EXISTENCE AND UNIQUENESS OF RIGID-BODY DYNAMICS WITH COULOMB FRICTION

Abstract

For motion planning and verification, as well as for model-based control, it is important to have an accurate dynamic system model. In many cases, friction is a significant component of the model. When considering the solution of the rigid-body forward dynamics problem in systems such as robots, we can often appeal to the fundamental theorem of differential equations to prove the existence and uniqueness of the solution. It is known, however, that when Coulomb friction is added to the dynamic equations, the forward dynamic solution may not exist and if it exists, it is not necessarily unique. In these situations, the inverse dynamic problem is ill-posed as well. Thus there is the need to understand the nature of the existence and uniqueness problems and to know under what conditions these problems arise. In this paper, we show that even single degree of freedom systems can exhibit existence and uniqueness problems. Next, we introduce compliance in the otherwise rigid-body model. This has two effects. First, the forward and inverse problems become well-posed. Thus we are guaranteed unique solutions. Secondly, we show that the extra dynamic solutions associated with the rigid model are unstable.