Sampling From the Medial Axis

Presented by Rahul Biswas

April 23, 2003CS326A: Motion Planning

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

Voronoi Diagram Consists of points equidistant from

obstaclesObstacles

Example Voronoi Diagram

Another Voronoi Diagram

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 Illustration

SampleNearest Neighbor

Retraction from Blocked Space

Find nearest point on obstacle boundary Retract from that point as before

Retraction Maps

Sampling Narrow Passages

Uniform vs MAPRM Samples

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

Algorithm

Algorithm 4.2

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

20

2.5

Experiment 1 Results

Authors surmise problem with corners

Experiment 2 Results

Conclusion Technique to sample intelligently for

PRMs Sample points from the medial axis

without computing Voronoi diagram Works for both rigid bodies and point

robots No extension to articulated robots Interesting concept but not useful in

practice