Integrating Topological and Metric Maps for Mobile Robot Navigation: a Statistical Approach
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From: AAAI-98 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved. Integrating Topological and Metric Maps for Mobile Robot Navigation: A Statistical Approach Sebastian Thrun1 Jens-Steffen Gutmann2 Dieter Fox3 Wolfram Burgard3 Benjamin J. Kuipers4 1Computer Science Department 2Institut f¨ur Informatik 3Institut f¨ur Informatik III 4Computer Science Department Carnegie Mellon University Universit¨at Freiburg University of Bonn University of Texas at Austin Pittsburgh, PA 15213 D-79110 Freiburg, Germany D-53117 Bonn, Germany Austin, TX 78712 Abstract array, and more directly suited to problem-solving algo- rithms (Kuipers & Byun 1991; Dudek et al. 1991). How- The problem of concurrent mapping and localization has re- ever, purely topological maps have difficulty distinguish- ceived considerable attention in the mobile robotics commu- ing adequately among different places, and have not, in nity. Existing approaches can largely be grouped into two dis- tinct paradigms: topological and metric. This paper proposes practice, been applied successfully to large environments. a method that integrates both. It poses the mapping problem as Recent progress in metric mapping (Lu & Milios 1997; a statistical maximum likelihood problem, and devises an ef- Thrun 1998) has made it possible to build useful and ac- ficient algorithm for search in likelihood space. It presents an curate metric maps of reasonably large-scale environments, novel mapping algorithm that integrates two phases: a topo- but memory and time complexity pose serious problems. logical and a metric mapping phase. The topological mapping In this paper, we propose and evaluate a new algorithmthat phase solves a global position alignment problem between po- integrates the topologicaland metric approach. We show that tentially indistinguishable, significant places. The subsequent both the topological and the metric mapping problems can metric mapping phase produces a fine-grained metric map of the environment in floating-point resolution. The approach be solved as different instances of a class of statistical es- is demonstrated empirically to scale up to large, cyclic, and timation problems, in which a robot seeks to find the most highly ambiguous environments. likely map from a set of observations and motion commands. These estimation problems are solved by a variant of the EM algorithm, which is an efficient hill-climbing method for Introduction maximum likelihood estimation in high-dimensional spaces. Over the past two decades, the problem of building maps in In the context of mapping, EM iterates two alternating steps: indoor environments has received significant attention in the a localizationstep, in which the robot is localized using a pre- mobilerobotics community. Theproblem of building maps is viously computed map, and a mapping step, which computes the problem determining the location of certain entities, such the most likely map based on the previously pose estimates. as landmarks or obstacles, in a global frame of reference. To The statistical framework is the foundation for two algo- build a map of its environment, a robot must know where rithmic phases, one that builds topological maps and one it is. Since robot motion is inaccurate, constructing maps that builds metric maps. Both components possess orthogo- of large indoor environments requires a robot to solve an nal strengths and weaknesses. The topological map builder inherent concurrent localization problem. is capable of solving global alignment problems that occur As of to date, there exist two major paradigms for mobile in datasets with unbounded odometric error and perceptual robot mapping: Metric and topological. Approaches in the ambiguity. Its result, however, is only approximate, partially metric paradigm generate fine-grained, metric descriptions because it ignores much of the sensor data. The metric ap- of a robot’s environment (Moravec 1988; Lu & Milios 1997). proach, on the other hand, builds fine-grained metric maps Approaches in the topological paradigm, on the other hand, in floating-point resolution. However, it can only compen- generate coarse, graph-like descriptions of environments, sate small odometric errors. By integrating both, topological where nodes correspond to significant, easy-to-distinguish and metric mapping, the algorithm can build high-resolution places or landmarks, and arcs correspond to actions or action- maps in large and highly ambiguous indoor environments. sequences that connect neighboring places (Matari´c 1990; Dudek et al. 1991). Examples of metric and topological Statistical Foundations methods are provided towards the end of this paper. It has