232 Voronoi diagram referred in this work is the VD generated for sets of points, where a generator is defined as a set of points. In this case, the re- gion segmentation based on the minimum Euclidean distance computation between a location and a set of points is called Generalized Voronoi Dia- gram (GVD). Definition 5. Generalized Voronoi iagram Given 12 {}  n Ggg g a collection of n point series in the plane which do not overlap 2 12  i gi n (6.19)  z ij gg ij (6.20) for every point 2 p , the minimum Euclidean distance from p to an- other point belonging to the series i g is called () i p g E d . ( ) min( ( ) )  iEiii p gdpppg E d (6.21) The Voronoi region will be defined as: 2 (){ ( ) ( ) } _  z iij Vg p p R pg pg j i EE dd (6.22) and the given sequence: 12 {( ) ( ) ( )}  n VVgVg Vg (6.23) will be the generalized Voronoi diagram generated by G . From now on, the GVD term is used when the generators are series of points instead of isolated points. Those series of points will be defined as generator group. The algorithm we present in this work is based on the GVD and is im- plemented in a sensor based way because it is defined in terms of a metric function ( () i p g E d ), that measure the Euclidean distance to the closest object ( i g ), represented as a set of points supplied by a sensor system. C. Castejón et al. D Voronoi-Based Outdoor Traversable Region Modelling 233 Supposedly the robot is modelled as a point operating in a subset belong- ing to the two-dimensional Euclidean space (in our particular case, al- though this is true with n-dimension too). The space W which is called Workspace, is obstacle populated 12 {}  n CC C that will be considered as a close set. The set of points where the robot can manoeuvre freely will be called free space and is defined in [5] as: 1 ½ ®¾ ¯¿  in i i FS W \ C (6.24) The workspace W is represented as a two-dimensional binary im- age (, ) B ij, where each position (, )ij has assigned a field value (0 or 1), that indicates if a pixel belong to a generator (field 0) o not (field 1). For each point belonged to the free space (, )ij FS is, at least, one point closest to the occupied space FS that will be called base point [50]. The LVD is obtained from the distances to the generated points which belong to the objects’ borders. To obtain the local model, the algorithm is executed in the three steps presented in next paragraphs. 1. Clustering Data representing borders between traversable and non-traversable regions are grouped in clusters. The cluster determination, in the three-dimensional information case, is not trivial. In other works, the sorted two-dimensional information, from a scanner laser, is easily separated in clusters, using the distances between successive scanned points [6], [27]. In this approach, the 3D information cannot be treated as other authors do, and a labelling tech- nique is used to obtain the clusters. The kernel applied is a circumference. The radius is the robot’s size. If there is a distance greater than the robot’s size between two points which belong to the non-traversable region border, then the robot can cross between them, and the points will be considered as belonging to different clusters. In figure 6.41 the data grouping of the real environment presented in 38 is presented. The result of this step is three generators called A, B and C. 6 6.4.3 TRM Algorithm 234 Fig. 6.41. Data grouping to obtain generators For the VD construction, the assignment rule is the Euclidean distance to generators. Generators are the clusters. For each free space cell (, )ij FS , we mean whose cells where the robot can evolve, the dis- tance between the cell and each point, which belongs to each clusters, is calculated. The minimum Euclidean distance between the cell and the points to one cluster will be considered to assign the cell to the correspond- ing Voronoi region. The label cell is evaluated, based on the minimum distance, as follows (see figure 6.42):  If the distance to an A cluster is less than the rest, the cell is evaluated as belonging to the Voronoi region associated to the cluster A.  If there are two equidistant clusters in a cell (for example A and B), then the cell is labelled as Voronoi edge.  For bigger equidistant it will be labelled as Voronoi node. C. Castejón et al. 2. Distance to the Generator Elements Computing 3. LVD Algorithm Voronoi-Based Outdoor Traversable Region Modelling 235 Fig. 6.42. Label cell evaluation The labelled of each cell belonged to the FS is carried out computing the distance between the centre of the cells. Nevertheless, the real meas- urement can be located in any position inside the cell. This must be taken into account when the distance to the generators is calculated. The maxi- mum discretized error is calculated, based on figure 6.43: 22 max 2 1 2 1 ()()dxxyy cc cc  (6.25) Fig. 6.43. Maximum distance between two cells In the same figure 43 can be seen that: 6 Error Caused by Discretizing 236 11 22 11 22 2 2 2 2 c c c c R xx R xx R yy R yy c  c  c  c  (6.26) and replacing 6.26 in 6.25 we obtain: 22 max 2 1 2 1 ()() cc dxxRyyR  (6.27) Therefore, max d is the maximum distance that we must consider when the labelling step is performed. We mean, the maximum error E when two cells are evaluated is: max Ed d  (6.28) And the LVD generation step is modified as follows: For each cell (, )ij belonging to the free space: 1. The distance to all the points belonged to each generator is computed. 