Int. Conf. on Intelligent Autonomous Systems July 2527, Venezia (IT). pp Planning Safe Paths for Nonholonomic Car-Like Robots Navigating

Transcription

1 Int. Conf. on Intelligent Autonomous Systems July 2527, Venezia (IT). pp Planning Safe Paths for Nonholonomic Car-Like Robots Navigating Through Computed Landmarks Th. Fraichard and A. Lambert Inria a Rh ne-alpes & Gravir b 655 av. de l'europe, Montbonnot, France July 2000 Abstract This paper addresses path planning with uncertainty for a car-like robot subject to conguration uncertainty. The robot estimates its conguration with odometry and an absolute localization device based on environmental feature matching. The issue is to compute safe paths that guarantee that the goal will be reached in spite of the uncertainty. The solution proposed relies upon the automatic construction of a set of landmarks characterized by (1) a region of the conguration space, (2) the `best' features for localization in this region, and (3) a perception uncertainty eld that measures how well a feature is perceived at each conguration in the region. The landmarks are used within an ecient roadmap-based path planning algorithm that returns a safe motion plan that alternates motion along safe paths and localization operations. Keywords mobile-robot, nonholonomic-system, path-planning, uncertainty, landmark. Acknowledgements This work was partially supported by the French programme La Route Automatis e: < a Institut National de Recherche en Informatique et en Automatique. b Lab. Graphisme, Vision et Robotique.

2

3 Planning Safe Paths for Nonholonomic Car-Like Robots Navigating Through Computed Landmarks Th. Fraichard and A. Lambert Inria Rh ne-alpes & Gravir y 655 av. de l'europe Montbonnot, France 1 Introduction Abstract This paper addresses path planning with uncertainty for a car-like robot subject to conguration uncertainty. The robot estimates its conguration with odometry and an absolute localization device based on environmental feature matching. The issue is to compute safe paths that guarantee that the goal will be reached in spite of the uncertainty. The solution proposed relies upon the automatic construction of a set of landmarks characterized by (1) a region of the conguration space, (2) the `best' features for localization in this region, and (3) a perception uncertainty eld that measures how well a feature is perceived at each conguration in the region. The landmarks are used within an ecient roadmap-based path planning algorithm that returns a safe motion plan that alternates motion along safe paths and localization operations. Navigating a mobile robot usually implies (a) planning a path, i.e. a continuous sequence of congurations, that takes the robot to its goal, and then (b) following this path. Path planning is thus a fundamental issue in Robotics and, as such, it has received a large attention in the past twenty years (cf. [11]). The primary concern of path planning is to compute collision-free paths and the underlying assumption of most path planners is that the robot should be able to follow the planned paths accurately enough. A challenging problem is that this assumption hardly holds for a mobile robot operating in the real world. To begin with, mobile robots usually rely on odometric techniques to estimate their conguration. As these techniques yield increasing and unbounded conguration uncertainty, failure to follow a planned path is bound to occur. To overcome this problem, mobile robots are often equipped with absolute localization devices. These devices usually rely on sensors identifying environmental features that are then matched against a priori models of the environment in order to estimate the robot's conguration 1 (cf. [1]). Combining the estimates provided by both odometric and feature matching techniques permits to reduce conguration uncertainty. However if a planned path does not allow the detection of the appropriate environmental features, the mobile robot may once again fail to reach its goal. A solution to these two problems (drift, feature detection), both related to the various sources of uncertainty aecting the robot, is to design a path planner that explicitly takes into account uncertainty so as to compute safe paths, i.e. paths that guarantee that the goal will be reached in spite of these uncertainties. Path planning with uncertainty for mobile robots has motivated a number of research works (cf. [8] and [11, chapter 10] for reviews on this topic). In most cases however, simplifying assumptions are made (point robots performing straight motions, perfect sensing, perfect control, etc. e.g. [2, 5, 13, 18]) that restrict their applicability to real robots. For instance, nonholonomy, which is a key issue in path planning for wheeled mobile robots (cf. [12]), has seldom been considered before in path planning with uncertainty. It is the purpose of this paper to address these challenging issues in a realistic context. The case of a nonholonomic car-like robot moving in a stationary workspace and subject to con- guration uncertainty is considered here. Odometry is the primary tool used to estimate the robot's conguration. The robot is also equipped with an absolute localization device based on feature matching. It is assumed that feature matching localization is costly and that its use should be minimized. Inst. Nat. de Recherche en Informatique et en Automatique. y Lab. Graphisme, Vision et Robotique. 1 Henceforth feature matching localization will denote the whole procedure: identication, matching, etc. 1

