A New Algebraic Rigid-Body Collision Law Based on Impulse Space Considerations

Abstract

We present a geometric representation of the set of three-dimensional rigid-body collisional impulses that are reasonably permissible by the combination of non-negative post-collision separation rate, non-negative collisional compression impulse, non-negative energy dissipation and the Coulomb friction inequality. The construction is presented for a variety of special collisional situations involving special symmetry or extremes in the mass distribution, the friction coefficient, or the initial conditions. We review a variety of known friction laws and show how they do and do not fit in the permissible region in impulse space as well as comment on other attributes of these laws. We present a few parameterizations of the full permissible region of impulse space. We present a simple generalization to arbitrary three-dimensional point contact collisions of a simple law previously only applicable to objects with contact-inertia eigenvectors aligned with the surface normal and initial relative tangential velocity component (e.g., spheres and disks). This new algebraic collision law has two restitution parameters for general three-dimensional frictional single-point rigid-body collisions. The new law generates a collisional impulse that is a weighted sum of the impulses from a frictionless but nonrebounding collision and from a perfectly sticking, nonrebounding collision. We describe useful properties of our law; show geometrically the set of impulses it can predict for several collisional situations; and compare it with existing laws. For simultaneous collisions we propose that the new algebraic law be used by recursively breaking these collisions into a sequence ordered by the normal approach velocities of potential contact pairs.