Minimum-Time Escape from the Conic Field of View of a Reeds-Shepp Pursuer: Classification and Game Value for a Slower or Equally Fast Evader
Abstract
We study minimum-time escape of an omnidirectional evader from the forward conic field of view of a Reeds-Shepp pursuer with minimum turning radius r. For speed ratio ν and cone half-angle 0<α<π/2, we prove a global escape/holding classification: straight reversal holds the whole cone for ν≤sinα, escape is confined to a barrier-bounded dart for sinα<ν≤1, and escape is possible from every state for ν>1. For the slower or equally fast evader, sinα<ν≤1, we construct the solution with Isaacs' characteristics and verify it in feedback form, using a Filippov pursuer relay and a one-sided evader branch. The finite escape time is the first root of one scalar equation, and the holding region is bounded by the nonusable parts of the rays and a barrier that closes with a convex corner on the front axis. Sample-and-hold counter-controls show that this escape time is the value of the game in nonanticipative strategies at every state except the corner. There the two strategy values differ: the pursuer's is infinite, whereas an evader that chooses which barrier branch to cross from the initial germ of the pursuer's control escapes before the barrier time τI, which is the evader's exact strategy value. An independent level-set solver, refined on fixed comparison points, agrees with the closed-form escape time and reproduces the holding region at the tested parameters.
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BibTeX
@misc{ruiz2026minimumtime,
title = {{Minimum-Time Escape from the Conic Field of View of a Reeds-Shepp Pursuer: Classification and Game Value for a Slower or Equally Fast Evader}},
author = {Ubaldo Ruiz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27629},
eprint = {2610.27629},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27629},
}
RIS
TY - UNPB
TI - Minimum-Time Escape from the Conic Field of View of a Reeds-Shepp Pursuer: Classification and Game Value for a Slower or Equally Fast Evader
AU - Ubaldo Ruiz
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27629
UR - https://arxiv.org/abs/2610.27629
AB - We study minimum-time escape of an omnidirectional evader from the forward conic field of view of a Reeds-Shepp pursuer with minimum turning radius r. For speed ratio ν and cone half-angle 0<α<π/2, we prove a global escape/holding classification: straight reversal holds the whole cone for ν≤sinα, escape is confined to a barrier-bounded dart for sinα<ν≤1, and escape is possible from every state for ν>1. For the slower or equally fast evader, sinα<ν≤1, we construct the solution with Isaacs' characteristics and verify it in feedback form, using a Filippov pursuer relay and a one-sided evader branch. The finite escape time is the first root of one scalar equation, and the holding region is bounded by the nonusable parts of the rays and a barrier that closes with a convex corner on the front axis. Sample-and-hold counter-controls show that this escape time is the value of the game in nonanticipative strategies at every state except the corner. There the two strategy values differ: the pursuer's is infinite, whereas an evader that chooses which barrier branch to cross from the initial germ of the pursuer's control escapes before the barrier time τ_I, which is the evader's exact strategy value. An independent level-set solver, refined on fixed comparison points, agrees with the closed-form escape time and reproduces the holding region at the tested parameters.
ER -