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6 - Dynamic systems: brain, body, and imitation

Published online by Cambridge University Press:  01 September 2009

Stefan Schaal
Affiliation:
University of Southern California, Los Angeles
Michael A. Arbib
Affiliation:
University of Southern California
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Summary

Introduction

The Mirror System Hypothesis (MSH) (see Arbib, Chapter 1, this volume) postulates that higher communication skills and language may be grounded in the basic skill of action recognition, enhanced by the ability of movement pantomime and imitation. This chapter will examine from a computational point of view how such recognition and imitation skills can be realized. Our goal is to take our formal knowledge of how to produce simple actions, also called movement primitives, and explore how a library of such movement primitives can be built and used to compose an increasingly large repertoire of complex actions. An important constraint in the development of this material comes from the need to account for movement imitation and movement recognition in one coherent framework, which naturally leads to an interplay between perception and action.

The existence of motor primitives (a.k.a. synergies, units of actions, basis behaviors, motor schemas, etc.) (Bernstein, 1967; Arbib, 1981; Viviani, 1986; Mataric, 1998; Miyamoto and Kawato, 1998; Schaal, 1999; Sternad and Schaal, 1999; Dautenhahn and Nehaniv, 2002) seems, so far, the only possibility for how one could conceive that biological and artificial motor systems are able to cope with the complexity of motor control and motor learning (Arbib, 1981; Schaal, 1999, 2002a, 2002b; Byrne 2003). This is because learning based on low-level motor commands, e.g., individual muscle activations, becomes computationally intractable for even moderately complex movement systems. Our computational approach to motor control with movement primitives is sketched as an abstract flowchart in Fig. 6.1 (Schaal, 1999).

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Publisher: Cambridge University Press
Print publication year: 2006

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References

Abend, W., Bizzi, E, and Morasso, P., 1982. Human arm trajectory formation. Brain 105: 331–348.CrossRefGoogle ScholarPubMed
Adamovich, S. V., Levin, M. F., and Feldman, A. G., 1994. Merging different motor patterns: coordination between rhythmical and discrete single-joint. Exp. Brain Res. 99: 325–337.CrossRefGoogle ScholarPubMed
Aggarwal, J. K., and Cai, Q., 1999. Human motion analysis: a review. Comput. Vision Image Understand. 73: 428–440.CrossRefGoogle Scholar
Alissandrakis, A., Nehaniv, C. L., and Dautenhahn, K., 2002. Imitating with ALICE: learning to imitate corresponding actions across dissimilar embodiments. IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans 32: 482–496.CrossRefGoogle Scholar
Amit, R., and Mataric, M., 2002. Learning movement sequences from demonstration. Proceedings International Conference on Development and Learning, Cambridge, MA, June 12–15 2002, pp. 302–306.
Andersen, R. A., Snyder, L. H., Bradley, D. C., and Xing, J, 1997. Multimodal representation of space in the posterior parietal cortex and its use in planning movements. Annu. Rev. Neurosci. 20: 303–330.CrossRefGoogle ScholarPubMed
Arbib, M. A. 1981. Perceptual structures and distributed motor control. In Brooks, V. B. (ed.) Handbook of Physiology, Section 2, The Nervous System, vol. 2, Motor Control, Part 1. Bethesda, MD: American Physiological Society, pp. 1449–1480.Google Scholar
Bagnell, J., Kadade, S., Ng, A., and Schneider, J. 2003. Policy search by dynamic programming. In Thrun, S., Saul, L. K. and Schölkopf, B. (eds.) Advances in Neural Information Processing Systems, vol. 16. Cambridge, MA: MIT Press, pp. 831–838.Google Scholar
Barto, A. G., and Mahadevan, S., 2003. Recent advances in hierarchical reinforcement learning. Discrete Event Dynamic Systems 13: 341–379.CrossRefGoogle Scholar
Bell, A. J., and Sejnowski, T. J., 1997. The “independent components” of natural scenes are edge filters. Vision Res. 37: 3327–3338.CrossRefGoogle ScholarPubMed
Bellman, R., 1957. Dynamic Programming. Princeton, NJ: Princeton University Press.Google ScholarPubMed
Bernstein, N. A., 1967. The Control and Regulation of Movements. London: Pergamon Press.Google Scholar
Bertsekas, D. P., and Tsitsiklis, J. N., 1996. Neuro-Dynamic Programming. Bellmont, MA: Athena Scientific.Google Scholar
Billard, A., and Mataric, M., 2001. Learning human arm movements by imitation: evaluation of a biologically inspired architecture. Robot. Auton. Syst. 941: 1–16.Google Scholar
Billard, A., and Schaal, S., 2002. Computational elements of robot learning by imitation. Proceedings Central Section Meeting, American Mathematical Society. Providence, RI, October 12–13, 2002.
