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1. Figure 2 6 Subcritical Bifurcation Phase Portrait 6 gt 0 and a lt 0 The phase portrait shows the displacement versus the velocity The middle point is asymptotically stable and the points to the left and right are unstable which will drive the system to positive or negative infinity This phase portrait is used to see how the system should behave 2 3 Euler Integration In order to handle linear and non linear functions and calculate the the future dis placement velocity and acceleration for the motor the acceleration is integrated over a small interval 7 At The acceleration is calculated by using the linear or non linear function of choice and using the force along the rod as the input force The 12 current displacement and velocity are also used See variables below e x current displacement e x current velocity e F current force applied along the rod of the motor x future displacement e t7 future velocity e Zi future velocity e f x ta F the linear or nonlinear function So the future acceleration is ir flute F 2 10 The future velocity is t r Tir T4 f tdt t Li Te TX 2 11 The future displacement is 27 Litr t f Tt rdt 0 Ti Tir 277 Es 2 12 13 2 4 Hardware Multiple devices come together together to make the complete design The block diagram is below Nano 17 Signal Conditioner Microcontroller LM1247 080 012. Figure 2 7 Complete Hardware Block Diagram 2 4 1 National Instruments Data Acquisition Board The data acquisition board used is the National Instruments NI PCI 6025E This board has 16 analog inputs at up to 200kS s at 12 or 16 bit resolution 9 It also has up to 2 analog outputs at 10kS s at 12 or 16 bit 9 The analog inputs of the NI PCI 6025E will be used to power and read the force values from the Nano17 One analog output is used to control the position of the linear motor For the application 14 in this thesis matlab will be used to control all DAQ operations 2 4 2 Linear Motor and Microcontroller The linear de motor used to for the S D test is the Faulhaber Quickshaft LM 1247 080 01 and is displayed below The Quickshaft LM1247 080 01 provides us with no residual static force and an excellent relationship between linear force and current The motor also has built in Hall sensors which allow position control 10 The MCLM 3006 S is provides precise positioning and very low speeds because it contains a high performance digital signal processor DSP The microcontroller will allow the following 10 Velocity control Velocity profiles e Positioning mode Stepper motor mode Analog positioning mode e Force control Analog Position Control is utilized to control the linear motor Initial settings as well query commands for getting the current position of the motor are sent over 15 RS232 communication T
3. 1 1 E t 5 ot she 2 4 The graph of the potential energy when 6 0 k gt 0 is pictured below Potential Ener 0 06 T T Iy 7 0 05 4 0 04 Energy o o e L 0 02 4 0 01 F 4 0 1 L L L L 0 04 0 03 0 02 0 01 0 0 01 0 02 0 03 0 04 Displacement m Figure 2 3 Potential Energy 2 2 Non Linear System 2 2 1 Duffing Oscillator The linear motor is meant to replace the spring in the experimental setup and bring the system to the edge of instability by replicating spring buckling The Duffing oscillator is a dynamical non linear system that is known to model chaotic behavior 7 The system will model different behaviors depending on the variables below 0 damping constant e 0 spring constant a duffing constant e forcing constant e w angle phase angle The Duffing equation is a non linear second order equation is seen below t Bx ax y cos wt 2 5 The Duffing Oscillator can take on different behaviors for instance changing 5 from positive to negative gives a supercritical pitchfork bifurcation If 5 gt 0 the restoring force is Frotor Bx ax a 2 6 When a gt 0 and for small values of x equation 2 7 is a hardening spring and when a lt 0 the equation is a softening spring 8 The case when 8 lt 0 creates a double well potential 6 The restoring force then becomes Emotor BT ax ak 2 7 The potentia
4. 1Nto 1N This is because 1N is the lowest force the motor would respond correctly at when told to move to a given displacement Also note that the motor always applies the continuous force and applies the peak force only if the force in the opposing direction is greater than the continuos force 3 2 Duffing Oscilator After the linear spring could be realized by the motor the next step was to replicate the Duffing oscillator that exhibits subcritical bifurcation The details on the potential energy graphs are found in section 2 2 1 The spring constant 6 was calculated using equation 3 1 The x value in the equation was changed to 0 01 because the desired equilibrium points were set to 0 01m 0 0m 0 0 01m 0 The a value was then calculated using equation 2 9 The results for the motor displacements and forces for different initial displacements are pictured below 35 E Target Position Peak Force q Actual Position Continuous Force 4 8 S 18 f 1 1 f 0 0 2 0 4 0 6 0 8 a 1 2 1 4 1 6 1 8 2 time a Motor displacement b Motor Force Figure 3 3 Stable Motor Response 5 314 a 3140000 6 10 5X 10 1 E Target Position Peak Force i Actual Position Continuous Force 0 5F ot 4 E 2 z E 14 J 1 5 y 10 1 1 3 a y a 7 2 1 1 1 1 fi 1 1
