Parameter Study of an Orbital Debris Defender using Two Team, Three Player Differential Game Theory

The United States Air Force and other national agencies rely on numerous space assets to project their doctrine. However, space is becoming an increasingly congested, contested, and competitive environment. A common risk mitigation strategy for the orbit debris problem is either performing evasive maneuvers, or placing additional shielding on the satellite before launch. Current risk mitigation strategies have significant consequences to satellite operators and may not produce sufficient risk mitigation. This research poses that an orbital debris defender, which would defend the primary satellite from orbital debris, may be a more effective risk mitigation strategy. By assuming the worst case scenario, an optimally performing pursuer, this research can show when and how often the defender can intercept debris. The results of this research provide the performance trade space for the orbital debris defender, and additional recommendations to future satellite designers. Additionally, this researched derived a way to generate a pseudo cooperation between defender and evader. This cooperation between evader and defender is a new way to solve differential games, and is not limited to the space domain considered herein

Guarding a Non-Maneuverable Translating Line with an Attached Defender

In this paper we consider a target-guarding differential game where the defender must protect a linearly translating line-segment by intercepting an attacker who tries to reach it. In contrast to common target-guarding problems, we assume that the defender is attached to the target and moves along with it. This assumption affects the defenders' maximum speed in inertial frame, which depends on the target's direction of motion. Zero-sum differential game of degree for both the attacker-win and defender-win scenarios are studied, where the payoff is defined to be the distance between the two agents at the time of game termination. We derive the equilibrium strategies and the Value function by leveraging the solution for the infinite-length target scenario. The zero-level set of this Value function provides the barrier surface that divides the state space into defender-win and attacker-win regions. We present simulation results to demonstrate the theoretical results.Comment: 8 pages, 8 figures. arXiv admin note: text overlap with arXiv:2207.0409

Risk-Minimizing Two-Player Zero-Sum Stochastic Differential Game via Path Integral Control

This paper addresses a continuous-time risk-minimizing two-player zero-sum stochastic differential game (SDG), in which each player aims to minimize its probability of failure. Failure occurs in the event when the state of the game enters into predefined undesirable domains, and one player's failure is the other's success. We derive a sufficient condition for this game to have a saddle-point equilibrium and show that it can be solved via a Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE) with Dirichlet boundary condition. Under certain assumptions on the system dynamics and cost function, we establish the existence and uniqueness of the saddle-point of the game. We provide explicit expressions for the saddle-point policies which can be numerically evaluated using path integral control. This allows us to solve the game online via Monte Carlo sampling of system trajectories. We implement our control synthesis framework on two classes of risk-minimizing zero-sum SDGs: a disturbance attenuation problem and a pursuit-evasion game. Simulation studies are presented to validate the proposed control synthesis framework.Comment: 8 pages, 4 figures, CDC 202

Can local single-pass methods solve any stationary Hamilton-Jacobi-Bellman equation?

The use of local single-pass methods (like, e.g., the Fast Marching method) has become popular in the solution of some Hamilton-Jacobi equations. The prototype of these equations is the eikonal equation, for which the methods can be applied saving CPU time and possibly memory allocation. Then, some natural questions arise: can local single-pass methods solve any Hamilton-Jacobi equation? If not, where the limit should be set? This paper tries to answer these questions. In order to give a complete picture, we present an overview of some fast methods available in literature and we briefly analyze their main features. We also introduce some numerical tools and provide several numerical tests which are intended to exhibit the limitations of the methods. We show that the construction of a local single-pass method for general Hamilton-Jacobi equations is very hard, if not impossible. Nevertheless, some special classes of problems can be actually solved, making local single-pass methods very useful from the practical point of view.Comment: 19 page

Games of Pursuit-Evasion with Multiple Agents and Subject to Uncertainties

Over the past decade, there have been constant efforts to induct unmanned aerial vehicles (UAVs) into military engagements, disaster management, weather monitoring, and package delivery, among various other applications. With UAVs starting to come out of controlled environments into real-world scenarios, uncertainties that can be either exogenous or endogenous play an important role in the planning and decision-making aspects of deploying UAVs. At the same time, while the demand for UAVs is steadily increasing, major governments are working on their regulations. There is an urgency to design surveillance and security systems that can efficiently regulate the traffic and usage of these UAVs, especially in secured airspaces. With this motivation, the thesis primarily focuses on airspace security, providing solutions for safe planning under uncertainties while addressing aspects concerning target acquisition and collision avoidance. In this thesis, we first present our work on solutions developed for airspace security that employ multiple agents to capture multiple targets in an efficient manner. Since multi-pursuer multi-evader problems are known to be intractable, heuristics based on the geometry of the game are employed to obtain task-allocation algorithms that are computationally efficient. This is achieved by first analyzing pursuit-evasion problems involving two pursuers and one evader. Using the insights obtained from this analysis, a dynamic allocation algorithm for the pursuers, which is independent of the evader's strategy, is proposed. The algorithm is further extended to solve multi-pursuer multi-evader problems for any number of pursuers and evaders, assuming both sets of agents to be heterogeneous in terms of speed capabilities. Next, we consider stochastic disturbances, analyzing pursuit-evasion problems under stochastic flow fields using forward reachability analysis, and covariance steering. The problem of steering a Gaussian in adversarial scenarios is first analyzed under the framework of general constrained games. The resulting covariance steering problem is solved numerically using iterative techniques. The proposed approach is applied to the missile endgame guidance problem. Subsequently, using the theory of covariance steering, an approach to solve pursuit-evasion problems under external stochastic flow fields is discussed. Assuming a linear feedback control strategy, a chance-constrained covariance game is constructed around the nominal solution of the players. The proposed approach is tested on realistic linear and nonlinear flow fields. Numerical simulations suggest that the pursuer can effectively steer the game towards capture. Finally, the uncertainties are assumed to be parametric in nature. To this end, we first formalize optimal control under parametric uncertainties while introducing sensitivity functions and costates based techniques to address robustness under parametric variations. Utilizing the sensitivity functions, we address the problem of safe path planning in environments containing dynamic obstacles with an uncertain motion model. The sensitivity function based-approach is then extended to address game-theoretic formulations that resemble a "fog of war" situation.Ph.D

Optimal interdiction of urban criminals with the aid of real-time information

Most violent crimes happen in urban and suburban cities. With emerging tracking techniques, law enforcement officers can have real-time location information of the escaping criminals and dynamically adjust the security resource allocation to interdict them. Unfortunately, existing work on urban network security games largely ignores such information. This paper addresses this omission. First, we show that ignoring the real-time information can cause an arbitrarily large loss of efficiency. To mitigate this loss, we propose a novel NEtwork purSuiT game (NEST) model that captures the interaction between an escaping adversary and a defender with multiple resources and real-time information available. Second, solving NEST is proven to be NP-hard. Third, after transforming the non-convex program of solving NEST to a linear program, we propose our incremental strategy generation algorithm, including: (i) novel pruning techniques in our best response oracle; and (ii) novel techniques for mapping strategies between subgames and adding multiple best response strategies at one iteration to solve extremely large problems. Finally, extensive experiments show the effectiveness of our approach, which scales up to realistic problem sizes with hundreds of nodes on networks including the real network of Manhattan