Understanding the difference between prox and complementarity formulations for simulation of systems with contact

To plan a robotic task involving intermittent contact, such as an assembly task, it is helpful to be able to simulate the task accurately and efficiently. In the past ten years, the prox formulation of the equations of motion has arisen as a competitive alternative to the well-known linear and nonlinear complementarity problem (LCP and NCP) formulations. In this paper, we compare these two formulations, showing through a set-based argument that the formulations are equivalent. Second, we provide simple examples to compare the most common approaches for solving these formulations. The prox formulation is solved by fixed-point iteration while the complementarity formulation is solved by a pivoting scheme, known as Lemke's algorithm. The well-known paradox of PAINLEVE¿ is used in a case where two solutions exist to illustrate that the fixed-point scheme can fail while the pivoting scheme will succeed.