Shaunak D. Bopardikar

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.

Brief paper: A cooperative Homicidal Chauffeur game

We address a pursuit-evasion problem involving an unbounded planar environment, a single evader, and multiple pursuers moving along curves of bounded curvature. The problem amounts to a multi-agent version of the classic homicidal chauffeur problem; we focus on parameter ranges in which a single pursuer is not sufficient to capture the evader. We propose a novel cooperative strategy in which the pursuers move in a daisy-chain formation and confine the evader to a bounded region. The proposed policy is inspired by certain hunting and foraging behaviors of various fish species. We characterize the required number of pursuers and the required value of the evader/pursuers speed ratio for which our strategy is guaranteed to lead to confinement.

Cooperative pursuit with sensing limitations

We address a discrete-time pursuit-evasion problem involving multiple pursuers and a single evader in an unbounded, planar environment in which each player has limited-range sensing. The evader appears at a random location in a bounded region and moves only when sensed. We propose a sweep-pursuit-capture strategy for a group of at least three pursuers and determine a lower bound on the probability of capture for the evader. This bound is a function of the pursuer formation and independent of the initial evader location and the evader strategy. We then propose a novel cooperative pursuit algorithm and show that the problem is reduced to one with unlimited sensing. We provide an upper bound on the time for our pursuit strategy to succeed. The final capture is achieved by using the established algorithm spheres. Our results show that on the basis of maximizing the probability of evader capture per pursuer, the pursuers should search the bounded region as a single group (conjoin) rather than to divide the region into smaller parts and search simultaneously in smaller groups (allocate).

A cooperative homicidal chauffeur game

We address a pursuit-evasion problem involving an unbounded planar environment, a single evader, and multiple pursuers moving along curves of bounded curvature. The problem amounts to a multi-agent version of the classic homicidal chauffeur problem; we focus on parameter ranges in which a single pursuer is not sufficient to capture the evader. We propose a novel cooperative strategy in which the pursuers move in a daisy-chain formation and confine the evader to a bounded region. The proposed policy is inspired by certain hunting and foraging behaviors of various fish species. We characterize the required number of pursuers and the required value of the evader/pursuers speed ratio for which our strategy is guaranteed to lead to confinement.

Sensing limitations in the Lion and Man problem

We address the discrete-time lion and man problem in a bounded, convex, planar environment in which both players have identical sensing ranges, restricted to closed discs about their current locations. The evader is randomly located inside the environment and moves only when detected. The players can step inside identical closed discs, centered at their respective positions. We propose a sweep-pursuit-capture strategy for the pursuer to capture the evader. The sweep phase is a search operation by the pursuer to detect an evader within its sensing radius. In the pursuit phase, the pursuer employs a greedy strategy of moving towards the last-sensed evader position. We show that in finite time, the problem reduces to a previously-studied problem with unlimited sensing, which allows us to use the established lion strategy in the capture phase. We give a novel upper bound on the time required for the pursuit phase to terminate using the greedy strategy and a sufficient condition for this strategy to work in terms of the value of the ratio of sensing to stepping radius of the players.

Active exploration using trajectory optimization for robotic grasping in the presence of occlusions

We consider the task of actively exploring unstructured environments to facilitate robotic grasping of occluded objects. Typically, the geometry and locations of these objects are not known a priori. We mount an RGB-D sensor on the robot gripper to maintain a 3D voxel map of the environment during exploration. The objective is to plan the motion of the sensor in order to search for feasible grasp handles that lie within occluded regions of the map. In contrast to prior work that generates exploration trajectories by sampling, we directly optimize the exploration trajectory to find grasp handles. Since it is challenging to optimize over the discrete voxel map, we encode the uncertainty of the positions of the occluded grasp handles as a mixture of Gaussians, one per occluded region. Our trajectory optimization approach encourages exploration by penalizing a measure of the uncertainty. We then plan a collision-free trajectory for the robot arm to the detected grasp handle. We evaluated our approach by actively exploring and attempting 300 grasps. Our experiments suggest that compared to the baseline method of sampling 10 trajectories, which successfully grasped 58% of the objects, our active exploration formulation with trajectory optimization successfully grasped 93% of the objects, was 1.3× faster, and had 3.2× fewer failed grasp attempts.