Bishop, C. M., 1995. Neural Networks for Pattern Recognition. New York: Oxford University Press.Google Scholar
Brooks, R. A., 1986. A robust layered control system for a mobile robot. IEEE J. Robot. Autom. 2: 14–23.CrossRefGoogle Scholar
Brown, T. G., 1914. On the nature of the fundamental activity of the nervous centres; together with an analysis of rhythmic activity in progression, and a theory of the evolution of function in the nervous system. J. Physiol. 48: 18–46.CrossRefGoogle Scholar
Bullock, D., and Grossberg, S., 1988. Neural dynamics of planned arm movements: emergent invariants and speed–accuracy properties during trajectory formation. Psychol. Rev. 95: 49–90.CrossRefGoogle ScholarPubMed
Bullock, D., Grossberg, S., and Guenther, F. H., 1993. A self-organizing neural model of motor equivalent reaching and tool use by a multijoint arm. J. Cogn. Neurosci. 5: 408–435.CrossRefGoogle ScholarPubMed
Burridge, R. R., Rizzi, A. A., and Koditschek, D. E., 1999. Sequential composition of dynamically dexterous robot behaviors. Int. J. Robot. Res. 18: 534–555.CrossRefGoogle Scholar
Byrne, R. W., 2003. Imitation as behaviour parsing. Phil. Trans. Roy. Soc. London B 358: 529–536.CrossRefGoogle ScholarPubMed
Craig, J. J., 1986. Introduction to Robotics. Reading, MA: Addison-Wesley.Google Scholar
Dautenhahn, K., and Nehaniv, C. L. (eds.), 2002. Imitation in Animals and Artifacts. Cambridge, MA: MIT Press.Google Scholar
Dayan, P., Hinton, G. E., Neal, R. M., and Zemel, R. S., 1995. The Helmholtz machine. Neur. Comput. 7: 889–904.CrossRefGoogle ScholarPubMed
Rugy, A., and Sternad, D., 2003. Interaction between discrete and rhythmic movements: reaction time and phase of discrete movement initiation during oscillatory movements. Brain Res. 994: 160–174.CrossRefGoogle ScholarPubMed
Demiris, J., and Hayes, G., 1999. Active and passive routes to imitation. Proceedings AISB 1999 Symposium of Imitation in Animals and Artifacts, pp. 81–87.Google Scholar
Dimitrijevic, M. R., Gerasimenko, Y., and Pinter, M. M., 1998. Evidence for a spinal central pattern generator in humans. Ann. N. Y. Acad. Sci. 860: 360–376.CrossRefGoogle ScholarPubMed
Dyer, P., and McReynolds, S. R., 1970. The Computation and Theory of Optimal Control. New York: Academic Press.Google Scholar
Fagg, A. H., and Arbib, M. A., 1998. Modeling parietal–premotor interactions in primate control of grasping. Neur. Networks 11: 1277–1303.CrossRefGoogle ScholarPubMed
Feldman, A. G., 1980. Superposition of motor programs. I. Rhythmic forearm movements in man. Neuroscience 5: 81–90.CrossRefGoogle ScholarPubMed
Flash, T., and Henis, E., 1991. Arm trajectory modification during reaching towards visual targets. J. Cogn. Neurosci. 3: 220–230.CrossRefGoogle ScholarPubMed
Flash, T., and Hogan, N., 1985. The coordination of arm movements: an experimentally confirmed mathematical model. J. Neurosci. 5: 1688–1703.CrossRefGoogle ScholarPubMed
Flash, T., and Sejnowski, T., 2001. Computational approaches to motor control. Curr. Opin. Neurobiol. 11: 655–662.CrossRefGoogle ScholarPubMed
Gallese, V., and Goldman, A., 1998. Mirror neurons and the simulation theory of mind-reading. Trends Cogn. Sci. 2: 493–501.CrossRefGoogle ScholarPubMed
Gavrila, D. M., 1997. The visual analysis of humant movements: a survey. Comput. Vision Image Understand. 73: 82–98.CrossRefGoogle Scholar
Georgopoulos, A. P., 1991. Higher order motor control. Annu. Rev. Neurosci. 14: 361–377.CrossRefGoogle ScholarPubMed
