On the Heuristics of A or A Algorithm in ITS and Robot Path-Planning

Transcription

1 On the Heuristics of A or A Algorithm in ITS and Robot Path-Planning Takayuki Goto, Takeshi Kosaka, and Hiroshi Noborio Division of Information and Computer Science, Graduate School of Engineering Osaka Electro-Communication University Hatsu-Cho 18-8, Neyagawa, Osaka , Japan nobori@noblab.osakac.ac.jp., Abstract Based on many large data set, we calibrate a nearoptimal heuristic as a constant function for guiding efficiently A or A algorithm. We claim that this magic number exists as a universal constant for several kinds of date sets. This is perhaps related to the nature of their data sets though it is not theoretically analyzed yet. In general, A and Dijkstra algorithms always pick up the optimal (e.g., the shortest) path between two nodes on a given graph. However, because they do not use any good heuristics, they spend much time to calculate. To overcome this drawback, we propose A or A algorithm with a smart heuristics. The algorithm quickly investigates an optimal or near-optimal path. Unfortunately, the heuristics does not always maintain the admissibility, and thus our algorithm does not sometimes pick up the optimal path. Finally, we ascertain superiority of the proposed algorithm to the classic A and Dijkstra algorithms by two kinds of extremely different data sets. 1 Introduction The network search is a classic problem, but it is still meaningful in ITS (Intelligent Transportation System) or a robot path planning. In ITS, we want to pick up the optimal (shortest) route between two locations in a huge road map. As contrasted with this, we need the optimal sequence of collision-free motions between two states of robotic manipulator in assembly and task planning. In these cases, size of search space is to be enormous. Therefore, we need an efficient algorithm to pick up the optimal or a near-optimal route until a destination quickly. Many types of graph search algorithms such as Dijk stra (A with no heuristics [1]) and A algorithms have been theoretically analyzed [2],[3]. Especially, several kinds of roles of heuristics have been deeply and mathematically analyzed. As contrasted with this, a lot of cheaper car navigation systems have been recently floating all over the world. Some systems adopt the algorithm Dijkstra or A in order to select the optimal path between two positions quickly [4]. Needless to say, all software in such commercial systems are closed, but we easily suppose the important terms are what kinds of graphs we construct and how are we search the graph. Since heuristics of Dijkstra and A are too conservative, they spend much calculation time in spite of selection of the optimal path. In this paper, we will investigate an adventurous heuristics that selects an optimal or near-optimal path in little calculation time. If a heuristics (an estimated cost from a present node to a target node) underestimates the optimal cost (shortest distance) along the optimal (shortest) path between two nodes (positions), it leads A and consequently selects the optimal path. Otherwise, it unfortunately leads A and consequently picks up a longer path. A algorithm equals to A algorithm except the quality of heuristics. Even though the heuristics overestimates the optimal cost (smallest distance) a little, we obtain a slightly longer path whose difference against the optimal path is bounded by a difference between near-optimal and the optimal costs (longer and the shortest distances) in heuristics. In addition to this, such A algorithm completely saves calculation time because its calculation cost also depends on the difference between near-optimal and the optimal costs in heuristics [2],[3]. On the observation, we calibrate an adventurous heuristics from many kinds of data, e.g., road maps for city and country sides, and configuration space of a robotic manipulator before searching. First of all, using Euclidean distance d(n) between an arbitrary node n and the goal node, we calibrate a nearoptimal heuristics evaluation function ˆd(n) byr(d)

2 d(n). In other words, we investigate an optimal ratio r(d) = ((Euclidean distance along the optimal path from one present node n of all start nodes to the goal node) / (Euclidean distance of straight segment between them)). For six quite different path-planning problems, the optimal ratios r(d) are calculated as log d , log d , log d , log d , log d , log d Observing many experimental results, we understand that r(d) gradually increases as d(n) increases such as r(d)=0.01log d+1.2. In general, the longer the distance between two positions is, the larger the difference between their straight Euclidean distance and the distance of their optimal route is. Thus far, we are believing the ratio r(d) is individually obtained for each path-planning problem. In this paper, we show this is not true, that is, a similar optimal ratio exists for many kinds of different path-planning problems, i.e., car navigation, motion planning of a robotic manipulator and so on. Secondly, we propose an algorithm A with a better heuristics