The motion of a disk rolling on a vibrating horizontal plane: Feasible control and path controllability

Let (X, Y, Z) be an inertial coordinate system and suppose that a horizontal plane is vibrating with an acceleration that is parallel to the (X, Y)-plane. A disk is rolling on the vibrating plane. Given two points A and B both fixed in the (X, Y)-plane. Open-loop strategies are computed for rolling the disk, on the vibrating plane, from A to a small neighbourhood of the point B, during a given time interval [0, tf] and subject to state and control constraints. In addition, the concept of path controllability is introduced and a condition is derived for the diskʹs motion path controllability. The derivation of this condition enables one to design closed-loop control laws for the diskʹs motion.