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Model-based control of a one-dimensional pendulum actuated with

1. 50 25 50 Exergy Valve action gt Exergy Valve action gt 20 4408 220 140 E Lei gt a 3 s E 9 15 1308 2 15 130 o o i kel 10 20 gt 8 10 420 gt a g 3 gt 3 gt 5 409 5 110 3 E E z E 0 0 0 0 7 9 4 11 8 14 2 16 6 19 13 15 8 18 6 214 24 2 27 p Nm p Nm Frequency 2 Hz Frequency 2 25 Hz 25 T 50 25 50 Exergy Valve action gt Exergy Valve action gt 20 4408 220 140 E Pa Pa Ba s gp g 15 1308 2 15 130 5 o E i 2 8 8 10 120 gt 10 420 gt 2 S 5 z g 2 8 id S 5 110 3 S 5 110 3 z a 5 E 0 l 0 0 i 0 25 28 2 31 4 34 6 37 8 41 37 39 4 41 8 44 2 46 6 49 p Nm p Nm Figure 4 8 Energy consumption and valve action in function of p for different trajectories In the previous discussion only sine wave trajectories were imposed For these trajectories the passive behaviour fits the imposed trajectory if p is set right In this case a constant p value is justified We have to remember that the goal of this thesis is to investigate the possibility to exploit natural dynamics for implementing it later into the biped Lucy In general the trajectories for such a biped are not just sine waves with only one frequency component For such a situation a constant p will not be suitable as shown in figure 4 9 In this figure a sine wave with a linear inc
2. sess 62 42 31 Experimenten eH TE 62 4 2 2 2 Results and discussions c ve reed dy cm e eyes t edendi eee 63 Simulation 1 the importance of setting a suitable stiffness 63 Simulation 2 evaluation of the mathematical formulation 67 Simulation 3 changing the stiffness online sss 69 437 he physical Bendert nente netstat 75 4 3 1 Evaluation of the joint trajectory tracking controller 75 4 3 2 Exploitation of the natural dynamics sss 80 4 3 2 1 General considerations eneen eenen eneen 80 4 3 2 2 Results and discussion sse 81 4x CONC IASI OM qi trinh dap ente toto asia kerde bo Ee pe te qu enar 86 General conclusions and future perspectives 88 Appendices A Dynamic model of the pendulum 9 B Safety OAL senen i a tede cuve CR AREA 94 BibiOSraphWasner dekens denna 95 NEN o0 c 95 Die Brostamint p snueteaame deseen eerte Ata PRO od tta a Ed re utes 97 Di Data aquisition Gard 4 tornen ate tede dist fo D uade 97 List of Figures Figure 1 1 Figure 1 2 Figure 1 3 Figure 1 4 Figure 2 1 Figure 2 2 Figure 2 3 Figure 2 4 Figure 2 5 Figure 2 6 Figure 2 7 Figure 2 8 Figure 2 9 Figure 2 10 Figure 2 11 Figure 2 12 Figure 2 13 Figure 2 14 Figure 2 15 Figure 2 16 Figure 2 17 Industrial ro
3. degrees degrees Figure 2 9 Static and closed muscle torque characteristics at different pressure levels with the design parameters set to the one used in the experimental set up 2 2 3 Constructing the experimental set up 2 2 3 1 The frame AS was argued previously a pull rod and leverage mechanism was selected to position two muscles in an antagonistic setup The basic frame in which this system is incorporated is depicted in figure 2 10 The modular unit is made of two slats at the side which are connected parallel to each other by two linking bars A joint rotary part provided with roller bearings is foreseen for the Chapter 2 Design 22 connection with an other modular unit The fixed base for the pull rods mechanism includes two rotary axes at which the muscles are attached The small rotations of these axes are guided by sliding bearings positioned in the frame All the parts of the basic frame are made of a high grade aluminium alloy AISiMgl apart from the bolts and nuts required to assemble the frame Sliding bearings 0 0 0 0 1 0 a 6 ule e Fixed base of Roller l i muscle pull rod beani Joint rotation mechanism earing points Figure 2 10 Exploded and assembled view of the basic frame Two more linking bars are connected to the joint rotary part of the basic frame This is the swinging part of the pendulum The mass is attached to the screw thread connecting th
4. but now exhaust valves will be opened If Peno is beyond level a 4 exhaust valves will be opened instead of 2 3 3 Mathematical formulation for compliance adaptation In section 2 2 2 3 a formulation of the compliance was given for closed muscles As shown a weighted sum of both pressures in the antagonistic muscle set up determines the joint compliance while pressure differences determine the generated torque and consequently also the joint position This means that compliance can be set while controlling the position This is interesting because by setting the appropriate joint stiffness control activity can be reduced while tracking a desired trajectory In the previous paragraph the parameter p used to influence the sum of pressures and consequently the joint stiffness was introduced The question 1s now what should be the value of p In other words what should be the joint stiffness so that the natural motion best fits the desired trajectories In this section a mathematical formulation developed by Verrelst 14 is given for estimating an appropriate value of ps The starting point for the estimation procedure is to fit the natural pressure slopes with the required ones The desired pressures depend on the desired trajectory The delta p unit in combination with the computed torque module determinates the required pressures as explained in the previous section Pressure changes with closed muscles are influen
5. ae 10 Figure 4 15 Angular velocity computed torque and absolute pressure for first tracking experiment In order to investigate if exploitation of the natural dynamics is possible a good working trajectory tracking controller has to be implemented In a practical setup some additional problems arise e g noise on the pressure sensor signal or small errors in the joint angle information An even more important problem is the measuring of the angular velocity The angular velocity is required by the computed torque module and in order to have a stable controller this parameter Chapter 4 Results and discussion 75 has to be well known and as much as possible free of noise As we cannot directly measure the angular velocity it should be calculated numerically starting from the joint angle acquired with the incremental encoder Therefore a first order numerical method was used Figure 4 15 shows the angular velocity the computed torque performance and the absolute pressure in muscle 1 for the first tracking experiment A sine wave with a frequency of 1 75 Hz was imposed and a constant p 14 Nm was used As we can see a lot of noise appears on the angular velocity signal This will have an impact on the tracking controller as the servo portion will have to cope with this noise We clearly see that the D action is trying to eliminate this noise The delta p unit transforms the noisy computed torque to pressure levels whic
6. fixed during the joint design process 2 2 2 3 Adaptable passive behaviour of a revolute joint A PPAM has two sources of compliance being gas compressibility and the dropping force to contraction characteristic 2 The latter effect is typical for pneumatic artificial muscles while the first is similar to standard pneumatic cylinders Joint stiffness the inverse of compliance for the considered revolute joint can be obtained by the angular derivative of the torque characteristic Chapter 2 Design 16 y 2T dT dT do do dp dt dp dt tede ee ee do do ao zde 2 12 dp l l The terms fg represent the share in stiffness of changing pressure with contraction which is determined by the action of the valves controlling the joint and by the thermodynamic processes taking place If the valves are closed and if we assume polytropic compression expansion the pressure changes inside a muscle are a function of volume changes 9 PV SE 2 13 with P Pim Pi 2 14 leading to dp V dV n ine F io i 2 15 With P V the absolute pressure and volume of musclei P the absolute initial pressure V the initial volume when the valves of muscle i were closed p and p the gauge pressure and initial gauge pressure respectively P the atm atmospheric pressure and n a polytropic exponent The polytropic exponent is introduced to describe deviations from the isentropic expansion compression An is
7. a passive spring element with an adaptable stiffness is created The stiffness is controlled by a weighted sum of both initial gauge pressures Since stiffness depends on a sum of gauge pressures while position is determined by differences in gauge pressure the angular position can be controlled while setting stiffness How to use this interesting property and how to incorporate it into the control of the pendulum in order to reduce energy consumption and valve control will be discussed in detail in the next chapter Chapter 2 Design 18 2 2 2 4 Design of the one dimensional pendulum All the equations described above in combination with the muscle force function are used to design the characteristics A description on how the design parameters of the former paragraph are chosen is now given As mentioned before the pendulum consists of one modular part of the biped Lucy Therefore some of the parameters are already fixed namely b 40mm lni 110mm l 2 350mm The central position of the pendulum was set to 9 0 For designing the remaining parameters some requirements should be met e We want an angle range between 20 and 20 or larger e Within the angle range the contraction has to be limited between 5 and 3596 A smaller contraction leads to stresses that are too high for the muscle fibres At higher contraction the forces drop too low for practical use e Enough torque should be provided by the anta
8. inflate and deflate the enclosed volume An epoxy resin fixes the membrane and the Kevlar fibres to the end fittings Figure 2 1 The Pleated Pneumatic Artificial Muscle Daerden 2 extensively discusses several characteristics concerning the PPAM for inelastic as well as elastic membranes In this work mainly two characteristics are important generated traction and enclosed volume for each contraction The first is used for joint torque dimensioning and control purposes while the latter is used to predict joint compliance with closed muscles Furthermore the maximum Chapter 2 Design diameter when the muscle is fully bulged should be taken into account when designing the different joints of the robot in order to provide enough space for the muscle to bulge When inflated with pressurised air the muscle generates a unidirectional pulling force along the longitudinal axis Generally if the number of fibres is large enough the generated muscle force depends on the applied gauge pressure p the contraction 2 and the two parameters initial muscle length and slenderness 14 The traction characteristic is given by l F plo f 6 72 Q 1 with f 5 2 the dimensionless force function as defined by Daerden 2 12000 1 bar 2 bar 3 bar 10000 8000 6000 Force N 4000 2000 0 5 10 15 20 25 30 35 40 45 Contraction Figure 2 2 Theoretical forces at pres
9. se est de combiner l exploitation du mouvement naturel de la pendule avec un contr le en position de type suivi de trajectoire Une trajectoire sp cifi e est suivie par un contr le en position pendant que la souplesse de la jointure est adapt e de telle facon que le r gime naturel correspond le plus possible la trajectoire de r f rence La pendule est actuellement assembl e et les composant test s Cette th se donne une d scription de la construction de la pendule L influence de la souplesse de la jointure sur l effort de contr le et la consommation d nergie est illustr e par des moyens de simulations et d exp rimentations sur la pendule r elle Une technique qui permet de d terminer la souplesse optimale tout au long d une trajectoire specifi e est tudi e La prochaine tape dans le domaine est d estimer correctement les param tres du syst me et d tendre ce syt me vers une pendule tridimensionelle De cette facon l incorporation de la technique pourra se faire sur le robot bip de Lucy Acknowledgements Na een jaar van pure fun in Barcelona heeft het mij veel moeite gekost om me weer in het leven van een ingenieursstudent te storten Tijdens de periode waar ik vorig jaar op n van de vele terrasjes aan het genieten was van het mooie weer heb ik nu de meeste tijd doorgebracht in het labo vloekend op de zoveelste klep die maar niet wilde werken Na hard werk te hebben geleverd ben ik dan toch geko
10. sup ply MEME Thermodynamic differential Hil Antagonistic equations muscle pou ERN Thermodynamic state equations Control unit Joint trajectory tracking Valve Exploiting Natural Dynamics Desired traj ectory control observer Sampling time 500 us Figure 3 4 Structure of the complete simulation model The first block contains the equations of motion given by 3 14 Since the equations of motion are second order these equations have to be transformed into a set of first order equations In order to do this the angular velocity is introduced w 3 21 Chapter 3 Control 52 In our case the equation of motion can now be rewritten as two first order equations 3 22 n D 0 r C 9 w G 0 0 o The thermodynamics of the joint is characterized by four first order differential equations on pressure 3 17 and air mass 3 20 2 first order equations for each muscle Finally the two thermodynamic state equations 3 19 complete the set The link between the equations of motion and the thermodynamic differential equations is given by the antagonistic muscle model In this model the generated torque c 3 7 of the joint is calculated with the pressure information of the thermodynamic block To calculate this torque the polynomial forces 2 2 must be known These are functions of the contraction 2 11 which can be calculated by knowing the angle information from the equations of motion Addi
11. 11 12 13 14 15 Pratt G Williamson M Dillworth P Pratt J andWright A 1995 Stiffness isn t everything Proceedings of the Fourth International Symposium on Experimental Robotics Stanford California USA pp 253 262 Rogers G and Mayhew Y 1992 Engineering Thermodynamics Work and Heat Transfer John Wiley amp Sons New York USA Tonietti G Schiavi R and Bicchi A 2004 Design and Control of a Variable Stiffness Actuator for Safe and Fast Physical Human Robot Interaction Proceedings of the 2004 International Conference on Robotics and Automation Interdepartmental Research Center E Piaggio Faculty of Engineering University of Pisa Italy Vanderborght B 2003 Quasi statische controle van een stappende robot aangedreven door actuatoren met regelbare stijheid Master Thesis Vrije Universiteit Brussel VUB van der Linde R 2001 Bipedal Walking with Active Springs Gait Synthesis and Prototype Design PhD thesis Technische Universiteit Delft Van Ham R Daerden F Verrelst B Lefeber D and Vandenhoudt J 2002 Control of pneumatic artificial muscles with enhanced speed up circuitry Proceedings of the 5th International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines Paris France pp 195 202 Verrelst B 2005 4 Dynamic Walking Biped Actuated By Pleated Pneumatic Artificial Muscles Basic concepts and control issu
12. 12 and Wisse 15 through implementation of McKibben s type of pneumatic artificial muscles 1 3 Approach Pneumatic artificial muscles have the possibility to adapt the stiffness while controlling position Exploitation of the natural dynamics of a walking robot by varying the compliance characteristics can be approached in two ways a more biological inspired approach and an analytical approach The first option is to Chapter 1 Introduction 5 design the robot taking into account all the inertial and compliance parameters in such a way that the system performs a motion close to walking without actuation by making the system oscillate at its natural frequency By controlling the joint compliance the motion characteristics are then adapted and as such adapt the walking pattern Concerning dynamic robot stability this is however a complex task and will probably result in a small range of feasible motion patterns A second approach is to design joint trajectories for a specific robot configuration so that dynamic stability is ensured if these trajectories are tracked by a trajectory tracking controller In order to reduce valve control activity the joint trajectory tracking unit then adapts the compliance of the different joints so that the natural motion best fits the given trajectories As a result the global stability is ensured due to the calculated trajectories while energy consumption is lowered by adapting the joint complia
13. A 25 Schematic overview of the pneumatic circuit aanne 26 View of the exhaust silencers aou o a S SR NNRE NGA RI NN ORE Eus 27 View of the supporting structure of the pendulum 28 Schematic overview of the control hardware 29 View of the speed up circuitry essere 30 Figure 2 18 Figure 2 20 Figure 2 21 Figure 2 22 Figure 2 23 Figure 3 1 Figure 3 2 Figure 3 3 Figure 3 4 Figure 4 1 Figure 4 2 Figure 4 3 Figure 4 4 Figure 4 5 Figure 4 6 Figure 4 7 Figure 4 8 Figure 4 9 Figure 4 10 Figure 4 11 Figure 4 12 View of the pressure Sensor pierre vesie even shes 31 View of the safety supply and transformation board 35 The Control tab of the GU Tu eese ta me e ee aues eia 37 The Model tab of the GUL aenema 37 The Valves Test tab of the GUI nnee vatbaar 38 Schematic overview of the control structure ne 4 Bang bang pressure control scheme see 44 The pendual memodel ace tre en Gee eeu item 48 Structure of the complete simulation model 52 The joint angle for perfect and erroneous parameter estimations Of th model rn ioni aa Gach aaa Ga iaki 59 Torques of the joint for perfect and erroneous parameter estimations OF the models dene tee e eR ee a ous RO e S 60 Muscle 1 pressure with valve action for perfect model 61 Actual
14. Desired angle Joint angle e AVV Figure 4 21 The uncontrolled oscillation of the pendulum for three different stiffness parameters p after a controlled period of 10 seconds As was mentioned before and shown in figure 4 20 setting a different stiffness changes the mean pressure This means that the stiffness is bounded between two values since the absolute pressure in the muscle is limited between 1 and 4 5 bar Varying the stiffness till those limits were reached showed that the range of Ps for this pendulum is bounded between 6 and 40 Nm In figure 4 19 the influence of the amplitude on the input exergy is shown The amplitude has little influence on the optimal stiffness like was expected but energy consumption seems to be higher Due to the nonlinear torque to angle relation the shape of the passive trajectory deviates from a pure sine wave The deviation from the sine wave increases for larger amplitudes Consequently more valve switching is needed for larger movements This clearly shows the importance of designing a joint with a linear torque characteristic when muscles are closed when sine wave trajectories with one frequency component are imposed Chapter 4 Results and discussion 84 In order to see whether the method of the pressure slopes can be used on the practical setup a resume of the optimal p values of the experimental setup is given in table 4 5 and compared with the calculated and simul
