Dynamic-Domain RRTs. Anna Yershova 1 Léonard Jaillet 2 Thierry Siméon 2 Steven M. LaValle 1

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1 Dynamic-Domain RRTs Anna Yershova 1 Léonard Jaillet 2 Thierry Siméon 2 Steven M. LaValle 1 1. University of Illinois at Urbana-Champaign 2. LAAS/CNRS Toulouse To appear: ICRA 2005 Barcelona Thanks to: US National Science Foundation, UIUC/CNRS funding, Kineo

2 Rapidly-exploring Random Trees (RRTs Introduced by LaValle and Kuffner, ICRA Applied, adapted, and extended in many works: 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; Branicky, Curtiss, 2002; Cortes, Simeon, 2004; Urmson, Simmons, 2003; Yamane, Kuffner, Hodgins, 2004; Strandberg, 2004;... Also, applications to biology, computational geography, verification, virtual prototyping, architecture, solar sailing, computer graphics,...

3 The RRT Construction Algorithm BUILD RRT(q init ) 1 T.init(q init ); 2 for k = 1 to K do 3 q rand RANDOM CONFIG(); 4 EXTEND(T, q rand ); EXTEND(T, q rand ) ε q new q init q near q Metric on C: ρ : C C [0, ) Nearest neighbors: Atramentov [Yershova], LaValle, 2002; Arya, Mount, 1997;... Incremental collision detection: Lin, Canny, 1991; Mirtich, 1997

4 A Rapidly-Exploring Random Tree (RRT)

5 Voronoi-Biased Exploration

6 Voronoi Diagram in R2

7 Voronoi Diagram in R2

8 Voronoi Diagram in R2

9 Refinement vs. Expansion Refinement Expansion Where will the random sample fall?

10 Limit Case: Pure Expansion Let X be an n-dimensonal ball, X = {x R n x r}, in which r is very large. The RRT will explore n + 1 opposing directions. The principle directions are vertices of a regular (n + 1)-simplex:

11 Determining the Boundary Expansion dominates Balanced refiniment and expansion The tradeoff depends on the size of the bounding box.

12 When is Expanson Bad? Refinement is good because because it is like multiresolution search. Expansion is bad when blocked by obstacles.

13 Bug Trap Small Bouding Box Large Bounding Box Which one will perform better?

14 Voronoi Bias for the Original RRT

15 Visibility-Based Clipping of the Voronoi Regions Nice idea, but how can this be done in practice? Even better: Voronoi diagram for obstacle-based metric

16 A Boundary Node Cobst (a) Cobst (b) Cobst (c) v v v (a) Regular RRT, unbounded Voronoi region (b) Visibility region (c) Dynamic domain

17 A Non-Boundary Node (a) (b) (c) C obst C obst C obst v v v R (a) Regular RRT, unbounded Voronoi region (b) Visibility region (c) Dynamic domain

18 Dynamic-Domain RRT Bias

19 DD-RRT Algorithm BUILD DYNAMIC DOMAIN RRT(q init ) 1 T.init(q init ); 2 for k = 1 to K do 3 repeat 4 q rand RANDOM CONFIG(); 5 q near NEAREST NEIGHBOR(q rand, T ); 6 until dist(q near, q rand ) < q near.radius 7 if CONNECT(T, q rand, q near, q new ) 8 q new.radius = ; 9 T.add vertex(q new ); 10 T.add edge(q near, q new ); 11 else 12 q near.radius = R; 13 Return T ;

20 Implementation Details MOVE3D (LAAS/CNRS) 333 Mhz Sunblade 100 with SunOs 5.9 (not very fast) Compiler: GCC 3.3 Fast nearest neighbor searching (Yershova [Atramentov], LaValle, 2002)

21 Experiments Two kinds: Controlled experiments for toy problems Challenging benchmarks from industry and biology

22 Shrinking Bug Trap (1) (2) (3) Large Medium Small

23 Shrinking Bug Trap Trap Size Statistic Dynamic-Domain bi-rrt bi-rrt Large time (1) 0.4 sec 0.1 sec no. nodes (1) CD calls (1) Medium time (2) 2.5 sec 379 sec no. nodes (2) CD calls (2) Small time (3) 1.6 sec > sec no. nodes (3) 1301 CD calls (3) 3022 The smaller the bug trap, the better the improvement.

24 (Example provided courtesy of KINEO) Wiper Motor

25 Wiper Motor Dynamic-Domain bi-rrt bi-rrt time 217 sec > sec no. nodes 219 CD calls [Movie]

26 Molecule

27 Molecule Dynamic-Domain bi-rrt bi-rrt time 70 sec 2926 sec no. nodes CD calls Improvement factor: 41.8 DD-RRT also solved in one hour a conformational search problem with 330 DOFs!

28 Labyrinth

29 Labyrinth Dynamic-Domain bi-rrt bi-rrt time 161 sec 237 sec no. nodes CD calls [Movie]

30 Conclusions Balancing the amount of refinement and expansion is important in RRTs. Provides dramatic performing improvements on some problems. Does not incur much penalty for unsuitable problems. Work in Progress: There is a radius parameter. Adaptive tuning is possible. Still investgating several variatons of the algorithm. Application to planning under differential constrants. Application to planning for closed chains.