Pursuit-Evasion on a Sphere and When It Can Be Considered Flat

Abstract

In classical works on a planar differential pursuit-evasion game with a faster pursuer, the intercept point resulting from the equilibrium strategies lies on the Apollonius circle. This property was exploited for the construction of the equilibrium strategies for two faster pursuers against one evader. Extensions for planar multiple-pursuer single-evader scenarios have been considered. We study a pursuit-evasion game on a sphere and the relation of the equilibrium intercept point to the Apollonius domain on the sphere. The domain is a generalization of the planar Apollonius circle set. We find a condition resulting in the intercept point belonging to the Apollonius domain, which is the characteristic of the planar game solution. Finally, we use this characteristic to discuss pursuit and evasion strategies in the context of two pursuers and a single slower evader on the sphere and illustrate it using numerical simulations.

Figures (5)

• Figure 1: Pursuit-evasion (P-E) of a pursuer (P) and a slower evader (E) on a sphere.
• Figure 2: Relative position between $P$ and $E$: The unit vector $\vec{n}_{GC}$ defines the orientation or the great circle passing through $P$ and $E$. The relative $P$-$E$ position is given by the angular distance $\alpha$ measured along the great circle in the direction defined by $\vec{n}_{GC}$ and the right hand rule. The $P$ control variable is the angle $u_P$, $E$ control variables are the angle $u_E$ and speed $v_E=|\vec{v}_E|$. The angles $u_E$ and $u_P$ are measured with respect to the great circle, i.e., with respect to the tangent vector $\vec{t}_E$ and $\vec{t}_P$.
• Figure 3: The Apollonius domain $\mathcal{A}$ is depicted by a set of points enclosed in the blue dashed curve connecting points $I_0$, $I_\lambda$ and $I_{\pi}$. The blue dashed curve is the boundary of Apollonius domain, $\partial \mathcal{A}$, and it belongs to the domain as well. The Apollonius domain includes $E$ and its boundary is a set of intercept points $I_{\lambda}$ satisfying $|\overset{\hbox{\large$\frown$}}{}\mathllap{PI}_{\lambda}|/\delta(\lambda)=\mu$, where $\delta(\lambda)$ and $|\overset{\hbox{\large$\frown$}}{}\mathllap{PI}_{\lambda}|$ are distances from $E$ and $P$ to $I_{\lambda}$, respectively. There is one-to-one correspondence between the angle $\lambda$ and the distance $\delta(\lambda)$.
• Figure 4: Apollonius domain for different $v_E$ (with $v_P = 1$) and $α$: The colors (also indicated by the subscripts) correspond to $\alpha_r < \alpha_C$, $\alpha_g = \alpha_C$, and $\alpha_b > \alpha_C$. The dashed lines indicate the Apollonius domains and the small $+$ indicates the equilibrium intercept point, $I_{P-E}$.
• Figure 5: Proposed solution for the two-pursuer, one-evader $\min$-$\max$ capture time game with $μ_1 = μ_2 = 0.5$, $α_1 = 0.9 π (1 - μ) \approx 1.41$, $α_2 = 0.8 α_1 \approx 1.13$, and with the pursuers' longitudinal offset as $λ_o = 0.4 π \approx 1.26$. The dashed lines denote the Apollonius domains with colors matching their respective pursuers.

Theorems & Definitions (14)

• Theorem 1
• proof
• Remark 1
• Remark 2
• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof