Target Defense against Sequentially Arriving Intruders: Algorithm for Agents with Dubins Dynamics

TL;DR

This work tackles a sequential target-defense problem where a single defender using non-holonomic Dubins dynamics protects a circular target against intruders arriving one-by-one on the TSR boundary. The authors formulate a two-phase engagement framework (partial-information and full-information), develop a geometric analysis based on Dubins paths, guarding arcs, and a capture circle to characterize capturability, and derive analytical expressions for capture probabilities and long-run capture rates via a two-state Markov chain. They show how the defender can strategically position at the target center or on the capture circle to maximize the expected capture percentage across a sequence, and validate the theory with Monte Carlo simulations that align with the analytic results. The findings provide a rigorous, scalable approach to sequential perimeter defense with realistic motion constraints, with potential implications for robotics-enabled security and surveillance systems.

Abstract

We consider a variant of the target defense problem where a single defender is tasked to capture a sequence of incoming intruders. Both the defender and the intruders have non-holonomic dynamics. The intruders' objective is to breach the target perimeter without being captured by the defender, while the defender's goal is to capture as many intruders as possible. After one intruder breaches or is captured, the next appears randomly on a fixed circle surrounding the target. Therefore, the defender's final position in one game becomes its starting position for the next. We divide an intruder-defender engagement into two phases, partial information and full information, depending on the information available to the players. We address the capturability of an intruder by the defender using the notions of Dubins path and guarding arc. We quantify the percentage of capture for both finite and infinite sequences of incoming intruders. Finally, the theoretical results are verified through numerical examples using Monte-Carlo-type random trials of experiments.

Figures (9)

• Figure 1: gh: Target region with radius $r_{_T}$, gh: intruders sensing region with radius $\rho_{_A}$, gh: Target's sensing anulus with radius $\rho_{_T}$. This figure was generated using AI-assisted image synthesis (OpenAI DALL·E).
• Figure 2: Left: Dubins path when the final orientation $\psi_f$ is not specified. Right: Dubins path when the final orientation $\psi_f$ is specified. The arrows indicate the agent's initial and final heading angles. The dashed circles are the turning circles with radius $R$ representing the minimum radius of curvature that the agent must maintain during its turns.
• Figure 3: The reachable set $\mathcal{R}(\mathbf{x}_{A},\psi_A,T)$ for $T=5$ is presented. Left: For $\nu_A=0.9$, $\mathcal{R}({\mathbf{x}_{A}}_0,{\psi_A}_0,T)$ is shown by varying $\omega_A=1, 1.5$ and $2$. Higher value of $\omega_A$ corresponds to a larger region. Right: For $\omega_A=1$, $\mathcal{R}({\mathbf{x}_{A}}_0,{\psi_A}_0,T)$ is shown by varying $\nu_A=0.3, 0.6$ and $0.9$.
• Figure 4: The blue and red dots represent the positions of the defender and the intruder, respectively. The black line represents $\mathcal{C}(\xi_A,\xi_D)$. The green region is a part of the circular target and the brown line represents the target sensing boundary.
• Figure 5: Critical configuration at time $t_A^*$.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• proof
• Remark 4