Simulation Lab #4: Dynamic Modeling and Simulation of Muscle
Contents
1. Vill Exploration Phase The objective of the following sets of exercises is for you to gain experience using and modifying SIMM Pipeline simulation code You will also begin to look at how various muscle model parameters affect the dynamic response of a muscle tendon actuator insight that should aid your design of your optimal muscle for the virtual tug of war VII A Exploration of SIMM Pipeline Code l As mentioned previously muscle length is a state variable that is solved for by numerical integration of the contraction dynamics equation However tendon force is what must be applied to the mechanical system during an actual simulation Investigate the pipeline code and document how the transformation from muscle length to muscle tendon force is made Write down all the routines that are called and explain how the curves shown in Figure 1 are actually implemented in the code to make the transformations from muscle length to tendon force By further exploring the code explain how muscle tendon forces are actually applied to the skeleton within a dynamic simulation Often we are interested in knowing the net muscle moment about a joint since that quantity can be compared with joint moments computed using inverse dynamics How would you go about computing net muscle moments about a joint using SD FAST and or Pipeline code VIII B Modification of SIMM Pipeline Code Currently formain c only outputs muscle activation and the time der2. Comment out muscle_1 from the muscle file and fully activate muscle 2 at time t 0 Perform isometric simulations with the following tendon to fiber length ratios s 0 5 1 0 2 0 For each tendon to fiber length ratio maintain the sum of the slack tendon length and optimal fiber length constant Is thy 0 3m Overly plots of the muscle force as a function of time from the three simulations Describe any differences that you see between the curves in terms of the magnitude and timing of peak force and the shape of the curve Explain why the simulations vary the way they do Your explanation should describe how the muscle force length velocity and tendon force strain properties combine to give the response you see 12 IX Design Phase Virtual Muscle Tug of War A single elimination muscle tug of war tournament will be held between members of the class A match will consist of two muscles competing head to head muscle to muscle in a one second winner take all tug of war You will be required to specify the muscle tendon parameters and excitation time history subject to the constraints described below In each tug of war whichever muscle has moved the block onto their side at the end of one second is declared the winner Both muscles will start a match at rest with zero activation Design variables you must specify the values of the following variables F maximum isometric muscle force Te inverse of the normalized maximu
3. and muscle_2 contracts isometrically Perform isometric simulations with a M range of tendon to fiber length ratios s o 0 5 1 0 2 0 4 0 8 0 For each tendon to fiber length ratio maintain the sum of the slack tendon length and optimal fiber length 0 3m constant ls h 11 Overly curves of the muscle force time histories for each tendon to fiber length ratio Describe any differences that you see between the curves in terms of the rate of force development and steady state force achieved Using what you know about muscle activation dynamics and muscle tendon mechanics explain why the muscle force responses differ the way they do with changes in tendon to fiber length ration Isokinetic Responses Isokinetic exercises constant velocity are often used to characterize the force generating properties of muscle For example isokinetic dynamometers are used to measure the maximum joint moment a human can produce during constant a constant angular velocity contraction In this set of simulations you will analyze how the output of a muscle tendon actuator varies with tendon to fiber length ratio during an isokinetic contraction You should modify your formain c driver program such the block translates at a constant 0 7 m s to the right after starting an initial position 0 05 m to the left of the mid position Use prescribed motion in SD FAST to control the motion of the block The initial block position can be varied in sdfor c
