A Stochastic Environment Modeling Method for Mobile Robot by using 2-D Laser scanner Young D. Kwon,Jin.S Lee Department of Electrical Engineering, Poh

Transcription

1 A Stochastic Environment Modeling Method for Mobile Robot by using -D Laser scanner Young D. Kwon,Jin.S Lee Department of Electrical Engineering, Pohang University of Science and Technology, Abstract A method of environment modeling is presented which is based on the stochastic approximation technique. The accurate modeling of geometric environment is in general dicult because of sensor noise, irregular shape of objects and possible change of its layout, and so on. To overcome the problem, the environment is represented in this paper as stochastic obstacle regions equipped with their own stochastic variables such as mean, variance and eigenvalues. The stochastic variables in each obstacle region are updated by using the distance information of the obstacles, which is acquired from the laser scanner every sampling time. The representation of the environment with the stochastic variables enables us to save CPU time and memory consumption in building the map. This technique can also detect if the obstacles are removed and is applicable to the quasi-static environment. If the eigenvalues of the covariance matrix in a certain region are known, then the feature of that region can be extracted as a line or as an ellipse. Thus, the presented algorithm can be used for the navigation and the localization of the mobile robot. The presented algorithm is successfully tested on our mobile robot ARES-II system equipped with the LADAR D-laser scanner. Key Words{ map building, mobile robot, stochastic,laser scanner 1 Introduction Having an accurate map is a prerequisite in making a successful navigation. Even in the case of mobile robot systems, the environment information is still crucial and helps them to maneuver safely and eciently. Generating an appropriate map, however, is generally dicult due to the following reasons. First, to build an accurate map, we must know the accurate positions and shapes of the obstacles. But due to the dead reckoning of the mobile robot and the sensor measurement errors, the acquired position information is not suciently accurate. To address this problem, Crowley[1] used the Kalman ltering technique to build the static environment models. As is well known, the Kalman ltering technique is one of the popular methods to extract system information from noisy and uncertain data. Leonard et al.[] extended the modelbuilding technique to the dynamically changing environment. According to this method, the next position of the mobile robot are estimated and then the beacons such as planes, corners or cylinders are predicted. If the predicted beacons match with the actual data at the next sampling time, then the credibility values increase. If not, they decrease. This way, the beacons and their credibility values determine the dynamic environment. It is a reliable technique that represent the obstacles by using the ultrasonic sensors. It also uses the Kalman lter to overcome the uncertainty of the sensor measurement and the dead reckoning. But this method requires a sensor model to estimate the geometric beacons. Further, because it is developed based on the ultrasonic sensor measurements, it can not represent many tiny obstacles with the restricted beacons such as plane,corner and cylinder. Also these obstacles are dicult to detect by using the ultrasonic sensors because of its resolution. Second, there are few laser scanner based techniques available how to represent arbitrary shaped obstacles. In general, most laser scanner based works represented the environment as lines, ellipses[3] or circles[4] according to their stochastic property. But they still can not represent highly scattered obstacles because they restrict the object feature to a circle or a line. Elfes [5], on the other hand, introduced a map building technique that represent the environment as a number of two dimensional grid cells. This concept is simple and useful for mobile robot navigation[7], but it is dicult to extract the obstacle features because it must check all the grid cells in the environment. The cell based map 1

2 Y X Figure 1: Laser scanner dispatch making technique is eective for navigation, but may not be adequate for localization problem. Moreover, it may consume signicant portion of memory space just to store safe cell information. In this paper, the map building technique based on the stochastic environment modeling is presented. A LADAR D laser scanner from the IBEO Lasertechnik Co.[10] is used to collect the data. Each sampling time, the laser scanner provides distance information of the obstacles about 70 degree range around the mobile robot with front side at the center of the range. Thus, various obstacle information around the front side can be simultaneously detected at one sampling time. Since the environment of the mobile robot work area may change, this fact is reected in the proposed map making algorithm. The proposed method represents the environment as stochastic regions with their own stochastic variables such as mean, variance and covariance. If the obstacles are represented as a line or an ellipse, it is quite common that a line is changed to an ellipse or an ellipse to a line. Then the extra procedure is required to detect the changes of the obstacle features. This is why many previous results is restricted to the map making for the static environment. The proposed method represent the obstacles as a region with its own stochastic variables and the obstacle features are updated by updating the stochastic variables. Thus, it is applicable to dynamically changing environment by constantly monitoring the validity of the obstacle regions that were stored in the map. The presented algorithm, however, is developed for the quasistatic environment, where the environment is changed occasionally and not for the dynamically changing environment, where the obstacles are constantly moving in real time. Another advantage of the method is its simple procedure for matching and updating the map with the laser scanner data. It simply checks if the detected obstacle region is new by comparing the mean and the eigenvalues of the obstacle regions. If the detected obstacle region comes from previously detected one, then the corresponding stored obstacle region is Figure : The laser scanner data updated with the new information. If not, then the new obstacle region is just added to the map by reecting the new information. This paper is organized as follows. Section describes the data clustering method. Section 3 explains the stochastic variables that describes an obstacle region and presents map matching and updating procedure. Section 4 presents experimental result and nally Section 5 makes conclusions and future works. Laser Scanner and Data Clustering Fig.1 describes the coordinate system that is attached to the laser scanner and the center position (x c ; y c ) of the mobile robot in terms of the coordinate system. The incoming data l j,j = 0; 1; : : : 70 are the distance values from the center position of the laser scanner to the objects. The object position (x j ; y j ) is determined from the measured distance l j as follows : (j? 45) (x j ; y j ) = ( x c +l j cos( + c? 180 )? l d cos c ; (j? 45) y c +l j sin( + c? 180 )? l d sin c ; ) where l d is distance between the center of the laser scanner and that of the mobile robot and c is the orientation of the mobile robot. This way, every object position is located as shown in Fig.. As noticed from Fig., dierent objects can be easily separated at a glance.

