Mobile Robots as Remote Sensors for Spatial Point Process Models

Spatial point process models are a commonly-used statistical tool for studying the distribution of objects of interest in a domain. We study the problem of deploying mobile robots as remote sensors to estimate the parameters of such a model, in particular the intensity parameter lambda which measures the mean density of points in a Poisson point process. This problem requires covering an appropriately large section of the domain while avoiding the objects, which we treat as obstacles. We develop a control law that covers an expanding section of the domain and an online criterion for determining when to stop sampling, i.e., when the covered area is large enough to achieve a desired level of estimation accuracy, and illustrate the resulting system with numerical simulations

Optimal obstacle placement with disambiguations

We introduce the optimal obstacle placement with disambiguations problem wherein the goal is to place true obstacles in an environment cluttered with false obstacles so as to maximize the total traversal length of a navigating agent (NAVA). Prior to the traversal, the NAVA is given location information and probabilistic estimates of each disk-shaped hindrance (hereinafter referred to as disk) being a true obstacle. The NAVA can disambiguate a disk's status only when situated on its boundary. There exists an obstacle placing agent (OPA) that locates obstacles prior to the NAVA's traversal. The goal of the OPA is to place true obstacles in between the clutter in such a way that the NAVA's traversal length is maximized in a game-theoretic sense. We assume the OPA knows the clutter spatial distribution type, but not the exact locations of clutter disks. We analyze the traversal length using repeated measures analysis of variance for various obstacle number, obstacle placing scheme and clutter spatial distribution type combinations in order to identify the optimal combination. Our results indicate that as the clutter becomes more regular (clustered), the NAVA's traversal length gets longer (shorter). On the other hand, the traversal length tends to follow a concave-down trend as the number of obstacles increases. We also provide a case study on a real-world maritime minefield data set.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS556 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal ship navigation and algorithms for stochactic obstacle scenes

Tezin basılısı İstanbul Şehir Üniversitesi Kütüphanesi'ndedir.This thesis is comprised of two different but related sections. In the first section, we consider the optimal ship navigation problem wherein the goal is to find the shortest path between two given coordinates in the presence of obstacles subject to safety distance and turn-radius constraints. These obstacles can be debris, rock formations, small islands, ice blocks, other ships, or even an entire coastline. We present a graph-theoretic solution on an appropriately-weighted directed graph representation of the navigation area obtained via 8-adjacency integer lattice discretization and utilization of the A∗ algorithm. We explicitly account for the following three conditions as part of the turn-radius constraints: (1) the ship’s left and right turn radii are different, (2) ship’s speed reduces while turning, and (3) the ship needs to navigate a certain minimum number of lattice edges along a straight line before making any turns. The last constraint ensures that the navigation area can be discretized at any desired resolution. We illustrate our methodology on an ice navigation example involving a 100,000 DWT merchant ship and present a proof- of-concept by simulating the ship’s path in a full-mission ship handling simulator at Istanbul Technical University. In