4. tendon contraction dynamics e Learn how to model and simulate dynamic musculo tendon actions using SIMM Pipeline routines e Become comfortable with modifying existing code that models muscle activation and mechanics e Explore the effect of various model parameters and simulation conditions on the dynamic response of muscle e Design your own optimal musculo tendon actuator to compete in a virtual muscle tug of war Deliverables Turn in computer files from your modeling and simulation work to home me382 username L4 where username is your workstation login You are the only person who will have read and write permission to this directory during the week or two that this lab is in progress In addition hand in a written report that summarizes your findings and addresses the questions that are posed Please turn in 1 A written report A Microsoft Word template for the report called lab4_ report doc is available on the BME workstations in software nmbl tutorials me382 L4 Deposit the following computer files to home me382 username L4 Muscle file to be used in virtual muscle tug of war username msl A where username is your BME login ID Input Files Needed SIMM Files software nmbl tutorials me382 L4 tug of war jnt SIMM joint file used to animate the translating block simulations software nmbl tutorials me382 L4 tug_ of war msl SIMM muscle file that contains parametric descriptions o
5. Makefile forward The makefile for a forward dynamics simulation It assumes that the name of your SD FAST system description file is model sd and the name of your model specific C file is sdfor c Can be run by entering make f Makefile forward Libraries software simm2 pipeline n32libs libacpp a General purpose routines used for parsing files on input software simm2 pipeline n32libs libwrap a Precompiled library that accounts for muscle wrapping in the calculation of muscle tendon length and velocity during a dynamic simulation V Getting Started Copy all of the input files into a directory that you create for this lab mkdir L4 cd L4 cp software simm2 pipeline src cp software nmbl tutorials me382 L4 cp software simm2 pipeline n32libs mkdir bones cd bones cp software nmbl tutorials me382 L4 bones Se SE SHE SHE SHE SHE H H Note There are different versions of the muscle tendon model routines assigns c derivs c inits c in two of the directories accessed above software nmbl tutorials me382 L4 software simm2 pipeline src In this lab you will be using the routines in software nmb1 tutorials me382 L4 which contains a slightly modified version of model 4 described in the Pipeline tutorial VI Muscle and Tendon Modeling Thorough review articles have been written on the development and use of Hill type musculo tendon models in dynamic simulations of moveme
6. q7 are normalized by optimal fiber length Tendon slack length is the length at which tendons begin to transmit force when stretched Velocities are normalized by the maximum contraction velocity of muscle v For a given muscle tendon length velocity and activation level the model computes muscle force F and tendon force F7 Four muscle specific parameters are commonly used to scale the dimensionless curves for individual muscles M e Fi maximum isometric muscle force M e b optimal muscle fiber length e i tendon slack length and e pennation angle There are different choices of state that can be used for the muscle tendon model Either muscle tendon force Zajac 1989 Anderson and Pandy 1999 or tendon length Winters 1990 is commonly used In this lab you will be using muscle length which has been shown to have the advantage of not requiring inversion of the tendon force strain curve Schutte 1992 Thus the state equation for musculo tendon dynamics can be given by iM _ Mrs gt M I v G9 al 3 where 7 represents the generalized coordinates of the system the vector of time derivatives of the generalized coordinates 4 is the muscle activation and is the current muscle fiber length During a forward dynamic simulation muscle length and activation are treated as states and solved for by numerically integrating equations 1 and 3 simultaneously with the system equations of mo
7. Simulation Lab 4 Dynamic Modeling and Simulation of Muscle Tendon Actuators Laboratory Developers Darryl Thelen Silvia Blemker Clay Anderson Scott Delp ME 382 Modeling and Simulation of Human Movement Professor Scott Delp Stanford University Spring 2001 l Introduction The force producing properties of muscle are complex highly nonlinear and can have substantial effects on movement See McMahon 1984 for review For simplicity lumped parameter dimensionless muscle models capable of representing a range of muscles with different architectures are commonly used in the dynamic simulation of movement Zajac 1989 In this tutorial we will review the differential equations that describe muscle activation and muscle tendon contraction dynamics when using a Hill type muscle model You will use SIMM Pipeline routines to implement muscle tendon models and conduct some simulations to investigate how various model parameters can affect the dynamic response of the actuators The lab will conclude with a Virtual Muscle Tug of War in which you will design an optimal muscle and compete directly against others in the class may the best muscle win ll Objectives The purpose of this tutorial is to learn how to use and modify SIMM Pipeline routines to model and simulate muscle tendon dynamic contractions By working through this tutorial you will e Become familiar with the differential equations describing muscle activation and muscle
