A Simple Path NonExistence Algorithm using Cobstacle Query

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A Simple Path Non-Existence Algorithm using
C-obstacle Query
http//gamma.cs.unc.edu/nopath
Liang-Jun Zhang University of North Carolina -
Chapel Hill Young J. Kim EWHA Womans
University, Korea Dinesh Manocha University
of North Carolina - Chapel Hill
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Motion Planning
To find a path
Goal
Robot
Initial
Obstacle
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Path Non-existence Problem
Obstacle
Obstacle
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Previous Work

• Exact Motion Planning
• Exact cell decomposition Schwartz et al. 83
• Roadmap Canny 88
• Criticality based method Latombe 99
• Implementation challenges
• Special and simple objects
• Ladders, sphere, convex shapes

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Previous Work

• Approximation Cell Decomposition
• Lozano-Pérez 83, Zhu et al. 91, Latombe 91
• Relatively easy to implement
• Combinatorial complexity of cell decomposition
• Computational issue for labelling the cells
  during cell decomposition

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Previous Work

• Probabilistic Sampling Based Approach
• Kavarki et al. 96 LaValle et al. 98, Choset
  et al. 05, LaValle 06
• Simple and widely used
• May not be terminated when non-path exists
• Difficult for narrow passage

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Previous Work

• Path non-existence for special cases
• Planar section, Basch et al. 01, Bretl et al.
  04

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Main Results

• Efficient cell labelling algorithm
• Workspace-based
• C-obstacle query using generalized penetration
  depth
• Improved cell decomposition algorithm
• Simple
• Efficient for path non-existence

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Path Non-existence Problem
Configuration space

• More difficult than finding a path
• To check all possible paths
• Identify a region in C-obstacle
• separating qinit and qgoal

qinit
qgoal
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C-obstacle Query

• Whether a primitive lies entirely in C-obstacle?
• Usually a cell
• Useful for path non-existence

qinit
qgoal
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Cell Decomposition for Path Non-existence

• Lozano-Pérez 83
• Zhu et al. 91
• Latombe 91

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Cell Decomposition for Path Non-existence
Connectivity Graph
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Cell Decomposition for Path Non-existence
Connectivity graph is not connected
No path!
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Previous Work on C-obstacle Query

• Explicit free space computation
• Exponential complexity Sacks 99, Sharir 97
• Hard in practice degeneracy
• Check against every C-surface
• Latombe 91, Zhu et al. 91
• C-surface enumeration
• To deal with non-linear C-surfaces
• Workspace distance computation
• Paden 89
• Overly conservative

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C-obstacle QueryA 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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Clearance VS Forbiddance

• Separation distance
• Clearance

• Penetration Depth
• Forbiddance

PD
d
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C-obstacle Query Algorithm

• Penetration Depth
• Extent of interpenetration between robot and
  obstacle
• Motion Bound
• Extent of the motion that robot can make.
• Is Penetration Depth gt Motion Bound?

Obstacle
Robot A(q)
Cell
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Translational Penetration Depth PDt
B

• Minimum translation to separate A, B
• Dobkin 93, Agarwal 00, Bergen 01, Kim 02
• PDt not applicable
• The robot is allowed to both translate and
  rotate.
• Undergoing rotation, A may escape from B more
  easily

A
A
B
A
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Generalized Penetration Depth PDg

• Take into account translational and rotational
  motion
• L. Zhang, Y. Kim, G. Varadhan, D. Manocha, ACM
  Solid and Physical Modeling 06
• Trajectory length
• Distance metric Dg
• Min/Max operations

A(q1)
A(q0)
Trajectory length
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PDg Computation

• Difficult for non-convex objects
• Theorem for convex objects, PDg PDt
• Convex/Convex
• Known efficient PDt algorithms directly
  applicable
• Dobkin 93, Agarwal 00, Bergen 01, Kim 02
• Non-Convex / Non-Convex
• A lower bound on PDg based on convex decomposition

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C-obstacle Query

• Is Penetration Depth gt Motion Bound?

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Motion Bound
Configuration space

• Schwarzer, Saha, Latombe 04
• Achieved by any diagonal line segment, e.g. qa,c

qa
Cell
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Free Cell Query

• Separation distance describes the clearance
• If Separation Distance Motion Boundthe robot
  can not intersect with the obstacle
• The cell lies inside free space

d
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Experimental Results C-obstacle Query
Computation
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Experimental Results Path Non-existence

• 2D rigid robots with 3-DOF
• 2 translational DOF and 1 rotational DOF

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Two-gear Example
Video
no path!
3.356s
Initial
Cells in C-obstacle
Roadmap in F
Goal
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Performance of Two-gear Example
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Five-gear Example
6.317s
No path!
Initial
Goal
Cells in C-obstacle
Roadmap in F
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Narrow PassageModified Five-gear Example
Initial
Goal
roadmap in free space
Video
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2D Puzzle
Removed
Goal
B1
B2
B3
B4
Initial
Narrow passage 15.8s
No path! 7.9s
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Conclusion

• C-obstacle query is essential for deciding path
  non-existence
• Efficient C-obstacle and free cell queries
• Workspace-based
• Using generalized penetration depth and
  separation distance computation
• Improved cell decomposition algorithm
• Simple
• Efficient for path non-existence

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Limitations

• C-obstacle free cell queries are conservative
• Can not deal with compliant motion planning
• Current implementation of cell decomposition is
  limited to 3-DOF robots

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Future Work

• Higher DOF motion planning
• 6 DOF rigid robot
• C-obstacle free cell queries are applicable
• Combinatorial complexity of cell decomposition
• Hybrid planner
• To combine with sampling based approach

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Acknowledgements

• Army Research Office, DARPA/REDCOM, NSF, ONR,
  Intel Corporation
• KRF, STAR program of MOST, Ewha SMBA consortium,
  ITRC program, Korea
• Mink2D, Tel Aviv University
• GAMMA Group, UNC Chapel Hill

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Thank you!Any Questions?

• http//gamma.cs.unc.edu/nopath

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Dg Metric in C-space
Trajectory length
Motion Paths in C-Space
Y
?
Dg(q0,q1)
X
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PDg definition
The minimum Dg distance over all possible
collision-free configurations
PDg
A
B
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Lower Bound on PDg

• Convex decomposition
• Eliminate non-overlapping pairs
• PDt for overlapping pairs
• LB(PDg) Max over all PDts

PDt
PDt
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Performance of Five-gear Example
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Compared with Star-shaped roadmap

• Pros
• Simpler than the star-shaped test
• Need not capture the intra-connectivity
• More likely to be extended for higher DOF
• Cons
• More conservative