{"title": "Simulation of Optimal Movements Using the Minimum-Muscle-Tension-Change Model", "book": "Advances in Neural Information Processing Systems", "page_first": 627, "page_last": 634, "abstract": null, "full_text": "Simulation  of Optimal Movements  Using the \n\nMinimum-Muscle-Tension-Change  Model. \n\nMenashe Dornay* \n\nYoji  Uno\" \n\nMitsuo Kawato * \n\nRyoji  Suzuki** \n\n\u00b7Cognitive  Processes  Department,  A TR  Auditory  and  Visual  Perception  Research \nLaboratories,  Sanpeidani,  Inuidani,  Seika-Cho,  Soraku-Gun,  Kyoto  619-02 Japan. \n\n\u00b7\u00b7Department  of  Mathematical  Engineering  and  Information  Physics,  Faculty  of \nEngineering,  University of Tokyo,  Hongo,  Bunkyo-ku, Tokyo,  113  Japan. \n\nAbstract \n\nThis  work  discusses  various  optimization  techniques  which  were \nproposed  in  models  for  controlling  arm  movements.  In  particular,  the \nminimum-muscle-tension-change  model  is  investigated.  A  dynamic \nsimulator of the  monkey's  arm,  including  seventeen  single  and double \njoint muscles,  is  utilized  to  generate  horizontal  hand  movements.  The \nhand  trajectories  produced by  this  algorithm  are discussed. \n\nINTRODUCTION \n\n1 \nTo  perform  a  voluntary  hand  movement,  the  primate  nervous  system  must  solve  the \nfollowing problems:  (A)  Which trajectory (hand path and velocity)  should be used while \nmoving  the  hand  from  the  initial  to the  desired position.  (lB)  What muscle forces  should \nbe generated.  Those  two  problems are  termed  \"ill-posed\"  because  they  can  be  solved in \nan  infinite  number  of ways.  The  interesting  question  to  us  is:  what  strategy  does  the \nnervous  system  use  while  choosing a  specific  solution for  these  problems? The  chosen \nsolutions  must  comply  with  the  known  experimental  data:  Human  and  monkey's  free \nhorizontal  multi-joint  hand  movements  have  straight  or  gently  curved  paths.  The  hand \nvelocity  profiles  are  always  roughly  bell  shaped  (Bizzi  &  Abend  1986). \n\n627 \n\n\f628 \n\nDamay,  Uno,  Kawato,  and Suzuki \n\n1.1  THE MINIMUM-JERK MODEL \nFlash  and  Hogan  (1985)  proposed  that  a  global  kinematic  optimization  approach,  the \nminimum-jerk model, defines a solution for the trajectory detennination problem (problem \nA).  Using this  strategy, the nervous  system is  choosing the  (unique) smoothest trajectory \nof the  hand  for  any  horizontal  movement,  without  having  to  deal  with  the  structure  or \ndynamics  of the  ann.  The minimum-jerk model  produces reasonable approximations  for \nhand  trajectories  in  unconstrained  point  to  point  movements  in  the  horizontal  plane  in \nfront  of the  body  (Flash  & Hogan  1985;  Morasso  1981;  Uno  et  al.  1989a).  It fails  to \ndescribe, however, some important experimental findings for human arm movements (Uno \net al.  1989a). \n\n1.2  THE EQUILIBRIUM-TRAJECTORY HYPOTHESIS \nAccording to  the  equilibrium-trajectory  hypothesis  (Feldman  1966),  the  nervous  system \ngenerates  movements  by  a gradual  change in  the  equilibrium posture of the  hand:  at all \ntimes  during  the  execution  of a  movement  the  muscle  forces  defines  a  stable  posture \nwhich  acts  as  a  point  of attraction  in  the  configurational  space  of the  limb.  