Finding points on C-objects 1.Determine a point o (the origin) inside s 2.Select m rays with origin o and directions uniformly distributed in C-space 3.For each ray identified above, use binary search to determine a point on s
Issues Selection of o in C- obstacle is crucial –To obtain uniform distribution of samples on the surface, would like to place origin somewhere near the center of C-object. – Still skewed objects would present a problem
Contribution over previous Obstacle–Based Methods Avoids sampling “uninteresting” obstacle boundaries. Local Approach: Does not need to “capture” the C-obstacle in any sense Complementary to the Uniform Sampling Approach
Issues Deciding the probability density (π B )around a point P, which has been chosen as first end- point. Combining Bridge Builder and Uniform Sampling –π =(1-w). π B +w.π v –π B : probability density induced by the Bridge Builder –π v : probability density induced by uniform sampling
Main Ideas Beneficial to have samples on the medial axis; however, computation of medial axis itself is costly. Retraction : takes nodes from free and obstacle space onto the medial axis w/o explicit computation of the medial axis. This method increases the number of nodes found in a narrow corridor –independent of the volume of corridor –Depends on obstacles bounding it
Approach for Free-Space Find x o (nearest boundary point) for each point x in Free Space. Search along the ray x o x and find arbitrarily close points x a and x b s.t. x o is the nearest point on the boundary for x a but not x b. Called canonical retraction map
Extended Retraction Map Doing only for Free-Space => Only more clearance. Doesn’t increase samples in Narrow Passages Retract points that fall in C obstacle also. Retract points in the direction of the nearest boundary point
Results for 2D case LEFT: Helpful: obstacle-space that retracts to narrow passage is large RIGHT: Not Helpful: Obstacle-space seeping into medial axis in narrow corridor is very low
Main Results Demonstrates an approach to use medial axis based ideas with random sampling Advantages: –Useful in cluttered environments. Where a great volume of obstacle space is adjacent to narrow spaces –Above Environment: Bisection technique for evaluating point on medial axis???
Limitations Additional primitive: “Nearest Contact Configuration”. For larger (complex) problems, this time may become significant…. Extension to higher dimensions difficult. Which direction to search for nearest contact?