[Paper Review] Rapidly computable viscous friction and no-slip rigid contact models

Evan Drumwright|arXiv (Cornell University)|Apr 3, 2015

Dynamics and Control of Mechanical Systems19 references3 citations

TL;DR

This paper presents a computationally efficient algorithm for simulating rigid contact with viscous friction or no-slip conditions, avoiding the high complexity of Coulomb friction models. By reformulating the problem as a positive-semidefinite linear complementarity problem (LCP), it achieves O(m³ + m²n) expected time complexity, reducing solver failures and enabling faster simulation in robotic locomotion and grasping tasks.

ABSTRACT

This article presents computationally efficient algorithms for modeling two special cases of rigid contact---contact with only viscous friction and contact without slip---that have particularly useful applications in robotic locomotion and grasping. Modeling rigid contact with Coulomb friction generally exhibits $O(n^3)$ expected time complexity in the number of contact points and $2^{O(n)}$ worst-case complexity. The special cases we consider exhibit $O(m^3 + m^2n)$ time complexity ($m$ is the number of independent coordinates in the multi rigid body system) in the expected case and polynomial complexity in the worst case; thus, asymptotic complexity is no longer driven by number of contact points (which is conceivably limitless) but instead is more dependent on the number of bodies in the system (which is often fixed). These special cases also require considerably fewer constrained nonlinear optimization variables thus yielding substantial improvements in running time. Finally, these special cases also afford one other advantage: the nonlinear optimization problems are numerically easier to solve.

Motivation & Objective

• Address the high computational cost of simulating rigid body contact with Coulomb friction, which scales as O(n³) and can reach 2^O(n) in worst-case scenarios.
• Develop computationally efficient alternatives for two special contact cases: pure viscous friction and no-slip rigid contact.
• Reduce the number of nonlinear optimization variables and improve numerical stability in contact solving.
• Enable faster, more reliable simulation for robotic locomotion and grasping by replacing complex Coulomb friction models.

Proposed method

• Formulate the contact problem as a mixed linear complementarity problem (MLCP), then reduce it to a positive-semidefinite linear complementarity problem (LCP) by eliminating unconstrained variables via matrix inversion.
• Use a modified projected primal-dual (PPM) algorithm to solve the resulting LCP, which is numerically more robust than copositive-plus LCPs used in standard Coulomb friction solvers.
• Apply the LCP formulation to two special cases: no-slip contact (zero relative tangential velocity) and viscous friction (friction force proportional to relative velocity).
• Derive the time complexity as O(m³ + m²n), where m is the number of independent coordinates and n is the number of contact points, decoupling complexity from the number of contact points.
• Implement the algorithm in the Moby simulator using symplectic Euler and Runge-Kutta integration for event-driven simulation with strict interpenetration avoidance.
• Optimize the PPM solver using efficient matrix operations and avoid MEX file compilation, yet still achieve significant speedups.

Experimental results

Research questions

• RQ1 Can rigid contact models with no-slip or viscous friction be solved with polynomial-time complexity, avoiding the exponential worst-case complexity of Coulomb friction?
• RQ2 Can the number of nonlinear optimization variables be significantly reduced in contact models without Coulomb friction?
• RQ3 Does replacing a copositive-plus LCP with a positive-semidefinite LCP improve numerical stability and solver convergence?
• RQ4 Can the proposed method achieve substantial runtime improvements in robotic locomotion and grasping simulations without sacrificing physical fidelity?

Key findings

• The proposed algorithm achieves O(m³ + m²n) expected time complexity, shifting computational burden from the number of contact points to the number of bodies in the system.
• The no-slip contact model simulation ran 28% faster than the baseline (301.796s vs. 416.284s), with mean contact points at 6 and standard deviation of 8.
• The viscous friction model achieved a 37% speedup (2139.73s vs. 2936.36s), though longer runtimes were observed due to sensitivity in ODEs causing rapid movements.
• The modified PPM algorithm outperformed quadprog by nearly 20% in speed, with only 5.5 mean pivot operations, despite no MEX optimization.
• The positive-semidefinite LCP formulation reduced solver failures and improved numerical robustness compared to standard Coulomb friction solvers.
• The method enables verifiable simulation with no interpenetration, making it suitable for high-fidelity robotic dynamics without sacrificing performance.