actuation Structural redesign Improved motors Wheel redesign Improved power supply d 2 Vertigo 2 1 Photograph e Prototype of Vertigo 2 1 Figure 2 6 Vertigo design generation 50 Jonathan Missel Design Iteration After Vertigo 1 0 was designed and manufactured the early stages of implementing simple feed forward control highlighted some of its shortcomings When operating on the ground for extended periods of time friction from the offset wheels tended to result in a net rotation Also the motors were too slow in practice even after modifying the gear box to eliminate one of the gear reductions It was decided that these problems were not debilitating for preliminary testing and experiments and fixing them could be delayed
192. rtigo using the Lagrangian formulation which followed the derivation of the Ballbot team at Carnegie Mellon University Their model was chosen based on its accuracy and their success with implementing it Jacobian linearization was applied to these equations to extract a linear state space model allowing linear systems analysis techniques to 96 Jonathan Missel Conclusions obtain locally accurate results The linearized model is also used throughout the control and estimation portions of this thesis The equations of motion for ground based Vertigo were then derived Their simplifying assumptions neglected nonlinear affects so their linearization was not necessary To understand the nature of the sphere based equations the controllability and observability of all feasible state conditions was explored and it was found that for ev ery combination of states the system is both controllable and observable To get more insightful information about how the controllability and observability are affected by these state combinations the inverse condition number was analyzed at each point Smaller inverse condition numbers meant that the matrix is nearing singularity and computation of its inverse or the solution to linear systems of equations is prone to numerical error The determinant of the controllability matrix was plotted and gave supporting evidence for the inverse condition number analyses For both controlla bility and observability
193. s are given 2 2 Conception This section discusses the engenderment of Vertigo as a concept It starts by es tablishing the design objective which is to be the motivation throughout and the reason behind every decision made in the project s maturity Various solutions to the given objective are then detailed and supporting arguments are presented for a platform that balances on a sphere Finally the challenges implied by this solution are explained and set to be addressed in the remaining design 2 2 1 Objective The primordial objective that seeded this project was to create a new ground platform with exceptional mobility and mechanical simplicity A machine capable of maneu vering with few constraints is attractive for experimentation and promises application in many areas Achieving this in a cleverly simplistic manner with no unnecessary parts means that weight and cost can decrease while reliability and performance increase Simplicity also aids in repair assembly disassembly manufacturing and mathematically modeling Therefore the goal was to create an inexpensive testbed that was mechanically efficient and had as many degrees of freedom as possible 18 Jonathan Missel Conception 2 2 2 Development To meet all aspects of the objective a drive concept that provided mobility was first generated and then an efficient design was created to embody this concept Looking to the work of others there were many existi
194. s expensive to design and manufacture because they only require that one two dimensional profile be sent to a CNC mill which can complete the entire cut in a single tool path Knowledge of the manufacturing procedure was used in aesthetic design as well To create tool paths for the mill to execute the SolidWorks prints had to be exported into MasterCAM a computer aided machining program that can write programs for the mill In MasterCAM splines are reduced from equations to a series of representative tangential lines There is a fixed number of lines per spline so short or conservative curves appear to be smooth but long profiles do not and distract from the overall appearance This is because light catches the flats like a disco ball and it is easy to 47 Jonathan Missel Design Iteration see that the shape is discontinuous For this reason the dramatic swooping curves of Vertigo s skeleton were composed of combinations of tangentially joined fixed radius arcs whose equations are readily passed between the programs and give a smooth finish to the profile The method of assembly also had to be accounted for in the design of the encoder mounts The initial design was feasible to manufacture and fit properly when as sembled but analysis of the CAD model revealed that it was not actually possible to assemble Fortunately this was spotted in the early stages of design before man ufacturing and served as a testament to the advantages
195. s of directly controlling the voltage to the motors Control is prescribed with a scalar reference value that is sent to Qwerk This value is between 50000 and 50000 and represents a unitless angular velocity factor Qwerk uses this value as a final value in a trapezoidal trajectory generator It then implements a PID controller with EMF feedback to follow this ve locity profile This entire process is embedded and the original trajectory generator and PID gains had an extremely slow response it took roughly 10 seconds to reach full speed from rest This is two or three orders of magnitude too slow when dealing with a system that samples around 25 Hz Mending this problem was discussed with the communication architecture but the issue was mentioned here because it was discovered through implementation and delayed progress by several months Yaw Control If yaw angle was not controlled during tracking the robot was free to rotate under external influences This meant forfeiting control of camera direction and was especially problematic when controlling from the inertial frame Any misalignment from yaw rotation with the two frames skewed all control application This was only a problem for ground based control as most sphere based methods only used Euler angles Eventually when translation is attempted this will also become an issue for sphere based control To address this problem two strategies for proportional yaw control were implement
196. s with the controllability Gramian Wg t W t eA BB ec dr 5 12 0 If W is nonsingular the pair A B is controllable It can also be shown that for 117 Jonathan Missel Sphere Based Control ti x t e tix 0 f AGM OBu 7 dr 5 13 0 a control defined by u t gt B e 1 9 wt t ex x1 5 14 will transfer the system from Xo to x in time t The development and proof of this can be found in 26 The input u t defines the minimal power control because for any input W t with the same state transfer in the same amount of time it holds that i a t a tat gt 1 und 5 15 to The proof of this is well known and can be found in 27 This method requires fixed final time for which the controller is designed and does not impose any constraints on the trajectory or input Figure 5 4 shows the results from a linear simulation for the Gramian based power optimal controller that takes Vertigo from zero initial conditions to x 2a 27 0 0 in 1 5 seconds Physically this means that the controller should move Vertigo from a stationary and upright position to a distance equal to one sphere circumference away while ending balanced upright with no velocity As shown in Figure 5 4 the linear simulation of this implemented controller takes all the states to their desired location in the prescribed time To do this in only 1 5 seconds is an impressive claim however close inspecti
