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the University of North Carolina at CHAPEL HILL
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
the University of North Carolina at CHAPEL HILL
Motion Planning
Initial
Goal
Obstacle
To find a path
Robot
72 DOFCourtesy of P. Isto and M. Saha, 2006
Goal
Initial
Obstacle
To report no path
Robot
the University of North Carolina at CHAPEL HILL
Path Non-existence Problem
ObstacleObstacle
GoalInitial
Robot
the University of North Carolina at CHAPEL HILL
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
the University of North Carolina at CHAPEL HILL
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
the University of North Carolina at CHAPEL HILL
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
the University of North Carolina at CHAPEL HILL
Previous Work
• Path non-existence for special cases♦ Planar section, [Basch et al. 01], [Bretl et al.
04]
the University of North Carolina at CHAPEL HILL
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
the University of North Carolina at CHAPEL HILL
Path Non-existence Problem
qgoal
• More difficult than finding a path♦ To check all possible
paths
• Identify a region in C-obstacle ♦ separating qinit and qgoal
qinit
Configuration space
the University of North Carolina at CHAPEL HILL
C-obstacle Query
• Whether a primitive lies entirely in C-obstacle?♦ Usually a cell
• Useful for path non-existence
qgoal
qinit
the University of North Carolina at CHAPEL HILL
full mixed
empty
• [Lozano-Pérez 83]• [Zhu et al. 91]• [Latombe 91]
Cell Decomposition for Path Non-existence
the University of North Carolina at CHAPEL HILL
Cell Decomposition for Path Non-existence
Connectivity Graph Guiding Path
the University of North Carolina at CHAPEL HILL
Cell Decomposition for Path Non-existence
Connectivity graph is not connected
No path!
the University of North Carolina at CHAPEL HILL
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
the University of North Carolina at CHAPEL HILL
C-obstacle QueryA Collision Detection Problem
•Does the cell lie inside C-obstacle?
• Do robot and obstacle intersect at all configurations?
Obstacle
WorkspaceConfiguration space
?Robot
the University of North Carolina at CHAPEL HILL
Clearance VS ‘Forbiddance’
• Separation distance
• Clearance
• Penetration Depth
• ‘Forbiddance’
PDd
the University of North Carolina at CHAPEL HILL
q
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 > Motion Bound?
Robot A(q)
PD
Cell
Obstacle
the University of North Carolina at CHAPEL HILL
Translational Penetration Depth: PDt
• 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
B
A
A’
A
B
the University of North Carolina at CHAPEL HILL
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
Trajectory length A(q0)
A(q1)
the University of North Carolina at CHAPEL HILL
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
the University of North Carolina at CHAPEL HILL
C-obstacle Query
Is Penetration Depth > Motion Bound?
the University of North Carolina at CHAPEL HILL
Motion Bound
• ♦ [Schwarzer, Saha, Latombe
04]
• ♦ Achieved by any diagonal
line segment, e.g. qa,c
bqaq,( , )a bq qMB A
Cell
,( , ) max ( , )a bq qMB A C MB A
bq
cq
Configuration space
qa
the University of North Carolina at CHAPEL HILL
Free Cell Query
• Separation distance describes the clearance• If Separation Distance ≥ Motion Bound
the robot can not intersect with the obstacle♦ The cell lies inside free space
d
the University of North Carolina at CHAPEL HILL
Experimental Results C-obstacle Query Computation
Faces # of A 28 8,452 304
Faces # of B 1,692 336 304
Per C-obstacle
query
1.901 (ms) 6.127 (ms) 4.112 (ms)
the University of North Carolina at CHAPEL HILL
Experimental Results Path Non-existence
• 2D rigid robots with 3-DOF♦ 2 translational DOF and 1 rotational DOF
B1B2
B3
B4
A
A'
the University of North Carolina at CHAPEL HILL
Two-gear Example
no path!
Cells in C-obstacle
Initial
Goal
Roadmap in F
Video 3.356s
the University of North Carolina at CHAPEL HILL
Performance of Two-gear Example
# of C-obstacle queries
30K
# of free cell queries
32K
# of iterations 41
Total timing 3.356s
Free cell queries 0.858s
C-obstacle queries 0.827s
Graph searching 0.466s
Subdivision 1.205s
# of total cells 28K
# of total free cells 2K
# of total c-obstacle cells
12K
# of total mixed cells 14K
the University of North Carolina at CHAPEL HILL
Five-gear Example
Cells in C-obstacle
Initial
Goal
Roadmap in F
6.317sNo path!
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Narrow PassageModified Five-gear Example
Total timing 85s
# of C-obstacle queries
176K
# of total cells 168K
Video
Initial
Goal
roadmap in free space
the University of North Carolina at CHAPEL HILL
2D Puzzle
No path!
7.9s
Narrow passage
15.8s
B1B2
B3
B4
Initial
GoalB1B2
B4
A
A'
VideoRemoved
the University of North Carolina at CHAPEL HILL
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
the University of North Carolina at CHAPEL HILL
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
the University of North Carolina at CHAPEL HILL
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
the University of North Carolina at CHAPEL HILL
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
the University of North Carolina at CHAPEL HILL
Thank you!
Any Questions?
http://gamma.cs.unc.edu/nopath
the University of North Carolina at CHAPEL HILL
Min over every path connecting q0 and q1
Max trajectory length for distinct points
Dg(q0, q1) =
Dg Metric in C-space
X
Yθ
q1q0
l1
l2
Motion Paths in C-Space Trajectory length
A(q0)
A(q1)
Dg(q0,q1)
the University of North Carolina at CHAPEL HILL
min({ ( , ) | int( ( )) })ggPD D A B 0q q q
PDg definition
The minimum Dg distance over all possible collision-free configurationsA
B
PDg
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Lower Bound on PDg
1. Convex decomposition 2. Eliminate non-overlapping pairs
3. PDt for overlapping pairs
4. LB(PDg) = Max over all PDts
PDt
PDt
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Performance of Five-gear Example
Total timing 6.317s
# of total cells 39K
# of C-obstacle queries
41K
the University of North Carolina at CHAPEL HILL
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
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