UOBPRM: A Uniformly Distributed Obstacle-Based PRM

Transcription

1 UOBPRM: A Uniformly Distributed Obstacle-Based PRM Hsin-Yi (Cindy) Yeh 1, Shawna Thomas 1, David Eppstein 2 and Nancy M. Amato 1 Abstract This paper presents a new sampling method for motion planning that can generate configurations more uniformly distributed on C-obstacle surfaces than prior approaches. Here, roadmap nodes are generated from the intersections between C-obstacles and a set of uniformly distributed fixed-length segments in C-space. The results show that this new sampling method yields samples that are more uniformly distributed than previous obstacle-based methods such as OBPRM, Gaussian sampling, and Bridge test sampling. UOBPRM is shown to have nodes more uniformly distributed near C-obstacle surfaces and also requires the fewest nodes and edges to solve challenging motion planning problems with varying narrow passages. I. INTRODUCTION The motion planning problem is to find a valid trajectory to move an object (e.g., a robot) from a start position to a goal position. Solutions to this problem can be applied to various domains such as computer animation [11], computeraided design (CAD) [7], and computational biology [2], [13]. Exact algorithms are computationally infeasible for many problems, particularly when the robot has many degrees of freedom [4]. Randomized algorithms can help address this issue. One randomized motion planning method is the probabilistic roadmap method (PRM) [9],[10],[12]. It works by sampling points randomly from configuration space (Cspace, which includes all placements of the robot, feasible or not) and connecting nearby points to form a roadmap. During query processing, a path is extracted from the roadmap using standard graph search techniques. Sampling near obstacles is important since it improves the configuration coverage in difficult areas of C-space (such as narrow passages). There are several PRM variants proposed to increase sampling in important regions of C-space, particularly near C-obstacle boundaries. OBPRM [3], [1], generates configurations near C-obstacles. They found that roadmaps with denser sampling near the C-obstacles helped to map the difficult regions in C-space. In [5], a Gaussian sampling strategy was proposed that guaranteed configurations with a Gaussian distribution around C-obstacle surfaces. The method has a parameter that controls the distribution and how The work of Yeh, Thomas and Amato supported in part by NSF Grants EIA , ACR , ACR , CCR , ACI , CRI , CCF , CCF , by the DOE, Chevron, IBM, Intel, HP, and by King Abdullah University of Science and Technology (KAUST) Award KUS-C The work of Eppstein supported in part by NSF Grants and , and by the Office of Naval Research under MURI grant N H.-Y.Yeh,S.Thomas,andN.M.AmatoarewiththeParasolLaboratory, Department of Computer Science and Engineering, Texas A&M University, {hyeh, sthomas, amato}@cse.tamu.edu 2 D. Eppstein is with the Computer Science Department, University of California, Irvine, eppstein@ics.uci.edu near the configurations are to the C-obstacles. Bridge test sampling [8] bridges the gap in order to increase the success rate for generating configurations in a narrow passage. These methods have been shown to work well experimentally, but they do not always cover surfaces uniformly. The node distribution of OBPRM is affected by the shape of the C-obstacle and is not guaranteed to be uniform. Gaussian samplingtakesalongtimetogeneratenodessinceitneedsto find two nodes with different validities. Bridge test sampling faces a similar challenge. In this paper, we propose a new obstacle-based sampler, Uniform OBPRM (UOBPRM), which guarantees a uniform distribution of samples near C-obstacle surfaces. The sampler first generates a set of uniformly distributed fixed-length segments and then identifies all intersections between the segments and the C-obstacles. This is done by testing intermediate points on the segment at some fixed resolution. Valid configurations adjacent to invalid configurations are retained as roadmap nodes. There are two important parameters which affect how close the configurations are to the obstacle surfaces: the length l of the line segments and the resolution t at which intermediate points are tested along the segments. Our experiments show that UOBPRM gives uniformly distributed configurations around obstacle surfaces. This new UOBPRM sampler can be 6 times faster than Bridge test sampling and 23 times faster than Gaussian sampling to find a path within a tunnel environment. It also has a better node distribution than both OBPRM and Gaussian sampling. II. RELATED WORK PRMs use sampling methods to generate configurations which become roadmap nodes. It is important for nodes to cover