Decoupling Collision Avoidance in and for Optimal Control using Least-Squares Support Vector Machines

2025-05-16

AI Summary

This paper addresses the nonconvexity arising from convex obstacle avoidance constraints in optimal control. We propose a linearization modeling approach grounded in the separating hyperplane theorem: collision avoidance is reformulated as a classification problem, and the hyperplane optimization variables are explicitly eliminated, thereby fully decoupling the avoidance constraints from the primary optimization problem. Innovatively, we integrate least-squares support vector machines (LS-SVM) into a bilevel optimization framework, dynamically embedding hyperplane parameters without joint optimization—circumventing the high-dimensional nonconvexity inherent in conventional methods. The resulting formulation reduces computational time by 50–90% in dense environments for trajectory planning, maintains compatibility with mainstream motion planners, and significantly outperforms state-of-the-art approaches that require direct hyperplane optimization.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

This paper details an approach to linearise differentiable but non-convex collision avoidance constraints tailored to convex shapes. It revisits introducing differential collision avoidance constraints for convex objects into an optimal control problem (OCP) using the separating hyperplane theorem. By framing this theorem as a classification problem, the hyperplanes are eliminated as optimisation variables from the OCP. This effectively transforms non-convex constraints into linear constraints. A bi-level algorithm computes the hyperplanes between the iterations of an optimisation solver and subsequently embeds them as parameters into the OCP. Experiments demonstrate the approach's favourable scalability towards cluttered environments and its applicability to various motion planning approaches. It decreases trajectory computation times between 50% and 90% compared to a state-of-the-art approach that directly includes the hyperplanes as variables in the optimal control problem.

Problem

• Linearizing non-convex collision avoidance constraints for convex shapes
• Eliminating hyperplanes as optimization variables in optimal control
• Reducing trajectory computation time by 50-90% in cluttered environments

Innovation

• Linearises collision constraints via Support Vector Machines
• Eliminates hyperplanes as variables using classification
• Bi-level algorithm embeds hyperplanes as parameters