Randomized sampling for large zero-sum games

This paper addresses the solution of large zero-sum matrix games using randomized methods. We formalize a procedure, termed as the sampled security policy (SSP) algorithm, by which a player can compute policies that, with a high confidence, are security policies against an adversary using randomized methods to explore the possible outcomes of the game. The SSP algorithm essentially consists of solving a stochastically sampled subgame that is much smaller than the original game. We also propose a randomized algorithm, termed as the sampled security value (SSV) algorithm, which computes a high-confidence security-level (i.e., worst-case outcome) for a given policy, which may or may not have been obtained using the SSP algorithm. For both the SSP and the SSV algorithms we provide results to determine how many samples are needed to guarantee a desired level of confidence. We start by providing results when the two players sample policies with the same distribution and subsequently extend these results to the case of mismatched distributions. We demonstrate the usefulness of these results in a hide-and-seek game that exhibits exponential complexity.

Multiobjective Path Planning: Localization Constraints and Collision Probability

We present a novel path planning algorithm that, starting from a probabilistic roadmap, efficiently constructs a product graph used to search for a near optimal solution of a multiobjective optimization problem. The goal is to find paths that minimize a primary cost, such as the path length from start to goal, subject to a bound on a secondary cost such as the state estimation error covariance. The proposed algorithm is efficient as it relies on a scalar metric, related to the largest eigenvalue of the error covariance, and adaptively quantizes the secondary cost, yielding a product graph whose number of vertices and edges provides a good tradeoff between optimality and computational complexity. We further show how our approach can be extended to handle constraints on the probability of collision avoidance specified at every vertex along the path. Numerical examples show 1) how the computed paths change as a function of the specified bound on the secondary costs, and 2) the tradeoff between accuracy and computational efficiency of the proposed approach compared with methods where the product graph is built by quantizing the secondary cost uniformly.

On Dynamic Vehicle Routing With Time Constraints

We consider the problem of dynamic vehicle routing under exact-time constraints on servicing demands. Demands are sequentially generated in an environment, and every demand needs to be serviced exactly after a fixed finite interval of time after it is generated. We design routing policies for a service vehicle to maximize the fraction of demands serviced at steady state. The main contributions are as follows. First, we demonstrate that this problem is described by an appropriate directed acyclic graph structure which leads to a computationally efficient routing algorithm based on a longest-path computation. Second, under the assumption of the demands being generated uniformly randomly in the environment and via a Poisson process in time, we provide two analytic lower bounds on the service fraction of the longest path policy. The first bound is relative to an optimal noncausal version of the policy, i.e., a policy based on knowledge of all future demand requests. The second bound is an explicit function of the vehicle dynamics and demand generation rate and, therefore, useful as a design tool. Finally, we present numerical results to support the analytic bounds.

On vehicle placement to intercept moving targets

We address optimal placement of vehicles with simple motion, to intercept a mobile target that arrives stochastically on a line segment. The optimality of vehicle placement is measured through a cost function associated with intercepting the target. With a single vehicle, we assume that the target either moves with fixed speed and in a fixed direction or moves to maximize the vertical height or intercept time. We show that each of the corresponding cost functions is convex, has smooth gradient and has a unique minimizing location, and so the optimal vehicle placement is obtained by any standard gradient-based optimization technique. With multiple vehicles, we assume that the target moves with fixed speed and in fixed direction. We present a discrete time partitioning and gradient-based algorithm, and characterize conditions under which the algorithm asymptotically leads the vehicles to a set of critical configurations of the cost function.