A Hybrid Systems Model for Simple Manipulation and Self-Manipulation Systems

Rigid bodies, plastic impact, persistent contact, Coulomb friction, and massless limbs are ubiquitous simplifications introduced to reduce the complexity of mechanics models despite the obvious physical inaccuracies that each incurs individually. In concert, it is well known that the interaction of such idealized approximations can lead to conflicting and even paradoxical results. As robotics modeling moves from the consideration of isolated behaviors to the analysis of tasks requiring their composition, a mathematically tractable framework for building models that combine these simple approximations with reliable results is overdue. In this paper we present a formal hybrid dynamical system model that introduces suitably restricted compositions of these familiar abstractions with the guarantee of a certain kind of consistency analogous to global existence and uniqueness in classical dynamical systems. While a real system will have continuous (though possibly very stiff and fast) dynamics through impacts, the hybrid system developed here provides a discontinuous but self--consistent approximation to the dynamics. The modeling choices sacrifice exact quantitative accuracy for qualitatively correct and analytically tractable results with certain formal guarantees.