The  actual \nhand movement is the realized trajectory. The realized hand trajectory is  usually different \nfrom  the  attracting  pre-planned  virtual  trajectory  (Hogan  1984).  Simulations  by  Flash \n(1987), have suggested that realistic multi-joint ann movements at moderate speed can be \ngenerated  by  moving  the  hand  eqUilibrium  position  along  a  pre-planned  minimum-jerk \nvirtual trajectory. The interactions of the dynamic properties of the ann and the attracting \nvirtual  trajectory  create  together  the  actual  realized  trajectory.  Flash  did  not  suggest  a \nsolution  to  problem lB. \n\nA static local optimization algorithm related to the  equilibrium-trajectory hypothesis and \ncalled backdriving was proposed by Mussa-Ivaldi et al.  (1991). This algorithm can be used \nto solve problem lB  only after the virtual  trajectory is  known. The virtual  trajectory is  not \nnecessarily a minimum-jerk trajectory. Driving the arm from a current equilibrium position \nto the next one on the virtual trajectory is  perfonned by two steps:  1)  simulate a passive \ndisplacement  of the  arm  to  the  new  position  and  2)  update  the  muscle  forces  so  as  to \neliminate  the  induced  hand  force.  A  unique active  change  (step  2)  is  chosen  by  finding \nthese  muscle  forces  which  minimize  the  change  in  the  potential  energy  stored  in  the \nmuscles.  Using  a  static  model  of the  monkey's  arm,  the  first  author  has  analyzed  this \nsequential  computational  approach,  including  a  solution  for  both  the  trajectory \ndetennination  (A)  and  the  muscle  forces  (lB)  problems  (Domay  1990,  1991a,  1991b). \n\nThe  equilibrium-trajectory  hypothesis  which  is  using  the  minimum-jerk  model  was \ncriticized by Katayama and Kawato (in preparation).  According to their recent findings, \nthe values of the dynamic stiffness used by Flash (1987) are too high to be realistic. They \nhave  found  that  a  very  complex  virtual  trajectory,  completely  different  from  the  one \npredicted by  the minimum-jerk model,  is  needed for  coding  realistic  hand  movements. \n\n\fSimulation of Optimal Movements Using the  Minimum-Muscle-Tension-Change Model \n\n629 \n\n2  GLOBAL  DYNAMIC  OPTIMIZATIONS \nA set of global dynamic optimizations have been proposed by Uno et al.  (1989a,  1989b). \nUno et al.  suggested  that  the  dynamic  properties  of the  arm  must be  considered  by  any \nalgorithm  for  controlling  hand  movements.  They  also proposed  that the  hand  trajectory \nand the motor commands (joint torques, muscle tensions, etc.,) are computed in  parallel. \n\n2.1  THE MINIMUM-TORQUE-CHANGE MODEL \nUno et al.  (1989a) have proposed the minimum-torque-change model. The model proposes \nthat  the  hand  trajectory  and  the  joint  torques  are  determined  simultaneously,  while  the \nalgorithm minimizes globally the rate of change of the joint torques. The minimum-torque(cid:173)\nchange model was  criticized by Flash  (1990),  saying  that the rotary  inertia used  was  not \nrealistic. If Flash's inertia values are  used then  the hand path predicted by  the  minimum(cid:173)\ntorque-change  model  is  curved  (Flash  1990). \n\n2.2  THE MINIMUM-MUSCLE-TENSION-CHANGE MODEL \nThe  minimum-muscle-tension-change model  (Uno  et al.  1989b,  Domay  et  al.  1991)  is  a \nparallel dynamic optimization approach in which the trajectory determination problem (A) \nand  the  muscle  force  generation  problem  (]B)  are  solved  simultaneously.  