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

Robotics (cs.RO); Systems and Control (eess.SY)

arXiv:2610.07371 [cs.RO]

(or arXiv:2610.07371v1 [cs.RO] for this version)

https://doi.org/10.48550/arXiv.2610.07371 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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