Robot Localization and Kalman Filters on finding Your Position in a Noisy World

Robot Localization and Kalman Filters on finding Your Position in a Noisy World

UTRECHT UNIVERSITY Robot Localization and Kalman Filters On finding your position in a noisy world by Rudy Negenborn A thesis submitted to the Institute of Information and Computing Sciences in partial fulfillment of the requirements for the degree of Master of Science, specialized in Intelligent Systems Thesis number: INF/SCR-03-09 September 1, 2003 iii \Every day I watched closely for the sun or stars to appear, to correct my chronometer, on the accuracy of which our lives and the success of the journey would depend." (F.A. Worsley, 1916 [53]) v Abstract The robot localization problem is a key problem in making truly autonomous robots. If a robot does not know where it is, it can be difficult to determine what to do next. In order to localize itself, a robot has access to relative and absolute measurements giving the robot feedback about its driving actions and the situation of the environment around the robot. Given this informa- tion, the robot has to determine its location as accurately as possible. What makes this difficult is the existence of uncertainty in both the driving and the sensing of the robot. The uncertain information needs to be combined in an optimal way. The Kalman Filter is a technique from estimation theory that combines the information of different uncertain sources to obtain the values of vari- ables of interest together with the uncertainty in these. The filter has been successfully applied in many applications, like missions to Mars, and auto- mated missile guidance systems. Although the concept of the filter is rel- atively easy to comprehend, the advantages and shortcomings can only be understood well with knowledge of the pure basics and with experience. In this work we provide a thorough discussion of the robot localization problem and Kalman Filter techniques. First, we look at current methods to obtain location information, pointing out advantages and disadvantages. We formalize how to combine this information in a probabilistic framework and discuss several currently used methods that implement it. Second, we look at the basic concepts involved in Kalman Filters and derive the equations of the basic filter and commonly used extensions. We create understanding of the workings, while discussing the differences between the extensions. Third, we discuss and experimentally show how Kalman Filters can be applied to the localization problem. We look at system and measurement models that are needed by the filter; that is, we model a driving system, a GPS-like sensor, and a landmark-based sensor. We perform simulations using these models in our own general Kalman Filter simulator showing different behaviors when applying the Kalman Filter to the localization problem. In order to use the landmark-based sensor when it can not uniquely identify landmarks, we extend the Kalman Filter to allow for multiple beliefs. The material presented in this work forms a basis for further studies in localization literature, application of Kalman Filters in any domain, and in particular practical application of Kalman Filters and localization on physical robots. Preface This Thesis This thesis is the result of the project that I performed in partial fulfillment for the degree of Master of Science, specialized in Intelligent Systems, to be acquired at the University of Utrecht, the Netherlands. I conducted the research involved in this project at the Datalogisk Institut, Københavns Universitet, Denmark, 2002-2003. In this thesis I look at two large research fields. First, the field of robotic research, which is a relatively new field, dynamic and full of interesting chal- lenges. The research in this field focuses at making robots more and more useful in practical settings. The potential applicability of robotic systems grows constantly, and with every new application, ideas for other applica- tions arise. I investigate a key problem in making robots capable of doing their own thing, the problem of Robot Localization. The second field which I investigate in this thesis is the field of estimation theory. I explore how we can deal with the Robot Localization problem using a certain estimation technique, the technique of Kalman Filters. Audience The audience intended to read this thesis are graduate computer science students with interests in robotics and state estimation. A major part of the theory involved in this thesis builds on probabilities and matrix calculations and therefore a basic understanding of statistics and linear algebra is an advantage. In particular knowledge of the properties of Gaussian distributed variables makes understanding this thesis easier. Introductionary texts on these topics can be found in [32, 39, 22]. viii Online At the website of this