no. 4
How humans fly
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1030-1054.

This paper is devoted to the general problem of reconstructing the cost from the observation of trajectories, in a problem of optimal control. It is motivated by the following applied problem, concerning HALE drones: one would like them to decide by themselves for their trajectories, and to behave at least as a good human pilot. This applied question is very similar to the problem of determining what is minimized in human locomotion. These starting points are the reasons for the particular classes of control systems 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 needed to reconstruct the cost, and two experiments are in general not enough. The method is constructive. 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.

DOI : 10.1051/cocv/2012043
Classification : 93C10, 49J15, 93C41, 93C15, 34K35
Mots clés : inverse optimal control, anthropomorphic control, transversality 
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Ajami, Alain; Gauthier, Jean-Paul; Maillot, Thibault; Serres, Ulysse. How humans fly. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1030-1054. doi : 10.1051/cocv/2012043. http://www.numdam.org/articles/10.1051/cocv/2012043/

[1] 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. | MR

[2] 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).

[3] 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.

[4] G. Arechavaleta, J.-P. Laumond, H. Hicheur, and A. Berthoz, On the nonholonomic nature of human locomotion. Autonomous Robots 25 2008 25-35.

[5] G. Arechavaleta , J.-P. Laumond, H. Hicheur, and A. Berthoz, An optimality principle governing human walking. Robot. IEEE Trans. on 24 2008) 5-14.

[6] 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 (2008) 25. | MR

[7] B. Berret, J.-P. Gauthier, and C. Papaxanthis, How humans control arm movements. Tr. Mat. Inst. Steklova 261 (2008) 47-60. | MR

[8] Y. Chitour, F. Jean, and P. Mason, Optimal control models of goal-oriented human locomotion. SIAM J. Control Optim. 50 (2012) 147-170. | MR

[9] H. Chitsaz and S. Lavalle, Time-optimal paths for a dubins airplane, in Decision and Control, 2007 46th IEEE Conference on (2007) 2379-2384.

[10] 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). | MR

[11] J.-P. Gauthier, B. Berret, and F. Jean. A biomechanical inactivation principle. Tr. Mat. Inst. Steklova 268 (2010) 100-123. | MR

[12] M. Golubitsky and V. Guillemin, Stable mappings and their singularities. Springer-Verlag, New York, Graduate Texts in Mathematics 14 (1973). | MR

[13] F. Jean, Optimal control models of the goal-oriented human locomotion, Talk given at the “Workshop on Nonlinear Control and Singularities”, Porquerolles, France (2010).

[14] W. Li, E. Todorov, and D. Liu, Inverse optimality design for biological movement systems. World Congress 18 (2011) 9662-9667.

[15] 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). | MR

[16] S.M. Rump, Verification of positive definiteness. BIT 46 (2006) 433-452. | MR

[17] R. Thom, Les singularités des applications différentiables. Ann. Inst. Fourier Grenoble 6 (1955-1956) 43-87. | MR

[18] R. Vinter, Optimal control, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA (2000). | MR

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