A Surveillance Game between a Differential Drive Robot and an Omnidirectional Agent: The Case of a Faster Evader

TL;DR

This work analyzes a surveillance game in which a Differential Drive Robot (DDR) with bounded sensing must keep an Omnidirectional Agent (OA) inside its detection region, while the OA is faster and seeks to escape. Using the zero-sum differential-game framework and Isaacs’ methodology, the authors derive time-optimal controls and characterize the solution, revealing four singular surfaces—Dispersal ($DS$), Transition ($TS$), Universal ($US$), and Focal ($FS$)—that govern optimal strategies as a function of the speed ratio $\rho_v=V_a^{\max}/V_r^{\max}$ and the geometry ratio $\rho_d=b/r_d$. They provide closed-form expressions for the optimal controls, construct the UP (OA escape region) in the reduced space, and develop a backward-integration approach to obtain primary trajectories and tributaries that fill the partitioned state space. Numerical simulations in both reduced and realistic coordinates illustrate how the OA’s radial escape tendency competes with complex surface-driven strategies, including rotations in place and tributary evolutions along US, TS, and FS. The results advance understanding of pursuit-evasion in robotics, informing surveillance strategies and sensor-robot design under speed differentials and nonholonomic constraints.

Abstract

A fundamental task in mobile robotics is to keep an agent under surveillance using an autonomous robotic platform equipped with a sensing device. Using differential game theory, we study a particular setup of the previous problem. A Differential Drive Robot (DDR) equipped with a bounded range sensor wants to keep surveillance of an Omnidirectional Agent (OA). The goal of the DDR is to maintain the OA inside its detection region for as much time as possible, while the OA, having the opposite goal, wants to leave the regions as soon as possible. We formulate the problem as a zero-sum differential game, and we compute the time-optimal motion strategies of the players to achieve their goals. We focus on the case where the OA is faster than the DDR. Given the OA's speed advantage, a winning strategy for the OA is always moving radially outwards to the DDR's position. However, this work shows that even though the previous strategy could be optimal in some cases, more complex motion strategies emerge based on the players' speed ratio. In particular, we exhibit that four classes of singular surfaces may appear in this game: Dispersal, Transition, Universal, and Focal surfaces. Each one of those surfaces implies a particular motion strategy for the players.

Figures (9)

• Figure 1: The pursuer DDR is represented by the white disc of radius $b$. The evader OA is represented by the red dot. The larger gray circle in the background represents the detection region of radius $r_d$.
• Figure 2: Partition of the playing space in reduced coordinates. The white disk represents the DDR and the large arrow its direction of motion. The outer circle corresponds to the Usable Part (UP), the silver line to the Transition Surface (TS), the vertical red line to the Universal Surface (US), blue vertical and orange horizontal lines are Dispersal Surfaces (DS). The zoom-in shows tributaries that emanate from the Focal Surface (FS).
• Figure 3: Diagram of $\rho_v$ vs $\rho_d$. A: Case of a slower evader $V_a^{\max}<V_r^{\max}$ studied in RUIZ-22. B: No critical point appears. C: Critical point $(0,y_c)$ appears. Symbols correspond to the cases shown in ($\star$) Fig. \ref{['fig:partition']}, ($\bullet$) Fig. \ref{['fig:VaryingVa_a']}, ($\blacksquare$) Fig. \ref{['fig:VaryingVa_b']}, ($\blacklozenge$) Fig. \ref{['fig:VaryingRd_a']}, ($\blacktriangle$) Fig. \ref{['fig:VaryingRd_b']}, and (+) Fig. \ref{['fig:VaryingRd_c']}.
• Figure 4: Partition of the playing space for fixed DDR's detection range $r_d=3m$ and different values of the OA's speed $V_a^{\max}$.
• Figure 5: Partition of the playing space for different values of the detection radius $r_d$.