5. duffing const 40 10 80 and hit enter 42 4 The motor will collect baseline data from the Nano17 Do not touch the motor while it is collecting this information 5 The system will beep telling you the test is running and to put your hand on the motor 6 The system will beep again when the test is complete and tell you remove your hand from the motor 7 To run the test again go back to step 3 8 Turn off the power supply when you are finished The first input to the function spring_type can be lin_neg_damp or duffing depending which equation you want the spring to use Further details of how this parameter is used can be found in section 2 5 2 The second input k_type can be const rodforce zforce or allforce Further details of how this parameter is used can be found in section 2 5 1 Further details on the motor_cntrl function can be found in section 2 5 Before running the program make sure the motor is place in the middle of the rod This will serve is the 0 position of the motor The program will tell you when to place and remove your hand from the motor 43 REFERENCES Venkadesan M Guckenheimer J Valero Cuevas FJ Manipulating the edge of instability Journal of Biomechanics 2007 40 1653 1661 Group Faulhaber Operating Instructions2nd ed 2008 Valero Cuevas FJ An integrative approach to the biomechanical function and neuromuscular control of the fingers Journal of Biomechanics 20
6. 25N and 35N 17 respectively 11 The transducer can also measure torque in the x y and z directions but this is not being used at the moment 2 4 4 Nanol7 Transducer Holder A custom device was designed in Solid Works and printed to hold the Nanol7 to the linear motor The device had to be made to both the specifications of the linear motor and the force transducer The printed device is pictured below Figure 2 9 Nano17 Transducer Holder 2 4 5 Linear Motor Holder A custom holder needed to be designed in order to holster the linear motor This holder needed to fit the specifications of the linear motor and also needed to provide stability since the user will be applying different forces to the linear motor The holder is made out of aluminum and is screwed into another aluminum device which will 18 be strapped down to a table The holder is versatile and allows various adjustments of the linear motor to provide comfort for the user The initial design of the linear motor holder is pictured below Figure 2 10 Linear Motor Holder 2 5 Software Code was written in Matlab C and C to control all tasks of the device will perform The matlab code controls the NIDAQ board which has inputs and outputs to the Nano17 and microcontroller The C and C code is only used for controlling the serial communication A state machine showing how the motor_cntrl function works is below 19 Setup DAQ Configure M
7. LPC parameter3 is second input Y Write Continuos Force State Call SerialWrite using LCC parameter4 is second input Output output position Error State Issue open port error Figure 2 16 wrc_serial Function State Diagram 31 Chapter 3 RESULTS Once the design of the system and program was completed tests were done to obtain behavior that mimicked the spring and damper system and the Duffing oscillator All of the results shown have the forcing input set to 0 3 1 Spring and Damper Equation To test whether the linear motor behaved like a spring that produced the correct amount of restoring force for the appropriate displacement the following calculation was done to compute the spring constant when no velocity is present k e Fe motor Tmaxdisplacement S 3 13N 0 033m k 95 3 1 Using the spring constant above tests out the max force of the motor at the max displacements It also gives us a starting point to increase the spring constant if a harder spring needs to be replicated The results for the motor displacements 32 simulated displacements and force delivered are pictured below 0 01 0 005 Gl 0 01 Distance m E Target Position Actual Position times 25 Displacement m 0 005 0 005 0 01 0 015 0 02 0 025 Response to Initial Con
8. the the current displacement current velocity future acceleration and time delay of the last time the loop was implemented The inputs are acc_calc x_val v_val delta_t The outputs arex_nert and v_next The variables are defined below e acc_calc future acceleration e x_val current displacement e v_val current velocity e delta t current time delay e x next future displacement e u_nert future velocity The function works by implementing the integration equations mentioned in section 2 3 First the equation multiplies delta_t by acc_calc and adds it to v_val based off of equation 2 11 to get v_next Then the function multiplies 2 by delta_t and v_val adds it to 2 multiplied by delta_t and acc_calc and adds that to x val to get x_ val based off of equation 2 12 The state machine is pictured below 26 V Next State v_next v_val delta_t acc_calc X Next State x_next x_val 2 delta_t v_val 2 dela_t2 Output x_next v_next Q Figure 2 14 propagate Function State Diagram The pseudo code for the spring function is below x_next v_next propagate acc_calc x_val v_val delta_t v_next v_val delta_t acc_calc x_next x_val 2x delta_t v_val 2x delta_t 2 acc_calc 2 5 4 open serial Function The open_serial Function is one of the two MEX functions created The function is responsible for opening and configuring the serial port The open serial Function has no inputs and
