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A biomechanical inactivation principle. (English) Zbl 1218.92012
Proc. Steklov Inst. Math. 268, 93-116 (2010) and Trudy Mat. Inst. Steklova 268, 100-123 (2010).
This paper is concerned with the mathematical side of the theory of inactivations in human biomechanics and addresses the question whether motor planning is optimal according to an identifiable criterion. In other words, the following inverse optimal control problem is posed: given recorded experimental data, infer a cost function with regard to which the observed behavior is optimal. In contrast to known methods, the authors suggest a new approach to optimal control problems which starts from the observation of simultaneous inactivations of opposing muscles during movements presumed as optimal. Using the Pontryagin maximum principle and transversality arguments from differential topology, it is proved that the minimization of a nonsmooth cost is a necessary condition to obtain inactivation phases along optimal trajectories. The periods of silence in the activation of muscles that are observed in practice during the motions of the arm can appear only if “energy expenditure” is minimized. On the other hand, minimization of a criterion accounting for “energy expenditure” ensures, for sufficiently short movements, existence of such periods of silence. Consequently, it is established that inactivation is a kind of necessary and sufficient conditions for the minimization of an absolute-work-like cost. The theory has been validated by practical experiments, including zero-gravity experiments.

MSC:
92C10 Biomechanics
49J15 Existence theories for optimal control problems involving ordinary differential equations
49N90 Applications of optimal control and differential games
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