deadlock-free and collision-free coordination of two robot manipulators patrick a. o’donnell and...
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![Page 1: Deadlock-Free and Collision-Free Coordination of Two Robot Manipulators Patrick A. O’Donnell and Tomas Lozano-Perez ’89 Presented by Vishal Srivastava](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d5d5503460f94a3b972/html5/thumbnails/1.jpg)
Deadlock-Free and Collision-Free Coordination of Two Robot Manipulators
Patrick A. O’Donnell and Tomas Lozano-Perez ’89
Presented by Vishal Srivastava
Slides by Huy Nguyen with additions and modifications by Vishal Srivastava
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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
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Assumptions
Environment is known by both robots
Individual paths are planned off-line prior to coordination
Paths are predictable; trajectories are less predictable
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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.
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Task-Completion (TC) Diagram
B
AsB sA
gB
gAsB
gB
sA
gA
Paths in C-Space Task-Completion Diagram
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Task-Completion (TC) Diagram
B
AsB sA
gB
gA
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Task-Completion (TC) Diagram
B
AsB sA
gB
gA
Axes represent robot path segments.
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Task-Completion (TC) Diagram
B
AsB sA
gB
gA
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.
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Task-Completion (TC) Diagram
B
AsB sA
gB
gA
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.
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Task-Completion (TC) Diagram
B
AsB sA
gB
gA
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.
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Task-Completion (TC) Diagram
A
BsA sB
gA
gB
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!
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Greedy Scheduler
B
AsB sA
gB
gA
procedure Greedy Scheduler; begin i:=0; j:=0; while i < m or j < n do begin if Ri,j is collision free then begin if i < m then begin Execute Ai; i:=i+1; end
if j < n then begin Execute Bj; j:=j+1; end end else if i < m and Ri,j-1 is collision free then begin Execute Ai; i:=i+1; end else if j < n and Ri-1,j is collision free then begin Execute Bj; j:=j+1; end Wait for any completion signals; end end
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Greedy Scheduler
B
AsB sA
gB
gA
procedure Greedy Scheduler; begin i:=0; j:=0; while i < m or j < n do begin if Ri,j is collision free then begin if i < m then begin Execute Ai; i:=i+1; end
if j < n then begin Execute Bj; j:=j+1; end end else if i < m and Ri,j-1 is collision free then begin Execute Ai; i:=i+1; end else if j < n and Ri-1,j is collision free then begin Execute Bj; j:=j+1; end Wait for any completion signals; end end
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Greedy Scheduler
B
AsB sA
gB
gA
procedure Greedy Scheduler; begin i:=0; j:=0; while i < m or j < n do begin if Ri,j is collision free then begin if i < m then begin Execute Ai; i:=i+1; end
if j < n then begin Execute Bj; j:=j+1; end end else if i < m and Ri,j-1 is collision free then begin Execute Ai; i:=i+1; end else if j < n and Ri-1,j is collision free then begin Execute Bj; j:=j+1; end Wait for any completion signals; end end
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Deadlock
B
AsB sA
gB
gA
Greedy Scheduler can become Deadlocked.
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SW-closure
.
B
AsB sA
gB
gA
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.
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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
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Increasing Parallelism
A
TC Diagram may have collision regions near diagonal because of original choice of paths.
B
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Increasing Parallelism
B
A
B
A
For a problematic collision region, replan the path of A treating the volume swept by B as an obstacle.
Replanned path of A will typically be longer.
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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?
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Robots A, B, C, …? Could this be extended for >2 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?
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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
???