h(d) = r(d) d(n) (d(n): Euclidean distance of straight segment between a present node n and the goal node). In this algorithm, we always select a nearoptimal path by small calculation time based on the near-optimal heuristics r(d) = 0.01 log d+1.2. Finally, in order to check superiority of the proposed algorithm against classic A and Dijkstra algorithms, we compare the former algorithm with the latter algorithms. As a result, calculation time is 0.23 times smaller but the driving distance is only 1.06 times longer on the average of six different problems. In addition, distributions of these results (time and distance) are fortunately too small for different problems such as car navigation in road maps of city and country sides, motionplanning for digital configuration space of a robotic manipulator. Nevertheless, we can find a magic number is universally used as a heuristics in many kinds of data for path-planning. As related papers, we focus on two. One is to use a randomized path planner [5],[6]. This can select a near-optimal or a slightly longer path from a huge search space. The planner is probabilistically resolution complete, namely, if the running time is not limited, the probability of finding a path (if there exists one) approaches 1.0. However, in practice, we do not want the planner to run indefinitely, and the planner is terminated after certain prescribed time bound. Experiments with various kinds of robots have been conducted, and the results show that, in most cases, this planner is very efficient in finding a path in high dimensional configuration space. Another is D algorithm [7],[9],[8],[10]. It starts with a map containing all known, assumed, and estimated mobility cost data for the environment. The map is an eight-connected two-dimensional grid. Using this map, an initial path is planned from the rover s starting location to the goal. As the robot drives the path, its sensors discover discrepancies between the initial map and environment. The map is updated, and the robot path is planned from its current location to the goal again. This process repeats until the robot reaches the goal or discovers that it cannot. The D algorithm is a dynamic version of the A algorithm. For large environments, it can plan paths many times faster than A. In section 2, we describe six kinds of graphs. The former two graphs are converted from road maps for a car and the latter four graphs are constructed from digital configuration space for a robotic manipulator. In section 3, we explain our algorithm between A and A algorithms. In section 4, we ascertain the superiority of our new algorithm against the classic algorithms Dijkstra, A and A. First of all, we investigate how much the distance of shortest path between two arbitrary nodes is longer than the distance of straight segment between the nodes. As a result, we can find a near-optimal heuristics as a magic number h(d) =d (0.01 log d +1.2) for all kinds of pathplanning problems. Secondly, we compare a new algorithm A using h(d) =d (0.01 log d +1.2) with A using h(d) = d. Then, we describe how much faster our proposed algorithm is and how much longer the path selected by our proposed algorithm is. Finally in section 5, we give a short concluding remark. 2 Seven Kinds of Huge Graphs In this section, we prepare six types of practical graphs whose sizes are too large. The graphs are classified into two categories: One is a graph converted from a road map. Another is a digital C-space (configuration space) of a three-degrees-of-freedom robotic manipulator. 2.1 Three kinds of Road Graphs In this paragraph, we explain how to construct a road graph. Firstly, a set of roadmaps in Japan is revised and published per year by CD-ROM from the geographical survey institute in ministry of land, infrastructure, and transport. In CD-ROM, a lot of relationships concerning to roads and crosses are stored, and therefore a road graph can be constructed from the database. In this research, we prepare two types of graphs converted from city and country areas, that is, Osaka, Fukuoka and Aomori prefectures in figures

3 1, 2 and 3, respectively. Therefore, these road graphs include many geometrical and topological differences. In each road map, there are many kinds of roads, for example, an (expensive) highway, roads constructed by nation, prefecture, city and so on. In these road graphs, a highway is colored by green, a national road is colored by red, a major prefecture road is colored by pink, a minor prefecture road is colored by yellow, a city road is colored by blue, and a narrow road is colored by black. In general, all crosses are expressed as nodes in a graph, and all roads are represented as arcs in the graph. Then, a cost of an arc is defined by a driving distance or time and so on. This always depends on a quality of a road, for example, a driving time on a wide road (especially a highway) is smaller than that on a narrow road. The weights are experimentally determined as follows: (highway):(national road):(main prefecture road):(prefecture road):(city road):(narrow road)=1 : 1.56 : 1.4 : 1.75 : 2.33 : 3.5 in our research. 