2. The minimum distance to each generator is obtained. We consider, for example in figure 6.42, the distance difference between two generator A and B as (,) ( ,) ( ,) cAB EAB g p g p  EE dd, where p is the centre of the cell (, )ij with coordinates (, ) x y , then:  If (,) 2 c EAB E!u for all ABz , the cell is considered as be- longed to the Voronoi region associated to the object A .  If (,) 2 c EAB Eu and (,)2 c EHB E!u for all HABzz, the cell is considered as Voronoi edge between two objects A and B .  For bigger equidistant the cell will be considered as Voronoi node. In figure 6.44 the result of the DVL computing from figure 6.41 is shown. In the figure the Voronoi edges and nodes are highlighted. C. Castejón et al. Voronoi-Based Outdoor Traversable Region Modelling 237 Fig. 6.44. DVL representation A priori, the cell size has influence on time computing and on the model accuracy. To test the influence, two cell sizes (suitable for the environ- ment, the robot size and the robot speed) have been chosen. The cell sizes are 20 cm and 50 cm. The experimental results have been carried out in an environment presented in figure 6.45 and the TNM in figures 6.46 and 6.47. Fig. 6.45. Environment with high obstacles density 6 6.4.4 Cell Size Influence Fig. 6.46. TNM. Top view Fig. 6.47. TNM. Side view The LVD obtained are presented in figures 6.50 where a 20 cm cell resolution is implemented, and 6.51 with a 50 cm resolution. And in fig- ures 6.48 and 6.49 the visibility maps are presented to compare the cell size. Fig. 6.48. Visibility Map. Cell size of 20 cm Fig. 6.49. Visibility Map. Cell size of 50 cm C. Castejón et al.238 Fig. 6.50. DVL for 20 cm cell size Fig. 6.51. DVL for 50 cm cell size Apparently, the graphical result does not change, we mean, the LVD form has not been loosen, only the number of points have decreased. In 50 cm cell size the number of nodes decreases and, above all, we notice that the difference between the two resolutions increases with the distance to the sensor system. Of course, the processing time in steps with digital im- age processing algorithms, is reduced because of the increases in cell size. The image changes the size from 100 200u to 40 80u (see table 6.1). In table 6.1 the time computing for 50 cm cell size is reduced to the 20% of the time for the 20 cm cell size. In order to explore a large unknown environment, the robot needs to build a global model, to know where it is at each instant, and where it can go. The model can be introduced a priori in the robot or it can be built while the robot is travelling, fussing local models. The model merge consist of assembling a set of data, obtained in different acquisitions in a unique model [17]. In this article, an incremental global map is built merging the different LVD while the robot is moving. 6 Voronoi-Based Outdoor Traversable Region Modelling 239 6.5 Incremental Model Construction Tab. 6.1. Different cell size time computing 240 To obtain an incremental model, based on the robot successive percep- tions, the following steps must be carried out [38]: 1. Environment information sensing: The robot, stopped, uses its exter- nal sensors (in this case a 3D scanner laser and a compass) to acquire all the needed information and build a local model (LVD). 2. The local model (LVD), useful for navigation, is transformed in a global coordinates system, with a differential GPS, and the local model is integrated with the global current model. 3. The robot moves to the next position, by path-planning or guiding, and goes back to the step 1. The exact global model construction, based on the successive percep- tions, is in general difficult to obtain due to the sensor uncertainty. Be- sides, in the majority of the cases, it is not interesting not only for the im- precision, but also the CPU time computing and the high memory storing makes the model not interesting for real-time objectives. For the integration, a transformation in the global Cartesian system is needed. A Global Positioning System (GPS) is used to perform the trans- formation for each LVD point (, ) ii x y . The algorithm is the following: For the first local map (transformed into global reference system): 1. To obtain a seed point (, ) CM CM Px y LVD , and to set it as Centre of Mass ( CM ). 2. Calculate the Euclidean distance between all the points (, ) iii Px y LVD and theCM . If the distance is less than an error value called max r , the point i P is included in the list belonging to the CM and a new CM is calculated. If the distance is greater than max r , then a new list is formed and it is set as another CM . For the rest of the LVD maps (transformed into global reference sys- tem): 1. For each point i PLVD we try to find the CM in the global map to which i P belongs (the Euclidean distance between i P and CM is less than max r ). 