9. 1 1 o 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2 0 0 2 0 4 0 6 0 8 1 1 2 14 1 6 1 8 2 time s time a Motor displacement b Motor Force Figure 3 4 Stable Motor Response 5 314 a 3140000 10 In Figure 3 3 the initial displacement was 0 009m and the initial velocity of 0m s In Figure 3 4 the initial displacement was 0 009m and the initial velocity of 0m s The system stayed stable and decreased with an exponential oscillation to the equilibrium 36 point 0 0 The behavior was similar to a spring since the motor was displaced in between 0 01m and 0 01m The behavior is different when the motor is displaced beyond 0 01m as pictured below 0 035 r r 10 1 1 E Target Position Peak Force Actual Position 9f Continuous Force 0 03 8l J 7 J g 0025 4 el g z S a 2s 2 oo J 4 J 3 0 015 J pl 4 J i Ooo o4 06 08 1 12 14 16 18 2 0 L q o z y ires 0 02 04 06 08 1 12 14 16 18 2 time a Motor displacement b Motor Force Figure 3 5 Unstable Motor Response 8 314 a 3140000 6 10 0 01 9 i i i i Actual Position Continuous Force 0 014 J 2 J 3 El 0 02 J ab Al E E od S E 7 2 LL A 0 024 4 e J 0 03 4 9 4 6088 i f f 1 i O 02 04 06 08 1 12 14 16 18 2 10 1 1 1 1 1 1 time s 0 02 04 06 08 1 12 14 16 18 2 time a Motor displacement b Motor
10. Baker A device was designed and implemented to replicate spring buckling using a linear motor The device was designed using an off the shelf motor microcontroller DAQ and force transducer Once the device was put together a program was written in matlab to control the motor Tests were done to replicate a spring damper system and once this system was realized tests were done to replicate the Duffing oscillater that exhibits subcritical pitchfork bifurcation The finished device can be used to investigate how vision and tactile integration play a role in dynamic manipulation TABLE OF CONTENTS Acknowledgments lt a A mn Ra So RRMA REE RR HS iii Abstract Eist or a eis ie A RAA A a a a ae viii T Tntroduction ra g eiie het Beers Aa Boos ae Be oh Ge Ces ee 1 2 Methods and Design stig ay a A AAA 5 Del Linear System wi e i a te DNA BUS ot dere AR a et aR ri Etes 6 2 1 1 Spring and Damper Equation us caes daa es 6 2 2 Non Linear SYST ME She ay Hee We MS Nes e EEE 8 2 2 1 Done Oscillator e ra La She oe GS tee a mu rte 8 Jr Huler Int gration saine d dh dis etats mia e Rate 12 24 Hardware cat oe ane GR ch ane GR va we ee Oey adh hs Hh ype BE ire 14 2 4 1 National Instruments Data Acquisition Board 14 2 4 2 Linear Motor and Microcontroller 15 2 4 3 Force Transducer la odes el oo ae AUS AA at 2 17 2 4 4 Nanol7 Transducer Holder 2 4 402 24 464 ee Seed 18 2 4 5 Linear Motor Holder 243 Sonde
11. Force Figure 3 6 Unstable Motor Response 5 314 a 3140000 6 10 37 In Figure 3 5 the initial displacement was 0 012m and the initial velocity was Om s In Figure 3 6 the initial displacement was 0 012m and the initial velocity was Om s The figure shows the system the motor attempting to go to positive and negative infinity for Figures 3 5 and 3 6 respectively The figures also show the motor applying the maximum force at each of the above displacements This force makes it nearly impossible to stabilize the system and get back to the equilibrium point at Om displacement The linear motor successfully modeled the linear spring equation and the Duffing oscillator Using the Duffing oscillator the motor is able to replicate spring buckling behavior The results show how the motor is attracted to its center region as long as it says within certain boundaries When the motor is moved outside of these boundaries or moved with a velocity that is too high the system will become unstable and will attempt to hit the end rails Further test of the parameters will need to be done to adjust the equilibrium points increase the force the motor applies applies in the stable region and to increase oscillations within the stable region Once the desired behavior is found the system can be used to conduct tests similar to the those in the paper Manipulating the edge of instability 1 38 Chapter 4 INSTRUCTION MANUAL I will go into d