3 But how we classify these data and separates the objects. If two successive data points are far away, then the two points may be classied into a dierent region. This way, the acquired data are mapped into a number of clustered regions. Let the clustered regions are denoted by R i 's for i = 1; ; : : : ; N. Then, whenever the following condition is satised, the two successive point (x j ; y j ) and (x j+1 ; y j+1 ) are located in same clustered region. (x j+1? x j ) + (y j+1? y j ) < D th The threshold value D th is appropriately selected taking the mobile robot size into account. D th is set to 0cm in our experiment. Thus, each R i consists of a set of data points from (x si ; y si ) to (x ei ; y ei ), where s i and e i are respectively the start and end indices. 3 Environment Modeling In general, the environment around the mobile robot could be quite complex. In the laboratory situation, it may be composed of many obstacles such as chairs, boxes, the legs of the tables, and so on. Thus, it is not practical to represent all those obstacles as either lines or ellipses. To remedy this problem, the map of the entire environment is represented by a series of stochastic obstacle regions M k 's,k = 1; ; : : : ; L, with their own stochastic variables. Each M k has the following data structure : M k : fm xk ; m yk ; xk ; yk ; xyk; n k g; where (m xk ; m yk ),( xk ; yk ), xyk and n k represent respectively the mean, the variance, the covariance of (x; y) and the number of scanning data used to determine the stochastic variables. In determining the stochastic variables of the region M k, the clustered regions R i 's acquired from the clustering procedure are used. 3.1 Stochastic Variables For every sampling time, a number of clustered regions R i 's is obtained and from all of the obstacle positions (x j ; y j ) included in R i, the stochastic variables of R i is computed as follows : P kr m xi = i x k ; P kr m yi = i y k ; xi = yi = = PkR i (x k? m xi ) ; PkR i (y k? m yi ) ; P kr i (x k? m xi )(y k? m yi ) ; where s i and e i are respectively the start and end indices of R i, (m xi ; m yi ), ( x ; y) and xy represent respectively the mean, the varaince and the covariance of the obstacle positions in R i. Additionally, the eigenvalues and the eigenvectors of the covariance matrix C i C i = xi yi are computed to characterize each clustered region R i. In general, the eigenvectors of the covariance matrix represent the orientation of the data set and the corresponding eigenvalues indicate the size of the data set. Fig. 3 shows the eigenvalues and the eigenvectors in the example data set. The largest eigenvalue is designated as the major eigenvalue 1 and the smallest eigenvalue as the minor eigenvalue. As shown in Fig. 3, these two eigenvalues clearly indicate how are the data scattered from the mean. This technique is quite common in image processing area[9]. If 1 is very large(fig.3 (a)), then the data are very much aligned which usually come from distinct obstacles such as wall. If 1 is close to 1 (Fig. 3 (b)), then the data are very much scattered which usually come from tiny obstacles closely located each other. The major eigenvalue 1i and the minor eigenvalue i for each R i are determined as q 1i = xi + yi + (xi? yi ) + 4 i = xi + yi? q( xi? yi ) + 4 and the angles 1i and i of the corresponding eigenvectors are computed as 1i = tan?1 ( 1i? xi ) i = tan?1 ( i? xi ): Now, these stochastic variables are to be used to update the map. The update procedure to be explained in the following subsection consist of two part : First, update M k with the information of matched R i, second, delete moved obstacle region M k by examining its validity. 3