2. If we do not find it a new list is formed with i CM P . C. Castejón et al. Voronoi-Based Outdoor Traversable Region Modelling 241 Equation 6.29 calculates the CM for a set of N points. The Euclid- ean distance is calculated in equation 6.30 to know if a point belongs to the CM . 11 , NN ii x y CM NN §· ¨¸ ¨¸ ©¹ ¦¦ (6.29)  22 iCM iCM rxx yy   (6.30) The max r value must be chosen considering the map cell resolution, and it always depends on the robot’s velocity movement. In figures 6.52, 6.53 and 6.54 the steps performed to build the global map are presented. Fig. 6.52. LVD Fig. 6.53. Data map clustering Fig. 6.54. Incremental Voronoi map 6 [...]... Yahja, S Singh, and A Stentz, An efficient on-line path planner for outdoor mobile robots, Robotics and autonomous systems 32 (2000), 129–143 52 D Van Zwynsvoorde, T Simeon, and R Alami, Incremental topological modeling using local voronoi-like graphs, EEE/RSJ International Conference on Intelligent Robots and Systems, vol 2, 2000, pp 897–902 7 Using Visual Features for Building and Localizing within Topological... than once, P E Rybski et al. : Using Visual Features for Building and Localizing within Topological Maps of Indoor Environments, Studies in Computational Intelligence (SCI) 8, 251–271 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com 252 P E Rybski et al there will be multiple nodes in the map for a single location These nodes will have to be identified and combined in order to generate... its position and the map structure The method is very general and does not require the environment to have uniquely distinctive features We analyze feature-based localization strategies and present experimental results in an indoor environment 7.1 Introduction We are interested in building maps of indoor environments using small robots that have limited sensing Since the robot must physically carry any... IEEE/RSJ International Conference on Intelligent Robots and Systems, 2000 26 In S Kweon and T Kanade, High resolution terrain map from multiple sensor data, IEEE International Workshop on intelligent robots and systems, 1990 27 Y.D Kwon and J.S Lee, A stochastic map building method for mobile robot using 2-d laser range finder., Autonomous Robots 7 (1999), 187–200 28 S Lacroix, I.K Jung, and A Mallet, Digital... no 6, 522–531 49 S Thrun, Learning metric-topological maps for indoor mobile robot navigation, Artificial Intelligence 99 (1998), no 1, 21–71 50 S Thrun and A Bücken, Integrating grid-based and topological maps for mobile robot navigation, Proceedings of the Thirteenth National Conference on Artificial Intelligence and the Eighth Innovative Applications of Artificial Intelligence Conference, vol 2,... the increase of new data and the different robot point of view With this algorithm, the global model can be built off-line, based on different local models or in real time, as the robot is travelling in autonomous or guiding mode 6.6 Conclusions A new methodology to model outdoor environments, based on threedimensional information and a topographical analysis, has been done The model can be built in. .. 199 7-0 296 and the DPYT project DPI 200 0-0 425 Further, we thank Angela Nombela for her assistance References 1 2 3 4 5 Howard A and Seraji H., Vision-based terrain characterization and traversability assessment, Journal of Robotic Systems 18 (2001), no 10, 577–587 S Betgé-Brezetz, R Chatila, and M Devi, Natural scene understanding for mobile robot navigation, IEEE International Conference on Robotics and. .. navigation in unknown terrains for a circular robot. , IEEE Transactions on Robotics and Automation 7 (1991), no 5, 699–707 45 H Seraji, Traversability index: a new concept for planetary rovers, Proceedings of the 1999 IEEE International Conference on Robotics & Automation, 1999 46 S Simhon and G Dudeck, A global topological map formed by local metric maps, International Conference on Intelligent robots and. .. 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Konukseven, and A Rizzi, Sensor based planning: a control law for generating the generalized voronoi graph, 8th International Conference on Advanced Robotics (ICAR’97) (New York, NY, USA), 1997, pp 333–338 H Choset, S Walker, K Eiamsa-Ard, and J Burdick, Sensor-based exploration: Incremental construction of the hierarchical generalized voronoi graph, International Journal of Robotics Research 19 (2000), no 2, . once, P.E.Rybskietal.: Using Visual Features for Building and Localizing within Topological Maps www.springerlink.com c  Springer-Verlag Berlin Heidelberg 2005 of Indoor Environments, Studies in Computational. additional information provided by the GPS that supply global meas- ures. The local model integration is easy and allows building topo- geometrical models in real time. Fig. 6.55. Real environment Fig using local voronoi-like graphs, EEE/RSJ Interna- tional Conference on Intelligent Robots and Systems, vol. 2, 2000, pp. 897–902. 6 7 Using Visual Features for Building and Localizing within Topological