important regions of C-space so that they can be connected later to represent the topology of C-space. Therefore, sampling methods play an important role in both the efficiency and success of PRMs. Configurations have one value for each degree of freedom. For example, for rigid bodies in 3D, configurations are 6- tuples, (x, y, z, α, β, γ), where the first three coordinates define the position and the last three are for orientation. A. PRM: Probabilistic Roadmap Method There are two phases in PRM: roadmap construction and query processing. Roadmap construction has 2 stages. First, it samples a set of configurations in C-space and retains the valid ones. Next, connections are attempted between neighboring configurations (as determined by some distance metric) using a local planner. Successful connections are stored as edges in the roadmap. After constructing the

2 roadmap, it can be used to process queries. The start and goal configurations are connected to the roadmap, and then a path is found through the roadmap between the two connected configurations. Many difficult planning problems can be solved in fractions of seconds after constructing the roadmap [10]. The original PRM used uniform sampling in C-space. This strategy had difficulties in sampling points in low-volume regions of C-space (so-called narrow passages). (a) (b) B. Obstacle-Based Sampling Strategies to generate configurations near C-obstacles may help sample in narrow passages. Here we discuss three approaches proposed to help address this issue. 1) OBPRM: OBPRM [3] uses a simple strategy to generate configurations close to the obstacle surfaces. It first finds an origin configuration c in that is invalid (e.g., colliding with the obstacle). A random ray is selected with its origin at c in and a free node, c out, is searched for on that ray. Then a bisection search is performed with the invalid node c in and thevalidnode c out.thisprocessterminateswhenaboundary configuration is found. The boundary points are retained as nodes in the roadmap. Pseudocode for generating a node in OBPRM is given in Algorithm 1. Algorithm 1 OBPRM: Obstacle-based PRM Sampler(n) Input. A maximum attempts n Output. A set of nodes V near obstacle surfaces 1: V = 2: for i = 1 n do 3: Find a point c in in a C-obstacle 4: Randomly select directions D originating from c in 5: for every direction d D do 6: Find a free node c out along d 7: Find a boundary point along c in c out by binary search 8: Add the boundary point to V 9: end for 10: end for 11: return V Although OBPRM generates configurations near C- obstacle surfaces, their distribution is sensitive to the shape of the C-obstacle and the position of the initial invalid configuration, c in, see Figure 1. There is no good position of the initial colliding configuration if the shape of the obstacle is not spherical. Even if the obstacle is spherical, the configuration distribution will not be uniform if the initial colliding configuration is not at the center of the C-obstacle, i.e., it will generate more configurations in the region closest to the initial colliding configuration. In an attempt to get a better configuration distribution, several heuristics have been proposed that use workspace information to select the initial colliding point [1]. Once the points representing both the robot and obstacle are selected, the selected points are placed in overlapping positions and a random orientation is selected to generate a random ray. The same bisection search between valid and invalid nodes Fig. 1. The configuration distribution of OBPRM is biased by (a) the shape of the C-obstacle and (b) the position of the initial colliding configuration. is applied continuously until a node close to the C-obstacle surface is found. Although these heuristics can help decrease the bias caused by either the shape of the C-obstacle or the position of the initial colliding configuration, they still cannot make any guarantees about the distribution of the configurations. 2) Gaussian Sampling: Gaussian sampling [5] attempts to generate configurations which are a Gaussian distance d away from obstacles. The first configuration is randomly generated and the second configuration is found a Gaussian distance d away from the first configuration, where d is userspecified. If one configuration is valid and the other is invalid, then the valid one is saved as a roadmap node; otherwise both configurations are discarded. Gaussian sampling can be costly because it is hard to get nodes with different validities, and many attempts are required to generate samples. At the same time, picking a good Gaussian distance d is important. If d is too small, the configurations will be too close to the obstacles and are hard to generate. When d is