Algorithmic Robotics and Motion Planning Fall 20067 Algorithmic
Algorithmic Robotics and Motion Planning Fall 2006/7 Algorithmic motion planning, an overview Dan Halperin Tel Aviv University
Motion planning: the basic problem Let B be a system (the robot) with k degrees of freedom moving in a known environment cluttered with obstacles. Given free start and goal placements for B decide whethere is a collision free motion for B from start to goal and if so plan such a motion.
![Configuration space of a robot system with k degrees of freedom [Lozano-Peréz ’ 79]](http://slidetodoc.com/presentation_image_h2/97cc9d3514cac77e045b9fb568e209b6/image-3.jpg "Configuration space of a robot system with k degrees of freedom [Lozano-Peréz ’ 79]")
Configuration space of a robot system with k degrees of freedom [Lozano-Peréz ’ 79] § the space of parametric representation § § § of all possible robot configurations C-obstacles: the expanded obstacles the robot -> a point k dimensional space point in configuration space: free, forbidden, semi-free path -> curve
Point robot www. seas. upenn. edu/~jwk/motion. Planning
Trapezoidal decomposition c 14 c 5 c 2 c 7 c 8 c 15 c 11 c 10 c 3 c 6 c 13 c 9 c 12 www. seas. upenn. edu/~jwk/motion. Planning
Connectivity graph c 14 c 5 c 2 c 7 c 8 c 15 c 11 c 10 c 3 c 6 c 9 c 12 c 4 c 2 c 13 c 14 c 7 c 15 c 8 c 11 c 10 c 3 c 6 c 9 www. seas. upenn. edu/~jwk/motion. Planning c 13 c 12
What is the number of Do. F’s? § § § a a a polygon robot translating in the plane polygon robot translating and rotating spherical robot moving in space spatial robot translating and rotating snake robot in the plane with 3 links
Two major planning frameworks § Cell decomposition § Road map § Motion planning methods differ along additional parameters
Hardness § The problem is hard when k is part of the input [Reif 79], [Hopcroft et al. 84], … § [Reif 79]: planning a free path for a robot made of an arbitrary number of polyhedral bodies connected together at some joint vertices, among a finite set of polyhedral obstacles, between any two given configurations, is a PSPACE-hard problem § Translating rectangles, planar linkages
![Complete solutions, I the Piano Movers series [Schwartz-Sharir 83], cell decomposition: a doubly-exponential solution,](http://slidetodoc.com/presentation_image_h2/97cc9d3514cac77e045b9fb568e209b6/image-10.jpg "Complete solutions, I the Piano Movers series [Schwartz-Sharir 83], cell decomposition: a doubly-exponential solution,")
Complete solutions, I the Piano Movers series [Schwartz-Sharir 83], cell decomposition: a doubly-exponential solution, O(nd)3^k) expected time assuming the robot complexity is constant, n is the complexity of the obstacles and d is the algebraic complexity of the problem
![Complete solutions, II roadmap [Canny 87]: a singly exponential solution, nk(log n)d. O(k^2) expected](http://slidetodoc.com/presentation_image_h2/97cc9d3514cac77e045b9fb568e209b6/image-11.jpg "Complete solutions, II roadmap [Canny 87]: a singly exponential solution, nk(log n)d. O(k^2) expected")
Complete solutions, II roadmap [Canny 87]: a singly exponential solution, nk(log n)d. O(k^2) expected time
And now to something completely different (temporarily diffrenet)
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- Slides: 13