2.2 Two kinds of Digital Configuration Space In this paragraph, we explain how to build a 2-D or 3-D digital configuration space. This has a flexibility for representing free shaped objects with three or more dimensions [11]. Therefore, it is suitable for pathplanning a three or more degrees-of-freedom robotic manipulator. (b) Figure 3 : (a) A set of cities at Aomori prefecture in CD-ROM published from a Japan government (the geographical survey institute), (b) its road map retrieved from the CD-ROM, and (c) its road graph converted from the road map. The goal is described as a red circle. In this paper, we define a digital configuration space C space as a collection of cells. First of all, the space C space with m dimensions is built in a joint space [θ 1,..., θ m ]. All the cells are built by dividing an available range of each degree into an arbitrary resolution: (θ 1,..., θ m )/θ 1 [θ 1,θ 1 ],..., θ m [θ m,θ m ]. The differences θ i θ i (i =1,..., m) are available ranges, and they are divided by 2 l times. l is given by a human operator, and the arbitrary resolution is determined by θ i θ i /2l (i =1,..., m). According to the division, a digital configuration space is built by 2 lm cells (nodes). For example, 2-D and 3-D digital maps are represented by arrangements [2 l, 2 l ] and [2 l, 2 l, 2 l ], respectively. (a) (c) Figure 1 : (a) A set of cities at Osaka prefecture in CD-ROM published from a Japan government (the geographical survey institute), (b),(c) its road maps retrieved from the CD-ROM, (d) its road graph converted from the road map. (a) (c) Figure 2 : (a) A set of cities at Fukuoka prefecture in CD-ROM published from a Japan government (the geographical survey institute), (b) its road map retrieved from the CD-ROM and (c) its road graph converted from the road map. The goal is colored as a red circle. (b) (a) Figure 4 : (a) A target robotic manipulator colored by green in a sparse 3-D environment with static obstacles. Another target tip position is colored by red. (b) Its 3-D configuration space C1. A purpose for using a digital map is to express three or more dimensional configuration space. First of all, if a robotic manipulator collides with its obstacles in a 3-D Cartesian space, a corresponding cell joins a set of Obs cells in a configuration space (C-space). Otherwise, the corresponding cell joins a set of Free cells in the C-space. Its dimension corresponds to the number of manipulator joints. Secondly, a robot is expressed by a cell (node), and is able to move one of neighbor cells (nodes) in a digi- (b)

4 (a) Figure 5 : (a) A target robotic manipulator in a dense 3-D environment with static obstacles, which is colored by green. Another target tip position of the manipulator is colored by red. (b) Its 3-D configuration space C2. tal C-space. If its dimension is m, 2 m 1 cells (nodes) are prepared for neighbors, respectively. Two neighbor cells (nodes) are implicitly connected by an edge. Furthermore, m different L 2 metrics between neighbor cells exist. For example, the L 2 metric between horizontal and vertical neighbor cells is defined by one, and the L 2 metric between diagonal neighbors is defined by a square route two ( 2). A total distance between start and goal cells (nodes n s and n g ) is calculated by summing these neighbor distances. Figure 6 : The ratio r(d) is optimally functioned by logd on the average for the road graph of Fukuoka Prefecture (d(n): Euclidean distance of straight segment between a present node n and the goal node). The black (dark gray) area means many nodes are located, and the yellow (light gray) means few nodes are located. The optimal function is denoted by blue (black) line. For this reason, the density of positions is represented by the color (gray) scale. In this research, we prepare four types of digital configuration space that a three-degrees-of-freedom manipulator works in two kinds of environment with a lot of static obstacles (Figures 4 and 5). Therefore, both graphs include many geometrical and topological differences. 3 Classic A-star or A Algorithm In this section, we explain the classic path-planning algorithm A or A. If the heuristics is admissible, this is to be A and it picks up the optimal path from a start node n s to a target node n g. Otherwise, it becomes A and consequently selects a longer path between the nodes. In A, a present node n p is numerically estimated by summing a past cost g(n p ) and a future cost (b) Figure 7 : The ratio r(d) is optimally functioned by log d on the average for the road graph of Aomori Prefecture (d(n): Euclidean distance of straight segment between a present node n and the goal node). Figure 8 : The ratio r(d) is optimally functioned by log d on the average in a digital configuration space C1 when a three-degrees-of-freedom robotic manipulator aims at a red final tip point in a sparse Euclidean space (d(n): Euclidean distance of straight segment between a present node n and the goal node). h(n p ) as the sum f(n p ). The past cost g(n p ) is calculated by summing a cost c(n l 1,n l ) of each edge