Reach-Avoid Differential game with Reachability Analysis for UAVs: A decomposition approach

TL;DR

This work tackles the challenge of reach-avoid games in three spatial dimensions for UAVs by introducing a dimensionality-reduction framework that decomposes the 9D joint dynamics into a horizontal 6D sub-game and a vertical 3D sub-game. Each sub-game is solved with Hamilton-Jacobi reachability analysis, and the authors define invariant capture sets to ensure that combining sub-game solutions preserves capture guarantees. A reach-track control scheme is developed to first reach the invariant capture regions and then track the attacker, with theoretical guarantees and a set of propositions and a theorem supporting defender victory under specified conditions. The approach is validated through extensive numerical simulations and Gazebo-based quadrotor experiments, demonstrating 3D interception capabilities and paving the way for real-world drone-on-drone scenarios. The results offer a tractable path to robust 3D RA planning for high-speed UAVs, highlighting the balance between model fidelity, computational tractability, and practical applicability.

Abstract

Reach-avoid (RA) games have significant applications in security and defense, particularly for unmanned aerial vehicles (UAVs). These problems are inherently challenging due to the need to consider obstacles, consider the adversarial nature of opponents, ensure optimality, and account for nonlinear dynamics. Hamilton-Jacobi (HJ) reachability analysis has emerged as a powerful tool for tackling these challenges; however, while it has been applied to games involving two spatial dimensions, directly extending this approach to three spatial dimensions is impossible due to high dimensionality. On the other hand, alternative approaches for solving RA games lack the generality to consider games with three spatial dimensions involving agents with non-trivial system dynamics. In this work, we propose a novel framework for dimensionality reduction by decomposing the problem into a horizontal RA sub-game and a vertical RA sub-game. We then solve each sub-game using HJ reachability analysis and consider second-order dynamics that account for the defender's acceleration. To reconstruct the solution to the original RA game from the sub-games, we introduce a HJ-based tracking control algorithm in each sub-game that not only guarantees capture of the attacker but also tracking of the attacker thereafter. We prove the conditions under which the capture guarantees are maintained. The effectiveness of our approach is demonstrated via numerical simulations, showing that the decomposition maintains optimality and guarantees in the original problem. Our methods are also validated in a Gazebo physics simulator, achieving successful capture of quadrotors in three spatial dimensions space for the first time to the best of our knowledge.

Figures (13)

• Figure 1: In three spatial dimensions, the defender needs to capture the attacker before it reaches $\mathcal{T}$ while avoiding $\Omega_{obs}$.
• Figure 2: Our approach breaks down the problem into horizontal and vertical game
• Figure 3: Our Reach-Track control procedure in vertical axis
• Figure 4: Plot of $z_{\text{rel}}$ against $v^{z}_{D}$, showing contour of relative distance function (light purple) and $V_{z, \infty}$ (bright yellow-orange)
• Figure 5: Winning region $\mathcal{W}_{D, z}$ for the defender (blue + green) at different durations $T$

Theorems & Definitions (2)

• Remark 1
• Remark 2