Journal of Mathematical Sciences

, Volume 195, Issue 3, pp 269–287 | Cite as

On Inverse Optimal Control Problems of Human Locomotion: Stability and Robustness of the Minimizers

Article

Abstract

In recent papers, models of human locomotion by means of optimal control problems have been proposed. In this paradigm, the trajectories are assumed to be solutions of an optimal control problem whose cost has to be determined. The purpose of the present paper is to analyze the class of optimal control problems defined in this way. We prove strong convergence results for their solutions, on the one hand, for perturbations of the initial and final points (stability), and, on the other hand, for perturbations of the cost (robustness).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. A. Agrachev and Yu. L. Sachkov, “Control theory from the geometric viewpoint,” in: Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag, Berlin (2004).Google Scholar
  2. 2.
    G. Arechavaleta, J.-P. Laumond, H. Hicheur, and A. Berthoz, “Optimizing principles underlying the shape of trajectories in goal oriented locomotion for humans,” in: IEEE/RAS International Conference on Humanoid Robots, Genoa (2006).Google Scholar
  3. 3.
    G. Arechavaleta, J.-P. Laumond, H. Hicheur, and A. Berthoz, “On the nonholonomic nature of human locomotion,” Autonomous Robots, 25, 25–35 (2008).CrossRefGoogle Scholar
  4. 4.
    G. Arechavaleta, J.-P. Laumond, H. Hicheur, and A. Berthoz, “An optimality principle governing human walking,” IEEE Trans. on Robotics, 24, No. 1, 5–14 (2008).CrossRefGoogle Scholar
  5. 5.
    A. V. Arutyunov and R. B. Vinter, “A simple ‘finite approximations’ proofs of the Pontryagin maximum principle under reduced differentiability hypotheses,” Set-Valued Anal., 12, Nos. 1–2, 5–24 (2004).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    T. Bayen, Y. Chitour, F. Jean, and P. Mason, “Asymptotic analysis of an optimal control problem connected to the human locomotion,” in: Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai (2009).Google Scholar
  7. 7.
    B. Berret, F. Jean, and J.-P. Gauthier, “A biomechanical inactivation principle,” Proc. Steklov Inst. Math., 268, 93–116 (2010).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    B. Bonnard and M. Chyba, “Singular trajectories and their role in control theory,” in: Math. Appl., 40, Springer-Verlag, Berlin (2003).Google Scholar
  9. 9.
    S. Boyd, L. E. Ghaoui, E. Feron, and Balakrishnan V., Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia (1994).CrossRefMATHGoogle Scholar
  10. 10.
    Y. Chitour, F. Jean, and P. Mason, “Optimal control models of the goal-oriented Human Locomotion,” SIAM J. Control Optim., 50, 147–170 (2012).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, New York (1989).CrossRefMATHGoogle Scholar
  12. 12.
    C. Darlot, J.-P. Gauthier, F. Jean, C. Papaxanthis, and T. Pozzo, “The inactivation principle: Mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements,” PLoS Comput. Biol., 4, No. 10 (2008).Google Scholar
  13. 13.
    R. Kalman, “When is a linear control system optimal?” J. Basic Eng., 86, 51–60 (1964).CrossRefGoogle Scholar
  14. 14.
    K. Mombaur, A. Truong, and J.-P. Laumond, “From human to humanoid locomotion—an inverse optimal control approach,” Auton. Robots, 28, 369–383 (2010).CrossRefGoogle Scholar
  15. 15.
    A. Y. Ng and S. Russell, “Algorithms for inverse reinforcement learning,” in: Proc. 17th International Conf. on Machine Learning, Morgan Kaufmann, San Francisco (2000), pp. 663–670.Google Scholar
  16. 16.
    L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley-Interscience, New York–London (1962).MATHGoogle Scholar
  17. 17.
    E. Todorov, “Optimal control theory,” in: Bayesian Brain: Probabilistic Approaches to Neural Coding (K. Doya et al., eds.), MIT Press, Cambridge (2006), pp. 269–298.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Laboratoire des Signaux et SystèmesGif-sur-YvetteFrance
  2. 2.ENSTA ParisTechParisFrance
  3. 3.Team GECO, INRIA Saclay — Île-de-FranceParisFrance

Personalised recommendations