Collision Detection with Analytical Derivatives of Contact Kinematics

TL;DR

This paper introduces iDCOL, a novel framework for differentiable collision detection using convex implicit surfaces, enabling smooth contact kinematics and derivatives essential for gradient-based robotics applications.

Contribution

It develops a regularization method for contact geometries and derives analytical derivatives via the Implicit Function Theorem, advancing differentiable collision detection techniques.

Findings

Robust collision detection in degenerate configurations

Fast Newton-based solver for implicit contact kinematics

Successful application in path planning and contact physics

Abstract

Differentiable contact kinematics are essential for gradient-based methods in robotics, yet the mapping from robot state to contact distance, location, and normal becomes non-smooth in degenerate configurations of shapes with zero or undefined curvature. We address this inherent limitation by selectively regularizing such geometries into strictly convex implicit representations, restoring uniqueness and smoothness of the contact map. Leveraging this geometric regularization, we develop iDCOL, an implicit differentiable collision detection and contact kinematics framework. iDCOL represents colliding bodies using strictly convex implicit surfaces and computes collision detection and contact kinematics by solving a fixed-size nonlinear system derived from a geometric scaling-based convex optimization formulation. By applying the Implicit Function Theorem to the resulting system residual, we derive analytical derivatives of the contact kinematic quantities. We develop a fast Newton-based solver for iDCOL and provide an open-source C++ implementation of the framework. The robustness of the approach is evaluated through extensive collision simulations and benchmarking, and applicability is demonstrated in gradient-based kinematic path planning and differentiable contact physics, including multi-body rigid collisions and a soft-robot interaction example.