12. of the spring rests on a force transducer that logged force at 1000Hz The motion capture cameras captured the 3D location and orientation of the reflective markers at 200Hz using a 4 camera mo tion capture system The palm of the hand never touched the spring or the vertical post and the subject can view the setup from any angle of their choosing Feedback was provided using an audible 500Hz tone that linearly decreased in volume as the vertical compressive spring force increased 1 The subjects were instructed to slowly compress the spring using only their thumpad to make the tone volume as faint as possible without letting the spring slip Once this point was reached the subjects needed to maintain this load force hear a constant tone for 10s without letting the spring slip and then slowly release the spring It did not matter if the spring oscillated or bent but that the volume remained constant and the spring did not slip 1 Experiments were performed over two days Day 1 was a training day used for acclimating the subjects to the system During this day the subjects performed 100 compressions and the performance was measured before and after the training On day 2 the performance of the subject was measured using normal thumbpad sensibility both with and without vision Thumbpad sensibility was then reduced when a hand surgeon administered 5 cc of 1 Lidocaine solution on the ulnar and radial sides of the base of the thumb just be
13. one output output 0 The MEX file for this function also has one additional sub function named SerialInit that has one input BaudRate and returns the serial object The variables are defined below 21 e output 0 the address of the serial port e BaudRate baudrate used to open the serial port The function works by calling the Seriallnit function which opens the serial port with the BaudRate passed to it by open serial Once the serial port is opened properly the function exits outputing the address of the open serial port object The state machine is below Initial State Declare all needed variables and call Seriallnit Open Port State Open Serial Port 5 Error State Issue open port error properly Set Port Parameters State Set default parameters Able to set parameters Error State Issue config port error Yes Qutput Serial Port Address Figure 2 15 open_serial Function State Diagram 28 2 5 5 wrc_serial Function The wrc_serial Function is the last of the MEX function created The function writes to reads from and configures the linear motor All of the commands are sent to and received from the microcontroller of the linear motor The function is very versatile because the inputs and outputs of the function change based off of its second parameter The wrc_serial takes a minimum of two inputs and does not need to output data The first pa
14. 05 38 673 684 Valero Cuevas FJ Smaby N Vendkadesan M T Wright The Strength Dexterity Test as a Measure of Dynamic Pinch Performance Journal of Biomechanics 2003 36 265 270 Guckenheimer J Holmes P Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields Springer 1983 Moon Francis C Chaotic Vibrations An Introduction for Applied Scientists and Engineers John Wiley and Sons 2004 Hilton Robert C Chaos and nonlinear dynamics an introduction for scientists and engineers Oxford University Press2nd ed 2000 Thompson JMT Stewart HB Nonlinear Dynamics and Chaos John Wiley and Sons2nd ed 2002 A4 9 Instruments National DAQ E Series E Series User Manual National Instru ments11500 North Mopac Expressway Austin Texas 787593504 USA 2007 10 Group Faulhaber Quickshaft Linear DC Servomotors 3 18N15th ed 2008 11 ATI ATI F T Controller Installation and Operation Manual Pinnacle Park and 1031 Goodworth Drive and Apex NC 27539 USA9610 05 1017 daq ed 2008 45
15. APPROVAL PAGE FOR GRADUATE THESIS OR PROJECT GS 13 SUBMITTED IN PARTIAL FULFILLMENT OF REQUIREMENTS FOR DEGREE OF MASTER OF SCIENCE AT CALIFORNIA STATE UNIVERSITY LOS ANGELES BY Corey Baker Candidate Electrical and Computer Engineering Department TITLE DESIGN AND IMPLEMENTATION OF A NON LINEAR DYNAMICAL SYSTEM REPLICATING SPRING BUCKLING BEHAVIOR APPROVED Deborah Won Faculty Member CSULA Signature Kamran Karimlou Faculty Member CSULA Signature Fred Daneshgaran Faculty Member CSULA Signature Fred Daneshgaran Department Chairperson Signature DATE June 17 2010 DESIGN AND IMPLEMENTATION OF A NON LINEAR DYNAMICAL SYSTEM REPLICATING SPRING BUCKLING BEHAVIOR A Thesis Presented to The Faculty of the Department of Electrical and Computer Engineering California State University Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science By Corey Baker June 2010 2010 Corey Baker ALL RIGHTS RESERVED il ACKNOWLEDGMENTS I would like thank Francisco Valero Cuevas and his Brain and Body Dynamics Lab at the University of Southern California for allowing me to work in their lab providing me with the supplies needed and guiding me in the project Without the help of all of you my work would have not been completed ill ABSTRACT Design and implementation of a non linear dynamical system replicating spring buckling behavior By Corey
16. Ne dat e 18 2 5 LSOWALE LL te Be ak QUE Dale gr eh et Be A A ne 19 2 5 1 spring const Function id LU Es LS A 20 2 5 2 spring Function 2 eos echo be ae ee re Ve M ete 23 2 5 3 propagate Function cis do dos DE Ra DR RAS OE 26 2 5 4 open_serial Function 2 hd ser So cep er ara 27 25 97 Wreserial PONCHO oo Ase te er ee Boe ee oe AS 29 3 RESTES e ae sig Raby agen Pay SM leds AR a ihe US nt 32 3 1 Spring and Damper Equation 30405444 a Rd pe A Ses 32 32 Dullme Oscilator as a 2 yee oe Sa LR a na oe 39 A Anstruchon Manual ass eg Ws ods oe eh Se teh at es Bo olen 39 AV initial Setpa teed ee a a et LAS Sy oo as E Bete 39 4 2 Code Modification ie ASE he Ge hice Sedat E Ad 40 4 3 Running the Program a Ne ae a ee ee aE 42 References A Bo Ek moled ete ka geek ba grand Beck Se 44 vi 1 1 2 1 Diz 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 11 2 12 2 13 2 14 2 15 2 16 LIST OF FIGURES Schematic of original experimental setup 1 2 Designed System Sumba la e ak deal en do a e NS ne 5 Simulated Spring Response T Potential Enero Las Ae oe a a Serpe Ure ate 8 Potential nero sor iio ol sio dog Ka At SG AE eR 10 Subcritical Bifurcation 8 gt 0 anda lt 0 lt 0 ese perra 11 Subcritical Bifurcation Phase Portrait 6 gt 0 anda lt 0 12 Complete Hardware Block Diagram 14 MCLM 3006 Circuit Design 2 a dan
