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# Probabilistic Roadmap Methods (PRMs) - PowerPoint PPT Presentation

Probabilistic Roadmap Methods (PRMs). Nancy M. Amato Parasol Lab,Texas A&M University. The Piano Movers Problem. (Basic) Motion Planning : Given a movable object and a description of the environment , find a sequence of valid configurations that moves it from the start to the goal. .

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Presentation Transcript

Nancy M. Amato

Parasol Lab,Texas A&M University

The Piano Movers Problem

(Basic) Motion Planning:

Given a movableobject and a description of the environment, find a sequence of valid configurations that moves it from the start to the goal.

start

The Alpha Puzzle

goal

obstacles

Motion Planning
Outline
• C-space, Planning in C-space (basic definitions)
• PRM variants (OBPRM, MAPRM)
• User-Input: Haptically Enhanced PRMs
• PRMs for more `complex’ robots
• deformable objects
• group behaviors for multiple robots
• protein folding

C-obst

C-obst

C-obst

C-obst

C-obst

g

b

a

3D C-space

(a,b,g)

Configuration Space (C-Space)

C-Space

• “robot” maps to a point in higher
• dimensional space
• parameter for each degree of freedom
• (dof) of robot
• C-space = set of all robot placements
• C-obstacle = infeasible robot placements

3D C-space

(x,y,z)

6D C-space

(x,y,z,pitch,roll,yaw)

2n-D C-space

(f1,y1, f2, y2, . . . , f n, y n)

simple 2D workspace obstacle

=>complicated 3D C-obstacle

C-obstacles

Figure from Latombe’91

Path is swept volume

Path is 1D curve

Motion Planning in C-space

Simple workspace obstacle transformed

Into complicated C-obstacle!!

C-space

Workspace

C-obst

C-obst

obst

obst

C-obst

C-obst

obst

obst

y

x

robot

robot

The Complexity of Motion Planning

Most motion planning problems of interest are

PSPACE-hard[Reif 79, Hopcroft et al. 84 & 86]

• The best deterministic algorithm known has running time that is exponential in the dimension of the robot’s C-space [Canny 86]
• C-space has high dimension - 6D for rigid body in 3-space
• simple obstacles have complex C-obstacles impractical to compute explicit representation of freespace for more than 4 or 5 dof
• So … attention has turned to randomized algorithms which
• trade full completeness of the planner
• for probabilistic completeness and a major gain in efficiency

goal

1. Randomly generate robot configurations (nodes)

- discard nodes that are invalid

2. Connect pairs of nodes to form roadmap

- simple, deterministic local planner (e.g., straightline)

- discard paths that are invalid

Query processing

start

1. Connect start and goal to roadmap

2. Find path in roadmap between start and goal

- regenerate plans for edges in roadmap

Probabilistic Roadmap Methods (PRMs)[Kavraki, Svestka, Latombe,Overmars 1996]

C-space

C-obst

C-obst

C-obst

C-obst

C-obst

• Primitives Required:
• Method for Sampling points in C-Space
• Method for `validating’ points in C-Space

goal

PRMs: The Good News

C-obst

1. PRMs are probabilistically complete

2. PRMs apply easily to high-dimensional C-space

3. PRMs support fast queries w/ enough preprocessing

Many success stories where PRMs solve previously unsolved problems

C-obst

C-obst

C-obst

C-obst

start

goal

C-obst

C-obst

• 1. PRMs don’t work as well for some problems:
• unlikely to sample nodes in narrow passages
• hard to sample/connect nodes on constraint surfaces
• Our work concentrates on improving PRM performance for such problems.

