Deadlock-Free and Collision-Free Coordination of Two Robot Manipulators

1
Deadlock-Free and Collision-Free Coordination of
Two Robot Manipulators

• Presented by Huy Nguyen
• April 28, 2003

2
Introduction

• Goal
• Coordinate the trajectories of two robot
  manipulators so as to avoid collisions and
  deadlock.
• Definitions
• path Curve in C-space.
• trajectory Time history of positions along
  path (curve in state space).

3
The Approach

• Patrick A. ODonnell and Tomas Lozano-Perez 89
• Decouple path specification step from trajectory
  specification step.
• Assume path of each manipulator has been planned
  off-line and is composed of a sequence of path
  segments.
• Assume that we can estimate the time required to
  execute each segment.
• Trajectory coordination problem is a scheduling
  problem where space is the shared resource.

4
Task-Completion (TC) Diagram
Task-Completion Diagram
Path in C-Space
gB
gB
B
gA
sA
sB
sA
gA
A
sB
5
Task-Completion (TC) Diagram
gB
B
sB
sA
gA
A
6
Task-Completion (TC) Diagram

• Axes represent segments of robot paths.

gB
B
sB
sA
gA
A
7
Task-Completion (TC) Diagram

• Axes represent segments of robot paths.
• Rectangle Rij is shaded if the swept volume of
  the ith path segment of A collides with swept
  volume of jth path segment of B.

gB
B
sB
sA
gA
A
8
Task-Completion (TC) Diagram

• Axes represent segments of robot paths.
• Rectangle Rij is shaded if the swept volume of
  the ith path segment of A collides with swept
  volume of jth path segment of B.
• A schedule is a non-decreasing curve that
  connects lower-left corner of diagram to
  top-right corner.

gB
B
sB
sA
gA
A
9
Task-Completion (TC) Diagram

• Axes represent segments of robot paths.
• Rectangle Rij is shaded if the swept volume of
  the ith path segment of A collides with swept
  volume of jth path segment of B.
• A schedule is a non-decreasing curve that
  connects lower-left corner of diagram to
  top-right corner.
• A safe schedule is a schedule that never
  penetrates interior of union of collision
  rectangles.

gB
B
sB
sA
gA
A
10
Task-Completion (TC) Diagram

• Axes represent segments of robot paths.
• Rectangle Rij is shaded if the swept volume of
  the ith path segment of A collides with swept
  volume of jth path segment of B.
• A schedule is a non-decreasing curve that
  connects lower-left corner of diagram to
  top-right corner.
• A safe schedule is a schedule that never
  penetrates interior of union of collision
  rectangles.
• Boundaries of collision rectangles are safe!

gA
A
sA
sB
gB
B
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Greedy Scheduler Demo
procedure Greedy Scheduler begin i0
j0 while i lt m or j lt n do
begin if Ri,j is collision free
then begin if i lt m
then begin Execute Ai
ii1 end if j lt n then
begin Execute Bj jj1 end
end else if i lt m and
Ri,j-1 is collision free then
begin Execute Ai ii1 end else
if j lt n and Ri-1,j is collision
free then begin Execute Bj
jj1 end Wait for any completion
signals end end
gB
B
sB
sA
gA
A
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Greedy Scheduler Demo
procedure Greedy Scheduler begin i0
j0 while i lt m or j lt n do
begin if Ri,j is collision free
then begin if i lt m
then begin Execute Ai
ii1 end if j lt n then
begin Execute Bj jj1 end
end else if i lt m and
Ri,j-1 is collision free then
begin Execute Ai ii1 end else
if j lt n and Ri-1,j is collision
free then begin Execute Bj
jj1 end Wait for any completion
signals end end
gB
B
sB
sA
gA
A
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Greedy Scheduler Demo
procedure Greedy Scheduler begin i0
j0 while i lt m or j lt n do
begin if Ri,j is collision free
then begin if i lt m
then begin Execute Ai
ii1 end if j lt n then
begin Execute Bj jj1 end
end else if i lt m and
Ri,j-1 is collision free then
begin Execute Ai ii1 end else
if j lt n and Ri-1,j is collision
free then begin Execute Bj
jj1 end Wait for any completion
signals end end
gB
B
sB
sA
gA
A
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Deadlock

• Greedy Scheduler can become Deadlocked.

gB
B
sB
sA
gA
A
15
SW-closure

• Can avoid deadlock by computing SW-closure of
  union of collision regions to fills in
  non-convexities.

.
gB
B
sB
sA
gA
A
16
Parallelism

• Previously, we could only execute one segment of
  A and/or B at each step.
• Parallelism decreases execution time.
• Axes correspond to expected execution time.
• Want a nearly diagonal path.

B
A
17
Increasing Potential Parallelism

• TC Diagram may have collision regions near
  diagonal because of original choice of paths.

B
A
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Increasing Potential Parallelism

• For a problematic collision region, replan path
  between the initial and final points of A by
  using swept volume of B as an obstacle.

B
B
A
A
New path may change collision rectangles and/or
execution times.
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Changing Tasks

• Allow one robot to deal with significant delay in
  the other.
• Construct TC diagram assuming each robot will
  carry out all tasks.
• Allow jumps to nonadjacent regions (assume end of
  one cycle is beginning of another.

B
A
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Conclusions

• Interesting Ideas
• Decoupling of path and trajectory planning.
• Interpretation as scheduling problem and use of
  the Task-Completion diagram.
• Only use space-time planning for collisions near
  diagonal to increase parallelism.
• Questions/Concerns
• Scalability. Computing all potential collisions
  is expensive.
• Actual results? Comparisons?

21
End

• Special thanks to Chris Clark and
• Guha Jayachandran for the diagrams!

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Previous Approaches

• Global Methods
• Construct complete collision-free trajectories.
• Example
• Construct Configuration Space-Time and compute
  trajectories for each robot one at a time, using
  swept volume (in space-time) of previous
  trajectories as obstacles.
• Pros
• Guarantee that robots will reach goal.
• Cons
• Depends on carefully controlled trajectories.

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Previous Approaches

• Local Methods
• Decide at each point in time the trajectory for
  each robot.
• Example
• At each point in time, define separating planes
  and ensure that objects stay on opposite sides.
• Pros
• Can accommodate unexpected variations in
  trajectories or unexpected obstacles.
• Cons
• May reach deadlock.
• Rely on changing paths to avoid collisions.