15. GUI consists of different tabs Each tab has some parameters to be filled in by the user The first tab to appear is the Control tab and is shown in figure 2 21 In this tab parameters like sampling rate acquisition time and PID gains used by trajectory tracking controller are set These values can be varied online A special attention is given to the compliance calculation More information on the definition of p is given in chapter 3 In the second tab shown in figure 2 22 the Model parameters are set As explained in the mechanical design we can alter the load of the pendulum by removing or adding discs By clicking on the check boxes the user can enter the discs used during the experiments Other masses free oscillation period and Chapter 2 Design 36 atmospheric pressure are also to be filled in so that the pendulum can be modelated EIL Bl x Model Trajectory Plotting Valves Test Acquisition Time s ao r Computed Torque Tuning Kp tuning H Sample Time s o 002 lo Initial DeltaP o Kituning J Ps Nm 32 0 27 Constant Ps Kd tuning Je Variable Ps 3 Valves closed at all time Figure 2 21 The Control tab of the GUI Fe SlingerUl Igi xi gesoonsonrenesogenegoecveng Control Model Trajectory Plotting Valves Test v Mass 1 White 1 0676 kg Mass Link 1 48286 v Mass 2 Purple 1 0407 k dos Lcog Link 0 147 v Ma
16. VEEP EEL A Sa UU EVE RAR UL 4 M og i dq d f l V V V i V y y V V V V V j n i I I I I I I I I 0 1 2 3 4 5 6 7 8 9 10 time s 3 5 T T T T T T I I Pressure muscle 1 i li l ll Desired pressure Jl LI ll Wil Wil Wl V Ho nm M l Valve action 2 um uw m l S E 25 4 oen E 5 VA 2 2A A A A A A A A A AAM A A A A A f EOM VWV NUM V J N a JV WW V 1 54 4 i i i i 0 1 2 3 4 5 6 7 8 9 10 time s Figure 4 17 Measured and desired joint angle and pressure course for muscle 1 Other solutions for this problem are increasing the outlet valves or setting the initial muscle pressure at a lower level The first option is important if high oscillation frequencies are needed since high pressure gradients over the valves will then be required The reason why the initial pressure is set at such a high level requires an explanation In the beginning the idea was to start at an initial angle of zero degrees and to use a sine function for the imposed trajectory Because of a discontinuity in the angular velocity t 20 0 O t 0 Aw instability occurred Therefore we opted to start at an initial angle equal to the amplitude of the imposed trajectory The trajectory used to bring the pendulum in its initial position as depicted in figure 4 18 By using this trajectory the discontinuity of the angular velocity is eliminated When the i
17. action Nm l action Nm E D action Nm naX 0 5 1 1 5 2 2 5 Time s Figure 4 2 Torques of the joint for perfect and erroneous parameter estimations of the model Figure 4 1 depicts the graphs of the angle for the perfect and erroneous model The first period was omitted because some settling time is needed and is not interesting for the purpose of this section As we see the tracking controller unit can still cope with the introduced errors In fact the tracking errors are very small The difference between both simulations and thus the influence of erroneous parameter estimation is better seen on figure 4 2 were the required torques calculated by the inverse dynamic control block and the actual applied torques are depicted The model based part and the servo portions are also plotted In the perfect model a little difference is observed between required and applied torque due to a minimum pressure error required for the bang bang controller to activate The desired torque is almost equal to the model part of the computed torque This is what we expected because the model part was calculated with exact parameters The servo portion deals with the difference in required and measured angle position and velocity The erroneous model shows the influence of the erroneous estimation of the model parameters First of all there is a bigger difference between the applied and required torque because of the influe
18. are two options For simple applications you can connect the encoder directly to the counter timer of the data acquisition card without any extra logic or signal conditioning In this configuration the counter will increment on state transitions on channel A Depending on the Figure 2 19 View ofthe incremental encoder state of B at those transitions the counter will count up or down While this method is very Chapter 2 Design 32 simple to implement it has a couple of potentially serious drawbacks If the encoder disk is not rotating but is vibrating enough back and forth to cause active transitions on channel A without changing the state of channel B then each movement will be incorrectly counted Another problem results when encoder outputs include noise or jitter that is large enough to be erroneously counted as a valid state transition Therefore a clock converter device LS7084 LSI Computer systems is used This device converts the A and B signals into a clock and up down signal that can be connected directly to the data acquisition card The UP DN output indicates the direction of rotation When in X4 mode resolution is increased four times as the CLK output will now pulse once on every transition of either the A or B signals The LS7084 includes low pass filters to prevent miscounts due to noise and jitter In addition the LS7804 uses dual one shots to prevent the miscounting produced by vibration or dither as descr
19. be Chapter 4 Results and discussion 70 seen as a noise signal disturbing K and is therefore neglected The resulting course of K is given in the third plot 2 S s E z T foc E E e X o 1 2 3 4 5 6 7 8 9 10 Time s 100 E 50 Z g S 50 z 100 0 1 2 3 4 5 6 7 8 9 10 2 Time s E 250 T T T T T T T T 5 S 200 150 T 100 E 50 i i L z 0 1 2 3 4 5 6 7 8 9 10 Time s Figure 4 10 Stiffness K and noise due to cotg term for the specific trajectory K is now given by K d o 2001 g cos 4 5 In the graph at the bottom of figure 4 10 we can observe that the required stiffness is increasing as the frequency increases This is what we expected because if the frequency of the imposed trajectory increases a higher stiffness of the antagonistic setup with closed muscles will be needed in order to exploit natural dynamics Using this equation running the simulation for an imposed trajectory with frequency varying between 1 5 Hz and 2 Hz and using a varying p gives us the pressure course as depicted in figure 4 11 To compare this to the situation where a constant p was used the pressure course from figure 4 9 was redrawn in figure 4 12 Comparing figure 4 11 and 4 12 we can conclude that valve Chapter 4 Results and discussion 71 action is extremely decreased The natural dynamics are now being exp
20. controller were described To evaluate whether the proposed controller can meet tracking requirements and deal with the natural dynamics of the system it was incorporated into the computer simulation and the physical pendulum This chapter will give results on this theme for both simulation and practical set up Results on the simulation model will be given in section 4 2 First of all the trajectory tracking control will be evaluated The influence of erroneous model parameter estimations will be shown Secondly the influence of the joint compliance on the energy consumption and valve actions will be discussed In this context special attention is given to the evaluation of the mathematical formulations given in the previous chapter for estimating the appropriate joint stiffness In section 4 3 tracking experiments on the physical set up will be discussed Successively the possibility to decrease control activity and energy consumption by setting a suitable stiffness will be shown on the physical pendulum 4 2 The simulation model 4 2 1 Evaluation of the joint trajectory tracking controller 4 2 1 1 General considerations As mentioned in chapter 3 the control architecture consists of 3 main blocks a feedback linearization module a delta p unit and a bang bang pressure controller The computed torque incorporates a model based module and a servo control The model based part requires a good estimation of the model parameter
21. not fit the desired trajectory but for Ps 32 Nm we can see that the base frequency is near to 2 Hz We can expect that for p 32 Nm the valve actions and energy consumption will be less than in the case of p 16 Nm as the natural dynamics fits best the imposed trajectory To investigate this assumption a simulation of an actuated oscillation was done for p 216 Nm and p 32 Nm Figure 4 5 depicts the joint position Chapter 4 Results and discussion 63 and angle velocity of the two simulations for one period There is no significant difference between the two simulations concerning the joint angle and angle velocity The tracking is slightly better for the case where p 32 Nm P 16 Nm 6 T T T 75 angle 3 6 desired angle 145 2 E 2 on 12 15 3 E Gi gt ga 15 5 E 36 angle velocity 45 E desired ang vel 75 5 0 6 0 7 0 8 0 9 1 Time s P 32 Nm 6 T T T 75 angle 3 6 desired angle 145 Ka A L E 2 ws 12 15 S E a gt 1 2 15 E 5 E 36 angle velocity 45 Z desired ang vel f l l ji I 75 45 0 6 0 7 0 8 0 9 1 Time s Figure 4 5 Actual and desired joint angle and angular velocity for Ps 16 Nm and p 32 Nm In figure 4 6 and 4 7 the gauge pressures and the valve actions are depicted To get a clearer view on the valve actions 2 periods are shown on the right
22. s During this period muscle 1 is closed and the natural dynamics of the system is exploited It s now clear that the dead zone of the bang bang controller will be very important to decrease the energy consumption and should be exploited as much as possible This will be discussed in the following section Chapter 4 Results and discussion 61 4 2 2 Exploitation of the natural dynamics 4 2 2 1 Experiments If we set some pressure values in both muscles of the antagonistic setup close them and then release the pendulum from a position different of the equilibrium the pendulum starts oscillating with a certain frequency If this frequency corresponds with the desired trajectory no valve action needs to be taken This means that if stiffness is set in such a way that the natural dynamics of the system suits the desired trajectory energy consumption will be minimized during tracking The first simulations were done in this context A constant p the parameter to influence the sum of the pressures and consequently the joint stiffness was set This will be done for two different values of ps in order to show the importance of setting a correct stiffness for minimizing the energy consumption The desired trajectory again is a sine wave with a frequency of 2 Hz and an amplitude of 5 In a second series of simulations the mathematical formulation proposed by Verrelst and described in chapter 3 will be evaluated for different sinusoid
23. to time Chapter 1 Introduction 1 1 Motivation and goal Nowadays robots are widely used in all kinds of production and assembly chains to assist in very hard repetitive and labour intensive tasks with optimum performance Most of the robots built use heavy electrical drives making these machines heavy and power demanding and not safe for humans Robots in assembly halls are completely isolated and can not be accessed when working As the field of robotic solutions is still expanding and more robots will be used in the immediate surroundings of people there is a need for safety against all possible accidents Therefore soft actuators are being developed Such soft actuators offer a natural compliance making them human friendly At the University of Pisa an electromechanical Variable Stiffness Actuation 10 motor has been developed and specifically designed for machines and robots physically interacting with humans Another example of such an actuator is the AMASC Actuator with Mechanically Adjustable Series Compliance 4 which uses two motors one for controlling the position and the other for controlling the stiffness the inverse of compliance Figure 1 1 Industrial robot with heavy electrical drives and the soft actuator AMASC For the last ten years the Robotics and Multibody Mechanics Research Group of the Vrije Universiteit Brussel is conducting research in the domain of legged bipedal robots Bipeds can be divided int
24. 0 40 0 10 20 30 40 p Nm p Nm Valve actions frequency 1 75 Hz Influence amplitude on total input exergy 200 25 n valves muscle 1 5 degrees pen n valves muscle 2 20 10 degrees 150 Out valves muscle 1 bd Out valves muscle 2 S 8 100 1 Ef M 2 10 E isl d E de 0 L 0 0 10 20 30 40 0 10 20 30 40 p Nm p Nm Figure 4 19 Energy consumption and valve action for one period of different trajectories in function of the stiffness parameter p We can clearly identify optimal values for p where energy consumption and valve actions are minimized This is also illustrated by figure 4 20 where the valve actions required to track a trajectory with a frequency of 1 75 Hz and an amplitude of 5 degrees is depicted In this figure two different values of p were set a wrong and the optimal value If we look between 1 32s and 1 7s no valve action is taken for p 9Nm because the natural pressure course fits the required one Comparing with p 30Nm we can conclude that it is important Chapter 4 Results and discussion 82 to set the right stiffness in order to exploit the natural dynamics of the pendulum This was already concluded in simulation but now it is confirmed in practice p 30 Nm I I Pressure muscle 1 Desired pressure Valve action e N T e a N to Pressur
25. 1 The controller sampling time is set to 2 ms and a valve delay time of 1 ms is introduced The valve delay time corresponds with the real data recorded by Verrelst As was mentioned in chapter 2 a controller sampling time of 2 ms is used for the control of the physical pendulum Next the trajectory tracking control shall be evaluated by imposing a desired sine wave trajectory with a frequency of 2 Hz and an amplitude of 5 The stiffness parameter p is set to 20 Nm This is chosen as an example to validate the trajectory tracking without taking into account the natural dynamics of the system 4 2 1 2 Results and discussion Perfect model Mean error on 0 094018 Maximum error on 0 46594 T T T q1 degrees q1Des degrees H i i 5 1 1 5 2 2 5 Time s Erroneous model Mean error on 0 1175 Maximum error on 0 35948 6 T T T q1 degrees q1Des degrees H S I 5 1 1 5 2 2 5 Time s Figure 4 1 The joint angle for perfect and erroneous parameter estimations of the model Chapter 4 Results and discussion 59 Perfect model Torque Nm Desired torque Nm friction CT model part Nm P action Nm l action Nm D action Nm 0 5 1 1 5 2 2 5 Time s Erroneous model Torque Nm Desired torque Nm friction CT model part Nm P
26. 296 0 02254 Table 2 2 Coefficients of the polynomial volume fitting approximation 10000 1bar 2bar 3 bar 5000 Force N 0 5 10 15 20 25 30 35 40 Contraction Volume ml N Qo A s s e e 10 15 20 25 30 35 40 Contraction e a Figure 2 4 Forces at pressure levels 1 2 and 3 bar and volume resulting from the polynomial fitting as a function of the contraction Chapter 2 Design 12 2 2 2 The one dimensional joint set up 2 2 2 1 Introduction Pneumatic artificial muscles can only exert a pulling force In order to have a bi directional working revolute joint two muscles are coupled antagonistically The antagonistic coupling of the two muscles is achieved with a pull rod and leverage mechanism as is depicted in figure 2 5 The lever arm can be varied in such a way that the highly non linear force length characteristic of the PPAM is transformed to a more flattened torque angle relation While one muscle contracts and rotates the joint in its direction the other muscle will elongate f Joint NV osition P i r f Joint Yx Stiffnes i Figure 2 5 The antagonistic muscle setup and the possibility to adapt both position and compliance independently The gauge pressure difference between the two muscles determines the generated torque 7 and consequently also angular position 0 Furthermore both press
27. 3HOSN3S ynya Electronic scheme of the safety supply and transformation board Figure B 1 94 Chapter B Safety board Bibliography Robotics 1 2 3 4 5 6 7 Craig J J 1986 Introduction to Robotics Mechanics amp Control Addison Wesley Publishing Company Daerden F 1999 Conception and Realization of Pleated Pneumatic Artificial Muscles and their Use as Compliant Actuation Elements PhD thesis Vrije Universiteit Brussel Garcia M Chatterjee A Ruina A and Coleman M 1998 The simplest walking model Stability complexity and scaling ASME Journal of Biomechanical Engineering 120 281 288 Hurst W Chestnutt J E and Rizzi A A 2004 An actuator with Mechanically Adjustable Series Compliance technical report CMU RI TR 04 24 Robotics Institute Carnegie Mellon University McGeer T 1990 Passive dynamic walking International Journal of Robotics Research Special Issue on Legged Locomotion 9 2 62 82 Pratt J E and Pratt G 1998 Exploiting natural dynamics in the control of a planar bipedal walking robot Proceedings of the 36 Annual Allerton Conference on Communication Control and Computing Monticello Illinois Pratt J E 2000 Exploiting Inherent Robustness and Natural Dynamics in the Control of Bipedal Walking Robots PhD Thesis Massachusetts Institute of Technology MIT Massachusetts USA 8 9 10
28. AA 81 Experimental calculated and simulated optimal values of p for different sine wave trajectories 85 Nomenclature Acronyms AICH DACOUT DIO PPAM USB Symbols na S8 S Analog Input Channel Analog Channel Ouptut Digital Input Output Pleated Pneumatic Artificial Muscle Universal Serial Bus controller coefficient denoting centre of gravity of the pendulum link angular design parameters of the joint muscle setup muscle contraction contraction of muscle i at chosen central position 0 air density at standard conditions contraction of muscle in a joint setup angular joint position chosen central position in the joint setup control variable of delta p control unit actuator torque reaction level of bang bang pressure controller reaction level of bang bang pressure controller distance between origin O and points B fixed base points of the joint muscle setup reaction level of bang bang pressure controller pneumatic valve flow constant centrifugal coriolis matrix distance between R and points D moving points of a joint muscle setup inertia matrix reaction level of bang bang pressure controller reaction level of bang bang pressure controller dimensionless force function of muscle i in the joint setup dimensionless muscle force function coefficients of a polynomial force fitting muscle traction force acceleration of gravity o kg m rad rad bar bar bar bar Std l min ba
29. DIUSOIE Too ocv Tas ctae bet ei Sede dS esi nates 79 Initial trajectory and imposed trajectory during experiments 80 Energy consumption and valve action for one period of different trajectories in function of the stiffness parameter p 82 Influence of the stiffness parameter p on the valve action and the absolute pressure in the muscle sse 83 The uncontrolled oscillation of the pendulum for three different stiffness parameters p after a controlled period of 10 seconds 84 Schematic overview of the studied model 9 Electronic scheme of the safety supply and transformation board 94 List of Tables Table 2 1 Table 2 2 Table 3 1 Table 4 1 Table 4 2 Table 4 3 Table 4 4 Table 4 5 Coefficients of the polynomial force function approximation 12 Coefficients of the polynomial volume fitting approximation 12 Inertial parameters of the pendulum model 49 Pressure reaction levels of the bang bang pressure controller ESSA SIAL AMON aren cedo ete te vand b tse qu dires 58 Air mass flow exergy and valve action during one period of a sine wave trajectory with frequency of 2 Hz for p 16 Nm and Calculated and simulated optimal values of p for different sine WAVES trajectories s dee aee ite ee tuu e s Shas ito 67 Pressure levels of the bang bang controller for the physical DEB QU TTE o eade on tee ec apa iaa waned o e pel du egi A