4 The solution proposed in this paper relies upon: (a) simulations of the robot's odometry and localization procedure in order to estimate in a conservative way how the conguration uncertainty evolves when the robot moves or localizes itself, and (b) a novel type of landmarks, i.e. environmental features (natural or articial) that the robot can perceive and use to localize itself. Landmarks have already been used in path planning with uncertainty. In most cases however, they tended to be binary (their associated features are visible or they are not) and to yield a constant localization uncertainty, e.g. [2, 6, 13]. This is far from being satisfactory because, if feature matching localization clearly requires the environmental features to be visible, it is also clear that the localization uncertainty is sensitive to the position of the environmental features with respect to the robot's sensors (e.g. the closer the feature, the better the localization). The landmarks introduced herein are characterized by: (1) a region of the conguration space, (2) a set of `best' features for localization in the region, and (3) a perception uncertainty eld that measures how well a given feature is perceived at each conguration in the region. They are actually derived from the local map concept developed by the second author. A local map associates to a conguration the set of environmental features that are most appropriate for localization. Local map validity regions are connected subsets of the conguration space with similar local maps [9]. Local maps were originally introduced to drive the feature matching procedure during navigation: by reducing the number of features to be matched and by selecting the best features, local maps simplify and improve localization. Ref. [18] introduced the concept of sensory uncertainty eld, that associates to each conguration of a robot its localization uncertainty, to address path planning with uncertainty. This concept is interesting. However, the conguration uncertainty of the robot is not taken into account when the eld is computed. The perception uncertainty eld is derived from this concept. It associates to each conguration of a landmark region the perception uncertainty of the dierent environmental features of the landmark. This eld is used, at a later stage, along with the current conguration uncertainty of the robot, to compute a conservative localization uncertainty which is better suited for safe path planning purposes. To begin with, the paper presents how the landmarks can be built automatically from a model of the localization sensor and a description of the environment, and then, how they can be used within an ecient safe path planning algorithm similar to the one developed by the rst author in [6]. The algorithm relies upon a roadmap, i.e. a graph in the conguration space whose edges are collisionfree paths that respects the nonholonomic constraints of a car-like vehicle. It searches the roadmap for the shortest path to the goal. Safeness along each edge-path is checked thanks to an odometry simulation. Whenever a node included in a landmark region is reached, feature matching localization takes place and conguration uncertainty is reduced accordingly. The algorithm returns a motion plan that alternates motions along safe paths and localization operations. Outline of the paper. First, the problem at hand is stated in 2 and the solution algorithm is outlined in 3. Details about odometry and feature matching localization are given in 4 while the method to build the landmarks is described in 5. Finally the planning scheme is presented ( 6) along with experimental results ( 7). 2 Problem Statement 2.1 Model of the Robot The robot considered is a car-like robot denoted by A. It moves on the IR 2 plane and its conguration is dened as q = (x; y; ) 2 IR 2 S 1 where (x; y) are the coordinates of the rear axle midpoint, and the orientation of A. It is subject to two nonholonomic kinematic constraints [12]: (1) it can only move forward and backward in a direction tangent to its main axis, and (2) its turning radius is lower bounded. Let C be the conguration space of A, a path for A is a continuous sequence of congurations, i.e. a continuous mapping : [0; 1]?! C. A path is feasible if it respects the nonholonomic constraints above. 2.2 Model of the Workspace Let W IR 2 denote the workspace of A. It is cluttered up with a set of stationary obstacles B = fb i ; i = 1 : : : bg dened as forbidden regions of W. A set of stationary features F = ff j ; j = 1 : : : fg is also dened. A feature is an element of the environment that can be detected by the localization sensor of A. A feature may be part of an obstacle. A typical indoor environment is depicted in Fig. 1. The obstacles are modeled as polygons. The bold lines represent features, i.e. part of the obstacles 2

