1. Multi-Robot Motion Planning #2 Jur van den Berg

2. Outline Recap: Composite Configuration Space Prioritized Planning Planning in Dynamic Environments Application: Traffic Reconstruction Reciprocal Velocity Obstacles

3. Composite Configuration Space Configuration space C = C 1  C 2  …  C N Dimension is sum of DOFs of all robots Very high-dimensional Cylindrical obstacles Composite Configuration Space 3 Robots, 1 DOF each

4. Prioritized Multi-Robot Planning Assign priorities to robots Plan path for robot in order of priorities Treat previously planned robots as moving obstacles Problematic Case 24 Robots

5. Dynamic Environments Moving Obstacles + Static Obstacles Frogger6 DOF Articulated Robot

6. Configuration-Time Space One additional dimension: time Obstacles are stationary in CT-space Configuration SpaceConfiguration-Time Space

7. Path Constraints Cannot go backward in time Maximum velocity 2D Configuration-Time Space3D Configuration-Time Space

8. Goal Specification Specific configuration and moment in time Specific configuration, as fast as possible g = (x, y, t)g = (x, y)

9. Possible Approaches Path-velocity decomposition First: plan path in configuration space Then: tune velocity along path Workspace2D Configuration-Time Space

10. Path-Velocity Decomposition Reduces problem to 2D Cell decomposition, visibility graph Cell decomposition(Adapted) Visibility Graph

11. Probabilistic Approaches PRM?

12. Probabilistic Approaches PRM? Directed Edges

13. Probabilistic Approaches PRM? Directed Edges Transitory Configuration Space Multiple-shot paradigm does not hold

14. Probabilistic Approaches (Rapid Random Trees) RRT Single-shot Build tree oriented along time-axis

15. Probabilistic Approaches Advantages – Any dimensional configuration-spaces – Any behavior of obstacles – Only requirement: is robot configured at c collision-free at time t ? Disadvantages – Narrow passages – All effort in query phase

16. Roadmap-based Approaches Roadmap-velocity decomposition First: build roadmap in configuration space Then: find trajectory on roadmap avoiding moving obstacles Roadmap in WorkspaceRoadmap-Time Space

17. Roadmap-based Approaches Discretize Roadmap- time space – Select time step  t – Constrain velocity to be {-v max, 0, v max } Find shortest path using A*

18. Roadmap-based Approaches

19. Prioritized Multi-Robot Planning Instead of planning in Nd-dimensional composite configuration space, plan N times in (d+1)-dimensional configuration-time space Finding a path is not guaranteed 12 Robots24 Robots

20. Application: Traffic Reconstruction Sensors A and B along a highway For each car: time, velocity and lane at position A and B What happened in between?

21. Approach Create roadmap encoding car’s kinematic constraints Plan trajectory between start and goal on roadmap encoding car’s dynamic constraints Plan in order of time at point A, and avoid previously planned cars

22. Video Link

23. References Erdmann, Lozano-Perez. On Multiple Moving Objects Kant, Zucker. Toward Efficient Trajectory Planning: the Path-Velocity Decomposition Van den Berg, Overmars. Prioritized Motion Planning for Multiple Robots Hsu, Kindel, Latombe, Rock. Randomized Kinodynamic Motion Planning with Moving Obstacles Van den Berg, Overmars. Roadmap-Based Motion Planning in Dynamic Environments Van den Berg, Sewall, Lin, Manocha. Virtualized Traffic: Reconstructing Traffic Flows from Discrete Spatio- Temporal Data