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

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

• Patrick A. ODonnell and Tomas Lozano-Perez 89
• Presented by Vishal Srivastava
• Slides by Huy Nguyen with additions and
  modifications by Vishal Srivastava

2
Introduction

• Goals
• Coordinate the trajectories of two robot
  manipulators so as to avoid collisions and
  deadlock.
• Minimize total execution time
• Definitions
• path Curve in C-space
• trajectory Time history of positions along a
  path

3
Assumptions

• Environment is known by both robots
• Individual paths are planned off-line prior to
  coordination
• Paths are predictable trajectories are less
  predictable

4
The Approach

• Decouple path specification step from trajectory
  specification step.
• Each individually-planned path is composed of a
  sequence of path segments.
• We estimate the time required to execute each
  segment.
• Trajectory coordination problem becomes a
  scheduling problem where space is the shared
  resource.

5
Task-Completion (TC) Diagram
Task-Completion Diagram
Paths in C-Space
gB
gB
B
gA
sA
sB
sA
gA
A
sB
6
Task-Completion (TC) Diagram
gB
B
sB
sA
gA
A
7
Task-Completion (TC) Diagram

• Axes represent robot path segments.

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

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

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

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

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

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

gB
B
sB
sA
gA
A
11
Task-Completion (TC) Diagram

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

gA
A
sA
sB
gB
B
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Greedy Scheduler
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
13
Greedy Scheduler
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
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
15
Deadlock

• Greedy Scheduler can become Deadlocked.

gB
B
sB
sA
gA
A
16
SW-closure

• Avoid deadlock by computing SW-closure of union
  of collision regions to fills in non-convexities.
• After taking the SW-closure
• A schedule exists if and only if both the origin
  and goal remain clear.

.
gB
B
sB
sA
gA
A
17
Increasing Parallelism

• Parallelism is the degree of concurrency with
  which the paths can be executed
• Assume segment lengths now corresponds to
  expected execution time
• Best-planned paths have high parallelism
• Strive for a path close to the diagonal

B
A
18
Increasing Parallelism

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

B
A
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Increasing Parallelism

• For a problematic collision region, replan the
  path of A treating the volume swept by B as an
  obstacle.

B
B
A
A
Replanned path of A will typically be longer.
20
Conclusions

• Main Ideas
• Decoupling of path and trajectory planning.
• Formulation of coordination as a scheduling
  problem, use of Task-Completion diagram, etc.
• Replans path to increase parallelism only in
  problem regions using space-time planning.
• Concerns
• How will it work for robots with multiple moving
  joints?
• Many approximations along the way. Too
  conservative? No precise coordination.
• No experimental data. Any implementations?

21
Robots A, B, C, ?

• Could this be extended for gt2 robots?
• N-dimensional TC-Diagrams
• Number of manipulators colliding in a region
  varies.
• Can make use of the degree of collision when
  deciding on which path segments to replan?
• More ideas?

22
Backtracking?
???

• Glaring omission ability to go backwards along
  the path
• Paths would be unchanged, but velocity of
  trajectory could be negative
• Search for safe schedule becomes more difficult
• SW-Closure would eliminate solutions
• Only worthwhile if such an interaction is
  anticipated

B
A