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

Geometric Pursuit Evasion

In this dissertation we investigate pursuit evasion problems set in geometric environments. These games model a variety of adversarial situations in which a team of agents, called pursuers, attempts to catch a rogue agent, called the evader. In particular, we consider the following problem: how many pursuers, each with the same maximum speed as the evader, are needed to guarantee a successful capture? Our primary focus is to provide combinatorial bounds on the number of pursuers that are necessary and sufficient to guarantee capture. The first problem we consider consists of an unpredictable evader that is free to move around a polygonal environment of arbitrary complexity. We assume that the pursuers have complete knowledge of the evader's location at all times, possibly obtained through a network of cameras placed in the environment. We show that regardless of the number of vertices and obstacles in the polygonal environment, three pursuers are always sufficient and sometimes necessary to capture the evader. We then consider several extensions of this problem to more complex environments. In particular, suppose the players move on the surface of a 3-dimensional polyhedral body; how many pursuers are required to capture the evader? We show that 4 pursuers always suffice (upper bound), and that 3 are sometimes necessary (lower bound), for any polyhedral surface with genus zero. Generalizing this bound to surfaces of genus g, we prove the sufficiency of (4g + 4) pursuers. Finally, we show that 4 pursuers also suffice under the "weighted region" constraints, where the movement costs through different regions of the (genus zero) surface have (different) multiplicative weights. Next we consider a more general problem with a less restrictive sensing model. The pursuers' sensors are visibility based, only providing the location of the evader if it is in direct line of sight. We begin my making only the minimalist assumption that pursuers and the evader have the same maximum speed. When the environment is a simply-connected (hole-free) polygon of n vertices, we show that Θ(n^1/2 ) pursuers are both necessary and sufficient in the worst-case. When the environment is a polygon with holes, we prove a lower bound of Ω(n^2/3 ) and an upper bound of O(n^5/6 ) pursuers, where n includes the vertices of the hole boundaries. However, we show that with realistic constraints on the polygonal environment these bounds can be drastically improved. Namely, if the players' movement speed is small compared to the features of the environment, we give an algorithm with a worst case upper bound of O(log n) pursuers for simply-connected n-gons and O(√h + log n) for polygons with h holes. The final problem we consider takes a small step toward addressing the fact that location sensing is noisy and imprecise in practice. Suppose a tracking agent wants to follow a moving target in the two-dimensional plane. We investigate what is the tracker's best strategy to follow the target and at what rate does the distance between the tracker and target grow under worst-case localization noise. We adopt a simple but realistic model of relative error in sensing noise: the localization error is proportional to the true distance between the tracker and the target. Under this model we are able to give tight upper and lower bounds for the worst-case tracking performance, both with or without obstacles in the Euclidean plane

Asymmetric Robot Motion Design for Pursuit-Evasion Games

Symmetric turning control is the typical design choice for most machines. However, historical examples of asymmetric machine design, as well as examples of asymmetry in nature, suggest that asymmetric turning may be a potential advantage in adversarial applications. For instance, aircraft of World Wars I and II were plagued by asymmetric turning controls as a result of gyroscopic forces from the rotating engine. Pilots of the time actually believed this to be a feature, not a bug, suggesting that the asymmetric turning improved strategic evasion and pursuit during battle. As autonomous robots become increasingly critical in military operations, it is imperative that we endow them with strategic designs for better performance. We seek to understand if asymmetric turning is an advantageous design. Using Karaman and Frazzoli's sample-based algorithm for pursuit-evasion games, software simulates robot motion planning in an asymmetric Dubins state space to observe how asymmetric turning influences agent success. We demonstrate mathematically that the Dubins interval path solutions are applicable to asymmetric Dubins vehicles, as both are utilized within the simulation. The Open Motion Planning Library (OMPL) is leveraged to implement the pursuit-evasion game algorithm. To simulate asymmetric action, agents are assigned varying degrees of asymmetric turning constraints, such that as one turn sharpens, the other broadens. Agents then compete in a pursuit-evasion game. Pursuit-evasion games are simulated across a range of asymmetric turning match-ups and agent starting positions. Results show that pursuer success increases as its asymmetry increases. Evader success remains constant, regardless of asymmetric turning influence. Furthermore, the advantages of asymmetric turning can be further augmented when considered in conjunction with relative agent starting position. The results of this research inform more intelligent machine design strategies for vehicles in dynamic spaces

Motion Strategies for Visibility based Target Tracking in Unknown Environments

Ph.DDOCTOR OF PHILOSOPH

Fast and robust generation of city scale urban ground plan

Since the introduction of the concept of Digital Earth, almost every major international city has been re-constructed in the virtual world. A large volume of geometric models describing urban objects has become freely available in public domain via software like Google Earth. Although mostly created for visualization, these urban models can benefit many applications beyond visualization including video games, city scale evacuation plan, traffic simulation and earth phenomenon simulations. However, these urban models are mostly loosely structured and implicitly defined and require tedious manual preparation that usually take weeks if not months before they can be used. In this paper, we present a framework that produces well-defined ground plans from these urban models, an important step in the preparation process. Designing algorithms that can robustly and efficiently handle unstructured urban models at city scale is the main technical challenge. In this work, we show both theoretically and empirically that our method is resolution complete, efficient and numerically stable. Based on our review of the related work, we believe this is the first work that attempts to create urban ground plans automatically from 3D architectural meshes at city level. With the goal of providing greater benefit beyond visualization from this large volume of urban models, our initial results are encouraging.published_or_final_versio

Humanoid gait generation via MPC: stability, robustness and extensions

Research on humanoid robots has made significant progress in recent years, and Model Predictive Control (MPC) has seen great applicability as a technique for gait generation. The main advantages of MPC are the possibility of enforcing constraints on state and inputs, and the constant replanning which grants a degree of robustness. This thesis describes a framework based on MPC for humanoid gait generation, and analyzes some theoretical aspects which have often been neglected. In particular, the stability of the controller is proved. Due to the presence of constraints, this requires proving recursive feasibility, i.e., that the algorithm is able to recursively guarantee that a solution satisfying the constraints is found. The scheme is referred to as Intrinsically Stable MPC (IS-MPC). A basic scheme is presented, and its stability and feasibility guarantees are discussed. Then, several extensions are introduced. The guarantees of the basic scheme are carried over to a robust version of IS-MPC. Furthermore, extension to uneven ground and to a more accurate multi-mass model are discussed. Experiments on two robotic platforms (the humanoid robots HRP-4 and NAO) are presented in the concluding section