the second section, we consider the stochastic obstacle scene problem wherein an agent needs to traverse a spatial arrangement of possible-obstacles, and the status of the obstacles may be disambiguated en route at a cost. The goal is to find an algorithm that decides what and where to disambiguate en route so that the expected length of the traversal is minimized. We present a polynomial-time method for a graph-theoretical version of the problem when the associated graph is restricted to parallel avenues with fixed policies within the avenues. We show how previously proposed algorithms for the continuous space version can be adapted to a discrete setting. We propose a gener- alized framework encompassing these algorithms that uses penalty functions to guide the navigation in realtime. Within this framework, we introduce a new algorithm that provides near-optimal results within very short execution times. Our algorithms are illustrated via computational experiments involving synthetic data as well as an actual naval minefield data set. Keywords: Graph theory, shortest path, ship navigation, probabilistic path planning, stochastic dynamic programming, Markov decision process, Canadian traveler’s problemContents Declaration of Authorship ii Abstract iv ¨ Oz v Acknowledgments vii List of Figures x List of Tables xi 1 Optimal Ship Navigation with Safety Distance and Realistic Turn Con- straints 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The Optimal Ship Navigation Problem . . . . . . . . . . . . . . . . . . . . 4 1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.1 Safety Distance Constraints . . . . . . . . . . . . . . . . . . . . . . 5 1.4.2 Lattice Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.3 Ship-Turn Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.4 The A∗ Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.5 Smoothing the Optimal Path . . . . . . . . . . . . . . . . . . . . . 13 1.5 Ice Navigation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Simulator Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.7 Summary, Conclusions, and Future Research . . . . . . . . . . . . . . . . 18 2 Algorithms for Stochastic Obstacle Scenes 21 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 The Stochastic Obstacle Scene Problem: Continuous vs. Discrete Settings 23 2.2.1 Deciding Where to Disambiguate: Single Disk Case . . . . . . . . 23 2.2.2 Deciding Where to Disambiguate: Two Disks Case . . . . . . . . . 25 2.2.3 Discretization of the Continuous Setting: An Example . . . . . . . 27 2.3 Definition of the Stochastic Obstacle Scene Problem . . . . . . . . . . . . 27 2.3.1 Continuous SOSP . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.2 Discrete SOSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.3 Discretized SOSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 A Polynomial Algorithm for Discrete SOSP on Parallel Graphs . . . . . . 29 2.5 Discrete Adaptation of the Simulated Risk Disambiguation Algorithm . . 30 2.5.1 Adaptation to Discrete SOSP . . . . . . . . . . . . . . . . . . . . . 30 2.5.2 Adaptation to Discretized SOSP . . . . . . . . . . . . . . . . . . . 32 2.6 Discrete Adaptation of the Reset Disambiguation Algorithm . . . . . . . . 33 2.7 Generalizing SRA and RDA: Penalty-Based Algorithms and DTA . . . . . 34 2.7.1 Illustration of the Algorithms . . . . . . . . . . . . . . . . . . . . . 36 2.8 Computational Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.8.1 Environment A (The COBRA Data) Experiments . . . . . . . . . 40 2.8.2 Environment B Experiments . . . . . . . . . . . . . . . . . . . . . 41 2.8.3 Environment C Experiments . . . . . . . . . . . . . . . . . . . . . 43 2.9 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 A Impact of Cost Change in Parallel Graphs 47 Bibliograph

