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
2093990123 1514487078 2050594090 1514218977
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]. 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. 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. 0. Background Notes.- 1. Continuous Partitions of Unity.- 2. Absolutely Continuous Functions.- 3. Some Compactness Theorems.- 4. Weak Convergence and Asymptotic Center of Bounded Sequences.- 5. Closed Convex Hulls and the Mean-Value Theorem.- 6. Lower Semicontinuous Convex Functions and Projections of Best Approximation.- 7. A Concise Introduction to Convex Analysis.- 1. Set-Valued Maps.- 1. Set-Valued Maps and Continuity Concepts.- 2. Examples of Set-Valued Maps.- 3. Continuity Properties of Maps with Closed Convex Graph.- 4. Upper Hemicontinuous Maps and the Convergence Theorem.- 5. Hausdorff Topology.- 6. The Selection Problem.- 7. The Minimal Selection.- 8. Chebishev Selection.- 9. The Barycentric Selection.- 10. Selection Theorems for Locally Selectionable Maps.- 11. Michael's Selection Theorem.- 12. The Approximate Selection Theorem and Kakutani's Fixed Point Theorem.- 13. (7-Selectionable Maps.- 14. Measurable Selections.- 2. Existence of Solutions to Differential Inclusions.- 1. Convex Valued Differential Inclusions.- 2. Qualitative Properties of the Set of Trajectories of Convex-Valued Differential Inclusions.- 3. Nonconvex-Valued Differential Inclusions.- 4. Differential Inclusions with Lipschitzean Maps and the Relaxation Theorem.- 5. The Fixed-Point Approach.- 6. The Lower Semicontinuous Case.- 3. Differential Inclusions with Maximal Monotone Maps.- 1. Maximal Monotone Maps.- 2. Existence and Uniqueness of Solutions to Differential Inclusions with Maximal Monotone Maps.- 3. Asymptotic Behavior of Trajectories and the Ergodic Theorem.- 4. Gradient Inclusions.- 5. Application: Gradient Methods for Constrained Minimization Problems.- 4. Viability Theory: The Nonconvex Case.- 1. Bouligand's Contingent Cone.- 2. Viable and Monotone Trajectories.- 3. Contingent Derivative of a Set-Valued Map.- 4. The Time Dependent Case.- 5. A Continuous Version of Newton's Method.- 6. A Viability Theorem for Continuous Maps with Nonconvex Images..- 7. Differential Inclusions with Memory.- 5. Viability Theory and Regulation of Controled Systems: The Convex Case.- 1. Tangent Cones and Normal Cones to Convex Sets.- 2. Viability Implies the Existence of an Equilibrium.- 3. Viability Implies the Existence of Periodic Trajectories.- 4. Regulation of Controled Systems Through Viability.- 5. Walras Equilibria and Dynamical Price Decentralization.- 6. Differential Variational Inequalities.- 7. Rate Equations and Inclusions.- 6. Liapunov Functions.- 1. Upper Contingent Derivative of a Real-Valued Function.- 2. Liapunov Functions and Existence of Equilibria.- 3. Monotone Trajectories of a Differential Inclusion.- 4. Construction of Liapunov Functions.- 5. Stability and Asymptotic Behavior of Trajectories.- Comments.