Abstract
Human motion studies have focused primarily on modeling straight point-to-point reaching movements. However, many goal-directed reaching movements, such as movements directed towards oneself, are not straight but rather follow highly curved trajectories. These movements are particularly interesting to study since they are essential in our everyday life, appear early in development and are routinely used to assess movement deficits following brain lesions. We argue that curved and straight-line reaching movements are generated by a unique neural controller and that the observed curvature of the movement is the result of an active control strategy that follows the geometry of one’s body, for instance to avoid trajectories that would hit the body or yield postures close to the joint limits. We present a mathematical model that accounts for such an active control strategy and show that the model reproduces with high accuracy the kinematic features of human data during unconstrained reaching movements directed toward the head. The model consists of a nonlinear dynamical system with a single stable attractor at the target. Embodiment-related task constraints are expressed as a force field that acts on the dynamical system. Finally, we discuss the biological plausibility and neural correlates of the model’s parameters and suggest that embodiment should be considered as a main cause for movement trajectory curvature.
Similar content being viewed by others
References
Admiraal MA, Kusters MJMAM, Gielen SCAM (2004) Modeling kinematics and dynamics of human arm movements. Motor Control 8(3): 312–338
Ajemian R, Bullock D, Grossberg S (2001) A model of movement coordinates in the motor cortex: posture-dependent changes in the gain and direction of single cell tuning curves. Cereb Cortex 11(12): 1124–1135
Atkeson CG, Hollerbach JM (1985) Kinematic features of unrestrained vertical arm movements. J Neurosci 5(9): 2318–2330
Bernstein NA (1947) On the construction of movements. Medgiz, Moscow
Bernstein NA (1996) Levels of construction of movements. In: Latash ML, Turvey MT(eds) Dexterity and its development. Lawrence Erlbaum Associates, Mahwah, pp 115–170
Biess A, Liebermann DG, Flash T (2007) A computational model for redundant human three dimensional pointing movements: integration of independant spatial and temporal motor plans simplifies movement dynamics. J Neurosci 27(48): 13045–13064
Bizzi E, Accornero N, Chapple W, Hogan N (1982) Arm trajectory formation in monkeys. Exp Brain Res 46(1): 139–143
Brenner E, Smeets JBJ (1995) Moving one’s finger to a visually specified position: target orientation influences the finger’s path. Exp Brain Res 105(2): 318–320
Brown SH, Cooke JD (1990) Movement-related phasic muscle activation. I. Relations with temporal profile of movement. J Neurophysiol 63(3): 455–464
Bullock D, Grossberg S (1988) Neural dynamics of planned arm movements: emergent invariants and speed-accuracy properties during trajectory formation. Psychol Rev 95(1): 49–90
Bullock D, Grossberg S, Mannes C (1993) A neural network model for cursive script production. Biol Cybern 70(1): 15–28
Bullock D, Cisek P, Grossberg S (1998) Cortical networks for control of voluntary arm movements under variable force conditions. Cereb Cortex 8(1): 48–62
Christel MI, Billard A (2002) Comparison between macaques’ and humans’ kinematics of prehension: the role of morphological differences and control mechanisms. Behav Brain Res 131(1–2): 169–184
Cisek P, Grossberg S, Bullock D (1998) A cortico-spinal model of reaching and proprioception under multiple task constraints. J Cogn Neurosci 10(4): 425–444
Clamman HP (1969) Statistical analysis of motor unit firing pattern in human skeletal muscle. Biophys J 9: 1233–1251
Clancy EA, Hogan N (1999) Probability density of the surface electromyogram and its relation to amplitude detectors. IEEE Trans Biomed Eng 46(6): 730–739
Conditt MA, Gandolfo F, Mussa-Ivaldi FA (1997) The motor system does not learn the dynamics of the arm by rote memorization of past experience. J Neurophysiol 78(1): 554–560
Cruse H, Brüwer M (1987) The human arm as a redundant manipulator: the control of path and joint angles. Biol Cybern 57(1–2): 134–144
d’Avella A, Saltiel P, Bizzi E (2003) Combinations of muscle synergies in the construction of a natural motor behavior. Nat Neurosci 6(3): 300–308
de Graaf JB, Sittig AC, Denier von der Gon JJ (1991) Misdirections in slow goal-directed arm movements and pointer-setting tasks. Exp Brain Res 84(2): 434–438
