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.

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