AO* and penalty based algorithms for the Canadian traveler problem

Tezin basılısı İstanbul Şehir Üniversitesi Kütüphanesi'ndedir.The Canadian Traveler Problem (CTP) is a challenging path planning problem on stochastic graphs where some edges are blocked with certain probabilities and status of edges can be disambiguated only upon reaching an end vertex. The goal is to devise a traversal policy that results in the shortest expected traversal length between a given starting vertex and a termination vertex. The organization of this thesis is as follows: In the first chapter we define CTP and its variant SOSP and present an extensive literature review related to these problems. In the second chapter, we introduce an optimal algorithm for the problem, based on an MDP formulation which is a new improvement on AO* search that takes advantage of the special problem structure in CTP. The new algorithm is called CAO*, which stands for AO* with Caching. CAO* uses a caching mechanism and makes use of admissible upper bounds for dynamic state-space pruning. CAO* is not polynomial-time, but it can dramatically shorten the execution time needed to find an exact solution for moderately sized instances. We present computational experiments on a realistic variant of the problem involving an actual maritime minefield data set. In the third chapter, we introduce a simple, yet fast and effective penalty-based heuristic for CTP that can be used in an online fashion. We present computational experiments involving real-world and synthetic data that suggest our algorithm finds near-optimal policies in very short execution times. Another efficient method for sub-optimally solving CTP, rollout-based algorithms, have also been shown to provide high quality policies for CTP. In the final chapter, we com- pare the two algorithmic frameworks via computational experiments involving Delaunay and grid graphs using one specific penalty-based algorithm and four rollout-based algo- rithms. Our results indicate that the penalty-based algorithm executes several orders of magnitude faster than rollout-based ones while also providing better policies, suggest- ing that penalty-based algorithms stand as a prominent candidate for fast and efficient sub-optimal solution of CTP.Declaration of Authorship ii Abstract iii Öz iv Acknowledgments v List of Figures viii List of Tables ix Abbreviations x 1 Introduction 1 1.1 Overview .................................... 1 1.2 The Canadian Traveler Problem ........................ 1 1.2.1 The Discrete Stochastic Obstacle Scene Problem .......... 2 1.3 Literature Review ................................ 3 1.4 Organization of the Thesis ........................... 4 2 An AO* Based Exact Algorithm for the Canadian Traveler Problem 5 2.1 Introduction ................................... 5 2.2 MDP and POMDP Formulations ....................... 6 2.2.1 MDP Formulation and The Bellman Equation ............ 7 2.2.2 Deterministic POMDP Formulation ................. 9 2.3 The CAO* Algorithm ............................. 11 2.3.1 AO Trees ................................ 11 2.3.2 The AO* Algorithm .......................... 14 2.3.3 The CAO* Algorithm ......................... 16 2.4 Computational Experiments .......................... 19 2.4.1 The BAO* and PAO* Algorithms ................... 19 2.4.2 Experimental Setup .......................... 21 2.4.3 Simulation Environment A ...................... 21 2.4.4 Simulation Environment B ....................... 22 2.4.5 Simulation Environment C....................... 24 2.4.6 Simulation Environment D ...................... 25 2.5 Summary and Conclusions ........................... 26 3 A Fast and Effective Online Algorithm for the Canadian Traveler Prob- lem 29 3.1 Introduction ................................... 29 3.2 The DT Algorithm ............................... 30 3.3 Computational Experiments .......................... 32 3.3.1 Environment 1 ............................. 32 3.3.2 Environment 2 ............................. 34 3.4 Conclusions and Future Research ....................... 34 3.4.1 Conclusions ............................... 34 3.4.2 Limitations and Future Research ................... 35 4 A Comparison of Penalty and Rollout-Based Policies for the Canadian Traveler Problem 36 4.1 Introduction ................................... 36 4.2 Algorithms for CTP .............................. 37 4.2.1 Optimism (OMT) ........................... 37 4.2.2 Hindsight Optimization (HOP) .................... 38 4.2.3 Optimistic Rollout (ORO) ....................... 39 4.2.4 Blind UCT (UCTB) .......................... 39 4.2.5 Optimistic UCT (UCTO) ....................... 40 4.3 Computational Experiments .......................... 41 4.3.1 Delaunay Graph Results ........................ 43 4.3.2 Grid Graph Results .......................... 45 4.4 Conclusions and Future Research ....................... 46 4.4.1 Conclusions ............................... 46 4.4.2 Limitations and Future Research ................... 46 A Problem Instances in Simulation Environments C and D 48 Bibliography 5

Disambiguation protocols based on risk simulation

Abstract — Suppose there is a need to swiftly navigate through a spatial arrangement of possibly forbidden regions, each region marked with the probability that it is indeed forbidden. In close proximity to any of these regions, you have the dynamic capability of disambiguating the region and learning for certain whether or not the region is forbidden—only in the latter case may you proceed through that region. The central issue is how to most effectively exploit this disambiguation capability to minimize the expected length of the traversal. Regions are never entered while they are possibly forbidden and, thus, no risk is ever actually incurred. Nonetheless, for the sole purpose of deciding where to disambiguate, it may be advantageous to simulate risk, temporarily pretending that possibly forbidden regions are riskily traversable, and each potential traversal is weighted with its level of undesirability, which is a function of its traversal length and traversal risk. Introduced in this paper is the simulated risk disambiguation protocol, which has you follow along a shortest traversal—in this undesirability sense—until an ambiguous region is about to be entered; at that location, a disambiguation is performed on this ambiguous region. (The process is then repeated from the current location, until the destination is reached.) We introduce the tangent arc graph as a means of simplifying the implementation of simulated risk disambiguation protocols, and we show how to efficiently implement the simulated risk disambiguation protocols which are based on linear undesirability functions. The effectiveness of these disambiguation protocols is illustrated with examples, including an example involving mine countermeasures path planning. Index Terms — probabilistic path planning, disambiguation protocol, random disambiguation path, Canadian Traveller Problem, visibility graph. I