Mathematics

Consider a mobile robot in the plane that can move forward and backward, and rotate around itself. The state of the robot is described by coordinates (x,y) of its center of mass and by angle of orientation theta. Given an initial and a terminal state of the robot, one should find the shortest path from the initial state to the terminal one, when the length of the path is measured in the space (x,y,theta). Such a problem is formalized as a left-invariant sub-Riemannian problem on the group of motions of a plane. The talk will be devoted to recent results on this optimal control problem obtained by geometric techniques: sub-Riemannian geodesics are parametrized by Jacobi's functions, the Maxwell points corresponding to discrete symmetries generated by reflections of pendulum are described, the first conjugate time is bounded by Maxwell times, the cut time is proved to be equal to the first Maxwell time, the global structure of the exponential mapping and optimal synthesis are described, the cut locus is parametrized.