17. ails on how this function works look at section 2 5 4 Make sure to compile this file if it is your first time using it or if you make changes to it To compile go to the matlab window and type mex open serial To configure the amount of time the program runs look in the motor_cntrl m file and look for the line stop_time 2 and change the value in red to the amount of time in seconds you want the program to run 40 5 To configure the inputs of the DAQ board look in the motor_cntrl m file Look for the line ai analoginput nidaq Dev5 and change the device to the DAQ you have the nano 17 connected to Look at the next line ch addchannel ai 0 1 2 3 4 5 and change the numbers in red to the channels the nano 17 is connected to To configure the output device of the DAQ to the microcontroller look for the line ao analogoutput nidaq Dev5 and change it values in red to the appropriate device Look at the next line ch addchannel ai 0 and change the number in red to the channel the microcontroller is connected to If you need to configure the default parameters that are sent to the motor open up the file wrc_serial c and look in the ConfigMotor HANDLE function Each line is a configuration parameter sent to the motor A brief description of each of the commands are in the comments For a full description of the commands look in the Operating Instructions of the microcontr
18. angle 23 phase angle e t_del current time delay acc future acceleration x_fut future displacement v_fut future velocity The function works by checking to see if spr_eq equals lin_damp or non_lin_damp If spr_eq equals lin_damp it sets the damping coefficient c to a constant value calculates the future acceleration using k m c v and f and plugging them into the linear spring damper equation 2 3 If spr_eq equals non_lin_damp it sets the damping coefficient c to a constant value calculates the current acceleration using B a Y w Q c v and f and plugging them into equation 2 5 If spr_eq is any other value the function issues an error The block diagram for spring is pictured below 24 Non Linear Damping State acc omega_not alpha x 3 delta v gamma cos ft phi Linear Damping State acc k m x c m v 1 m f Propogat State Call propagate function Output acc x_future v_future Figure 2 13 spring Function State Diagram The pseudo code for the spring function is below acc x_fut v_fut spring spr_eq m k x v f t_del switch spr_eq lin_damp c 10 acc k m c m v 1 m f call propagate function duffing will be implemented later otherwise error unrecognized spring constant type 25 2 5 3 propagate Function The propagate Function calculates the future displacement and velocity based off of
19. ditions 0 0 2 0 4 0 6 0 8 1 1 2 Time sec b Simulated motor displacement Peak Force Continuous Force c Motor Force k 95 9 10 Figure 3 1 Comparison of Motor Response to Simulated Response k 95 9 10 33 0 025 r r r r r r r r r dut E Target Position 0 021 q Actual Position n 1 0 005 0 015 Response to Initial Conditions 0 01 0 005 0 005 Distance m 1 o o 2 0 01 Displacement m 0 01 0 015 0 019 0 02 0 0 4 0 025 0 02 L 4 L L 1 L L 0 0 2 0 4 0 6 0 8 1 12 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2 Time sec time s a Motor displacement b Simulated motor displacement Peak Force Continuous Force o AAA time c Motor Force Figure 3 2 Comparison of Motor Response to Simulated Response k 295 0 10 In both of the figures the initial displacement was 0 024m and the initial velocity was Om s The motor response and the simulated response were similar for the two test trials above A closer response to the simulated response can be realized by adjusting the PID in the linear motor For these test cases and the ones to follow the PID values were set to default values When considering the motor force plots 34 Figures 3 1 c and 3 2 c is the minimum value of the continuous force oscillates from
20. e of gravity and other external loads 4 Identifying how vision assists in grasping objects along with identifying time delays can aid in clinical efforts To see how vision and time delays play a role experiments where done and the results are in the paper Manipulating the edge of instability 1 Results from bifurcation theory suggests that many nonlinear dynam ical systems exhibit low dimensional dynamics at the edge of of instability 5 Based on the center manifold theorem a low dimensional form on a center manifold can represent the dynamics of high dimentional systems at the edge of instability 5 In the paper the thumbpad was used to compress a slender spring as far as possible just before slipping This brought the thumb spring nervous system to the edge of instability 1 The experiments were done on 12 consenting young adults 9 male 3 female who were all right handed and had no recent hand impairments All subjects had no prior experience with the experimental setup The experimental setup looks like the figure below Motion capture Reflective camera marker l y O e S a 414 in e i is nol gano 7 Prt gr catty E Fy 30 2 N u Sample 2 0 Compression TK i ls l 2 6 10 Loadcell Time sec Figure 1 1 Schematic of original experimental setup 1 Subjects wrapped their fingers around the vertical post and placed their thumb pad on top of the endcap in Figure 1 1 The bottom
