The Classical Homicidal Chauffeur Game

Abstract

The Homicidal Chauffeur differential game is often mentioned in the literature, but the game’s complete solution is not readily available. In this work, the complete solution process of the Homicidal Chauffeur differential game is illustrated for the parameter range in the heart of the speed ratio-capture radius parameter space initially investigated by Isaacs, and referred to by Breakwell and Merz as the classical Homicidal Chauffeur differential game. Some salient features of the solution of this foundational differential game are highlighted, and some popular misconceptions are dispelled. This tutorial paper fills a gap in the literature on pursuit-evasion differential games.

Appendix: The Evader’s Tributary Option

If the Evader contacts point B, if he chooses, he can enter the region of the tributary trajectories. This is shown by repeating the methodology used in Sect. 4.2. The normal at B, pointing into the region of the tributary trajectories, is

$$\begin{aligned} \mathbf {n}=(\sqrt{1-\mu ^2},-\mu ) \end{aligned}$$

Since the Evader is entering the region of the tributary trajectories from point B, the forward time dynamics are examined. Recalling the forward time dynamics from Eq. (4), the vector

$$\begin{aligned} \mathbf {f}=\begin{pmatrix} -yu+\mu \sin \varPsi \\ xu-1+\mu \cos \varPsi \end{pmatrix} \end{aligned}$$

When the Evader is located at B, the Pursuer applies the control $u^*=-1$. To enter the region of the tributary trajectories from B, the Evader applies his tributary trajectory control given by Eq. (24). At B, when these controls are applied, the dynamics are

$$\begin{aligned} \mathbf {f}=\begin{pmatrix} y_B+\mu \sin \tau _1\\ -x_B-1+\mu \cos \tau _1 \end{pmatrix} \end{aligned}$$

where $(x_B, y_B)$ is the Evader’s position at the end point B of the barrier—see Eqs. (21) and (22). For the Pursuer to push the Evader off of B, into the region of the tributary trajectories, the following must hold true

$$\begin{aligned} \mathbf {n}\cdot \mathbf {f}\;\Big |_B>0 \end{aligned}$$

Evaluating the above inequality yields

$$\begin{aligned} \mathbf {n}\cdot \mathbf {f}=\sqrt{1-\mu ^2}y_B+\mu \sqrt{1-\mu ^2} \sin \tau _1+\mu x+\mu -\mu ^2\cos \tau _1 \end{aligned}$$

$\Rightarrow$

$$\begin{aligned} \mathbf {n}\cdot \mathbf {f}=\mu \sqrt{1-\mu ^2} \sin \tau _1-\mu ^2 \cos \tau _1+\mu \end{aligned}$$

From Sect. 3.4, it is known that there is a critical tributary trajectory that contacts B at the retrograde time $\tau _1$—see Eq. (33).

A numerical evaluation of the equation above reveals that, at the critical retrograde time specified by Eq. (33), $\mathbf {n}\cdot \mathbf {f}>0$, thereby verifying that when the Evader applies his tributary region control at B, he can move from B into the region of the tributary trajectories.