A Complementarity Problem Approach for Multibody Contact Dynamics with Regularized Friction

Contact modelling, simulation, and analysis of mechanical systems with friction need proper treatments due to the non-smooth nature of the contact phenomenon. Two main approaches to deal with the contact problems are constraint and compliant based formulations [3]. In constraint based approaches, the problem is formulated by imposing the kinematic unilateral constraints between bodies in contact. This forms a complementarity condition, meaning that either normal contact velocity or force exists at each time instance. By adding the Coulomb friction, this phenomenon can be represented by a complementarity problem with a friction cone. To solve the resulting complementarity problem, the actual cone may be tackled via appropriate iterative techniques [2]. Alternatively, the friction cone can be linearized [1]. In the latter case, various available pivoting or iterative algorithms can be applied to solve the resulting linear complementarity problem (LCP) [4]. On the other hand, in the compliant based approaches, the complementarity conditions disappear by relaxing the normal direction and characterizing the normal force based on the stiffness and normal penetration. A selected friction model is then incorporated into the equations. This leads us to a set of differential equations in which appropriate numerical techniques should be used to solve the equations. While selecting high stiffness values increases the accuracy of the model, it can imply a set of stiff differential equations which encompasses challenges and drawbacks. A novel complementarity problem formulation based on the combination of the above approaches is introduced in this paper. In this approach, contacts are characterized based on the kinematic constraints while the friction forces are simultaneously regularized and incorporated into the formulation. As a result, the mathematical formulation representing the model is a complementarity problem whose variables are the normal forces. The friction forces are eliminated because of their dependencies to the normal forces, positions, or velocities. More specifically, the friction forces are expressed in terms of the normal forces in the kinetic phase, and they depend on the positions or velocities in the static phase (depending on the selected friction model). It can be shown that if the equations of motion are discretized, the resulting LCP in our proposed formulation reads as