4 Thus, condition 1 examines if the distance d i is smaller than the minor eigenvalue k of M k : d i < k (a) (b) Figure 3: Eigenvalues and Eigenvectors Figure 4: Example of matching case 3. Matching and Updating of the Map To check if R i comes from M k or not, we use following two conditions. 1. Is the mean of R i located within k distance from the major axis of M k?.. Are R i and M k overlapped or located suciently close each other?. Also, to check the validity of the M k, we consider following condition. 3. Is R i located closer to the mobile robot than M k? If the answers to condition 1 and are armative, then M k is updated and if the answer to condition 3 is armative, then the obstacle is deleted. Fig. 4 shows the case when the answers to condition 1 and are armative. When these two conditions are met, R i is called matched with M k. As mentioned before, the characteristics of the obstacle regions are presented with its major and minor eigenvalues. The minor eigenvalue k is used to test condition 1 and the major eigenvalue 1i and 1k are used to test condition. The matching condition 1 means that the mean position of the clustered region R i must be located suciently close to the major axis of the previously detected obstacle region M k. As shown in Fig. 3, the major axis is determined from the major eigenvector. Thus, the major axis of M k is a line that satises (sin 1k )(x? m xk )? (cos 1k )(y? m yk ) = 0; where the 1k repersent the angle of the major eigenvector. Also, the distance d i between the mean position of R i and major axis of M k is computed as d i = (sin 1k )(m xi? m xk )? (cos 1k )(m yi? m yk ): which means that k is used as an uncertain bound of M k. Condition examines if R i contains part of M k. This condition is checked by using the major eigenvalue 1i and 1k. As shown in Fig. 4, if the sum of the major eigenvalues of M k and the projected major eigenvalue of R i into the major axis of the M k is larger than p d M? d i, then the M k and R i are called overlapped, where d M is the distance between mean position of R i and M k. Mathematically, condition examines if, is satised, where d M? d i < 1i cos + 1k d M = (m xi? m xk ) + (m yi? m yk ) ; = j 1i? 1k j; and indicate the angle dierence between the major axes of M k and R i. Therefore, if condition 1 and are satised, then M k and R i are matched and the stochastic variables of M k is updated as follows : where m xk = k 1 m xi + k m xk ; m yk = k 1 m yi + k m yk ; xk = k 1 xi + k xk + k 1 k (m xi? m xk ) ; yk = k 1 yi + k yk + k 1 k (m yi? m yk ) ; xyk = k 1 + k xyk + k 1 k (m xi? m xk )(m yi? m yk ); and n k = n k + ; k 1 = k = n k + n k n k + and In parallel with the above procedure, we examine condition 3 to determine if the previously built M k is valid or not. This condition is examined when the environment is dynamically changing. As of now, the presented algorithm is applicable not to the environment with constantly moving obstacles but to the quasi-static environment when some of the obstacles get displaced and change their locations. For example, the location of the chair can be intentionally changed in the laboratory and the algorithm notices that the chairs have been moved by testing the conditions and update the map by deleting the previous stochastic region of the moved chairs. As shown in Fig. 5, this condition can be checked by comparing the distance l o from 4

