Sensors and Methods for Mobile Robot Positioning

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210 Part II Systems and Methods for Mobile Robot Positioning In the second case, in which k vertical edges are indistinguishable from each other, the location algorithm finds all the solutions by investigating all the possibilities of correspondences. The algorithm first chooses any four rays, say r , r , r , and r . For any ordered quadruplet (p , p , p , p ) out of n 1 2 3 4 i j l m mark points p 1,...,p n, it solves for the position based on the assumption that 1 2r , 3r , r , 4and r correspond to p, p, p, and p , respectively. For n(n-1)(n-2)(n-3) different quadruples, the algorithm i j l m 4 3 can solve for the position in O(n ) time. Sugihara also proposed an algorithm that runs in O(n log 3 2 n) time with O(n) space or in O(n ) time with O(n ) space. In the second part of the paper, he considers the case where the marks are distinguishable but the directions of rays are inaccurate. In this case, an estimated position falls in a region instead of a point. p1 p2 p1 p2 p3 p1 p2 θ θ + δθ camera camera camera θ − δθ sang03.cdr, .wmf Figure 9.3: a. Possible camera locations (circular arc) determined by two rays and corresponding mark points. b. Unique camera position determined by three rays and corresponding mark points. c. Possible camera locations (shaded region) determined by two noisy rays and corresponding mark points. (Adapted from [Sugihara 1988; Krotkov 1989]). Krotkov [1989] followed the approach of Sugihara and formulated the positioning problem as a search in a tree of interpretation (pairing of landmark directions and landmark points). He developed an algorithm to search the tree efficiently and to determine the solution positions, taking into account errors in the landmark direction angle. According to his analysis, if the error in angle measurement is at most *2, then the possible camera location lies not on an arc of a circle, but in the shaded region shown in Fig. 3c. This region is bounded by two circular arcs. Krotkov presented simulation results and analyses for the worst-case errors and probabilistic errors in ray angle measurements. The conclusions from the simulation results are: C the number of solution positions computed by his algorithm depends significantly on the number of angular observations and the observation uncertainty *2. C The distribution of solution errors is approximately a Gaussian whose variance is a function of *2 for all the angular observation errors he used: a. uniform, b. normal, and c. the worst-case distribution. Betke and Gurvits [1994] proposed an algorithm for robot positioning based on ray angle measurements using a single camera. Chenavier and Crowley [1992] added an odometric sensor to landmark-based ray measurements and used an extended Kalman filter for combining vision and odometric information.
Chapter 9: Vision-Based Positioning 211 9.2.2 Two-Dimensional Positioning Using Stereo Cameras Hager and Atiya [1993] developed a method that uses a stereo pair of cameras to determine correspondence between observed landmarks and a pre-loaded map, and to estimate the twodimensional location of the sensor from the correspondence. Landmarks are derived from vertical edges. By using two cameras for stereo range imaging the algorithm can determine the twodimensional locations of observed points — in contrast to the ray angles used by single-camera approaches. Hager and Atiya's algorithm performs localization by recognizing ambiguous sets of correspondences between all the possible triplets of map points p i, p j, p k and those of observed points o a, o b, o c. It achieves this by transforming both observed data and stored map points into a representation that is invariant to translation and rotation, and directly comparing observed and stored entities. The permissible range of triangle parameters due to sensor distortion and noise is computed and taken into account. 3 For n map points and m observed points, the off-line initialization stage consumes O(n log n) time to compute and sort all triangle parameters from the map points. At run time, the worst case 3 3 complexity is O(m (n + log n)). However, an efficient strategy of marking and scanning reduces the search space and real-time performance (half a second) is demonstrated for five observed and 40 stored landmarks. 9.3 Camera-Calibration Approaches The camera-calibration approaches are more complex than the two-dimensional localization algorithms discussed earlier. This is because calibration procedures compute the intrinsic and extrinsic camera parameters from a set of multiple features provided by landmarks. Their aim is to establish the three-dimensional position and orientation of a camera with respect to a reference coordinate system. The intrinsic camera parameters include the effective focal length, the lens distortion parameters, and the parameters for image sensor size. The computed extrinsic parameters provide three-dimensional position and orientation information of a camera coordinate system relative to the object or world coordinate system where the features are represented. The camera calibration is a complex problem because of these difficulties: C All the intrinsic and extrinsic parameters should be computed from the two-dimensional projections of a limited number of feature points, C the parameters are inter-related, and C the formulation is non-linear due to the perspectivity of the pin-hole camera model. The relationship between the three-dimensional camera coordinate system (see Fig. 1) T X = [X, Y, Z] (9.2) and the object coordinate system T X W = [X W, Y W, Z W] (9.3) is given by a rigid body transformation
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Acknowledgments This research was s
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INTRODUCTION Leonard and Durrant-Wh
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Part I Sensors for Mobile Robot Pos
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14 Part I Sensors for Mobile Robot
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CHAPTER 2 HEADING SENSORS Heading s
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CHAPTER 5 ODOMETRY AND OTHER DEAD-R
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Page 236 and 237: REFERENCES 1. Abidi, M. and Chandra
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262 Index AC ..............47, 59,
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