Thèse de doctorat en Systèmes informatiques
Sous la direction de Jean-Paul Laumond.
Soutenue en 2007
à Toulouse, INSA , en partenariat avec Laboratoire d'Analyse et d'Architecture des Systèmes L.A.A.S. (Toulouse) (laboratoire) .
Modélisation de la locomation humaine par la commande optimale
Pas de résumé disponible.
This work seeks to analyze human walking at the trajectory planning level from an optimal control perspective. Our approach emphasizes the close relationship between the geometric shape of human locomotion in goal-directed movements and the simplified kinematic model of a wheeled mobile robot. This kind of system has been extensively studied in robotics community. From a kinematic perspective, the characteristic of this wheeled robot is the nonholonomic constraint of the wheels on the floor, which forces the vehicle to move tangentially to its main axis. In the case of human walking, the observation indicates that the direction of the motion is given by the direction of the body (due to some anatomical, mechanical body constraints. . . ). This coupling between the direction q and the position (x,y) of the body can be summarized by tanq = ˙ y ˙ x. It is known that this differential equation defines a non integrable 2- dimensional distribution in the 3-dimensional manifold R2×S1 gathering all the configurations (x,y,q). The controls of a vehicle are usually the linear velocity (via the accelerator and the brake) and the angular velocity (via the steering wheel). The first question addressed in this study can be roughly formulated as : where is the “steering wheel” of the human body located ? It appears that the torso can be considered as a kind of a steering wheel that steers the human body. This model has been validated on a database of 1,560 trajectories recorded from seven subjects. In the second part we address the following question : among all possible trajectories reaching a given position and direction, the subject has chosen one. Why ? The central idea to understand the shape of trajectories has been to relate this problem to an optimal control scheme : the trajectory is chosen according to some optimization principle. The subjects being viewed as a controlled system, we tried to identify several criteria that could be optimized. Is it the time to perform the trajectory ? the length of the path ? the minimum jerk along the path ?. . . We argue that the time derivative of the curvature of the locomotor trajectories is minimized. We show that the human locomotor trajectories are well approximated by the geodesics of a differential system minimizing the L2 norm of the control. Such geodesics are made of arcs of clothoids. The clothoid is a curve whose curvature grows with the distance from the origin. The accuracy of the model is supported by the fact that 90 percent of trajectories are approximated with an average error < 10cm. In the last part of this work we provide the partition of the 3-dimensional configuration space in cells : 2 points belong to a same cell if and only if they are reachable from the origin by a path of the same type. Such a decomposition is known as the synthesis of the optimal control problem. Most of the time when the target changes slightly the optimal trajectories change slightly. However, some singularities appear at some critical frontiers between cells. It is noticeable that they correspond to the strategy change for the walking subjects. This fundamental result is another poof of the locomotion model we have proposed