An Implicit Time-Stepping Method for Multibody Systems with Intermittent Contact Nilanjan Chakraborty, Stephen Berard, Srinivas Akella, and Jeff Trinkle Department of Computer Science Rensselaer Polytechnic Institute (To appear in Robotics Science and Systems 2007)

Introduction Dexterous manipulation/Grasping Mechanical design Computer Games Robonaut threading nut onto bolt Example Grasping experiment where the Circular lock piece must be grasped by the parallel jaw grippers as they close. (Brost and Christensen, 1996)

Related Work Differential Algebraic Equation (DAE) (Haug et al. 1986) –Differential Equations (Motion model) + Algebraic Constraints –Requires knowledge of contact interactions (sliding, rolling or separating) –Contact Interactions not known apriori! Differential Complementarity Problem (DCP) (Stewart and Trinkle 1996; Anitescu, Cremer and Potra 1996; Pfeiffer and Glocker 1996; Trinkle et al. 1997; Trinkle, Tzitizouris and Pang 2001) –Differential Equations (Motion model) + Complementarity constraints (Contact Model, Friction Model) + Algebraic Constraints

Motivation: Integrate Collision Detection Collision Detection Solve Dynamics Update Position

Motivation: Disc rolling on plane Eliminate sources of instability and inaccuracy: –Polyhedral approximations –decoupling of collision detection from dynamics –approximations of quadratic Coulomb friction model 10 vertices 100 vertices

Motivation: Disc rolling on plane Discretization of geometry and linearization of the distance function lead to a loss of energy in current simulators Results of simulating the rolling disc using the Stewart-Trinkle algorithm for varying number of edges and varying step-size. The top horizontal line is the computed value obtained by our geometrically implicit time-stepper using an implicit surface representation of the disc. (Error Tolerance = 1e-06)

Motivation: Integrate Collision Detection Collision Detection Solve Dynamics Update Position Goal: Integrate collision detection with equations of motion Assumption: Convex objects described as an intersection of implicit surfaces.

Complementarity Problem

Continuous Time Dynamics Model Newton-Euler equations: Kinematic map: Contact constraints: Mass MatrixContact forces and momentApplied forceCoriolis force Friction Model: Set of Complementarity Constraints Joint Constraints: Set of Algebraic Constraints

Discrete Time Dynamics Model Discrete Time Model: (Stewart and Trinkle 1996, uses linearized friction cone, subproblem at each time step is LCP) (Tzitzouris 2001 for closed form distance functions, subproblem at each time Step is NCP)

Contact Constraints Contact Constraint in workspace: Closest Points are given by: Contact Constraint in Configuration Space:

Contact Constraints From KKT Conditions:

Discrete Time Dynamics Model The mathematical model is a Mixed Nonlinear Complementarity Problem.

Results

Conclusion We presented a geometrically implicit time-stepper for multi-body systems that combines the collision detection and dynamic time step to deal with a source of inaccuracy in dynamic simulation.

Future Work Address the question of existence and uniqueness of solutions Implementation for intersection of surfaces. Extend to nonconvex implicit surface objects described as an union of convex objects as well as parametric surfaces. Precisely quantify the tradeoffs between the computation speed and physical accuracy

THANK YOU!