Citation Motion Planning for a Rigid Body Using
Random Networks on the Medial Axis of the Free Space Steven A. Wilmarth, Nancy M. Amato, and
Peter F. Stiller Proceedings of the 15th Annual ACM
Symposium on Computational Geometry, 1999, pp. 173-180
Wilmarth is a Math Ph.D. from Texas A&M MAPRM = Medial Axis PRM
Why Medial Axes are Useful
Medial axis = lines in voronoi diagram Represent maximal clearance paths for
robots Excellent vertices for PRM in narrow
passages
Sampling from the Medial Axis
Very difficult to compute medial axis explicitly
Main idea:“retract a configuration, free or not, onto the medial axis of the free space without having to compute the medial axis explicitly”
retract = map a point onto another point
Retraction to the Medial Axis
Two types of points: simple point – one nearest neighbor multiple point – two nearest neighbors
Want to retract simple points Find nearest neighbor of simple point Move away from nearest neighbor until
additional nearest neighbor arises
Retraction from Blocked Space
Find nearest point on obstacle boundary Retract from that point as before
Rigid Body Robots More complicated problem
Not assuming convex robots, obstacles Collision checking more expensive Must account for both rotation and
translation
Robot
SE(3) Configuration Space SE(3)
Translation: tx, ty, tz (T) Rotation: rx, ry, rz (R) 6-dim, as opposed to 3-dim point robot
Collision checking transformed point q becomes Rq + p transforming robot yields set of points O(n) collision checking is now much more
Distance Metric Want Riemannian (distance) metric on
SE(3) Two criteria:
Shortest path between (R,p1),(R,p2) is wholly translational
Shortest path from free configuration to contact configuration is also wholly translational
Achieved by weighted sum of T and R R is weighted more so that movement via
rotation is more expensive than translation
Complexity Analysis for Algorithm 4.2 Must check all features of robot and all
features of obstacles O(nU*nV*log(nUnV) + nU*nV*tcd(nU,nV)) tcd(n,m) is
collision detection time for objects of size m and n constant for polygonal robots and obstacles
Finding nodes is substantially more expensive
Test Scenario Must pass block through narrow pipe Rest of the block is solid
Two experiments 1: Cube Width = 2 2: Cube Width = 1.5
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2.5