large, the configurations are not close to the obstacles. Algorithm 2 outlines Gaussian sampling. Algorithm 2 Gaussian Sampler(n, d) Input. A maximum attempts n and a distance d Output. A set of nodes V a Gaussian distance d away from obstacle surfaces 1: V = 2: for i = 1 n do 3: Randomly select a configuration c 1 4: Generate configuration c 2 a Gaussian distance d along a random ray from c 1 5: if the validity of c 1 and c 2 are different then 6: Add the valid one to V 7: end if 8: end for 9: return V 3) Bridge Test Sampling: Bridge test sampling [8] also uses the collision information of the configurations to bias sampling toward difficult regions, such as narrow passages. It first randomly generates an invalid configuration, c 1, and then randomly generates a second one, c 2, a distance d away. If c 2 is also invalid, the midpoint between c 1 and c 2 will be checked for validity and kept if valid. Bridge test sampling

3 takes longer than Gaussian sampling since it must have a sequence of three nodes such that the endpoints are invalid and the midpoint is valid. Algorithm 3 outlines the approach. Algorithm 3 Bridge Test Sampler(n, d) Input. A maximum attempts n and a distance d Output. A set of nodes V near obstacle surfaces 1: V = 2: for i = 1 n do 3: Randomly select a configuration c 1 4: if c 1 is invalid then 5: Generate configuration c 2 a Gaussian distance d away along a random ray from c 1 6: if c 2 is invalid then 7: if the midpoint between c 1 and c 2 is valid then 8: Add the midpoint to V 9: end if 10: end if 11: end if 12: end for 13: return V 4) Example: Figure 2 illustrates the differences between OBPRM, OBPRM with heuristics, Gaussian sampling and Bridge test sampling. The environment includes a wall with a small hole in it (see Figure 2(a)). The robot is a small stick. The heuristic used for OBPRM sampling was to randomly select one object vertex to represent the object (robot or obstacle). This strategy is biased towards the regions with more vertices. Figures 2(b) 2(e) show the roadmaps generated by each sampling method. From the results, neither OBPRM nor Gaussian sampling can generate configurations inside the hole. Bridge test sampling has some configurations inside the hole but it takes long time to get them. All configurations are around the obstacle. OBPRM with heuristics is also able to generate some configurations inside the hole. None of the distributions appear very uniform. III. UOBPRM: UNIFORM OBPRM Our goal is to obtain a uniform distribution of configurations near C-obstacle surfaces. In UOBPRM, only node generation is different. The connection and query are done as in other PRM approaches. The basic idea of node generation in UOBPRM is to uniformly distribute a set of fixed length line segments in C-space and to retain the intersection points of the line segments with C-obstacle surfaces as roadmap nodes. Since we usually want free configurations, we can retain a free configuration adjacent to the C-obstacle. See Figure3foranexample.Aswewillshow,thismethodresults in a more uniform distribution than OBPRM and is more efficient than Gaussian sampling. A. Node generation in UOBPRM The pseudocode for node generation in UOBPRM is sketched in Algorithm 4. We first uniformly sample a node c in C-space, regardless of validity. Next a random segment of length l is generated with one endpoint at c by selecting Fig. 3. UOBPRM generates uniformly distributed samples by collecting the intersections between uniformly distributed segments and obstacle surfaces. a random direction d, and then extending the segment in that direction. We then check for intersections between the segment and C-obstacles. Algorithm 4 UOBPRM Node Generation(n, l, t) Input. A maximum attempts n, a line segment of length l, and a step size t Output. A set of nodes V uniformly distributed near obstacle surfaces 1: Set a bounding box whose margin is l away from the obstacles (see Algorithm 6) 2: V = 3: for i = 1 n do 4: Uniformly sample configuration c 5: Generate a random direction d 6: Extend a segment s from c distance l in direction d 7: V Intersect(s,l,t) 8: end for 9: return V Thelength, l,ofthesegmentandtheresolution, t,atwhich we check the segment, play important roles in generating nodes for UOBPRM. They determine the closeness of the resulting configurations to the C-obstacle surfaces and the time the sampler takes to generate nodes. The configuration is guaranteed to be within t of C-obstacle surface, so the smaller t, the closer to the C-obstacle. Another consideration is that t is chosen to be appropriate for the environment. Typically, this is the same size as the resolution for collision detection. If l is large, the cost is higher because there are more intermediate configurations to check for intersection. 