197. se of Observability Condition O phi radians theta radians Figure 3 6 Inverse condition number of observability matrix while varying 0 and 4 shifts to the right or left giving the symmetrical nature of these plots In Figure 3 6 if the constant 0 profile is considered again the results show that the system is more easily observed near the equilibrium This suggests that given the input and output history it is easier to determine the initial condition when it started closer to vertical Again the angle definition is the reason for the symmetrical shape of the plot in Figure 3 6 For both plots the scaling shows that these peaks are small so the state variations do not have an overwhelming impact As explained before the inverse of the controllability matrix must exist to be put in to CCF therefore the determinant must be nonzero To further illustrate this the determinant of the controllability matrix was calculated and plotted for analysis as well Figure 3 7 shows how these results agree with the findings of the inverse condition number This analysis was not done for observability because it does not always take the form of a square matrix and its determinant is not always calculable however techniques exist that account for this 83 Jonathan Missel Analysis Determinant of Controllability Matrix WAM oy ARRERA J E phi radians 1 theta radians Figure 3 7 Determinant of controllability
198. ses causing the body to fall more slowly Additionally the lever arm from the control to the center of gravity also increases Both of these tend to decrease the amount of control effort required to recover This explains the concave characteristics with increasing length It is important to note that this fixed angle example assumes the same initial angle of departure but in practice larger rotational inertia cases take longer to fall and will not experience as much departure in a given period of time So when time is included a longer is actually perceived as easier to control For the same plot if we inspect how the condition changes as Vertigo s mass 87 Jonathan Missel Analysis Controllability Rank Vertigo Mass kg 15 01 rn Figure 3 10 Rank of controllability matrix while varying 6 and d Observability Rank Wertigo Mass kg 15 01 m Figure 3 11 Rank of observability matrix while varying and 88 Jonathan Missel Analysis Inverse of Controllability Condition 0 016 0 015 0 014 0 013 0 012 25 0 1 rn Vertigo Mass kg Ls Figure 3 12 Inverse condition number of controllability matrix while varying Vertigo mass and Inverse of Observability Condition 0 2 0 15 mm Vertigo Mass kg 15 01 Figure 3 13 Inverse condition number of observability matrix while varying Vertigo mass and 89 Jonathan Missel Analysis increases a definite
199. sized sphere and operate on the ground was desirable This would also help with robustness against uncertainty in the sphere size Recognizing that yaw control is not required for omni directionality and wanting to avoid over actuation concept B was initially chosen for Vertigo over concept D To solve the support issue two passive omni wheels were arranged on the other side of the sphere this also preserved the symmetry of the system eliminating that flaw As the project was under way and concept B was under construction a continued literature survey uncovered the recently published paper on Carnegie Mellon s Ballbot This robot used the inverse computer mouse drive detailed by concept A Although Ballbot required a precisely sized sphere and 27 Jonathan Missel Components could not be placed on the ground it had the same degrees of freedom and number of actuators as Vertigo A drive for complete originality pushed the project into an early design iteration that implemented concept D This method had the same decoupled directional control as Ballbot but had the additional advantage of yaw control and the ability to function on any size sphere or on the ground This iteration came after the first version Vertigo 1 0 was built so the development of both designs is included in this chapter 2 4 Components With the objective and general method for accomplishment established further de velopment of the structure necessitated a
200. sphere at the bottom The figure shows just one configuration of the assembly variation will be explained in the discussion Throughout the development SolidWorks was used to design the robot in CAD as components were ordered and received the model was updated with exact measurements The time and effort to make precise working CAD models is well worth the benefit they are extremely convenient for design because the new ideas can be tried and analyzed with time being the only cost The model was used to fit all components and confirm that the design was feasibility before the final prints went to the machine shop Renderings from this model are used interchangeably with actual photographs to help illustrate the various assemblies being discussed Perched on top of Vertigo is a very light weight plastic spherical webcam that includes a cover to protect the lens and an image capture button on the side It came with a mount for the sphere to snap in to but it was designed to grasp computer monitors and was not suitable for adhering to Vertigo Therefore a new mount had to be designed and built to withstand crashes and possibly help protect the camera as well The final mount design scavenged the clip from the original mount so the camera could be disconnected easily This also protected the camera by allowing it to break free in collisions rather than bear the impulse of the entire robot A ball and socket joint was incorporated into the design and h
201. sponse experienced slightly different yaw perturbations each experiment and its accuracy varied accordingly If the accumulated disturbances ever exceeded 2 5 complete divergence occurred because control was applied along the wrong axis To demonstrate some of the disturbances Figure 7 6 shown the test trajectory being traversed 4 times while holding yaw to zero with a proportional controller By completing the circle multiple times it is clear where the track consistently departed from the trajectory These represent uneven ground that caused slip The lower lower left and right portions of the circle show examples of this There are also two pronounced departures near the top these were caused by temporary communication failure between Vicon and Simulink While delayed control was held constant re sulting in the signature tangential wander The departure to the right happened on an even part of the floor with surefooted actuation so the recovery was accordingly swift The tangential departure to the left was on uneven ground that was prone to slip as seen by the other three tracks this resulted in a slightly longer recovery time The only other common obstacle that is not seen here is a Vicon object tracking failure This happens when Vicon loses sight of one or more markers and does not 164 Jonathan Missel Ground Based Implementation LQR ground track with P yaw control 4 laps 0 5 0 4 0 31 0 2 y position
