Complete Motion Planning

1
Complete Motion Planning

• Liang-Jun Zhang
• Robotics, Comp790-072
• Oct 26, 2006

2
Outline

• Motivation/Challenge
• Approaches
• Exact Motion Planning
• Approximation Cell Decomposition
• Hybrid Planner

3
Motion Planning
To find a path
Goal
Robot
Initial
Obstacle
4
Why Complete Motion Planning?

• Complete motion planning
• Always terminate
• Not efficient
• Not robust even for low DOF

• Probabilistic roadmap motion planning
• Efficient
• Work for complex problems with many DOF
• Difficult for narrow passages
• May not terminate when no path exists

5
Path Non-existence Problem
Obstacle
Obstacle
6
Main Challenge

• Exponential complexity nDOF
• Degree of freedom DOF
• Geometric complexity n
• More difficult than finding a path
• To check all possible paths

Obstacle
7
Approaches

• Exact Motion Planning
• Based on exact representation of free space
• Approximation Cell Decomposition (ACD)
• A Hybrid planner

8
Configuration Space 2D Translation
Workspace
Configuration Space
Goal
Free
Robot
y
x
Start
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Configuration Space Computation

• Varadhan et al, ICRA 2006
• 2 Translation 1 Rotation
• 215 seconds

Obstacle
?
y
x
Robot
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Exact Motion Planning

• Approaches
• Exact cell decomposition Schwartz et al. 83
• Roadmap Canny 88
• Criticality based method Latombe 99
• Voronoi Diagram
• Star-shaped roadmap Varadhan et al. 06
• Not practical
• Due to free space computation
• Limit for special and simple objects
• Ladders, sphere, convex shapes
• 3DOF

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Approaches

• Exact Motion Planning
• Based on exact representation of free space
• Approximation Cell Decomposition (ACD)
• A Hybrid Planner Combing ACD and PRM

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Approximation Cell Decomposition (ACD)

• Not compute the free space exactly at once
• But compute it incrementally
• Relatively easy to implement
• Lozano-Pérez 83
• Zhu et al. 91
• Latombe 91
• Zhang et al. 06

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Approximation Cell Decomposition
Configuration Space

• Full cell
• Empty cell
• Mixed cell
• Mixed
• Uncertain

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Connectivity Graph
Gf Free Connectivity Graph
Gf is a subgraph of G
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Finding a Path by ACD
Initial
Goal
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Finding a Path by ACD

• First Graph Cut Algorithm
• Guiding path in connectivity graph G
• Only subdivide along this path
• Update the graphs G and Gf

L Guiding Path
Described in Latombes book
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First Graph Cut Algorithm
L
Only subdivide along L
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Finding a Path by ACD
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ACD for Path Non-existence
Initial
Goal
C-space
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ACD for Path Non-existence
Connectivity Graph
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ACD for Path Non-existence
Connectivity graph is not connected
No path!
Sufficient condition for deciding path
non-existence
22
Two-gear Example
Video
no path!
3.356s
Initial
Cells in C-obstacle
Roadmap in F
Goal
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Cell Labeling

• Free Cell Query
• Whether a cell completely lies in free space?
• C-obstacle Cell Query
• Whether a cell completely lies in C-obstacle?

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Free Cell Query A Collision Detection Problem

• Does the cell lie inside free space?

• Do robot and obstacle separate at all
  configurations?

Robot
Obstacle
?
Configuration space
Workspace
25
Clearance

• Separation distance
• A well studied geometric problem
• Determine a volume in C-space which are
  completely free

d
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C-obstacle QueryAnother Collision Detection
Problem

• Does the cell lie inside C-obstacle?

• Do robot and obstacle intersect at all
  configurations?

Robot
?
Obstacle
Configuration space
Workspace
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Forbiddance

• Forbiddance dual to clearance
• Penetration Depth
• A geometric computation problem less investigated
• Zhang et al. ACM SPM 2006

PD
28
Limitation of ACD

• Combinatorial complexity of cell decomposition
• Limited for low DOF problem
• 3-DOF robots

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Approaches

• Exact Motion Planning
• Based on exact representation of free space
• Approximation Cell Decomposition (ACD)
• A Hybrid Planner Combing ACD and PRM

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Hybrid Planning

• Probabilistic roadmap motion planning
• Efficient
• Many DOFs
• Narrow passages
• Path non-existence

• Complete Motion Planning
• Complete
• Not efficient

Can we combine them together?
31
Hybrid Approach for Complete Motion Planning

• Use Probabilistic Roadmap (PRM)
• Capture the connectivity for mixed cells
• Avoid substantial subdivision
• Use Approximation Cell Decomposition (ACD)
• Completeness
• Improve the sampling on narrow passages

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Connectivity Graph
Gf Free Connectivity Graph
G Connectivity Graph
Gf is a subgraph of G
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Pseudo-free edges
Initial
Goal
Pseudo free edge for two adjacent cells
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Pseudo-free Connectivity Graph Gsf
Gsf Gf Pseudo-edges
Initial
Goal
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Algorithm

• Gf
• Gsf
• G

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Results of Hybrid Planning
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Results of Hybrid Planning
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Results of Hybrid Planning

• 2.5 - 10 times speedup

3 DOF 3 DOF 4 DOF 4 DOF 4 DOF 4 DOF
timing cells timing cells timing cells
Hybrid 34s 50K 16s 48K 102s 164K
ACD 85s 168K ? ? ? ?
Speedup 2.5 3.3 10 ? 10 ?
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Summary

• Difficult for Exact Motion Planning
• Due to the difficulty of free space configuration
  computation
• ACD is more practical
• Explore the free space incrementally
• Hybrid Planning
• Combine the completeness of ACD and efficiency of
  PRM

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Future Work

• Complete motion planning for 6DOF rigid robots
• More accurate PDg computation
• Efficient C-Obstacle representation and
  computation
• Extend for articulated robots

41
Reference Exact Motion Planning

• J. Canny. The Complexity of Robot Motion
  Planning. ACM Doctoral Dissertation Award. MIT
  Press, 1988.
• F. Avnaim and J.-D. Boissonnat. Practical exact
  motion planning of a class of robots with three
  degrees of freedom. In Proc. of Canadian
  Conference on Computational Geometry, page 19,
  1989.
• J. T. Schwartz and M. Sharir. On the piano movers
  probelem ii, general techniques for computing
  topological properties of real algebraic
  manifolds. Advances of Applied Maths, 4298351,
  1983.
• Gokul Varadhan, Dinesh Manocha, Star-shaped
  Roadmaps - A Deterministic Sampling Approach for
  Complete Motion Planning, Robotics Science and
  Systems 2006

42
Reference Approximation Cell Decomposition

• T. Lozano-Perez and M. Wesley. An algorithm for
  planning collisionfree paths among polyhedral
  obstacles. Comm. ACM, 22(10)560570, 1979.
• R. A. Brooks and T. Lozano-Perez. A subdivision
  algorithm in configuration space for findpath
  with rotation. IEEE Trans. Syst, SMC-15224233,
  1985.
• D. Zhu and J. Latombe. Constraint reformulation
  in a hierarchical path planner. Proceedings of
  International Conference on Robotics and
  Automation, pages 19181923, 1990.
• L. Zhang, Y. Kim, and D. Manocha. A simple path
  non-existence algorithm using c-obstacle query.
  In Proc. of WAFR, 2006.
• L. Zhang, Y.J. Kim, and D. Manocha, A Hybrid
  Approach for Complete Motion Planning, UNC-CS
  Tech Report 06-022