Motion Planning for Multiple Robots
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Transcript of 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.
• Problem—Dimension of planning space is very large.
• Solution—Constrain 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(nD+1), 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
D1=2
D2=2
Total DOF = D1+D2 = 4
y
yx
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Constrained Planning (1)
Example: Translational Motion, Enforced Contact
D1=2
Total DOF = D1+D2,c = D1+D2-1 = 3
y
x
D2,c=1
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Constrained Planning (2)
Example: Translational Motion, One Robot Stationary
Total DOF = D1+D2,s = D1+D2-2 = 2
D1=2
y
xD2,s=0
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Constrained Planning (3)
• Define a permissible multi-configuration as…– Robot 1 stationary at start or goal (DOF=D2)
– Robot 2 stationary at start or goal (DOF=D1)
– Robots 1 and 2 in contact (DOF=D1+D2-1)
• Maximum DOF is D1+D2-1
• 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
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Coordination Diagram
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Nominal Multi-Path
Arbitrary FeasibleMulti-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 positionCS1,g = Robot 1 stationary at goal positionCS2,s = Robot 2 stationary at start positionCS2,g = Robot 2 stationary at goal positionCScontact = Robots moving in contact
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Transitions (1)
CS1,s
CS1,g CS2,g
CS2,s
CScontact
Easy
Hard
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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
1. Divide CS1,s into connected cells
2. Each cell is bounded by a number of patches
3. 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(nD1+D2) time
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Extension to Three Robots (1)
Example: Translational Motion, Enforced Contact
D1=2
Total DOF = D1+D2,c+D3,c = D1+D2+D3-2 = 4
y
x
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D2,c=1
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D3,c=2
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Extension to Three Robots (2)
• Permissible Multi-Configurations:– (k=0,1,2) robots moving in contact– (2-k) robots stationary at either start or goal
positions
kki i
DDOF 0
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Extension to Three Robots (2)
• Main result is analogous — O(nD1+D2+D3-1)
• 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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CC
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Low-Density
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 λ.
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High-Density
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Not Bounded-Reach
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.
Bounded-Reach
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Bounded-Reach
Multi-Robot Reach (1)
• Problem—A multi-robot does not have bounded-reach
Not Bounded-Reach
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Multi-Robot Reach (2)
• Solution—Permissible 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)