This article describes an absolutely stable, first-order constraint solverfor multi-rigid body systems that calculates (predicts) constraint forces for typical bilateral and unilateral constraints, contact constraints with friction, and many other constraint types. Redundant constraints do not pose numerical problems or require regularization. Coulomb friction for contact is modeled using a true friction cone, rather than a linearized approximation. The computational expense of the solver is dependent upon the types of constraints present in the input. The hardest (in a computational complexity sense) inputs are reducible to solving convex optimization problems, i.e., polynomial time solvable. The simplest inputs require only solving a linear system. The solver is L-stable, which will imply that the forces due to constraints induce no computational stiffness into the multi-body dynamics differential equations. This approach is targeted to multibodies simulated with coarse accuracy, subject to computational stiffness arising from constraints, and where the number of constraint equations is not large compared to the number of multibody position and velocity state variables. For such applications, the approach should prove far faster than using other implicit integration approaches. I assess the approach on some fundamental multibody dynamics problems.
The contact aspects of the constraint model introduced in this article are effectively that of the @cite_25 contact model, though I introduce numerous computational and modeling advantages over the approach they describe. Mine does not require categorizing contact into sticking and sliding. My approach permits using realistic stiffness values derived via engineering tables and without needing to consider whether the choice of modeling parameters might make the simulation unstable.
@cite_25
2040973163
The use of Coulomb's friction law with the principles of classical rigid-body dynamics introduces mathematical inconsistencies. Specifically, the forward dynamics problem can have no solutions or multiple solutions. In these situations, compliant contact models, while increasing the dimensionality of the state vector, can resolve these problems. The simplicity and efficiency of rigid-body models, however, provide strong motivation for their use during those portions of a simulation when the rigid-body solution is unique and stable. In this paper, we use singular perturbation analysis in conjunction with linear complementarity theory to establish conditions under which the solution predicted by the rigid-body dynamic model is stable. We employ a general model of contact compliance to derive stability criteria for planar mechanical systems. In particular, we show that for cases with one sliding contact, there is always at most one stable solution. Our approach is not directly applicable to transitions between rolling and sliding where the Coulomb friction law is discontinuous. To overcome this difficulty, we introduce a smooth nonlinear friction law, which approximates Coulomb friction. Such a friction model can also increase the efficiency of both rigid-body and compliant contact simulation. Numerical simulations for the different models and comparison with experimental results are also presented.