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

Abstract

Abstract. In recent papers models of the human locomotion by means of an optimal control problem 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 result 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).

Geometric modeling of the movement based on an inverse optimal control approach

“…It is rather standard that such a property implies the continuity of the associated value function (see for instance [7,Prop. 2]).…”

“…A natural cost can for instance correspond to a compromise between time minimization and minimization of an energy term. Inspired by this idea, in [6], [7] we introduced the general class of costs of the following form…”

“…Nevertheless, for some specific systems, some preliminary results are obtained; we refer to [12], where the issue of the reconstruction of the cost is analyzed, and to [7], which is concerned with continuity properties of the minimizers, with respect to variations of the cost in some particular class. paolo.mason@lss.supelec.fr.…”

“…For every positive integer k, L k denotes the class of costs that depend on the (k+1)-th jets of the geometric curves, and that satisfy some coercivity and convexity hypotheses. The class of admissible costs is given by the union of L k , for all positive k. With the purpose of investigating the continuity of the minimizers with respect to variations of the cost, we formalize and extend the robustness result of [7] to this more general case, in the following sense: given a suitable family (L ε ) ε ⊂ L k of costs that converge to some L 0 ∈ L k as ε → 0, then the optimal trajectories associated with L ε converge to the set of minimizers for the cost L 0 (see Proposition 2.6 and Corollary 2.7).…”

“…In [7], we focused on the classes L 1 and L 2 , being them the most appropriate to model goal-oriented human locomotion (see [6]). In particular, we proved a robustness result for sequences of minimizers associated with a sequence of costs belonging to L 2 and converging to some ℓ 0 ∈ L 1 .…”