The problem of optimal approach in locally Euclidean spaces

A differential game of optimal approach with simple motions when players move in locally Euclidean spaces is studied. The game-end moment is fixed, and the game payment is a distance between the pursuer and the evader at the game-end moment. The value of game is obtained in the explicit form for any initial positions of players. Moreover, the differential game of optimal approach for the denumerable number of pursuers and one evader in the Euclidean space is solved. All pursuers are controlled by one parameter.     

A discrete dynamic game modelling anti-missile defense scenarios

A discretized game model of an anti-missile defense scenario is presented. Such an engagement is a pursuit of a blind evader, committed to satisfy a terminal constraint. It can be formulated as an incomplete information zero-sum pursuit-evasion game with state constraint. Such a game becomes non trivial if the evader has higher maneuverability than the pursuer. In this case the game admits a saddle-point solution in mixed strategies. Two simple examples illustrate the major elements of the solution.     

A linear differential game with a fixed termination time and restrictions on resources

A linear game with a fixed termination time and integral restrictions on player controls is considered. The pursuer constructs his control with knowledge of the control of the evader, and the evader uses information on all previous actions of his opponent at each instant of time. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 178–183, July–August, 2000.     

多追捕者-单-逃跑者追逃问题实现成功捕获的约束条件

针对包含有n个追捕者及1个逃跑者的2维平面多机器人追逃问题,对实现成功捕获的约束条件进行了研究.经过理论分析得出:在机器人拥有全局视野的情况下,即使单一逃跑者性能优于每个追捕者,只要满足追捕者与逃跑者的速率比大于sin(π/n),逃跑机器人落在追捕机器人所构成的凸多边形内部且逃跑者和追捕者构成的相邻追-逃阿波罗尼奥斯圆满足两两相交(相切)这2个约束条件,则追捕者通过选择合适的追捕策略就一定可以实现成功抓捕.此外,还给出了在此约束条件下的追捕者和逃跑者的追逃策略.多组仿真实验同样证明了本文提出的约束条件是正确的.     

A cooperative pursuit-evasion game in wireless sensor and actor networks

This paper studies the problem of the pursuit-evasion game under the wireless sensor and actor networks (WSANs). In order to plan paths for pursuers to capture an evader in the pursuit-evasion game, a novel multi-step cooperative strategy is presented. Under this strategy, the pursuit-evasion game is studied in two stages. In the first stage we assume that the evader is always static in the workplace, and in the second stage the evader will move once it senses the existence of pursuers. A Daisy-Chain Formation algorithm and a sliding mode-based method are presented to control the pursuit. Based on Lyapunov stability theory, the proposed algorithm is proved to be convergent. At last, simulation results are provided to demonstrate the effectiveness of the proposed method.     

Two weak pursuers in a game against a single evader

We consider an antagonistic differential game where the first player controls the actions of two pursuers that aim to minimize, at a given time instant their miss with respect to an evader. The second (maximizing) player is identified with the evader. We study the case when dynamic capabilities of pursuers are less than the capabilities of the evader. We propose a quasioptimal control method for the first player with switching lines. We also show modeling results.     

Tracking an omnidirectional evader with a differential drive robot

In this paper we consider the surveillance problem of tracking a moving evader by a nonholonomic mobile pursuer. We deal specifically with the situation in which the only constraint on the evader’s velocity is a bound on speed (i.e., the evader is able to move omnidirectionally), and the pursuer is a nonholonomic, differential drive system having bounded speed.     

Filtration of a random process in a statistically uncertain linear stochastic differential system

Minimax filtration of a process in a stochastic linear differential system with uncertain perturbation intensities for dynamics and observation models is studied. The filter is optimized by an integral quality criterion. Minimax filtering equations are derived from the solution of the dual optimization problem. A numerical filter designing method is described and its convergence is proved. Results of numerical experiments are given.Translated from Avtomatika i Telemekhanika, No. 1, 2005, pp. 59–71.Original Russian Text Copyright © 2005 by Miller, Pankov.This work was supported by the Russian Foundation for Basic Research, project no. 02-01-00361.     

Periodic motion planning for virtually constrained Euler–Lagrange systems

The paper suggests an explicit form of a general integral of motion for some classes of dynamical systems including n-degrees of freedom Euler–Lagrange systems subject to (n-1) virtual holonomic constraints. The knowledge of this integral allows to extend the classical results due to Lyapunov for detecting a presence of periodic solutions for a family of second order systems, and allows to solve the periodic motion planning task for underactuated Euler–Lagrange systems, when there is only one not directly actuated generalized coordinate. As an illustrative example, we have shown how to create a periodic oscillation of the pendulum for a cart–pendulum system and how then to make them orbitally exponentially stable following the machinery developed in [A. Shiriaev, J. Perram, C. Canudas-de-Wit, Constructive tool for an orbital stabilization of underactuated nonlinear systems: virtual constraint approach, IEEE Trans. Automat. Control 50 (8) (2005) 1164–1176]. The extension here also considers time-varying virtual constraints.     

