probabilistic roadmap methods prms l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Probabilistic Roadmap Methods (PRMs) PowerPoint Presentation
Download Presentation
Probabilistic Roadmap Methods (PRMs)

Loading in 2 Seconds...

play fullscreen
1 / 26

Probabilistic Roadmap Methods (PRMs) - PowerPoint PPT Presentation


  • 428 Views
  • Uploaded on

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. .

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Probabilistic Roadmap Methods (PRMs)' - Antony


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
probabilistic roadmap methods prms

Probabilistic Roadmap Methods(PRMs)

Nancy M. Amato

Parasol Lab,Texas A&M University

motion planning

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

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)

c obstacles7
simple 2D workspace obstacle

=>complicated 3D C-obstacle

C-obstacles

Figure from Latombe’91

motion planning in c space

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
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
probabilistic roadmap methods prms kavraki svestka latombe overmars 1996

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

Roadmap Construction (Pre-processing)

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
prms pros cons

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

PRMs: The Bad News

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
Related Work (selected)
  • Probabilistic Roadmap Methods
    • 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
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
obprm an obstacle based prm ieee icra 96 ieee icra 98 wafr 98

goal

goal

OBPRM Roadmap

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...
OBPRM: An Obstacle-Based PRM[IEEE ICRA’96, IEEE ICRA’98, WAFR’98]
  • To Navigate Narrow Passages we must sample in them
  • most PRM nodes are where planning is easy (not needed)

PRM Roadmap

obprm finding points on c obstacles

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 vs obprm roadmaps
PRM vs OBPRM Roadmaps
  • PRM
  • 328 nodes
  • 4 major CCs
  • OBPRM
  • 161 nodes
  • 2 major CCs
obprm for paper folding
OBPRM for Paper Folding

Box: 12 => 5 dof

Periscope: 11 dof

  • Roadmap Statistics
  • 218 nodes, 3 sec
  • Roadmap Statistics
  • 450 nodes, 6 sec
slide18

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

repairing approximate paths

repaired

path

approximate

path

Repairing Approximate Paths
  • Create initial roadmap
  • 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

hybrid human planner system ieee icra 01 ieee icra 00 pug 99

PHANToM

Hybrid Human/Planner System[IEEE ICRA’01, IEEE ICRA’00, PUG’99]
  • 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
    • Intelligent CAD Applications
    • Molecule Docking in drug design
    • Animation w/ Deformable Models
applications
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
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
deformable objects repairing path using bounding box deformation

Deformed ChainMail

Bounding Box

Apply FFD

ChainMail Box

Obstacle

Deformable ObjectsRepairing Path usingBounding Box Deformation
  • 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
Approximate Solution

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

Query