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Unformatted text preview: a path in the space by finding a path between cells
Source: O'Rourke Moving a Ladder (continued)
Find Free Path through Configuration Space Retraction method Construct "Voronoi diagram" of obstacles for (fixed orientation of) ladder L Set of free points x such that, when ladder's reference point is placed at x, L is equidistant from >= 2 obstacle points Distance of point p to ladder L is minimum length of any line segment from p to a point on L. "Stack" Voronoi diagrams for successive angles to form twisted "sheets " Perform path planning in "network" of diagram formed by ribs between Voronoi sheets rib is place where 2 sheets meet "Voronoi vertex" is equidistant from at least 3 obstacle points. Moving L so its reference point stays on diagram edges places L as far from nearby obstacles as possible.
obstacles a "Voronoi diagram" edge Source: O'Rourke Moving a Ladder (continued)
Find Free Path through Configuration Space
2D Time Complexity
Authors
Shwartz, Sharir O'Dunlaing et al. Leven , Sharir Sifrony, Sharir Vegter O'Rourke
Source: O'Rourke Date
1983 1987 1987 1987 1990 1985b Time Complexity
O(n 5 ) O( n 2 log n log* n) O( n 2 log n) O( n 2 log n)
O(n 2 ) ( n 2 ) 3D Time Complexity
Authors
Shwartz, Sharir Ke, O'Rourke Canny Ke, O'Rourke Date
1984 1987 1987 1988 Time Complexity
O( n11 ) O( n 6 log n) O( n 5 log n)
( n 4 ) Canny (1987): Any motion planning problem in which the robot has d degrees of motion freedom can be solved in O(ndlogn) time. Robot Arm Motion Planar, multilink arm links L1, L2,.., Ln, connected at joints J0, J1, J2,.., Jn joint J0 anchored at origin no obstacles arm may selfintersect
L1 L2 can tip of arm reach this? L1 can reach all points on this circle L2 can reach all points on each such circle centered on a point of L1's circle origin = J0 J1 tip of arm Reachable region for an nlink arm is an annulus centered on the origin Robot Arm Motion: Reachability Region Two cases showing reachability region for a 2link arm is an annulus centered on the origin Source: O'Rourke Robot Arm Motion: Reachability Region Reachability region is independent of order in which links are arranged. justify using parallelogram determined by link vector sum (commutativity of vector a...
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 Spring '10
 DANIELS
 Algorithms

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