de Graaf JB, Sittig AC, Denier von der Gon JJ (1994) Misdirections in slow, goal-directed arm movements are not primarily visually based. Exp Brain Res 99(3): 464–472
Dalby JT, Gibson D, Grossi V, Schneider RD (1980) Lateralized hand gesture during speech. J Motor Behav 12(4): 292–297
De Renzi E, Lucchelli F (1988) Ideational apraxia. Brain 111(Pt 5): 1173–1185
Desmurget M, Grafton S (2000) Forward modeling allows feedback control for fast reaching movements. Trends Cogn Sci 4(11): 423–431
Desmurget M, Jordan M, Prablanc C, Jeannerod M (1997) Constrained and unconstrained movements involve different control strategies. J Neurophysiol 77(3): 1644–1650
Fitts PM (1954) The information capacity of the human motor system in controlling the amplitude of movement. J Exp Psychol 47(6): 381–391
Flash T (1987) The control of hand equilibrium trajectories in multi-joint arm movements. Biol Cybern 57(4–5): 257–274
Flash T, Hogan N (1985) The coordination of arm movements: an experimentally confirmed mathematical model. J Neurosci 5(7): 1688–1703
Gandolfo F, Mussa-Ivaldi FA, Bizzi E (1996) Motor learning by field approximation. Proc Natl Acad Sci USA 93(9): 3843–3846
Gielen CCAM, van den Oosten K, Pull ter Gunne F (1985) Relation between EMG activation patterns and kinematic properties of aimed arm movements. J Mot Behav 17(4): 421–442
Goldenberg G, Hagmann S (1997) The meaning of meaningless gestures: a study of visuo-imitative apraxia. Neuropsychologia 35(3): 333–341
Graziano MSA, Cooke DF, Taylor CSR (2000) Coding the location of the arm by sight. Science 290(5497): 1782–1786
Graziano MSA, Taylor CSR, Moore T (2002) Complex movements evoked by microstimulation of precentral cortex. Neuron 34(5): 841–851
Graziano MSA, Aflalo TNS, Cooke DF (2005) Arm movements evoked by electrical stimulation in the motor cortex of monkeys. J Neurophysiol 94(6): 4209–4223
Guigon E, Baraduc P, Desmurget M (2007) Computational motor control: redundancy and invariance. J Neurophysiol 97(1): 331–347
Hamilton AFC, Wolpert DM (2002) Controlling the statistics of action: obstacle avoidance. J Neurophysiol 87(5): 2434–2440
Harris CM, Wolpert DM (1998) Signal-dependent noise determines motor planning. Nature 394(6695): 780–784
Hersch M, Billard A (2007) Reaching with multi-referential dynamical systems. Auton Robot 25(1–2): 71–83
Hoos H, Stützle T (2004) Stochastic local search: Foundations and applications. Elsevier/Morgan Kaufmann, San Francisco
Klein Breteler MD, Meulenbroek RGJ, Gielen SCAM (1998) Geometric features of workspace and joint-space paths of 3D reaching movements. Acta Psychol 100(1–2): 37–53
Lacquaniti F, Terzuolo C, Viviani P (1983) The law relating the kinematic and figural aspects of drawing movements. Acta Psychol 54(1–3): 115–130
Lacquaniti F, Soechting JF, Terzuolo SA (1986) Path constraints on point-to-point arm movements in three-dimensional space. Neuroscience 17(2): 313–324
Lavergne J, Kimura D (1987) Hand movement asymmetry during speech: no effect of speaking topic. Neuropsychologia 25(4): 689–693
Liebermann DG, Biess A, Gielen CCAM, Flash T (2006) Intrinsic joint kinematic planning. II: Hand-path predictions based on a Listing’s plane constraint. Exp Brain Res 171(2): 155–173
Liebermann DG, Krasovsky T, Berman S (2008) Planning maximally smooth hand movements constrained to nonplanar workspaces. J Mot Behav 40(6): 516–531
Matthews PBC (1996) Relationship of firing intervals of human motor units to the trajectory of post-spike after-hyperpolarization and synaptic noise. J Physiol 492(Pt 2): 597–628
McIntyre J, Gurfinkel EV, Lipshits MI, Droulez J, Gurfinkel VS (1995) Measurements of human force control during a constrained arm motion using a force-actuated joystick. J Neurophysiol 73(3): 1201–1222
Miall RC, Haggard PN (1995) The curvature of human arm movements in the absence of visual experience. Exp Brain Res 103(3): 421–428
Moran DW, Schwartz AB (1999) Motor cortical activity during drawing movements: population representation during spiral tracing. J Neurophysiol 82(5): 2693–2704
Morasso P (1981) Spatial control of arm movements. Exp Brain Res 42(2): 223–227
Nakano E, Imamizu H, Osu R, Uno Y, Gomi H, Yoshioka T, Kawato M (1999) Quantitative examinations of internal representations for arm trajectory planning: minimum commanded torque change model. J Neurophysiol 81(5): 2140–2155
Neter J, Kutner MH, Nachtsheim CJ, Wasserman W (1996) Applied linear statistical models. McGraw-Hill/Irwin, Chicago
Nishikawa KC, Murray ST, Flanders M (1999) Do arm postures vary with the speed of reaching?. J Neurophysiol 81(5): 2582–2586
Okadome T, Honda M (1999) Kinematic construction of the trajectory of sequential arm movements. Biol Cybern 80(3): 157–169