202. ssemble to com prise this system the sphere and the body Vertigo To explore the sensitivity to these components a controllability and observability analysis is done for both while varying their dimensional and mass parameters As before the determinant and con dition number are also considered for further evaluation 78 Jonathan Missel Analysis 3 3 2 State Observability and Controllability Analysis For a system to be controllable an input must exist that will transfer any state to any other state in a finite amount of time It assumes that the linear system has infinite control magnitude available and that any trajectory may be taken This is an extremely important preliminary calculation for system design to ensure that it is possible for the objectives to be met The controllability is determined by checking the rank of the controllability matrix C rank C rank B AB A B 3 26 If the controllability matrix has full row rank then the system is controllable In this case there are four states so the controllability matrix must have rank four If a system is observable it means that for any initial state there exists a finite positive time where a record of the input and output is enough to uniquely determine that initial state To determine observability the observability matrix O must be found to have full column rank rank O rank C CA CA 3 27 Just as in controllability for this system
203. st one other point of contact must be placed on the sphere to apply force against the rollers Lastly this dependence on the sphere parameters means that each system has to be designed for a very specifically sized sphere This is undesirable especially when such trivial things as temperature and wear can change the diameter of the sphere enough to influence or incapacitate the system The second solution concept B came from a different train of thought all together Traditional omni directional robots overcome the problem of multiple actuators on the same surface with omni wheels Based on this Figure 2 2 b shows how this design s Euclidian perpendicularity can be adapted to conform to the contours of a sphere to give the same affect As with the inverse mouse ball method this has two actuators oriented perpendicular to each other and in a single plane that is parallel to the ground However by using omni wheels rather than rollers the actuator plane is no longer constrained to intersect the equator of the sphere This is because all components of motion opposing the direction of control are passively permissible as slip and the perpendicular orientation of the wheels decouples the directional control Since control no longer needs to be applied at the equator the sphere diameter is irrelevant as long as it is large enough to make contact with both actuators This also means that the robot could be placed directly on the ground analogues to a
204. ster click find and search for the keyword gain After changing them click save NOTE The original gains were 100 0 500 0 These with the original tra 215 Jonathan Missel Changing Qwerk s Embedded Code jectory required about 10 seconds for the motors to accelerate Much too slow of a response to mitigate this the P and D gains should be increased and the I gain should remain low The combination 500 0 1000 0 was too high and caused spastic motor behavior This was lowered until they worked properly again at 150 0 600 0 Here the response took about 6 seconds Still too slow For changing the acceleration type cd terk embed trunk share src libqwerk then gedit qemotortraj cxx Then multiply the acceleration by a constant to dilate it s affect It is currently multiplied by 200 to achieve a slope that is near to a step function save The following section picks up in the QDG under the section Building the Qwerk Source In terminal run the commands one at a time under Building the Qwerk Source cd terk embed trunk share make cd terk embed trunk cirrus arm linux 1 0 4 make edb9302 NOTE may take about 8 mins to run cd terk embed trunk cirrus arm linux 1 0 4 edb9302 for i in ramdisk gz zlmage optfs out do sudo cp backup t i var www done Then just type in the password password DOESN T ALWAYS ASK FOR THIS At this point the new firmware has been compiled but is not
205. talk to Qwerks Connecting to the Qwerk Using Minicom Run minicom sudo minicom Welcome to minicom 2 1 OPTIONS History Buffer F key Macros Search History Buffer 118n Compiled on Nov 5 2005 15 45 44 Press CTRL A Z for help on special keys We re now ready to turn on the Qwerk Once we do so it ll spit out some system info and then pause for 1 second before executing its boot script We don t want it to execute the boot script yet so we re going to type CTRL C to abort it 229 Jonathan Missel Changing Qwerk s Embedded Code Go ahead and turn on the Qwerk but type CTRL C immediately afterwards It should look something like this Ethernet eth0 MAC address 00 dc 6c 7d 6b 25 IP 192 168 1 120 255 255 255 0 Gateway 192 168 1 1 Default server 192 168 1 103 DNS server IP 192 168 1 1 RedBoot tm bootstrap and debug environment ROMRAM Non certified release version v2_0 built 13 09 18 Mar 21 2006 Platform Cirrus Logic EDB9302 Board ARM920T Rev A Copyright C 2000 2001 2002 Red Hat Inc RAM 0x00000000 0x02000000 0x00041fa8 0x01fdd000 available FLASH 0x60000000 0x60800000 64 blocks of 0x00020000 bytes each Executing boot script in 1 000 seconds enter C to abort C RedBoot gt You should now see a RedBoot gt prompt Pay attention to the IP Gateway and Default Server IP addresses The IP address is the address of the Qwerk the Gateway is your router and the Default Serve
206. te bias lt 1 sec Resolution lt 025 sec Bandwidth gt 25 Hz Random walk lt 2 25 hr 34 Jonathan Missel Components e Acceleration Range 4 g Bias lt 012 g Resolution lt 00006 g Bandwidth gt 75 Hz Random walk lt 1 m s hr e Operating temperature 40 to 71 C e Electrical Voltage 9 to 30 VDC Current lt 250 mA Power consumption lt 3 W Output format RS 232 e Weight lt 1 4 lbs From the specifications it is clear that this unit is extremely precise and does not take much power to operate It also fits within the voltage limits of Qwerk and outputs the supported RS 232 data format so it is more than adequate for measuring the dynamics of the body Additionally the physical dimensions are close to a cube which helps maintain the symmetry assumptions in modeling For the sphere the actuation method detailed in the previous section has ef fectively decomposed its rotation Encoders that measure the rotation of the omni wheels would be able to provide information on the motion of the sphere with respect 35 Jonathan Missel Components to the body Encoders are inexpensive compact and very common so implementa tion is well documented Quadrature encoders are particularly attractive because of their high precision and they are supported by Qwerk They have four slot patterns in their encoder disks and combine the readings from