21. etails on how to setup the microcontroller and linear motor for usage There are some additional needed devices in order to get the system up and running such as a data acquisition board and power supply 4 1 Initial Setup 1 Wire up the DAQ and the Nanol7 2 Connect one wire from the AOO output channel of the DAQ to the analog input of the microcontroller 3 Connect one wire from the AO ground channel of the DAQ to the analog ground of the microcontroller 4 Hook up the RS232 cable to the microcontroller and the host computer 5 Hook the power supply up to the microcontroller the voltage range is 12V 30V I typically operate the system at 24V 39 4 2 Code Modification Once all connections are made modifications to the matlab code need to be done in order to get the correct functionality out of the system The steps below will help make these modifications 1 2 Set the path in matlab to the folder with all of the files Open up the open_serial c file and look in the function SerialInit int BaudRate and look for the line hSerial CreateFile _T COM5 and change the value in red to the communications port the microcontroller is hooked up to on the host computer If the baudrate needs to be changed look in the open_serial c file look in the mexFunction int nihs and look for the line ptr Seriallnit 57600 and change the value in red to a baudrate of your choosing To get further det
22. he maximum baudrate the microcontroller can be set to is 115 kBaud but 56 kbaud was the highest baudrate the motor was able to operate properly at The maximum force the motor can exert is 9 36N and can deliver this force for 2 seconds or until the motor reaches a temperature cut off After this the motor drops down to a maximum continuous force of 3 13N and doesn t go over this force until the temperature drops below the cut off temperature The maximum speed and acceleration the motor can provide are 2 71 m s and 91 57 m s respectively Variables that are used to calculate various equations for the motor are e Force constant kr e Continuous current Ie e Peak current I The equations used to calculate the force of the motor at any given time are e Constant force Fos kel 2 13 e Peak force Fos kpl max 2 14 The circuit design for the microcontroller is below 16 Figure 2 8 MCLM 3006 Circuit Design 2 2 4 3 Force Transducer To interpret the force a patient is exerting on the linear motor an ATI Nano17 force transducer is used Force is detected and fed back into a data acquisition board and used as in input for the linear or non linear equation The Nano17 measures force in the x y and z directions This allows not only the vertical force of a patient to be measured but also the force along the rod of the linear motor The maximum force the Nanol7 can endure in the x y and z directions are 25N
23. l energy when there is no forcing y 0 can be found for the oscillator by integrating equation 2 5 The solution is below E t soi ha E y 2 8 The potential energy graph is pictured below Potential Energy Potential Energy Energy o o 5 a 0 005 o i y y f i E 0 04 f 7 s i 7 0 OOS SEE OI o 001 70 02 50 08 0 04 0 04 0 03 0 02 0 01 o 001 0 02 003 0 04 Displacement m Displacement m a 8 gt 0 6 0 a lt 0 b 6 lt 0 6 0 a lt 0 Figure 2 4 Potential Energy In order to manipulate the edge of instability a subcritical pitchfork bifurcation needs to be replicated This done by changing and a so that Figure 2 4 b is inverted The graph of the subcritical bifurcation is pictured below 10 Potential Ener 0 01 T T 2y T 0 005 F 4 0 005 4 Energy 0 01 4 0 015 t J 0 02 L L L 1 L i L 0 04 0 03 0 02 0 01 0 0 01 0 02 0 03 0 04 Displacement m Figure 2 5 Subcritical Bifurcation 5 gt 0 and a lt 0 There will always be an equilibrium point at 0 0 To calculate the other two equilibrium points the equation below is used ge da 2 9 Q The phase portrait for the subcritical pitchfork bifurcation created using pplane8 which was designed by John C Polking at Rice University The phase portrait is pictured below 11 W 1 1 1 1 0 1 0 08 0 06 0 04 0 02 0 0 02 0 04 0 06 0 08 0 1