30. E E E valve action FA valve action 2 pressure bar g pressure bar An required pressure bar An required pressure bar 1 5 1 5 1 2 3 4 5 4 42 44 4 6 4 8 5 Time s Time s 2 5 r r 2 5 T T 5 3 2 2 N N o o E E 5 2 2 o o E E FA valve action E valve action 2 pressure bar 2 pressure bar An required pressure bar An required pressure bar 1 5 i z 1 5 5 1 2 3 4 5 4 42 44 4 6 4 8 5 Time s Time s Figure 4 7 Actual and required pressure in muscle 1 and 2 for p 32 Nm is taken by the valves of muscle 1 or between 2 45 and 3 51s The reason for this is that the pressure slope induced by the natural dynamics of the system fits the pressure slope required to track the desired trajectory We can also Chapter 4 Results and discussion 65 understand why the pressure levels of the bang bang controller are so important during the exploitation of the natural dynamics The pressure error of muscle 1 near to 4 75 seconds has a maximum value of 59 mbar which means that p is still situated in the dead zone A smaller dead zone would have required the bang bang controller to take action A larger dead zone will increase the exploitation of the natural dynamics but a larger deviation on the trajectory tracking will be observed A compromise has to be made between control action and tracking error Table 4 2 gives a summary of
31. This tracking controller will be incorporated in a simulation model and in the physical set up Section 3 3 gives a mathematical formulation for compliance adaptation in order to reduce energy consumption and control activity during the joint trajectory tracking A simulation model of a one dimensional pendulum with one antagonistic muscle pair actuating the joint is presented in section 3 4 This model is used to show the importance of appropriate stiffness selection in order to reduce control activity A joint tracking controller is incorporated in a simulation model Finally some energy considerations concerning the proposed control strategy are given in section 3 5 3 2 Joint tracking controller The joint tracking controller has to command the valves of the two muscles in order to track an imposed desired trajectory The complete system incorporates several nonlinearities such as the non linear behaviour of the pendulum configuration and the nonlinearities introduced by the antagonistic muscle set up The tracking controller is designed in a modular way which means that the controller can be easily adapted for an application with another mechanical configuration but with an analogue antagonistic muscle actuator set up Therefore the controller is multistage of which each stage deals with the different nonlinearities separately A schematic overview of the proposed control structure is given in figure 3 1 The controller consists of th
32. Visual C The minimum control period during which all control calculations are done is 2 ms Therefore a controller sampling time of 2 ms is used A cable connector is used to connect the data acquisition card with the sensors and valves islands The pressure sensor outputs a non referenced single ended NRSE analog signal The left muscle pressure sensor is connected to analog input 6 PIN 15 the right muscle pressure sensor to analog input 7 PIN 17 As we mentioned before analog signals can be disturbed by noise A proper wiring system and correct grounding of the signal circuit can strongly reduce this problem To measure a grounded signal source with a single ended configuration the DAQ card was configured in the NRSE input configuration The signal is then connected to the desired analog input ACH6 for left pressure sensor and ACH7 for right pressure sensor of the DAQ card and the signal local ground reference is connected to AIGND The ground point of the signal should now be connected to the AISENSE pin Any potential difference between the DAQ card ground and the signal ground appears as a common mode signal at both the analog input ACH6 ACH7 and AIGND and this difference is rejected by the internal electronics of the DAQ card If this method is not used a difference in ground potentials would appear as an error in the measured voltage As mentioned before the incremental encoder has 3 output channels but only 2 will be used T
33. a lot of noise A silencer consists of a closed permeable tube which makes the pressurised air leave the volume slowly resulting in a strongly reduced noise generation But generally a silencer also obstructs the dynamic performance of muscle deflation since a pressure rise in the silencer lowers the CE 3 NM B 4 y use large silencers with good permeable Figure 2 14 View of the exhaust material 14 As the basic frame of the silencers exhaust airflow It is therefore important to pendulum is not moving and the valves very large exhaust silencers For implementation on Lucy it should be taken into account that the valves island are connected to the moving robot so smaller exhaust silencers are used 2 2 3 5 The support As pneumatic muscles can easily exert forces up to 5000 N and the pendulum is swinging at frequencies where dynamic forces are being important a strong supporting structure has to be provided to hang up the pendulum Besides this Chapter 2 Design 27 the supporting structure has to be as such that the pendulum can be easily dismounted and inverted Several configurations can be used for this purpose Figure 2 15 shows a picture of the used supporting structure It has been decided to use a rectangular pole that is clamped to the wall One side of the slats of the basic frame is pressed against the pole with screw thread and rectangular tubes used as washers Holes were made into the pole in ord
34. acquired This output voltage is proportional to the applied pressure in the muscle so by means of simple linear fitting of the measured data voltage offset and sensitivity can be found 2 3 3 Incremental encoder The angular position is measured with an incremental encoder HEDM6540 Agilent Such a device converts rotary displacements into digital pulse signals This type is an optical encoder which consist of a rotating disk a light source and a photodetector light sensor The disk which is mounted on the rotating shaft has pattern of opaque and transparent sectors coded into the disk As the disk rotates these patterns interrupt the light emitted onto the photodetector generating a digital or pulse signal output The encoder is incremental which means that the encoder does not output absolute position To determine direction and reference position 3 channels are used Using two code tracks A and B with sectors positioned 90 if A leads B for example the disk is rotating in a clockwise direction If B leads A then the disk is rotating in a counter clockwise direction The third output channel is used as a zero or reference signal which supplies a single pulse per revolution In our case only channel A and B are used We assume that the position where no muscle is inflated corresponds with zero degrees and is therefore taken as reference position When using a quadrature encoder with the data acquisition card there
35. al trajectories Still a constant p will be used Finally the possibility to track a random trajectory with exploitation of the natural dynamics will be investigated in the third series of simulations The stiffness is changed online in order to set the most suitable stiffness at every control sample time No parameter deviations are introduced in the simulations The controller sampling time and the valve delay time are still 2 ms respectively 1 ms Chapter 4 Results and discussion 62 4 2 2 2 Results and discussions Simulation 1 the importance of setting a suitable stiffness We can illustrate the influence of the stiffness by considering the unactuated oscillation with closed muscles of the pendulum as was explained in the previous section and compare it with the desired trajectory In figure 4 4 one period of the unactuated oscillation is shown for 2 different values of p as well as the 2 Hz sine wave that has to be tracked p 16 Nm I I q1 degrees q1Des degrees Joint angle e I I I I I I 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4 0 45 0 5 Time s P 32 Nm I I q1 degrees q1Des degrees Joint angle e I I I I I I 0 0 05 0 1 0 15 0 2 0 25 0 3 0 35 0 4 0 45 0 5 Time s Figure 4 4 Actual and desired joint angle with closed muscles for Ps 16 Nm and p 32 Nm For both stiffness the oscillation does
36. al dynamics in combination with trajectory tracking Two main steps should be taken to improve the performance the stiffness control e A correct estimation of the pendulum s parameters The muscle volumes the polytropic exponent and the friction are the most important parameters to be estimated e Fine tuning of the pressure levels of the bang bang controller The dead zone should be exploited as much as possible without losing tracking accuracy Before using this technique on Lucy a study should be done on a double pendulum in order to investigate the coupling effects which can lead to chaotic behaviour of the system Finally the same investigation should be done on an inverted pendulum To conclude in this master thesis the first steps in the direction of exploitation of natural dynamics by using PPAM s were made Chapter 5 General conclusions and future perspectives 90 Appendix A Dynamic model of the pendulum G m 1 1 G m 1 Figure A 1 Schematic overview of the studied model In this section the equation of motion for the model depicted in figure A 1 is derived A Newton Lagrange formulation of the equation of motion is used for this purpose The model has only one DOF 80 Ed OK AU Q A 1 dt o 00 600 K is the total kinetic energy and U the potential energy Q is the generalized force associated with the pendulum The kinematic expressions for the positions and velociti
37. and desired joint angle with closed muscles for p 16 Nm Actual and required pressure in muscle 1 and 2 for p 16 Nm 65 Actual and required pressure in muscle 1 and 2 for p 32 Nm 65 Energy consumption and valve action in function of ps for different trajectories cc Aico sales eei ur t a S bri ead 68 Joint angle pressure and frequency for a trajectory with linear varying regency uus s esee tede eas te bea te ta ee ire Dt Sede ties 69 Stiffness K and noise due to cotg term for the specific trajectory Actual and required pressure and valve action for a trajectory with linear varying frequency and p online changed 72 Actual and required pressure and valve action for a trajectory with linear varying frequency and a constant ps nennen eenen 72 Figure 4 13 p in function of the frequency of the imposed trajectory 73 Figure 4 14 Figure 4 15 Figure 4 16 Figure 4 17 Figure 4 18 Figure 4 19 Figure 4 20 Figure 4 21 Figure A 1 Figure B 1 Effect on the valve actions due to deviation on the online changing parameter De NUN ERE NUN E M ETE MERE 74 Angular velocity computed torque and absolute pressure for first tracking experiment ert cee t Ro er Red deae tips 75 Angular velocity computed torque and absolute pressure for a tracking experiment by using a filter on the angular velocity 76 Measured and desired joint angle and pressure course for
38. apter 3 Control 47 p and an erroneous stiffness will be set The influence of the end fittings and tubing volumes will be explained in more detail in the next chapter As mentioned in the previous chapter special attention was given to the compliance calculation during the design of the GUI On the Control tab the user can choose between three different options he can set a constant p by giving it a certain value or by calculating it with the method described above or use a variable p also calculated by the latter method 3 4 The simulation model of the pendulum 3 4 Description of the pendulum set up Figure 3 3 The pendulum model The pendulum model consists of 2 parts the link and the mass The mass is modelled as a point mass and is positioned at a distance of the rotation point The length of the link is its mass m and the moment of inertia about its centre of mass G is The location of the centre of mass G of the link is given by OG aL with a 0 769 The parameter values are given in table 3 Chapter 3 Control 48 i 1 m m kg I kgm 1 04 5327 0 852 2 045 1483 0 0362 Table 3 1 Inertial parameters of the pendulum model An accurate estimation of those parameters is crucial as well for the computer simulation as for the joint tracking controller In general the model of the pendulum will not be perfect Think for example of the f
39. ation values Sine wave frequency Hz sar Nm pe Nim pier Nm 1 5 12 12 6 12 5 1 75 19 21 4 21 2 29 30 31 7 31 Table 4 5 Experimental calculated and simulated optimal values of p for different sine wave trajectories The experimental values are near to the calculated and simulated values From this we can conclude that the method of the pressure slope could be used but only as an approximation of the optimal stiffness The stiffness will not be set at his optimal value but still energy consumption and valve action will be decreased and natural dynamics will be exploited The error that is made by using this method can be decreased by a better parameter estimation e g leaks in the tubing and muscle and friction The most important parameters to be estimated are the volume of the muscle and the polytropic exponent The volume is calculated using the estimated polynomial fitting described in section 2 2 1 which does not take into account the volumes of the end fittings and tubing s Besides this the estimated polynomial fitting was done on a theoretical model of the muscle For the mathematical description of the enclosed volume the pleated polyester membrane is approximated by considering at each parallel section a circular membrane pattern instead of the pleated structure No experimental data on the volume has ever been acquired The polytropic exponent is introduced to des
40. bot with heavy electrical drives and the soft actuator AMASC cs ope denten 1 The active walker Asimo and the passive walker Denise 2 The bipedal walking robot Lucy assu eee tiec oett 3 CAD drawing and photograph of the physical pendulum 4 The Pleated Pneumatic Artificial Muscle esse 9 Theoretical forces at pressure levels 1 2 and 3 bar as a function of the contractio oes eter titer secti oder ieri edet es 10 Theoretical maximum muscle diameter as function of contraction 11 Forces at pressure levels 1 2 and 3 bar and volume resulting from the polynomial fitting as a function of the contraction 12 The antagonistic muscle setup and the possibility to adapt both position and compliance independently ss 13 Basic configuration of the pull rod and leverage system 14 Influence on muscle contraction by changing d and o 20 Static and closed muscle torque characteristics 21 Static and closed muscle torque characteristics at different pressure levels with the design parameters set to the one used in the experimental SCAB seco a e pacto eatis atte ded n dle npa dius 22 Exploded and assembled view of the basic frame 23 View of the connection plate and the position limiter 24 View of the valveasland os ea see vr
41. ced by ps Chapter 3 Control 45 The required pressure slopes are calculated by deriving equations 3 6 with respect to the trajectory 0 EE LOAD te 3 9a d t d0 d0 dp __ Ps dt dAp 40 a6 do 220 Taking into account equation 3 8 the derivatives can be expanded by dAp 1 E dt F Ht 4 K T 3 10 d t L 2 d d ves with K equation 2 12 representing the stiffness associated with the desired trajectory and 7 the torque calculated by the computed torque module On the other hand combining equation 2 15 valid for closed muscles with equations 3 6 yields dp n Vy dV A Fs dV ds Dt Pom AP nas 3 11 Boin y do l p y de VM dp n Vy dV EN Vy dV 2 PAD pe 3 11b 7 e IE yan A P V d ely The idea is to match for each muscle the required pressure slope with the slope associated with the natural dynamics by selecting an appropriate p value Once the desired trajectory is known expressions 3 9 and 3 11 can be evaluated in every point 0 Subsequently a p value is searched in order to match as much as possible both pressure slopes For each 0 equations 3 9a and 3 11a and equations 3 9b and 3 11b are thus respectively combined and each solved for a value pj J Chapter 3 Control 46 E 1 dt 1l n dV LY n dV dAp TITRE Pim tA p do 1 V s 2b 2 dB 6412 1 B Ps an PELU AA A t
42. co en 3 12b t dO t V d0 V do d Note that the initial volume V when closing a muscle is set equal to the actual volume V From here on there are two options We can use a constant p as explained by Verrelst The p are calculated in each point separated by equal time intervals along the desired trajectory and a mean is then taken to select one ps Ps db ip 3 13 with z the number of points chosen to evaluate equations 3 12 This means that a constant p is used along the whole trajectory Probably a constant p will only be suitable for some simple trajectories As we want a method for any trajectory a variable p is needed One solution is to use the same idea of comparing the pressure slopes for the complete trajectory at once but for the calculation a differential formulation on p has to be solved To avoid complicated calculations it was chosen for evaluating the equations 3 12 at every sample time during the control of the pendulum At every control sample time the angle 0 is acquired and both Ps and Ps are calculated The mean i i for both muscles is then calculated with p ng so that at every sample time a different p is set As we see in the equations 3 12a and 3 12b a correct estimation of the volume is needed So the volumes of the end fittings and tubing s should be added If this is not done the equations 3 12a and 3 12b will not give the most suitable Ch
43. cribe deviations from the isentropic expansion compression taking place when muscles are closed and the pendulum is oscillating The value of the exponent should be experimentally estimated and may depend on the specific process The estimation of these parameters was not done during this thesis but is very importance if a varying stiffness has to be Chapter 4 Results and discussion 85 used in combination with trajectory tracking But still promising results were produced 4 4 Conclusion In this chapter the trajectory tracking control structure as described in chapter 3 was evaluated on a computer simulation model and a physical pendulum The simulation model showed that the proposed control architecture copes with the nonlinearities introduced by the actuator characteristics and dynamic model of the pendulum The importance of a correct estimation of the model parameters was demonstrated by introducing deviations on the inertia the mass the centre of gravity and on the force function The implementation of the control unit into the physical pendulum required the introduction of a filter to reduce the noise on the calculated angular velocity A tracking experiment illustrates that the control unit is able to follow a sine wave The main differences with the simulation model were also discussed In the context of the exploitation of the natural dynamics 3 experiments were performed on the computer simulation model and one on the p