5 that can be detected while the thin lines represent the part of the obstacles that cannot or are dicult to detect (windows, etc.). 2.3 Conguration Uncertainty The knowledge that A has of its current conguration is always uncertain, i.e. with a limited accuracy. A probabilistic representation was chosen to model the uncertainty [16]. Accordingly an uncertain conguration is represented by its estimate ^q and its covariance matrix C. For collision checking purposes, it is necessary to know (^q; C), i.e. the set of congurations that are potentially occupied by A with a given condence threshold. This set is dened by the equiprobable contour of the probability distribution represented by (^q; C), it is a conguration space ellipsoid: (q? ^q) T C?1 (q? ^q) 2 (1) where is the condence threshold. For eciency reasons, it was further decided to decouple the position and orientation uncertainties and to approximate (^q; C) by an elliptical cylinder dened by the xy-projection of the ellipsoid (1), and by its -projection [9]. Accordingly an uncertain conguration (^q; C) is safe i: 8i 2 f1; : : : ; bg; 8q p 2 (^q; C); A(q p ) \ B i = ; where A(q) is the region of W occupied by A when in the conguration q. 2.4 Uncertainty Evolution, Safe Path The conguration uncertainty of A changes over time: it increases when A moves around (odometry) and it decreases when feature matching localization takes place. Hence the denition of the two following functions: The rst one, Od, models the evolution of the conguration uncertainty when A moves along a given path : Od(; C; s) returns C O, i.e. the conguration uncertainty at position (s) along the path assuming that A starts at conguration (0) with uncertainty C. The second one, Lo, computes the localization uncertainty, i.e. the conguration uncertainty resulting from feature matching localization. Lo takes as input the current uncertain conguration and the set of environmental features, it returns a covariance matrix: Lo(^q; C; F) = C L. Using Od, it is possible to dene a safe path as a path such that: 2.5 Safe Path Planning 8s 2 [0; 1]; ((s); Od(; C; s)) is safe. Safe path planning can now be stated as follows: let (^q s ; C s ) and (^q g ; C g ) be two uncertain congurations. ^q s is the start conguration and C s the corresponding uncertainty, whereas ^q g is the goal con- guration and C g the maximum nal uncertainty allowed. The sequence f 1 ; Lo; 2 ; Lo; : : : ; Lo; n g constitutes a safe path from (^q s ; C s ) to (^q g ; C g ) i (the Lo indicate when feature matching localization occurs): 1 starts at ^q s with uncertainty C s. n ends at ^q g with a nal uncertainty C n such that: (^q g ; C n ) (^q g ; C g ) 8i 2 f1; : : : ; ng; i is feasible and safe. 3 The Approach The solution scheme proposed relies upon a roadmap, i.e. a graph in the conguration space whose edges are feasible and collision-free paths for the robot A. The origin of the roadmap is not relevant. It may be automatically generated or manually built, but it must capture the connectivity of the collision-free conguration space. The Probabilistic Path Planner [17] is a general planning scheme that constitutes an automatic and ecient way to build roadmaps. In a previous work, the rst author has shown how to use the Probabilistic Path Planner in order to compute roadmaps that satisfy the requirements above in the case of a car-like robot [7]. 3