24. low the metacarpophalangeal MCP joint of the thumb but away from the thenar eminence to obtain a digital nerve block without affecting any musculature and associated sensors 1 In order to perform experiments like the one mentioned above the spring will be taken out of the system and replaced by a linear motor If successful the device developed should provide us with a different set of testing abilities since we can manipulate the edge of instability with a known low dimensional non linear dynamical system The system of choice will be the Duffing oscillator which is known to model chaotic behavior depending on the parameters chosen If configured correctly the Duffing oscilator could resemble the subcritical pitchfork bifurcation 6 that models spring buckling as seen in the paper Manipulating the edge of instability 1 Chapter 2 METHODS AND DESIGN In order to design the system off the shelf components such as a linear motor data acquisition board and a force transducer will be needed to make the new device used to manipulate the edge of stability Also parts designed in solid works such as a transducer holder and linear motor holder are designed to help combine all of the devices together Before making the motor behave like a non linear system we first attempt a linear case using the Spring and Damper Equation After implementing and testing the linear case we move on to a non linear system using the Duffing oscilla
25. oller A further description of how the wrc_serial c file works can be found in section 2 5 5 Make sure to compile this file using the mex command after making changes to it The default value used for the spring constant can be changed by looking into the spring_const m file Change the value of std_spr_const to change the default 41 value of the spring constant Additional variations of the spring constant can be added by adding more case statements to this function or editing the current case statements Details on how the spring_const m function works can be found in section 2 5 1 11 The linear and non linear equations the motor replicates can be modified by editing the spring m file Additional equations can be added by adding more case statements to the function and filling in the equation solved for the accelera tion Each equation should always call the propagate function after acceleration is computed Further details on this function can be found in section 2 5 2 4 3 Running the Program Everything is now properly setup for usage The function that controls the program for the system is motor_cntrl c This function has five input parameters which are spring_type k_type por vi pp To run the program do the following 1 Place the motor in the middle of the rod 2 Turn on the power supply and make sure the applied voltage is correct 3 Type the following command in the matlab window motor_cntrl
26. otor While current_time lt stop_time Y Set default values for the motor y Call spring constant function Plot all saved values Y Call spring function Get force from Load Cell Turn off motor and force transducer Set force of the motor Set displacement of motor l Get displacement of motor y Save all values 2 Figure 2 11 motor_cntrl Function Block Diagram The motor_cntrl functions makes calls to many other Matlab C and C func tions These functions are described in the following sections 2 5 1 spring const Function This function sets the spring constant variable for the linear or non linear function It has three inputs which are spr_const_type xy_force z_ force and outputs the spring 20 constant to be used k The variable meanings are defined below e spr_const_type string that will dictate how the spring constant is calculated xy_force horizontal force that is applied in the direction of the rod e z_force vertical force applied k the spring constant value that will be outputted It works by first setting a default value to k_const The function checks spr_const_type to see whether its value is const or zforce If spr_const_type is const then k is set to k_const If spr_const_type is zforce then k becomes the product of k_const and z_fo
27. rameter is always the address of the serial port object The second parameter is always an integer value from 1 to 4 that tells the function how to behave Every case below uses the first parameter to read or write to the serial port so the use of the first parameter will be omitted for the rest of the discussion When every case below is entered the function checks to make sure the right amount of inputs and outputs have been passed to the function before it proceeds If the wrong amounts of inputs or outputs are passed the function issues an error and exits The input and output checking is omitted from below but is seen in the function state machine If the second parameter is 1 the function needs two inputs and no outputs The second parameter value of 1 is used to configure the motor by calling the subfunction ConfigMotor The ConfigMotor calls another subfunction SerialWrite to write all the configuration values to the motor If the second parameter is 2 the function needs two inputs and one output The 29 second parameter value of 2 is used to set the position of the motor by using the SerialWrite function to write the get position command POS to the linear motor and then using the SerialRead subfunction to receive the position value of the motor If the second parameter is 3 the function has four inputs and no outputs The second parameter value of 3 sets the peak and continuous fo