8. e animation slower for the animation to proceed at a speed that you can visualize The sizes and colors of the muscles will change to reflect their activation levels Use the motion curve gt command in the Plot Maker to make graphs of the generalized coordinate x and the muscle activations You should get results that look like those shown in Figure 3a and b 7 Chapter 2 of the Pipeline manual describes how the muscle excitation time relationship can be specified as either an open loop step function or spline fit Sketch out the excitations of muscles A and B that were used to generate the previous simulation 8 A useful feature of defining the muscle parameters and excitations within the muscle file is the ability to make changes to the model parameters or inputs without recompiling the program You will be using these capabilities extensively in designing your muscle for the tug of war Practice changing the excitation functions and rerun the simulations to demonstrate how the simulations respond to difference excitation patterns Motion Curves r 0 84 S muscle_1 muscle_2 weal Ie 0 0 0 2 04 0 6 08 1 0 sd_motion 2 sd_motion 2 muscle_1 sd_motion 2 muscle_2 a b c Figure 3 Muscle activations a block translation b and muscle forces c for initial demonstration simulation 10
9. e you started SIMM e model sd SD FAST system description file that is used to generate code that describes the system equations of motion e sdfor c Model specific C code that contains the body segment parameters and joint kinematics Set the default parameter values that appear in forparams txt Forparams txt contains the names of the input and output files that the simulation needs The following file names should be set e muscle file tug_of war msl e output_motion file forward mot Compile the simulation program using the makefile provided The makefile includes a SD FAST call that processes your system description file Therefore as in the last tutorial you need to be logged on to Hill for the makefile to be able to run SD FAST make f Makefile forward Don t be surprised to see a few warnings the first time you compile your code The SD FAST library sdlib c does not have an accompanying header file and consequently you will get warnings about functions not being previously declared Running the makefile will create an executable file called sdfor By default the makefile compiles the code using the debugging option g Once you have confirmed that the simulation is running properly you can change this option to 02 to make the program run faster 5 Run the simultion sdfor 6 View the simulation results in SIMM Load the motion file forward mot and animate the motion You may need to set the gear speed of th
10. ed to be 15 and 50 ms respectively Zajac 1989 Winters 1990 VI B Muscle tendon contraction dynamics The force producing properties of muscle are complex and nonlinear See McMahon 1984 for review For simplicity lumped parameter dimensionless muscle models capable of representing a range of muscles with different architectures are commonly used in dynamic simulation of movement Zajac 1989 In this model the muscle force length and force velocity and tendon force strain relationships are represented by dimensionless curves Figure 1 o MT a 1 IMcos a 000000 Vy OA Wad A Em normalized tendon force normalized muscle force normalized muscle force normalized i M normalized muscle fiber 1 A muscle fiber V M 4M Vmax Lo strain 1d S Li a tendon l ls length velocity Figure 1 Dimensionless model of muscle and tendon Muscle properties are represented by an active contractile element CE in parallel with a passive elastic element top Muscle force is dependent on muscle fiber length middle plot and velocity right plot Muscle is in series with tendon which is represented by a nonlinear elastic element left plot Pennation angle is the angle between the muscle fibers and the tendon The forces in muscle and tendon are normalized by peak isometric muscle force F Muscle fiber length and tendon length
11. etric force F and time is normalized to the inverse M ee normalized maximum contraction velocity Te l ix After normalization the contraction dynamics state equation is written as re g a where the normalized muscle length is given by Pes TT Solving the state equation requires inversion of the force velocity curve which can be problematic when either muscle activation is near zero or the muscle fibers are very long or very short which results in a small active force length factor This difficulty is overcome in the model by including a small amount of passive damping in parallel with the contractile element such that the force velocity curve remains invertible at low activation Schutte 1992 The Pipeline code assigns c derivs c inits c can include up to 10 different muscle tendon models numbered 1 through 10 In this lab you will be using muscle model 7 which is muscle model 4 which is described in the Pipeline manual with a slight change in model parameters to make them more intuitive The dynamic muscle parameters that must be defined in the muscle file are ee e timescale inverse of the normalized maximum contraction velocity te Vax max is af Sey ae expressed in fiber lengths per second max max 0 e activation_timeconstant muscle activation time constant fect e deactivation_timeconstant muscle deactivation time constant Teac rc b e damping normalized
