DeadlockFree and CollisionFree Coordination of Two Robot Manipulators Patrick A. ODonnell and Toms L

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Deadlock-Free and Collision-Free Coordination of
Two Robot ManipulatorsPatrick A. ODonnell and
Tomás Lozano-Pérez
A presentation on

• by
• Guha Jayachandran
• April 22, 2002

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Whats it all about?
A method for coordinating the trajectories of two
robots so that they dont collide -- and dont
deadlock either.
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Whats it all about?
A method for coordinating the trajectories of two
robots so that they dont collide -- and dont
deadlock either.
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Whats it all about?
A method for coordinating the trajectories of two
robots so that they dont collide -- and dont
deadlock either.
Im waiting for you!
Im waiting for you!
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Goals
Goal is to do this in such a way that - Each
robots motions can be planned nearly
independently, - Outputted trajectories
guarantee that the robots reach their goals, -
Trajectories can be executed without precise time
coordination between manipulators, and - Safety
of the manipulators should not depend on accurate
trajectory control of individual manipulators.
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Assumptions
The manipulators are loosely coupled, not tightly
coupled.
The path of each robot can be planned ahead of
time in such a way that it avoids collisions with
all obstacles (except the other robot). Note that
this doesnt mean trajectories are predictable.
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Idea Make into a Scheduling Problem
We have pre-planned path for each manipulator.
Each of these paths is composed of a sequence of
path segments. We roughly estimate the time to
execute each path segment. With this
information, the trajectory coordination problem
has become a scheduling problem with space as the
shared resource! Attack the problem as such...
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Using a Task Completion Diagram
On horizontal axis, segments of the path for
robot B. On vertical axis, segments of the path
for robot A. If volume swept out by robot B while
travelling path segment i overlaps with volume
swept out by robot A while travelling segment j,
then shade in square (i, j). The union of shaded
squares are the collision regions.
B
Diagram from Chris Clarks 2001 presentation.
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A Safe Schedule
Schedule is any non-decreasing curve that
connects lower left corner to top right corner.
Safe schedule is a schedule that doesnt
penetrate the interior of the union of the
collision rectangles.
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Greedy Algorithm to Fill in Diagram
g
A
s
s
g
B
Clearly, gives collision free path...
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Preventing Deadlocks
but need to do more to prevent deadlock.
SW closure of collision regions fills in non
convexities in connected components of the union
of the collision regions.
A
A schedule exists iff neither the origin nor the
goal is part of any collision or SW closure
region.
B
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Making Schedule from T.C. Diagram
Few approaches - Local method. Assumes there is
a central controller that initiates the motions
for both manipulators. (Like Greedy shown
previously) - Decentralized (also could be with
Greedy). A shaded square is a lock that can be
indivisibly tested and set. - Others.
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Parallelism
Parallelism gt Smaller execution time Modify T.C.
diagram so that axes correspond to expected
execution time. Each path segments expected time
determines dimension in diagram.
New segment
A
A
B
B
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More Parallelism
Replan path for a particular segment which is
problematic. Pick a collision square (or larger
collision region formed from union of collision
squares) such that 1. Region is shaded because
of a collision and not because of SW closure. 2.
Initial and final positions of path segments
giving rise to collision region are free of
collisions. 3. Region is large enough that it
causes a significant increase in total time of
the best schedule to go around it.
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Changing the Task
What about when task assignments not fixed? Like
if one robot gets delayed and want another to
take over. Use expanded diagram containing all
combinations of assignments to task steps to
different robots. Start position for each task
must be same. In simple case, end same as start.
A
1
3
2
1
1
2
3
B
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Even Older Approaches
Two types Global ones constructed complete
trajectory for each robot one at a time. Depended
on carefully controlled trajectories and
coordination and was expensive. Local ones (like
using collision avoidance and coordination)
decide trajectory as go, based on measurements of
position. Deadlock a real possibility and not
well suited to where paths tightly
constrained. Big difference Decoupling of path
specification from trajectory specification.
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How Probabilistic Roadmaps Could be Used
Not in the paper (1989), but P.R.M.s could
probably be used to improve approach. Two
possible ways...
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Use P.R.M. with All D.O.F.
Using probabilistic roadmap, we could just
consider the two manipulators as one. That is,
construct roadmap that incorporates all degrees
of freedom of both robots plus time. Since
probabilistic roadmaps handle high dimensionality
well, this should work fine.
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Make P.R.M. Instead of Using Grid
Instead of using greedy algorithm to find path
along grid segments, ignore the grid lines and
construct a P.R.M. in 2D space to find a path
from start to goal in the diagram.
Avoid drawbacks of discretization.
g
A
s
s
g
B
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Final Thoughts
A simple useful algorithm. Key pluses paths can
be planned independently, does not need lock step
coordination, not computationally expensive,
avoids deadlock. Scalability, importance of good
path segmentation are issues to consider. Can be
improved by techniques developed over the past 13
years.