44. e Because the research on this topic will be used for further implementation on Lucy the investigation is done on one modular part of Lucy With such a modular part a single pendulum set up shown in figure 1 4 was created in order to investigate energy consumption reduction with imposed swinging motion of the pendulum while the stiffness of the joint is changed The idea in this work is to change the natural frequency in combination with pure trajectory control in such a way that control efforts are minimized This strategy is tested on both a physical and mathematical computer simulation model Figure 1 4 CAD drawing and photograph of the physical pendulum 1 2 Background information The first official recordings of passively walking toys goes back till 1888 e g Fallis patent Inspired by the observation of human data in which the muscles of the swing leg are activated only at the beginning and the end of the swing phase Morawski amp Wojcieszak were the first to give a mathematical formulation Chapter 1 Introduction 4 for these toys They concluded that in human locomotion the motion of the swing leg is merely a result of gravity acting on an unactuated double pendulum Inspired by this calculations it was McGeer 5 who introduced the concept of Passive Dynamic Walking McGeer showed that a simple planar mechanism with two legs could be made to walk in a stable way down a slight slope with no other energy input
45. e 1 bar N N N I I Pressure muscle 1 Desired pressure H Valve action Pressure 1 bar N Figure 4 20 Influence ofthe stiffness parameter p on the valve action and the absolute pressure in the muscle Another way to illustrate how p influences the natural dynamics is the unactuated oscillation of the pendulum for the two different stiffness parameters This is given in figure 4 21 After 10 seconds of control the muscles are closed If the stiffness is suitable when p 19Nm the base frequency of the uncontrolled oscillation approximates 1 75 Hz during the two first periods While when p 30Nm the base frequency is situated around 2 Hz since the stiffness of the joint is higher When p 10Nm the base frequency is about 1 4 Hz since the stiffness of the joint is lower We only consider the two first periods because due to friction and leakage of the pneumatic system the uncontrolled oscillation will damp and frequency will change Chapter 4 Results and discussion 83 Desired angle MIT I I I I Measured Angle P 30Nm Joint angle e 5 6 7 8 9 10 11 12 13 14 15 time s I I I I T T T T T 5H Measured Angle p Poot 19Nm Desired angle Joint angle e TW a 5 6 10 11 12 13 14 15 time s I I I 5H Measured Angle P 10Nm
46. e two linking bars The mass consists of simple discs recuperated from dumbbells In order to attach the discs and because the diameters of the discs centre holes are different to the diameter of the screw thread centering rings were fabricated The discs are colour marked so that the mass of each disc is known They can be easily changed to alter the load The muscles are positioned crosswise to allow complete bulging At one side they are attached to the frame via the fixed rotary base and at the other side the interface to the next modular unit is provided via the leverage mechanism Two connection plates showed in figure 2 11 are fixed to the next modular unit and incorporate the leverage mechanism Again sliding bearings are used to guide the rotations of both rotary axes The position of the rotation points determines the dimensions of the leverage mechanism and consequently joint torque characteristics The mathematical formulation of the torque as a function of force relation was given in section 2 2 2 2 The connection plates incorporate the parameters a a d and d of the leverage mechanism for both muscles Since Chapter 2 Design 23 these parameters have a large influence the connection plate system is the one which can be changed easily besides muscle dimensions to alter joint torque characteristics Therefore the two plates have to be replaced with only different positioned holes for the sliding bearings Position
47. ed by 14 As we need these equations to understand how the computer simulation explained in the next section works those equations will be repeated here The thermodynamic processes in the two muscles valve systems are described by four first order differential equations Two equations determine the pressure changes in both muscles of the pendulum and the remaining two describe conservation of mass in the respective muscle volume In assumption that the pressurised air behaves like a perfect gas the perfect gas law completes the set of equations required to run the simulation The first law of thermodynamics while neglecting the fluid s kinetic and potential energy and assuming a polytropic process can be written for each muscle as 14 Appendix B DP rmi TT i Mar Pn p Y 3 17 Sh E Chapter 3 Control 50 with the dry air gas constant 7 is the temperature of the supply air and T y air g p p ppiy air the temperature in muscle i The total orifice flow through the opened inlet valves and exhaust valves of muscle i are given by m and m respectively The latter two can be calculated with the following equations which represents a normalized approximation of a valve orifice flow defined by the International Standard ISO6358 1989 293 P P bY _ CP 1 as LL 3 182 M air uP 0 I lis b if P Y 293 P 1 CP i lt b 3 18b M gir u P 0 TE if P with o the air den
48. eeded Nm 150 a 60 719 and P40 P2073 bar I I I Closed muscles torque Nm d 35mm d 30mm d 25mm d 20mm L 0 degrees degrees Figure 2 8 Static and closed muscle torque characteristics Based on these plots the design parameters were set to the following values d 30mm a 60 E 19 As we can see on figure 2 7 for these values the contraction stays between the limits of 5 and 35 for an angle range between 35 degrees and 35 degrees In figure 2 9 the static torque characteristics and torque characteristics with closed muscle were redrawn with the chosen design parameter values This was done for different pressure levels Chapter 2 Design 21 Within the angle range Figure 2 9 shows that enough torque can be provided to lift the mass If we now look at the torque characteristics of the pendulum with closed muscles we see that there is an almost linear evolution of the torque between 20 degrees and 20 degrees The angle range is now fixed and all requirements are met o 60 d 30mm 9719 a 60 d 30mm 719 150 I I I 150 7 1 bar bar 2 bar 2 bar 3 bar 4 bar Minimal Torque Needed Nm 100 Static torque 5 7 2 Nm Closed muscles torque Nm
49. ent As the valves are controlled by the control unit the valve actions will give us an idea of the control action The control signals to action the valves have two possible values 1 for an open valve and 0 for a closed valve To have an idea of the amount of control action the control signals for the input as well as for the output valves are accumulated during the experiment or simulation The time per period during which the input output valves are open is then calculated with sum Valve Signals Valve Action ms period SampleTime 1000 3 23 sine Wave rm With SampleTime the sample time at which data is acquired and not the controller sample time In the case of the physical pendulum the sample time at which data is acquired and the controller sample time are the same The thermodynamic conditions of the pressurized air also determine energy consumption So apart from valve actions it is interesting to consider actual air mass entering and leaving the total system The energy consumption depends not only on the air mass flows but is related to the thermodynamic conditions of the compressed air supply source It is not straightforward to calculate the actual energy needed to power the pendulum since this depends on how the pressurized air of the pneumatic supply source has been created One way to give an idea of energy consumption is to calculate the exergy associated with the particular pneumatic air mass flow Exerg
50. entropic process assumes reversible adiabatic thermodynamic conditions and the exponent becomes in this case 1 2 y c Jes 4 for dry air 9 During the joint design process it is ensured that the torque to angle and the volume to angle characteristics of a joint are monotonous functions Meaning that the derivatives dt d and dV d0 keep the same sign within the range of Chapter 2 Design 17 motion for which the joint was designed Thus referring to figure 2 6 increasing the joint angle will increase the torque function 7 0 while the volume of the respective muscle will decrease Indeed the larger 0 the lesser muscle 1 is contracted Consequently the generated force is bigger On the contrary less contraction means that the muscle gets thinner and that volume decreases Thus dt d0 0 and dV d0 lt 0 For the other muscle 2 in the antagonistic setup the actions are opposite dt d0 0 and dV d0 0 Combining 2 12 2 13 and 2 15 with this information gives K k 0 p k A p gt K am OYE se 2 16 with k O t 8 n ho eds vdo Vv d0 KO t yn 2 l Pos sg V d0 V do dt dt kn 0 K O k 0 ITE am 9 ki 0 k 8 d0 do The coefficients k 0 k 0 k 0 are determined by the geometry of the atm joint and its muscles From equation 2 16 we can conclude that when muscles in an antagonistic joint setup are closed
51. er to fix both inverted and non inverted pendulum To avoid oscillations of the basic frame due to dynamical forces generated by the swing motion the other slat is attached to a rectangular frame The pendulum is now resting in a kind of symmetrical cage and can easily be disassembled and inverted In the future a second rectangular frame should be used because oscillations were still observed at higher frequencies of the swing motion Pole Washer Plate Rectangular frame Figure 2 15 View of the supporting structure of the pendulum Each valve island together with its electronic circuit is mounted on a plate Each plate is clamped at one side of to the pole It can be easily positioned where desired Chapter 2 Design 28 2 3 Electronic design In figure 2 16 an overview of the control hardware used to control joint position and stiffness is given A multifunction I O data acquisition card from NI DAQ is used to exchange data with a central PC used to control the whole pendulum User Interface Central PC ISA databus AT MIO 16E 10 Data acquisition card Emergency lt AAA Pressure Stop supply unit Speed up Speed up circuitry circuitry Valve Safety Suppl y Valve Transformation E island island Board Pressure Pressure extensor flexor muscle muscle im Joint Angle T me Alarm Joint Stiffness signal Figure 2 16 Schematic overview of the control hardware Pressures are measured wi
52. es Phd Thesis Vrije Universiteit Brussel VUB Wisse M 2004 Essentials of Dynamic Walking Analysis and Design of Two Legged Robots PhD thesis Technische Universiteit Delft Bibliography 96 16 http mms tudelft nl dbl 17 http world honda com ASIMO 18 http www sony net SonyInfo ORIO 2 Programming http www codeguru com Visual studio 6 0 MSDN 19 http www mathworks com 3 Data acquisition card AT E Series User Manual Multifunction I O Devices for the PC AT National Instruments 2002 20 NI DAQ Function Reference Manual for PC Compatibles Version 6 6 National Instruments 1999 http www ni com Bibliography 97
53. es are then transformed into desired muscle pressure levels by a delta p unit coping with the nonlinearities introduced by the muscle actuation system Finally a bang bang pressure controller commands the valves in order to set the required pressures in the muscles The exploitation of the natural dynamics is important as it will decrease energy consumption and control action In this context two slightly different mathematical formulations were given to predict an adequate stiffness setting First a suitable constant stiffness was discussed In the second formulation the stiffness is changed online which makes it more suitable for complicated trajectories A simulation model which incorporates this control architecture as well as the modelling of the pendulum dynamics and the thermodynamic processes which take place in the muscles were described This simulation model will be used to investigate exploitation of the natural dynamics and particularly to test the mathematical formulation described for that purpose Finally some energy considerations were made in order to have an idea about the amount of energy consumption and valves action during the simulation or experiment Chapter 3 Control 56 Chapter 4 Results and discussion 4 1 Introduction In the previous chapters the experimental setup and simulation model as well as a control architecture which combines the exploitation of the natural dynamics with a trajectory tracking
54. es of the centers of mass for the different links are given by OG 1 sin 9 cos 0 A2 OG al sin 0 cos 0 A 3 vo L cos 6 sin 6 6 A 4 vo al cos0 sin 0 A 5 The total potential energy of the pendulum is equal to U mgY tm gY Which results in U g ml e m l cos The partial derivative becomes g m am l sin The kinetic energy K of part i is given by with the angular velocity for part i For the two parts this becomes 1 K m1 8 2 1 K genet L p The total kinetic energy is the sum of these two terms 1 K 5nd m a l np Now the different derivatives of the kinetic energy can be calculated ZE n1 m al 1 p dt 60 OK 0 00 Chapter A Dynamic model of the pendulum A 6 A 7 A 8 A 9 A 10 A 11 A 12 A 13 92 The torque 7 applied in the pivot point represents the generalized force So the equation of motion for this model can be summarized as followed r D 0 C 0 0 0 G 0 A 14 With D 0 m l m a7l I C 0 0 0 A 15 G 0 g m l m a L sin 0 Chapter A Dynamic model of the pendulum 93 Appendix B Safety Board 0 L 3 9 L O W tr TALLNAA SONLLHOMTILNO N3dd3 5 px 14018 N3dd3 Tt TALLNAA SSNIQ3O LININYY ONILHONTLNO zor LOOPNE i m AICUNVV SNIGSOA M31H3ANOOD 32019 13534 T bLbins OrS9NG3H M3005N3 23V001232 N
55. etween measured data and theoretical model of the force was observed For dimensioning purpose the theoretical model can be used However for the trajectory tracking control an accurate estimation of the real force function is required Since the force functions of different muscles are very similar a 4 order polynomial function fit on the pressure scaled measured data was performed in order to achieve a better force estimation The force function can be expressed as F pl f e pK E he foet fi foe 2 2 Chapter 2 Design 11 Finally for reason of programming convenience a polynomial fitting was performed on the theoretical data for the enclosed muscle volume V e Ev e lo v46 46 v4e v E ve v 2 3 Those equations are much easier to handle than the numerical solution derived from the mathematical model as derived by Daerden and Verrelst In table 2 1 and table 2 2 the coefficients of respectively the force and volume fitting are given The values are valid when the generated force F is expressed in N the initial muscle length in m the pressure expressed in bar the volume given in ml and the contraction expressed in Both polynomial fittings are shown in Figure 2 4 fo fi h f fa 146099 128611 6 7178 93 171 623 2 0413 Table 2 1 Coefficients of the polynomial force function approximation Vo Vi V5 V3 V4 Vs 71728 30080 2386 3 113 82 2 6
56. eviously are too complex and too heavy for an application such as an autonomous bipedal robot Therefore an interesting alternative is the pleated pneumatic artificial muscle PPAM developed at the Robotics and Multibody Mechanics Research Group It is believed that these pneumatic actuators have interesting characteristics which can be used in the field of legged robotics In this context the development of a planar walking robot actuated with pleated pneumatic artificial muscles PPAM was started The robot has been given the name Lucy figure 1 3 It weighs 30 kg and is 150 cm tall It uses 12 muscles to actuate 6 pin joints as such Lucy is only able to walk in the sagittal plane A sliding mechanism prevents the robot from falling side wards The goal of this project is to create a lightweight biped which is able to walk in a dynamical stable way while exploiting the adaptable passive behaviour of the pleated pneumatic artificial muscles in order to reduce energy consumption and control efforts The robot is currently able to move in a quasi static way but stiffness in the joints is still set constant 11 14 Figure 1 3 The bipedal walking robot Lucy The main purpose of this master thesis is to investigate the adaptable passive behavior of the Pleated Pneumatic Artificial Muscle which allows the stiffness Chapter 1 Introduction 3 of a joint actuated by two antagonistically coupled muscles to be varied onlin
57. g a variable p for tracking trajectories with not only one frequency component has been showed Using a variable p will result in a variable mean of the pressure in each muscle separately Therefore the antagonistic setup should be designed in such a way that p stays between a certain range in order to limit the gauge pressure in the muscles between 0 bar and 3 5 bar Or in the case of the biped Lucy the generated trajectories should take into account those limits if natural dynamics should be exploited Chapter 4 Results and discussion 74 4 3 The physical pendulum 4 3 1 Evaluation of the joint trajectory tracking controller In the previous section the joint trajectory tracking controller was evaluated by means of computer simulations The physical pendulum was built in order to perform some tracking experiments and to adapt the simulation model according to the experimental data In this section the performance will be discussed and an overview will be given of which parameters differs from the simulation model A correct estimation of these parameters was not done but the importance should not be neglected He SERERE euin Measured Desired Angular velocity s Total torque CT model part P action action D action Torque Nm iet EE NAW AANA IAA eae s Pressure 1 bar N a
58. gd traject terwijl de stijfheid van de structuur zodanig wordt ingesteld dat de slingerbeweging volledig binnen de natuurlijke dynamica van het systeem valt De slinger is gebouwd en de bijhorende hardware componenten zijn getest Deze thesis geeft een beschrijving van het ontwerp en de bouw van de slinger De invloed van de variabele stijfheid op het energieverbruik en de kleppenacties wordt ge llustreerd aan de hand van simulaties en experimenten op de werkelijke slinger Een techniek om de optimale stijfheid langsheen een opgelegd traject te bepalen en te vari ren werd onderzocht De volgende stappen binnen dit onderzoeksgebied zijn de correcte bepaling van de parameters van het systeem en een uitbreiding naar een driedimensionale slinger om de opgedane kennis te kunnen gebruiken voor verdere ontwikkeling van de tweepotige stappende robot Lucy R sum Contr le bas sur un mod le d une pendule actionn e par des muscles artificiels pneumatiques pli s avec une raideur adaptable Gr ce la compressibilit de l air et de la caract ristique force contraction d croissante du muscle une jointure actionn e par deux muscles antagonistes pr sente un comportement compliant dont la souplesse peut tre control e ind pendamment de la position Cette souplesse adaptive permet de b n ficier du r gime naturel du syst me pour ainsi diminuer la consommation d nergie et l effort de contr le L objectif de cette th
