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

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

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

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

Rapidly-exploring random tree approachesGENERATE_RRT(xinit, K, t)1. T.init(xinit);

2. For k = 1 to K do

3. xrand RANDOM_STATE();

4. xnear NEAREST_NEIGHBOR(xrand, T);

5. u SELECT_INTPUT(xrand, xnear);

6. xnew NEW_STATE(xnear, u, t);

7. T.add_vertex(xnew);

8. T.add_edge(xnear, xnew, u);

9. Return T;

xnear

xrand

xinit

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:

GoalsExisting 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

• 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 FormulationGiven 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

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Algorithm presentation

Overview of the kd-tree algorithm

Modification of kd-tree algorithm to handle topology

Analysis of the algorithm

Experimental results

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.

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Kd-trees. Construction

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Kd-trees. Query

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Algorithm Presentation

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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

For 50,000 data points 100 queries were made:

dim topology new algorithm new algorithm brute force

construction time

100 queries time

100 queries time

3 S1 x S1 x S1 5.6s 0.1s 29.2s

6 R3 x P3 7.1s 3.2s 68.9s

13 R3 x (S1)3 x (P3)2 8.8s 9.2s 153.1s

ExperimentsPRM method

topology

new algorithm brute force

nodes time(s) nodes time(s)

R3 x P3 11,684 96.83 11,698 247.29

27,076 147.58 27,259 740.42

Success 42,630 173.43 42,714 1,229.70

58,117 215.16 58,308 1,796.17

ExperimentsRRT method

topology

new algorithm brute force

nodes time(s) nodes time(s)

R2 x S1 37,177 584.9 39,610 7,501.4

topology

new algorithm brute force

nodes time(s) nodes time(s)

(R3 x P3)8 17,271 4,631.9 18,029 8,461.6

ExperimentsRRT method

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