Multi-Robot Motion Planning #2
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Transcript of Multi-Robot Motion Planning #2
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Multi-Robot Motion Planning #2
Jur van den Berg
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Outline
• Recap: Composite Configuration Space• Prioritized Planning• Planning in Dynamic Environments• Application: Traffic Reconstruction• Reciprocal Velocity Obstacles
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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
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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
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Dynamic Environments
• Moving Obstacles + Static Obstacles
Frogger 6 DOF Articulated Robot
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Configuration-Time Space
• One additional dimension: time• Obstacles are stationary in CT-space
Configuration Space Configuration-Time Space
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Path Constraints
• Cannot go backward in time• Maximum velocity
2D Configuration-Time Space 3D Configuration-Time Space
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Goal Specification
• Specific configuration and moment in time• Specific configuration, as fast as possible
g = (x, y, t) g = (x, y)
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Possible Approaches
• Path-velocity decomposition• First: plan path in configuration space• Then: tune velocity along path
Workspace 2D Configuration-Time Space
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Path-Velocity Decomposition
• Reduces problem to 2D• Cell decomposition, visibility graph
Cell decomposition (Adapted) Visibility Graph
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Probabilistic Approaches
• PRM?
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Probabilistic Approaches
• PRM?• Directed Edges
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Probabilistic Approaches
• PRM?• Directed Edges• Transitory
Configuration Space• Multiple-shot
paradigm does not hold
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Probabilistic Approaches
• (Rapid Random Trees) RRT• Single-shot• Build tree oriented along time-axis
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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
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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
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Roadmap-based Approaches• Discretize Roadmap-
time space– Select time step t– Constrain velocity to be
{-vmax, 0, vmax}
• Find shortest path using A*
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Roadmap-based Approaches
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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
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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?
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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
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Video
• Link
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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