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. O’Donnell 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 B A sBsB sAsA gBgB gAgA sBsB gBgB sAsA gAgA Path in C-Space Task-Completion Diagram

5. Task-Completion (TC) Diagram B A sBsB sAsA gBgB gAgA

6. B A sBsB sAsA gBgB gAgA  Axes represent segments of robot paths.

7. Task-Completion (TC) Diagram B A sBsB sAsA gBgB gAgA  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.

8. Task-Completion (TC) Diagram B A sBsB sAsA gBgB gAgA  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.

9. Task-Completion (TC) Diagram B A sBsB sAsA gBgB gAgA  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.

10. Task-Completion (TC) Diagram A B sAsA sBsB gAgA gBgB  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!

11. Greedy Scheduler Demo B A sBsB sAsA gBgB gAgA procedure Greedy Scheduler; begin i:=0; j:=0; while i < m or j < n do begin if R i,j is collision free then begin if i < m then begin Execute A i ; i:=i+1; end if j < n then begin Execute B j ; j:=j+1; end end else if i < m and R i,j-1 is collision free then begin Execute Ai; i:=i+1; end else if j < n and R i-1,j is collision free then begin Execute Bj; j:=j+1; end Wait for any completion signals; end

12. Greedy Scheduler Demo B A sBsB sAsA gBgB gAgA procedure Greedy Scheduler; begin i:=0; j:=0; while i < m or j < n do begin if R i,j is collision free then begin if i < m then begin Execute A i ; i:=i+1; end if j < n then begin Execute B j ; j:=j+1; end end else if i < m and R i,j-1 is collision free then begin Execute Ai; i:=i+1; end else if j < n and R i-1,j is collision free then begin Execute Bj; j:=j+1; end Wait for any completion signals; end

13. Greedy Scheduler Demo B A sBsB sAsA gBgB gAgA procedure Greedy Scheduler; begin i:=0; j:=0; while i < m or j < n do begin if R i,j is collision free then begin if i < m then begin Execute A i ; i:=i+1; end if j < n then begin Execute B j ; j:=j+1; end end else if i < m and R i,j-1 is collision free then begin Execute Ai; i:=i+1; end else if j < n and R i-1,j is collision free then begin Execute Bj; j:=j+1; end Wait for any completion signals; end

14. Deadlock B A sBsB sAsA gBgB gAgA  Greedy Scheduler can become Deadlocked.

15. SW-closure. B A sBsB sAsA gBgB gAgA  Can avoid deadlock by computing SW-closure of union of collision regions to fills in non-convexities.

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

18. Increasing Potential Parallelism B A B A  For a problematic collision region, replan path between the initial and final points of A by using swept volume of B as an obstacle. New path may change collision rectangles and/or execution times.

19. 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

20. 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!

22. 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.

23. 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.