Minimum-Time Escape from the Conic Field of View of a Reeds--Shepp Pursuer: the Faster Evader

Ubaldo Ruiz

cs.ROOct 28, 2026

Minimum-Time Escape from the Conic Field of View of a Reeds--Shepp Pursuer: the Faster Evader

Authors: Ubaldo Ruiz

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

Abstract

We continue the study of minimum-time escape of an omnidirectional evader from the forward conic field of view of a Reeds--Shepp pursuer, now for a faster evader, and solve the game for every speed ratio $ν>1$ and every half-angle $0<α<π/2$ : the escape time is characterized by explicit characteristics and the selection of roots of a scalar equation, and it is the value of the game in nonanticipative strategies at every state. Above the end $E$ of the evader's dispersal line on the front axis, the evader chases the reversing pursuer along a universal line fed by tributaries. The characteristic from the boundary of the usable part is an involute of a circle of radius $ν$ ending at a cusp, and two explicit thresholds decide whether it reaches the axis before the cusp and whether the jump of the escape time across it ends before the cusp or runs to it; in the latter case primaries fold back next to the barrier. The pursuer's optimal strategy is the relay of the slower case. The evader follows the tributaries and switches once to a primary, or commits in advance to a heading certified by the scalar equation of the primaries. Level-set and semi-Lagrangian computations are compared with the value.

Figures & tables

Figure 1: Regimes of the conic escape game. Solid curves:

\nu=\sin\alpha

and

\nu=1

(Part I). Dashed:

\alpha_{F}(\nu)

, at or below which the

\mathrm{BUP}

characteristic reaches the front axis at or before its cusp (regime F1, Section 4 ); above it lies regime F2 (Section 5 ). Dotted:

\alpha_{Z}(\nu)

, which separates F2a, where the jump ends before the cusp, from F2b. The value is proved in F1, F2a and F2b (Sections 4 , 5.2 and 5.3 ), and on the curve

\alpha_{Z}

. The inset enlarges

1<\nu\leq 1.6

. Crosses: examples used here and in Part I.

Figure 2: Regime F1,

(\nu,\alpha)=(1.1,10^{\circ})

. (a) The synthesis of Theorem 4.2 . (b) Junction near the axis:

\mathcal{D}_{PT}^{\pm}

bound the shaded tributary channel and meet the barrier at

Z_{\pm}

; the dotted continuations of the

\mathrm{BUP}

characteristics reach

I

without a jump. (c) Mean signed error against the proved value

W_{T}

on the axis column

0.575\leq y\leq 0.60

, for fixed-domain WENO computations with a grid node on the axis or a half-cell shift, and for semi-Lagrangian computations with either player first (Section 6 ).

Regime	Condition	Jump across the $\mathrm{BUP}$ characteristic	Evader’s strategy	Result
F1	$\nu\sin\theta_{c}\leq 1$ ( $\alpha\leq\alpha_{F}$ )	ends at $Z_{\pm}$ , before the axis contact $I$	tributary feedback, one switch to the primary feedback	Thms. 4.1 , 4.2
F2a	$\nu\sin\theta_{c}>1$ , $\widetilde{W}_{T}(P_{c})<\tau_{c}$ ( $\alpha_{F}<\alpha<\alpha_{Z}$ )	ends at $Z_{\pm}$ , before the cusp	as in F1	Thm. 5.1
F2b	$\widetilde{W}_{T}(P_{c})>\tau_{c}$ ( $\alpha>\alpha_{Z}$ )	runs to the cusps $P_{c\pm}$ ; folded primaries on the far side near them	committed primary heading, or tributary feedback then a committed core heading	Thm. 5.2
$\alpha=\alpha_{Z}$	$\widetilde{W}_{T}(P_{c})=\tau_{c}$	runs to the cusps; tributary on the far side near them	as in F2b	Thm. 5.2

Table 1: Regimes of the faster-evader game,

\nu>1

0<\alpha<\pi/2

. In every regime the pursuer’s optimal strategy is the relay

(v_{p},\omega_{p})=(-1,-\operatorname{sgn}x)

, and each player can guarantee

W

to arbitrary precision by sampled state feedback.

Figure 3: Synthesis in regime F2b (Theorem 5.2 ). (a)

(\nu,\alpha)=(1.5,30^{\circ})

; (b) enlargement of the lens next to the barrier; (c)

(1.5,60^{\circ})

, with

\mathcal{D}_{PT}

ending on the

\mathrm{NUP}

. Forward-time arrows: primaries blue, tributaries teal. Universal line

\mathcal{U}

: purple; dispersal lines: orange dashed; barrier: magenta, from

B_{\pm}

(squares) to the cusps (dots);

\mathrm{UP}

: green;

\mathrm{NUP}

: gray dashed. Branches are retained where their values are smallest.

