cs.ROOct 20, 2026

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

Authors: Ubaldo Ruiz

Organizations: Centro de Investigación Científica y de Educación Superior de Ensenada (CICESE), Ensenada, Baja California, Mexico

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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 rr. For speed ratio νν and cone half-angle 0<α<π/20<α<π/2, we prove a global escape/holding classification: straight reversal holds the whole cone for ν≤sin⁡αν\le\sinα, escape is confined to a barrier-bounded dart for sin⁡α<ν≤1\sinα<ν\le1, and escape is possible from every state for ν>1ν>1. For the slower or equally fast evader, sin⁡α<ν≤1\sinα<ν\le1, 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 τIτ_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.

Figures & tables

Figure 1: Conic field of view and coordinates. (a) Global frame: pursuer heading θp\theta_{p} , evader position pe\mathbf{p}_{e} , range ρ\rho and clockwise bearing φ\varphi . The cone is drawn truncated but has unbounded range. (b) Pursuer-fixed frame: xx points right and yy along the heading. The usable segments UP (green), of length ρB=r(ν−sin⁡α)\rho_{B}=r(\nu-\sin\alpha) , end at B±B_{\pm} (magenta); the remaining rays are NUP (gray dashed). Both panels use α=30∘\alpha=30^{\circ} ; (b) uses ν=0.9\nu=0.9 .
Figure 2: Game of kind. For ν≤sin⁡α\nu\leq\sin\alpha straight reversal holds the whole cone; for sin⁡α<ν≤1\sin\alpha<\nu\leq 1 escape is confined to a dart bounded by a barrier (Theorems 6.1 and 6.2 ); for ν>1\nu>1 escape is possible from every state (Proposition 3 (c)). Crosses: the examples of Figs. 3 and 4 .
Figure 3: Characteristic synthesis for the slower evader, verified in feedback form by Theorem 6.1 . (a) ν=0.7\nu=0.7 , α=10∘\alpha=10^{\circ} ; (b) ν=0.9\nu=0.9 , α=25∘\alpha=25^{\circ} . Blue: primary trajectories, with arrowheads in forward time and open circles at their exits on the UP (green); magenta: barrier from the BUP points B±B_{\pm} (squares) closing on the axis at (0,yI)(0,y_{I}) (diamond); orange dashed: the evader’s axis dispersal line De\mathcal{D}_{e} ; gray dashed: NUP; gray: holding region H\mathcal{H} ( W=∞W=\infty ).
Figure 4: Level sets of the escape time WW of Theorem 6.1 (solid blue) and of the independent WENO5 solution at h=0.002h=0.002 (dashed orange) in the right half of the cone, at the contour interval ΔW=0.025\Delta W=0.025 ; labels give the time on selected curves. Gray: nodes not reached by the solver within the horizon, numerical evidence of the holding region. The horizontal axes are stretched. (a) ν=0.7\nu=0.7 , α=10∘\alpha=10^{\circ} ; (b) ν=0.9\nu=0.9 , α=25∘\alpha=25^{\circ} .
(ν,α)(\nu,\alpha)hhmedian rel.max abs.meanmisclassified
(0.9,25∘)(0.9,25^{\circ})0.0047.9×10−77.9\times 10^{-7}1.3×10−61.3\times 10^{-6}1.1×10−71.1\times 10^{-7}0/5996
 0.0021.5×10−71.5\times 10^{-7}4.2×10−84.2\times 10^{-8}1.5×10−81.5\times 10^{-8}0/5996
 0.0014.4×10−84.4\times 10^{-8}9.3×10−99.3\times 10^{-9}4.2×10−94.2\times 10^{-9}0/5996
(0.7,10∘)(0.7,10^{\circ})0.0042.1×10−62.1\times 10^{-6}4.1×10−64.1\times 10^{-6}1.5×10−81.5\times 10^{-8}0/1675
 0.0023.8×10−73.8\times 10^{-7}5.5×10−85.5\times 10^{-8}2.7×10−82.7\times 10^{-8}0/1675
 0.0011.0×10−71.0\times 10^{-7}1.4×10−81.4\times 10^{-8}7.0×10−97.0\times 10^{-9}0/1675
Table 1: Level-set solver against the closed-form escape time WW at fixed comparison points: median relative error ∣T−W∣/max⁡(W,0.05)|T-W|/\max(W,0.05) , largest absolute error, signed mean error, and misclassified escape/holding nodes among those compared.
 DCRS (this paper)
Pursuer speedvp=Vpmax⁡v_{p}=V_{p}^{\max}∣vp∣≤Vpmax⁡|v_{p}|\leq V_{p}^{\max}
Usable partρ<r(ν+sin⁡α)\rho<r(\nu+\sin\alpha)ρ<r(ν−sin⁡α)\rho<r(\nu-\sin\alpha)
Apex usableall ν\nuν>sin⁡α\nu>\sin\alpha
Sliding arc on NUPyesno
Holding regionnone reportedKK if ν≤sin⁡α\nu\leq\sin\alpha ; all but a dart if sin⁡α<ν≤1\sin\alpha<\nu\leq 1 ; none if ν>1\nu>1
Closed barrierabsent for 0<α<π/20<\alpha<\pi/2sin⁡α<ν≤1\sin\alpha<\nu\leq 1
Table 2: Conic escape game for 0<α<π/20<\alpha<\pi/2 : Dubins pursuer (DC) of Lewin and Olsder (1979) and Reeds–Shepp pursuer (RS, this paper); ν=1/w\nu=1/w in their notation. Entries list features that occur in the respective syntheses for some parameter values, not at every pair (ν,α)(\nu,\alpha) .
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Holding region for ν=0.9\nu=0.9 , α=25∘\alpha=25^{\circ} (normalized units). (a) Escape region E\mathcal{E} (light), holding states off the barrier H∗\mathcal{H}_{*} (gray), UP\mathrm{UP} (green), NUP\mathrm{NUP} (gray dashed), BUP points B±B_{\pm} (squares), axis dispersal line De\mathcal{D}_{e} (orange dashed) and barrier branches B±\mathcal{B}_{\pm} , colored by the escape-side trace Wc=τPW_{c}=\tau_{P} , which decreases from τI\tau_{I} at the corner cc to 00 at B±B_{\pm} ; W=+∞W=+\infty on B\mathcal{B} . The box marks the enlargement in (b). (b) The corner cc , with the inward normals λR\boldsymbol{\lambda}_{R} , λL\boldsymbol{\lambda}_{L} of ( 25 ) and the sliding direction of the relay on the axis inside H\mathcal{H} .
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