1. Artificial Intelligence Lab
Finding “Narrow Passages” with Probabilistic Roadmaps: The Small-Step Retraction Method
Mitul Saha and Jean-Claude Latombe
Hello/Hi,
My name is Mitul and I am from Stanford AI lab.
Research supported by
NSF, ABB and GM
Artificial Intelligence Lab
Stanford University

2. Probabilistic Roadmaps (PRM)
milestone
local path
Roadmap
components
goal
configuration
start
configuration
We have the problem of finding a collision free paths between start and goal configurations.
The configuration space is usually unknown as it would take exponential amount of time to build one.
Instead we construct a roadmap to capture/represent/approximate the freespace of the robot.
Then paths are found by query-ing the roadmap.
c-obstacle
free-space
Configuration-space
components
[Kavraki, Svetska, Latombe, Overmars, 1996]

3. PRM planners solve complicated problems
Environment with Complex Geometry and robots with large number of DOFs
Can find a path from a fraction of seconds to few minutes IF c-space is FREE from narrow passages
Without explicitly computing the configuration space of the robot
Complex geometries:
obstacles: polygons
Robot: 4053 polygons
High dimensional

4. Main Issue: “Narrow Passages”
low density of
free samples
high density
of free samples
narrow passage
free samples
It is difficult to capture the free space associated with the narrow passages
Because in the narrow passage, volume associated with the free space is relatively low.
colliding samples
colliding local path
The efficiency of PRM planners drops dramatically
in spaces with narrow passages

5. Main Issue: “Narrow Passages”
Problems with “narrow passages” are commonly encountered
Narrow passage problems are not fictitious!!
Narrow passages are quite common in path planning problems. For example this is a very
Common industrial problem where the robot must place its bulky end-effector or welding gun
Deep inside the car through the opening in the door window. A typical collision free path would
Go through configurations which have very low clearance between the window edges and the welding
Gun.
In industrial scenarios consideration of compactness during design of workcells gives rise to narrow
Passages in the configuration space of the operating robots.

6. Main Issue: “Narrow Passages”?
Proposed strategies:
Filtering strategies,
e.g., Gaussian sampling [Boor et al. ‘99]
and bridge test [Hsu et al. ‘03]
 rely heavily on rejection sampling
Retraction strategies,
e.g., [Wilmart et al. ‘99][Lien et al. ‘03]
 waste time moving many configurations out of collision
Narrow passages pose a difficult challenging to PRM based motion planning.
The intriguing nature of “Narrow passages” has attracted a lot of attention in the PRM based motion planning community.
A lot of interesting techniques have been proposed.
They can be classified into two techniques: Filtering and Retraction strategies.
Filtering strategies consist of over-sampling c-space and retaining much fewer collision-free
Confs which are more likely to be narrow passages. Due to heavy rejection sampling,
they have a significant overhead in running time. Hence they can be inefficient for problems without
Difficult narrow passages.
We do not reject samples, instead we increase the density of samples in narrow passages
By retracting barely colliding confs into narrow passages.
Retraction based methods retract colliding configurations into the free space. We have observed that
Attempting to retracting deeply embedded confs can make the planner inefficient.
We only retract barely colliding samples. They are relatively easier to retract than deeply embedded colliding confs
And significantly improve the density of free samples in the narrow passages.

7. Motivating Observation
easy
narrow passages
difficult
planning
time
We observed that slightly widening difficult narrow passages dramatically improves the efficiency of PRM planning.
We also observe not much gain for easy narrow passages, but anyway we do not add significant overhead.
decreasing width of
the narrow passage

8. Small-Step Retraction Method
free space
c-obstacle
start
goal
Roadmap
construction
and repair
fattened
free space
widened
passage
Fattening
(1)
(2 & 3)
Slightly fatten the robot’s free space
Construct a roadmap in fattened free space
Repair the roadmap into original free space

9. Small-Step Retraction Method
free space
c-obstacle
start
goal
Roadmap
construction
and repair
Fattening
widened
passage
fattened
free space
Free space can be “indirectly” fattened by
reducing the scale of the geometries (usually of
the robot) in the 3D workcell with respect to
their medial axis
-This can be pushed into the pre-processing
phase

10. Small-Step Retraction Method
free space
c-obstacle
start
goal
fattened
free space
widened
passage
Roadmap
construction
and repair
Fattening
start
Repair during
construction
Pessimist
Strategy
Optimist
Strategy
goal
Repair after
construction
fattened
free space

11. Small-Step Retraction Method
free space
c-obstacle
start
goal
fattened
free space
widened
passage
Roadmap
construction
and repair
Fattening
start
Optimist may fail due to
“false passages” but Pessimist
is probabilistically complete
Hence Optimist is less reliable,
but much faster due to its
lazy strategy
Repair during
construction
Pessimist
Strategy
Optimist
Strategy
goal
Repair after
construction
fattened
free space

12. Small-Step Retraction Method
free space
c-obstacle
start
goal
fattened
free space
widened
passage
Roadmap
construction
and repair
Fattening
start
Repair during
construction
Pessimist
Strategy
Integrated planner:
1. Try Optimist for N time.
2. If Optimist fails,
then run Pessimist
Optimist
Strategy
goal
Repair after
construction
fattened
free space

13. Upto 3 orders of magnitude improvement in the planning time
Quantitative Results
Fattening “preserves” topology/ connectivity of the free space
Fattening “alters” the topology/ connectivity of the free space
(h)
(a)
(b)
(c)
Alpha 1.1
(g)
Alpha 1.0
(d)
(f)
(e)
Time
SSRP
(secs)
SBL
(g)
386
572
(h)
3365
>100000
Time
SSRP
(secs)
SBL
(a)
9.4
12295
(b) 32
5955
(c)
2.1 41
(d)
492
863
(e) 65
631
(f)
13588
>100000
Our
planner
A recent
PRM planner
SBL: an efficient and recent single query planner
All times are in seconds
Gains upto 3 order or magnitudes seen
Upto 3 orders of magnitude
improvement in the planning time
was observed

14. Quantitative Results Test environments “without” narrow passages
SSRP and SBL have similar performance
In test environments without narrow passages, SSRP is as efficient as SBL.
This is also a significant achievement because previous planners designed for problems with narrow passages
have significant overheads, which makes them inefficient for problems without narrow passages.
Recall that we repair only barely colliding configurations and hence do not have significant overhead.
(i)
(j)
Time
SSRP
SBL
(i)
1.68
1.60
(j)
2.59
2.40

15. Conclusion
SSRP is very efficient at finding narrow passages and still works well when there is none
The main drawback is that there is an additional pre-computation step

16. Finding “Narrow Passages” with Probabilistic Roadmaps: The Small-Step Retraction Method
Hello/Hi,
My name is Mitul and I am from Stanford AI lab.