We consider a planar pursuit-evasion problem in which a Dubins pursuer seeks to capture a slower first-order point-mass evader that can change its heading instantaneously. First, we propose a turn-rate feedback law with a single tunable gain and derive an explicit admissible gain interval that guarantees finite-time capture against any admissible evader strategy while satisfying the turn-rate constraint. Next, we obtain an explicit capture-time bound and characterize the turning authority required for the gain interval to be non-empty. Finally, we show that the dependence on the evader's speed bound can be removed by deriving a speed ceiling, computable solely from the pursuer's parameters, below which the guarantees hold at the maximum allowable gain. Numerical simulations illustrate the effectiveness of the proposed approach.
Figures & tables
Figure 1: Relative geometry between a Dubins pursuer and a point-mass evader, illustrating the range ( ρ ) separating them, and the angles β and α .
Figure 2: Schematic for Theorem 1 showing the evolution of the heading error under the proposed control law with k>kmin . The heading error α(t) is driven to the set P in finite time and approaches the set M at least exponentially fast thereafter.
Figure 3: Evolution of the pursuer and the evader trajectories under the representative evader policies and the HCG game in which both players use their equilibrium strategies. The pursuer’s and evader’s initial positions are marked by a filled circle and a filled square, respectively, and their terminal positions at capture by stars. In each case, the game terminates with the evader getting captured at tc=1.14 s (LOS-normal evasion), 1.42 s (Random evasion), and 1.61 s (HCG evasion), against the certified bound tˉc=7.89 s. In the HCG game, the evader gets captured comparatively faster at 0.78 s.
Figure 4: Evolution of the heading error α(t) under the representative evader policies and the HCG pursuit game. The boundary ∂P of the set P is shown by solid black lines and the boundary ∂M of the set M by grey dash-dotted lines. The entry times into P are τp=0.634 s (LOS-normal Evasion), 0.519 s (Random evasion), and 0.522 s (HCG evasion), against the certified bound τˉp=3 s. In the HCG game, α decreases in magnitude almost linearly and reaches α=0 at around 0.35 s.
Figure 5: ρ(t) under the representative evader policies and in the HCG pursuit game, from a common initial configuration. The horizontal line marks the capture radius rc=0.125 m. In all cases, the evader is captured well within the bound tˉc=7.89 s certified by Theorem 1 . Against the same HCG evader, the time-optimal HCG pursuer captures faster ( 0.78 s) than the proposed law ( 1.61 s).
Figure 6: Pursuer turn rate ωp(t) under the representative evader policies and in the HCG pursuit game, with the actuator limits ±ωm=±10 rad/s shown as horizontal lines. Under the proposed law, ∣ωp∣≤ωm holds at all times, with a peak of 6.1 rad/s due to the anti-aligned initial heading. The HCG pursuer’s command is discontinuous, i.e., it saturates at −ωm until α=0 and then switches to ωp=0 .
Figure 7: Empirical distribution, mean, and standard deviation of the ratio τ of the numerical capture time tsim to the theoretical upper bound tˉc obtained from ( 14 ), under the HCG evasion policy of the evader with the pursuer initialized at p(0)=(2,0) m with θ(0)=0 and v=0.5 m/s, while the evader’s initial position is drawn uniformly at random from an annulus centered at the pursuer. It is found that τ<1 ( tsim<tˉc ) in each case, and that the bound is more conservative for μm=0.2 m/s compared to μm=0.05 m/s in line with the theoretical findings.