207. tely dependent on the application of a single control input as the rider manipulates the throttle to power the back wheel and directly maintain pitch When the bike naturally falls to one side it turns in circles like a freely rolling coin To avoid falling over the lean angle roll must be kept in check This is done through centripetal force by balancing speed with the roll angle while turning To change the direction of the circular turns increasing speed will cause the roll angle to surpass vertical and lean to the other side Yaw is determined by how much of the turn is completed before straightening out Speed of rotation can be increased by taking on a more aggressive roll angle with a shorter turning radius To recap pitch is directly controlled through powering the rear wheel and the coupled motion of roll and yaw 13 Jonathan Missel System Description is governed by centripetal force and how long the stunt is carried out for This is a simplified explanation of the dynamics at work phenomena like gyroscopic motion and momentum are not considered Additionally the coupling between translation and pitch is not stressed when fully detailed a total of four resultant degrees of freedom are being managed Though simplified this is a close example of how Vertigo 2 1 could remain operational with actuator failure As long as any two are intact it can be controlled analogously to a motorcycle on one powered wheel Even though it is p
208. tention however it confined research to the two wheeled differential drive configurations despite its many drawbacks Their most significant shortcomings are coupling between yaw and directional control and their tendency to become dynamically unstable when perturbed in the coronal plane Not until 2006 were the full benefits of dynamic stability exploited in ground robotics with Dr Ralph Hollis Ballbot created at Carnegie Mellon s Robotic Institute 3 Ballbot is a true biaxial inverted pendulum that balances on a ball Its chief advantage is its omni directionality it does not have to turn in order to change directions It is also dynamically stable in both directions so the same control methodology could be applied in both the sagittal and coronal plane Ballbot was designed as a robot for researching motion in interactive human environments so it took on the rough dimensions of the human frame Its smooth motion has potential to assuage the resistance to assimilation of human and robotic cohabitation at home and in the work place Though it is very slender it is human height and difficult to transport so testing requires a large space and is risky This serves as evidence that it was not intended to be an experimental platform Despite all its grandeur Ballbot s configuration sacrifices direct control in yaw a critical aspect of many trajectories In addition functionality is somewhat limited because it cannot be driven on the
209. termine its progress MP3 PCM and WAV audio clips of 15 seconds or less could be sent with the second method These audio files had to be saved in the associated directory and were called by name and extension in the function The third and final audio command sent text and synthesized it into speech enabling the robot to say any desired phrase This function was quite entertaining but also proved valuable when coupled with the battery voltage command This partnership formed a safety net by sending an audible alert to warn the user to replace the battery when its terminal voltage dropped low enough to impede Vertigo s response This prevented crashing due to low power There were initial concerns with using the audio commands that the code would pause whilst the sound was being played however there was no trouble in practice as the code continued to proceed while projecting sound Additionally overtaxing of these commands simply built up a queue for the sound bites to be played in order of assignment The DC motor commands were the most sought after functions in the library Their application required some manipulation because they were written specifically to drive a particular two wheeled robot however a logic convention was established to independently control all four motors The functions accepted scalar values between 50 000 and 50 000 to represent the angular velocity of the motors Since this board was built to work with a myriad of m
210. th proportional yaw control norm error 1 8976 162 Jonathan Missel Ground Based Implementation LQR ground track with inertial control and no yaw control 04 03 02 y position m 0 1 0 21 04 1 1 L 1 1 L 1 1 1 1 1 05 04 03 02 01 0 01 02 03 04 05 x position m Figure 7 4 Circle trajectory tracking without yaw control norm error 0 8614 LQR ground track with P yaw control 0 5 0 4 0 3 0 2 0 1 y position m o 1 1 L 1 1 L i 1 1 0 5 0 4 0 3 02 01 0 01 02 03 04 05 x position m Figure 7 5 Circle trajectory tracking with proportional yaw control norm error 0 3866 163 Jonathan Missel Ground Based Implementation track started at the bottom of the circle and progressed counterclockwise Shortly after starting both experiments encountered an uneven section of floor which caused slip in the y direction In Figure 7 4 where no yaw control was applied this slip and recovery undesirably induced a slight yaw rotation Because control was being applied from the inertial frame this rotation skewed the control input for the re mainder of the trajectory track In Figure 7 5 the same slip occurred but the fixed yaw control accounted for any undesired yaw that may have been induced and the track was quickly resumed The yaw controlled response had a norm error about 2 5 times smaller than the uncontrolled response These results capture just one exam ple the uncontrolled re
211. th the sphere and ground based configurations of Vertigo so only ground based control is considered here because it is easier to visualize Figure A 4 Ground based trace of translation with rotation Figure A 4 shows the trace as ground based Vertigo travels along the straight 190 Jonathan Missel Control Superposition green trajectory and executes a 5 yaw rotation with constant angular velocity The red X represents a simplified top view of Vertigo at its initial position As its center of mass progresses along the straight green line and yaws at a constant rate periodic snapshots are taken and traces of the four corners of the X the motor locations are drawn These traces represent the paths that each individual actuator takes Since the actuators only have to control motion perpendicular to their axle trajectory pro jections can be drawn onto the plane of the wheel for each respective actuator trace In this way each actuator need only control the projected motion For continuous control this is equivalent to superposing the control More specifically the control for translation transformed into the body frame and yaw rotation can be determined separately then added together to compose a complete control profile for the ma neuver To transform the translation control into the body frame information about the yaw angle is needed In practice this is a problem because continuous knowledge of yaw and continuously transform
212. the body was nearly vertical as it slowly crawled to the final states By increasing the weighting on the angle states and reducing it on the rates the performance can be improved by allowing the angles to change more quickly to their final values Figures 5 16 and 5 17 show the response with these weightings and it is clear that the final states were reached more abruptly However this improvement came at the expense of a more chattery response and greater demands from the control input This is a tradeoff of performance to accuracy and power and is common in con 131 Jonathan Missel Sphere Based Control 4 6 radians radians 1 o 2 5 10 15 20 8 5 10 15 20 Time Time o 6 dot rad s o o dot rad s x 1 oN a 1 N 10 15 20 5 10 15 20 Time Time Qo Figure 5 14 State response with tracking LQR controller heavily weighted rates LOR Track Response 2pi rotation o for o o o P o N Magnitude wrt Vertical radian units o 0 2 0 41 0 6 ang 0 8 rate control 1 T 1 1 fi J 0 2 4 6 8 10 12 14 16 18 20 Figure 5 15 Response with tracking LQR controller wrt vertical heavily weighted rates 132 Jonathan Missel Sphere Based Control 10 5 x Al y Cc S S ha TF 3 3 o 0 Z 5