long been recognized (Chatila & Laumond 1985; This paper poses the problem of mapping as a statistical max- Kuipers & Byun 1991) that either paradigm alone, met- imum likelihood estimation problem (Thrun, Fox, & Burgard ric or topological, has significant drawbacks. In principle, 1998). To generate a map, we assume that a robot is given a topological maps should scale better than metric maps to stream of data, denoted large-scale environments, because a coarse-grained, graph- T T T 1 1 2 2 1 = fo ;u ;o ;u ;:::o ;u ;o g; structured representation is much more compact than a dense d (1) (a) (b) (c) The Map Likelihood Function In statistical terms, the problem of mapping is the problem of finding the most likely map given the data = P mjd: m argmax (2) m mjd Figure 1: Basic probabilistic models: (a) Robot motion model. The probability P can be written as Shown here is the accumulated uncertainty after moving 40 meter, Z starting at a known pose. (b-c) Perceptual model. (b) shows a mjd = P mj; d P jd d : P (3) map with two indistinguishable landmarks, and (c) displays the uncertainty after sensing a landmark in 5 meter distance. Here the variable denotes the set of all poses at times T t 1 t ;:::; g ; ;:::;T = f where o stands for an observation that the robot made 1 2 ,thatis, : ,where denotes t t u at time t,and for an action command that the robot the robot’s pose at time . By virtue of Bayes rule, the P mj; d t + T executed between time t to time 1. denotes the total probability on the right hand side of Equation (3) number of time steps in the data. Without loss of generality, can be re-written as we assume that the data is an alternated sequence of actions P djm; P mj mj; d = and observations. P (4) dj In statistical terms, the problem of mapping is the prob- P t lem of finding the most likely map given the data. Maps will t Based on the observation that o at time depends only on m = fm g x;y be denoted by x;y . A map is an assignment t the map m and the corresponding location ,thefirstterm m x y of “properties” x;y to each - -locations in the world. In topological approaches to mapping, the properties-of- on the right hand side of Equation (4) can be transformed interest are usually locations of landmarks (Chown, Kaplan, into & Kortenkamp 1995) or, alternatively, location of signifi- T Y t t djm; = P o jm; cant places (Kuipers & Byun 1991; Choset 1996). Metric P (5) approaches, on the other hand, usually use the location of ob- = t 1 stacles as properties-of-interest (Chatila & Laumond 1985; mj = P m Moravec 1988; Lu & Milios 1997). Furthermore, P in Equation (4), since in P m Our approach assumes that the robot is given two basic, the absence of any data, m does not depend on . probabilistic models, one that describes robot motion, and is the Bayesian prior over all maps, which henceforth will one that models robot perception. be assumed to be uniformly distributed. Finally, the term P jd 0 in Equation (3) can be re-written as P ju; The motion model, denoted , describes the 0 probability that the robot’s pose is , if it previously exe- T 1 Y t t t+ cuted action u at pose .Here is used to refer to a pose, 1 ju ; jd = P P (6) y that is the x- -location of a robot together with its head- = t 1 ing direction. Figure 1a illustrates the motion model, by 0 showing the probability distribution for upon executing The latter transformation is based on the observation that the t+ t 1 the action “go forward 40 meters.” Notice that in these robot’s pose depends only on the robot’s pose one t x y and other diagrams, poses are projected into - -space time step earlier and the action u executed there. Putting (the heading direction is omitted). all this together leads to the likelihood function Q Z P ojm; The perceptual model, denoted , models the T t t P o jm; P m t= likelihood of observing o in situations where both the 1 P mjd = P dj world m and the robot’s pose are known. For low- dimensional sensors such as proximity sensors, perceptual T 1 Y t+ t t 1 P u ; d : models can readily be found in the literature (Burgard et j (7) al. 1996; Moravec 1988). Figure 1b&c illustrates a per- = t 1 ceptual model for a robot that can detect landmarks and mjd that can measure, with some uncertainty, their relative ori- Since we are only interested in maximizing P , not in entations and distances. Figure 1b shows an example map computing an exact value, we can safely drop the constants P m P dj m, in which the dark spots indicate the locations of two in- and . The resulting expression, ojm; distinguishable landmarks. Figure 1c plots P for Z T T 1 Y Y different poses , for the specific observation that the robot t t t+ t t 1 P o jm; P u ; d ; observed a landmark ahead in five meters distance. The argmax j (8) m t= = darker a pose, the more likely it is under this observation.