project, an HTML, pdf, and postscipt version of this thesis can be found. The first two of these include direct links to referenced articles and an interactive table of contents. Besides this, the website also contains the source codes of the used simulator, experimental settings, and the final presentation of the project. The website of the project can be found at http://www:negenborn:net/kal loc/. Feel free to contact me through e-mail with any comments regarding this work at rudy@negenborn:net. Acknowledgments The help and support of a number of people has contributed to the achieve- ments of this project. First of all, I want to thank Prof. Dr. Phil. Peter Johansen for his hospitality and for offering me the opportunity to perform my research at the Department of Computer Science at the Copenhagen University. I want to thank him and Dr. Marco Wiering for supervising and supporting my work. For proof-reading and commenting on preliminary drafs of this work I want to thank Michael Folkmann, Frederik Rønn, and Stine Søndergaard. For interesting discussions I also want to thank Jacob Blom Andersen and Anders Christian Kølle. Finally I want to thank Dr. Michael Egmont-Petersen for the help in establishing the connection with Denmark. Rudy Negenborn Copenhagen, August 2003 Contents 1 Introduction 1 1.1 Autonomous Mobile Robots ................... 2 1.2 Robot Navigation ......................... 4 1.3 Errors and Uncertainty ...................... 5 1.4 Kalman Filters .......................... 7 1.5 Problem and Outline ....................... 7 2 Robot Localization 9 2.1 Localization Problem ....................... 9 2.2 Available Information ...................... 10 2.3 Relative Position Measurements ................. 12 2.4 Absolute Position Measurements ................ 13 2.5 Multi-Sensor Fusion ....................... 16 2.6 Summary ............................. 17 3 A Probabilistic Framework 19 3.1 Probabilistic Localization .................... 19 3.2 Localization Formula ....................... 22 3.3 Complexity Issues ........................ 27 3.4 Implementations ......................... 28 3.5 Summary ............................. 31 4 Kalman Filters 33 4.1 Kalman Filters .......................... 33 4.2 Example .............................. 34 4.3 Concepts ............................. 37 4.4 Assumptions ........................... 39 4.5 Gaussian Implications ...................... 42 4.6 KF Equations ........................... 44 4.7 Linear Kalman Filter ....................... 49 x CONTENTS 4.8 Minimum Variance Estimator .................. 53 4.9 Additional System Dependencies ................ 55 4.10 Summary ............................. 58 5 Kalman Filter Extensions 59 5.1 Nonlinear Dynamic Systems ................... 59 5.2 Perturbation Kalman Filter ................... 60 5.3 Extended Kalman Filter ..................... 66 5.4 Iterated Extended Kalman Filter ................ 70 5.5 Additional Dependencies Revisited ............... 71 5.6 Related Work ........................... 74 5.7 Summary ............................. 75 6 System and Measurement Models 77 6.1 Driving System .......................... 77 6.2 Full State Sensor ......................... 83 6.3 Landmark Detection Sensor ................... 85 6.4 Modeling Remarks ........................ 89 6.5 Summary ............................. 89 7 Kalman Localization 91 7.1 Analyzing Kalman Filters .................... 91 7.2 Predictive Position Tracking ................... 93 7.3 Corrective Position Tracking ................... 101 7.4 Kidnapped Robot ......................... 110 7.5 Global Localization ........................ 112 7.6 Related Work ........................... 116 7.7 Summary ............................. 116 8 Landmark Kalman Localization 119 8.1 Landmarks and Localization ................... 119 8.2 Unique Landmark Localization ................. 120 8.3 Type Based Landmark Localization .............. 128 8.4 Summary ............................. 134 9 Conclusion 135 9.1 Future Work ........................... 137 Bibliography 139 A Simulator 145 Chapter 1 Introduction Imagine yourself walking down a street. The street is filled with obstacles, like houses, trees, and other people. You want to go to the supermarket. With your eyes closed. You will most probably find it rather difficult to find your way to the supermarket. Even if you know exactly which route to walk and even if there are no other people blocking the route. Fortunately, most of us have eyes that help us in finding our way. At every movement we make, our eyes tell us where that movement brought us in the world. They constantly correct the imperfections of our movements. You do not realize how inaccurate the movements that you make are, until you try to walk a straight line with eyes closed. Even though you think that

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