Minimum Time Strategies for a Differential Drive Robot Escaping from a Circular Detection Region

TL;DR

This work models the problem of a Differential Drive Robot (DDR) escaping from a circular detection region as a zero-sum differential game between an evader and a pursuer. Using retro-time Hamiltonian methods, it derives time-optimal controls in closed form, revealing two singular surfaces: the Transition Surface (where the evader switches controls) and the Dispersal Surface (where dual optimal choices exist). The analysis shows that the solution structure depends on the speed ratio $\rho_v=\frac{V_d^{\max}}{V_r^{\max}}$ and the radius ratio $\rho_l=\frac{r_d}{b}$, partitioning the reduced space into regions that favor primary translation or rotation-in-place strategies. Numerical simulations validate the theoretical findings, illustrating how optimal strategies shift with $\rho_v$ and $\rho_l$ and confirming that, despite the evader’s speed advantage, the dynamics exhibit rich behavior beyond simple outward translation. These results provide practical insights for designing surveillance and evasion strategies in mobile robotics when a DDR interacts with a moving or stationary detection region.

Abstract

A Differential Drive Robot (DDR) located inside a circular detection region in the plane wants to escape from it in minimum time. Various robotics applications can be modeled like the previous problem, such as a DDR escaping as soon as possible from a forbidden/dangerous region in the plane or running out from the sensor footprint of an unmanned vehicle flying at a constant altitude. In this paper, we find the motion strategies to accomplish its goal under two scenarios. In one, the detection region moves slower than the DDR and seeks to prevent escape; in another, its position is fixed. We formulate the problem as a zero-sum pursuit-evasion game, and using differential games theory, we compute the players' time-optimal motion strategies. Given the DDR's speed advantage, it can always escape by translating away from the center of the detection region at maximum speed. In this work, we show that the previous strategy could be optimal in some cases; however, other motion strategies emerge based on the player's speed ratio and the players' initial configurations.

Figures (6)

• Figure 1: Game representation.
• Figure 2: Partition of the reduced space for different values of $\rho_v$. The black circle represents the detection region, the bold arcs indicate the UP, and the dashed circle is the evader's radius. The blue region corresponds to the configurations where the system follows a trajectory of the primary solution. Similarly, the golden region indicates those configurations where the evader performs a rotation in place. The red curve corresponds to the Transition Surface, those configurations where the evader switches its control. The green line indicates the Dispersal Surfaces at the $x$-axis.
• Figure 3: Partition of the reduced space for different values of $\rho_l$.
• Figure 4: Trajectory of the system in the reduced space. The black circle represents the detection region, the bold arcs indicate the UP, and the dashed circle is the evader's radius. The blue line corresponds to the portion belonging to the primary solution, and the golden curve to the portion where the evader performs a rotation in place.
• Figure 5: Trajectory of the players in the realistic space. The blue squares indicate the pursuer's positions, and the blue arrows its motion directions. Similarly, the red dots correspond to the evader's positions, and the red arrows its motion directions. The circles indicate the detection region at the initial and final players' configurations.