Expressiveness, Equivalence, and Uncertainty in Velocity Obstacles and Closest Point of Approach Metrics

2026-10-05

AI Summary

This study addresses the unclear relationships and lack of equivalence among Time to Closest Point of Approach (TCPA), Distance at Closest Point of Approach (DCPA), and Velocity Obstacles (VO) under uncertainty in autonomous navigation. We first establish equivalence conditions for these metrics under perfect information and extend them to convex relative state sets with bounded uncertainty. Methodologically, we demonstrate that independently computing bounds disrupts the joint relationships among these metrics. Accordingly, a set-valued, uncertainty-aware VO representation is proposed to preserve these relationships, integrating convex geometric analysis for uncertainty modeling in motion planning. Our results prove that the proposed VO representation maintains the joint characteristics of collision-inducing velocities under uncertainty, outperforming independently computed CPA-based metrics.

Abstract

Time to Closest Point of Approach (TCPA), Distance to Closest Point of Approach (DCPA), and Velocity Obstacles (VOs), are widely used to assess and mitigate collision risk in autonomous navigation, yet their relationship and behavior under uncertainty remain largely unexplored. Assuming perfect state information, we establish a relationship between these representations over finite and infinite prediction horizons and derive conditions under which they provide equivalent characterizations of collision risk. Under bounded uncertainty, we extend the Closest Point of Approach (CPA) metrics and VO to convex relative-state sets. We show that in this setting, independently computed TCPA and DCPA bounds lose the joint relationship required for VO membership, while uncertainty-aware VOs preserve this relationship through a set-valued representation of collision-inducing velocities.

Problem

• Velocity Obstacles
• Closest Point of Approach
• Collision Risk
• Uncertainty
• Autonomous Navigation

Innovation

• Velocity Obstacles
• Closest Point of Approach
• Uncertainty
• Set-valued Representation
• Collision Risk