Existence and Uniqueness of Rigid Body Dynamics with Coulomb Friction

For motion planning and veri cation as well as for model based control it is important to have an accurate dynamic system model In many cases friction is a signi cant 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 di erential 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 e ects 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