Sampling Strategies for PRMs modified from slides of T.V.N. Sri Ram Basic PRM algorithm Issue • Narrow passages OBPRMs •A randomized roadmap method for path and manipulation planning (Amato,Wu ICRA’96) •OBPRM: An obstacle-based PRM for 3D workspaces (Amato,Bayazit, Dale, Jones and Vallejo) Roadmap candidate points chosen on Cobstacle surfaces Basic Ideas Given Algorithm 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 Cobstacle 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 Issues (contd) • Paths touch C-obstacle Main Advantage • Useful in manipulation planning where the robot has to move along contact surfaces • Useful when C-space is very cluttered. Results Bridge Test The Bridge Test for Sampling Narrow Passages with Probabilistic Roadmap Planners (Hsu, Jiang, Reif, Sun ICRA’03) Main Idea • Accept mid-point as a new node in roadmap graph if two end-points are in collision and midpoint is free • Constrain the length of the bridge: Favourable to build these in narrow passages Algorithm 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 (πB )around a point P, which has been chosen as first endpoint. • Combining Bridge Builder and Uniform Sampling – π =(1-w). πB +w.πv – πB : probability density induced by the Bridge Builder – πv : probability density induced by uniform sampling Results Nmil Nclear Ncon Medial-Axis Based PRM MAPRM: A Probabilistic Roadmap Planner with Sampling on the Medial Axis of the Free Space (Wilmarth, Amato, Stiller ICRA’99) Definitions 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 xo (nearest boundary point) for each point x in Free Space. • Search along the ray xox and find arbitrarily close points xa and xb s.t. xo is the nearest point on the boundary for xa but not xb. • 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 Cobstacle 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 MAPRM for 3D rigid bodies Example 2 Example 3 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?