Narges Noori

Lion and man with visibility in monotone polygons

In the original version of the lion and man game, a lion tries to capture a man who is trying to escape in a circular arena. The players have equal speeds. They can observe each other at all times. We study a new variant of the game in which the lion has only line-of-sight visibility. That is it can observe the man’s position only if the line segment connecting them does not intersect the boundary. We show that despite this limitation, the lion can capture the man in any monotone polygon in finite time.

Searching for a one-dimensional random walker: Deterministic strategies with a time budget when crossing is allowed

We present deterministic strategies for capturing a target performing a discrete random walk on a discretized line segment. The searcher has a limited time budget. Its goal is to maximize the probability of capturing the target within the budget. A challenging aspect of our model is that the target can cross the searcher without being captured when they take the same edge at the same time in opposite directions. We present a Partially Observable Markov Decision Process (POMDP) approach for finding the optimal search strategy. We also present an efficient approximate solution to the POMDP. The strategies found by this approach reveal structural properties of the efficient search strategies which we exploit to solve the problem efficiently without running the POMDP.

The lion and man game on polyhedral surfaces with boundary

We study the lion-and-man game in which a group of lions (the pursuers) try to capture a man (the evader). The players have equal speed. They can observe each other at all times. While the game is well-studied in planar domains such as polygons, very little is known about its properties in higher dimensions. In this paper, we study the lion and man game when played on the surface of a genus-zero polyhedron with boundary. We show that three lions with non-zero capture distance δ can capture the man in time O((A/δ2 + L/δ)2 δ/2) where A is the area of the surface, and L is the total edge length of the surface.

The role of target modeling in designing search strategies

This paper studies the problem of searching for an unknown moving target in a bounded two-dimensional convex area with a mobile robot. A key component of designing a search strategy is the target motion model, which is often unknown in practical scenarios. When designing search strategies, researchers either (1) ignore the target motion and treat the target as a stationary object with unknown location, (2) treat the target as an adversary and model the search task as a game, or (3) use a stochastic model such as a random walk. For each of these models we analyze possible search paths with the objective of minimizing the expected capture time. Our intent is to investigate how the choice of the model influences the choice of the strategy and consequently how the capture time will depend on this choice. In addition to a theoretical analysis, we compare the strategies in simulation.

The Lion and Man Game on Convex Terrains

We study the lion-and-man game in which a lion (the pursuer) tries to capture a man (the evader). The players have equal speed and they can observe each other at all times. In this paper, we study the game on surfaces of convex terrains. We show that the lion can capture the man in finite number of steps determined by the terrain geometry.

Finding and tracking targets in the wild: Algorithms and field deployments

We describe our efforts on building a robotic system for detecting and tracking radio-tagged invasive fish using teams of autonomous ground and surface vehicles. In addition to system building and field experiments, our efforts clustered around three fundamental problems: (1) Search: how to find the target as quickly as possible, (2) Active localization: how to actively choose measurement locations to accurately estimate target locations, and (3) Long-term autonomy through energy-efficiency and harvesting. We present specific problem formulations and a summary of our results so far. We conclude the paper with a discussion on our progress and next steps.

Navigation around an unknown obstacle for autonomous surface vehicles using a forward-facing sonar

A robotic boat is moving between two points when it encounters an obstacle of unknown size. The boat must find a short path around the obstacle to resume its original course. How should the boat move when it can only sense the proximity of the obstacle, and does not have prior information about the obstacles size? We study this problem for a robotic boat with a forward-facing sonar. We study two versions of the problem. First, we solve a simplified case when the obstacle is a rectangle of known orientation but unknown dimensions. Second, we study a more general case where an arbitrarily shaped obstacle is contained between two known parallel lines. We study the performance of the algorithms analytically using competitive analysis and present results from field experiments. The experimental setup is relevant for harbor patrol or autonomous navigation in shallow water.

Searching for a one-dimensional random walker: Randomized strategy with energy budget

In this paper we study the problem of designing search strategies to find a target whose motion is described by a random walk along a one-dimensional bounded environment. The sensing model and the characteristic of the environment require the searcher and the target to be on the same site at the same time to guarantee capture. The objective is to optimize the searchers motion, given by a sequence of actions (move right, left or remain stationary), so that the probability of capturing the target is maximized. Each action is associated with an energy cost. The searcher strategy is constrained by a total energy budget. We propose a class of randomized strategies for which we provide an analytical expression for the capture probability as a function of a single parameter. We then use this expression to find the best strategy within this class. In addition to theoretical results, the algorithms are analyzed in simulation and compared with other intuitive solutions.

Constrained Probabilistic Search for a One-Dimensional Random Walker

This paper addresses a fundamental search problem in which a searcher subject to time and energy constraints tries to find a mobile target. The targets motion is modeled as a random walk on a discrete set of points on a line segment. At each time step, the target chooses one of the adjacent nodes at random and moves there. We study two detection models. In the no-crossing model, the searcher detects the target if it is on the same node or if it takes the same edge at the same time. In the crossing model, detection happens only if the target lands on the same node at the same time. For the no-crossing model, where move and stay actions may have different costs, we present an optimal search strategy under energy and time constraints. For the crossing model, we formulate the problem of designing an optimal strategy as a partially observable Markov decision process (POMDP) and solve it using methods that reduce the state-space representation of the belief. The POMDP solution reveals structural properties of the optimal solution. We use this structure to design an efficient strategy and analytically study its performance. Finally, we present preliminary experimental results to demonstrate the applicability of our model to our tracking system, which is used for finding radio-tagged invasive fish.

Pursuit-Evasion: A Toolkit to Make Applications More Accessible [Tutorial]

A pursuit-evasion game takes place between two players. The pursuer is charged with capturing the evader while the evader tries to avoid getting caught. Many robotics applications such as search, tracking, and surveillance can be modeled as pursuit-evasion games. Equally important, these games can be modeled as fun mathematics problems to inspire newcomers to the field of robotics. We have witnessed this firsthand during summer Research Experiences for Undergraduates programs at the Institute for Mathematics and its Applications, located at the University of Minnesota. The subject is accessible, with many open problems that require creativity, insight, and strong algorithmic thinking. Our summer students digested the basics of the field and developed results that evolved into research publications [1], [2].