28. rce If spr_const_type is any other value the function issues an error The block diagram for spring_const is pictured below 21 Initial State k_const const Z Force State k k_const z_force Constant State k k_const Figure 2 12 spring const Function State Diagram The pseudo code for spring_const is below k spring_const spr_const_type xy_force z_force k_const const switch spr_const_type const k k_const zforce k k_const z_force otherwise error unrecognized spring constant type The xy_force input variable is not used in the block diagram or the pseudo code This is because it currently has no affect on the spring constant but this feature can be added at a later date 22 2 5 2 spring Function This function determines what type of spring equation to use and makes calls to calculate the future displacement velocity and acceleration The function spring has the following inputs spr_eq m k x v f t del The outputs are acc x_fut and v fut These variables are defined below e spr_eq string that will dictate which equation will be used to calculate the future acceleration future displacement and future velocity e m mass of the system e k spring constant e c damping coefficient e x current displacement e v current velocity damping constant a duffing constant e current horizontal force applied to the system e w
29. rce of the motor using the values of the third and fourth input parameters respectively This is done by calling the subfunction Serial Write to write each value to the motor If the second parameter is 4 the function has two inputs and no outputs The second parameter value of 4 is used to turn off the motor The second parameter value of 4 uses SerialWrite subfunction to send the disable command DI to the motor After the motor has been disabled the wrc_serial Function calls the subfunction CloseHandle to close the serial port The state machine for the wrc_serial Function is pictured below 30 Declare all needed variables Initial State Has the minimum inputs and outputs been passed Get Inputs State Get the first and second input y Confi otor State Call ConfigMotor function No Error State Issue open port error Check Second Parameter Error State Issue open port error y Get Motor Position State y Y Set Motor Force State Disable State Call SerialWrite using DI is second input Has the correct amount of inputs and outputs been passed Yes Has the correct amount of inputs and outputs been passed No Yes y Want Position State Call SerialWrite using POS is second input y Write Peak Force State Call SerialWrite using
30. t 17 Nano17 Transducer Holder a Pe ed ee ae pee 18 Linear Motor Holder 2 62 aS Br mue de 64 eke A Ses 19 motor_cntrl Function Block Diagram 20 spring_const Function State Diagram 22 spring Function State Diagram tsar ah a RAE ee 25 propagate Function State Diagram elena We ln 27 open_serial Function State Diagrams is e e a 28 wrc_serial Function State Diagram 31 vii 3 1 3 2 3 3 3 4 3 9 3 6 Comparison of Motor Response to Simulated Response k 95 0 10 33 Comparison of Motor Response to Simulated Response k 295 0 10 34 Stable Motor Response 5 314 a 3140000 0 10 36 Stable Motor Response 314 a 3140000 d 10 36 Unstable Motor Response 6 314 3140000 6 10 37 Unstable Motor Response 6 314 3140000 6 10 37 vill Chapter 1 INTRODUCTION The nervous system is key to integrating and managing dynamic sensorimotor in formation Understanding neural control can be complicated because the intricate behavior is non linear and highly dimensional 3 Grasping objects with the fingers and thumb is a simple task that humans perform daily Vision is often not needed to grasp an object but when sensory feedback such as feeling in fingers is absent vision is more relied upon 1 Force vectors applied by the fingers on an object must be of sufficient magnitude to prevent slipping in the presenc
31. tor The designed system setup is pictured below a Complete View Figure 2 1 Designed System 2 1 Linear System 2 1 1 Spring and Damper Equation Hooke s Law of elasticity is known to approximate the extension of a spring The variables used in Hooke s Law in terms of our application are below e Fmotor restoring force of the motor e x displacement of the motor e k spring constant The equation is below Frs kx 2 1 To approximate an oscillating spring whose oscillations reduce due to friction we add damping to the equation above The following variables are added e c damping coefficient e x velocity The equation turns into Emotor kx ct 2 2 Using Newton s second law of motion F mz on the left side of the equation and adding a forcing input J on the right side the equation becomes m kx ci 1 k p Sa GS Bar 2 3 m m The simulated response to an initial displacement of 0 03m k 1N m c 0 5 I 0 is below Response to Initial Conditions 0 03 T 1 0 025 0 02 7 0 015 7 0 005 5 Displacement m 0 005 7 0 01 7 0 015 1 f 0 5 10 15 20 25 Time sec Figure 2 2 Simulated Spring Response The potential energy can be determined from equation 2 3 by putting all terms on the left hand side and having no forcing input J 0 and integrating equation The potential energy equation is below
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