Complementarity Techniques for Minimal Coordinate Contact Dynamics

In this paper, nonsmooth contact dynamics of articulated rigid multibody systems is formulated as a complementarity problem. Minimal coordinate (MC) formulation is used to derive the dynamic equations of motion as it provides significant computational cost benefits and leads to a smaller-sized complementarity problem when compared with the frequently used redundant coordinate (RC) formulation. Additionally, an operational space (OS) formulation is employed to take advantage of the low-order structure-based recursive algorithms that do not require mass matrix inversion, leading to a further reduction in these computational costs. Based on the accuracy with which Coulomb's friction cone is modeled, the complementarity problem can be posed either as a linear complementarity problem (LCP), where the friction cone is approximated using a polygon, or as a nonlinear complementarity problem (NCP), where the friction cone is modeled exactly. Both formulations are studied in this paper. These complementarity problems are further recast as nonsmooth unconstrained optimization problems, which are solved by employing a class of Levenberg–Marquardt (LM) algorithms. The necessary theory detailing these techniques is discussed and five solvers are implemented to solve contact dynamics problems. A simple test case of a sphere moving on a plane surface is used to validate these solvers for a single contact, whereas a 12-link complex pendulum example is chosen to compare the accuracy of the solvers for the case of multiple simultaneous contacts. The simulation results validate the MC-based NCP formulations developed in this paper. Moreover, we observe that the LCP solvers deliver accuracy comparable to that of the NCP solvers when the friction cone direction vectors in the contact tangent plane are aligned with the sliding contact velocity at each time step. The theory and simulation results show that the NCP approach can be seamlessly recast into an MC OS formulation, thus allowing for accurate modeling of frictional contacts, while at the same time reducing overall computational costs associated with contact and collision dynamics problems in articulated rigid body systems.