12. f the two muscles included in the translating block simulations software nmbl tutorials me382 L4 bones small box asc bone file for a 10 cm x 10 cm box software nmbl tutorials me382 L4 bones floorls asc software nmbl tutorials me382 L4 bones floor2s asc bone files for the floor Fh Fh Pipeline Code Main Driver Routine for Forward Dynamic Simulations software simm2 pipeline src formain c This file contains the main driver routine for a forward dynamic simulation and several other utility routines that are independent of the musculoskeletal model in the simulation This file can be generated by SIMM Pipeline You will probably want to generate it once and then customize it to your specific simulation This file contains the subroutines main sdumotion sduforce and init _motion General Purpose Source Files software simm2 pipeline src gmc c General muscle code a collection of routines to calculate musculo tendon model quantities e g passive muscle force tendon force and pennation angle from current state information software simm2 pipeline src mathtools c General purpose mathematics routines software simm2 pipeline src pipetools c Utility routines for conducting dynamic simulations software simm2 pipeline src readmuscles c Routines to read in the musculo tendon model information from a muscle file Attachment points wrapp
13. ing surfaces and muscle specific parameters e g maximum isometric force tendon slack length etc are input and use during the simulation Muscle files are read at runtime allowing the changing of muscle parameters and muscle excitations without altering the simulation code software simm2 pipeline src readtools c General purpose utility routines used to read in data from input muscle files and kinetics data files Muscle Tendon Model Source Files software nmbl tutorials me382 L4 assigns c software nmbl tutorials me382 L4 derivs c software nmbl tutorials me382 L4 inits c Code implementing dynamic muscle tendon models The code is set up such that you can use one of the existing models described in the Pipeline manual or can alternatively use templates that are provided to create your own model Header Files software simm2 pipeline src basic h Contains defines and enum that are used in many source files software simm2 pipeline src functions h Contains prototypes for many of the functions in the source files software simm2 pipeline src structs h Contains definitions of all of the structures that are used in the Dynamics Pipeline software simm2 pipeline src universal h Contains includes for all of the standard header files This file should be included at the top of every source file in the Pipeline Make Files software simm2 pipeline src
14. inters and SL Woo Springer Verlag New York Zajac FE 1989 Muscle and tendon properties models scaling and application to biomechanics and motor control CRC Critical Reviews in Biomedical Engineering 17 359 411 14
15. ivatives of muscle activation to the motion file However you might well be interested in plotting muscle tendon force Modify the driver routine formain c to output headers for and the values of muscle tendon forces In the header name the muscle forces muscle _1 force and muscle 2 force To access muscle forces you may want to look in the file structs h to see how muscle force is saved in a muscle structure Using the simulation of the previous section you should get muscle tendon forces that look like those shown in Figure 3c VII C Affect of Model Parameters on Dynamic Response T M Zajac 1989 showed that the tendon to fiber length ratio s Il can have substantial effects on the mechanical response of a muscle tendon actuator The following sets of simulations are designed to have you vary the simulation code and parameters in order to examine the effect of tendon to fiber length ratio on the mechanical response of a muscle during isometric and isokinetic contractions For each of these simulations comment out muscle_1 in the muscle file such that the simulation includes only one muscle muscle_2 l Isometric Responses Perform a set of simulations in which you vary the tendon to fiber length ratio and look at how this affects the force time response in an isometric simulation For these simulations fully activate muscle_2 at time t 0 Modify the formain c program such that the block does not move use prescribed motion in SD FAST