213. the concept of omni directional motion In addition it also gave many opportunities for experimental variation when implemented such as sinusoidal waves and ellipses During the LQR controller development discrete time equations of motion with T 0 04 were used These were derived earlier in the Mathematical Modeling and Analysis chapter and follow the planar model of the ground based configuration T 1 0 04 E 0 0240 x 0 1 x 1 1976 olfj 00 The discrete time LQR tracking problem is designed for full state feedback how ever as seen from the equations this measurement model does not have such luxuries In order to obtain full state measurement numerical derivatives were taken to ap proximate the rate states for implementation By definition LQR works by forcing the states to zero In the tracking problem 112 Jonathan Missel Ground Based Control N 4 N 4 N 4 Figure 5 1 Ground based free body diagram it is desired that the states be forced to a reference trajectory rather than zero To accomplish this a new system is defined where the states are the errors between the current and reference positions Therefore when the LQR controller acts to minimize the new states it is minimizing the error and forcing the system to follow the desired trajectory The LQR controller was designed by taking advantage of MATLAB s built in Iqrd function This solves for the optimal controller gains for a prescribed system a
214. the task of opening a door Without prior knowledge of the exact size and location of the door two wheeled self balancing robots cannot accomplish this While they do have a niche two wheeled self balancing robots can be viewed as an incomplete idea because they maintain the disadvantages of their constituents they require active control to maintain stability yet are still prone to tipping over Bipedal walkers like Honda s ASIMO Figure 1 1 e have managed to exploit 4 Jonathan Missel Motivation Conventional four wheeled robots Conventional three wheeled robots N On level ground d Two wheeled self balancing robots g Y On level ground h Note Self balancing robot maintains good traction by shifting its CG above its wheels at all time Figure 1 2 Statically and dynamically stable vehicles encountering terrain dynamic stability in both directions and are omni directional to boot These ma chines have taken advantage of biomechanical design by letting nature come up with the solution Modeling robots after humans promises extremely agile performance ASIMO can run carry objects and even climb stairs With continued development these platforms will even be able to outperform humans However all this grandeur comes at a tremendous cost both fiscal and in robustness Each of the 48 ASIMO units cost nearly 1 000 000 just to manufacture This does not include the fund Jonathan Missel Motivati
215. tigo 1 0 s motion with the addi tion of direct yaw control This was successfully accomplished by utilizing actuation method D Just as importantly this iteration was used as an opportunity to further improve upon the overall design by taking simple and well reasoned measures that had substantial impact on performance usability and robustness Though the new legs were made in two parts and the motor mounts were no longer merged with the legs 60 Jonathan Missel Design Iteration Figure 2 9 Vertigo 2 1 motor encoder wheel exploded subassembly CAD both generations had the exact same number of unique modified and manufactured parts This was mostly because all four limbs of the new design were identical where the original had two with motors and two with passive rollers Many safety features were added or improved upon making Vertigo 2 1 an exceptionally robust piece of hardware The new legs offer enhanced protection for every component including electrical connections Wiring was made easier with larger cutouts in the legs more clearance and better positioning of the encoder leads The new legs were designed to not only be durable enough to survive impacts but were sized to protect the other components 61 Jonathan Missel Future Improvements Small features like the speaker rounded cap screws and filleted corners made working with the robot much easier By allowing the motors to be adjusted even faster flexibili
216. tion Vertigo s mobility makes it capable of modeling these more expensive and complex systems which would help expedite their development in a cost effective manner Vertigo also has the potential to make original contributions in the fields of rover locomotion and human and machine cognition among others This new form of mobility promises to affordably stretch the bounds of what ground vehicles are capable of 1 2 Background Recognizing the heritage from which Vertigo evolved is an important part of fully un derstanding it and appreciating the legacy that it carries on The inherent instability of inverted pendulums has proved critical to their contribution toward technological advancement They have been used in such diverse applications as seismograph in strumentation 5 biological neuron modeling 6 and testing control methodology Although this configuration has been around for a long time the development of Jonathan Missel Background mobile inverted pendulums is in relative infancy The geometry of these systems has a strong likeness to that of the human frame for this reason terminology from anatomy is frequently used Figure 1 3 shows how the anatomical planes of the human body are defined This chapter briefly outlines the background history of Vertigo s most unique feature dynamic stability More specifically it covers the progression of mobile inverted pendulums and their directionality Brukner and Ka
217. to highlight Serial port setup and type ENTER to choose it You should now see the following menu A Serial Device dev tty8 B Lockfile Location var lock C Callin Program D Callout Program E Bps Par Bits 38400 8N 1 F Hardware Flow Control Yes G Software Flow Control No Change which setting First type A to set the serial device The device name to use depends on your computer For example one of my machines only has one serial port so I set the device to dev ttyS0 Another a MacBook running Ubuntu under VMWare Fusion doesn t have any serial ports so I use a GWC UC320 USB to serial adapter which Ubuntu recognizes as dev ttyUSB0 The name of your serial port device may be 226 Jonathan Missel Changing Qwerk s Embedded Code different When you re done editing the device type ENTER to save it A Serial Device dev ttyS0 B Lockfile Location var lock C Callin Program D Callout Program E Bps Par Bits 38400 8N1 F Hardware Flow Control Yes G Software Flow Control No Change which setting Now type E to set the Bps Par Bits Doing so will open a new menu that should look like this Comm Parameters Current 38400 8N1 Speed Parity Data A 300 L None 5 B 1200 M Even T 6 C 2400 N Odd U 7 D 4800 O Mark V 8 E 9600 P Space F 19200 Stopbits