No  explicit \ntrajectory is  imposed on  the  hand,  but that it must reach the  final  desired  state  (position, \nvelocity, etc.) in a pre-specified time.  The numerical solution used is a \"penalty\" method, \nin which  the  controller minimizes  globally  by iterations an  energy function  E  : \n\n(1) \n\nE  is  the  energy  that  must  be  minimized  in  iterations.  ED \nis  a  collection  of  hard \nconstraints, like, for example that the hand must reach the desired position at the specified \ntime.  Es  is  a smoothness  constraint, like the  minimum-muscle-tension-change model.  \"(cid:173)\nis  a regularization  function,  that needs  to  become  smaller and  smaller as  the  number of \niterations  increases.  This  is  a  key  point  because  the  hard  constraints  must  be  strictly \nsatisfied  at  the  end  of the  iterative  process.  \u00a3  is  a  small  rate  term.  The  smoothness \nconstraint  Es  ' is  the  minimum-muscle-tension-change model,  defined as: \n\n(2) \n\n!; is  the  tension  of muscle  i,  n  is  the  total  number of muscles,  to  is  the  initial  time and \ntrut  is  the  final  time  of the  movement. \nPreliminary  studies  have  shown  (Uno  et  al.  1989b)  that  the  minimum-muscle-tension(cid:173)\nchange model  can  simulate  reasonable  hand movements. \n\n\f630 \n\nDamay,  Uno,  Kawato,  and Suzuki \n\n3  THE MONKEY'S  ARM  MODEL \nThe  model  used  was  recently  described  (Domay  1991a;  Domay et al.  1991).  It is  based \non  anatomical  study  using  the  Rhesus  monkey.  Attachments  of 17  shoulder.  elbow  and \ndouble  joint  muscles  were  marked  on  the  skeleton.  The  skeleton  was  cleaned  and \nreassembled  to  a  natural  configuration  of a  monkey  during  horizontal  arm  movements \n(Fig.  1).  X-ray analysis  was  used to create a simplified horizontal model  of the arm  (Fig. \n1).  Effective  origins  and  insertions  of  the  muscles  were  estimated  by  computer \nsimulations to ensure the postural stability ofthe hand at equilibrium (Domay 1991a). The \nsimplified dynamic  model  used  in  this  study is  described in  Domay et al.  (1991). \n\nFigure  1:  The Monkey's  Arm  Model.  Top left is  a  ventral view  of the  skeleton.  Middle \nright  is  a dorsal  view.  The  bottom  shows  a  top-down  X-ray  projection  of the  skeleton. \nwith  the  axes  marked on it.  The photos  were taken  by Mr.  H.S.  Hall.  MIT. \n\n\fSimulation of Optimal Movements Using the  Minimum-Muscle-Tension-Change Model \n\n631 \n\n4  THE BEHAVIORAL TASK \nWe  tried  to  simulate  the  horizontal  arm  movements  reported  by  Uno  et al.  (1989a)  for \nhuman  subjects,  using  the  monkey's  model.  Fig.  2  (left)  shows  a  top  view  of the  hand \nworkspace  of the  monkey  (light  small  dots).  We  used  7  hand  positions  defined  by  the \nfollowing  shoulder and elbow relative angles  (in  degrees):  Tl  (14,122); Tz {67,100};  T3 \n{75,64};  T4  {63,45};  Ts  {35,54};  T6  {-5,lOl}  and T7  {-25,45}. The joint angles  used  by \nUno  et al.  (1989a)  for  T4  and  T7,  {77.22}  and  {O,O},  are  out  of the  workspace  of the \nmonkey's  hand  (open  circles  in  Fig  2.  left).  We  approximated  them  by  our  T4  and  T7 \n(filled circles).  