Nonholonomic Rolling Nonprehensile Manipulation Primitive

This chapter reviews the problem of nonholonomic rolling in nonprehen- sile manipulation tasks through two challenging and illustrative examples: the robotic hula-hoop and the ballbot system. The hula-hoop consists of an actuated stick and an unactuated hoop. First, the corresponding kinematic model is derived. Second, the dynamic model is derived through the Lagrange-D’Alembert equations. Then a control strategy is designed to rotate the hoop at some desired constant speed whereas positioning it over a desired point on the stick surface. A stability analysis, which guarantees ultimate boundedness of all signals of interest, is carried out. The ball-bot is an underactuated and nonholonomic constrained mobile robot whose upward equilibrium point must be stabilised by active controls. Coordinate-invariant equations of motion are derived for the ballbot. The linearised equations of motion are then derived, followed by the detailed controllability analysis. Excluding the rotary degree of freedom of the ball in the inertial vertical direction, the linear system turns out to be controllable. It follows that the nonlinear system is locally controllable, and a proportional-derivative type controller is designed to locally exponentially stabilise the upward equilibrium point and the translation of the ball. Numerical simulations for these two examples illustrate the effectiveness of the proposed methods. This chapter is based on the works presented in [1–4].