3차원 무인항공기 방어를 위한 수평 수직 도달 회피 게임 차원축소 프레임워크
📝 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. Nomenclature
💡 Analysis
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. Nomenclature
📄 Content
Reach-avoid games are a class of differential games in which an agent (attacker) aims to reach a target set while avoiding predefined obstacles and an adversarial opponent (defender). In this game, the attacker will try to arrive at the target goal region while the defender tries to intercept it before it can achieve its goal. As technologies such as unmanned aerial vehicles (UAVs) and unmanned underwater vehicles (UUV) become more prevalent, reach-avoid games in three spatial dimensions have become more relevant than ever before.
Since the seminal work on differential games by Rufus Isaacs [1], there has been substantial effort devoted to developing methods for solving these games. Many approaches rely on geometric methods [1][2][3][4], constructing dominance regions such as Voronoi cells and Apollonius circles for players with single integrator dynamics, where optimal trajectories are straight lines. Other studies [5,6] construct these regions more analytically with convex regions for multi-agent target defense games. In [7,8] the method of characteristics is used to solve a system of ordinary differential equations whose solutions satisfy the Hamilton-Jacobi-Isaac (HJI) partial differential equation. While these methods have primarily focused on differential games with two spatial dimensions, they have also been extended to games with three spatial dimensions [9][10][11], also for single integrator dynamics. Three spatial dimensions introduce additional complexity by adding an extra spatial dimension for each attacker and defender, amplifying the challenge of analysis.
These existing methods often yield feasible results because of the homogeneous single-order integrator dynamics assumption for all players, which exhibit straight-line optimal trajectories. However, this assumption on the model is not reflective of real-world autonomous systems, especially ones capable of high speed such as quadrotors, aircraft, etc.
Consequently, the applicability of these approaches to realistic systems with high-dimensional state spaces remains unclear.
To overcome the constraints of simplified dynamics, reachability analysis has emerged as a powerful method, and involves directly solving the Hamilton-Jacobi (HJ) partial differential equation (PDE) to obtain viscosity solutions via level set methods [12]. The biggest advantages of this approach are consideration of complex non-linear dynamical systems beyond single-integrator models, representation of complete winning regions, and guarantees of optimality in the resulting feedback strategies for each player [13]. HJ reachability analysis has also been applied effectively for safety verification of complex real-world robotics systems [14][15][16][17][18], as well as multi-agent reach-avoid games for single integrator dynamics [19][20][21]. The main drawback of this approach is that the HJ PDE is solved numerically on a grid and hence has exponential time and memory complexity, rendering reachability analysis for systems with high-dimensional state spaces intractable.
In this paper, we set out to address this challenge and apply HJ reachability analysis for reach-avoid game in three spatial dimensions. Our goal is to develop a tractable decomposition-based algorithm that approximates the solution to the original problem and under the certain conditions, can guarantee a winning capture strategy for the defender. In particular, we are interested in guaranteeing the performance of the defender in both capturing the attacker and avoiding the obstacles to win the game, while considering applicability of the numerical solution to fast real-world systems.
Such criteria require a higher-fidelity model of the defender that captures second-order dynamics in all three spatial dimensions. Our paper employs HJ reachability to approximate the winning region for both the attacker and the defender.
For the first time, we obtain an approximative solution for the reach-avoid game in three spatial dimensions while accounting for second order dynamics of the defender, allowing us to validate the numerical solution on quadrotors simulated in Gazebo. In addition to ensuring capture, the proposed method also enables tracking of the target thereafter, which paves the way towards real-world drone-on-drone interception. This is achieved by the following contributions:
Dynamics modeling simplification done in a conservative way: quadrotor dynamics involving a 12-dimensional (12D) state space are simplified into 6D double integrator dynamics for the defender and 3D single integrator dynamics for the attacker, giving the attacker advantage by over-approximating its physical capabilities.
Decomposition of horizontal and vertical spatial dimensions: The resulting problem involving a 9D state is broken down into a 6D horizontal sub-game and 3D vertical sub-game that can be tractably solved using HJ reachability.
The reach-track control algorithm, allowing solutions of the subgames to
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