1) Finding intersections between segments and C- obstacles: We find intersections between a segment and C- obstacles by checking the validity of configurations along the segment at a fixed resolution. Each intersection will have one valid configuration and one invalid configuration bounding it. These valid configurations will be retained as roadmap nodes. Algorithm 5 sketches the process. Validity checking is applied to all intermediate nodes along the segment since there may be multiple intersections of the segment with the obstacle, see Figure 4. 2) Refine bounding box if needed: In practice, motion planning problems are defined with respect to a bounding box that sets the boundaries of the environment. For UOBPRM,

4 (a) Wall environment (b) OBPRM (c) OBPRM with heuristics (d) Gaussian sampling (e) Bridge test sampling Fig. 2. The roadmap generated by different samplers in a wall environment. Algorithm 5 Intersect(s, l, t) Input. A line segment s of length l and a step size t Output. A set of intersections I 1: for i = 1 (l/t) do 2: Generate node c i along s 3: if validity(c i ) validity(c i+1 ) then 4: Add the valid one to I 5: end if 6: end for 7: return I Fig. 4. Finding intersections between the line segment and the obstacle by checking the validity of intermediate configurations along segment. The valid node is retained at every validity change. Here, the valid nodes that are retained are solid. if the bounding box is closer than l, the length of the segment, to a C-obstacle boundary, then segments that would yield points on the C-obstacle surface may be disqualified. Figure 5 shows an example of such a case. Hence, we temporarily expand the bounding box for UOBPRM node generation (line 1 in Algorithm 4). Note that although we temporarily adjust the bounding box for UOBPRM node generation, we do not generate nodes outside the original bounding box, because we only retain samples contained in the original bounding box as roadmap nodes. In this work, we adjust the bounding box as follows. We first find the bounding box for each workspace obstacle and then take the union of the workspace obstacle bounding boxes. After getting the new workspace obstacle based bounding box, we expand each dimension by l + r where r is the maximum robot diameter. (This accounts for different robot orientations.) This updated bounding box will give us enough space to generate line segments of length l around C-obstacles. Algorithm 6 outlines the approach. Fig. 5. Let the robot be a point in the plane. The original bounding box (solid line) is too close to the obstacles in the upper left and bottom right corners, restricting the line segments that can be placed in the red regions. The bounding box is extended to the dashed line to allow all segments of length l that could intersect the obstacles. Algorithm 6 Reset Bounding Box Input. The length l of the line segments used in sampling, maximum robot diameter r, and a set of obstacles O Output. A new bounding box denoted by min {x,y,z} and max {x,y,z} 1: original bounding box = {min{x,y,z},max{x,y,z}} 2: l = l + r 3: for every obstacle o O do 4: bbx(o) = {min o {x,y,z},max o {x,y,z}} 5: end for 6: min O {x,y,z} = min(min o {x,y,z}; o O) 7: max O {x,y,z} = max(max o {x,y,z}; o O) 8: min {x,y,z} = max(min O {x,y,z} l,min{x,y,z}) 9: max {x,y,z} = min(max O {x,y,z}+l,max{z,y,z}) B. Proof of UOBPRM Node Distribution In this section, we sketch an outline of the proof that the points computed by UOBPRM node generation are distributed uniformly on C-obstacle boundaries. Lemma 1: Given an environment E with a finite set of obstacles and a bounding box B whose boundaries are at least l away from any C-obstacle, the probability of finding an intersection point p from a line segment of length l chosen uniformly at random and the environment obstacles is constant throughout B. Proof: Let p be a point on the surface of the C- obstacles. Let S be the sphere centered at p with radius l. Let (c, d ) be a detecting line of length l that starts from a point c and follows the unit vector û d. p is on (c, d ) if and only if c = p or c p, S contains c, and û pc = û d where