Harris, C. M., and Wolpert, D. M., 1998. Signal-dependent noise determines motor planning. Nature 394: 780–784.CrossRefGoogle ScholarPubMed
Haruno, M., Wolpert, D. M., and Kawato, M., 2001. Mosaic model for sensorimotor learning and control. Neur. Comput. 13: 2201–2220.CrossRefGoogle ScholarPubMed
Hasan, Z., 1986. Optimized movement trajectories and joint stiffness in unperturbed, inertially loaded movements. Biol. Cybernet. 53: 373–382.CrossRefGoogle ScholarPubMed
Henis, E. A., and Flash, T., 1995. Mechanisms underlying the generation of averaged modified trajectories. Biol. Cybernet. 72: 407–419.CrossRefGoogle Scholar
Hinton, G. E., Dayan, P., Frey, B. J., and Neal, R. M., 1995. The wake–sleep algorithm for unsupervised neural networks. Science 268: 1158–1161.CrossRefGoogle ScholarPubMed
Hodgkin, A. L., and Huxley, A. F., 1952. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117: 500–544.CrossRefGoogle ScholarPubMed
Hoff, B., 1994. A model of duration in normal and perturbed reaching movement. Biol. Cybernet. 71: 481–488.CrossRefGoogle Scholar
Hoff, B., and Arbib, M. A. 1992. A model of the effects of speed, accuracy, and perturbation on visually guided reaching. In Caminiti, R. M., Johnson, P. B. and Burnod, Y. (eds.) Experimental Brain Research Series, vol. 22. Berlin: Springer-Verlag, pp. 285–306.Google Scholar
Hogan, N., 1984. An organizing principle for a class of voluntary movements. J. Neurosci. 4: 2745–2754.CrossRefGoogle ScholarPubMed
Hollerbach, J. M., 1984. Dynamic scaling of manipulator trajectories. Trans. Am. Soc. Mech. Eng. 106: 139–156.Google Scholar
Ijspeert, A., Nakanishi, J., and Schaal, S., 2001. Trajectory formation for imitation with nonlinear dynamical systems. Proceedings IEEE Int. Conference on Intelligent Robots and Systems, Weilea, HI, pp. 729–757.
Ijspeert, A., Nakanishi, J., and Schaal, S. 2002. Movement imitation with nonlinear dynamical systems in humanoid robots. Proceedings Int. Conference on Robotics and Automation, Washington, May 11–15, 2002.
Ijspeert, A., Nakanishi, J., and Schaal, S. 2003. Learning attractor landscapes for learning motor primitives. In Becker, S., Thrun, S. and Obermayer, K. (eds.) Advances in Neural Information Processing Systems, vol. 15. Cambridge, MA: MIT Press, pp. 1547–1554.Google Scholar
Imamizu, H., Miyauchi, S., Tamada, T, et al., 2000. Human cerebellar activity reflecting an acquired internal model of a new tool. Nature 403: 192–195.CrossRefGoogle ScholarPubMed
Inamura, T., Toshima, I., and Nakamura, Y., 2002. Acquisition and embodiment of motion elements in closed mimesis loop. Proceedings Int. Conference on Robotics and Automation, Washington, May 11–15, 2002, pp. 1539–1544.
Inamura, T., Iwaki, T., Tanie, H., and Nakamura, Y., 2004. Embodied symbol emergence based on mimesis theory. Int. J. Robot. Res. 23: 363–377.CrossRefGoogle Scholar
Ivry, R. B., Spencer, R. M., Zelaznik, H. N., and Diedrichsen, J., 2002. The cerebellum and event timing. Ann. N. Y. Acad. Sci. 978: 302–317.CrossRefGoogle ScholarPubMed
Jordan, I. M., and Rumelhart, D., 1992. Supervised learning with a distal teacher. Cogn. Sci., 16: 307–354.CrossRefGoogle Scholar
Kalaska, J. F., Scott, S. H., Cisek, P., and Sergio, L. E., 1997. Cortical control of reaching movements. Curr. Opin. Neurobiol. 7: 849–859.CrossRefGoogle ScholarPubMed
Kawamura, S., and Fukao, N., 1994. Interpolation for input torque patterns obtained through learning control. Proceedings Int. Conference on Automation, Robotics and Computer Vision, Singapore, pp. 183–191.