6 Given such a roadmap and assuming that the start and goal congurations are roadmap nodes 2, safe path planning can be turned into graph search. The graph is explored until the goal node is reached. In the course of the exploration, two important tasks are carried out: When an edge path is visited: the algorithm checks whether the path is safe (it cannot be checked a priori since it depends on the robot conguration uncertainty at the beginning of the path). When a node is visited: the algorithm checks whether feature matching localization can take place. If so, it estimates the output of the localization. In theory, feature matching localization can take place anywhere. However, the resulting localization uncertainty (which is a direct measure of the quality of the localization) depends heavily on two factors: what features are visible and how well can the robot's sensors perceive them? To answer these questions and improve the performance of the feature matching localization, the solution proposed relies upon a novel type of landmarks that are characterized by: (1) a region of the conguration space, (2) a set of `best' features for localization in the region, and (3) a perception uncertainty eld that measures how well a given feature is perceived at each conguration in the region. The region associated with a landmark is used to determine whether the robot has reached a landmark (it takes into account the robot's uncertainty). If so, then the best features are then used to simulate the feature matching localization. Once again, the robot's uncertainty is taken into account; this is achieved thanks to the perception uncertainty eld. The next three sections respectively describes how odometry and feature matching localization are simulated for the purpose of safe path planning ( 4), how the landmarks are built ( 5) and how safe path planning is achieved ( 6). 4 Odometry, Localization 4.1 Odometry (Od) To decide whether a given path is safe, it is necessary to know how the conguration uncertainty changes along the path, i.e. to dene the Od function (cf. 2.4). A relies on odometric techniques to estimate its current conguration and it was decided to use an uncertainty evolution model derived from the classical odometric models used in navigation [3]. Let (^q(t); C(t)) be the uncertain conguration of A at time t. At time t + t, odometry returns (s; ), i.e. the estimates of the translational and rotational motion performed by A. Then, using the prediction part of an Extended Kalman Filter, it is possible to compute the new uncertain conguration (^q(t + t); C(t + t)) at time t + t (cf. [9] for more details). Od is dened by discretizing, estimating (s; ) at each step and by iteratively feeding the values obtained to the Extended Kalman Filter. 4.2 Localization (Lo) To determine the conguration uncertainty after a localization, it is necessary to simulate the feature matching localization procedure, i.e. to dene the Lo function (cf. 2.4). To do so, a model of the sensor is required rst. It is assumed that A is equipped with a rotating range scanner that returns the distance to the nearest obstacle in a given direction. As per [4], the sensor uncertainty is modeled by a circular covariance. Using the set of environmental features F and the sensor model, it is possible to estimate the sensor measurements and their uncertainty at a given conguration (cf. [9]). Finally, the simulated sensor measurements are fed into a classical feature matching localization procedure based upon Extended Kalman Filtering technique [14]. 5 Landmarks Recall that a feature is an environmental object that can be detected by the robot's sensors and used in feature matching localization. As soon as a feature is detectable, feature matching localization can take place. However, the resulting localization uncertainty depends on what features are visible and how well can the robot's sensors perceive them. To simplify and improve localization, a novel type of landmark is introduced. Such a landmark is characterized by: (1) R, a region of the conguration space, (2) F, a set of `best' features for localization in R, and (3) puf, a perception uncertainty eld that measures how well each feature of F is perceived at each conguration in R. 2 This assumption can be lifted easily. 4

7 F and R derive from the concepts of local map and local map validity regions developed by the second author in [9]. A local map is dened as the minimum set of features that can be used for a good localization in a region of C (called the local map validity region). Whenever A enters a region R, i.e. when (^q; C) R, localization based upon the features of F takes place. By reducing the number of features to be matched and by selecting the best features, local maps and local map validity regions simplify and improve localization. They were originally introduced to drive the feature matching procedure at navigation time. To be used at planning time, they are complemented with the perception uncertainty eld puf that takes as input a conguration q and a feature F i and returns the best sensor uncertainty (determined thanks to the simulated sensor measurements) associated with F i when observed from q. puf is computed globally over the conguration space C. As it will be seen further down, puf permits to compute a conservative localization uncertainty that does take into account the conguration uncertainty of A (cf. 6.4). In the meantime, the next two sections describe how the landmarks are automatically built from the model of the environment. 5.1 From Features to Local Maps In order to select the best features, A is placed at a given conguration q and a sensor simulation is done. Localization is then performed on each feature F j separately. The result is a set of covariance matrices C j representing the localization uncertainty when using feature F j. Each matrix C j determines an ellipsoid (q; C j ), the analysis of the intersection between the (q; C j ) permits to determine the `best' features for localization at q or rather to eliminate the features that are useless in the sense that, taking them into account into the localization process, does not improve signicantly the quality of the localization. The features F k to be eliminated are those for which (q; C k ) is strictly greater than the volume dened by \ j (q; C j ) (cf. [9]). This process is repeated over the discretized conguration space C. Its output denes the local maps for each discrete conguration q, i.e. the set of best features to be used for localization at q. 5.2 From Local Maps to Landmarks Using a clustering algorithm such as the one presented in [10], the connected congurations sharing the same local maps are grouped together to dene the local map validity regions. Along with the perception uncertainty eld puf, a local map validity region R and its local map F dene a landmark. 6 Safe Path Planning The roadmap, which is a graph, is explored using the classical Dijkstra algorithm [15]. This algorithm is guaranteed to return the path that optimizes a given cost function. Path length is a straightforward optimality criterion. Like [18], it proved interesting though to dene a cost function combining path length and path reliability. The cost function is detailed in 6.1. In the course of the graph search, each path connecting two nodes must be checked for safeness. This point is presented in 6.2. In addition, when a node is visited, the algorithm has to check whether feature matching localization can take place and if so, it must compute the corresponding localization uncertainty. These two points are respectively addressed in 6.3 and Cost Function Let be a path in the roadmap; it is a sequence of congurations (nodes of the graph) connected by feasible paths (edges of the graph). In the course of the roadmap exploration, the conguration uncertainty of A is computed and propagated. Accordingly an uncertain conguration (^q i ; C i ) is associated with each conguration-node of. The cost function J combines path length and path reliability. It is dened as: J =X 1 L(q i ; q i+1 ) + 2 R(q i ; q i+1 ) (2) i where 1 and 2 are two weighing coecients. L(q i ; q i+1 ) denotes the length of the edge-path connecting q i to q i+1. R(q i ; q i+1 ) is a measure of the reliability of the edge-path connecting q i to q i+1, it is dened as the product of the average uncertainty along the edge-path with the length of the edge-path: R(q i ; q i+1 ) = L(q i ; q i+1 ) V (q i) + V (q i+1 ) 2 where V (q i ) denotes the volume of (^q i ; C i ). 5