59. gher the electric power consumption price and weight will be Simulations of the pressure control on a constant volume were done by Van Ham et al in the context of the development of Lucy and this led to the compromise of 2 inlet and 4 outlet valves The different number between inlet and outlet comes from the asymmetric pressure conditions between inlet and outlet and the aim to create equal muscle s inflation and deflation flows For detailed information on the simulations is referred to 13 Figure 2 12 View ofthe valve island Chapter 2 Design 25 The 6 valves are brought together in a valve island with special designed inlet and outlet collectors after removing parts of the original housing material A photograph of the valve island is given in figure 2 12 The total weight of this device is less than 150 g 14 Particles of epoxy present in the muscles and other dirtiness can affect the good working of the valves To protect the valves a filter gauze is placed at the supply connection of the valve island and between the muscle and the valve island Leaks were detected there where the electric wiring is passing through the valve island To prevent the air to escape through these holes a supplementary rubber joint is placed between valve and valve island 2 2 3 3 The pneumatic circuit Figure 2 13 gives an overview of the pneumatic circuit which is used to regulate the supply pressure of the different mu
60. gonistic setup to hold a mass of 10 kg at the maximum of the angle range Therefore static torque characteristics of the joint should be compared with the torque needed to hold a mass of 10 kg e Stability has to be insured The torque characteristic of the joint with closed muscles is important here As the stiffness of the joint is given by K Z this characteristic has to be increasing K lt 0 In that case the joint stiffness is positive If that is not the case the swing motion will be amplified and instability will occur Besides this within the angle range we want this torque characteristic to be as linear as possible The joint stiffness will then be almost constant within our working domain Chapter 2 Design 19 e The diameter of the muscles should be taken into account during the design in order to provide enough space for the muscle to bulge The joint range is determined via the minimum and maximum angles at the minimum and maximum contraction of the muscles Angle range and torque characteristics are determined as a function of the joint application The design of all these functionalities is complex and is linked to the specific motion of the pendulum A small Matlab program has been written to investigate the influence on the joint characteristics by changing the design parameters The characteristics are shown in Figure 2 7 and 2 8 Figure 2 7 shows the contraction of one muscle in function of the joint angle Ma
61. gy consumption More information on how energy is being calculated will be given in the next chapter e SlingerUl Figure 2 23 The Valves Test tab of the GUI Chapter 2 Design 38 As we experienced a lot of problems making the valves island to work properly a last tab 1s used to test the valves Input and output valves can be tested by choosing the desired valve and to start the test An overview of the valves test tab is given in figure 2 23 2 5 Conclusion In this chapter the hard and software needed to construct the pendulum was described By analyzing the joint characteristics the muscles and actuators connection parameters were determined To acquire the pressures in the muscles and the joint angle and to control the valves a data acquisition card is used The control of the pendulum is done by a PC To make the experimental setup user friendly a GUI was implemented Chapter 2 Design 39 Chapter 3 Control 3 1 Introduction The main purpose of this work is to incorporate the exploitation of the natural dynamics by adapting joint stiffness in combination with trajectory tracking In section 3 2 the main focus is on the development of a tracking controller which incorporates the actuator characteristics and dynamic model of the robot The control strategy consists of a multilevel construction of several essential blocks trying to cope with the non linear structure using model based feedforward techniques
62. h means that a lot of valve action will be taken by the bang bang controller as shown at the bottom of figure 4 15 The valves are switching a lot and most of the time the 4 outlet or 2 inlet valves are opened in order to track the pressure course With regard to energy consumption this is unacceptable First tracking experiment 50 i i F B Measured A A A A A A A f A iu e an AIV PINE VV EET VVVvvVvuvvVvuUNVJVV VV S lt E e P action 5 laction E D action l 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 10 time s 3 5 T T T T T T T I I T g j i i i Pressure muscle 1 I 2 3r hp gr op Hg gr d go yp mr m ar Desired pressure Valve action g 2 5 7 F1 A EN A AN FA 2 2H A AN AA AA AA AN AA A AA AA ANN A PNA NN EWM PVPDPA gt PAPIA PAPI IIS 1 5 i i i i 1 i i i 0 1 2 3 4 5 6 7 8 9 10 time s Figure 4 16 Angular velocity computed torque and absolute pressure for a tracking experiment by using a filter on the angular velocity To avoid such problems a first order digital filter was used on the angular velocity A bilinear transformation was done in order to convert an analog to a Chapter 4 Results and discussion 76 digital filter The bilinear transformation is a mathematical mapping of variables In digital filtering it is a standard method of mapping the s or analog plane into the z or digital pla
63. he clock converter transforms these two signals into CLK and UP DOWN The pulse signal CLK is connected to Programmable Function Input PFI GPCTRO SOURCE and the direction signal UP DOWN is connected to the UP_DOWN DIO6 input The DAQ card is configured for a simple event count with hardware up down control In this configuration the DAQ card will control the counter in such a way that it will count up down on every transition of either A or B Each valve island has 6 valves to be controlled so 8 digital signals are needed Because one of the digital channels is already occupied by one of the signals of Chapter 2 Design 34 the encoder an analog signal is used The valves are connected through DIO1 DIO8 except from DIO6 The remaining valve is connected to DACOOUT DACIOUT is used to determine the control time of the whole set up by means of a scope The remaining 24 bit counter timer is configured in such a way that an internal base clock of 100 kHz is used as timer The time the encoder counter and sensors values are read at the beginning of every control loop and valve states are changed if necessary 2 3 5 Safety supply and transformation board Emergency stop Pressure connection sensors connection wate DE e connection P Clock converter 2 2 valve supply 5 V Speed Up circuitry supply cix 3 J JT 3 2 valve supply 24 V Speed Up circuitry supply pei e 5 V board sup
64. he same as used for Lucy except from the AD converter Since this sensor is inside the muscle volume an absolute pressure sensor is provided In order to pass through the entrance of a muscle the size of the sensor and its electronics has to be small 12 mm An absolute pressure sensor CPC100AFC from Honeywell has been selected for this purpose The sensor measures absolute pressure values up to 100 psi 6 9 bar and has an accuracy of about 20 mbar The output of the pressure sensor is amplified by a differential amplifier As the pressure sensor outputs an analogue signal special attention should be given to possible noise interferences High precision and accuracy is required so a cable consisting of three wire signal leads is used The third signal lead or shield is necessary and is grounded at the signal source to reduce common mode noise In section 2 2 4 more information on how this noise was reduced is given AD Converter Differential Pressure not used Amplifier Sensor Figure 2 18 View of the pressure sensor Because the electronics of the pressure sensors are handmade they differ from each other That s why a calibration has to be done By doing this offset and sensitivity of each pressure sensor are determined so that at the same pressure Chapter 2 Design 31 level the same voltage is output for both sensors For several pressure levels the output voltage of the left and right pressure sensors has been
65. hysical pendulum In the first experiment the simulation results showed the importance of setting the right joint stiffness in order to reduce control effort and energy consumption In the second experiment the mathematical formulation described in section 3 3 was evaluated Therefore the constant stiffness obtained by this method was compared with the optimal stiffness found by tuning the stiffness manually We concluded that the mathematical formulation can indeed be used to estimate the optimal stiffness when a sine wave with one frequency component is imposed In order to exploit the natural dynamics for trajectories with more than one frequency component the stiffness was changed online in experiment 3 A sine wave trajectory with linear varying frequency was imposed By using the method of varying stiffness the control unit was able to track the desired trajectory in combination with the exploitation of the natural dynamics The experiment on the physical pendulum was performed to find out if energy consumption and control action can practically be minimized by setting the appropriate stiffness and to determine the value of the optimal stiffness For Chapter 4 Results and discussion 86 different trajectories the optimal stiffness was found By comparing the experimental values of the stiffness parameter with those found in the simulation we were able to conclude that the method of the pressure slopes can be used to approximate the
66. ibed previously 2 3 4 Data acquisition card The AT MIO 16E 10 data acquisition card has 16 single ended or 8 differential analog inputs with a sampling rate of 100 kSamples s 2 analog output channels and 8 digital I O channels built in A Data Acquisition System Timing Controller DAQ STC is available The DAQ STC includes 10 counter timer devices eight of which are designed to control the timing of analog input and analog output operations The remaining two counter timers are 24 bit up down counter timers available for a wide variety of timing and counting applications and will be used for retrieving information on the joint angle and also used as an internal clock Each of the two 24 bit counter timers of the DAQ STC includes three input signals SOURCE GATE and UP DOWN and two output signals OUT and INTERRUPT The DAQ card is installed in a Pentium 500 Mhz computer When programming the DAQ hardware any application development environment ADE can be used but in either case a NI DAQ driver must be installed This driver has an extensive library of functions that can be called Chapter 2 Design 33 from the ADE Those functions can be found in 20 The driver also carries out many of the complex interactions such as programming interrupts between DAQ and computer In the beginning Matlab was used to program the DAQ card but complicated programming and low sampling rate made us change our mind and we finally decided to use
67. kinematics of a one dimensional joint setup controlled by two antagonistically positioned muscles are reported Special attention is devoted to the design of the actuator connection The electronics needed to control the pressures in the muscles will be explained in the second paragraph electronic design In the third paragraph software a user manual of the guided user interface will be given so that the experimental set up can be easily used by other persons Constructing the pendulum by just one person would have taken too much time in the context of a master thesis Therefore my colleague Pieter Beyl and I worked as a team As a consequence all the work except from the GUI referred to in this chapter has been carried out together 2 2 Mechanical Design 2 2 1 Pleated Pneumatic Artificial Muscle The PPAM consist of a membrane made of an aromatic polyamide such as Kevlar to which a thin liner of polypropylene is attached in order to make the membrane airtight The membrane of this muscle is arranged into radially laid out folds that can unfurl free of radial stress when inflated Reinforcing high tensile Kevlar fibers are positioned in each crease The high tensile longitudinal fibres of the membrane transfer tension while the folded structure allows the muscle to expand radially which avoid energy losses and hysteresis The folded membrane is positioned into two end fittings which close the muscle and provide tubing to
68. limiter Connection plate Figure 2 11 View ofthe connection plate and the position limiter At the joint rotation side an angular position limiter is provided This device is equipped with two screws which can be regulated separately in order to set the joint rotation range The limiter is used for the two following reasons e Avoid singular joint configurations in the pull rod and leverage mechanism This happens when the axis of the muscle is aligned with the joint rotation point and the muscle attachment point in the leverage mechanism In this situation the muscle can seriously damage the leverage mechanism when increasing pressures would by required by the controller e Limit the contraction between 5 and 35 94 Chapter 2 Design 24 2 2 3 2 The valve system The force developed by the muscles is proportional to the pressure into the muscle Therefore a rapid and accurate pressure control is needed to set stiffness and position Fast switching on off valves are used The pneumatic solenoid valve 821 2 2 NC made by Matrix weighs only 25 g They have a reported switching time of about 1 ms and flow rate of 180 Std l min The permitted pressure difference over the valve ranges for each type between 0 6 bar and 2 8 barrespectively To pressurise and depressurise the muscle which has a varying volume up to 400 ml it is best to place a number of these small on off valves in parallel Obviously the more valves used the hi
69. llowing expression for the computed torque is thus obtained F C 0 0 0 G 0 Do K 0 6 K 0 6 K oyd 3 5 The required pressures values to be set in the muscles are then calculated by the delta p control unit These two gauge pressures of both muscles are generated as follows zobs yA 3 6a B Fg Re In Ps AJ P 73 6 Ap 3 6b with p a parameter that is used to influence the sum of pressures and consequently the joint stiffness Ap influences the difference in pressure of the two muscles and consequently the generated torque The functions 7 and 0 are calculated with estimated values of the muscle force functions and geometrical parameters of the pull rod and leverage mechanism of the actuation system Expression 2 5 allows linking the required torque to the required pressure values in the muscles T p 0 p t 8 5 0 0 Ap 3 7 If the calculated pressure values p and p of equations 3 6 are set in the muscles the generated torque depends only on Ap and is independent of the joint stiffness parameter p in case the modelling would be perfect This means that joint stiffness is changed without affecting the joint angular position Chapter 3 Control 43 Feeding back the knee angle and introducing the torque 7 expression 3 7 can be used to determine the required Ap 7 PROFEO n The delta p unit is actually a feedforward calculation from torque level to
70. loited during the whole simulation this by making p variable 10 4 I I Frequency I 1 55 1 6 1 55 17 1 75 1 8 1 85 19 1 55 2 Variable P A I 3 5 A d 1 yH f LIN A N Y A v vi i Y T hj v s A y 2 ir 25 B N ENA S R IW Y E TENA E A LN j X m ug gi LLLI EL HE ER E d M AZ MEL IE E E EE ELE IEIEENEILIHS 5 A fw 1 j LAT A A Mu f v amp A v a f f y 1 5 fy b m n v NSS N X 1 e 4 valve action pressure required pressure 0 5 I I I I I 1 2 3 4 5 6 7 8 9 Time s Figure 4 11 Actual and required pressure and valve action for a trajectory with linear varying frequency and p online changed 4 Frequency 1 55 1 6 1 65 1 7 1 75 1 8 1 85 1 9 1 95 2 Constant P 21 Nm 3 5 4 3h 4 T p 25r 4 2 Wt 11 _ l l MU L HL MAL O w O Me LA 5 21 1 m m m WW u UT Lr ul Tata ll Uu Ue J A A B AA AAT A A Aj f N N h wae l f V fh J WEM ATA I 1 i V V VU VU VEMM AMETE M v V V V V YY n V oy V 4h Voy v y V y valve action m pressure required pressure 0 5 I I 1 2 3 4 5 6 7 8 9 10 Time s Figure 4 12 Actual and required pressure and valve action for a trajectory with linear varying frequency and a constant ps Chapter 4 Results and discussion 72 Because the required oscillation freque
71. lve f mo 10 Openallinlet valves Table 4 4 Pressure levels of the bang bang controller for the physical pendulum 4 3 2 2 Results and discussion In the context of the exploitation of the natural dynamics one experiment was done on the physical pendulum The main goal is to find out if energy consumption and control action can be minimized by setting the appropriate stiffness and to determine the value of the optimal stiffness Therefore the stiffness parameter p was varied and for different trajectories energy consumption and valve action were calculated The same experiment was done Chapter 4 Results and discussion 81 in the computer simulation model of which results were already shown in figure 4 8 Remember that in the simulation model the optimal p and the p found with the mathematical formulation given by Verrelst were compared We concluded that the method of the pressure slopes was suitable for sine trajectories with one frequency component as the two values were almost the same In the case of the physical pendulum energy consumption and valve action defined by 3 23 for one period of different trajectories are given in figure 4 19 Total input exergy vs P Total output exergy vs P 25 12 1 5Hz 1 5 Hz 20 1 75 Hz 10 1 75 Hz 2 Hz 2Hz a Exergy J Exergy J o a 0 10 20 3
72. men tot een zelf bevredigend resultaat Dit afstudeerwerk is uiteraard niet zonder de hulp en steun van anderen tot stand gekomen en daarom dit dankwoord Ten eerste wil ik mijn promotor Prof Lefeber en co promotor Prof Kool bedanken om mij de mogelijkheid te geven tot deze eindverhandeling Bram Vanderborght en Bj rn Verrelst zou ik willen bedanken voor de begeleiding de goede raad de hulp bij praktische problemen en vooral voor het nalezen van de eindverhandeling Een groot deel van deze thesis werd verwezenlijkt in samenwerking met mijn klasgenoot Pieter Beyl We hebben dan ook vele uren samen doorgebracht in het labo en zou hem willen bedanken voor de vlotte samenwerking Voor de technische hulp kon ik steeds terecht bij Ronald Van Ham Jean Paul Schepens Andr Plasschaert Thierry Lenoir en Gabri l Van den Nest en dank hen hiervoor Joris Naudet en Michael Vandamme wil ik bedanken voor hun hulp bij het programmeren Verder zou ik nog mijn vrienden willen bedanken voor hun aanwezigheid en het plezier dat we doorheen de jaren aan de VUB hebben beleefd Thomas ik ben je enorm dankbaar voor je hulp bij het maken van de User Interface Nathalie jou wil ik speciaal bedanken niet alleen voor het nalezen van mijn proefschrift maar ook voor je steun en de vele koffiepauzes die mij er steeds weer bovenop hielpen Papi y Mami os quiero agradecer por vuestro inter s apoyo y amor durante toda mi carrera universitaria Sin vosotros n
