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 C-obstacle surfaces

Basic Ideas

Algorithm

Given

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

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 mid-point 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 end-point.

• 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?