218. tor mounts into one part is that the wheels had to be offset from the center of the sphere It was understood that this would influence the motion slightly in practice but theoretically it was not very significant The dynamics of this system ideally prefer control along the axis of symmetry and through the center of gravity To accomplish this the structure would need to be off set to accommodate the wheels This would further weaken the assumption of symmetrical inertias Since the omni directional wheels always act through the center of the ball anyway this is physically similar to having the legs in the plane of symmetry and offsetting the wheels Therefore straight legs were just as affective and were made to greatly simplifying their design and manufacturing The biggest repercussion of 42 Jonathan Missel Structure straight legs is that the body will in affect be rotated with respect to the reference frame that control is applied in slightly coupling the dynamics The lower portion of the leg was designed to permit some flex to help all four wheels maintain contact with the sphere functioning similarly to the suspension of a car Additionally permitting flex absorbs some of the shock from impact which helps to protect the robot as a whole As an adaptable vehicle determining exact specifications for this permitted flexing was impossible because it required knowledge of the mass and inertia which are variable Therefore from expe
219. track a prescribed trajectory Simulations with this controller were then set up to explore the omni directionality of the system and create a standard benchmark trajectory for accuracy assessment on the implemented control Several other controllers such as PID were designed for the ground based mode but most of their work was completed as part of their implementation and will therefore be covered in that chapter For the standard candle LQR simulation a consistent measure of accuracy was established as the norm error and a goal for required precision set norm error lt 1 norm error norm norm errx norm erry This criterion for error calculation accounted for the accumulated error in both x and y position and in time It did so by calculating the difference in position from 111 Jonathan Missel Ground Based Control the trajectory at each time step normalizing the vectors of x error and y error then normalizing the vector of these values and taking the absolute value Because this value was dependent on both space and time a standardized trajectory had to be used for evaluating performance to maintain uniform and comparable results This trajectory was chosen to be a circular path with a radius of 0 5 meters centered at the origin that traversed the circumference once at an angular velocity of 0 3 radians second A circular trajectory was chosen to study the response of both sets of actuators and validate
220. trolling the robot and GUI RobotClient java Contains all methods for controlling and accessing the robot as well as the GUI RobotClientGUI java Specifies the properties of the GUI SimpleRobotClient java Wrapper class for RobotClient java can be used in terchangeably with RobotClient java CreateClient java Class containing methods for controlling the iRobot Create or iRobot Roomba See the Create recipe at www terk ri cmu edu for more information RSSReaders Folder containing classes for reading RSS feeds from the internet RSSReader java Generic class for reading feeds WeatherReader java Class for reading weather data for any US city courtesy of www wunderground com feeds TTS Folder containing class for generatic speech from text TTS java Generic class for generating text to speech Method 3 Through MATLAB Instantiating Objects from Java Classes in Matlab 208 Jonathan Missel Connection to Vertigo NOTE Steps 2 and on may not be needed if you are using a dynamic Java path see Versions used Windows Vista Business Java jdk1 6 0_06 Matlab 7 4 0 R2007a 1 Matlab comes with a version of Java that it uses instead of the one installed on your computer You can check the version with the version java command in command window You should update Matlab s java version to the one you have been compiling your classes with a This can be done by right clicking on My Compute
221. ty and theoretical principles These would serve as a foundation for future work in balancing and navigating on the sphere In many ways early implementation mir rored the efforts of establishing the communication architecture Accordingly the first controllers were implemented through the Java based Terk programs and inter faced through the provided GUIs Complexity grew as feed forward programs were coded in JCreator These programs performed simple tasks and were written to help troubleshoot hardware This practice developed an understanding of the library of commands that accessed Qwerk s functions and would later be used to send input signals to Vertigo This Java interface was also used to calibrate measurements and 153 Jonathan Missel Implementation Challenges develop conversion factors for signals between components When it was discovered that the current version of firmware on Qwerk did not actually support the onboard sensors it was necessary to establish the external sensing communication loop that tied in Vicon Simulink MATLAB Java and Qwerk The variable stored for connecting to Qwerk did not allow controllers to be implemented in Simulink therefore MATLAB had to serve as the central hub for coding With most of the kinks ironed out from this network basic control implementation began Many challenges threatened to jeopardize implementation several were anticipated but the majority of them were unforeseen These wi
222. ty in the inertial and center of mass properties was achieved At least one spare was made of every manufactured part to eliminate downtime if something broke and two sets of batteries were purchased so that experiments would never be halted due to low power Aesthetics also played a role in the design with knowledge that this may one day be marketed as a commercial platform it will need every edge it can get for sales Ver tigo s open design balances simplicity and detail to show off its functionality without appearing austere or purpose built even though it is The materials geometry and components were all chosen to form a durable system that matched its mathematical model well In summary the new design managed to provide solutions to Vertigo 1 0 s flaws maintain or improve upon all of its advantages and avoid accruing new drawbacks 2 7 Future Improvements As work progressed with building and implementing Vertigo 2 1 even more ideas were formed that would alter or improve the hardware In this section these ideas are covered and the consequences of their application discussed Some of these concepts were devised prior to manufacturing the current design but were omitted as their tradeoffs did not make them sensible for inclusion at this early stage As discussed previously in the selection of omni wheels this system required that only one row of rollers be used For all existing wheels this meant that gaps be tween the ro
