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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

x

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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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15 7

06

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4

1

2

3

60

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Example (2)

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15 7

06

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3

4

1

2

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60

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Coordination Diagram

0 21 4 53 6 70

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1

4

5

3

6

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Coordination Diagram

0 21 4 53 6 70

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1

4

5

3

6

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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

1

D2,c=1

2

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

C

BC

C

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.

C

C CB

C

C

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)