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How humans fly

Published online by Cambridge University Press:  03 June 2013

Alain Ajami ,
Jean-Paul Gauthier ,
Thibault Maillot  and
Ulysse Serres
Alain Ajami
Affiliation:
Université du Sud-Toulon-Var, LSIS, UMR CNRS 7296, B.P 20132, 83957 La Garde Cedex, France. [email protected]; [email protected]; [email protected]
Jean-Paul Gauthier
Affiliation:
Université du Sud-Toulon-Var, LSIS, UMR CNRS 7296, B.P 20132, 83957 La Garde Cedex, France. [email protected]; [email protected]; [email protected] INRIA GECO Project 
Thibault Maillot
Affiliation:
Université du Sud-Toulon-Var, LSIS, UMR CNRS 7296, B.P 20132, 83957 La Garde Cedex, France. [email protected]; [email protected]; [email protected]
Ulysse Serres
Affiliation:
Université de Lyon, 69 622 Lyon, France Université Lyon 1, Villeurbanne; LAGEP, UMR CNRS 5007, 43 bd du 11 novembre 1918, 69100 Villeurbanne, France; [email protected]
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Abstract

This paper is devoted to the general problem of reconstructing the cost from theobservation of trajectories, in a problem of optimal control. It is motivated by thefollowing applied problem, concerning HALE drones: one would like them to decide bythemselves for their trajectories, and to behave at least as a good human pilot. Thisapplied question is very similar to the problem of determining what is minimized in humanlocomotion. These starting points are the reasons for the particular classes of controlsystems and of costs under consideration. To summarize, our conclusion is that in general,inside these classes, three experiments visiting the same values of the control are neededto reconstruct the cost, and two experiments are in general not enough. The method isconstructive.

The proof of these results is mostly based upon the Thom’s transversality theory.

This study is partly supported by FUI AAP9 project SHARE, and by ANR Project GCM, program“blanche”, project number NT09-504490.

Keywords

Inverse optimal controlanthropomorphic controltransversality
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 1030 - 1054
Copyright
© EDP Sciences, SMAI, 2013

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References

A.A. Agrachev and Y.L. Sachkov, Control theory from the geometric viewpoint. Springer-Verlag, Berlin, Encyclopaedia of Mathematical Sciences 87 (2004). Control Theory and Optimization, II.
A. Ajami, T. Maillot, N. Boizot, J.-F. Balmat, and J.-P. Gauthier. Simulation of a uav ground control station, in Proceedings of the 9th International Conference of Modeling and Simulation, MOSIM’12 (2012). To appear, Bordeaux, France (2012).
G. Arechavaleta, J.-P. Laumond, H. Hicheur, and A. Berthoz, Optimizing principles underlying the shape of trajectories in goal oriented locomotion for humans, in Humanoid Robots, 2006 6th IEEE-RAS International Conference on (2006) 131–136.
Arechavaleta, G., Laumond, J.-P., Hicheur, H., and Berthoz, A., On the nonholonomic nature of human locomotion. Autonomous Robots 25 2008 25–35. Google Scholar
Arechavaleta, G., Laumond, J.-P., Hicheur, H., and Berthoz, A., An optimality principle governing human walking. Robot. IEEE Trans. on 24 2008) 5–14. Google Scholar
Berret, B., Darlot, C., Jean, F., Pozzo, T., Papaxanthis, C., and Gauthier, J.P., The inactivation principle: mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements. PLoS Comput. Biol. 4 (2008) 25. Google Scholar
Berret, B., Gauthier, J.-P., and Papaxanthis, C., How humans control arm movements. Tr. Mat. Inst. Steklova 261 (2008) 47–60. Google Scholar
Chitour, Y., Jean, F., and Mason, P., Optimal control models of goal-oriented human locomotion. SIAM J. Control Optim. 50 (2012) 147–170. Google Scholar
H. Chitsaz and S. LaValle, Time-optimal paths for a dubins airplane, in Decision and Control, 2007 46th IEEE Conference on (2007) 2379–2384.
F. Chittaro, F. Jean, and P. Mason. On the inverse optimal control problems of the human locomotion: stability and robustness of the minimizers. J. Math. Sci. (To appear).
Gauthier, J.-P., Berret, B., and Jean, F.. A biomechanical inactivation principle. Tr. Mat. Inst. Steklova 268 (2010) 100–123. Google Scholar
M. Golubitsky and V. Guillemin, Stable mappings and their singularities. Springer-Verlag, New York, Graduate Texts in Mathematics 14 (1973).
F. Jean, Optimal control models of the goal-oriented human locomotion, Talk given at the “Workshop on Nonlinear Control and Singularities”, Porquerolles, France (2010).
Li, W., Todorov, E., and Liu, D., Inverse optimality design for biological movement systems. World Congress 18 (2011) 9662–9667. Google Scholar
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mishchenko, The mathematical theory of optimal processes, Translated from the Russian by K.N. Trirogoff, edited by L.W. Neustadt. Interscience Publishers John Wiley & Sons, Inc.  New York-London (1962).
Rump, S.M., Verification of positive definiteness. BIT 46 (2006) 433–452. Google Scholar
Thom, R., Les singularités des applications différentiables. Ann. Inst. Fourier Grenoble 6 (1955–1956) 43–87. Google Scholar
R. Vinter, Optimal control, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA (2000).