An effective paradigm for simulating the dynamics of robots that locomote and manipulate is multi-rigid body simulation with rigid contact. This paradigm provides reasonable tradeoffs between accuracy, running time, and simplicity of parameter selection and identification. The Stewart-Trinkle Anitescu-Potra time stepping approach is the basis of many existing implementations. It successfully treats inconsistent (Painleve-type) contact configurations, efficiently handles many contact events occurring in short time intervals, and provably converges to the solution of the continuous time differential algebraic equations (DAEs) as the integration step size tends to zero. However, there is currently no means to determine when the solution has largely converged, i.e., when smaller integration steps would result in only small increases in accuracy. The present work describes an approach that computes the event times (when the set of active equations in a DAE changes) of all contact impact events for a multi-body simulation, toward using integration techniques with error control to compute a solution with desired accuracy. We also describe a first-order, variable integration approach that ensures that rigid bodies with convex polytopic geometries never interpenetrate. This approach permits taking large steps when possible and takes small steps when contact is complex.
Stewart and Trinkle @cite_10 and Anitescu and Potra @cite_0 described an approach (based on Moreau's time stepping'' @cite_13 discretization of the rigid body dynamics and differential inclusion theory) that provably mitigated the problem of inconsistent configurations. These authors proved that the complementarity problem @cite_30 based approaches used in these works do non-positive work and always possess a solution if the contact constraints are linearly independent. Shortly after introducing this method, Stewart proved convergence of the approach to the solution to the continuous time dynamics @cite_17 as the integration step tends to zero. Stewart-Trinkle @cite_2 and Anitescu-Potra @cite_14 later described approaches that, respectively, prevent and minimize interpenetration; Figures and illustrate the similarities and differences. Proofs of convergence and constraint stabilization rely upon assumptions that the complementarity problem has unique solutions @cite_2 and the function @cite_14 is differentiable.
@cite_30 @cite_13 @cite_14 @cite_0 @cite_2 @cite_10 @cite_17
2000151369 2335533 2050594090 1514487078 2113115376 2085897137 2093990123
Preface to the Classics Edition Preface Glossary of notation Numbering system 1. Introduction 2. Background 3. Existence and multiplicity 4. Pivoting methods 5. Iterative methods 6. Geometry and degree theory 7. Sensitivity and stability analysis Bibliography Index. This paper is devoted to mechanical systems with a finite number of degrees of freedom; let q1,...,qn denote (possibly local) coordinates in the configuration manifold Q. In addition to the constraints, bilateral and frictionless, which have permitted such a finite-dimensional parametrization of Q, we assume the system submitted to a finite family of unilateral constraints whose geometrical effect is expressed by v inequalities @math (1.1) defining a closed region L of Q. As every greek index in the sequel, α takes its values in the set 1,2,...,v . The v functions fα are supposed C1, with nonzero gradients, at least in some neighborhood of the respective surfaces fα = 0; for the sake of simplicity, we assume them independent of time. We present a method for achieving geometrical constraint stabilization for a linear-complementarity-based time-stepping scheme for rigid multibody dynamics with joints, contact, and friction. The method requires the solution of only one linear complementarity problem per step. We prove that the velocity stays bounded and that the constraint infeasibility is uniformly bounded in terms of the size of the time step and the current value of the velocity. Several examples, including one for joint-only systems, are used to demonstrate the constraint stabilization effect. Copyright © 2004 John Wiley & Sons, Ltd. A linear complementarity formulation for dynamic multi-rigid-body contact problems with Coulomb friction is presented. The formulation, based on explicit Euler integration and polygonal approximation of the friction cone, is guaranteed to have a solution for any number of contacts and contact configuration. A model with the same property, based on the Poisson hypothesis, is formulated for impact problems with friction and nonzero restitution coefficients. An explicit Euler scheme based on these formulations is presented and is proved to have uniformly bounded velocities as the stepsize tends to zero for the Newton–Euler formulation in body co-ordinates. In this paper a new time-stepping method for simulating systems of rigid bodies is given. Unlike methods which take an instantaneous point of view, our method is based on impulse-momentum equations, and so does not need to explicitly resolve impulsive forces. On the other hand, our method is distinct from previous impulsive methods in that it does not require explicit collision checking and it can handle simultaneous impacts. Numerical results are given for one planar and one three dimensional example, which demonstrate the practicality of the method, and its convergence as the step size becomes small. In this paper a new time-stepping method for simulating systems of rigid bodies is given which incorporates Coulomb friction and inelastic impacts and shocks. Unlike other methods which take an instantaneous point of view, this method does not need to identify explicitly impulsive forces. Instead, the treatment is similar to that of J. J. Moreau and Monteiro-Marques, except that the numerical formulation used here ensures that there is no inter-penetration of rigid bodies, unlike their velocity-based formulation. Numerical results are given for the method presented here for a spinning rod impacting a table in two dimensions, and a system of four balls colliding on a table in a fully three-dimensional way. These numerical results also show the practicality of the method, and convergence of the method as the step size becomes small. This paper gives convergence theory for a new implicit time‐stepping scheme for general rigid‐body dynamics with Coulomb friction and purely inelastic collisions and shocks. An important consequence of this work is the proof of existence of solutions of rigid‐body problems which include the famous counterexamples of Painleve. The mathematical basis for this work is the formulation of the rigid‐body problem in terms of measure differential inclusions of Moreau and Monteiro Marques. The implicit time‐stepping method is based on complementarity problems, and is essentially a particular case of the algorithm described in Anitescu & Potra [2], which in turn is based on the formulation in Stewart & Trinkle [47].