16. m contraction velocityl 1 Veal Ip optimal muscle fiber length o muscle fibre pennation angle corresponding to muscle fiber at optimal length x t muscle excitation time history Definition of other model parameters you will use M A physiological cross sectional area of muscle in cm V muscle volume A iy o gt maximum isometric stress of muscle assumed equal to 35 N cm Zajac 1989 Design constraints Bia doe V lt 100 cm 0 05m lt 1 lt 0 2 m K gt 0 1m 0 0 lt a lt 30 2 lt lt 10 max FY IM lt 175 W max x at lt 0 5 10 ms lt t lt 20 ms act 40 MS ST ga E 60 ms 30 MS S Tercr Tace S40 ms act Additional Note Passive muscle forces will not be included in the tug of war In performing simulations as part of your muscle design process you are responsible for turning off muscle passive force To do this edit the file derivs c Under musc_deriv_func 7 replace the following line passive force calc nonzero passive force ms normstate fiber length 0 0 with 13 passive force 0 0 Design Process You are individually responsible for devising and implementing a process to use in designing the muscle tendon actuator For this class the process and results should be clearly documented in a brief report The report excluding figures and tables should not be more than 2 pages It should include the following sections e Introduction o Objectives of you
17. nt Zajac 1989 Winters 1990 Following is a brief review of the Hill type model put forth in those papers as it has been implemented within SIMM pipeline If you are interested in greater detail you should consult the references directly VI A Activation dynamics A muscle is not capable of generating force or relaxing instantaneously The development of force is a complex sequence of events which begins with the firing of motor units and culminates in the formation of actin myosin cross bridges within the myofibrils of the muscle When the motor units of a muscle depolarize action potentials are elicited in the fibers of the muscle and cause calcium ions to be released from the sarcoplasmic reticulum The increase in calcium ion concentrations then initiates the cross bridge formation between the actin and myosin filaments See Guyton 1986 for review In isolated muscle twitch experiments the delay between a motor unit action potential and the development of peak force has been observed to vary from as little as 5 milliseconds for fast ocular muscles to as much as 40 or 50 milliseconds for muscles comprised of higher percentages of slow twitch fibers The relaxation of muscle depends on the re uptake of calcium ions into the sarcoplasmic reticulum This re uptake is a slower process than the calcium ion release and so the time required for muscle force to fall can be considerably longer than the time for it to develop In the muscle simula
18. passive damping in parallel with contractile element b FUL 0 0 max Vil Using SIMM Pipeline to Simulate Muscle Tendon Dynamics VII A SIMM Pipeline Basics The basic steps involved in creating a muscle driven simulation using SIMM Pipeline and SD FAST involve 1 Using SIMM Pipeline to generate an SD FAST system description and model specific code sdfor c from a joint file 2 Running SD FAST to process the system description file and generate code that represents the equations of motion of the system 3 Incorporate SIMM Pipeline and SD FAST routines with any additional custom code you desire to simulate your particular application Often custom code is used to simulate external contact forces to run a sensitivity study or to perform numerical optimization A driver program formain c is generated by SIMM Pipeline and can be used as a starting point for creating an application specific driver routine for your simulation AS a preview please read the following sections of the Dynamics Pipeline manual Chapter 1 Introduction Chapter 2 Input Files Chapter 3 Muscle Modeling Chapter 4 Forward Dynamics Analysis Note that Chapter 2 describes the additional information that must be included in a SIMM joint and muscle file in order to create a dynamic simulation In a joint file this additional information includes the mass and moments of inertia of body segments in your model In a muscle file dynamic muscle tendon model parameter
19. r design e Methods o Outline of steps used in your design process e Results o Should include a combination of mathematical analysis parameter sensitivity studies and results of prototype muscle simulations o Description of the final muscle design e Discussion o Justification of your final design o Evaluation of the strengths and potential weaknesses of your design e g under what conditions do you expect your muscle to perform well X References Anderson FC and Pandy MG 1999 A dynamic optimization solution for jumping in three dimensions Computer Methods in Biomechanics and Biomedical Engineering 2 201 231 Delp SL Loan P 1995 A graphics based software system to develop and analyze models of musculoskeletal structures Comput Biol Med 1 21 34 Hatze H 1976 The complete optimization of human motion Mathematical Biosciences 28 99 135 McMahon TA 1984 Muscles Reflexes and Locomotion Princeton University Press Princeton New Jersey Schutte LM 1992 Using Musculoskeletal Models to Explore Strategies for Improving Performance in Electrical Stimulation Induced Leg Cycle Ergometry Ph D Dissertation Mechanical Engineering Department Stanford University Symbolic Dynamics Inc 1996 SD FAST User s Manual Version B 2 Mountain View CA Winters JM 1990 Hill based muscle models a systems engineering perspective in Multiple Muscle Systems Biomechanics and Movement Organization edited by JM W