Anticipated velocity based guidance strategy for wheeled mobile evader amidst stationary and moving obstacles in bounded environment

This paper is concerned with a class of pursuit‐evasion game problems amidst stationary and moving obstacles in a bounded environment. We concentrate on evader's strategy taking into account the following challenges: (i) pursuer and evader are nonholonomic wheeled mobile robots and the evader is slower than the pursuer; (ii) pursuer follows a proportional navigation law; and (iii) geometry of the environment is not known to the players, a priori. We propose an efficient evader‐centric anticipated velocity based guidance strategy. Pursuer's trajectory is anticipated at each step by the evader using quadratic polynomial interpolation. The aim of the evader is to escape interception with the pursuer for maximum possible time. To deal with static obstacles, a technique based on a well‐known tangent bug algorithm is presented. While dealing with dynamic obstacles, a recently introduced reciprocal orientation method is employed to avoid collision in situations when the dynamic obstacle also cooperates in the process. In case dynamic obstacles do not participate in the process of collision avoidance, a well‐known velocity obstacle method is employed for planning safe collision‐free paths. Efficiency of the proposed algorithms is analyzed with respect to the interception time and the distance traveled by the players. Copyright © 2014 John Wiley & Sons, Ltd.     

Sufficient condition of completion of a pursuit in a nonlinear differential game

The sufficient condition of pursuit completion is proved for a nonlinear differential game on a smooth manifold with integral constraints on the players’ controls. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 134–140, November–December, 1999.     

Cooperative Multiple Pursuers against a Single Evader

This paper considers a pursuit-evasion game for non-holonomic systems where a group of pursuers attempts to capture an evader in a bounded connected domain. The problem is challenging because all vehicles have the same maneuvering capability in terms of speed and turn radius constraint. The paper initially discusses a simple approach for holonomic systems that is based on the minimization of the safe-reachable area (the area containing the set of points to where an evader can travel without being caught). This idea is then extended to develop a pursuit-evasion strategy for non-holonomic systems. However, solving such a problem is computationally intractable. Therefore, we propose a computationally efficient algorithm to obtain approximate solutions. This paper also proposes an alternative approach to obtain a simple yet effective solution to the cooperative pursuit problem that is based on missile guidance laws. As there is no analytical proof of capture, we empirically evaluate the performance of the algorithms and perform a comparative study using solutions obtained from umpteen simulations. A total of four different cooperative pursuit strategies and three different evader strategies are taken into account for the comparative study. In the process, an evader strategy which is superior to that based on the optimization of safe-reachable area is also identified.     

On Discrete-Time Pursuit-Evasion Games With Sensing Limitations

In this paper, we address discrete-time pursuit-evasion games in the plane where every player has identical sensing and motion ranges restricted to closed disks of given sensing and stepping radii. A single evader is initially located inside a bounded subset of the environment and does not move until detected. We propose a sweep-pursuit-capture pursuer strategy to capture the evader and apply it to two variants of the game. The first involves a single pursuer and an evader in a bounded convex environment, and the second involves multiple pursuers and an evader in a boundaryless environment. In the first game, we give a sufficient condition on the ratio of sensing to stepping radius of the players that guarantees capture. In the second, we determine the minimum probability of capture, which is a function of a novel pursuer formation and independent of the initial evader location. The sweep and pursuit phases reduce both games to previously studied problems with unlimited range sensing, and capture is achieved using available strategies. We obtain novel upper bounds on the capture time and present simulation studies that address the performance of the strategies under sensing errors, different ratios of sensing to stepping radius, greater evader speed, and a different number of pursuers.      

Maintaining strong mutual visibility of an evader moving over the reduced visibility graph

In this paper, we address the problem of determining whether a mobile robot, called the pursuer, is able to maintain strong mutual visibility (a visibility notion between regions over a convex partition of the environment) of an antagonist agent, called the evader. We frame the problem as a non cooperative game. We consider the case in which the pursuer and the evader move at bounded speed, traveling in a known polygonal environment with or without holes, and in which there are no restrictions as to the distance that might separate the agents. Unlike our previous efforts (Murrieta-Cid et al. in Int J Robot Res 26:233–253, 2007), we give special attention to the combinatorial problem that arises when searching for a solution through visiting several locations in an environment with obstacles. In this paper we take a step further, namely, we assume an antagonistic evader who moves continuously and unpredictably, but with a constraint over its set of admissible motion policies, as the evader moves in the shortest-path roadmap, also called the reduced visibility graph (RVG). The pursuer does not know which among the possible paths over the RVG the evader will choose, but the pursuer is free to move within all the environment. We provide a constructive method to solve the decision problem of determining whether or not the pursuer is able to maintain strong mutual visibility of the evader. This method is based on an algorithm that computes the safe areas (areas that keep evader surveillance) at all times. We prove decidability of this problem, and provide a complexity measure to this evader surveillance game; both contributions hold for any general polygonal environment that might or not contain holes. All our algorithms have been implemented and we show simulation results.     