Kawato, M., 1990. Computational schemes and neural network models for formation and control of multijoint arm trajectory. In Miller, W. T., Sutton, R. S. and Werbos, P. J. (eds.) Neural Networks for Control. Cambridge, MA: MIT Press, pp. 197–228.CrossRefGoogle Scholar
Kawato, M. 1996. Bi-directional theory approach to integration. In Konczak, J. and Thelen, E. (eds.) Attention and Performance, vol. 16. Cambridge, MA: MIT Press, pp. 335–367.Google Scholar
Kawato, M. 1999. Internal models for motor control and trajectory planning. Curr. Opin. Neurobiol. 9: 718–727.CrossRefGoogle ScholarPubMed
Kawato, M., and Wolpert, D., 1998. Internal models for motor control. Novartis Found. Symp. 218: 291–304.Google ScholarPubMed
Kawato, M., Gandolfo, F., Gomi, H., and Wada, Y., 1994. Teaching by showing in kendama based on optimization principle. Proceedings Int. Conference on Artificial Neural Networks, vol. 1, pp. 601–606.Google Scholar
Keating, J. G., and Thach, W. T., 1997. No clock signal in the discharge of neurons in the deep cerebellar nuclei. J. Neurophysiol. 77: 2232–2234.CrossRefGoogle ScholarPubMed
Kelso, J. A. S., 1995. Dynamic Patterns: The Self-Organization of Brain and Behavior. Cambridge, MA: MIT Press.Google Scholar
Kugler, P. N., and Turvey, M. T., 1987. Information, Natural Law, and the Self-Assembly of Rhythmic Movement. Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Lamb, T., and Yang, J. F., 2000. Could different directions of infant stepping be controlled by the same locomotor central pattern generator? J. Neurophysiol. 83: 2814–2824.CrossRefGoogle ScholarPubMed
Latash, M. L., 1993. Control of Human Movement. Champaign, IL: Human Kinetics.Google Scholar
Liberman, A. M., 1996. Speech: A Special Code. Cambridge, MA: MIT Press.Google Scholar
Lohmiller, W., and Slotine, J. J. E., 1998. On contraction analysis for nonlinear systems. Automatica 6: 683–696.CrossRefGoogle Scholar
Marder, E., 2000. Motor pattern generation. Curr. Opin. Neurobiol. 10: 691–698.CrossRefGoogle ScholarPubMed
Mataric, M., 1998. Behavior-based robotics as a tool for synthesis of artificial behavior and analysis of natural behavior. Trends Cogn. Sci. 2: 82–86.CrossRefGoogle ScholarPubMed
Meltzoff, A. N., and Moore, M. K. 1995. Infant's understanding of people and things: from body imitation to folk psychology. In Bermúdez, J. L., Marcel, A. and Eilan, N. (eds.) The Body and the Self. Cambridge, MA: MIT Press, pp. 43–69.Google Scholar
Miyamoto, H., and Kawato, M., 1998. A tennis serve and upswing learning robot based on bi-directional theory. Neur. Networks 11: 1331–1344.CrossRefGoogle ScholarPubMed
Miyamoto, H., Schaal, S., Gandolfo, F., et al., 1996. A Kendama learning robot based on bi-directional theory. Neur. Networks 9: 1281–1302.CrossRefGoogle Scholar
Mohajerian, P., Mistry, M, and Schaal, S., 2004. Neuronal or spinal level interaction between rhythmic and discrete motion during multi-joint arm movement. Abstracts 34th Meeting Soci. Neuroscience, San Diego, CA, October, 23–27.
