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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Complete Motion Planning Liang-Jun Zhang Robotics, Comp Oct 26, 2006

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 2 Outline Motivation/Challenge Approaches Exact Motion Planning Approximation Cell Decomposition Hybrid Planner

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 3 Motion Planning Initial Goal Obstacle To find a path Robot 72 DOF Courtesy of P. Isto and M. Saha, 2006 Goal Initial Obstacle To report no path Robot

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 4 Why Complete Motion Planning? Probabilistic roadmap motion planning Efficient Work for complex problems with many DOF Difficult for narrow passages May not terminate when no path exists Complete motion planning Always terminate Not efficient Not robust even for low DOF

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 5 Path Non-existence Problem Obstacle Goal Initial Robot

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 6 Main Challenge Obstacle Goal Initial Robot Exponential complexity: n DOF Degree of freedom: DOF Geometric complexity: n More difficult than finding a path To check all possible paths

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 7 Approaches Exact Motion Planning Based on exact representation of free space Approximation Cell Decomposition (ACD) A Hybrid planner

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 8 Configuration Space: 2D Translation Workspace Configuration Space x y Robot Start Goal Free Obstacle C- obstacle

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 9 Configuration Space Computation [Varadhan et al, ICRA 2006] 2 Translation + 1 Rotation 215 seconds Obstacle Robot x y

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 10 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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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 11 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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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 12 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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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 13 fullmixed empty Approximation Cell Decomposition Full cell Empty cell Mixed cell Mixed Uncertain Configuration Space

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 14 Connectivity Graph G f : Free Connectivity GraphG: Connectivity Graph G f is a subgraph of G

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 15 Finding a Path by ACD Goal Initial G f : Free Connectivity Graph

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 16 Finding a Path by ACD L: Guiding Path First Graph Cut Algorithm Guiding path in connectivity graph G Only subdivide along this path Update the graphs G and G f Described in Latombe’s book

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 17 First Graph Cut Algorithm Only subdivide along L L

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 18 Finding a Path by ACD

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 19 ACD for Path Non-existence C-space Goal Initial

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 20 Connectivity Graph Guiding Path ACD for Path Non-existence

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 21 ACD for Path Non-existence Connectivity graph is not connected No path! Sufficient condition for deciding path non-existence

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 22 Two-gear Example no path! Cells in C-obstacle Initial Goal Roadmap in F Vide o 3.356s

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 23 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? fullmixed empty

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 24 Free Cell Query A Collision Detection Problem Does the cell lie inside free space? Do robot and obstacle separate at all configurations? Obstacle Workspace Configuration space ? Robot

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 25 Clearance Separation distance A well studied geometric problem Determine a volume in C-space which are completely free d

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 26 C-obstacle Query Another Collision Detection Problem Does the cell lie inside C-obstacle? Do robot and obstacle intersect at all configurations? Obstacle Workspace Configuration space ? Robot

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 27 ‘Forbiddance’ ‘Forbiddance’: dual to clearance Penetration Depth A geometric computation problem less investigated [Zhang et al. ACM SPM 2006] PD

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 28 Limitation of ACD Combinatorial complexity of cell decomposition Limited for low DOF problem 3-DOF robots

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 29 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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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 30 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?

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 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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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 32 Connectivity Graph G f : Free Connectivity GraphG: Connectivity Graph G f is a subgraph of G

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 33 Pseudo-free edges Pseudo free edge for two adjacent cells Goal Initial

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 34 Pseudo-free Connectivity Graph: G sf Goal Initial G sf = G f + Pseudo-edges

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 35 Algorithm G f G sf G

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 36 Results of Hybrid Planning

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 37 Results of Hybrid Planning

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 38 Results of Hybrid Planning times speedup 3 DOF4 DOF timingcells #timingcells #timingcells # Hybrid34s50K16s48K102s164K ACD85s168K???? Speedup2.53.3≥10? ?

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 39 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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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 40 Future Work Complete motion planning for 6DOF rigid robots More accurate PD g computation Efficient C-Obstacle representation and computation Extend for articulated robots

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 41 Reference: Exact Motion Planning J. Canny. The Complexity of Robot Motion Planning. ACM Doctoral Dissertation Award. MIT Press, 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, 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, Gokul Varadhan, Dinesh Manocha, Star-shaped Roadmaps - A Deterministic Sampling Approach for Complete Motion Planning, Robotics: Science and Systems 2006

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 42 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, 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, D. Zhu and J. Latombe. Constraint reformulation in a hierarchical path planner. Proceedings of International Conference on Robotics and Automation, pages 1918– 1923, L. Zhang, Y. Kim, and D. Manocha. A simple path non- existence algorithm using c-obstacle query. In Proc. of WAFR, L. Zhang, Y.J. Kim, and D. Manocha, A Hybrid Approach for Complete Motion Planning, UNC-CS Tech Report

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