Simultaneous control of position and orientation for ball-plate manipulation problem based on time-State control form

This paper deals with the ball-plate manipulation problem considered as a typical but complicated model of a driftless nonholonomic system. Due to a strong nonlinearity of the ball-plate system, a state equation of the kinematic model cannot be transformed into a chained form, which is known to be effective in constructing a feedback control law for some driftless nonholonomic systems. To address this problem, we utilize a time-state control form, a kind of canonical form which covers a broader class of systems than the chained form. This form is first applied to two separate subproblems, position control, in which the planar position of the ball is controlled but not the orientation, and orientation control, in which the orientation is controlled without changing the positional relation between the ball and the plates. It turns out that there exists a linearly uncontrollable subspace in the transformed subsystem, which turns into controllable by a change of coordinates. This implies that the system has the structure of a system with two generators. We propose a control strategy using iterative changes of coordinates, ensuring convergence in the neighborhood of the origin. Finally, we unify the subproblems into simultaneous control of position and orientation, i.e., the whole configuration of the system. The important idea in the simultaneous control is the coordinate transformation, which enables us to avoid a singular point. Results of simulations show that the proposed method achieve robustness to a measurement noise and perturbation of radius of the ball.