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## Probabilistic Roadmap Methods (PRMs)

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(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 PlanningOutline

- 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

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)

Path is 1D curve

Motion Planning in C-spaceSimple 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

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: 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 & ConsRelated 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

- 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

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...

- To Navigate Narrow Passages we must sample in them
- most PRM nodes are where planning is easy (not needed)

PRM Roadmap

3

4

5

1

OBPRM: Finding Points on C-obstaclesBasic 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

OBPRM for Paper Folding

Box: 12 => 5 dof

Periscope: 11 dof

- Roadmap Statistics
- 218 nodes, 3 sec

- Roadmap Statistics
- 450 nodes, 6 sec

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

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]

- 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

- 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

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

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

Query

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