Oldfield RC (1971) The assessment and analysis of handedness: the Edinburgh inventory. Neuropsychologia 9(1): 97–113
Osu R, Uno Y, Koike Y, Kawato M (1997) Possible explanations for trajectory curvature in multijoint arm movements. J Exp Psychol Hum Percept Perform 23(3): 890–913
Osu R, Kamimura N, Iwasaki H, Nakano E, Harris CM, Wada Y, Kawato M (2004) Optimal impedance control for task achievement in the presence of signal-dependant noise. J Neurophysiol 92(2): 1199–1215
Paine RW, Grossberg S, Van Gemmert AWA (2004) A quantitative evaluation of the AVITEWRITE model of handwriting learning. Hum Mov Sci 23(6): 837–860
Papaxanthis C, Pozzo T, Popov KE, McIntyre J (1998) Hand trajectories of vertical arm movements in one-G and zero-G environments. Evidence for a central representation of gravitational force. Exp Brain Res 120(4): 496–502
Pellegrini JJ, Flanders M (1996) Force path curvature and conserved features of muscle activation. Exp Brain Res 110(1): 80–90
Petreska B, Adriani M, Blanke O, Billard AG (2007) Apraxia: a review. Prog Brain Res 164: 61–83
Sabes PN, Jordan MI (1997) Obstacle avoidance and a perturbation sensitivity model of motor planning. J Neurosci 17(18): 7119–7128
Sauser E, Billard A (2006) Dynamic updating of distributed neural representations using forward models. Biol Cybern 95(6): 567–588
Schaal S, Sternad D (2001) Origins and violations of the 2/3 power law in rhythmic three-dimensional arm movements. Exp Brain Res 136(1): 60–72
Schmidt RA, Zelaznik H, Hawkins B, Frank JS, Quinn JT (1979) Motor-output variability: a theory for the accuracy of rapid motor acts. Psychol Rev 47(5): 415–451
Scott SH, Sergio LE, Kalaska JF (1997) Reaching movements with similar hand paths but different arm orientations. II. Activity of individual cells in dorsal premotor cortex and parietal area 5. J Neurophysiol 78(5): 2413–2426
Sha D, Patton J, Mussa-Ivaldi FA (2006) Minimum jerk reaching movements of human arm with mechanical constraints at endpoint. Int J Comput Syst Signal 7(1): 41–50
Shadmehr R, Brashers-Krug T (1997) Functional stages in the formation of human long-term motor memory. J Neurosci 17(1): 409–419
Shadmehr R, Mussa-Ivaldi FA (1994) Adaptive representation of dynamics during learning of a motor task. J Neurosci 14(5 Pt 2): 3208–3224
Soechting JF, Lacquaniti F (1981) Invariant characteristics of a pointing movement in man. J Neurosci 1(7): 710–720
Soechting JF, Buneo CA, Herrmann U, Flandres M (1995) Moving effortlessly in three dimensions: does Donders’ law apply to arm movement?. J Neurosci 15(9): 6271–6280
St-Amant Y, Rancourt D, Clancy EA (1998) Influence of smoothing window length on electromyogram amplitude estimates. IEEE Trans Biomed Eng 45(6): 795–800
Sutton GG, Sykes K (1967) The variation of hand tremor with force in healthy subjects. J Physiol 191(3): 699–711
Thoroughman KA, Shadmehr R (2000) Learning of action through adaptive combination of motor primitives. Nature 407(6805): 742–747
Todorov E (2004) Optimality principles in sensorimotor control. Nat Neurosci 7(9): 907–915
Todorov E, Jordan MI (2002) Optimal feedback control as a theory of motor coordination. Nat Neurosci 5(11): 1226–1235
Torres EB, Zipser D (2002) Reaching to grasp with a multi-joined arm. I. Computational model. J Neurophysiol 88(5): 2355–2367
Uno Y, Kawato M, Suzuki R (1989) Formation and control of optimal trajectory in human multijoint arm movement. Minimum torque-change model. Biol Cybern 61(2): 89–101
Vetter P, Flash T, Wolpert DM (2002) Planning movements in a simple redundant task. Curr Biol 12(6): 488–491
Wada Y, Kaneko Y, Nakano E, Osu R, Kawato M (2001) Quantitative examinations for multi joint arm trajectory planning—using a robust calculation algorithm of the minimum commanded torque change trajectory. Neural Netw 14(4–5): 381–393
Wang W, Chan SS, Heldman DA, Moran DW (2007) Motor cortical representation of position and velocity during reaching. J Neurophysiol 97(6): 4258–4270
Wolpert DM, Ghahramani Z, Jordan MI (1994) Perceptual distortion contributes to the curvature of human reaching movements. Exp Brain Res 98(1): 153–156
Wolpert DM, Ghahramani Z, Jordan MI (1995) Are arm trajectories planned in kinematic or dynamic coordinates? An adaptation study. Exp Brain Res 103(3): 460–470
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Petreska, B., Billard, A. Movement curvature planning through force field internal models. Biol Cybern 100, 331–350 (2009). https://doi.org/10.1007/s00422-009-0300-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00422-009-0300-2