Anytime computation algorithms for stochastically parametric approach-evasion differential games

We consider an approach-evasion differential game where the inputs of one of the players are upper bounded by a random variable. The game enjoys the order preserving property where a larger relaxation of the random variable induces a smaller value function. Two numerical computation algorithms are proposed to asymptotically recover the expected value function. The performance of the proposed algorithms is compared via a stochastically parametric homicidal chauffeur game. The algorithms are also applied to the scenario of merging lanes in urban transportation.National Science Foundation (U.S.) (Grant 1239182)United States. Air Force Office of Scientific Research (Grant FA8650-07-2-3744

Receding Horizon Control for Uncertain Pursuit-Evasion Games

A robust technique for handling parameter and strategy uncertainty in a pursuit-evasion framework is developed. The method is a receding horizon controller valid for problem classes with singularly perturbed trajectories that approximates the optimal feedback solution with small loss in optimality. The receding horizon method is used to ensure the controller is robust to incorrect or extraneous information about an opposing player's dynamics or strategy. A simple analytic pursuit-evasion game motivates the method by demonstrating that the receding horizon solution closely approximates the optimal solution and may be solved much faster. Simulations of a nonlinear game show that the receding horizon controller is especially useful when it is unknown whether the opposing player is performing an active or passive maneuver. In several cases, the receding horizon controller is shown to become more effective than a game-optimal controller acting with an incorrect strategy estimate. The major limitation of the technique for a nonlinear system is the expensive solution time; therefore, the optimal control problem is translated to a nonlinear programming problem and the test cases are repeated. Finally, the test cases are run on hardware to validate the method for real-time practical operation. The singular-perturbation algorithm applied herein is valid only for a small subset of all pursuit and evasion games. Nonetheless, the methods developed here can in theory be used for any generic game scenario, given that sufficient computing power is available to find the numerical solutions.

Numerical solution methods for differential game problems

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2009.Includes bibliographical references (p. 95-98).Differential game theory provides a potential means for the parametric analysis of combat engagement scenarios. To determine its viability for this type of analysis, three frameworks for solving differential game problems are evaluated. Each method solves zero-sum, pursuit-evasion games in which two players have opposing goals. A solution to the saddle-point equilibrium problem is sought in which one player minimizes the value of the game while the other player maximizes it. The boundary value method is an indirect method that makes use of the analytical necessary conditions of optimality and is solved using a conventional optimal control framework. This method provides a high accuracy solution but has a limited convergence space that requires a good initial guess for both the state and less intuitive costate. The decomposition method in which optimal trajectories for each player are iteratively calculated is a direct method that bypasses the need for costate information. Because a linearized cost gradient is used to update the evader's strategy the initial conditions can heavily influence the convergence of the problem. The new method of neural networks involves the use of neural networks to govern the control policy for each player. An optimization tool adjusts the weights and biases of the network to form the control policy that results in the best final value of the game. An automatic differentiation engine provides gradient information for the sensitivity of each weight to the final cost.(cont.) The final weights define the control policy's response to a range of initial conditions dependent upon the breadth of the state-space used to train each neural network. The neural nets are initialized with a normal distribution of weights so that no information regarding the state, costate, or switching structure of the controller is required. In its current form this method often converges to a sub-optimal solution. Also, creative techniques are required when dealing with boundary conditions and free end-time problems.by Philip A. Johnson.S.M

Construction of Barrier in a Fishing Game With Point Capture

This paper addresses a particular pursuit-evasion game, called as “fishing game” where a faster evader attempts to pass the gap between two pursuers. We are concerned with the conditions under which the evader or pursuers can win the game. This is a game of kind in which an essential aspect, barrier, separates the state space into disjoint parts associated with each player's winning region. We present a method of explicit policy to construct the barrier. This method divides the fishing game into two subgames related to the included angle and the relative distances between the evader and the pursuers, respectively, and then analyzes the possibility of capture or escape for each subgame to ascertain the analytical forms of the barrier. Furthermore, we fuse the games of kind and degree by solving the optimal control strategies in the minimum time for each player when the initial state lies in their winning regions. Along with the optimal strategies, the trajectories of the players are delineated and the upper bounds of their winning times are also derived

Pursuit and evasion games: semi-direct and cooperative control methods

Pursuit and evasion games have garnered much research attention since the class of problems was first posed over a half century ago. With wide applicability to both civilian and military problems, the study of pursuit and evasion games showed much early promise. Early work generally focused on analytical solutions to games involving a single pursuer and a single evader. These solutions generally assumed simple system dynamics to facilitate convergence to a solution. More recently, numerical techniques have been utilized to solve more difficult problems. While many sophisticated numerical tools exist for standard optimization and optimal control problems, developing a more complete set of numerical tools for pursuit and evasion games is still a developing topic of research. This thesis extends the current body of numeric solution tools in two ways. First, an existing approach that modifies sophisticated optimization tools to solve two player pursuer and evasion games is extended to incorporate a class of state inequality constraints. Several classical problems are solved to illustrate the e±cacy of the new approach. Second, a new cooperation metric is introduced into the system objective function for multi-player pursuit and evasion games. This new cooperation metric encourages multiple pursuers to surround and then proceed to capture an evader. Several examples are provided to demonstrate this new cooperation metric

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

Influence of maneuverability on helicopter combat effectiveness

A computational procedure employing a stochastic learning method in conjunction with dynamic simulation of helicopter flight and weapon system operation was used to derive helicopter maneuvering strategies. The derived strategies maximize either survival or kill probability and are in the form of a feedback control based upon threat visual or warning system cues. Maneuverability parameters implicit in the strategy development include maximum longitudinal acceleration and deceleration, maximum sustained and transient load factor turn rate at forward speed, and maximum pedal turn rate and lateral acceleration at hover. Results are presented in terms of probability of skill for all combat initial conditions for two threat categories