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

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

Defender-assisted Evasion and Pursuit Maneuvers

Motivated by the possibilities afforded by active target defense, a 3-agent pursuit-evasion differential game involving an Attacker/Pursuer, a Target/Evader, and a Defender is considered. The Defender strives to assist the Target by intercepting the Attacker before the latter reaches the Target. A barrier surface in a reduced state space separates the winning regions of the Attacker and Target-Defender team. In this thesis, attention focuses primarily on the Attacker\u27s region of win where, under optimal Attacker play, the Defender cannot preclude the Attacker from capturing the Target. Both optimal and suboptimal strategies are investigated. This thesis uses several methods to breakdown and analyze the 3-player differential game

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

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

A Pursuit Evasion Game Approach to Obstacle Avoidance

We propose an event triggered obstacle avoidance control scheme that guarantees the safety of a mobile robot in a material transport task at a warehouse. This collision avoidance strategy is based on a pursuer static obstacle (P.S.O.) game which is a modification of the reversed homicidal chauffeur game. In the PSO game, the evader represents the mobile robot and is modelled by a Dubins vehicle while the pursuer represents the human working at the warehouse and is modelled as an omnidirectional agent. Additionally, we consider the presence of a static obstacle in the work space. Based on our analysis, the evader has two options to avoid both the pursuer and the obstacle. The first option is to turn hard in the opposite direction of both the pursuer and the obstacle. The second option is to go in between the obstacle and the pursuer. The work of the reversed homicidal chauffeur game shows that the first evasive strategy guarantees the mobile robot's safety. However, a clear advantage of implementing the second evasive strategy is that the mobile robot can delay its evasive maneuver and hence the mobile robot can follow its predetermined path for a longer period of time. Therefore, the mobile robot actively checks the feasibility of executing the second evasive strategy rather than executing the first evasive strategy right away. Based on a numerical study of our collision avoidance strategy, we argue that both the task duration and the path error due to the evasive action are reduced when compared to simply turning hard in the opposite direction of both the pursuer and the obstacle. Our event triggered obstacle avoidance controller is also compared to strategies based on ISO 13482 and the solution of the velocity obstacle set which are commonly used in the current robotic industry. Based on this comparison, we argue that there is a clear trade-off between the reduction in duration and path error and the guarantee of safety during a material transport task

Contributions To Pursuit-Evasion Game Theory.

This dissertation studies adversarial conflicts among a group of agents moving in the plane, possibly among obstacles, where some agents are pursuers and others are evaders. The goal of the pursuers is to capture the evaders, where capture requires a pursuer to be either co-located with an evader, or in close proximity. The goal of the evaders is to avoid capture. These scenarios, where different groups compete to accomplish conflicting goals, are referred to as pursuit-evasion games, and the agents are called players. Games featuring one pursuer and one evader are analyzed using dominance, where a point in the plane is said to be dominated by a player if that player is able to reach the point before the opposing players, regardless of the opposing players' actions. Two generalizations of the Apollonius circle are provided. One solves games with environments containing obstacles, and the other provides an alternative solution method for the Homicidal Chauffeur game. Optimal pursuit and evasion strategies based on dominance are provided. One benefit of dominance analysis is that it extends to games with many players. Two foundational games are studied; one features multiple pursuers against a single evader, and the other features a single pursuer against multiple evaders. Both are solved using dominance through a reduction to single pursuer, single evader games. Another game featuring competing teams of pursuers is introduced, where an evader cooperates with friendly pursuers to rendezvous before being captured by adversaries. Next, the assumption of complete and perfect information is relaxed, and uncertainties in player speeds, player positions, obstacle locations, and cost functions are studied. The sensitivity of the dominance boundary to perturbations in parameters is provided, and probabilistic dominance is introduced. The effect of information is studied by comparing solutions of games with perfect information to games with uncertainty. Finally, a pursuit law is developed that requires minimal information and highlights a limitation of dominance regions. These contributions extend pursuit-evasion game theory to a number of games that have not previously been solved, and in some cases, the solutions presented are more amenable to implementation than previous methods.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/120650/1/dwoyler_1.pd