MultiRobot Motion Planning

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
  (d1)-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 cars kinematic
  constraints
• Plan trajectory between start and goal on roadmap
  encoding cars 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