The behavioral task that we simulated using the minimum-muscle-tension(cid:173)\nchange model consisted of the  4  trajectories  shown  in Fig.  2  (right). \n\n5  SIMULATION RESULTS \nFigure 2 (right) shows  the paths (Tz->T6 ), (T3->T6),  (T4->T1), and  (T7->Ts)' The paths T2-\n>T6'  T3->T6 and  T7->Ts  are  slightly  convex.  Slightly  convex  paths  for  Tz->T6  were \nreported in  human  movements  by Flash  (1987),  Uno et al.  (l989a) and Morasso  (1981). \nHuman T3->T6 paths have a small tendency to be slightly convex (Uno et al.  1989a; Flash \n(1987).  In  our simulations,  Tz->T6  and T3->T6 have  slightly  larger curvatures  than  those \nreported in humans. Human large movements from  the side of the body to the front of the \nbody  similar  to  our  T7->Ts  were  reported  by  Uno  et  al.  (1989a).  The  path  of  these \nmovements is convex and similar to our simulation results. The simulated path of T 4-> T 1 \nis slightly curved  to  the  left and  then  to  the  right,  but roughly  straight.  The human's T4-\n> TI  paths look slightly straighter than  in  our simulations  (Uno et at.  1989a;  Flash  1987). \n\n0.( \n\n,.--..... \nif) \n~  0:3 \n(l) \n~ \n(l)  0.2 \nE \n' - '  o. i \n>-' \n\n0 .0 \n\nT4 \nT3 ___  , \nT5 \n\"-\n-..:....... \n. ... \n~ .......... :::.:::.. \n.... \nT6 \n\nT 1 \n\n...... \n\n{.;#:!.. \n\n\u2022 \n\n'. \n\n\\\" \n\n\\\\ \n\nT7 \n\n+ \n\n-01  ! \n\n-C.2 \n\n-01 \n\n0.0 \n\n0.1 \n\n0.2 \n\n03 \n\n04 \n\nX  (meters) \n\n-O.38m \n\nX \n\n-O.2m \n\nFigure 2.  The Behavioral Task. The left side shows the hand workspace (small dots).  The \nshoulder  position  and  origin  of coordinates  (0,0)  is  marked  by  +.  The  elbow  location \nwhen  the  hand  is  on  position  TI  is  marked  by  E.  The  right  side  shows  4  hand  paths \nsimulated by the minimum-musc1e-tension-change model.  Arrows  indicate the directions \nof the  movements. \n\n\f632 \n\nDomay,  Uno,  Kawato,  and Suzuki \n\nFig.  3  shows  the  corresponding  simulated  hand  velocities.  The  velocity  profiles  have  a \nsingle peak and are roughly bell shaped,  like those reported for  human  subjects.  The left \nside of the  velocity profile  of T 4->T t  looks  slightly irregular. \n\nThe  hand  trajectories  simulated  here  are  in  general  closer  to  human  data  than  those \nreported  by  us  in  the  past  (Domay  et al.  1991).  In  the  current  study  we  used  a  much \nslower protocol for reducing A than in the previous study, and we think that we are closer \nnow  to  the  optimal  solution  of  the  numerical  calculation  than  in  the  previous  study. \nIndeed,  the hand velocity profiles and muscle tension profiles look smoother here than in \nthe previous  study.  It is in general very difficult to  guarantee that the optimal solution is \nachieved,  unless an  unpractical large number of iterations is  used.  Fig.  4  (top,left) shows \nthe way  ED  and  Es  of equation  1 are changing as  a function  A for the trajectory T7->T5\u2022 \nIdeally.  both  should  reach  a  plato  when  the  optimal  solution  is  reached.  The  muscle \ntensions  simulated  for  T1->T5  are  shown in  Fig.  4.  They  look quite  smooth . \n\nf :;:-\n\n. \n.. \n'/\"\" \n. \n. \n-. \n. \n. \n: \n\u2022 1:: \nj \n: \n~ \n.... \no  ./' \n\n\u2022\u2022 \n\n\u2022  J \n\n/\" .. \n~ . \n:/ \n. \n' .. \n\n. / \n\n\u2022\u2022 \n\n02 \n\n\" \n\n.t \n\nII \n\n\u2022\u2022 \n\ne. \n\n.,  \" \n\nII  \u2022\u2022 \n\n.\u2022  T .. T \n11 \n\n4 \n\nI. .. \n\n..  