5 û pc is the unit vector from p c, see Figure 6. Therefore, the probability that p intersects the detecting line segment is P(c = p) + P(c S c p) P(û d = û pc ). Because c and û d are both selected uniformly at random and S is enclosed in B, the probability that a point on the boundary of the C-obstacles intersects with the detecting line segment is a uniform across all C-obstacle boundary points. Fig. 6. There is a uniform probability that an obstacle point p intersects a line segment (c, d ) of length l when all obstacles are at least l away from the boundary. Corollary 1: For n randomly generated line segments of fixed length l, the probability of finding intersection points with obstacle boundaries is constant throughout E. Since the probability of occurrence is the same, the distribution of the intersection points found by UOBPRM is uniform. IV. EXPERIMENTS We would like to study both the configuration distribution and the cost for different sampling methods in several environments. If the configurations are uniformly distributed, the number of configurations is proportional to the obstacle surface area. The cost depends on how sampling method generates roadmap nodes. OBPRM is affected by the step size. Gaussian sampling and Bridge test sampling are sensitive to d and the cost of UOBPRM is related to l/t. A. Planners Studied We compare 5 different sampling strategies: PRM, OBPRM, Gaussian, Bridge test and UOBPRM. We also study the affects of different UOBPRM parameters. All sampling methods are implemented in the C++ motion planning library developed in the Parasol Lab at Texas A&M University, which contains a number of PRM variants and uses the STAPL parallel C++ library [6]. Both OBPRM and UOBPRM have a step size t which affects roadmap quality. In OBPRM, the step size is used to determine the increment of configurations towards or away from obstacles after finding the initial colliding configuration. Similarly, t determines how many intermediate configurations are checked along a segment in UOBPRM. UOBPRM has an extra parameter, the length l of the line segment. For Gaussian sampling, the Gaussian distance d is the parameter which influences the distance of the second configuration from the first one. Bridge test sampling also has an input parameter d whose valid midpoint is collected as the roadmap configuration. We study different values for t, l, and d for these samplers. B. Environments Studied We studied the performance of each sampler in the different environments shown in Figure 7. Figure 7(a) has a unit radius ball obstacle at the center of an environment whose bounding box is Figure 7(b) has 4 unit balls placed on a grid in a bounding box that is Figure 7(c) is a variant of Figure 7(b). It has a mixture of 2 unit balls and 2 cubes which are Finally, we study a tunnel environment with narrow passages of different widths (Figure 7(d)). C. Configuration Distribution In this section, we study the distribution of configurations obtained by PRM, Gaussian sampling, Bridge test sampling, OBPRM and UOBPRM. For each environment in Figure 7, we use PRM, Gaussian sampling, Bridge test sampling, OBPRM, and UOBPRM to sample configurations. We then partition the environment into several subregions and count the number of configurations in each subregion. If the nodes are uniformly distributed, then the number of nodes should be proportional to the surface area for every region. 1) A Single Ball Environment: We generate 1000 nodes with each sampler and compute the node distribution by computing the number of samples generated in each cell of a regular grid covering the environment. Results are averaged over 10 runs. The grid equally partitions the environment into 16 cubical cells. Starting from 1, the cells are indexed from left to right, from top to bottom. Since the ball symmetrically occupiesthecenter4cells(number6,7,10and11),asimilar number of configurations in these four cells is expected if the distribution is uniform around obstacle surfaces. Figure 8 compares the node distribution. The red bars in Figure 8 show the percentage of configurations within the regions that the ball occupies and the blue ones represents the free space. An ideal node distribution around obstacle surfaces will result with each red bar at 25% and blue bar at 0%. Bridge test sampling, OBPRM and UOBPRM all generate configurations near obstacle and PRM and Gaussian sampling still have configurations scattered in the free space, especially PRM. The configurations generated by PRM are dispersed in the whole environment, as in Figure 9(a). Figure 9(b) shows that Gaussian sampling still has some configurations not close enough to the obstacle surfaces. It shows that UOBPRM with line segment of length equal to 1 gives us a more uniform distribution around obstacle surfaces than the others. The results here used step size of 0.1 and Gaussian distance 0.2. The Bridge test distance is 0.2. Figure 9(c) shows the configurations generated by OBPRM and Figure 9(d) is for UOBPRM with l = 1. As shown, UOBPRM gives a more uniform distribution, and the configurations are closer to the obstacle surface. Table I shows how the configuration distribution around ball and free space are for each sampler. UOBPRM has the most uniformly distributed configurations among all sampling methods since its standard deviation is the lowest.