along the selected optimal route from n s to n p, whose endpoints are defined by nodes n l 1 and n l. On the other hand, the future cost h(n p ) is defined as heuristic evaluation function. The heuristics is calculated as r(d) h(n p ). (h(n p ): heuristics of n p, e.g., Euclidean distance of straight segment between n p and n g, r(d): an arbitrary function of heuristics). In A, the node selected by expansion is the one having the lowest h(), and when two paths lead to the same node, the one with the higher h() is discarded. As a result, A always gets an optimal (shortest) path between n s and n p. Figure 9 : The ratio r(d) is optimally functioned by log d on the average in a digital configuration space C1 when a three-degrees-of-freedom robotic manipulator aims at a green final state in a sparse Euclidean space (d(n): Euclidean distance of straight segment between a present node n and the goal node). 1. Put a start node n s on OPEN. 2. If OPEN is empty, exit with failure. 3. Remove from OPEN and place on CLOSED

5 Figure 10 : The ratio r(d) is optimally functioned by log d on the average in another digital configuration space C2 when a three-degrees-of-freedom robotic manipulator aims at a red final tip point in a dense Euclidean space (d(n): Euclidean distance of straight segment between a present node n and the goal node). Figure 12 : Concerning to calculation time, we obtain a ratio set A with the normal heuristics d(n)) in the road graph of Fukuoka Prefecture. The black (dark gray) area means many nodes are located, and the yellow (light gray) means few nodes are located. For this reason, the density of nodes is represented by the color (gray) scale. The total average ratio is Figure 11 : The ratio r(d) is optimally functioned by log d on the average in another digital configuration space C2 when a three-degrees-of-freedom robotic manipulator aims at a green final state in a dense Euclidean space (d(n): Euclidean distance of straight segment between a present node and the goal node). anoden p which f(n p ) is minimum (break ties arbitrary). 4. If n p is a goal node n g, exit successfully with a shortest (optimal) path obtained by tracing back the pointers from n p to n s. 5. Otherwise, expand n p for selecting all Free and unvisited nodes n i neighboring n p, and attach to them pointers back to n p. For every neighbor node n i, (5a) If n i was not already on OPEN or CLOSED, estimate h(n i ) (a lower bound of the shortest route between n i and n g ), and calculate f(n i )=g(n i )+ h(n i ) where g(n i )=g(n p )+c(n p,n i ) and g(n s ) =0. (5b) If n i was already on OPEN or CLOSED, direct its pointer along the path yielding the lowest g(n i ). (5c) If n i required pointer adjustment and was found on CLOSED, remove from CLOSED and place on OPEN anoden i. 6. Go to step 2. Whenever h(n p )isanoptimistic estimate of h (n p ), the algorithm A always returns the optimal (shortest) path. h (n p ) is the length of optimal route between n p and n g, and h(n p ) is an underestimated length of the route. This motivates naming this class of estimates: Admissible heuristics [3]. 4 Simulation Results In this section, we ascertain that our proposed A with a better heuristics selects a near-optimal path ef- Figure 13 : Concerning to calculation time, we obtain a ratio set A with the normal heuristics d(n)) in the road graph of Aomori Prefecture. The total average ratio is Figure 14 : Concerning to calculation time, we obtain a ratio set A with the normal heuristics d(n)) for a red robotic manipulator in a digital configuration space C1. The black (dark gray) area means many nodes are located, and the yellow (light gray) means few nodes are located. For this reason, the density of nodes is represented by the color (gray) scale. The total average ratio is Figure 15 : Concerning to calculation time, we obtain a ratio set A with the normal heuristics d(n)) for a green robotic manipulator in a digital configuration space C1. The total average ratio is

6 ficiently in many kinds of large graphs. From these experimental results, we understand the proposed algorithm is used for ITS and robot path planning. 4.1 Comparison between A-star and A In this paragraph, by changing the heuristics from zero times to three times against the distance of straight segment between two nodes (positions), we seek for a wonderful ratio function r(d). If the ratio r(d) is zero, the algorithm becomes the Dijkstra s algorithm, and also if the ratio r(d) is one, the algorithm is regarded as the classic A algorithm. For this purpose, we investigate our algorithm s performance when r(d) is gradually changed between one and three in a road graph whose area is Osaka prefecture (Fig.1). Figure 16 : Concerning to calculation time, we obtain a ratio set A with the normal heuristics d(n)) for a red robotic manipulator in a digital configuration space C2. The black (dark gray) area means many nodes are located, and the yellow (light gray) means few nodes are located. For this reason, the node density is represented by the color (gray) scale. The total average ratio is otherwise, it becomes A. In