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Efficient Nearest Neighbor Searching for Motion Planning: 

Efficient Nearest Neighbor Searching for Motion Planning Anna Atramentov Dept. of Computer Science Iowa State University Ames, IA, USA Steven M. LaValle Dept. of Computer Science University of Illinois Urbana, IL, USA Support provided in part by an NSF CAREER award.

Motivation: 

Motivation Statistics Pattern recognition Machine Learning Nearest neighbor searching is a fundamental problem in many applications: PRM-based methods RRT-based methods In motion planning the following algorithms rely heavily on nearest neighbor algorithms:

Basic Motion Planning Problem: 

Basic Motion Planning Problem Given: 2D or 3D world Geometric models of a robot and obstacles Configuration space Initial and goal configurations Task: Compute a collision free path that connects initial and goal configurations

Slide4: 

Probabilistic roadmap approaches (Kavraki, Svestka, Latombe, Overmars, 1994) The precomputation phase consists of the following steps: Generate vertices in configuration space at random Connect close vertices Return resulting graph Obstacle-Based PRM (Amato, Wu, 1996); Sensor-based PRM (Yu, Gupta, 1998); Gaussian PRM (Boor, Overmars, van der Stappen, 1999); Medial axis PRMs (Wilmarth, Amato, Stiller, 1999; Psiula, Hoff, Lin, Manocha, 2000; Kavraki, Guibas, 2000); Contact space PRM (Ji, Xiao, 2000); Closed-chain PRMs (LaValle, Yakey, Kavraki, 1999; Han, Amato 2000); Lazy PRM (Bohlin, Kavraki, 2000); PRM for changing environments (Leven, Hutchinson, 2000); Visibility PRM (Simeon, Laumond, Nissoux, 2000). The query phase: Connect initial and goal to graph Search the graph

Rapidly-exploring random tree approaches: 

Rapidly-exploring random tree approaches GENERATE_RRT(xinit, K, t) T.init(xinit); For k = 1 to K do xrand  RANDOM_STATE(); xnear  NEAREST_NEIGHBOR(xrand, T); u  SELECT_INTPUT(xrand, xnear); xnew  NEW_STATE(xnear, u, t); T.add_vertex(xnew); T.add_edge(xnear, xnew, u); Return T; xnear xrand xnew LaValle, 1998; LaValle, Kuffner, 1999, 2000; Frazzoli, Dahleh, Feron, 2000; Toussaint, Basar, Bullo, 2000; Vallejo, Jones, Amato, 2000; Strady, Laumond, 2000; Mayeux, Simeon, 2000; Karatas, Bullo, 2001; Li, Chang, 2001; Kuffner, Nishiwaki, Kagami, Inaba, Inoue, 2000, 2001; Williams, Kim, Hofbaur, How, Kennell, Loy, Ragno, Stedl, Walcott, 2001; Carpin, Pagello, 2002. The result is a tree rooted at xinit:

Goals: 

Goals Existing nearest neighbor packages: ANN (U. of Maryland) Ranger (SUNY Stony Brook) Problem: They only work for Rn. Configuration spaces that usually arise in motion planning are products of R, S1 and projective spaces. Theoretical results: Problem: Difficulty of implementation P. Indyk, R. Motwani, 1998; P. Indyk, 1998, 1999; Our goal: Design simple and efficient algorithm for finding nearest neighbor in these topological spaces

Literature on NN searching: 

Literature on NN searching It is very well studied problem Kd-tree approach is very simple and efficient T. Cover, P. Hart, 1967 D. Dobkin, R. Lipton, 1976 J. Bentley, M. Shamos, 1976 S. Arya, D. Mount,  1993, 1994 M. Bern, 1993 T. Chan,  1997 J. Kleinberg,  1997 K. Clarkson, 1988, 1994, 1997 P. Agarwal, J. Erickson, 1998 P. Indyk, R. Motwani, 1998 E. Kushilevitz, R. Ostrovsky, Y. Rabani, 1998 P. Indyk, 1998, 1999 A. Borodin, R. Ostrovsky, Y. Rabani,  1999

Problem Formulation: 

Problem Formulation Given a d-dimensional manifold, T, represented as a polygonal schema, and a set of data points in T. Preprocess these points so that, for any query point q  T, the nearest data point to q can be found quickly. The manifolds of interest: Euclidean one-space, represented by (0,1)  R. Circle, represented by [0,1], in which 0  1 by identification. P3, represented by [0, 1]3 with antipodal points identified. Examples of 4-sided polygonal schemas: cylinder torus projective plane

Example: a torus: 

Example: a torus 4 7 6 5 1 3 2 9 8 10 11 q

Algorithm presentation: 

Algorithm presentation Overview of the kd-tree algorithm Modification of kd-tree algorithm to handle topology Analysis of the algorithm Experimental results

Kd-trees: 

Kd-trees The kd-tree is a powerful data structure that is based on recursively subdividing a set of points with alternating axis-aligned hyperplanes. The classical kd-tree uses O(dn lgn) precomputation time, O(dn) space and answers queries in time logarithmic in n, but exponential in d. l1 l8 l2 l3 l4 l5 l7 l6 l9 l10

Kd-trees. Construction: 

Kd-trees. Construction l5 l9 l6 l3 l10 l7 l4 l8 l2 l1 l8 l2 l3 l4 l5 l7 l6 l9 l10

Kd-trees. Query: 

Kd-trees. Query 4 7 6 5 1 3 2 9 8 10 11 l5 l9 l6 l3 l10 l7 l4 l8 l2 l1 l8 l2 l3 l4 l5 l7 l6 l9 l10

Algorithm Presentation: 

Algorithm Presentation

Analysis of the Algorithm: 

Analysis of the Algorithm Proposition 1. The algorithm correctly returns the nearest neighbor. Proof idea: The points of kd-tree not visited by an algorithm will always be further from the query point then some point already visited. Proposition 2. For n points in dimension d, the construction time is O(dn lgn), the space is O(dn), and the query time is logarithmic in n, but exponential in d. Proof idea: This follows directly from the well-known complexity of the basic kd-tree.

Experiments: 

Experiments For 50,000 data points 100 queries were made:

ExperimentsPRM method: 

Experiments PRM method

ExperimentsRRT method: 

Experiments RRT method

ExperimentsRRT method: 

Experiments RRT method

Conclusion: 

Conclusion We extended kd-tree to handle topology of the configuration space We have presented simple and efficient algorithm We have developed software for this algorithm which will be included in Motion Strategy Library (http://msl.cs.uiuc.edu/msl/) Future Work Extension to more efficient kd-trees Extension to different topological spaces Extension to different metric spaces

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