73. nce of Chapter 4 Results and discussion 60 the force function errors The model based part of the computed torque differs substantially of the calculated torque due to the introduction of deviations on the mass and inertia The servo portion has to take more action as the model based part is using wrong parameters Perfect model muscle 1 2 2 T T I valve action 2 pnr ee a poy E required pressure bar 1 L 0 5 1 1 5 2 2 5 Time s Detail perfect model muscle 1 2 2 T T T T T T I I I valve action 2 a a pressure bar required pressure bar 1 l i l I i i 0 5 0 55 0 6 0 65 0 7 0 75 0 8 0 85 0 9 0 95 1 Time s i L Figure 4 3 Muscle 1 pressure with valve action for perfect model Figure 4 3 depicts required and actual gauge pressures and valve actions taken by the bang bang controller for muscle 1 A closed valve is represented by a horizontal line at the 2 bar pressure level An open inlet valve is represented by a small peak upwards while an open outlet valve is represented by a downwards peak Larger peaks means that 2 inlets or 4 outlets are opened From figure 4 3 we can conclude that the bang bang controller is able to track the desired pressure quite accurately No action taken by the valves means that the pressure error Poo is situated in the dead zone of the bang bang controller This is the case between 0 708 s and 0 826
74. nces But of course setting the ranges in which the natural motion corresponds to the calculated trajectories required for dynamic stability asks for a proper design of all inertial and joint design parameters So that in the end a combination of these two different approaches will give interesting results In this context two master theses were proposed by the Robotics and Multibody Mechanics Research Group of the Vrije Universiteit Brussel In this report the model based approach will be discussed My colleague Pieter Beyl will carry out a more biological approach by using Fuzzy Logic controllers artificial neural networks ANN central pattern generators CPG Look Up Tables or combinations of all strategies mentioned above As we both needed an experimental set up to check our simulation results the first part of our thesis consisted of designing and constructing the pendulum Therefore the following steps were executed e Determination of the dimensions of the leverage mechanism and consequently the joint torque characteristics e Building testing and debugging the electronic components e Building a program library used to send and receive data by use of a data acquisition board Chapter 1 Introduction 6 e Assembling the pendulum e Implementation of the joint trajectory controller e Writing a user interface to communicate with the pendulum and extract the data information e Implementation of a controller for setti
75. ncy is increasing energy has to be injected into the system This explains why in figure 4 11 more inlet valve actions are taken during the whole trajectory Once the frequency remains constant the inlet valve actions will decrease since no additional energy has to be injected Note that the pressure is increasing as the frequency of the imposed trajectory increases As mentioned before the required stiffness raises as the frequency raises which means that the sum of pressures in the muscles has to increase This also means that p increases as it represents a parameter to influence the sum of the pressures The evolution of p along the trajectory is shown in figure 90 I I T T T T T P calculated Nm P Nm 807 p Nm NN T P calculated 50 deviation Nm j 70 P calculated 50 deviation Nm J AM 60 M NM IL AAV E a Nii Vy AO ee yy i V 50r Ae a RU adi La E a AAA EIT ARA Z AANT RY m z A hi m 1 h h A J you amp 40r v d AN IV M AY id HARI MS 4 4 AAL YI AAV VS s k UE Ley Wir VIA V LO 4 VIRA AN YY VA AIV E AJV Vv pr peer 0 L L L L ji L ji 1 5 1 55 1 6 1 65 1 7 1 75 1 8 1 85 1 9 1 95 2 Frequency Hz Figure 4 13 Ps in function of the frequency of the imposed trajectory Deviations on p were introduced to investigate the effect on the valve control when the online calculation of p is not giving the correct value This can be interesting t
76. ne It transforms analog filters designed using classical filter design techniques into their discrete equivalents 19 The first order analog filter used on the physical pendulum is given by 1 H s Fa 4 6 QO Pre warping the filter design gives 2 o IT tan 4 7 T 2 with T the sample time and the cut off pulsation of the desired digital filter given by iU Om icy 4 72 fuc 1047 4 7b T 2ms 4 7c The bilinear transformation is given by H z H E 4 8 4 9 Chapter 4 Results and discussion 77 With Ay e ps 4 9a TQ TO b cb sl 2 poer i Ws To 4 9b The introduction of a filter greatly improved the performance of the joint trajectory controller Figure 4 16 shows the angular velocity computed torque and the absolute pressure when a filter is used Very remarkable is the effect of the filter on the valve actions and is a first step in the correct direction concerning energy consumption The gains used by the servo portion of the computed torque were manually tuned in such a way that stability is still ensured and control actions are minimized without affecting the tracking accuracy As was explained in chapter 3 it is important to have a good estimation of the model parameters in order to reduce the servo portion to a minimum No friction model was used during the experiment Also the existing of small air leaks in the muscles and tube connection
77. nergy consumption A method to estimate the energy consumption was given in chapter 3 Of course this will not give us the correct value but it is interesting to have an idea of the energy consumption and control action taking place during the experiment If a correct estimation is needed Chapter 4 Results and discussion 80 measurement of the airflow air temperature and muscle volume will be needed In our case we only want to investigate whether the PPAM can be used to exploit the natural dynamics of the system and therefore the formulation from chapter 3 will do The calculation of the energy is only started after two seconds of acquisition when the pendulum is in regime As mentioned before the dead zone of the bang bang controller has an important influence on energy consumption Changing the pressure levels of the bang bang controller will increase or decrease valve actions In order to investigate only the influence of the stiffness on the energy consumption and valve actions these pressure levels were set to constant values given by table 4 4 These values were manually tuned so that control effort is minimized by taking into account the trajectory tracking accuracy P 2 5bar P gt 2 5 bar Valve action Pon mbar p mbar a 120 120 Open all exhaust valves b 60 30 Open one exhaust valve c 40 25 Close all exhaust valves d 40 25 Closeall inlet valves e 60 30 Open one inlet va
78. ng the joint stiffness in order to incorporate natural dynamics e Measuring and analyzing the results The first five tasks were done together with my colleague Pieter Beyl The remaining tasks were carried out by myself since a completely different approach was chosen 1 4 Outline This thesis proceeds as follow Chapter 2 Design gives an extensive description of the mechanical design with the joint torque characteristics as an important discussion The electronic design needed to control the valves which set the pressures in the muscles is also given Those pressures define the joint stiffness and position A manual guide of the user interface is given in order that further research on this topic can be done by other people Chapter 3 Control reports on the controller used for tracking a desired trajectory in combination with exploitation of the natural dynamics The tracking control strategy consists of a multilevel construction of several essential blocks trying to cope with the nonlinear structure by using model based feedforward techniques The method used to take into account the natural dynamics will be formulated In order to evaluate this control architecture a computer simulation model was built A description of the modelling of the robot dynamics and the thermodynamics of the muscle valve system is given followed by an overview of the complete simulator Some energy consideration will be given so that the method pro
79. nitial angle is reached a certain settling time is needed for the pendulum to stay at rest The settling time can seriously be decreased by increasing the stiffness of the joint This means that Chapter 4 Results and discussion 79 p has to be increased during the tracking of the initial trajectory and as a consequence the initial pressure will be at a higher level I I Initial trajectory Required trajecory experiment Joint angle e time s Figure 4 18 Initial trajectory and imposed trajectory during experiments During the experiments an increasing deviation on the joint angle was observed as the time increased due to some miscounts of the encoder signal This was not a problem in the context of this thesis because the acquisition time during the experiments was relatively short But in the future the third signal of the incremental encoder should also be incorporated in order to reset the encoder counter when the pendulum passes trough his reference angle Based on this section we can conclude that the proposed control architecture is able to track sine wave trajectories In order to reduce energy consumption the exploitation of the natural dynamics is required This is shown in the next section 4 3 2 Exploitation of the natural dynamics 4 3 2 1 General considerations We want to investigate the exploitation of the natural dynamics of the system in order to reduce e
80. nsists of a multilevel construction of several essential blocks which try to cope with the system s nonlinearities at separate levels A model based feedforward controller calculates the required torques to track the desired joint trajectories These calculated torques are then transformed into desired muscle pressure levels by a delta p unit coping with the nonlinearities introduced by the muscle actuation system Finally a bang bang pressure controller with dead zone commands the valves in order to set the required pressures in the muscles A mathematical formulation is given in order to incorporate the natural dynamics in the control unit The starting point for finding an appropriate constant stiffness is to fit the natural pressure slopes this is when both muscles are closed with the required ones The better these two fit the less action will be taken by the bang bang controller Consequently energy consumption and control effort are decreased Based on this formulation an additional method is given to change the stiffness online In order to evaluate the tracking controller and the pressure slope method a simulator based on the one used for Lucy is developed Chapter 4 gives an overview of the results that are obtained with the computer simulation model and the practical setup For both tracking experiments are performed in order to evaluate the robustness of the trajectory tracking controller From these experiments the conclusi
81. o 2 main categories on the one hand the full actuated robots which don t use natural or passive dynamics at all like Asimo 17 and QRIO 18 on the other hand the Passive walkers who don t require actuation at all to walk down a sloped surface or only use a little actuation just enough to overcome energy losses due to friction and impact effects when walking over level ground An example of such a robot is the Dutch robot Denise Figure 1 2 The active walker Asimo and the passive walker Denise The main advantage of passive walkers is that they are highly energy efficient but unfortunately they are of little practical use They have difficulties to start can t change their speed and cannot stop in the same way a completely actuated robot can But on the other hand a completely actuated robot consumes a lot of energy As energy consumption is an important issue for bipeds it remains important to exploit the natural dynamics by trying to incorporate the unforced motion of a system instead of ignoring or avoiding it The goal of the Robotics and Multibody Mechanics Research Group is to develop a robot which combines these two categories This robot should be able to adapt the natural dynamics as a function of the imposed walking motion For Chapter 1 Introduction 2 this research a joint with controllable compliance is needed in order to influence the natural dynamics But the variable stiffness actuators mentioned pr
82. o evaluate the method of using a varying p or stiffness as described in chapter 3 This was done by imposing the same trajectory as in figure 4 9 First some comments have to be given on the equations that were used during this simulation In chapter 3 a complete description of the mathematical formulation was given Using these equations led to unsatisfying results the stiffness K associated with the desired trajectory became infinitive in some points Chapter 4 Results and discussion 69 The explanation for this phenomenon can be found by starting from the dynamically required stiffness derived from the equation of motion dT d d8 5 K Belen d dt d dO d p rm 8000 ai 2 ees 4 1 with Acos o t 4 22 w 0 ot 4 2b o Quis Omin 4 2c t simulation The derivatives of the desired trajectory 6 are given by Z A sin o t ot Ko 201 eno Por Acoso vor jo 204 2osineyd vor Ab Ti A coslo t 4 vt Kc 2wt 2aA sin o t oF 4 3b d Asin o I ot Yo 204 40A coslo t ot Vo 2o dt 0 0 0 0 4 3c 2o4cos o cot Yo 2o Thus we have for K K d o t 241 6c cotglo t t e t g cos 0 4 4 We see that a cotg functions arises for this specific trajectory The course of K is depicted in the first plot in figure 4 10 In the second plot the cotg term is given The latter explains why K is being infinitive The cotg term can
83. o hubiera sido lo mismo Os quiero mucho Tenslotte zou ik mijn ouders willen bedanken voor het vertrouwen de steun en de vrijheid die ze me door alle jaren heen gegeven hebben Aan hen heb ik dan ook alles te danken Brussel Bruno Juni 2005 Contents ACO GUC TION ais qu rei ois var Ve dioc ed A e eR uela edd l 1 1 Motivati n and goal wonenden steen TA A ii 1 1 2 Background informationele ede Re Yr er naa 4 137 NPPLOACN eee eiit e od Gc eon ce a Pai abge d 5 I OUIS us e Edessa ear cena Mas Dada it avv RO ete Padus 7 VOSS 1210 EH Tm 8 2 1 IntrodueHon euis reae aee pesi ee An dd eee 8 24s Mechameal DESIBDL o doo erra omes itte des inu UH UR ue quas 9 22 Pleated Pneumatic Artificial Muscle neee 9 2 22 The one dimensional joint set up sssssseeeneee 13 2 2251 TNO CG TOU S ne HOS EA eR tto n nes 13 220 Kinematics of the one dimensional pendulum 13 222 3 Adaptable passive behaviour of a revolute joint 16 2224 Design of the one dimensional pendulum 19 2 23 Constructing the experimental set up nanne eneen 22 2 23 1 The fr me ausa eode eve ante d ith ave ois te a P ORA DU d 22 2 2 2 5 DHS WAVE Sy Stems sisse dea ise el 25 2 2 3 3 The pneumatic CIT CUI se eoe disce bey hat be dieta ues 26 2 2 3 4 The exhaust siletiCeEa soe uo env Stu bt a eiis 27 2 2 3 5 TEA Ta 001 arare ee E 21 2 3 BIGetronie destpilc ans etend 29 2 3 1 Valve
84. o know since we can already expect that the online calculation of the optimal p for the simulation will not necessarily give the optimal p for the physical pendulum The influence on the valve actions is given in figure 4 14 On the left we can see the valve actions during the whole simulation and on the Chapter 4 Results and discussion 73 right a detailed view is given We can draw the same conclusion as before a wrong variable value of p will increase the valve actions Therefore it will be important to have a good model of the pendulum setup such that the method with variable stiffness can be used during experiments Frequency Frequency T 1 6 1 7 1 8 1 9 2 225 173 1735 174 1745 175 E m E 2 LIE EEE EELDE EEE LEE 8 2 JF Df s LI E z Lord DH a p uer d 2 2 a a gt gt 1 5 1 5 2 4 6 8 10 4 5 4 6 47 4 8 4 9 5 Fz i 1 6 17 18 19 2 2 Wes 173 1735 174 1745 175 Hd MOM M M N N A N NN g LLL E MMI e AEE S S rrrrrrETEUTETITIEI B Cpu 0 enm Dui a amp gt gt a 15 a 25 gt 2 4 6 8 10 gt 45 4 6 47 4 8 4 9 5 250716 17 18 19 Wize 133 1735 174 1745 175 H EEEN NNT H TP CN W E cV Ee ann ein a E E E E d 2 4 6 8 10 gt 025 4 6 47 48 49 5 Time s Time s Figure 4 14 Effect on the valve actions due to deviation on the online changing parameter p With this simulation the importance of usin
85. of each figure For the same reason as before the first period is not shown In figure 4 7 the mean pressure is twice as high as in figure 4 6 this because of the difference in value of p A very clear difference can be observed between the two simulations Looking at the valve action we can immediately conclude that the case where p 32 Nm is the most appropriate one to track a sine wave with a frequency of 2 Hz That is what we already suggested before For example between 4 29 and 4 91 seconds no action Chapter 4 Results and discussion 64 p 16Nm p 16Nm 1 5 r r 1 5 T T a a 2 2 v Q Q E z a 3 1 g E M E g valve action g valve action g pressure g pressure An required pressure An required pressure 0 5 0 5 1 2 3 4 5 4 42 44 4 6 4 8 5 Time s Time s 1 5 r r 1 5 T T a A A a 2 j 2 N N vo vo E E s 1 a3 yA E v v i M M x g valve action g valve action g pressure g pressure An required pressure An required pressure 0 5 i 0 5 1 2 3 4 5 4 42 44 4 6 4 8 5 Time s Time s Figure 4 6 Actual and required pressure in muscle 1 and 2 for p 16 Nm p 32Nm p 32Nm 2 5 T r 2 5 r T T 5 Ss 2 2 o o E E 5 3 2 E E
86. on can be drawn that the proposed control structure is indeed suitable to track imposed trajectories Great importance is attached to the incorporation of the natural dynamics of the system First of all the simulation model is used to show the influence of setting a constant suitable stiffness on the energy consumption and control effort when a sine wave with only one frequency component is imposed The optimal stiffness values are found for different frequencies of the imposed sine wave These values are then compared to the constant values obtained with the method of pressure slopes Based on these results we can consider that the method can be used for setting the suitable stiffness We can draw the same conclusion for the method of changing the stiffness online This is interesting for further research on Lucy Chapter 5 General conclusions and future perspectives 89 since the required trajectories are not pure sine waves and thus an online varying stiffness will be needed in order to exploit natural dynamics of the complete robot Experiments on the physical pendulum are performed in order to find the optimal stiffness for imposed sine wave trajectories with one frequency component These optimal values are compared with the simulation values The results are very promising as both values are situated near to each other This means that a method is found to control the stiffness of the physical pendulum in order to exploit the natur