addition, if the heuristic is zero, the algorithm becomes Dijkstra s algorithm. The A algorithm (underestimates the optimal cost as heuristics) picks up the optimal path from an arbitrary node to a target node. As contrasted with this, the A algorithm (overestimates the optimal cost as heuristics) selects a longer path between the nodes. In general, calculation cost of our algorithm (switching between A and A) is strongly depending on the heuristics. To save the calculation cost, we should estimate an uncertain cost along the optimal path between two nodes (positions). In each of many experimental results, we calculate all ratios ((the cost on the optimal path from an arbitrary node to a target node)/(the cost of straight segment between two nodes)) for all pairs of two nodes in order to obtain a near-optimal ratio function r(d). The cost of straight segment between two nodes is defined in a road graph or a digital C-space as follows: In the road map, we consider the straight segment between present and target positions (nodes n p and n g ), and define the cost as a driving time if the segment is constructed by the highway. On the other hand, concerning to the C- space, we define the cost as the L 2 metric between present and target positions (nodes n p and n g ) Figure 17 : Concerning to calculation time, we obtain a ratio set A with the normal heuristics d(n)) for a green robotic manipulator in a digital configuration space C2. The total average ratio is In conclusion, as illustrated in Table 1, the algorithm with r(d) = selects a good nearoptimal path efficiently. In conclusion, we select r(d) between 1.2 and A Near-Optimal Heuristics Evaluated from Six Kinds of Graphs In general, if the heuristic is admissible (if an estimated cost from a present node to a target node is smaller than or equals to the true cost on the optimal path between the nodes), our algorithm becomes A, Figure 18 : Concerning to path length, we obtain a ratio set for all the better heuristics d(n) (0.01 log d(n)+1.2))/(that of A with the normal heuristics d(n)) in the road graph of Fukuoka Prefecture. The dark gray area means many nodes are located, and the light gray means few nodes are located. For this reason, the density of nodes is represented by color (gray) scale. The total average ratio is Here, h(n) = d(n p ) denotes the cost on straight segment between two nodes n p and n g. As contrasted with this, ĥ(n) denotes the sum of costs on the optimal path between the nodes n p and n g. In this problem, we want to know how much time ĥ(n) is larger than h(n) =d(n p ). If we estimate a ratio r(d), our algorithm becomes faster to find an optimal or near-optimal path until a target node by defining the heuristics as r(d) h(n). After investigating r(d) for all node pairs in many types of graphs, we can see

7 Table 1 : If r(d) is given as a constant value, we obtain several results of A or A algorithm in a road graph whose arc costs are constant. The r(d) ratio Small graph Large graph Calculation Cost sum on Number of Node num- Calculation Cost sum on Number of Node numtime [sec] a selected expanded ber on a selected time [sec] a selected expanded path nodes path path nodes ber on a selected path Figure 19 : Concerning to path length, we obtain a ratio set for all the better heuristics d(n) (0.01 log d(n)+1.2))/(that of A with the normal heuristics d(n)) in the road graph of Aomori Prefecture. The total average ratio is that a near optimal ratio r(d) against the distance d is almost the same in all kinds of search space (Fig.6, 7, 8, 9, 10 and 11). Each ratio r(d) is calculated by ((the cost along the optimal route between two positions)/(the cost on straight segment between two positions)). Here, d changes from one to zero by ((the distance of straight segment between a destination and an arbitrary position)/(the maximum distance on straight segment between a destination and its farthest position)). 9, 10 and 11, we can investigate the optimal heuristics on the average, whose function is denoted by the distance of straight segment between start and goal positions. In this paper, we approximate the average function as a logd + b, and therefore we select the best pair of a and b to minimize the sum of square differences from estimated values a logd + b to all true values on every distance d. Excel software automatically selects this function a logd+ b, that is, this optimal pair of coefficients a and b. As a result, we can get the optimal heuristic functions log d , log d , log d , log d , log d , log d , respectively. They are similar and consequently can be summarized by a near-optimal function like as 0.01log d+1.2. After this, we should describe that A with heuristics 0.01log d+1.2 picks up a near-optimal path very fast. Figure 20 : Concerning to path length, we obtain a ratio set for all the better heuristics d(n) (0.01 log d(n)+1.2))/(that of A with the normal heuristics d(n)) for a red robotic manipulator in a digital configuration space C1. The total average ratio is From these data described in the figures 6, 7, 8, Figure 21 : Concerning to path length, we obtain a ratio set for all the better heuristics d(n) (0.01 log d(n)+1.2))/(that of A with the normal heuristics d(n)) for a green robotic manipulator in a digital configuration space C1. The total average ratio is First of all, we calculate a comparative ratio of calculation time by ((calculation time of A with d (0.01 log d +1.2))/(that of A with d)) for six types of large graphs (Fig.12, 13, 14, 15, 16 and 17). As shown in six kinds of different experimental results, all ratios,

