Control of Nonprehensile Manipulation with Ball-Beam System: Palm Circle of Contact Juggling

This paper deals with the realization of “Palm circle” task for a ball-beam system in a 2-dimensional space. The palm circle means the situation where the spatial position of the ball is stationary at a point while the beam is moving cyclicly. First, the equations of motion of the ball and the beam are derived via the Lagrange equation with the constraints that stand for the rolling contact between the ball and the beam. Second, the equations of motion are transformed to the ones in terms of the control variables, i.e. the position of the ball, the relative position of the ball on the beam and the orientation angle of the beam. Third, the LQR servo controller is derived based on the linear system around the origin of the equations of motion for the control. Finally, the effectiveness of the controller with a cyclic trajectory of ball on the beam is verified by numerical simulations.