Multi-Robot Motion Planning #2

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 spaceC = C1 C2  …  CN • 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 24 Robots Problematic Case

5. Dynamic Environments • Moving Obstacles + Static Obstacles Frogger 6 DOF Articulated Robot

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

7. Path Constraints • Cannot go backward in time • Maximum velocity 2D Configuration-Time Space 3D 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 Workspace 2D 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 Workspace Roadmap-Time Space

17. Roadmap-based Approaches • Discretize Roadmap-time space • Select time step Dt • Constrain velocity to be {-vmax, 0, vmax} • 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 Robots 24 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