Parameterized pursuit-evasion games

AbstractWe study the parameterized complexity of four variants of pursuit-evasion on graphs: Seeded Pursuit Evasion, Short Seeded Pursuit Evasion, Directed Pursuit Evasion and Short Directed Pursuit Evasion. Both Seeded Pursuit Evasion and Short Seeded Pursuit Evasion are played on undirected graphs with given starting positions for both the cops and the robber. Directed Pursuit Evasion and its short variant are played on directed graphs, with the players free to choose their starting positions. We show for Seeded Pursuit Evasion and Directed Pursuit Evasion that finding a winning strategy for the cops is AW[*]-hard when we parameterize by the number of cops. Further, we show that the short (k-move) variants of these problems (Short Seeded Pursuit Evasion and Short Directed Pursuit Evasion) are AW[*]-complete when we parameterize by both the number of cops and turns

MPC-based humanoid pursuit-evasion in the presence of obstacles

We consider a pursuit-evasion problem between humanoids in the presence of obstacles. In our scenario, the pursuer enters the safety area of the evader headed for collision, while the latter executes a fast evasive motion. Control schemes are designed for both the pursuer and the evader. They are structurally identical, although the objectives are different: the pursuer tries to align its direction of motion with the line- of-sight to the evader, whereas the evader tries to move in a direction orthogonal to the line-of-sight to the pursuer. At the core of the control architecture is a Model Predictive Control scheme for generating a stable gait. This allows for the inclusion of workspace obstacles, which we take into account at two levels: during the determination of the footsteps orientation and as an explicit MPC constraint. We illustrate the results with simulations on NAO humanoids

Two-Dimensional Pursuit-Evasion in a Compact Domain with Piecewise Analytic Boundary

In a pursuit-evasion game, a team of pursuers attempt to capture an evader. The players alternate turns, move with equal speed, and have full information about the state of the game. We consider the most restictive capture condition: a pursuer must become colocated with the evader to win the game. We prove two general results about pursuit-evasion games in topological spaces. First, we show that one pursuer has a winning strategy in any CAT(0) space under this restrictive capture criterion. This complements a result of Alexander, Bishop and Ghrist, who provide a winning strategy for a game with positive capture radius. Second, we consider the game played in a compact domain in Euclidean two-space with piecewise analytic boundary and arbitrary Euler characteristic. We show that three pursuers always have a winning strategy by extending recent work of Bhadauria, Klein, Isler and Suri from polygonal environments to our more general setting.Comment: 21 pages, 6 figure

Deep Reinforcement Learning for Swarm Systems

Recently, deep reinforcement learning (RL) methods have been applied successfully to multi-agent scenarios. Typically, these methods rely on a concatenation of agent states to represent the information content required for decentralized decision making. However, concatenation scales poorly to swarm systems with a large number of homogeneous agents as it does not exploit the fundamental properties inherent to these systems: (i) the agents in the swarm are interchangeable and (ii) the exact number of agents in the swarm is irrelevant. Therefore, we propose a new state representation for deep multi-agent RL based on mean embeddings of distributions. We treat the agents as samples of a distribution and use the empirical mean embedding as input for a decentralized policy. We define different feature spaces of the mean embedding using histograms, radial basis functions and a neural network learned end-to-end. We evaluate the representation on two well known problems from the swarm literature (rendezvous and pursuit evasion), in a globally and locally observable setup. For the local setup we furthermore introduce simple communication protocols. Of all approaches, the mean embedding representation using neural network features enables the richest information exchange between neighboring agents facilitating the development of more complex collective strategies.Comment: 31 pages, 12 figures, version 3 (published in JMLR Volume 20

Pursuit-evasion predator-prey waves in two spatial dimensions

We consider a spatially distributed population dynamics model with excitable predator-prey dynamics, where species propagate in space due to their taxis with respect to each other's gradient in addition to, or instead of, their diffusive spread. Earlier, we have described new phenomena in this model in one spatial dimension, not found in analogous systems without taxis: reflecting and self-splitting waves. Here we identify new phenomena in two spatial dimensions: unusual patterns of meander of spirals, partial reflection of waves, swelling wavetips, attachment of free wave ends to wave backs, and as a result, a novel mechanism of self-supporting complicated spatio-temporal activity, unknown in reaction-diffusion population models.Comment: 15 pages, 15 figures, submitted to Chao

Analyzing helicopter evasive maneuver effectiveness against rocket-propelled grenades

It has long been acknowledged that military helicopters are vulnerable to ground-launched threats, in particular, the RPG-7 rocket-propelled grenade. Current helicopter threat mitigation strategies rely on a combination of operational tactics and selectively placed armor plating, which can help to mitigate but not entirely remove the threat. However, in recent years, a number of active protection systems designed to protect land-based vehicles from rocket and missile fire have been developed. These systems all use a sensor suite to detect, track, and predict the threat trajectory, which is then employed in the computation of an intercept trajectory for a defensive kill mechanism. Although a complete active protection system in its current form is unsuitable for helicopters, in this paper, it is assumed that the active protection system’s track and threat trajectory prediction subsystem could be used offline as a tool to develop tactics and techniques to counter the threat from rocket-propelled grenade attacks. It is further proposed that such a maneuver can be found by solving a pursuit–evasion differential game. Because the first stage in solving this problem is developing the capability to evaluate the game, nonlinear dynamic and spatial models for a helicopter, RPG-7 round, and gunner, and evasion strategies were developed and integrated into a new simulation engine. Analysis of the results from representative vignettes demonstrates that the simulation yields the value of the engagement pursuit–evasion game. It is also shown that, in the majority of cases, survivability can be significantly improved by performing an appropriate evasive maneuver. Consequently, this simulation may be used as an important tool for both designing and evaluating evasive tactics and is the first step in designing a maneuver-based active protection system, leading to improved rotorcraft survivability

Rubber Bands, Pursuit Games and Shy Couplings

In this paper, we consider pursuit-evasion and probabilistic consequences of some geometric notions for bounded and suitably regular domains in Euclidean space that are CAT(kappa) for some kappa > 0. These geometric notions are useful for analyzing the related problems of (a) existence/nonexistence of successful evasion strategies for the Man in Lion and Man problems, and (b) existence/nonexistence of shy couplings for reflected Brownian motions. They involve properties of rubber bands and the extent to which a loop in the domain in question can be deformed to a point without, in between, increasing its loop length. The existence of a stable rubber band will imply the existence of a successful evasion strategy but, if all loops in the domain are well-contractible, then no successful evasion strategy will exist and there can be no co-adapted shy coupling. For example, there can be no shy couplings in bounded and suitably regular star-shaped domains and so, in this setting, any two reflected Brownian motions must almost surely make arbitrarily close encounters as t tends to infinity.Comment: 48 pages, 8 figure

Optimal feedback strategies for pursuit-evasion and interception in a plane

Variable-speed pursuit-evasion and interception for two aircraft moving in a horizontal plane are analyzed in terms of a coordinate frame fixed in the plane at termination. Each participant's optimal motion can be represented by extremal trajectory maps. These maps are used to discuss sub-optimal approximations that are independent of the other participant. A method of constructing sections of the barrier, dispersal, and control-level surfaces and thus determining feedback strategies is described. Some examples are shown for pursuit-evasion and the minimum-time interception of a straight-flying target