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

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

Assumptions

Environment is known by both robots

Individual paths are planned off-line prior to coordination

Paths are predictable; trajectories are less predictable

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.

Task-Completion (TC) Diagram

B

AsB sA

gB

gAsB

gB

sA

gA

Paths in C-Space Task-Completion Diagram

Task-Completion (TC) Diagram

B

AsB sA

gB

gA

Task-Completion (TC) Diagram

B

AsB sA

gB

gA

Axes represent robot path segments.

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.

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.

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.

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!

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

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

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

Deadlock

B

AsB sA

gB

gA

Greedy Scheduler can become Deadlocked.

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.

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

Increasing Parallelism

A

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

B

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.

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?

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?

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

???