This paper considers a reach-avoid game on a rectangular domain with two defenders and one attacker. The attacker aims to reach a specified edge of the game domain boundary, while the defenders strive to prevent that by capturing the attacker. First, we are concerned with the barrier, which is the boundary of the reach-avoid set, splitting the state space into two disjoint parts: 1) defender dominance region (DDR) and 2) attacker dominance region (ADR). For the initial states lying in the DDR, there exists a strategy for the defenders to intercept the attacker regardless of the attacker’s best effort, while for the initial states lying in the ADR, the attacker can always find a successful attack strategy. We propose an attack region method to construct the barrier analytically by employing Voronoi diagram and Apollonius circle for two kinds of speed ratios. Then, by taking practical payoff functions into considerations, we present optimal strategies for the players when their initial states lie in their winning regions, and show that the ADR is divided into several parts corresponding to different strategies for the players. Numerical approaches, which suffer from inherent inaccuracy, have already been utilized for multiplayer reach-avoid games, but computational complexity complicates solving such games and consequently hinders efficient on-line applications. However, this method can obtain the exact formulation of the barrier and is applicable for real-time updates.

Reach-Avoid Games With Two Defenders and One Attacker: An Analytical Approach

Multi-player pursuit-evasion games with one superior evader

Stochastic optimal control via forward and backward stochastic differential equations and importance sampling

In this paper, a reachability-based approach is adopted to deal with the pursuit–evasion differential game between one evader and multiple pursuers in the presence of dynamic environmental disturba...

Multiple-Pursuer/One-Evader Pursuit–Evasion Game in Dynamic Flowfields

Pursuit-evasion is fascinating in both nature and artificial world. Typically, a pursuer runs faster than its targeted evader while with less agile maneuverability. Naturally, there is a wonder that how an evader escapes from a faster pursuer or how faster a pursuer is able to capture an agile evader? This is not yet answered from the dynamics (i.e., Lagrangian or Newtonian) perspective. In this paper, we first provide a concise dynamics formulation from a bio-inspired perspective, in which the evader's escape strategy consists of two simplest possible yet efficient ingredients integrated as an organic whole, i.e., the suddenly turning-left or turning-right propelling maneuver, together with the early alert condition for starting and maintaining this maneuver. Then, we characterize the dynamic properties of the system at two different levels: 1) the maneuvers and non-trivial escape of the evader, at the level of individual runs of the system; and further 2) the non-trivial escape zones, the sharp phase-transitions and the phase-transition lines of the gaming outcome, at the level of the running results with respect to different ranges of the system parameters. The results are consistent with natural observations and may disclose some clues of natural laws, as well as imply applications in competition of autonomous mobile robots.

A Dynamics Perspective of Pursuit-Evasion: Capturing and Escaping When the Pursuer Runs Faster Than the Agile Evader

In this paper we propose a new methodology for decision-making under uncertainty using recent advancements in the areas of nonlinear stochastic optimal control theory, applied mathematics, and machine learning. Grounded on the fundamental relation between certain nonlinear partial differential equations and forward-backward stochastic differential equations, we develop a control framework that is scalable and applicable to general classes of stochastic systems and decision-making problem formulations in robotics and autonomy. The proposed deep neural network architectures for stochastic control consist of recurrent and fully connected layers. The performance and scalability of the aforementioned algorithm are investigated in three non-linear systems in simulation with and without control constraints. We conclude with a discussion on future directions and their implications to robotics.

Learning Deep Stochastic Optimal Control Policies Using Forward-Backward SDEs

We consider the following differential game of pursuit and evasion involving two participating players: an evader, which has limited maneuverability, and an agile pursuer. The agents move on the Euclidean plane with different but constant speeds. Whereas the pursuer can change the orientation of its velocity vector arbitrarily fast, that is, he is a “pedestrian” a la Isaacs, the evader cannot make turns having a radius smaller than a specified minimum turning radius. This problem can be seen as a reversed Homicidal Chauffeur game, hence the name “Suicidal Pedestrian Differential Game.” The aim of this paper is to derive the optimal strategies of the two players and characterize the initial conditions that lead to capture if the pursuer acts optimally, and areas that guarantee evasion regardless of the pursuer’s strategy. Both proximity-capture and point-capture are considered. After applying the optimal strategy for the evader, it is shown that the case of point-capture reduces to a special version of Zermelo’s Navigation Problem (ZNP) for the pursuer. Therefore, the well-known ZNP solution can be used to validate the results obtained through the differential game framework, as well as to characterize the time-optimal trajectories. The results are directly applicable to collision avoidance in maritime and Air Traffic Control applications.

On the Suicidal Pedestrian Differential Game

Stochastic L1-optimal control via forward and backward sampling

We present a fast and feature-complete differentiable physics engine, Nimble (nimblephysics.org), that supports Lagrangian dynamics and hard contact constraints for articulated rigid body simulation. Our differentiable physics engine offers a complete set of features that are typically only available in non-differentiable physics simulators commonly used by robotics applications. We solve contact constraints precisely using linear complementarity problems (LCPs). We present efficient and novel analytical gradients through the LCP formulation of inelastic contact that exploit the sparsity of the LCP solution. We support complex contact geometry, and gradients approximating continuous-time elastic collision. We also introduce a novel method to compute complementarity-aware gradients that help downstream optimization tasks avoid stalling in saddle points. We show that an implementation of this combination in an existing physics engine (DART) is capable of a 87x single-core speedup over finite-differencing in computing analytical Jacobians for a single timestep, while preserving all the expressiveness of original DART.

http://www.roboticsproceedings.org/rss17/p034.pdf

Ioannis Exarchos

Papers

Stochastic optimal control via forward and backward stochastic differential equations and importance sampling

Learning Deep Stochastic Optimal Control Policies Using Forward-Backward SDEs

On the Suicidal Pedestrian Differential Game

Stochastic L1-optimal control via forward and backward sampling

Fast and Feature-Complete Differentiable Physics Engine for Articulated Rigid Bodies with Contact Constraints