Modeling the kinematics and dynamics of compliant contact

In this presentation, we discuss an intuitive model of the kinematics and dynamics of spatial compliant contact. The kinematic model gives a relation between the relative velocity of two rigid bodies and the velocities of two points on their surfaces that have smallest distance. The dynamic model uses a 6D spatial spring element to model the elastic forces between the bodies resulting from the deformation of the surfaces during the contact situations. Simulation shows the application of this model for two objects bouncing on the floor and on each other. 1 Problem Description In the modeling of walking machines, we would like to have a relatively simple model of the interaction between the feet and the ground that still captures the forces and torques in all directions. Current contact models are either too simple (e.g. only in two dimensions, or based on instantaneous dissipation on impact as described for example in [1]) or too complex (e.g. FEM) for our purposes and in general not very intuitive. The robotics framework of twists and wrenches to model generalized velocities and forces in six dimensions can be used to obtain a simple, intuitive contact model that still can handle the full 3D behavior. 2 Kinematic Model The kinematic part models the velocities of the two closest points p1 and p2 (one on each body), as shown in Figure 1. It is easily obtained from two trivial observations: (a) The two surface normals at the contact points are equal but opposite (when expressed in the same coordinate frame). In equation form: g1 = −H 1 2 g2 (1) (b) The points are separated by a line with length ∆ and direction g1 (we could also choose g2 of course). H 2 p2 − p1 = ∆g1 (2) where ∆ = 〈