Figure 4: An optimal play for

\nu=1.5

\alpha=30^{\circ}

, in forward time. (a) Reduced frame: tributary phase to the universal line at

U_{0}

(teal), straight approach along

\mathcal{U}

E

(purple), and primary phase from

E

to the escape point on the

\mathrm{UP}

(blue). (b) Global frame: the reversing pursuer (solid) and the evader (dashed), with the line of sight and the field of view at the start, at

U_{0}

, at

E

and at escape.

Figure 5: Regime F2b: level sets of the value of Theorem 5.2 (solid) and of the independent WENO5 solution (dashed) in the right half of the cone, at levels spaced by

0.1

. (a)

\nu=1.5

\alpha=30^{\circ}

, grid spacing

h=0.0025

; (b)

\nu=1.5

\alpha=60^{\circ}

h=0.005

\mathrm{UP}

thick,

B_{+}

marked; barrier from

B_{+}

to the cusp

P_{c}

(dot), across which both solutions jump.

$(\nu,\alpha)$	$h$	prim. median	prim. max	trib. mean	trib. max
$(1.5,30^{\circ})$	0.0025	$2.1\times 10^{-8}$	$3.8\times 10^{-5}$	$-0.0026$	$0.0026$
$(1.5,60^{\circ})$	0.005	$3.2\times 10^{-8}$	$4.6\times 10^{-5}$	$-0.0020$	$0.0020$
$(2,15^{\circ})$	0.005	$5.9\times 10^{-8}$	$6.0\times 10^{-5}$	$-0.0038$	$0.0038$

Table 2: Level-set comparisons in regime F2b away from rays, the axis, barriers and kinks. Primary columns give the median relative error

|W_{\rm LS}-W|/\max(W,0.05)

, with

W_{\rm LS}

the level-set value, and the maximum absolute error; tributary columns give the signed mean and the maximum absolute error.

$h$	axis, P / E	trib. mean, P / E	prim. max
0.002	$-0.122$ / $+0.036$	$-0.116$ / $+0.041$	$0.047$
0.001	$-0.050$ / $+0.026$	$-0.046$ / $+0.030$	$0.011$
0.0005	$-0.020$ / $+0.017$	$-0.016$ / $+0.021$	$0.004$
extrap.	$+0.009$ / $+0.008$	$+0.013$ / $+0.012$

Table 3: Full-game semi-Lagrangian computation for

\nu=1.1

\alpha=10^{\circ}

; P, E: pursuer or evader first. Axis: mean of

V-W_{T}

on the axis column

0.575\leq y\leq 0.60

. Tributaries: mean of

V-W_{T}

at 214 fixed tributary points. Primaries: largest

|V-W|

at 536 fixed primary points. Last row: first-order extrapolation of the mean errors.

regime	initial states	$\max(t_{f}-W)$ , evader	$\min(t_{f}-W)$ , relay
F1	270	$1.9\times 10^{-3}$	$+5.8\times 10^{-6}$
F2a	421	$2.0\times 10^{-3}$	$+5.6\times 10^{-6}$

Table 4: Closed-loop tests of Theorems 4.1 (F1, five pairs including

\alpha=\alpha_{F}(1.1)-10^{-8}

) and 5.1 (F2a, nine pairs, four of them close to

\alpha_{Z}

): largest overshoot of the evader feedback against the tested pursuers and smallest margin of the relay against the tested evaders, over all pairs. Per-pair results are in the Supplementary Material, Table S12.

	DC	RS, $\nu>1$
Usable part	$\rho<r(\nu+\sin\alpha)$	$\rho<r(\nu-\sin\alpha)$
Holding region	none (no barrier for $\Theta\leq\pi/2$ )	none (Part I, Prop. 3(c))
Point dividing the axis	graze point $G$ (a primary is tangent to the axis)	$E$ (subsolution test fails, $\lambda_{y}=1/(\nu-1)$ )
Axis below it	evader’s dispersal line	evader’s dispersal line $\mathcal{D}_{e}$
Axis above it	pursuer’s dispersal line	universal line $\mathcal{U}$ , pursuer backs away ( $\omega_{p}=0$ )
Surface leaving the axis there	equivocal surface, to the ray or to a switch curve	evader’s dispersal line $\mathcal{D}_{PT}$ , to the barrier or to the $\mathrm{NUP}$

Table 5: Conic escape game for convex cones,

0<\alpha<\pi/2

: Dubins pursuer (DC; Lewin and Olsder, 1979 , Regions I–II, every

w=1/\nu

) and Reeds–Shepp pursuer with a faster evader (RS,

\nu>1

, this paper). RS entries are proved for every

\alpha

(Theorems 4.2 , 5.1 and 5.2 ).

Game Theory

Game theory studies how interacting decision-makers choose strategies, form agreements, and shape outcomes under competition and cooperation.

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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

Ubaldo Ruiz

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

Game Theory

Open paper AskPDF arXiv

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\le\sinα

, escape is confined to a barrier-bounded dart for

\sinα<ν\le1

, and escape is possible from every state for

ν>1

. For the slower or equally fast evader,

\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

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.

Abstract

Game Theory

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