6 (a) Single Ball (b) 4 Ball (c) Mixed (d) Tunnel Fig. 7. Environment studied. The cube shown in (b) and (c) is the robot and is used in all environments. (a) PRM (b) Gaussian sampler d = 0.2 Fig. 8. Distribution comparison of ball (red) and free (blue) regions. Ideal percentage for ball is 25% and free is 0%. TABLE I AVERAGE AND STANDARD DEVIATION OF BALL AND FREE SPACE IN SINGLE BALL ENVIRONMENT FOR DIFFERENT SAMPLERS. THE IDEAL AVERAGE FOR BALL IS 0.25 AND FREE IS 0. ERROR IS CALCULATED AS THE % DIFFERENCE TO IDEAL. Sampler Ball Free Avg Std Error Avg Std Error PRM n/a Gaussian n/a Bridge test OBPRM UOBPRM l = 0.5 UOBPRM l = ) Environment With 4 Balls of Equal Size: In this experiment, t = and l = 1. We generate 2000 nodes with each sampler and average the results over 10 runs. We partition the space into 4 identical regions, and we separate each ball into 4 same sized regions. Thus, we have a total of 16 regions which have the same obstacle surface area. Figure 10 shows the node distribution comparison. Each color represents a different ball. If the distribution is uniform, each region will have 6.25% of the nodes. The standard deviations of configuration percentage for PRM, Gaussian sampling, Bridge test sampling, OBPRM and UOBPRM with l = 1 are , , , , and Figure 11(a) and 11(b) show the samples generated by (c) OBPRM t = 0.1 (d) UOBPRM t = 0.1, l = 1 Fig. 9. Sample distribution example in the single ball environment. UOBPRM has the most uniformly distributed samples around the surface. OBPRM and UOBPRM, respectively. Here UOBPRM and Bridge test sampler produce a distribution that is close to uniform distribution than the other samplers. As seen in Figure 11(a) and 11(b), OBPRM has fewer nodes on the boundary side than it should for a uniform distribution. 3) Environment With a Mixture of Balls and Cubes: The step size t here is 0.05 and l is 1. After generating 2000 nodes, we separate the environment into four regions where one obstacle is either a ball or cube only. The results are averaged over 10 runs. The node distribution should be proportional to the surface where the unit ball is 4π and the cube is 24. So there should be about 1.9 times more nodes in the cube regions than in the ball regions if the nodes are distributed uniformly. We separate each obstacle into 4 same sized regions to get 16 regions for the whole environment. Figure 12 shows the node distributions for each sampler. The configuration percentage of the ball is colored in red and the cube is colored in blue. The number of nodes generated by UOBPRM and OBPRM reflects the obstacle surface area. 4.31% is the ideal percentage of the nodes for the regions containing the balls and 8.19% is the ideal percentage for the regions containing the cubes. The node distribution for UOBPRM is better than other sampling meth-

7 (a) OBPRM t = 0.05 (b) UOBPRM t = 0.05, l = 1 Fig. 13. Sample distribution example in the mixture environment. Fig. 10. Distribution comparison, each ball is a different color. Ideal percentage is 6.25%. UOBPRM and Bridge test sampling generate more uniformly distributed samples than the others. (a) OBPRM t = (b) UOBPRM t = 0.025, l = 1 Fig. 11. Sample distribution example in the 4 balls environment. ods. Figure 13(a) and 13(b) shows that UOBPRM generates more uniformly distributed configurations within each region than OBPRM especially in the area close to the boundary. the roadmap by using fewer nodes and edges. Results are averaged over 10 runs. Table II shows how well the sampling methods perform. UOBPRM with t = 1 