5 Figure 5: The case condition 3 is met (x c ; y c ) to (m xi ; m yi ) and the distance l m from (x c ; y c ) to (m xk ; m yk ). Also, the slope interval which is determined from the lines joining the center position of the mobile robot (x c ; y c ) and the start and end position of each region is required. The start position (x si ; y ei ) and the end position (x ei ; y ei ) of R i can be determined as follows : Figure 6: The mobile robot systemares-ii (x si ; y ei ) = (m xi? 1i cos 1i ; m yi? 1i sin 1i ); (x ei ; y ei ) = (m xi + 1i cos 1i ; m yi + 1i sin 1i ) Then, the slope interval [ si ; ei ] for R i are determined as : [ si ; ei ] = [tan?1 y si? y c x si? x c ; tan?1 y ei? y c x ei? x c ] Similarly the start position (x sk ; y sk ) and the end position (x ek ; y ek ) come from (x sk ; y sk ) = (m xk? 1k cos 1k ; m yk? 1k sin 1k ) (x ek ; y ek ) = (m xk + 1k cos 1k ; m yk + 1k sin 1k ) and the slope interval [ sk ; ek ] for M k are determined as : [ sk ; ek ] = [tan?1 y sk? y c x sk? x c ; tan?1 y ek? y c x ek? x c ] When the two intervals [ si ; ei ] and [ si ; ei ] are compared, and if any of these two intervals fully cover the other one and l m < l o, then the obstacle region M k is not valid. That is, because l m < l o, the obstacle region R i can not be detected if M k does not moved. So, M k is deleted from the map. Finally, if R i does not satisfy any of the above conditions for any M k, then R i is the newly detected obstacle region and is included in the map. 4 Experimental Result Fig. 6 shows our mobile robot system ARES-II equipped with a D-laser scanner mounted on the mobile platform. The width of the ARES-II is about 36 Figure 7: The picture of part of our laboratory cm and the length is about 60 cm. We test the presented algorithm while the mobile robot moves around in our laboratory. Fig. 7 shows part of our laboratory consisting of many chairs, tables and other hardware parts. Fig. 8 shows the initial scanning data and the corresponding stochastic obstacle regions. Here, each stochastic obstacle regions are represented with a line or an ellipse just to simplify the display. At the rst sampling time, there does not exist any matching or update procedure and the clustered regions R i 's are simply the stochastic obstacle regions M k 's. Fig. 9 shows the result after the 10-th sampling time. As shown in Fig. 9, the stochastic obstacle regions M k 's are updated or a new region is assigned as the navigation proceeds. Fig. 10 shows the nal stochastic obstacle regions after navigation is completed. As shown in Fig. 10, the proposed algorithm is able to classify the legs of the tables and the chairs. The nal map consists of total 41 stochastic obstacle regions for this experiment. About 1K bytes memory is required to store the information of about 50cm x 350cm work area. Thus, the memory requirement is not at all severe. If the cell based map making techniques are employed, then about 1.8K bytes of memory was required when each cell is stored in integer type and cell size is about 10cm x 10cm. Also, the presented method is simple and appropriate for real time computation. In reality, one full procedure of map matching and updating took about 00 msec. If we use the ultrasonic sensors[8], then the classica- 5

6 Figure 9: The raw position data and the stochastic obstacle regions after the 10-th sampling time Figure 8: The raw position data and the stochastic obstacle regions for the rst sampling time tion of the legs or the chairs is very dicult, because of poor resolution of the ultrasonic sensors. But the laser scanner guarantees a set of high quality obstacle data each sampling time. The LADAR -D laser scanner is able to provide 8 scan data set every second with 0:6 resolution but we used collected 5 scan data set every second with 1 resolution. 5 Conclusion A stochastic environment modelling technique is presented for the mobile robot navigation. The presented algorithm represent the obstacle as one of the stochastic obstacle regions that are consructed using the stochastic parameters. The algorithm is exible and ecient in representing the various shaped obstacles. The presented algorithm is simple and applicable to the real time navigation under quasi-static environment. Integrating the constantly moving obstacles into the presented algorithm remains as a future work. Acknowledgment Financial support from Korea Science and Engineering Foundation through the Automation Research Center at POSTECH is greatefully acknowledged. Figure 10: The nal result of the presented algorithm References [1] J.L Crowley,\World modelling and position estimation for a mobile robot using ultra sonic ranging,"proceedings of the IEEE International Conference on Robotics and Automation, pp , May 1989 [] John J. Leonard, Hugh F. Durrant-Whyte, and Ingemar J. Cox,\Dynamic Map Building for an Autonomous Mobile Robot,"The International Journal of Robotics Research,Vol.11,No.4,pp.86-98,Aug. 199 [3] R.M. Taylor and P.J. Probert,\Range Finding and Feature Extraction by Segmentation of Images for Mobile Robot Navigation", Proceedings of the IEEE International Conference on Robotics and Automation,pp95-100,April,1996 [4] J. Vandorpe, H. Van Brussel, and H.Xu, \Exact Dynamic Map Building for a Mobile Robot using Geometrical Primitives Produced by a D Range 6

7 Finder," Proceedings of the IEEE International Conference on Robotics and Automation,pp ,April,1996 [5] A.Elfes,\Occupancy grids: A stochastic spatial representation for active robot perception,"autonomous Mobile Robot, IEEE Computer Society Press,pp ,1991 [6] A.Elfes,\Sonar based real mapping and navigationsystem," Proceedings of the IEEE International Conference on Robotics and Automation, pp ,1986 [7] J.Borenstein and Y.Koren,\The vector eld histogram fast obstacle avoidance for mobile robot," IEEE Transaction on Robotics and Automation, Vol.7,No3,pp.78-88,Jun.1991 [8] Young. D. Kwon and Jin.S. Lee, \An Obstacle Avoidance Algorithm For Mobile Robot:The Inproved Weighted Safety Vector Field Method," Proceedings of the IEEE 10th International Symposium on Intelligent Control, pp ,Aug 1995 [9] Rafael C. Gonzalez, Richard E. Woods, \ Digital Image Processing" [10] \LADAR D Controller Software Programmers Manual", IBEO Lasertechnik,199 7