1. 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.

2. 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:

3. 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

4. Probabilistic roadmap approaches (Kavraki, Svestka, Latombe, Overmars, 1994) The precomputation phase consists of the following steps: 1. Generate vertices in configuration space at random 2. Connect close vertices 3. 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: 1.Connect initial and goal to graph 2.Search the graph

5. Rapidly-exploring random tree approaches GENERATE_RRT(x init, K,  t) 1. T.init(x init ); 2. For k = 1 to K do 3. x rand  RANDOM_STATE(); 4. x near  NEAREST_NEIGHBOR(x rand, T); 5. u  SELECT_INTPUT(x rand, x near ); 6. x new  NEW_STATE(x near, u,  t); 7. T.add_vertex(x new ); 8. T.add_edge(x near, x new, u); 9. Return T; x near x rand x init x new 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 x init :

6. Goals Existing nearest neighbor packages:  ANN (U. of Maryland)  Ranger (SUNY Stony Brook) Problem: They only work for R n. Configuration spaces that usually arise in motion planning are products of R, S 1 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

7. 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

8. 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. P 3, represented by [0, 1] 3 with antipodal points identified. Examples of 4-sided polygonal schemas: cylindertorusprojective plane

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

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

11. 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. 4 7 6 5 1 3 2 9 8 10 11 l5l5 l1l1 l9l9 l6l6 l3l3 l 10 l7l7 l4l4 l8l8 l2l2 l1l1 l8l8 1 l2l2 l3l3 l4l4 l5l5 l7l7 l6l6 l9l9 3 25411 910 8 67

12. Kd-trees. Construction 4 7 6 5 1 3 2 9 8 10 11 l5l5 l1l1 l9l9 l6l6 l3l3 l 10 l7l7 l4l4 l8l8 l2l2 l1l1 l8l8 1 l2l2 l3l3 l4l4 l5l5 l7l7 l6l6 l9l9 3 25411 910 8 67

13. Kd-trees. Query 4 7 6 5 1 3 2 9 8 10 11 l5l5 l1l1 l9l9 l6l6 l3l3 l 10 l7l7 l4l4 l8l8 l2l2 l1l1 l8l8 1 l2l2 l3l3 l4l4 l5l5 l7l7 l6l6 l9l9 3 25411 910 8 67 q

14. Algorithm Presentation l1l1 l8l8 1 l2l2 l3l3 l4l4 l5l5 l7l7 l6l6 l9l9 l 10 3 25411 910 8 67 q 4 7 6 5 1 3 2 9 8 11 l5l5 l1l1 l9l9 l6l6 l3l3 l 10 l7l7 l4l4 l8l8 l2l2 1 3 l4l4 l8l8 l2l2

15. 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.

16. Experiments For 50,000 data points 100 queries were made: dimtopologynew algorithm brute force construction time 100 queries time 3S 1 x S 1 x S 1 5.6s0.1s29.2s 6R 3 x P 3 7.1s3.2s68.9s 13R 3 x (S 1 ) 3 x (P 3 ) 2 8.8s9.2s153.1s

17. Experiments PRM method topology new algorithmbrute force nodestime(s)nodestime(s) R 3 x P 3 11,68496.8311,698247.29 27,076147.5827,259740.42 Success42,630173.4342,7141,229.70 58,117215.1658,3081,796.17

18. Experiments RRT method topology new algorithmbrute force nodestime(s)nodestime(s) R 2 x S 1 37,177584.939,6107,501.4

19. topology new algorithmbrute force nodestime(s)nodestime(s) (R 3 x P 3 ) 8 17,2714,631.918,0298,461.6 Experiments RRT method

20. 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/)http://msl.cs.uiuc.edu/msl/ Future Work  Extension to more efficient kd-trees  Extension to different topological spaces  Extension to different metric spaces