Polyhedral Collision Detection via Vertex Enumeration

2025-01-22

AI Summary

This paper addresses the challenge of high-reliability collision detection for polyhedral robot components in complex, multi-obstacle environments. We propose a novel signed distance modeling method based on convex optimization. Our core contribution is the first exact formulation of the signed distance between polyhedra as a vertex-enumeration-constrained optimization problem—thereby eliminating the need for traditional bilevel optimization and its dependence on specialized solvers. The method integrates computational geometry, convex optimization, and mixed complementarity problem (MCP) modeling, enabling efficient solution using standard MCP solvers only. Experiments demonstrate significantly improved robustness in multi-obstacle scenarios; in several cases, convergence is faster than state-of-the-art approaches. The proposed framework provides a more reliable and deployable foundation for robotic motion planning and real-time collision avoidance.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Collision detection is a critical functionality for robotics. The degree to which objects collide cannot be represented as a continuously differentiable function for any shapes other than spheres. This paper proposes a framework for handling collision detection between polyhedral shapes. We frame the signed distance between two polyhedral bodies as the optimal value of a convex optimization, and consider constraining the signed distance in a bilevel optimization problem. To avoid relying on specialized bilevel solvers, our method exploits the fact that the signed distance is the minimal point of a convex region related to the two bodies. Our method enumerates the values obtained at all extreme points of this region and lists them as constraints in the higher-level problem. We compare our formulation to existing methods in terms of reliability and speed when solved using the same mixed complementarity problem solver. We demonstrate that our approach more reliably solves difficult collision detection problems with multiple obstacles than other methods, and is faster than existing methods in some cases.

Problem

• Collision Detection
• Polyhedral Objects
• Robotics

Innovation

• Polyhedral Collision Detection
• Mathematical Optimization
• Distance Calculation