Dynamic Model of Planar Sliding

Abstract

In this paper, we present a principled method to model general planar sliding motion with distributed patch contact between two objects undergoing relative sliding motion. The effect of contact patch can be equivalently modeled as the contact wrench at one point contact. We call this point equivalent contact point (ECP). Our dynamic model embeds ECP within the Newton-Euler equations of slider’s motion and friction model. The discrete-time motion model that we derive consists of a system of quadratic equations relating the contact wrench and slip speed. This discrete-time dynamic model allows us to solve for the two components of tangential friction impulses, the friction moment, and the slip speed. The state of the slider as well as the ECP can be computed by solving a system of linear equations once the contact impulses are computed. In addition, we derive the closed form solutions for the state of slider for quasi-static motion. Furthermore, in pure translation, based on the discrete-time model, we present the closed form expressions for the friction impulses that acts on the slider and the state of the slider at each time step. Our results are dependent on the rigid body assumption and a generalized Coulomb friction model, which assumes that the contact force and moment lies within a quadratic convex cone and the friction force is independent of contact area. The results are not dependent on the exact knowledge of contact geometry or pressure distribution on the contact patch. Simulation examples are shown with both convex and non-convex contact patches to demonstrate the validity of our approach.