Complete Motion Planning LiangJun Zhang Robotics Comp 790
Complete Motion Planning Liang-Jun Zhang Robotics, Comp 790 -072 Oct 26, 2006 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Motion Planning To find a path 72 DOFno path To report Courtesy of P. Isto and M. Saha, 2006 Goal Robot Initial Obstacle 3 Obstacle The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Why Complete Motion Planning? Probabilistic roadmap motion planning 4 Complete motion planning Efficient Work for complex problems with many DOF Always terminate Difficult for narrow passages May not terminate when no path exists Not efficient Not robust even for low DOF The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Path Non-existence Problem Robot Initial 5 Obstacle Goal The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Main Challenge Exponential complexity: n. DOF Degree of freedom: DOF Geometric complexity: n More difficult than finding a path To check all possible paths Robot Initial 6 Obstacle Goal The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Approaches Exact Motion Planning Based on exact representation of free space Approximation Cell Decomposition (ACD) A Hybrid planner 7 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Configuration Space: 2 D Translation Workspace Configuration Space Goal Obstacle Robot Start 8 C-obstacle Free y x The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Configuration Space Computation [Varadhan et al, ICRA 2006] 2 Translation + 1 Rotation 215 seconds Obstacle y x Robot 9 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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 • 3 DOF 10 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Approaches Exact Motion Planning Based on exact representation of free space Approximation Cell Decomposition (ACD) A Hybrid Planner Combing ACD and PRM 11 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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] 12 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Approximation Cell Decomposition Full cell Configuration Space full mixed Empty cell Mixed Uncertain empty 13 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Connectivity Graph Gf : Free Connectivity Graph G: Connectivity Graph Gf is a subgraph of G 14 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Finding a Path by ACD Gf : Free Connectivity Graph Initial Goal 15 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Finding a Path by ACD First Graph Cut Algorithm L: Guiding Path Guiding path in connectivity graph G Only subdivide along this path Update the graphs G and Gf Described in Latombe’s book 16 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
First Graph Cut Algorithm L Only subdivide along L 17 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Finding a Path by ACD 18 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
ACD for Path Non-existence Initial Goal C-space 19 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
ACD for Path Non-existence Connectivity Graph 20 Guiding Path The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
ACD for Path Non-existence Connectivity graph is not connected No path! Sufficient condition for deciding path non-existence 21 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Two-gear Example Video no path! Initial 3. 356 s Cells in C-obstacle Roadmap in F Goal 22 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Cell Labeling full Free Cell Query mixed Whether a cell completely lies in free space? C-obstacle Cell Query Whether a cell completely lies in C-obstacle? 23 empty The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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 24 Workspace The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Clearance Separation distance A well studied geometric problem d Determine a volume in C-space which are completely free 25 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
C-obstacle Query Another Collision Detection Problem • Does the cell lie inside C-obstacle? • Do robot and obstacle intersect at all configurations? Robot ? Obstacle Configuration space 26 Workspace The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
‘Forbiddance’: dual to clearance Penetration Depth A geometric computation problem less investigated PD [Zhang et al. ACM SPM 2006] 27 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Limitation of ACD Combinatorial complexity of cell decomposition Limited for low DOF problem 3 -DOF robots 28 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Approaches Exact Motion Planning Based on exact representation of free space Approximation Cell Decomposition (ACD) A Hybrid Planner Combing ACD and PRM 29 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Hybrid Planning Probabilistic roadmap motion planning Complete Motion Planning + Efficient + Many DOFs + Complete - Narrow passages - Path non-existence - Not efficient Can we combine them together? 30 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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 31 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Connectivity Graph Gf : Free Connectivity Graph G: Connectivity Graph Gf is a subgraph of G 32 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Pseudo-free edges Initial Goal Pseudo free edge for two adjacent cells 33 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Pseudo-free Connectivity Graph: Gsf = Gf + Pseudo-edges Initial 34 Goal The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Algorithm Gf Gsf G 35 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Results of Hybrid Planning 36 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Results of Hybrid Planning 37 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Results of Hybrid Planning 2. 5 - 10 times speedup 3 DOF 4 DOF timing cells # Hybrid 34 s 50 K 16 s 48 K 102 s 164 K ACD 85 s 168 K ? ? Speedup 2. 5 3. 3 ≥ 10 ? 38 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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 39 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Future Work Complete motion planning for 6 DOF rigid robots More accurate PDg computation Efficient C-Obstacle representation and computation Extend for articulated robots 40 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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, 4: 298– 351, 1983. Gokul Varadhan, Dinesh Manocha, Star-shaped Roadmaps - A Deterministic Sampling Approach for Complete Motion Planning, Robotics: Science and Systems 2006 41 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Reference: Approximation Cell Decomposition • T. Lozano-P´erez and M. Wesley. An algorithm for planning collisionfree paths among polyhedral obstacles. Comm. ACM, 22(10): 560– 570, 1979. • R. A. Brooks and T. Lozano-P´erez. A subdivision algorithm in configuration space for findpath with rotation. IEEE Trans. Syst, SMC-15: 224– 233, 1985. • D. Zhu and J. Latombe. Constraint reformulation in a hierarchical path planner. Proceedings of International Conference on Robotics and Automation, pages 1918– 1923, 1990. L. Zhang, Y. Kim, and D. Manocha. A simple path nonexistence 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 42 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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