Dynamic-Domain%20RRTs

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Dynamic-Domain RRTs
Anna Yershova, Steven M. LaValle 03/08/2006
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Basic Motion Planning Problem

• Given
• 2D or 3D world
• Geometric models of obstacles
• Geometric models
• Configuration space
• Initial and goal configurations
• Task
• Compute a collision free path that connects
  initial and goal configurations

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Rapidly-exploring Random Trees (RRTs)

• Introduced by LaValle and Kuffner, ICRA 1999.
• 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,
  ...

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The RRT Construction Algorithm

• GENERATE_RRT(xinit, K, ?t)
• T.init(xinit)
• For k 1 to K do
• xrand ? RANDOM_STATE()
• xnear ? NEAREST_NEIGHBOR(xrand, T)
• if CONNECT(T, xrand, xnear, xnew)
• T.add_vertex(xnew)
• T.add_edge(xnear, xnew, u)
• Return T

xnear
xnew
xinit
The result is a tree rooted at xinit
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A Rapidly-exploring Random Tree (RRT)
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Voronoi Biased Exploration
Is this always a good idea?
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Voronoi Diagram in R 2
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Voronoi Diagram in R 2
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Voronoi Diagram in R 2
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Refinement vs. Expansion
refinement
expansion
Where will the random sample fall? How to control
the behavior of RRT?
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Limit Case Pure Expansion

• Let X be an n-dimensonal ball,
• in which r is very large.
• The RRT will explore n 1 opposite directions.
• The principle directions are vertices of a
  regular (n 1)-simplex

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Determining the Boundary
Expansion dominates
Balanced refinement and expansion
The tradeoff depends on the size of the bounding
box
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Controlling the Voronoi Bias

• Refinement is good when multiresolution search is
  needed
• Expansion is good when the tree can grow and not
  blocked by obstacles
• Main motivation
• Voronoi bias does not take into account obstacles
• How to incorporate the obstacles into Voronoi
  bias?

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Bug Trap
Small Bounding Box
Large Bounding Box

• Which one will perform better?

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Voronoi Bias for the Original RRT
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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
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A Boundary Node

• (a) Regular RRT, unbounded Voronoi region
• (b) Visibility region
• (c) Dynamic domain

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A Non-Boundary Node

• (a) Regular RRT, unbounded Voronoi region
• (b) Visibility region
• (c) Dynamic domain

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Dynamic-Domain RRT Bias
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Dynamic-Domain RRT Construction
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Dynamic-Domain RRT Bias
Tradeoff between nearest neighbor calls and
collision detection calls
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Recent Efforts

• Adaptive tuning of the radius
• the radius is not fixed but is increased with
  every extension success and is decreased with
  every failure
• Nearest neighbor calls
• kd-tree based implementation
• O(log n) instead of naïve O(n) query time
• Uniform sampling from dynamic domain
• Rejection-based method is not efficient for high
  dimensions
• Uniform distribution should be generated directly

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Adaptive Tuning of Parameter
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Adaptive Tuning of Parameter
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Nearest Neighbor Calls Uniform Sampling

• Efficient implementation using kd-trees
• O(log n) query time instead of naïve O(n) query
  time

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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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Shrinking Bug Trap
Large Medium
Small
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Shrinking Bug Trap
The smaller the bug trap, the better the
improvement
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Wiper Motor (courtesy of KINEO)

• 6 dof problem
• CD calls are expensive

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Molecule

• 68 dof problem was solved in 2 minutes
• 330 dof in 1 hour
• 6 dof in 1 min. 30 times improvement comparing to
  RRT
• CD calls are expensive

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Labyrinth

• 3 dof problem
• CD calls are not expensive

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3D grid

• 6 dof problem
• CD calls are not expensive

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Spiral

• 6 dof problem
• CD calls are not expensive

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Conclusions

• Controlling Voronoi bias is important in RRTs.
• Provides dramatic performance improvements on
  some problems.
• Does not incur much penalty for unsuitable
  problems.
• Work in Progress
• Application to planning under differential
  constraints.
• Application to planning for closed chains.