8 i.e., , , , , , are located around 0.2 (at most smaller than 0.5). As a result, we can see that the proposed algorithm is 5 times faster than A in different kinds of enormous graphs. Especially, this characteristic is stably ascertained in two completely different problems on road map for a car and configuration space for a robotic manipulator. For this reason, we believe that the superiority of proposed heuristics can be universally used for the other problems. Figure 22 : Concerning to path length, we obtain a ratio set for all the better heuristics d(n) (0.01 log d(n)+1.2))/(that of A with the normal heuristics d(n)) for a red robotic manipulator in a digital configuration space C2. The total average ratio is Figure 23 : Concerning to path length, we obtain a ratio set for all the better heuristics d(n) (0.01 log d(n)+1.2))/(that of A with the normal heuristics d(n)) for a green robotic manipulator in a digital configuration space C2. The total average ratio is Secondly, we calculate a comparative ratio of path length by ((path length of A with d (0.01 log d + 1.2))/(that of A with d)) for six types of large graphs (Fig.18, 19, 20, 21, 22 and 23). As shown in six different results, all ratios, i.e., , , , , , are located around 1.0 (at most smaller than 1.25). As a result, we can see that a near-optimal path selected in the proposed algorithm is quite similar to the optimal path selected in A in enormous graphs. Especially, this characteristic is stably ascertained in two completely different problems on road map for a car and configuration space for a robotic manipulator. For this reason, we believe that the superiority of proposed heuristics can be universally used for the other problems. Overall, almost all paths selected by our proposed algorithm under an excellent heuristics are nearly optimal although speed of the algorithm is five times faster. The longer the Euclidean distance between two places is, the larger a difference between distances of the straight path and the shortest path between two places on a given graph is. The smart heuristics corrects the difference, and consequently our algorithm including the heuristics picks up an optimal or nearoptimal path by short calculation time. Moreover, this characteristic is universally used for many pathplanning tasks. Finally, all programs are running in Windows2000 OS (Japanese Edition) in PC (CPU Pentium4 2.2GHz, Memory 1GB). 5 Conclusions In this paper, we propose an efficient path-planning algorithm for selecting the optimal or a near-optimal path from a huge graph. The basic idea is to seek for a near-optimal heuristics in advance. In almost all the different graphs, the optimal function r(d) isto be around In addition, if a given heuristic is not admissible, our algorithm becomes A and cannot select the optimal path. Even in this case, since the heuristics approximates the optimal cost until a target point, the fast A algorithm picks up a nearoptimal path. As a result, the path length is not so bad. Therefore in a practical use, we save calculation time extremely but the path quality is invariant in our proposed algorithm. References [1] E.W.Dijkstra, A Note on Two Problems in Connecting with Graphs, Numerische Mathematik, vol.1, pp , [2] N.J.Nilsson, Principles of Artificial Intelligence Tioga Publishing Co., [3] J.Pearl, Heuristics - Intelligent Search Strategies for Computer Problem Solving, Addison-Wesley Series in Artificial Intelligence, [4] S.Nishimura, S.Tsuda, S.Ohashi and K.Kagawa, Navigation System, J. of the Robotics Society of Japan, Vol.17, No.3, pp , 1999 (in Japanese). [5] L.Kavraki, M.Kolountzakis, and J.-C. Latombe, Analysis of Probabilistic Rooadmaps for Path Planning, Proc. of the Int. Conf. on Robotics and Automation, pp , [6] L.E.Kavraki, M.N.Kolountzakis, and J.C. Latombe, Analysis of Probabilistic Roadmaps for Path Planning, IEEE Trans. on Robotics and Automation, Vol.14, No.1, pp , [7] A.Stentz, Optimal and Efficient Path Planning for Partially- Known Environments, Proc. of the Int. Conf. on Robotics and Automation, pp , [8] A.Stentz, The focussed D algorithm for real-time replanning, Proc. of the 14th Int. Joint Conf. on Artificial Intelligence, pp , [9] A.Stentz and M.Hebert, A complete navigation system for goal acquisition in unknown environments, J. of Autonomous Robots, Vol.2, No.2, August [10] S.Singh, R.Simmons, T.Smith, A.Stentz, V.Verma, A.Yahja and K.Schwehr, Recent Progress in Local and Global Travers ability for Planetary Rovers, Proc. of the Int. Conf. on Robotics and Automation, pp , [11] J.C.Latombe, Robot motion planning, Kluwer Academic Publishers., 1991.