87. optimal stiffness A better parameter estimation of the pendulum s model and the bang bang controller s pressure level will decrease the error made on the estimation of the stiffness Once these parameters are correctly estimated we should be able to vary the stiffness along the trajectory Therefore adapting the pendulum model to all experimental data and tuning the bang bang controller is the next step to take in order to completely exploit the natural dynamics Chapter 4 Results and discussion 87 Chapter 5 General conclusions and future perspectives This thesis reports on the design construction and control of a one dimensional pendulum actuated by a pair of pleated pneumatic artificial muscles PPAM Pneumatic artificial muscles have very interesting characteristics towards exploitation of natural dynamics of the system Due to the compressibility of air and the dropping force to contraction characteristic a joint actuated by two antagonistically positioned muscles has a compliant behaviour which can be adapted while controlling position The main purpose is to use this adaptable joint compliance to combine exploitation of natural dynamics combined with joint trajectory control in order to reduce control effort and energy consumption significantly The joint trajectory controller tracks the imposed trajectory while changing the joint stiffness as such that the natural trajectory corresponds as much as possible to the reference t
88. or control This system acts like two coupled pendula The stance leg acts like an inverted pendulum and the swing leg acts like a free pendulum attached to the stance leg at the hip Given sufficient mass at the hip the system will have a stable trajectory that repeats itself and will return to this trajectory even if perturbed slightly McGeer also constructed a physical example His research was continued by the group of Andy Ruina 3 They studied the models of McGeer in more detail and extended the two dimensional model to three dimensions while building several Passive Walkers Using natural frequencies of the robot is a useful concept for energy efficient walking However the natural frequency of a system is defined by inherent system properties like dimensions mass distribution collision behavior and joint compliance As a result the eigenfrequency of these models is set during construction which limits the different passive walking patterns and walking speeds In order to change the natural frequency of the system passive elements with variable compliance were implemented In this context the group of Takanishi developed the two legged walker WL 14 where a complex non linear spring mechanism makes changes in stiffness possible A more elegant way to implement variable compliance is to use pneumatic artificial muscles Stiffness can easily be changed by changing the applied pressures Research on this topic was done by van der Linde
89. ot oriented and always positive e l is the actual length of muscle i e is the length of the base suspension bar measured between the origin O and the pivot point R e represents the rotation angle measured between RC and the Y axis is oriented counter clockwise is positive The generated torque can be found by combining the traction force generated by the muscles and the orthogonal leverage arms 7 and 7 T 8 T 0 T 0 F 0 n 0 F 0 r 0 Pilo f 0 0 Palo f 0 r 0 2 4 Pit A Pot 0 2 5 with and 0 the torque functions depending on the angular position of the joint To calculate the force functions we will use the polynomial function fitted on the measured force data as described in the first paragraph Chapter 2 Design 15 The vectors B D and RD are being calculated with following equations B D b d sin a 0 l d cos a 6 2 6 B D d sin a 0 b 1 d cos a 0 2 7 RD d sin a 0 d cos a 2 8 RD d sin a 0 d cos a 0 2 9 The expression for 7 can then be found as B D x RD BD The muscle contraction as function of the rotation angle is given by D REN BD B D 0 1 ef gf 2 11 lo li Li The contraction is defined with respect to 2 which is the contraction of muscle i at a chosen central position In our case 0 0 The parameters are
90. ply y r pd ves AlSense p1 p2 12 V pressure sensors supply UP D WN Alarm CLK Signal Figure 2 20 View of the safety supply and transformation board Chapter 2 Design 35 The maximum absolute pressure in the muscles without damaging them is about 4 5 bar The supply pressure however is 7 bar If working properly the pressure is limited to 4 bar by the controller If a problem occurs with the controller the pressure can exceed this limit and the muscles will be damaged To avoid such a situation and to protect the muscles against high pressure we implemented a safety circuit which handles all the alarm signals originating from the pressure sensor inside the muscle and an emergency stop Whenever an alarm signal is activated the 3 2 supply valve of the two pressure regulating pneumatic circuits is closed while the depressurising valve is opened in order to deflate all the muscles The safety supply and transformation board showed in figure 2 20 consists of the safety circuit the connection to all voltage supply sources and to the signals from the pressure sensors and encoder On this board the encoder signals are also transformed by the clock converter 2 4 Software As mentioned before Visual C is used to program the DAQ card To make experiments and testing more user friendly a graphical user interface GUI was developed A brief explanation on how to use the GUI will now be given The
91. posed for incorporating the natural dynamics can be evaluated Chapter 1 Introduction 7 Results and discussions on both the simulation model and the physical pendulum are given in chapter 4 For both cases the performances of the joint trajectory tracking controller will be discussed In the context of the exploitation of the natural dynamics simulation and experimental results will be given Finally a comparison between simulation and experiments on the real pendulum will be discussed in order to conclude whether or not an online changing stiffness can be used for further implementation on Lucy General conclusions are drawn in chapter 7 followed by the planning of future research in the domain of exploitation of natural dynamics Chapter 1 Introduction 8 Chapter 2 Design 2 Introduction The main goal of the pendulum is to investigate whether the PPAM can be an interesting actuator for exploitation of natural dynamics in combination with trajectory tracking The pendulum set up has been designed for that purpose and should be suitable for experiment on the following main items e Evaluation of a trajectory tracking control strategy with the specific pneumatic system e Evaluation of the adaptable passive behaviour of the PPAM and the influence on the natural dynamics of the system This chapter starts with a description of the mechanical design of the pendulum The PPAM and its characteristics will be discussed The
92. pressure level using the cinematic model of the muscle actuation system The calculated Ap affects the torque required to track the desired trajectory while Ps is introduced to determine the sum of the pressures which influences the stiffness of the joint as was discussed in section 2 2 2 3 Increasing p lowers the compliance of the joint In the last control block the desired gauge pressures calculated by the delta p unit are compared with the measured gauge pressure values after which appropriate valve actions are taken by a bang bang pressure controller We define the pressure error as p p p with p the desired pressure calculated by the delta p unit and p the pressure measured in the muscle The bang bang pressure control scheme is given in figure 3 2 Action Perror Figure 3 2 Bang bang pressure control scheme If the pressure difference stays between c and d no valve action takes place dead zone and the muscle stays closed In this situation the muscle is acting as a compliant passive element If the pressure error is increasing and reaches level e one inlet valve is opened in order to make the pressure rise to the desired Chapter 3 Control 44 level If one opened inlet valve is not enough and p is still increasing and becomes larger than f a second inlet valve is opened The inlet valves are closed again when the difference drops below level d The same approach is used for negative values of p
93. r bar bar OSCUM E Subscript air atm d i centre of gravity of part i of the pendulum gravitational torque force vector moment of inertia of part i joint stiffness function for a joint setup joint stiffness kinetic energy initial muscle length base suspension bar length of the muscle joint setup length from pivot to part i actual length of muscle 7 in the joint setup air mass flow through valves of muscle i mass of part 7 number of fibres polytropic exponent gauge pressure stiffness parameter of delta p control unit initial gauge pressure when muscle 7 was closed initial absolute pressure when muscle i was closed generalized coordinate generalized torque associated to q leverage arm of muscle i in the joint setup rotation point of the joint muscle setup time torque function of muscle 7 in a joint setup joint torque temperature torque generated by muscle 7 in the joint setup potential energy dimensionless volume function coefficients of polynomial muscle volume fitting enclosed volume of muscle initial volume when muscle i was closed work of compressed air atmospheric derivative muscle number kgm Nm bar BERBERS bar bar bar m isotherm S Superscript part number at isothermal conditions sample analog digital of exhaust valve of inlet valve of compressed air supply required value calculated with estimated parameter values derivative with respect
94. rajectories The investigation on this theme is done on both a computer simulation model and a physical pendulum The latter together with its soft and hardware had to be designed and constructed A lot of time was spent on testing and debugging the different components Chapter 2 discusses both mechanical and electronic design required for a well working physical setup The mechanical design focuses on the actuator connection as it defines the joint characteristics of the antagonistic muscle setup and consequently the compliance Therefore a formal description of the joint kinematics is given and it is shown that the joint position is influenced by differences in both muscle pressures while the compliance of the joint with closed muscles is set by a weighted sum of pressures The basic frame supporting structure and pneumatic circuit are assembled to form one modular part of Lucy In order to control the pressure in a muscle fast switching on off valves are used The electronics required to action these valves together with the pressure sensors and the incremental encoder are tested and described A data acquisition card serves as a data transfer agent between the experimental setup and a computer on which the complete control of the pendulum is implemented The control architecture used to combine joint trajectory tracking with exploitation of the natural dynamics is explained in chapter 3 The joint trajectory tracking controller co
95. reasing frequency from 1 5 Hz to 2 Hz is imposed Only the pressure course for one muscle is shown here The trajectory can still be tracked but we can clearly see that the natural dynamics are only exploited in the time interval 2 2s 3s or in other words for a certain frequency range of the imposed trajectory Ata frequency higher than 1 92 Hz even 4 exhaust valves are opened in order to increase the flow rate and consequently track the required pressure course Note also that the mean pressure inside the muscle remains constant during the whole simulation due to a constant p 21Nm Chapter 4 Results and discussion 68 Frequency eL 155 1 6 1 65 17 1 75 1 8 1 85 19 1 95 2 4b 22r a Or Bs S 4 q1 degrees g q1Des degrees I L L I L 1 I I 1 2 3 4 5 6 7 8 9 10 Time s Frequency 2 5 1 55 1 6 1 55 17 1 75 1 8 1 85 1 9 1 95 2 E e 2 2 3 5 E E 15 g valve action 2 pressure m required pressure 1 I V U W IU LJ 7 1 2 3 4 5 6 7 8 9 10 Time s Figure 4 9 Joint angle pressure and frequency for a trajectory with linear varying frequency The conclusion of this simulation is that a varying p will be more suitable for trajectories different from a pure sine wave This will be the discussion of the next simulation Simulation 3 changing the stiffness online The purpose of this simulation is t
96. ree parts a feedback linearization module a delta p unit and a bang bang pressure controller The feedback linearization or computed torque module is a standard nonlinear control technique which deals with the nonlinear behaviour of the mechanical pendulum configuration 1 The delta p unit translates calculated torques into desired muscle pressure levels coping with the nonlinearities introduced by the muscle actuation system Finally the bang bang pressure controller commands the valves in order to set the required pressures in the muscles This control structure will be implemented in a computer simulation of the pendulum as well as in the control of the physical set up As is said before the pendulum consists of one modular part of Lucy which means that the used tracking controller will have the same structure as described in 14 For the sake of convenience the same discussion on the control strategy will be given Some of the modules were slightly changed an integrator is added to the computed torque method and of course the model dynamics and parameters are not the same Special attention is also given to the estimation of these parameters Dis P Valves 1 3 E Actions Desired Pressure Model Trajectory Control Valves 2 Actions Figure 3 1 Schematic overview of the control structure Chapter 3 Control 41 In the first module the computed torque 7 is calculated The computed torque method consi
97. riction model this is not taken into account in our model To illustrate the influence of the parameter errors on the joint tracking controller we can first look at the real pendulum dynamics T D 0 C 0 0 0 G 0 3 14 Although for our control law we use equation 3 2 r at B with a D 0 and B C 0 0 0 G 0 where in this case D 0 G 0 and C 6 6 are the estimated inertia gravity and coriolis centrifugal term Decoupling and linearizing will not therefore be perfectly accomplished when parameters are not known exactly Starting from equation 3 5 the closed loop equations for the system are given by E K E K E K Ed b lp bg c 6p c 6 cas We clearly see that if the parameters are exactly estimated the right hand disappears and no control error will be made As mentioned before this will not be the case in our physical setup Therefore some deviations on these model parameters were incorporated in the simulation Chapter 3 Control 49 3 4 2 The mechanical equations of the model The equation of motion is given by 3 14 r D 0 C 0 0 0 G 0 Assuming the model as being frictionless we can now calculate the elements of the inertia coriolis and gravity terms Appendix A D 0 2 m m a7l I 3 162 G 0 g m m al sin 3 16c 3 4 3 The thermodynamical equations of the model The equations describing the thermodynamic processes in the pneumatic system were already discuss
98. s as was explained before In order to investigate the influence of a wrong estimation of the model parameters deviations on the mass inertia centre of gravities and force function were introduced into the simulation A simulation with 0 deviation which means that the parameter estimations are exact will be compared with a simulation with 10 deviation on the inertia 7 5 on the centre of gravity and 5 on the mass and the force function The servo portion of the computer torque module requires a correct tuning of the gains This was done intuitively but in such a way that stability is still ensured and control actions are minimized without affecting the tracking accuracy D mbar Valve action a 120 Open all exhaust valves b 60 Open one exhaust valve c 50 Close all exhaust valves d 50 Close all inlet valves e 60 Open one inlet valve f 120 Open all inlet valves Table 4 1 Pressure reaction levels of the bang bang pressure controller used in simulation Chapter 4 Results and discussion 58 The bang bang controller compares the required pressure to be set into the muscle with the measured gauge pressures and takes the appropriate valve actions depending on the various reaction levels These levels were also manually tuned in order to minimize energy consumption by taking into account the trajectory tracking accuracy The values of the pressure reaction levels are given in table 4
99. s were not taken into account In the future a fine tuning of the model should be made The joint angle is given in figure 4 17 We can see that the control unit is suitable to track this trajectory During the first period tracking is not so accurate An explanation can be found in the pressure course also depicted in figure 4 15 In the beginning the pendulum is at rest at an angle of 5 The muscles are inflated and a certain pressure is set into the muscle Once the experiment starts a pressure much lower than the actual pressure is needed All the outlet valves are opened in order to track the pressure course but this is not enough The pressure gradient over the valves is insufficient so that the time constant of deflation is too large in comparison to the requirements associated with the imposed pressure course This can cause the pendulum to become unstable A reason for this unwanted phenomena is the use of an exhaust silencer as it obstructs the dynamic performance of the muscle deflation The Chapter 4 Results and discussion 78 pressure rise in the silencer lowers the exhaust airflow During the construction this silencer was made as large as possible to avoid such situations 6 T T T T T T T T I A 1 1 A A A Measured al A hi N i i f A j i A f Required H l T T1 x IB II I AL EL AL LLY TT IT z MN LEL TL T L IBN BN VELEN Bont LEENE EEE LENEN Ek Ba UE
100. scles of the pendulum The pneumatic scheme shows the 2 identical pneumatic circuits of which each drives one antagonistic muscle The valve island is depicted with separately inlet and exhaust which each of them is represented by two 2 2 electrically actuated valve symbols These two symbols represent the 2 reaction levels of the valve system The number of actual valves which are included in each configuration are depicted as well The right and left muscle are connected to the supply pressure regulating unit by separate tubes The Chapter 2 Design Supply pressure regulating unit Q High pressure regulator 2 Valve High pressure valve 12 Inlet valves Muscle 2 Pressure sensor Pressure sensor Muscle 1 Exhaust valves Exhaust valves Figure 2 13 Schematic overview of the pneumatic circuit 26 pressure regulating unit consists of a mechanical unit that determines the pressure in the circuit The circuit is interrupted with an electrically actuated 3 2 valve The exhaust of this valve is connected to an electrically actuated 2 2 depressurizing valve in order to deflate the complete pendulum 2 2 3 4 The exhaust silencer Additionally a silencer is added at the exhaust of each valve island Without a Exhaust silencers silencer the immediate expansion to atmospheric conditions of the compressed air at the exhaust creates