Morasso, P., 1981. Spatial control of arm movements. Exp. Brain Res. 42: 223–227.CrossRefGoogle ScholarPubMed
Mussa-Ivaldi, F. A., 1988. Do neurons in the motor cortex encode movement direction? An alternative hypothesis. Neurosci. Lett. 91: 106–111.CrossRefGoogle ScholarPubMed
Nakanishi, J., Morimoto, J., Endo, G., et al., 2004. Learning from demonstration and adaptation of biped locomotion. Robot. Auton. Syst. 47: 79–91.CrossRefGoogle Scholar
Nakanishi, J., Cory, R., Mistry, M., Peters, J., and Schaal, S., 2005. Comparative experiments on task space control with redundancy resolution. Proceedings IEEE Int. Conference on Intelligent Robots and Systems, pp. 1575–1582.
Olshausen, B. A., and Field, D. J., 1996. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381: 607–609.CrossRefGoogle ScholarPubMed
Osentoski, S., Manfredi, V., and Mahadevan S., 2004. Learning hierarchical models of activity. Proceedings IEEE Int. Conference on Intelligent Robots and Systems, Sendai, Japan.
Oztop, E., Wolpert, D., and Kawato, M., 2005. Mental state inference using visual control parameters. Brain Res. Cogn. Brain Res. 22: 129–151.CrossRefGoogle ScholarPubMed
Paine, R. W., and Tani, J., 2004. Motor primitive and sequence self-organization in a hierarchical recurrent neural network. Neur. Networks 17: 1291–1309.CrossRefGoogle Scholar
Perrett, D., Harries M., Mistlin, A. J., and Chitty, A. J. 1989a. Three stages in the classification of body movements by visual neurons. In Barlow, H. B. (ed.) Images and Understanding. Cambridge, UK: Cambridge University Press, 94–107.Google Scholar
Perrett, D. I., Harries, M. H., Bevan, R., et al., 1989b. Frameworks of analysis for the neural representation of animate objects and actions. J. Exp. Biol. 146: 87–113.Google ScholarPubMed
Peters, J., Vijayakumar, S., and Schaal, S., 2003. Reinforcement learning for humanoid robotics. Proceedings, 3rd IEEE–RAS Int. Conference on Humanoid Robots, Karlsruhe, Germany, September 29–30, 2003.
Picard, N., and Strick, P. L., 2001. Imaging the premotor areas. Curr. Opin. Neurobiol. 11: 663–672.CrossRefGoogle ScholarPubMed
Pinter, M. M., and Dimitrijevic, M. R., 1999. Gait after spinal cord injury and the central pattern generator for locomotion. Spinal Cord 37: 531–537.CrossRefGoogle ScholarPubMed
Pook, P. K., and Ballard, D. H., 1993. Recognizing teleoperated manipulations. Proceedings IEEE Int. Conference on Robotics and Automation, Atlanta, GA, pp. 913–918.
Richardson, M. J., and Flash, T., 2002. Comparing smooth arm movements with the two-thirds power law and the related segmented-control hypothesis. J. Neurosci. 22: 8201–8211.CrossRefGoogle ScholarPubMed
Rizzi, A. A., and Koditschek, D. E., 1994. Further progress in robot juggling: solvable mirror laws. Proceedings IEEE Int. Conference on Robotics and Automation, San Diego, CA, pp. 2935–2940.
Rizzolatti, G., Fadiga, L., Gallese, V., and Fogassi, L., 1996. Premotor cortex and the recognition of motor actions. Cogn. Brain Res. 3: 131–141.CrossRefGoogle ScholarPubMed
Roberts, P. D., and Bell, C. C., 2000. Computational consequences of temporally asymmetric learning rules. II. Sensory image cancellation. J. Comput. Neurosci. 9: 67–83.CrossRefGoogle ScholarPubMed
Sabes, P. N., 2000. The planning and control of reaching movements. Curr. Opin. Neurobiol. 10: 740–746.CrossRefGoogle ScholarPubMed
Samejima, K., Doya, K., and Kawato, M., 2003. Inter-module credit assignment in modular reinforcement learning. Neur. Networks 16: 985–994.CrossRefGoogle ScholarPubMed
Schaal, S., 1997. Learning from demonstration. In Mozer, M. C., Jordan, M. and Petsche, T. (eds.) Advances in Neural Information Processing Systems, vol. 9. Cambridge, MA: MIT Press, pp. 1040–1046.Google Scholar
Schaal, S. 1999. Is imitation learning the route to humanoid robots? Trends Cog. Sci. 3: 233–242.CrossRefGoogle ScholarPubMed
Schaal, S. 2002a. Arm and hand movement control. In Arbib, M. A. (ed.) The Handbook of Brain Theory and Neural Networks, 2nd edn. Cambridge, MA: MIT Press, pp. 110–113.Google Scholar
Schaal, S. 2002b. Learning robot control. In Arbib, M. A. (ed.) The Handbook of Brain Theory and Neural Networks, 2nd edn. Cambridge, MA: MIT Press, pp. 983–987.Google Scholar
Schaal, S., and Atkeson, C. G., 1993. Open loop stable control strategies for robot juggling. Proceedings IEEE Int. Conference on Robotics and Automation, Atlanta, GA, pp. 913–918.