T  O+:1j \n5 \n\n7 \n\n.. \n\nt. \n\n-() ---'--'-'--,-\n\nTlme(.' \n\nG' \n\n00 \n\n/~\"\" \n. ... \n. \n. \n.:. \n. \n. \n. \n. \n.. \n-, \n\\  ... \n\u2022\u2022 \ne.. \n\n...  a. \n\n! \n\n, \n\n, \n\n.\u2022 \n\n~  I \n\n, \n\n.,,/ \n\n\u2022\u2022 \n\nFigure 3.  The Hand Tangential  VelocIty. \n\n6  DISCUSSION \nVarious  control  strategies  have  been  proposed  to  explain  the  roughly  straight  hand \ntrajectory  shown  by  primates  in  planar reaching  movements.  The  minimum-jerk  model \n(Flash  &  Hogan  1985)  takes  into  account  only  the  desired  hand  movement.  and \ncompletely ignores the dynamic properties of the arm. This simplified approach is a good \napproximation for many movements, but cannot explain some experimental evidence (Uno \net al.  1989a).  A more demanding  approach,  the  minimum-torque-change  model  (Uno et \nal.  1989a), takes  into account the  dynamics  of the arm,  but emphasizes only  the torques \nat  the  joints,  and  completely  ignores  the  properties  of  the  muscles.  This  model  was \ncriticized  to  produce  unrealistic  hand  trajectories  when  proper  inertia  values  are  used \n(Flash 1990). A third and more complicated model is the minimum-muscle-tension-change \nmodel (Uno et a1.  1989b. Domay et al.  1991). The minimum-muscle-tension-change model \nwas  shown here to produce gently curved hand movements, which although not identical, \nare quite close to  the primate behavior.  In the current study the initial  and final  tensions \nof the muscles  were  assumed to be  zero.  This  is  not a  realistic  assumption  since  even  a \nstatic  hand  at an  equilibrium  is  expected  to  have  some  stiffness.  Using  the  minimum(cid:173)\nmuscle-tension-change model with non-zero initial and final  muscle tensions  is  a logical \n\n\fSimulation of Optimal  Movements Using the  Minimum-Muscle-Tension-Change Model \n\n633 \n\nMuscle  Forces \n\n1 '  Se2 \n\nSe3 \n\nN \n\nSel \n\n/,I:~ .... \n\nOJ ~ \u2022 \n\n_-,-. _ _ _ \n\n\u2022\u2022 \n\n\u2022\u2022 \n\n\u2022\u2022 \n\n\u2022\u2022 \n\n\u2022\u2022 \n\n\u2022  II \n\n\u2022  , \n\n\u2022  '! \n\n0 I \n\n\u2022  ~ \n\ne I \n\nSe4 \n\nSe5 \n\n~\\ .. \n\nSfG \n\n.. / \\  \n\\ \n\\ \n\n.... \n..: \n\n\u2022 t \n\nII \n\n~-\nIJ \n\nI '  \n\nII \n\nII \n\n\u2022 \n\nSf7 \n\n./\\\\ \n\n,  : \n\n.1 \n\nI I   I '   I .  \n\nII \n\nII \n\n0& \n\nII \n\nI .   I I   I '   \u2022\u2022 \n\n.1 \n\n\u2022 \u2022 \n\nI I . .   ,.  I I   '1 \n\n' I  \n\nSfB \n\n.~. \n\nSf9 \n\n~\\ \n\nEelO \n\nJ  Eel! \n\n,  : \n\n'-:-:--~-:---:'7-~ \n,  0 \n\n'I \n\nII \n\n'- .. \n\nIe \n\nII \n\nI '   I'  I '  \n\n\u2022  It \n\n. . \"   II \n\nII  I '  \n\n'\n\n. \n\nlOOt  I' \n\nII \n\nII \n\n'I \n\nEf12 \n\nEf13 \n\nEf14 \n\nDe15 \n\n\u2022\u2022 \n\n01 \n\nII \n\nII \n\nI .   . ,   I .   I I  \n\n\u2022\u2022 \n\nI I  \n\n... \n\n\u2022\u2022 \n\n.1  J' \n\nI I  \n\nIJ \n\n, .   I I   I '   I I \n\nDf16 \n\n.r--.. \n\nDf17 \n\n,./\"\\ \n\n,. \n\n'--\no \n1 \nTime(s) \n\nFigure  4.  Numerical  Analysis  and  Muscle  Tensions  For  T7->Ts.  S=shoulder,  E=elbow, \nD=double-joint muscle,  e=extensor,  f=flexor. \n\nstudy which we intend to test in the near future. Still, the minimum-muscle-tension-change \nmodel  considers  only  the  muscle  moment-arms  (Il) and  momvels  (olll ae) and ignores \nthe muscle length-tension curves.  