Planning of smooth motions for a ball-plate system with limited contact area

The paper deals with the motion planning for a rolling system with limited contact area. The system under consideration is represented by a hemispherical object that can roll without slipping or spinning on the plane. Under the constraints imposed on the size of the contact area, the construction of motion can be regarded as a problem of parallel parking in a finite number of movement steps. A motion planning algorithm, realizing the movement steps by tracing smooth figure eights on the hemisphere, is introduced. To generate asymmetric figure eights, a generalization of the Viviani curve is proposed. An exceptional case of the algorithm, corresponding to a spinning maneuver, is constructed with the use of the Cassini curve. The convergence of the algorithm is analyzed and its computational feasibility is verified under simulation.