223. vase downloads index jsp Follow their instructions for installing it on your machine Compiling and Running the MyFirstRobot Program in JCreator RECOMMENDED METHOD JCreator is a freeware Java IDE A project file MyFirstRobot jcp is available in the MyFirstRobot folder with all of the correct configuration settings to quickly begin compiling and running the MyFirstRobot program Install JCreator LE from http www jcreator com download htm Open the file MyFirstRobot jcp The JCreator IDE should start In the File View of the IDE open MyFirstRobot java for editing by double clicking on it Compile the project by either hitting F7 or going to Build gt Compile Project in the top menu Run the project by going to Build gt Execute File in the top menu Note that you must execute the file NOT the project Running the program will open a graphical interface To connect to a robot click on the Connect button A connection dialog will pop up and you will have to select whether you are connecting to the robot via the CMU based relay relay mode 204 Jonathan Missel Connection to Vertigo or if you will connect over a local network by entering the robot s IP address direct connect For more information about these two modes of operating your qwerk visit http www terk ri cmu edu projects qwerk overview php Make your selection and click Next If you entered direct conn
224. vation EA a n a de ee Be ee A ee a 1 12 Backer ound e bi peaked cl talado oe A gaY 7 1 3 System DescripHon ys soa tara a ea o a BAGS 11 2 Design 17 2 1 IO ICO ss e A a a A 17 22 Conceptio mee Kye ts o ad a A e 18 22 Objective A a See a a eee a A 18 2 2 2 Deyelopment e Dira sie a a ke aes le A ay Se AA 19 2 2 3 Design Challenges ooa aaa ie e a 21 2 3 Act ation Method o i s se ASA e A ERA E 22 2A COMPONENTS e sa ci at ae li a AR oe 28 2 4 1 Onboard Computo a toe dpe die dl BO ed ee oa 24 27 Actuation saene e ia NA ZAI CCSS a A e POO 2AA POWEL te an tere kt hah eal oh Shot Nees oh ei a Bg A etane hele add a e hs GA he WS RE HS HS Ee SS ES ai 260 esis Iteration gt anida ath wed Os oe oO dra A AA 2 7 Future Improvements see ee hk ed A hd Mathematical Modeling and Analysis onl TIntroducti n La A A a ae a oe Be s 3 2 Mathematical Model ors tea ud este A ee S220 Introductions LE ra a a Se oe Bh ey 3 2 2 Model Derivation 9 prepa de a Ne oe 92 9 Linearization Le ed a ek ee ad 3 2 4 Ground Based Derivation 2er Aa 3 3 Analysis 5 2 Ge 4G hha we Bh AO RDG e e LS Be we Le aay N state se aora Cay e eae es Ae aoe Se ae le Pee ia 3 3 2 State Observability and Controllability Analysis 3 3 3 Design Parameter Observability and Controllability Analysis SA Conclusions ras 33 AS A a A eee Communication Architecture Ad AMO ANCONS e e o A AA A ADS 4 2 Onboard Sensing Architecture o
225. verse condition values spans only about 0004 and no maxima or minima are present Accordingly the observability is not as critical for design because it is not affected greatly Just as before the counteracting principles result in a contoured surface The mesh of controllability matrix determinants in Figure 3 14 confirms the results from the inverse condition number As the length increases the determinant approaches zero monotonically for this range The contour of this surface ensues from the same counteracting phenomena mention earlier 90 Jonathan Missel Analysis 8 Determinant of Controllability Matrix 0 15 Vertigo Mass kg 15 01 mn Figure 3 14 Determinant of controllability matrix while varying Vertigo mass and Inverse of Controllability Condition 0 16 0 14 0 12 D 1 D 08 0 06 0 04 0 02 2 5 Vertigo Mass kg Figure 3 15 Inverse condition number of controllability matrix while varying Vertigo mass and with fixed rotational inertias 91 Jonathan Missel Analysis As proof of concept this design analysis was also carried out with fixed rotational inertia to isolate the principles at work Figure 3 15 shows the inverse controllability condition mesh for this By comparing these results with the original findings it is clear that these design parameters most significantly impact controllability through rotational inertia when mass is increased Without the increasing inertia a
226. with its mobility However there is one remaining feature that lingers unaddressed yaw control Ballbot is omni directional so it can move and change direction without having to turn but it also has no choice it is not able to turn This thesis introduces a robot named Vertigo a proposed solution to the final obstacle in achieving a dynamically stable system with all three planar degrees of freedom As it was in the contributory stages this evolution in mobility was an exercise in innovative mechanical design and was rooted in punctuations of dynamics Jonathan Missel Background and control theory In many ways Vertigo is similar to Ballbot with its primary departure being the method of actuating the spherical base It does so in a way that fully liberates all three forms of planar motion by directly controlling yaw in addition to Ballbots two directional degrees of freedom It was also designed to maximize adaptability by accepting different sized spheres reconfiguring to change center of mass and inertial characteristics and can even operate without the sphere as a statically stable omni directional ground vehicle This allows Vertigo to carry out maneuvers that no other vehicle can giving it the flexibility to model and test control theory for a wide range of other systems Rocket guidance under actuated spacecraft and in flight refueling are among the challenging control problems that require extensive research and experimenta
227. www directory You can either do so in the usual way or by doing the following which creates backups of any images which currently exist this doesn t matter for this initial copy of course but you may find it nice to use in the future cd terk embed trunk cirrus arm linux 1 0 4 edb9302 for i in ramdisk gz zImage optfs out do sudo cp backup t i var www done Network Setup Configure your network as described in the Network Setup section of the Upgrading Firmware document on the TeRK web site Since your computer will be serving the firmware images to the Qwerk you must have your network configured as shown in figures 4 or 5 of the Network Setup document Installing Firmware Images To install Linux on the Qwerk we ll connect to it via the serial cable using Minicom load in three images and then create a boot script 223 Jonathan Missel Changing Qwerk s Embedded Code Installing and Configuring Minicom We first need to install minicom sudo apt get install minicom Reading package lists Done Building dependency tree Done Recommended packages lrzsz The following NEW packages will be installed minicom 0 upgraded 1 newly installed 0 to remove and 0 not upgraded Need to get 155kB of archives After unpacking 913kB of additional disk space will be used Get 1 http us archive ubuntu com dapper main minicom 2 1 10 155kB Fetched 155kB in 4s 34 0kB s Selecting previously deselecte
228. y bundled the four encoder wires but tied in the DC motor leads as well The ribbon terminated with a port connection that allowed an orga nized harness to be made that extended the leads from Qwerk so the ribbon could simply be plugged in The location of the encoders worked perfectly with this assem bly because it nestled them in the center with Qwerk where they were protected and near to their electrical connections It should be noted here that between the 1 0 and 2 1 designs it was discovered that the current version of Qwerk s firmware did not support the IMU or quadrature encoders However the circuitry and terminals were all present and the manufacturer was in the process of developing new firmware that promised to incorporate the seamless use of these devices Therefore design proceeded with these components knowing that in the future all forms of onboard sensing would be made functional For the time being the responsibility of sensing would be placed on Vicon and external motion capture system that only required small retro reflective markers be placed on Vertigo Detailed descriptions of the sensing methods can be found in Chapter 4 The new actuation method also required single row omni wheels allowing the originals to bemodified for use At their new orientation the unmodified wheels were still affective at controlling the sphere but did not permit ground based control This was because the rollers were set in a cumbersome thic