101. sity at standard conditions C and b are two flow constants characterising the valve system The constant C is associated with the amount of air flowing through the valve orifice while b represents the critical pressure ratio at which orifice air flows become maximal P and P are the upstream and downstream absolute pressures while 77 air is the upstream temperature When choking occurs equation 3 18b is valid otherwise equation 3 182 is used The muscles are controlled by a number of fast switching on off valves Once the actions opening or closing of the valves are known all the air flows can be calculated in order to be substituted in 3 17 The temperature in the muscle is calculated with the perfect gas law Ta 3 19 with P p P atm the absolute pressure in muscle i The total air mass m is given by integration of the net mass flow entering muscle i M s m B m 3 20 air air Chapter 3 Control 51 3 4 4 Complete simulation model The structure of the complete simulation model is given in figure 3 4 and is based on the simulation model of Lucy The kernel of these simulations is based on three equation blocks which integrates first order differential equations only The differential equations are numerically integrated using a 4 order Runge Kutta method with integration time step of 50 us Runge Kutta numerical integration Integration timestep 50 us Tis ply
102. ss 3 Pink 1 0646 kg M Mass 4 Green 1 0736 kg mesi ms 1 101 052 v Mass 5 2 x Whit 1 045 k Sinus 3 Lcod 0 4 IV Mass B 2 x Purple E 0612 kg V Mass 7 2 x Pink 1 0667 kg Atmospheric 101130 Pressure Pal v Mass 8 2 x Green E 0757 kg Figure 2 22 The Model tab of the GUI Chapter 2 Design 37 The Trajectory tab is used to specify the initial angle the offset and the frequency of the desired trajectory we want to be tracked by the tracking controller In the Plotting tab the user can specify the file where data has to be written to Because plotting data in Visual C is too complicated we rather use Matlab Therefore data acquired in Visual C has to be sent to Matlab The Matlab engine is loaded in C to access Matlab s prebuilt graphics functions By using this engine we are able to send the data to the Matlab workspace and plot them directly from Visual C After each experiment the data is automatically sent to Matlab and plotted The option to plot different type of data during the experiment or test can also be enabled Plotting the data during the experiment takes a lot of time so it should only be done when no control action is needed Once the data is plotted Matlab workspace will open so that the data can be manipulated if necessary To calculate the energy consumption during the experiment a Matlab M file is made Typing CalcEnergy into the workspace will give the valve action and ener
103. sts in decomposing the controller into two parts One part is model based in that it makes use of the modelled dynamics of the particular system under control The second part of the control is error driven in that it forms error signals by differencing desired and actual variables and multiplies this by gains It s also called the servo portion 1 To find the model based part the rigid body dynamics can be written down r D 0 6 C 0 0 0 G 0 3 1 where D 0 is the inertia matrix C 0 is the centrifugal and coriolis term G 0 is the gravity term and the joints positions of the real system In our case the system is one dimensional Consequently those matrices will consist of only one term The symbol denotes that the respective expressions are calculated with estimated parameter values The main problem is to find an optimal model of the pendulum We can linearize this equation by choosing r at B with a D 0 and B C 0 0 0 G 0 3 2 Filling this into the rigid body mechanics 3 1 gives 6 3 3 a I As we see the resulting equation is that of unit mass system The design of the servo position is now very simple gains are chosen as if we were controlling systems composed of single unit masses r 0 K 0 6 K 0 6 K O dt 3 4 Chapter 3 Control 42 The symbol represents required values The gains K K and K are manually tuned in order to influence controller performance The fo
104. sure levels 1 2 and 3 bar as a function of the contraction Using 110 mm R 11 5 mm and n 40 results in the theoretical force characteristics depicted in figure 2 2 14 The traction as a function of contraction is drawn for different applied gauge pressures 1 2 and 3 bar The graph shows the non linear character of the generated muscle force For small Chapter 2 Design 10 contractions the forces are extremely high while for large contractions the forces drop to zero For the pendulum contractions will be bounded somewhere between 5 and 35 The first limit is set to bound the stresses on the fibres and consequently extends the lifetime of the muscle And beyond 35 contraction forces drop too low to be of practical use Figure 2 3 depicts the theoretical maximum diameter to contraction for the considered muscle 2 This approximated maximum diameter should be taken into account during the design of the joints of the pendulum in order to provide enough space for the muscle to bulge 0 11 0 10 0 09 0 08 0 07 0 06 0 05 0 04 0 03 Maximum diameter m o on e lt 2 30 35 40 45 Contraction en Figure 2 3 Theoretical maximum muscle diameter as function of contraction To compare the theoretical characteristics with those of the physical PPAM static load tests were carried out by Verrelst A brief discussion on the results will be given For more information we refer to 14 A deviation b
105. system speed up circuitry eee entr reon doeet 30 2 3 2 Pressure SeDnSOE ao eee nt i aaan 31 233 Ineremental encoder sensira ar tiin ain aari 32 2 3 4 Data acquisition Card a e erige utter t e e totas 33 2 3 5 Safety supply and transformation board 35 2 4 SoftWare M 36 2 5 Conclusion dos dated tar aatees Ae Sis te aso te beds rapit 39 CONO EE 40 Bly TOrOHUCHOR sers overneemt nenten derden 40 3 2 Joint tracking controller uie eiecti ene 40 3 3 Mathematical formulation for compliance adaptation 45 3 4 The simulation model of the pendulum sss 48 3 4 1 Description of the pendulum set up nennen erneer 48 3 4 2 The mechanical equations of the model 50 3 4 3 The thermodynamical equations of the model 50 3 4 4 Complete simulation model eterne 52 DOs Energy considerations creet iin tds ne addi 54 JG SCONCIUSI Oe a ee E E a Ea see E 55 Results and discussion naaar neerensnseerenneeerenn 57 db InftodBetlott uet eroe i or et hau o P oes 57 42 The sumilatom models osea eq edax epe Ds 58 4 2 1 Evaluation of the joint trajectory tracking controller 58 4 2 1 1 General considerations esee 58 4 2 1 2 Results and discussion dise dat a HER ERR uS RARI SERIE 59 4 2 2 Exploitation of the natural dynamics
106. th absolute pressure sensors and the angular position is captured with an incremental encoder The valves of the two islands are Chapter 2 Design 29 controlled by digital micro controller signals after being transformed by the speed up circuitry in order to enhance switching speed of the valves In the next sections detailed information is given about the different elements of the control hardware 2 3 1 Valve system speed up circuitry To have a rapid and accurate pressure control switching time of the valves should be as little as possible In order to enhance the opening time of the Matrix valves the manufacturer proposes a speed up in tension circuitry shown in photograph 2 17 Several practical tests for which is referred to 13 have resulted in an opening and closing times of about 1 ms An opening voltage of 32 V is being applied during 1 ms Figure 2 17 View of the speed up circuitry The data acquisition card commands the valves via discrete 5 V on off signals Two circuits control separately the two inlet valves and another two control the Chapter 2 Design 30 exhaust valves Hereby three valves are controlled simultaneously by one circuit For the electronic scheme of the speed up circuitry we refer to 14 D 2 2 3 2 Pressure Sensor To have a good dynamic pressure measurement the pressure sensor shown in figure 2 18 is positioned inside the muscle The pressure sensor and its electronics are t
107. the air mass flow the exergy and the valve action for both simulations over one period The valve actions are defined as in 3 23 Ps 16 Nm ps 32 Nm Muscle 1 Muscle 2 Total Muscle 1 Muscle2 Total Airflow input 120 1 mg 107 9 mg 228 0 mg 21 7 mg 13 9 mg 35 6 mg Airflow output 111 4mg 116 6 mg 228 0 mg 14 1 mg 22 1 mg 36 2 mg Exergy inlet 19 7 17 7 J 37 4 J 3 6 J 2 37 5 9 J Exergy outlet 6 8 J 1 04 13 8 J 1 4J 2 1J 334 Inlet valve action 39 8 ms 36 0ms 75 8 ms 7 7 ms 4 9ms 12 6 ms Outlet valve action 134 4 ms 141 8 ms 276 2 ms 10 5 ms 16 7 ms 27 2 ms Table 4 2 Air mass flow exergy and valve action during one period of a sine wave trajectory with frequency of 2 Hz for p 216 Nm and Ps 32 Nm The most interesting value is the exergy inlet because this will give an indication of the energy consumption of the system As we can see the total input exergy in the case of p 32 Nm equals 5 9 J which is much lower than 37 4 J in the case of p 16 Nm Therefore from this experiment we can conclude that a suitable stiffness affects in a positive way the energy consumption and control action during the trajectory tracking Chapter 4 Results and discussion 66 Simulation 2 evaluation of the mathematical formulation The main goal is to see whether the mathematical formulation given by Verrelst is usable Therefore in this simula
108. tion joint sine wave trajectories with different frequencies are imposed Table 4 3 gives an overview of the different sine wave trajectories for which the mathematical formulation was evaluated In the second column the calculated values of ps are given The values in the third column give the optimal ps for the corresponding trajectory For each trajectory the latter values were obtained by varying p and evaluate where the energy consumption and valve actions are minimized Sine wave frequency Hz pz Nm pz Nm 1 5 12 6 12 5 1 75 Aa F 21 2 317 31 2 25 43 2 42 5 Table 4 3 Calculated and simulated optimal values of p for different sine waves trajectories Figure 4 8 depicts the energy consumption and valve actions per period for the different sine waves in function of p For each trajectory there isa p where energy consumption and valve action is minimum This is the most optimal p By comparing the calculated and the optimal p we can conclude that the mathematical formulation is indeed suitable for setting the stiffness The optimal D is increasing as the frequency is increasing This was expected because p is influencing the stiffness and a higher oscillation frequency requires a higher stiffness Chapter 4 Results and discussion 67 Frequency 1 5 Hz Frequency 1 75 Hz
109. tion of the pendulum The effect on valve control activity and energy consumption by changing the compliance characteristics are shown by means of a simulation and experiments on the physical pendulum A technique to change the stiffness online has been evaluated The next steps in the study are a correct estimation of the system s parameters and the extension of the system to a three dimensional pendulum in that way that it can be used for implementation on the walking biped Lucy from the Vrije Universiteit Brussel samenvatting Studie van een modelgebaseerde controle van een slinger aangedreven door balgactuatoren met regelbare stijfheid Een scharniergewricht aangedreven door een stel antagonistisch opgestelde spieren vertoont een soepel en elastisch gedrag te wijten aan de samendrukbaarheid van lucht en de dalende kracht contractie karakteristiek van de spier Bovendien kan deze soepelheid ingesteld worden onafhankelijk van de positie van het gewricht De variabele stijfheid het omgekeerde van de soepelheid be nvloedt de eigenfrequentie van het systeem en biedt dus de mogelijkheid om de natuurlijke dynamica van de slinger uit te buiten om zo het energieverbruik en de controleactiviteit te verminderen Het hoofddoel van dit project is te komen tot een controle van de slinger waarbij de uitbuiting van de natuurlijke dynamica van het systeem gecombineerd wordt met een trajectsturing De trajectsturing volgt een opgele
110. tionally to determine the pressure changes in the thermodynamic differential equation block muscle volume and volume changes are needed They are calculated with the polynomial volume functions 2 3 Consequently angle and angular velocity are required from the equations of motion Starting from the desired trajectory and by taking the natural dynamics into account the control unit determines the appropriate valve actions and sends valve control signals to the delay observer The valves have an observed opening and closing time delay of about 1ms 13 This is incorporated in the delay observer In the simulation a delay time of opening or closing the valve of 1 ms is used The valve system model will then calculate the air mass flow rates for each muscle Therefore temperature and pressure in the muscles are required from the thermodynamic differential equations block Additionally deviations can be introduced on the mass centre of gravity and inertia parameters and on the force functions in order to check the robustness of the controller as was explained in the previous section Chapter 3 Control 53 3 5 Energy considerations The main purpose of this thesis is to investigate the exploitation of the natural dynamics of the system in order to reduce energy consumption and control action Therefore it will be interesting to have an idea of the energy consumption and control action that takes place during the simulation or experim
111. ures can be increased decreased in such a way that joint stiffness K raises lowers without affecting angular position Thus both position and compliance can be controlled independently 2 2 2 2 Kinematics of the one dimensional pendulum The kinematics of the pendulum have the same structure as one modular part of Lucy For the sake of convenience the same discussion will be given as in 14 Chapter 2 Design 13 The basic configuration of the pull rod and leverage mechanism is depicted in figure 2 6 Figure 2 6 Basic configuration of the pull rod and leverage system The two muscles are connected at one side of the system to a fixed base in the points Bl and B2 respectively The other ends of the muscles are attached to a Chapter 2 Design 14 pivoting part at the points D1 and D2 of which the rotation axis passes through a point R The rods are assumed to be rigid An orthogonal X Y coordinate system is defined The X axis is aligned with the base points B and B while the vertical Y axis intersects the physical pivoting point A and lies along the base suspension bar of the pull rod mechanism The essential parameters to be determined during the design process of the joint are the following e bis the distance between the origin O and the point B e d is the distance between the pivoting point R and the point D e a is the angle between the vector RD and RC point on the rotating part a is n
112. v Vrije Universiteit Brussel Faculteit Ingenieurswetenschappen Vakgroep Werktuigkunde Model based control of a one dimensional pendulum actuated with Pleated Pneumatic Artificial Muscles with adaptable stiffness Bruno Meira y Duran Proefschrift ingediend tot het behalen van de academische graad van Burgerlijk Werktuigkundig Elektrotechnisch Ingenieur Academiejaar 2004 2005 Promoter Prof dr ir Dirk Lefeber Co Promotor Prof dr ir Patrick Kool Abstract Model based control of a one dimensional pendulum actuated with Pleated Pneumatic Artificial Muscles with adaptable stiffness Due to the compressibility of air and the dropping force to contraction characteristic of the muscle a joint actuated by two of these muscles set in antagonistically position shows a compliant behaviour and can be adapted while controlling position This compliance adaptation enhances the possibilities of exploitation of natural dynamics so that energy consumption and control activity is decreased The main purpose of the project is to investigate the exploitation of natural dynamics in combination with the joint trajectory control The joint tracking controller tracks the desired trajectory while adapting the joint compliance as such that the natural regime corresponds as much as possible to the reference trajectory Currently the pendulum is assembled and its hardware components have been tested This thesis reports on the design and construc
113. ximum and minimum limits of the contraction are also plotted In the left plot d is changed In the right plot the contractions for different values of are given 40 35 30 25 20 a 60 en 719 I I Contraction 4 96 Minimal contraction Maximal contraction T T 40 d 30 mm en 971976 35 30 25 20 I I Contraction c 4dop puL pReL22 d 20mmyg 1 1 a 45 d 25mm d 30mm a 60 35mm 5 i a 70 90 0 0 40 20 0 20 40 40 20 0 20 40 degrees degrees Figure 2 7 Influence on muscle contraction by changing d and a Figure 2 8 shows two different torque characteristics On the left side the static torque of the joint is plotted for a maximum pressure of 3 bar in both muscle 1 and 2 The influence on the static torque characteristic by changing d is also Chapter 2 Design 20 shown On the right side the torque characteristic of the joint with closed muscles is plotted to analyze the joint stiffness when muscles are closed This characteristic is found by setting 3 bar into the muscle and closing the muscle at zero degrees By moving the pendulum out of is equilibrium a repelling torque is generated This torque characteristic is plotted on the right 150 a 60 s519 and p p 3 bar I I I Static Torque Nm Minimal torque n
114. y is in fact the maxi mum amount of energy with respect to the surrounding environment which can be transformed into useful work For a compressor the minimal work Chapter 3 Control 54 needed to compress air from pressure level p to p is done at isothermal conditions and can be calculated as follows 9 m rT In Zt 3 24 P Wssdas hereby assuming the air to behave as a perfect gas The symbol m represents the total air mass flowing through the compressor at pressure level p r is the dry air gas constant and T is the temperature the air at pressure level p expressed in Kelvin Using this equation we can calculate the exergy of the supply source as well as the exergy of the air leaving the muscles The exergy of the supply source for muscle i is given by oec wor P Pent y 3 25a atm while the output exergy can be calculated with Wina ctr In y 3 25b Pam During the simulation the absolute supply level is set at 7 bar the atmospheric absolute pressure at 1 bar and the atmospheric temperature is 293 K 3 6 Conclusion In this chapter a control architecture structure for the pendulum was discussed The structure covers the exploitation of the natural dynamics in combination with joint trajectory tracking The joint trajectory tracking controller uses a computed torque method to cope with the nonlinear behavior of the pendulum Chapter 3 Control 55 configuration The calculated torqu

Model-based control of a one-dimensional pendulum actuated with

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