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.
A rigid [point] contact'', i.e., a point of contact between two completely rigid bodies, corresponds to a unilateral rigid constraint and several more equations that describe the frictional interaction at that point (see, e.g., @cite_19 ). Rigid contact makes reasoning about the contact geometry simple, but the problems from redundant forces described above (also called indeterminacy'') remain. The conceptual problem of inconsistent configurations'', like Painlev ' e 's famous paradox (described in detail in @cite_11 ) emerges with rigid contact also. And rigid contact is non-smooth: force is discontinuous when two bodies first contact, for example. In addition to these challenges that arise from the mathematics (and likely the real physics as well, see, e.g. @cite_20 ), the computational demands of implementing such models are significant.
@cite_19 @cite_20 @cite_11
2007677977 2163862895
This paper is a summary of a comprehensive study of the problem of predicting the possible acceleration(s) of a set of rigid, three-dimensional bodies in contact in the presence of Coulomb friction. We begin with a brief introduction to this problem and a survey of related work and previous approaches. This is followed by the introduction of two novel complementarity formulations for the contact problem under two friction laws: Coulomb''s Law and an analogous law in which Coulomb''s quadratic friction cone is approximated by a pyramid. Under a full co umn rank assumption on the system Jacobian matrix, we establish the existence and uniqueness of a solution to our new models in the case where the friction coefficients are nonnegative and sufficiently small. For the model based on the friction pyramid law, we also show that the classical Lemke almost-complementary pivot algorithm and our new feasible interior point method are guaranteed to compute a solution. Extensive computational results are presented to demonstrate the effectiveness and characteristics of these solution methods for solving the model with friction pyramids. Finally, results are summarized and conclusions are drawn. Rigid-body dynamics with unilateral contact is a good approximation for a wide range of everyday phenomena, from the operation of car brakes to walking to rock slides. It is also of vital importance for simulating robots, virtual reality, and realistic animation. However, correctly modeling rigid-body dynamics with friction is difficult due to a number of discontinuities in the behavior of rigid bodies and the discontinuities inherent in the Coulomb friction law. This is particularly crucial for handling situations with large coefficients of friction, which can result in paradoxical results known at least since Painleve [C. R. Acad. Sci. Paris, 121 (1895), pp. 112--115]. This single example has been a counterexample and cause of controversy ever since, and only recently have there been rigorous mathematical results that show the existence of solutions to his example. The new mathematical developments in rigid-body dynamics have come from several sources: "sweeping processes" and the measure differential inclusions of Moreau in the 1970s and 1980s, the variational inequality approaches of Duvaut and J.-L. Lions in the 1970s, and the use of complementarity problems to formulate frictional contact problems by Lotstedt in the early 1980s. However, it wasn't until much more recently that these tools were finally able to produce rigorous results about rigid-body dynamics with Coulomb friction and impulses.