20. s are added that are used in the activation and dynamic contraction equations In addition the muscle file can include a description of the muscle inputs or excitations Either open loop step patterns or spline fits can be used to specify excitation as a function of time or as a function of a generalized coordinate VII B Muscle Tug of War Model In this lab you will be using a simple mechanical model to investigate the dynamics properties of muscle tendon actuators The model consists of a block translating on a frictionless surface under the action of two opposing muscle tendon actuators i e a muscle tug of war Figure 2 gt muscle 1 Figure 2 Model consists of a translating block on a frictionless surface being acted upon by two opposing muscles The block is 0 1 x 0 1 m and has a mass of 20 kg The distance between fixed supports is 0 7 meters With the block centered the muscle tendon lengths are 0 3 m The following steps should be followed to create a dynamic muscle driven simulation of the translating block model l Start SIMM and load the joint and muscle files Change to lab 4 directory cd L4 Start SIMM simm2 Load the muscle and joint files Joint file tug_of_war jnt Muscle file tug_of_war msl Use SIMM Pipeline to write dynamics files for your system Open the File Writer tool and click on the forward dyn button SIMM Pipeline will create two files in your current working directory the directory wher
21. tion Within a simulation the muscle tendon force F id acting on the skeleton is computed from the system states muscle length activation generalized coordinates and generalized speeds The generalized coordinates and generalized speeds of the system are first used to numerically compute the muscle tendon length T and muscle tendon velocity l For example muscle tendon length is computed by adding up the incremental lengths between muscle path points as defined in a joint file Checks are included to see if wrapping surfaces are being contacted by the muscle and if so the additional length required to wrap around a surface is included in the computation of the muscle tendon length Muscle tendon velocities are computed similarly by adding up the incremental velocities between muscle path points as defined in a joint file After computing E and L7 the muscle activation is used along with the dimensionless curves shown in Figure 1 to compute the muscle tendon force VI C SIMM Pipeline Implementation of Muscle Tendon Model Various versions of the Hill type muscle tendon models as discussed previously have been implemented in the SIMM Pipeline code There are a few notes that should be made with regards to the implementation First a normalized form of the contraction dynamics state equation is used All length quantities are normalized to optimal muscle fiber length lo force quantities are normalized to maximum isom
22. tions you will conduct in this lab activation dynamics is modeled using a first order differential equation with a variable time constant This equation relates the rate of change in activation i e the concentration of calcium ions within the muscle to excitation i e the firing of motor units j 0 0 T a x 1 where 4 is the activation level of a muscle is the excitation level and T is a variable time constant which is given by Face T deact X T deact x gt a t a x 2 T deact x lt a In the above equation the parameters describe the rates of rise of activation act and deactivation Taeact in response to muscle excitation In the model activation is allowed to vary continuously between zero no contraction and one full contraction In the body the excitation level of a muscle is a function both of the number of motor units recruited and the firing frequency of the motor units Some models for excitation contraction coupling distinguish these two control mechanisms Hatze 1976 but it is often not computationally feasible to use such models when conducting complex dynamic simulations In the simulation the muscle excitation signal is assumed to represent the net effect of both motor neuron recruitment and firing frequency and like muscle activation is also allowed to vary continuously between zero no excitation and one full excitation The activation and deactivation time constants can be assum
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