Frédéric Jean

The inactivation principle: mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements.

An important question in the literature focusing on motor control is to determine which laws drive biological limb movements. This question has prompted numerous investigations analyzing arm movements in both humans and monkeys. Many theories assume that among all possible movements the one actually performed satisfies an optimality criterion. In the framework of optimal control theory, a first approach is to choose a cost function and test whether the proposed model fits with experimental data. A second approach (generally considered as the more difficult) is to infer the cost function from behavioral data. The cost proposed here includes a term called the absolute work of forces, reflecting the mechanical energy expenditure. Contrary to most investigations studying optimality principles of arm movements, this model has the particularity of using a cost function that is not smooth. First, a mathematical theory related to both direct and inverse optimal control approaches is presented. The first theoretical result is the Inactivation Principle, according to which minimizing a term similar to the absolute work implies simultaneous inactivation of agonistic and antagonistic muscles acting on a single joint, near the time of peak velocity. The second theoretical result is that, conversely, the presence of non-smoothness in the cost function is a necessary condition for the existence of such inactivation. Second, during an experimental study, participants were asked to perform fast vertical arm movements with one, two, and three degrees of freedom. Observed trajectories, velocity profiles, and final postures were accurately simulated by the model. In accordance, electromyographic signals showed brief simultaneous inactivation of opposing muscles during movements. Thus, assuming that human movements are optimal with respect to a certain integral cost, the minimization of an absolute-work-like cost is supported by experimental observations. Such types of optimality criteria may be applied to a large range of biological movements.

Singular Trajectories of Control-Affine Systems

When applying methods of optimal control to motion planning or stabilization problems, we see that some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, minimizing singular trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of control-affine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic system—with respect to the Whitney topology—all nontrivial singular trajectories are of minimal order and of corank one. These results, established both for driftless and for control-affine systems, extend results of [Y. Chitour, F. Jean, and E. Trelat, Comptes Rendus Math., 337 (2003), pp. 49-52 (in French); Y. Chitour, F. Jean, and E. Trelat, J. Differential Geom., 73 (2006), pp. 45-73]. As a consequence, for generic control-affine systems (with or without drift) defined by more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. Besides, we prove that, given a control-affine system satisfying the Lie algebra rank condition (LARC), singular trajectories are strictly abnormal, generically with respect to the cost. We then show how these results can be used to derive regularity results for the value function and in the theory of Hamilton-Jacobi equations, which in turn have applications for stabilization and motion planning, from both theoretical and implementational points of view.

Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning

Nonholonomic systems are control systems which depend linearly on the control. Their underlying geometry is the sub-Riemannian geometry, which plays for these systems the same role as Euclidean geometry does for linear systems. In particular the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. The aim of these notes is to present these notions of approximation and their application to the motion planning problem for nonholonomic systems.

Complexity of nonholonomic motion planning

The complexity of motion planning amidst obstacles is a well modeled and understood notion. What is the increase of the complexity when the problem is to plan the trajectories of a nonholonomic robot? We show that this quantity can be seen as a function of paths and of the distance between the paths and the obstacles. We propose various definitions of it, from both topological and metric points of view, and compare their values. For two of them we give estimates which involve some E-norm on the tangent space to the configuration space. Finally we apply these results to compute the complexity needed to park a car-like robot with trailers.

Nonhomogeneous nilpotent approximations for nonholonomic systems with singularities

Nilpotent approximations are a useful tool for analyzing and controlling systems whose tangent linearization does not preserve controllability, such as nonholonomic mechanisms. However, conventional homogeneous approximations exhibit a drawback: in the neighborhood of singular points (where the system growth vector is not constant) the vector fields of the approximate dynamics do not vary continuously with the approximation point. The geometric counterpart of this situation is that the sub-Riemannian distance estimate provided by the classical Ball-Box Theorem is not uniform at singular points. With reference to a specific family of driftless systems, we show how to build a nonhomogeneous nilpotent approximation whose vector fields vary continuously around singular points. It is also proven that the privileged coordinates associated to such an approximation provide a uniform estimate of the distance.

Uniform Estimation of Sub-Riemannian Balls

AbstractA fundamental result of sub-Riemannian geometry, the ball-box theorem, states that small sub-Riemannian balls look like boxes

Measures of transverse paths in sub-Riemannian geometry

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First-Order Theory

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