A more complicated model which we are studying now \nis the minimum-motor-command-change model, which includes the length-tension curves. \n\n\f634 \n\nDomay,  Uno,  Kawato.  and Suzuki \n\nAcknowledgements \nM.  Domay and M.  Kawato would like to thank Drs. K.  Nakane and E.  Yodogawa,  ATR, \nfor  their  valuable  help  and  support.  Preparation  of the  paper  was  supported  by  Human \nFrontier Science Program grant to  M.  Kawato. \n\nReferences \n1  E  Bizzi  &  WK  Abend (1986)  Control  of multijoint movements.  In  MJ.  Cohen and \nF.  Strumwasser (Eds.)  Comparative Neurobiology:  Modes  of Communication  in  the \nNervous  System,  John  Wiley  &  Sons, pp.  255-277 \n\n2  M  Dornay  (1990)  Control  of movement  and  the  postural  stability  of the  monkey's \narm.  Proc. 3rd International Symposium  on Bioelectronic and Molecular Electronic \nDevices,  Kobe,  Japan,  December  18-20,  pp.  101-102 \n\n3  M Domay (1991 a) Static analysis of posture and movement, using a 17 -muscle model \n\nof the  monkey's  arm. ATR  Technical Report TR-A-0109 \n\n4  M  Domay  (1991b)  Control  of  movement,  postural  stability,  and  muscle  angular \nstiffness.  Proc.  IEEE Systems,  Man  and Cybernetics,  Virginia,  USA,  pp.  1373-1379 \n5  M Dornay, Y Uno, M Kawato &  R  Suzuki (1991) Simulation of optimal movements \nusing a  17-muscle model of the  monkey's arm. Proc.  SICE 30th Annual Conference, \nES-1-4,  July  17-19,  Yonezawam  Japan,  pp.  919-922 \n\n6  AG  Feldman  (1966)  Functional  tuning  of  the  nervous  system  with  control  of \n\nmovement or maintenance  of a  steady posture.  Biophysics, li, pp.  766-775 \n\n7  T Flash  &  N Hogan  (1985)  The coordination of arm  movements:  an experimentally \n\nconfIrmed mathematical  model.  J.  Neurosci., 2,.,  pp.  1688-1703 \n\n8  T  Flash  (1987)  The  control  of  hand  equilibrium  trajectories  in  multi-joint  arm \n\nmovements.  Biol.  Cybern.,  57, pp.  257-274 \n\n9  T Flash (1990)  The organization of human arm  trajectory control.  In J.  Winters  and \nS.  Woo  (Eds.)  Multiple muscle systems: Biomechanics and movement organization, \nSpringer-Verlag,  pp.  282-301 \n\nION Hogan  (1984)  An  organizing  principle  for  a  class  of voluntary  movements.  J. \n\nNeurosci., i, pp.  2745-2754 \n\n11  P  Morasso (1981)  Spatial control of arm  movements.  Experimental Brain Research, \n\n42,  pp.  223-227 \n\n12  FA Mussa-Ivaldi, P Morasso, N Hogan & E Bizzi (1991)  Network models  of motor \nsystems  with  many  degrees  of freedom.  In  M.D.  Fraser  (Ed.)  Advances  in  control \nnetworks and large  scale parallel distributed processing models,  Albex  Publ.  Corp. \n13  Y Uno,  M Kawato &  R  Suzuki  (1989a) Formation and control of optimal trajectory \nin human multijoint arm movement - minimum-torque-change model. Biol.  Cybern., \ng, pp.  89-101 \n\n14  Y  Uno,  R  Suzuki  &  M  Kawato  (1989b)  Minimum  muscle-tension  change  model \nwhich  reproduces  human  arm  movement.  Proceedings  of the  4th  Symposium  on \nBiological and Physiological Engineering,  pp.  299-302,  (in  Japanese) \n\n\fPART X \n\nApPLICATIONS \n\n\f\f", "award": [], "sourceid": 487, "authors": [{"given_name": "Menashe", "family_name": "Dornay", "institution": null}, {"given_name": "Yoji", "family_name": "Uno", "institution": null}, {"given_name": "Mitsuo", "family_name": "Kawato", "institution": null}, {"given_name": "Ryoji", "family_name": "Suzuki", "institution": null}]}