On Frictional Mechanical Systems and Their Computational Power

In this paper we define a class of mechanical systems consisting of rigid objects (defined by linear or quadratic surface patches) connected by frictional contact linkages between surfaces. (This class of mechanisms is similar to the Analytical Engine developed by Babbage in 1800s except that we assume frictional surfaces instead of toothed gears.) We prove that a universal Turing Machine (TM) can be simulated by a (universal) frictional mechanical system in this class consisting of a constant number of parts. Our universal frictional mechanical system has the property that it can reach a distinguished final configuration through a sequence of legal movements if and only if the universal TM accepts the input string encoded by its initial configuration. There are two implications from this result. First, the robotic mover’s problem is undecidable when there are frictional linkages. Second, a mechanical computer can be constructed that has the computational power of any conventional electronic computer and yet has only a constant number of mechanical