Motion Planning for Multiple Robots

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Motion Planning forMultiple Robots

• B. Aronov, M. de Berg, A. Frank van der Stappen,
  P. Svestka, J. Vleugels
• Presented by Tim Bretl

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Main Idea

• Want to use centralized planning because it is
  complete.
• ProblemDimension of planning space is very
  large.
• SolutionConstrain relative positions of robots
  to reduce the dimension of the planning space
  while maintaining completeness.

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Assumptions (1)

• n Number of obstacles in the workspace.
• N Number of robots in the workspace.
• All robots and obstacles have constant complexity.

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Assumptions (2)

• Using an existing, deterministic path planner
  (Basu et al.) to generate roadmaps with
  complexity O(nD1), where D is the total number
  of dimensions of the configuration space.

Reduce D to reduce planning complexity!
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Outline

• Two-Robot Planning
• Three-Robot Planning
• N-Robot Planning
• Bounded-Reach Robots
• Summary and Problems

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Two-Robot Planning
Example Translational Motion, Arbitrary Relative
Position
y
y
x
D12
x
D22
Total DOF D1D2 4
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Constrained Planning (1)
Example Translational Motion, Enforced Contact
y
?
x
D12
D2,c1
Total DOF D1D2,c D1D2-1 3
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Constrained Planning (2)
Example Translational Motion, One Robot
Stationary
y
D2,s0
x
D12
Total DOF D1D2,s D1D2-2 2
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Constrained Planning (3)

• Define a permissible multi-configuration as
• Robot 1 stationary at start or goal (DOFD2)
• Robot 2 stationary at start or goal (DOFD1)
• Robots 1 and 2 in contact (DOFD1D2-1)
• Maximum DOF is D1D2-1
• If we could plan using only permissible
  multi-configurations, DOF could be reduced by one

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Lemma

• If a feasible plan exists for two robots, then a
  feasible plan exists using only permissible
  multi-configurations.

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Example (1)
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Example (2)
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Coordination Diagram
0
2
1
4
5
3
6
7
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Coordination Diagram
Nominal Multi-Path
Arbitrary Feasible Multi-Path
Multi-Paths Using Only Permissible
Multi-Configurations
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Example (1)
(Using only permissible multi-configurations)
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One Subtlety

• Still need to connect the spaces of permissible
  multi-configurations with discrete transitions

CS1,s Robot 1 stationary at start
position CS1,g Robot 1 stationary at goal
position CS2,s Robot 2 stationary at start
position CS2,g Robot 2 stationary at goal
position CScontact Robots moving in contact
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Transitions (1)
CS1,s
CS2,s
Easy
CS1,g
CS2,g
Hard
CScontact
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Transitions (2)

• Calculating transitions to/from CScontact is
  hard, because there is a continuum of possible
  transitions.

• Example Solution Method for CS1,s
• Divide CS1,s into connected cells
• Each cell is bounded by a number of patches
• For each patch that corresponds to contact
  configurations, take an arbitrary point on the
  patch as a transition point

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Main Result

• Algorithm
• Compute a roadmap for each of the five
  permissible multi-configuration spaces
• Compute a complete representative set of
  transitions between these spaces
• Gives a roadmap for the complete problem
• Can be computed in order O(nD1D2) time

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Extension to Three Robots (1)
Example Translational Motion, Enforced Contact
y
?1
?2
x
D12
D2,c1
D3,c2
Total DOF D1D2,cD3,c D1D2D3-2 4
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Extension to Three Robots (2)

• Permissible Multi-Configurations
• (k0,1,2) robots moving in contact
• (2-k) robots stationary at either start or goal
  positions

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Extension to Three Robots (2)

• Main result is analogous O(nD1D2D3-1)
• More difficult to prove

Coordination diagram now has three dimensions.
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Extension to N Robots

• Divide the robots into three groups
• 2 single robot groups
• 1 multi-robot group containing N-2 robots
• Now the result for three robots applies, reducing
  DOF by two
• It is not known whether a stronger result
  (analogous to that for two and three robots) can
  be shown (reducing DOF by N)

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Bounded-Reach Robots

• Low-density environment
• Bounded-reach robot

Total planning time is O(n log n) (Van der
Stappen et al.)
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Low-Density Environment

• For any ball B, the number of obstacles C of size
  bigger than B that intersect B is at most some
  small number ?.

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Bounded-Reach Robot

• The reach R of a robot is the radius of the
  smallest ball completely containing the robot
  regardless of configuration.
• A robot with bounded-reach has a reach that is a
  small fraction of the minimum obstacle size.

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Multi-Robot Reach (1)

• ProblemA multi-robot does not have bounded-reach

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Multi-Robot Reach (2)

• SolutionPermissible multi-configurations do have
  bounded-reach and can represent the entire
  planning space

Total planning time (for two or three robots) is
O(n log n)
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Summary

• Paper gives a useful algorithm for a small
  reduction in DOF for complete, centralized
  multi-robot planning
• The results are even better for bounded-reach
  robots in low-density environments

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Problems

• Mainly useful for answering yes/no connectivity
  questions for real robots, you probably want to
  avoid contact configurations
• Plans are not optimal (in fact, are usually far
  from optimal)