Zachariah E. Fuchs

Cooperative defense within a single-pursuer, two-evader pursuit evasion differential game

This paper is motivated by a desire to develop analytical formulations for cooperative defensive strategies against predator(s).We formulate a single-pursuer, two-evader differential game with a novel cost functional. Each of the three agents are modeled as massless particles that move with constant velocity. The pursuer attempts to capture either of the evaders while minimizing its cost. Simultaneously, the evaders strive to maximize the pursuers cost. The proposed cost functional represents the increased cost to the pursuer when presented with multiple, potentially dangerous targets. It captures the effect of cooperation between the evaders. In order to solve the game, we develop the optimality conditions for the equilibrium strategies. We then integrate the resulting system of ordinary differential equations backwards in time from the terminal conditions to generate the optimal trajectories of the three agent system. The resulting trajectories display cooperative behaviors between the two evaders, which are qualitatively similar to behaviors found in predator-prey interactions in nature. Brief description of singular surfaces is also included.

Cooperative target defense differential game with a constrained-maneuverable Defender

This paper addresses the active target defense differential game where a Target aircraft is pursued by an Attacker missile, and a Defender missile is employed in order to intercept the Attacker and protect the evading Target. This paper extends the results concerning the active target defense differential game by allowing for the Defenders turning rate to be constrained. The restriction imposed on the Defenders turning rate is of great operational relevance since initially the Defenders heading might be different than the otherwise optimal heading when simple motion is assumed.

Singular analysis of a multi-agent, turn-constrained, defensive game

We examine a two-team, differential game, which models an active target defense scenario. One team represents an attacking force and consists of a single Attacker. The opposing team represents a defensive force consisting of a high-value Target and a mobile Defender. The Attacker strives to get as close to the Target as possible before it is intercepted by the Defender. Conversely, the defensive team attempts to maneuver so that the Defender intercepts the Attacker as far from the Target as possible. The Attacker and Target move with simple motion, while the Defender has a constrained turn rate. We discuss two types of singular surfaces that appear within the equilibrium solution of this game, namely the universal surface and the dispersal surface. Several numerical examples are presented to illustrate the characteristics of the equilibrium solutions that interact with these surfaces.

Generalized Engage or Retreat Differential Game with Escort Regions

This paper is motivated by the desire to develop optimal defensive control strategies that discourage an attacker from engaging in attack while simultaneously encouraging retreat. We develop a general, two-player, differential game in which one player represents an attacker and the opposing player represents the defender. The attacker possesses superior dynamics such that it is capable of terminating the game either in engagement or retreat as it so chooses. The defender is incapable of directly preventing engagement. Instead, the defender uses the manipulation of the attacker’s utility function as a form of indirect control in an attempt to make retreat a more attractive option over engagement. The solution to the overall engage or retreat differential game is found by solving two related optimization problems: the differential game of engagement and the optimal constrained retreat. The equilibrium open-loop control strategies and resulting game values of the attack or retreat game are expressed in terms of the solutions to these subproblems. Within the optimal constrained retreat problem, a value function constraint is imposed in order to prevent the attacker from moving into regions where engagement becomes optimal. This leads to regions of constrained retreat which we refer to as escort regions.

An engage or retreat differential game with an escort region

This paper is motivated by the desire to develop optimal defensive strategies that discourage an attacker from engaging a high-value target. We analyze an Engage or Retreat Differential Game in which one player represents an attacker and the other player represents a defender. The attacker is capable of terminating the game in engagement or retreat. The defender attempts to manipulate the attackers choice through the strategic maximization or minimization of the attackers cost function. Both players are free to switch strategies at any point during the game. We show that for certain conditions, the defender should cooperate with the attacker so that retreat becomes the most attractive option. A value function constraint prevents the attackers retreat trajectory from passing into a region in which engagement would become optimal. This leads to certain regions of constrained retreat which may be thought of as escort regions. In these escort regions, the defender cooperates with the attacker so long as the attacker maintains appropriate separation from the regions of engagement. The theory is illustrated with a numerical example.

Cooperative Missile Guidance for Active Defense of Air Vehicles

In air combat, an active countermeasure against an attacking missile homing into a Target aircraft entails the launch of a defending missile. The Target is protected by the Defender, which aims to intercept the Attacker before the latter reaches the Target aircraft. A differential game is presented where the Target–Defender team strives to maximize the terminal separation between the Target and the Attacker at the time instant where the Attacker is intercepted by the Defender, whereas the Attacker strives to minimize the said separation. This paper discusses the case where the Defender is endowed with a positive capture radius. Optimal strategies for the three agents are derived and simulation examples illustrate the effectiveness of the proposed approach.

Evolutionary design of engagement strategies for turn-constrained agents

We examine the use of an evolutionary algorithm to design a feedback controller for a dual pursuit-evasion problem. In this problem, two players, Player A and Player B, move about an obstacle-free, two-dimensional plane with constant speeds and bounded turn rates. Player A strives to capture Player B by maneuvering behind and closing within a defined capture distance. Simultaneously, Player B is attempting to capture Player A while avoiding being captured itself. Although the general form of this problem is two-sided, we examine the design of strategies for Player A against a collection of possible adversarial strategies implemented by Player B. We pose a nearest neighbor switching control structure that is represented using a parameterized matrix. An evolutionary algorithm is utilized to evolve these parameters in order to develop a feedback controller for Player A to efficiently capture Player B while evading capture itself.

Zero-sum turret defense differential game with singular surfaces

We examine a two-player, zero-sum, differential turret defense game, which consists of two players, a mobile Attacker and a stationary Defender. The Attacker is modeled as a mobile agent moving with a constant speed about a infinite, two-dimensional plane, and it is capable of making instantaneous changes in direction. The Defender is modeled as a stationary target located at the origin, and it is capable of steering a direction of focus toward the Attacker. The game terminates when the Attacker intercepts the Defender. Over the course of the game, the Attacker incurs an integral cost. The cost functional consists of a constant time penalty and a Defender generated cost based on the relative agents positions. The goal of the Attacker is to terminate the game with minimal cost, while the Defender strives to maximize the Attackers cost. We show that the global equilibrium solution contains three singular surfaces that divide the state space into regions with qualitatively different equilibrium control strategies for the two players.

Multispectral target recognition using adaptive radar and infrared data integration

We report a RF and IR data-integration strategy based on a probabilistic (or a distribution) model. At the heart of our approach is the ability to extract the probability density functions (pdfs) from the sensed dataset for RF and IR respectively followed by the detection or target identification process based on posterior fusion (i.e., the product of individual pdfs) and Bayesian decision process. The pdf-acquisition processes in RF and IR modules have been further refined with clutter models and data-compression techniques.