Impending Motion Direction of Contacting Rigid Bodies

Abstract

Three different formulations are presented for expressing the initial motion direction of a system of contacting frictionless rigid bodies under gravity;. The bodies are assumed to have no initial velocity. The first formulation expresses the accelerations of the bodies in terms of the contact forces between the bodies. The contact forces are themselves expressed as the solution to a quadratic programming problem. The second formulation expresses the accelerations of the bodies according to Gauss' ``principle of least constraint.'' This principle is well-known to apply to systems with holonomic motion constraints (such as joints or hinges); in this paper, we show that the principle extends to the nonholonomic constraints that arise due to contact between bodies. The third formulation is conceptually the simplest; it says simply that the initial acceleration of the system is in the direction that most quickly decreases the gravitational potential energy of the system without violating the contact constraints between the bodies.