A biomechanical inactivation principle

  • Jean-Paul Gauthier
  • Bastien Berret
  • Frédéric Jean
Article

Abstract

This paper develops the mathematical side of a theory of inactivations in human biomechanics. This theory has been validated by practical experiments, including zero-gravity experiments. The theory mostly relies on Pontryagin’s maximum principle on the one side and on transversality theory on the other side. It turns out that the periods of silence in the activation of muscles that are observed in practice during the motions of the arm can appear only if “something like the energy expenditure” is minimized. Conversely, minimization of a criterion taking into account the “energy expenditure” guaranties the presence of these periods of silence, for sufficiently short movements.

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References

  1. 1.
    W. Abend, E. Bizzi, and P. Morasso, “Human Arm Trajectory Formation,” Brain 105(Part 2), 331–348 (1982).CrossRefGoogle Scholar
  2. 2.
    C. G. Atkeson and J. M. Hollerbach, “Kinematic Features of Unrestrained Vertical Arm Movements,” J. Neurosci. 5(9), 2318–2330 (1985).Google Scholar
  3. 3.
    S. Ben-Itzhak and A. Karniel, “Minimum Acceleration Criterion with Constraints Implies Bang-Bang Control as an Underlying Principle for Optimal Trajectories of Arm Reaching Movements,” Neural Comput. 20(3), 779–812 (2008).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    N. Bernstein, The Co-ordination and Regulation of Movements (Pergamon Press, Oxford, 1967).Google Scholar
  5. 5.
    B. Berret, C. Darlot, F. Jean, T. Pozzo, C. Papaxanthis, and J.-P. Gauthier, “The Inactivation Principle: Mathematical Solutions Minimizing the Absolute Work and Biological Implications for the Planning of Arm Movements,” PLoS Comput. Biol. 4(10), e1000194 (2008).CrossRefMathSciNetGoogle Scholar
  6. 6.
    B. Berret, J.-P. Gauthier, and C. Papaxanthis, “How Humans Control Arm Movements,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 261, 47–60 (2008) [Proc. Steklov Inst. Math. 261, 44–58 (2008)].MathSciNetGoogle Scholar
  7. 7.
    J. J. Boessenkool, E. J. Nijhof, and C. J. Erkelens, “A Comparison of Curvatures of Left and Right Hand Movements in a Simple Pointing Task,” Exp. Brain Res. 120(3), 369–376 (1998).CrossRefGoogle Scholar
  8. 8.
    B. Bonnard, “Invariants in the Feedback Classification of Nonlinear Systems,” in New Trends in Nonlinear Control Theory (Springer, Berlin, 1989), Lect. Notes Control Inf. Sci. 122, pp. 13–22.CrossRefGoogle Scholar
  9. 9.
    S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, PA, 1994), SIAM Stud. Appl. Math. 15.MATHGoogle Scholar
  10. 10.
    F. H. Clarke, Optimization and Nonsmooth Analysis (J. Wiley and Sons, New York, 1983).MATHGoogle Scholar
  11. 11.
    T. Flash and N. Hogan, “The Coordination of Arm Movements: An Experimentally Confirmed Mathematical Model,” J. Neurosci. 5(7), 1688–1703 (1985).Google Scholar
  12. 12.
    J.-P. Gauthier and I. Kupka, Deterministic Observation Theory and Applications (Cambridge Univ. Press, Cambridge, 2001).MATHCrossRefGoogle Scholar
  13. 13.
    J.-P. Gauthier and V. Zakalyukin, “On the One-Step-Bracket-Generating Motion Planning Problem,” J. Dyn. Control Syst. 11, 215–235 (2005).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    R. Gentili, V. Cahouet, and C. Papaxanthis, “Motor Planning of Arm Movements Is Direction-Dependent in the Gravity Field,” Neuroscience 145(1), 20–32 (2007).CrossRefGoogle Scholar