Schaal, S., and Sternad, D., 1998. Programmable pattern generators. Proceedings 3rd Int. Conference on Computational Intelligence in Neuroscience, Research Triangle Park, NC, pp. 48–51.
Schaal, S., and Sternad, D. 2001. Origins and violations of the 2/3 power law in rhythmic 3D movements. Exp. Brain Res. 136: 60–72.CrossRefGoogle Scholar
Schaal, S., Atkeson, C. G., and Botros, S., 1992. What should be learned? Proceedings 7th Yale Workshop on Adaptive and Learning Systems, New Haven, CT, pp. 199–204.
Schaal, S., Ijspeert, A., and Billard, A., 2003a. Computational approaches to motor learning by imitation. Phil. Trans. Roy. Soc. London B 358: 537–547.CrossRefGoogle ScholarPubMed
Schaal. S., Peters, J., Nakanishi, J., and Ijspeert, A., 2003b. Control, planning, learning, and imitation with dynamic movement primitives. Proceedings Workshop on Bilateral Paradigms on Humans and Humanoids, IEEE Int. Conference on Intelligent Robots and Systems, Las Vegas, NV, October 27–31, 2003.
Schaal, S., Sternad, D., Osu, R., and Kawato, M., 2004. Rhythmic movement is not discrete. Nature Neurosci. 7: 1137–1144.Google Scholar
Schweighofer, N., Arbib, M. A., Spoelstra, J., and Kawato, M., 1998. Role of the cerebellum in reaching movements in humans. II. A neural model of the intermediate cerebellum. Eur. J. Neurosci. 10: 95–105.CrossRefGoogle Scholar
Shams, L., and Malsburg, C., 2002. The role of complex cells in object recognition. Vision Res. 42: 2547–2554.CrossRefGoogle ScholarPubMed
Shidara, M., Kawano, K., Gomi, H., and Kawato, M., 1993. Inverse-dynamics model eye movement control by Purkinje cells in the cerebellum. Nature 365: 50–52.CrossRefGoogle ScholarPubMed
Smits-Engelsman, B. C., Galen, G. P., and Duysens, J., 2002. The breakdown of Fitts' law in rapid, reciprocal aiming movements. Exp. Brain Res. 145: 222–230.CrossRefGoogle Scholar
Soechting, J. F., and Terzuolo, C. A., 1987. Organization of arm movements in three-dimensional space: Wrist motion is piecewise planar. Neuroscience 23: 53–61.CrossRefGoogle ScholarPubMed
Sosnik, R., Hauptmann, B., Karni, A., and Flash, T., 2004. When practice leads to co-articulation: the evolution of geometrically defined movement primitives. Exp. Brain Res. 156: 422–438.