8 6.2 Edge-Path Safeness Potential collisions between the robot and the environment are detected by computing A((^q; C)) and by comparing it with the location of the obstacles. The odometry function Od is used to determine the conguration uncertainty along a path. For the sake of eciency, collision checking is not performed in the three dimensional conguration space C, but in the two dimensional workspace W by computing the intersection between the obstacles B i ; i 2 f1; : : : ; bg, and the region swept by A (integrating its uncertainty) when it moves along a given path. The recursive and resolution-dependent collision checker developed in [6] is used to that purpose. 6.3 Landmark Perceptibility When a node of the roadmap is visited, the algorithm must determine whether feature matching localization can take place. Recall that the landmarks have been introduced to guide feature matching localization. The landmark regions dene a partition of the conguration space C and each node of the roadmap belong to a given landmark region. Accordingly, when the node q of the roadmap is visited, the algorithm rst checks whether, the landmark associated with q, can be used for localization. In other words, it checks that A((q; C)) R where C is the current conguration uncertainty of A (to do so, R is discretized). 6.4 Localization Uncertainty Once it has been determined that feature matching localization can take place at a given node q with a given landmark, the algorithm must compute an estimate of the localization uncertainty and use it to update the conguration uncertainty of A. Recall that a set of best features F is associated to. The features of F are used for localization purposes. Let C be the current conguration uncertainty of A, a straightforward manner to estimate the localization uncertainty C l would be to use Lo(q; C; F ). However, since the quality of the localization with respect to a given feature F j clearly depends on the conguration of A at localization time, it was decided to use the following function instead: Loc-Unc (q; C; F ) C l = C forall F j 2 F q max = max qp 2(^q;C) puf(q p; F j ) C l = Lo(q max ; C l ; ff j g) endforall return C l By processing one feature at a time in the worst possible situation, a more conservative estimate of the localization uncertainty is obtained. 7 Experimental Results The left-hand side of Fig. 1 depicts the landmark regions that have been automatically determined in a m. indoor environment that contains 10 obstacles and 17 features. 23 landmark regions have been computed (dierent regions may have similar grey shades). Two examples of landmarks are depicted on the right-hand side of Fig. 1. For each example are represented (1) the validity region of the local map, and (2) the set of best features that have been retained for localization purposes. Two results of the safe path planning algorithm are depicted in Fig. 2. The rectangles represent the robot at dierent congurations (corresponding to nodes of the roadmap). The curves between the nodes are edges of the roadmap, i.e. paths verifying the nonholonomic constraints of A. The ellipses inside the rectangles represent the conguration uncertainties (before and after localization when localization can take place). In the situation depicted on the left hand side of Fig. 2, A starts from q s with an initial conguration uncertainty C s. Its goal is to reach q g with a conguration uncertainty less than C g. Since A((q s ; C s )) R A, localization can take place there and the initial conguration uncertainty ellipse is reduced. Later, the planning process has returned the sequence fq s ; q a ; q b ; q g g as being a safe path to q g with localization taking place in q s, q a and q g. The situation depicted on the right hand side of Fig. 2 is similar. The safe path is now fq s ; q a ; q b ; q c ; q d ; q g g with localization taking place in q s, q b and q d. 6