  15. 15.
    E. Guigon, P. Baraduc, and M. Desmurget, “Computational Motor Control: Redundancy and Invariance,” J. Neurophysiol. 97(1), 331–347 (2007).CrossRefGoogle Scholar
  16. 16.
    M. Hallett and C. D. Marsden, “Ballistic Flexion Movements of the Human Thumb,” J. Physiol. 294, 33–50 (1979).Google Scholar
  17. 17.
    C. M. Harris and D. M. Wolpert, “Signal-Dependent Noise Determines Motor Planning,” Nature 394, 780–784 (1998).CrossRefGoogle Scholar
  18. 18.
    F. Hermens and S. Gielen, “Posture-Based or Trajectory-Based Movement Planning: A Comparison of Direct and Indirect Pointing Movements,” Exp. Brain Res. 159(3), 340–348 (2004).CrossRefGoogle Scholar
  19. 19.
    M. J. Hollerbach and T. Flash, “Dynamic Interactions between Limb Segments during Planar Arm Movement,” Biol. Cybern. 44(1), 67–77 (1982).CrossRefGoogle Scholar
  20. 20.
    R. Kalman, “When Is a Linear Control System Optimal?,” Trans. ASME, Ser. D: J. Basic Eng. 86, 51–60 (1964).Google Scholar
  21. 21.
    E. B. Lee and L. Markus, Foundations of Optimal Control Theory (J. Wiley and Sons, New York, 1967).MATHGoogle Scholar
  22. 22.
    P. Morasso, “Spatial Control of Arm Movements,” Exp. Brain Res. 42(2), 223–227 (1981).CrossRefGoogle Scholar
  23. 23.
    A. Y. Ng and S. Russell, “Algorithms for Inverse Reinforcement Learning,” in Proc. 17th Int. Conf. on Machine Learning (Morgan Kaufmann Publ., San Francisco, CA, 2000), pp. 663–670.Google Scholar
  24. 24.
    J. Nishii and T. Murakami, “Energetic Optimality of Arm Trajectory,” in Proc. Int. Conf. on Biomechanics of Man (Charles Univ., Prague, 2002), pp. 30–33.Google Scholar
  25. 25.
    K. C. Nishikawa, S. T. Murray, and M. Flanders, “Do Arm Postures Vary with the Speed of Reaching?,” J. Neurophysiol. 81(5), 2582–2586 (1999).Google Scholar
  26. 26.
    C. Papaxanthis, T. Pozzo, and M. Schieppati, “Trajectories of Arm Pointing Movements on the Sagittal Plane Vary with both Direction and Speed,” Exp. Brain Res. 148(4), 498–503 (2003).Google Scholar
  27. 27.
    C. Papaxanthis, T. Pozzo, and P. Stapley, “Effects of Movement Direction upon Kinematic Characteristics of Vertical Arm Pointing Movements in Man,” Neurosci. Lett. 253(2), 103–106 (1998).CrossRefGoogle Scholar
  28. 28.
    L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Pergamon Press, Oxford, 1964).Google Scholar
  29. 29.
    J. F. Soechting, “Effect of Target Size on Spatial and Temporal Characteristics of a Pointing Movement in Man,” Exp. Brain Res. 54(1), 121–132 (1984).CrossRefGoogle Scholar
  30. 30.
    J. F. Soechting and F. Lacquaniti, “Invariant Characteristics of a Pointing Movement in Man,” J. Neurosci. 1(7), 710–720 (1981).Google Scholar
  31. 31.
    E. Todorov, “Optimal Control Theory,” in Bayesian Brain: Probabilistic Approaches to Neural Coding, Ed. by K. Doya et al. (MIT Press, Cambridge, MA, 2007), Ch. 12, pp. 269–298.Google Scholar
  32. 32.
    Y. Uno, M. Kawato, and R. Suzuki, “Formation and Control of Optimal Trajectory in Human Multijoint Arm Movement. Minimum Torque-Change Model,” Biol. Cybern. 61(2), 89–101 (1989).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  • Jean-Paul Gauthier
    • 1
  • Bastien Berret
    • 2
  • Frédéric Jean
    • 3
  1. 1.Institut Universitaire de TechnologieUniversité de ToulonLa GardeFrance
  2. 2.INSERM U887 Motricité-Plasticité, DijonUniversité de BourgogneDijonFrance
  3. 3.École Nationale Supérieure de Techniques AvancéesParisFrance

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