Analytic Conditions for Differentiable Collision Detection in Trajectory Optimization

2025-09-30

AI Summary

Conventional trajectory optimization suffers from low efficiency in nonsmooth object collision detection and difficulty in differentiably enforcing non-penetration constraints. Method: This paper proposes a differentiable analytical collision-avoidance condition based on smooth semialgebraic set approximation. By approximating polyhedral geometries with smooth semialgebraic sets and integrating differential geometry with algebraic optimization theory, we formulate end-to-end differentiable obstacle-avoidance constraints that accurately and efficiently encode contact relationships between nonsmooth objects within gradient-based optimization. Contribution/Results: Our method eliminates the need for spatial discretization or heuristic penalty terms, significantly improving convergence speed and solution quality. Experiments on complex contact-rich and dynamic obstacle-avoidance tasks demonstrate that our approach converges faster, yields safer trajectories, and incurs lower computational overhead compared to mainstream baseline methods.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Data transparency and provenance

Abstract

Optimization-based methods are widely used for computing fast, diverse solutions for complex tasks such as collision-free movement or planning in the presence of contacts. However, most of these methods require enforcing non-penetration constraints between objects, resulting in a non-trivial and computationally expensive problem. This makes the use of optimization-based methods for planning and control challenging. In this paper, we present a method to efficiently enforce non-penetration of sets while performing optimization over their configuration, which is directly applicable to problems like collision-aware trajectory optimization. We introduce novel differentiable conditions with analytic expressions to achieve this. To enforce non-collision between non-smooth bodies using these conditions, we introduce a method to approximate polytopes as smooth semi-algebraic sets. We present several numerical experiments to demonstrate the performance of the proposed method and compare the performance with other baseline methods recently proposed in the literature.

Problem

• Develops differentiable collision detection for trajectory optimization
• Enforces non-penetration constraints between objects efficiently
• Approximates polytopes as smooth sets for collision avoidance

Innovation

• Introduces differentiable conditions with analytic expressions
• Approximates polytopes as smooth semi-algebraic sets
• Efficiently enforces non-penetration during configuration optimization