229. y on the ground In this configuration the system acts as a more traditional statically stable omni directional ground vehicle rather than an inverted pendulum as with sphere based mode In both ground and sphere based modes the four spherically orthogonal motors drive omni directional wheels This actuation method along with a symmetrical design provides Vertigo with one of its most convenient features decoupled control in x y and yaw For planar translation an opposite pair of actuators controls the motion while the other perpendicular pair slips freely in the direction of motion due to the omni wheels Accordingly the two pairs of actuators are managed independently from one another however yaw control 110 Jonathan Missel Ground Based Control mandates that all four motors collectively work to rotate the robot The equations of motion reflect this decoupled behavior and it is the objective of this chapter to develop controllers that govern the response of these equations It covers the derivation and simulation of controllers for both ground and sphere based configurations These controllers function as verification for the physical concept and build a foundation for the early stages of implementation by managing the basic aspects of sphere based motion balance and translation 5 2 Ground Based Control For the ground based control of Vertigo the primary focus was on developing Linear Quadratic Regulator LQR control to
230. y to balance eliminated the need for differential drives to require a third passive caster that functioned solely as a static stabilizer Later they introduced the first one wheeled balancing design It rolled on the major axis of an ellipsoid to balance in the sagittal plane and hinged at the hips to generate a control torque to balance in the coronal plane 10 By balancing on an ellipsoid the advantages of having a differential drive were lost and rather than being able to turn in place the robot was limited to wide arcs This was also the first attempt at a mobile biaxial inverted pendulum however only one axis was controlled by the base while the other was analogous to the reaction wheel control In 1999 inventor Dean Kamen started a company devoted to making clean trans portation that focused on utilizing dynamic stability The mobile inverted pendulum experienced its greatest boost in popularity in late 2001 when this burgeoning com pany unveiled the Segway Personal Transporter to the public for the first time 11 It had a two wheeled differential drive that utilized the same principles as its Japanese Jonathan Missel Background predecessors but its sensing and control systems differed The most noteworthy as pect of the Segway is that it was designed to be a vehicle for humans a very dynamic and unpredictable payload The attractiveness of zero turning radius speed and the novelty of this design drew a lot of at
231. yet installed on 216 Jonathan Missel Changing Qwerk s Embedded Code Qwerk Installing Firmware All programs and windows from above should still be open Cable in to router Both Vertigo Qwerk and your computer MUST be cabled in Connect to serial port You must connect your computer to Qwerk through the serial port UART1 on Qwerk Plug the USB end of the blue USB Serial Adapter into your computer Connect the serial end to the gray serial to RJ11 phone jack adapter then connect that to the UART1 jack on Qwerk See figure below for visual reference the red line shows the lines of connection under consideration WA Figure D 1 Wiring for access to Qwerk 217 Jonathan Missel Changing Qwerk s Embedded Code Type sudo minicom in the Terminal window This should bring up a Welcome to minicom 2 1 prompt Turn off Qwerk Click on Terminal to make it the active window Turn on Qwerk AND hit ctrl C when you see text appear in Terminal There is a one second window you have to do this in Hitting it a few times to make sure you got it won t hurt anything Press enter if instructed If this part has been executed properly it would end up giving you the RedBoot gt command prompt in terminal This is where you will enter the following commands In terminal run the commands in the QDG under Loading the Linux Images load r v b 0x1000000 m http optfs out load r v b 0x800000 m http ramdis
232. ysis Therefore all of the state varying characteristics were captured in the angle state analysis As a result of the controllability and observability being invariant to rate states the four remaining state combination analyses yielded no new results and were not included for this reason 3 3 3 Design Parameter Observability and Controllability Anal ysis The physical parameters of a system can greatly affect the mathematical modeling Vertigo is a testbed for dynamics and controls experiments and for that reason was designed to be reconfigurable permitting changes to the center of mass location and magnitude In addition new design iterations are always underway so it is important to determine how these physical changes in the system affect its performance The two main components of Vertigo s assembly are the sphere and the body To explore the sensitivity to these components a controllability and observability analysis was done for both constituents while varying their dimensional and mass parameters This analysis also looks into the determinant and inverse condition number for fur ther evaluation Performing this analysis sheds light on how the equations of motion are affected by uncertainty in these parameters which will help in the estimation and filtering to come Vertigo Design Analysis First controllability and observability are checked by plotting the rank of these two matrices while the design parameters body
233. ystem is then able to determine the position and attitude of this object All time derivative states must be calculated from these measurements externally Track the motion of the sphere was not possible with this method because the markers interfered with the actuators and ground If the entire surface of the sphere were coated in retro reflective material Vicon would recognize it as a single marker and could track its location However it would have no inclination about rotation which is the desired state Figure 6 1 illustrates the planar Vicon measurement model is the angle of the Vicon object with respect to vertical and d is its distance from the origin These readings can be related to the states of the system by defining h p PEL 6 5 dy 2110 lysin 0 d where r is the radius of the sphere and l is the distance from the center of the sphere to the center of the defined object Vicon is a very accurate system despite this there is a great deal of uncertainty in these measurements and it all stems from the way it defines objects In Figure 6 1 the object is shown to be centered in the upper half of the IMU this is because the IMU and its mounting plate are the most convenient and protected places to attach markers The difficulty is in knowing exactly how this object is defined Vicon does their best to simplify the procedure by creating objects with Euler angles that initially align with the inertial reference fr

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