# ON THE OPTIMALITY OF DUBINS PATHS UNDER NONSMOOTH CONSTRAINTS.

Topic: Optimal control | All

27 octobre 2008, Salle 214 (Batiment P, Couperin), au Centre Automatique et Systèmes, Fontainebleau.

14h00 : Ricardo SANFELICE, CAS, Mines ParisTech.

In his seminal work more than 50 years ago, Dubins derived necessary conditions characterizing minimum-length continuously differentiable paths with bounded curvature of a particle (or vehicle) traveling at unit speed from given initial and final points with specified velocity vectors. This celebrated result states that such minimum-length paths are given by curves consisting of no more than three pieces, where each piece is either a straight line segment or an arc corresponding to a turn with minimum radius. In this talk, we present a nonsmooth version of Dubins problem: the paths develop across a partitioned state space with different forward velocity and turning radius constraints. We show that tools from nonsmooth analysis can be employed to derive optimality conditions for the paths. In particular, we recast this problem as a hybrid optimal control problem and solve it using optimality principles for hybrid systems. Among the optimality conditions, we derive a "refraction" law at the boundary of the regions, which generalizes Snell's law of refraction in optics to the case of paths with bounded maximum curvature. The results have applications in robotics, in particular, in motion planning of autonomous vehicles with obstacles, different terrain properties, and other topological constraints. Briefly, we will present another control algorithm for nonholonomic vehicles that can be recast in a hybrid system framework. We will outline the application of tools recently developed for such systems - these will be presented in detail in the upcoming minicourse on hybrid systems on November 12.

14h00 : Ricardo SANFELICE, CAS, Mines ParisTech.

In his seminal work more than 50 years ago, Dubins derived necessary conditions characterizing minimum-length continuously differentiable paths with bounded curvature of a particle (or vehicle) traveling at unit speed from given initial and final points with specified velocity vectors. This celebrated result states that such minimum-length paths are given by curves consisting of no more than three pieces, where each piece is either a straight line segment or an arc corresponding to a turn with minimum radius. In this talk, we present a nonsmooth version of Dubins problem: the paths develop across a partitioned state space with different forward velocity and turning radius constraints. We show that tools from nonsmooth analysis can be employed to derive optimality conditions for the paths. In particular, we recast this problem as a hybrid optimal control problem and solve it using optimality principles for hybrid systems. Among the optimality conditions, we derive a "refraction" law at the boundary of the regions, which generalizes Snell's law of refraction in optics to the case of paths with bounded maximum curvature. The results have applications in robotics, in particular, in motion planning of autonomous vehicles with obstacles, different terrain properties, and other topological constraints. Briefly, we will present another control algorithm for nonholonomic vehicles that can be recast in a hybrid system framework. We will outline the application of tools recently developed for such systems - these will be presented in detail in the upcoming minicourse on hybrid systems on November 12.