performs the best since it needs the least configurations and edges to find the query path. PRM is not good at solving this kind of difficult problem. For UOBPRM, since small step size t generates configurations close to the surfaces, it needs more configurations and longer time to solve the problem. TABLE II TIME REQUIRED TO SOLVE THE TUNNEL ENVIRONMENT QUERY FOR VARIOUS SAMPLERS AND INPUT PARAMETERS. Sampler # Nodes # Edges Time (sec) CD Calls PRM Gaussian Bridge test OBPRM UOBPRM t = UOBPRM t = UOBPRM t = UOBPRM t = UOBPRM t = Fig. 12. Distribution comparison of ball(red) and cube(blue) regions. Ideal percentage for ball is 4.31% and cube is 8.19%. The sample distribution for UOBPRM is the most uniform. 4) A Heterogeneous Tunnel Environment: One real planning problem we study is a heterogeneous tunnel environment as in Figure 7(d). It has narrow passage which is a difficult region to solve. We try to use different sampling methods to find a path between the start and the goal configurations. They reside in the free space at the two ends of the environment. The more uniform the configurations are, the faster the sampler will be able to find a path in D. Cost In addition to the sample distribution, we are also interested in the cost of generating samples. PRM is fast when the C- space is free, but it does not work well in difficult problems. Gaussian sampling takes longer to generate samples because it must sample two configurations with different validities. Similarly, Bridge test sampling must find a sequence of three samples such that the endpoints are invalid and the midpoint is valid. The cost for OBPRM is largely related to the step size. The smaller the step size, the longer it takes to generate nodes. For UOBPRM, node generation time depends on both the length of the line segment l and the step size t. If the segment is long and the step size is longer, then few intermediate nodes need to be tested. When the segment is short but the step size is small, then more intermediate nodes need to be tested. Therefore, the main factor determining the cost for UOBPRM is l/t. We examine cost in two environments: the single ball environment (Figure 7(a)) and the tunnel environment (Figure 7(d)). Tables III and IV display the time for each

8 sampler to generate 1000 nodes in both environments. Both environments yield similar performance trends. As expected, PRM takes the least amount of time. However, as shown previously, these samples are not distributed on the obstacle surfaces and PRM s performance degrades with increased problem difficulty. Bridge test sampling takes the longest time and is sensitive to d. As mentioned above, it is more difficult to randomly sample two invalid points whose midpoint is valid. Depending on the parameters chosen, OBPRM and UOBPRM take the shortest amount of time. OBPRM increases generation time as the step size is decreased. The time of UOBPRM is related to l/t, with generation time increasing as l/t increases. Note that only UOBPRM makes any claim as to the distribution of samples on the obstacle surfaces, and it can do so with similar or less computational resources than the other methods. TABLE III GENERATION TIME FOR VARIOUS SAMPLERS AND INPUT PARAMETERS IN THE SINGLE BALL ENVIRONMENT. Sampler Parameter Time (sec) PRM n/a 0.07 Gaussian d = d = Bridge test d = d = t = OBPRM t = t = t = l/t = l/t = UOBPRM l/t = l/t = l/t = l/t = TABLE IV GENERATION TIME FOR VARIOUS SAMPLERS AND INPUT PARAMETERS IN THE TUNNEL ENVIRONMENT. Sampler Parameter Time (sec) PRM n/a 0.39 d = Gaussian d = d = Bridge test d = d = t = OBPRM t = t = t = l/t = l/t = l/t = UOBPRM l/t = l/t = l/t = l/t = l/t = V. CONCLUSION We propose a new sampler, UOBPRM, which generates more uniformly distributed configurations close to the obstacles as compared to prior methods including Gaussian sampling, Bridge test sampling and OBPRM. UOBPRM achieves this by finding all intersections between obstacles and a set of uniformly distributed line segments. UOBPRM is faster than the Gaussian sampler and Bridge test sampler and its distribution is better than both OBPRM and Gaussian sampling. The cost of UOBPRM is dependent on the length of the segments l and the step size t and is directly proportional to l/t. UOBPRM is a viable option for challenging motion planning problems as demonstrated by its performance in the tunnel environment with narrow passages of varying size. Inthefuture,weplantoinvestigatehowtofindthebalance between sample quality and generation cost and how this can be tuned for different applications. We also want to study how UOBPRM performs in environments with concave obstacles. We plan to explore UOBPRM s performance on other types of robots, such articulated linkages. REFERENCES [1] N. M. Amato, O. B. Bayazit, L. K. Dale, C. V. Jones, and D. Vallejo. OBPRM: An obstacle-based PRM for 3D workspaces. In Robotics: The Algorithmic Perspective, pages , Natick, MA, A.K. Peters. Proc. Third Workshop on Algorithmic Foundations of Robotics (WAFR), Houston, TX, [2] N. M. Amato and G. Song. Using motion planning to study protein folding pathways. J. Comput. Biol., 9(2): , Special issue of Int. Conf. Comput. Molecular Biology (RECOMB) [3] N. M. Amato and Y. Wu. A randomized roadmap method for path and manipulation planning. In Proc. IEEE Int. Conf. Robot. Autom. (ICRA), pages , [4] J. Barraquand and J. C. Latombe. Robot motion planning: A distributed representation approach. Int. J. Robot. Res., 10(6): , [5] V. Boor, M. H. Overmars, and A. F. van der Stappen. The Gaussian sampling strategy for probabilistic roadmap planners. In Proc. IEEE Int. Conf. Robot. Autom. (ICRA), volume 2, pages , May [6] Antal Buss, Harshvardhan, Ioannis Papadopoulos, Olga Pearce, Timmie Smith, Gabriel Tanase, Nathan Thomas, Xiabing Xu, Mauro Bianco, Nancy M. Amato, and Lawrence Rauchwerger. STAPL: Standard template adaptive parallel library. In Proc. Annual Haifa Experimental Systems Conference (SYSTOR), pages 1 10, New York, NY, USA, ACM. [7] H. Chang and T. Y. Li. Assembly maintainability study with motion planning. In Proc. IEEE Int. Conf. Robot. Autom. (ICRA), pages , [8] D. Hsu, T. Jiang, J.H. Reif, and Z. Sun. Bridge test for sampling narrow passages with probabilistic roadmap planners. In Proc. IEEE Int. Conf. Robot. Autom. (ICRA), pages , [9] L. Kavraki and J. C. Latombe. Randomized preprocessing of configuration space for fast path planning. In Proc. IEEE Int. Conf. Robot. Autom. (ICRA), pages , [10] L. E. Kavraki, P. Švestka, J. C. Latombe, and M. H. Overmars. Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Trans. Robot. Automat., 12(4): , August [11] Y. Koga, K. Kondo, J. Kuffner, and J.C. Latombe. Planning motions with intentions. In Proc. ACM SIGGRAPH, pages , [12] M. Overmars and P. Švestka. A probabilistic learning approach to motion planning. In Proc. Int. Workshop on Algorithmic Foundations of Robotics (WAFR), pages 19 37, [13] X.Tang,B.Kirkpatrick,S.Thomas,G.Song,andN.M.Amato. Using motion planning to study RNA folding kinetics. J. Comput. Biol., 12(6): , Special issue of Int. Conf. Comput. Molecular Biology (RECOMB) 2004.