1. Probabilistic Roadmap Methods (PRMs)
Nancy M. Amato
Parasol Lab,Texas A&M University

2. Motion Planning The Alpha Puzzle The Piano Movers Problem start goal
obstacles
(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.
Have movies for both cases - you can play them as you speak.

3. Applications of PRM-based Motion Planning (Amato et al)

4. 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

5. Configuration 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
C-obst
3D C-space
(a,b,g) a b g
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)

6. C-obstacles workspace obstacles transformed to C-obstacles
Figure from Latombe’91

7. C-obstacles => complicated 3D C-obstacle
simple 2D workspace obstacle
=> complicated 3D C-obstacle
Figure from Latombe’91

8. Motion Planning in C-space
Simple workspace obstacle transformed
Into complicated C-obstacle!!
C-space
Workspace
C-obst
C-obst
Path is 1D curve
Path is swept volume
obst
obst
C-obst
C-obst
obst
obst y x
robot
robot

9. 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

10. Roadmap Construction (Pre-processing)
Probabilistic Roadmap Methods (PRMs) [Kavraki, Svestka, Latombe,Overmars 1996]
C-space
Roadmap Construction (Pre-processing)
Query processing
start
goal
1. Randomly generate robot configurations (nodes)
- discard nodes that are invalid
1. Connect start and goal to roadmap
C-obst
2. Connect pairs of nodes to form roadmap
- simple, deterministic local planner (e.g., straightline)
- discard paths that are invalid
2. Find path in roadmap between start and goal
- regenerate plans for edges in roadmap
C-obst
C-obst
C-obst
C-obst
Primitives Required:
Method for Sampling points in C-Space
Method for `validating’ points in C-Space

11. PRMs: Pros & Cons PRMs: The Good News PRMs: The Bad News
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
start
goal
PRMs: The Bad News
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.
start
goal
C-obst

12. 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]

13. 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

14. 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)
start
goal
C-obst
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 Roadmap
PRM Roadmap
start
goal
C-obst

15. OBPRM: Finding Points on C-obstacles2 4
Basic Idea (for workspace obstacle S) 5 3
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 1
C-obst

16. PRM vs OBPRM Roadmaps PRM 328 nodes 4 major CCs OBPRM 161 nodes

17. OBPRM for Paper Folding
Box: 12 => 5 dof
Periscope: 11 dof
Roadmap Statistics
218 nodes, 3 sec
Roadmap Statistics
450 nodes, 6 sec

18. MAPRM: Medial-Axis PRM Steve 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.
Medial axis
C-obstacle

19. Repairing Approximate Paths
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
repaired
path
Create initial roadmap
Extract approximate path P
Repair P (push to C-free)
Focus search around P
Use OBPRM-like techniques
C-obstacle
approximate
path
C-obstacle

20. Hybrid Human/Planner System [IEEE ICRA’01, IEEE ICRA’00, PUG’99]
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
Intelligent CAD Applications
Molecule Docking in drug design
Animation w/ Deformable Models

21. Hybrid Human/Planner System Haptic Hints Results
0.85
0.95 1
Flange Problem

22. Hybrid Human/Planner System Haptic Hints Results
0.85
0.95 1
Flange Problem

23. 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

24. 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

25. Deformable Objects Repairing Path using 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.
Apply FFD
ChainMail Box
Deformed ChainMail
Bounding Box
Obstacle

26. Approximate Solution Query
The approximate path returned is not the shortest path but the energetically most feasible path (less deformation required)
Query