9 PSfrag replacements PSfrag replacements R A R B A Region Features R E R B C R D C R C R D R D E R E Region Features PSfrag replacements PSfrag replacements R A R B R C Region Features R B R A R D R E Region Features Figure 1: landmark regions in an indoor environment (left); two regions (R A associated features (right). and R D ) and their q g q a q b PSfrag replacements q d q g q s q a q b q g q c rag replacements q s q b q a q s Figure 2: two examples of safe paths. 8 Discussion and Conclusion This paper has presented a safe path planner for a car-like robot that estimates its conguration with odometry and an absolute localization device based on environmental feature matching. The nonholonomic kinematics of such a robot is taken into account thanks to a roadmap-based path planner. Respective simulations of the robot's odometry and localization procedure permit to estimate in a conservative way how the conguration uncertainty evolves when the robot moves or localizes itself. These simulation are used to explore the roadmap and compute a safe motion plan that alternates motion along safe paths and localization operations. The safeness issue is mainly addressed through landmarks that are characterized by: (1) a region of the conguration space, (2) a set of `best' features for localization in the region, and (3) a perception uncertainty eld that measures how well a given 7

10 feature is perceived at each conguration in the region. Their main purpose is to simplify and improve localization by reducing the number of features to be matched and by selecting only the best features. Future developments will include experiments with a real car-like robot in order to validate the soundness of our uncertainty evolution models. Acknowledgements. This work was partially supported by the French programme La Route Automatis e: References [1] J. Borenstein, H. R. Everett, and L. Feng. Where am I? Sensors and methods for autonomous mobile robot positioning 1995 edition. Technical Report, University of Michigan, [2] B. Bouilly, T. Sim on, and R. Alami. A numerical technique for planning motion strategies of a mobile robot in presence of uncertainty. In Proc. of the IEEE Int. Conf. on Robotics and Automation, volume 2, pages , Nagoya (JP), May [3] F. Chenavier and J. L. Crowley. Position estimation for a mobile robot using vision and odometry. In Proc. of the IEEE Int. Conf. on Robotics and Automation, pages , Nice (FR), May [4] J. L. Crowley. World modeling and position estimation for a mobile robot using ultrasonic ranging. In Proc. of the IEEE Int. Conf. on Robotics and Automation, pages , Scottsdale, AZ (US), May [5] F. De la Rosa, C. Laugier, and J. N jera. Robust path planning in the plane. IEEE Trans. Robotics and Automation, 12(2):347352, April [6] Th. Fraichard and R. Mermond. Path planning with uncertainty for car-like robots. In Proc. of the IEEE Int. Conf. on Robotics and Automation, volume 1, pages 2732, Leuven (BE), May [7] Th. Fraichard, A. Scheuer, and R. Desvigne. From Reeds and Shepp's to continuous-curvature paths. In Proc. of the IEEE Int. Conf. on Advanced Robotics, pages , Tokyo (JP), October [8] K. Y. Goldberg, M. T. Mason, and A. Requicha. Geometric uncertainty in motion planning: summary report and bibliography. Technical Report IRIS-297, University of Southern California, [9] A. Lambert and N. Le Fort-Piat. Safe task planning integrating uncertainties and local maps federations. Int. Journal of Robotics Research, To appear. [10] A. Lambert, M. Van Dang, and G. Govaert. Generating validity regions using Markovian spatial clustering. In Proc. of the IEEE Int. Conf. on Advanced Robotics, pages , Tokyo (JP), October [11] J.-C. Latombe. Robot motion planning. Kluwer Academic Press, [12] J.-P. Laumond, editor. Robot motion planning and control, volume 229 of Lecture Notes in Control and Information Science. Springer, [13] A. Lazanas and J.-C. Latombe. Motion planning with uncertainty: a landmark approach. In Articial Intelligence, volume 76, pages , [14] J. J. Leonard and H.F. Durant-Whyte. Mobile robot localization by tracking geometric beacons. IEEE Trans. Robotics and Automation, 7(3):376382, [15] N. J. Nilsson. Principles of articial intelligence. Morgan Kaufmann, Los Altos, CA (USA), [16] R. C. Smith and P. Cheeseman. On the representation and estimation of spatial uncertainty. Int. Journal of Robotics Research, 5(4):5668, Winter [17] P. Svestka and M. H. Overmars. Probabilistic path planning. In J.-P. Laumond, editor, Robot motion planning and control, volume 229 of Lecture Notes in Control and Information Science, pages Springer, [18] H. Takeda, C. Facchinetti, and J.-C. Latombe. Planning the motions of a mobile robot in a sensory uncertainty eld. IEEE Trans. on Pattern Analysis and Machine Intelligence, 16(10): , October