C-obst

C-obst

start

PRMs: Pros & Cons
Related Work (selected)
• Uniform Sampling (original) [Kavraki, Latombe, Overmars, Svestka, 92, 94, 96]
• Obstacle-based PRM (OBPRM) [Amato et al, 98]
• PRM Roadmaps in Dilated Free space[Hsu et al, 98]
• Gaussian Sampling PRMs[Boor/Overmars/van der Steppen 99]
• PRM for Closed Chain Systems[Lavalle/Yakey/Kavraki 99, Han/Amato 00]
• PRM for Flexible/Deformable Objects[Kavraki et al 98, Bayazit/Lien/Amato 01]
• Visibility Roadmaps[Laumond et al 99]
• Using Medial Axis [Kavraki et al 99, Lien/Thomas/Wilmarth/Amato/Stiller 99, 03, Lin et al 00]
• Generating Contact Configurations[Xiao et al 99]
• Single Shot [Vallejo/Remmler/Amato 01]
• Bio-Applications: Protein Folding [Song/Thomas/Amato 01,02,03, Apaydin et al 01,02]
• Lazy Evaluation Methods: [Nielsen/Kavraki 00 Bohlin/Kavraki 00, Song/Miller/Amato 01, 03]
• Related Methods
• Ariadnes Clew Algorithm [Ahuactzin et al, 92]
• RRT (Rapidly Exploring Random Trees) [Lavalle/Kuffner 99]
Other Sampling Strategies
• Obstacle Sensitive Sampling Strategies
• Goal: Sample nodes in Narrow Passages
• OBPRM: Obstacle-Based PRM
• sampling around obstacles
• MAPRM: Medial-Axis PRM
• sampling on the medial axis
• Repairing/Improving approximate paths
• transforming invalid samples into valid samples

goal

goal

C-obst

C-obst

C-obst

C-obst

C-obst

C-obst

C-obst

C-obst

start

start

• Idea: Can we sample nodes near C-obstacle surfaces?
• we cannot explicitly construct the C-obstacles...
• we do have models of the (workspace) obstacles...
• To Navigate Narrow Passages we must sample in them
• most PRM nodes are where planning is easy (not needed)

2

3

4

5

1

OBPRM: Finding Points on C-obstacles

Basic Idea (for workspace obstacle S)

1. Find a point in S’s C-obstacle

(robot placement colliding with S)

2. Select a random direction in C-space

3. Find a free point in that direction

4. Find boundary point between them

using binary search (collision checks)

Note: we can use more sophisticated heuristics to try to cover C-obstacle

C-obst

• PRM
• 328 nodes
• 4 major CCs
• OBPRM
• 161 nodes
• 2 major CCs
OBPRM for Paper Folding

Box: 12 => 5 dof

Periscope: 11 dof

• 218 nodes, 3 sec
• 450 nodes, 6 sec

Medial axis

MAPRM: Medial-Axis PRMSteve Wilmarth, Jyh-Ming Lien, Shawna Thomas [IEEE ICRA’99, ACM SoCG’99, IEEE ICRA’03]
• Key Observation: We can efficiently retract almost any configuration, free or not, onto the medial axis of the free space without computing the medial axis explicitly.

C-obstacle

repaired

path

approximate

path

Repairing Approximate Paths
• Extract approximate path P
• Repair P (push to C-free)
• Focus search around P
• Use OBPRM-like techniques
• Problem: Even with the best sampling methods, roadmaps may not contain valid solution paths
• may lack critical cfgs in narrow passages
• may contain approximate paths that are ‘nearly’ valid

Repairing/Improving Approximate Paths

C-obstacle

C-obstacle

PHANToM

• 1. User collects approximate path using haptic device
• User insight identifies critical cfgs
• User feels when robot touches obstacles and adjusts trajectory
• 2. Approximate path passed to planner and it fixes it
• Current Applications
• Molecule Docking in drug design
• Animation w/ Deformable Models
Applications
• Same Method: Diverse range of problems
• Deformable Objects
• Simulating Group Behaviors
• homing, covering, shepherding
• Modeling Molecular Motions
• Protein folding, RNA folding, protein/ligand binding
Deformable Objects[IEEE ICRA’02]
• Problem: Find a path for a deformable object that can deform to avoid collision with obstacles
• deformable objects have infinite dof!
• Our Approach:
• Build roadmap relaxing collision free requirement (allow penetration)
• Extract approximate path for a rigid version of robot (select path with smallest collision/penetration)
• Follow path and deform the robot to avoid the collisions

Deformed ChainMail

Bounding Box

Apply FFD

ChainMail Box

Obstacle

• Bounding Box Deformation
• build a 3D voxel bounding box.
• Convert it to ChainMail bounding box (3D grid of springs)
• Deform ChainMail Bounding box.
• Deform objects using Free Form Deformation based on
• deformation of the bounding box.
Approximate Solution

The approximate path returned is not the shortest path but the energetically most feasible path (less deformation required)

Query