Displacements by Successive rotations for Vehicles Subject to Given Constraints

The displacements of a vehicle on a plane can be subject to constraints depending on the nature of the vehicle. One can, for instance, think of the existence of a smallest turning circle for a car. In this paper our purpose is to show, on a simple example, how such constraints can be handled. We, in fact, consider the case of a vehicle the motions of which consist of a finite sequence of rotations, each rotation being subject to the following constraints. 1) The radius of the circles along which the displacements of the vehicle take place are larger than a critical radius. 2) The centers of the successive rotations are located along a straight line defined by the geometry of the vehicle. The mathematical analysis of this problem relies on a suitable choice of frames of reference in which the expression of the constraints is particularly simple. It is then shown that, under the above constraints, an arbitrary displacement can always be achieved by three approriate rotation