On a class of pursuit-evasion games

A class of pursuit-evasion games is studied. The problem is to find an optimal control pair which is a saddle point of the terminal miss distance functional, minimizes cost functionals of the pursuer and the evader, and steers the pursuer and the evader to their respective targets. Necessary conditions for an optimal control pair are obtained.     

Evasion strategies of a three-player lifeline game

This study examines a multi-player pursuit-evasion game, more specifically, a three-player lifeline game in a planar environment, where a single evader is tasked with reaching a lifeline prior to capture. A decomposition method based on an explicit policy is proposed to address the game qualitatively from two main aspects: (1) the evader’s position distribution to guarantee winning the game (i.e., the escape zone), which is based on the premise of knowing the pursuers’ positions initially, and (2) evasion strategies in the escape zone. First, this study decomposes the three-player lifeline game into two two-player sub-games and obtains an analytic expression of the escape zone by constructing a barrier, which is an integration of the solutions of two sub-games. This study then explicitly partitions the escape zone into several regions and derives an evasion strategy for each region. In particular, this study provides a resultant force method for the evader to balance the active goal of reaching the lifeline and the passive goal of avoiding capture. Finally, some examples from a lifeline game involving more than one pursuer are used to verify the effectiveness and scalability of the evasion strategies.     

Some aspects of two-on-one pursuitevasion

$\text{Pursuit}\text{evasion}$ problems featuring two pursuers and one evader are examined in the context of simple vehicle models. A type of coalition tactic which exploits the difference between open-loop and closed-loop conjugate-points is introduced and illustrated. The approach would permit identification of some two-on-one team tactics opportunities from families of one-on-one differential game solutions.     

On the optimal pure strategy sets for a mixed missile guidance lawsynthesis

The terminal phase of a missile-versus-aircraft engagement in a noise-corrupted environment is formulated as an imperfect information zero-sum differential game. The payoff of the game is the single shot kill probability of the missile to be maximized by its designer and minimized by the pilot of the evading aircraft. In the past, it was shown that in such a game the optimal strategy of the evader is mixed. The proposed formulation allows a mixed strategy for the pursuer. The mathematical basis for such an analysis is provided, and a constructive methodology for the synthesis of missile guidance laws based on the concept of mixed strategies is outlined     

Multiple-objective risk-sensitive control and its small noise limit

This paper is concerned with a (minimizing) multiple-objective risk-sensitive control problem. Asymptotic analysis leads to the introduction of a new class of two-player, zero-sum, deterministic differential games. The distinguishing feature of this class of games is that the cost functional is multiple-objective in nature, being composed of the risk-neutral integral costs associated with the original risk-sensitive problem. More precisely, the opposing player in such a game seeks to maximize the most ‘vulnerable’ member of a given set of cost functionals while the original controller seeks to minimize the worst ‘damage’ that the opponent can do over this set. It is then shown that the problem of finding an efficient risk-sensitive controller is equivalent, asymptotically, to solving this differential game. Surprisingly, this differential game is proved to be independent of the weights on the different objectives in the original multiple-objective risk-sensitive problem. As a by-product, our results generalize the existing results for the single-objective risk-sensitive control problem to a substantially larger class of nonlinear systems, including those with control-dependent diffusion terms.     

OUTPUT FEEDBACK GUARANTEED COST CONTROL OF UNCERTAIN SYSTEMS ON AN INFINITE TIME INTERVAL

This paper considers the problem of optimal guaranteed cost control of an uncertain system via output feedback. The uncertain system under consideration contains an uncertainty block subject to an integral quadratic constraint. The cost function considered is a quadratic cost function defined over an infinite time interval. The main result of the paper gives a necessary and sufficient condition for the existence of a guaranteed cost controller guaranteeing a specified level of performance. This condition is given in terms of the existence of suitable solutions to an algebraic Riccati equation and a Riccati differential equation. The resulting guaranteed cost controller is in general time-varying. © 1997 by John Wiley & Sons, Ltd.