Staude, G, and Wolf, W., 1997. Quantitative assessment of phase entrainment between discrete and cyclic motor actions. Biomed. Tech. 42: 478–481.CrossRefGoogle ScholarPubMed
Stein, R. B., Ogusztöreli, M. N., and Capaday, C. 1986. What is optimized in muscular movements? In Jones, N. L., McCartney, N. and McComas, A. J. (eds.) Human Muscle Power. Champaign, Il: Human Kinetics, pp. 131–150.Google Scholar
Sternad, D., and Dean, W. J., 2003. Rhythmic and discrete elements in multi-joint coordination. Brain Res. 989: 152–171.CrossRefGoogle ScholarPubMed
Sternad, D., and Schaal, D., 1999. Segmentation of endpoint trajectories does not imply segmented control. Exp. Brain Res. 124: 118–136.CrossRefGoogle Scholar
Sternad, D., Dean, W. J., and Schaal, S. 1999. Interaction of discrete and rhythmic dynamics in single-joint movements. In Grealy, M. and Thomson, J. (eds.) Studies in Perception and Action, vol. 5, Hillsdale, NJ: Lawrence Erlbaum, pp. 282–287.Google Scholar
Sternad, D., Dean, W. J., and Schaal, S. 2000. Interaction of rhythmic and discrete pattern generators in single joint movements. Hum. Mov. Sci. 19: 627–665.CrossRefGoogle Scholar
Sternad, D., Rugy, A., Pataky, T., and Dean, W. J., 2002. Interaction of discrete and rhythmic movements over a wide range of periods. Exp. Brain Res. 147: 162–174.CrossRefGoogle Scholar
Strogatz, S. H., 1994. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Reading, MA: Addison-Wesley.Google Scholar
Sutton, R. S., and Barto, A. G., 1998. Reinforcement Learning: An Introduction. Cambridge, MA: MIT Press.Google Scholar
Sutton, R. S., Precup, D., and Singh, S., 1999. Between MDPs and semi-MDPs: a framework for temporal abstraction in reinforcement learning. Artif. Intell. 112: 181–211.CrossRefGoogle Scholar
Tani, J., and Nolfi, S., 1999. Learning to perceive the world as articulated: an approach for hierarchical learning in sensory-motor systems. Neur. Networks 12: 1131–1141.CrossRefGoogle ScholarPubMed
Tesauro, G. 1992. Temporal difference learning of backgammon strategy. In Sleeman, D. and Edwards, P. (eds.) Proceedings 9th International Workshop Machine. Morgan Kaufmann, San Mateo, CA, pp. 9–18.Google Scholar
Tevatia, G., and Schaal, S., 2000. Inverse kinematics for humanoid robots. Proceedings Int. Conference on Robotics and Automation, San Fransisco, CA, pp. 294–299.
Thach, W. T., 1998. A role for the cerebellum in learning movement coordination. Neurobiol. Learn. Mem. 70: 177–188.CrossRefGoogle ScholarPubMed
Todorov, E., and Jordan, M. I., 2002. Optimal feedback control as a theory of motor coordination. Native Neurosci. 5: 1226–1235.CrossRefGoogle ScholarPubMed
Turvey, M. T., 1990. Coordination. Am. Psychol. 45: 938–953.CrossRefGoogle ScholarPubMed
Uno, Y., Kawato, M., and Suzuki, R., 1989. Formation and control of optimal trajectory in human multijoint arm movement: minimum torque-change model. Biol. Cybernet. 61: 89–101.CrossRefGoogle ScholarPubMed
Vijayakumar, S., and Schaal, S., 2000. Locally weighted projection regression: an O(n) algorithm for incremental real time learning in high dimensional spaces. Proceedings 17th Int. Conference on Machine Learning, vol. 1, Stanford, CA, pp. 288–293.
Viviani, P., 1986. Do units of motor action really exist? In Experimental Brain Research Series, vol. 15. Berlin: Springer-Verlag, pp. 828–845.
Wada, Y., and Kawato, M., 1994. Trajectory formation of arm movement by a neural network with forward and inverse dynamics models. Systems Comput. in Japan 24: 37–50.Google Scholar
Wada, Y., and Kawato, M. 1995. A theory for cursive handwriting based on the minimization principle. Biol. Cybernet. 73: 3–13.CrossRefGoogle ScholarPubMed
Wei, K., Wertman, G., and Sternad, D., 2003. Interactions between rhythmic and discrete components in a bimanual task. Motor Control 7: 134–155.CrossRefGoogle Scholar
Williamson, M., 1998. Neural control of rhythmic arm movements. Neur. Networks 11: 1379–1394.CrossRefGoogle ScholarPubMed
Wolpert, D. M., 1997. Computational approaches to motor control. Trends Cogn. Sci. 1: 209–216.CrossRefGoogle ScholarPubMed
Wolpert, D. M., and Kawato, M., 1998. Multiple paired forward and inverse models for motor control. Neur. Networks 11: 1317–1329.CrossRefGoogle ScholarPubMed
Wolpert, D. M., Miall, R. C., and Kawato, M., 1998. Internal models in the cerebellum. Trends Cogn. Sci. 2: 338–347.CrossRefGoogle ScholarPubMed

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