[PPT] - k -Color Multi-Robot Motion Planning Kiril Solovey Tel-Aviv PowerPoint Presentation

SLIDE 1

k-Color Multi-Robot Motion Planning

Kiril Solovey

Tel-Aviv University, Israel

WAFR, 2012

* Joint work with Dan Halperin

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 1 / 23

SLIDE 2

Multi-Robot Problems

Classic : Every robot has start and target positions

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 2 / 23

SLIDE 3

Multi-Robot Problems

Classic : Every robot has start and target positions

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 2 / 23

SLIDE 4

Multi-Robot Problems

Classic : Every robot has start and target positions

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 2 / 23

SLIDE 5

Multi-Robot Problems

Classic : Every robot has start and target positions Unlabeled :

◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 2 / 23

SLIDE 6

Multi-Robot Problems

Classic : Every robot has start and target positions Unlabeled :

◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 2 / 23

SLIDE 7

Multi-Robot Problems

Classic : Every robot has start and target positions Unlabeled :

◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 2 / 23

SLIDE 8

Multi-Robot Problems

Classic : Every robot has start and target positions Unlabeled :

◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 2 / 23

SLIDE 9

Multi-Robot Problems

Classic : Every robot has start and target positions Unlabeled :

◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 2 / 23

SLIDE 10

Multi-Robot Problems

Classic : Every robot has start and target positions Unlabeled :

◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied

k-Color :

◮ Several groups of identical robots ◮ Interchangeable positions in each

group

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 2 / 23

SLIDE 11

Multi-Robot Problems

Classic : Every robot has start and target positions Unlabeled :

◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied

k-Color :

◮ Several groups of identical robots ◮ Interchangeable positions in each

group

Unlabeled = 1-Color Classic = Fully-Colored

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 2 / 23

SLIDE 12

Contribution

UPUMP: novel algorithm for the unlabeled problem Tailor-made for multi-robot General Simple Technique:

◮ Samples of amplified configurations ◮ Unlabeled problem reduced to several discrete problems Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 3 / 23

SLIDE 13

Contribution

UPUMP: novel algorithm for the unlabeled problem Tailor-made for multi-robot General Simple Technique:

◮ Samples of amplified configurations ◮ Unlabeled problem reduced to several discrete problems

KPUMP: A straightforward extension of UPUMP to the k-color case

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 3 / 23

SLIDE 14

Previous Related Work (Partial List)

Schwartz & Sharir - Piano Movers III ’83

◮ First complete multi-robot algorithm ◮ Disk robots

Hopcroft et al. - Hardness of the warehouse problem ’84

◮ Rectangular robot in the plane is PSPACE-hard

van den Berg et al. - Optimal decoupling into sequential plans ’09

◮ Problem decomposed into sequential subproblems

Sampling based methods

◮ Svestka & Overmars - Coordinated path planning ’98 ◮ Hirsch & Halperin - Hybrid motion planning ’02

Composite robot approach (see next slide)

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 4 / 23

SLIDE 15

Composite Robot Approach

Group of robots considered as a single robot Apply known sampling-based techniques

◮ PRM - Kavraki et al. ◮ RRT - Kuffner & LaValle ◮ EST - Hsu et al.

Each sample is a collection of positions—one for every robot Disadvantages:

◮ Ignores properties of the multi-robot problem ◮ Performs all operations in high-dimensional configuration space ◮ Increase in the number of robots drastically increases running time Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 5 / 23

SLIDE 16

Observation

Movements of individual robots can be easily produced if the other robots are considered as obstacles

×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 6 / 23

SLIDE 17

Observation

Movements of individual robots can be easily produced if the other robots are considered as obstacles

×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 6 / 23

SLIDE 18

Exploiting the Observation

Our approach: Sample a large collection of non-overlapping positions Construct a graph where an edge represents valid movement of individual robot

V ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 7 / 23

SLIDE 19

Exploiting the Observation

Our approach: Sample a large collection of non-overlapping positions Construct a graph where an edge represents valid movement of individual robot

V ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 7 / 23

SLIDE 20

Exploiting the Observation

Our approach: Sample a large collection of non-overlapping positions Construct a graph where an edge represents valid movement of individual robot

G ×3

UPUMP considers multi-robot movements as well!

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 7 / 23

SLIDE 21

Sampling Vertices

Sample n non-overlapping single-robot positions where n > m m: number of robots n: number of samples

V

m = 3, n = 0

×3

Such V is called a pumped configuration

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 8 / 23

SLIDE 22

Sampling Vertices

Sample n non-overlapping single-robot positions where n > m m: number of robots n: number of samples

V

m = 3, n = 1

×3

Such V is called a pumped configuration

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 8 / 23

SLIDE 23

Sampling Vertices

Sample n non-overlapping single-robot positions where n > m m: number of robots n: number of samples

V

m = 3, n = 2

×3

Such V is called a pumped configuration

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 8 / 23

SLIDE 24

Sampling Vertices

Sample n non-overlapping single-robot positions where n > m m: number of robots n: number of samples

V

m = 3, n = 3

×3

Such V is called a pumped configuration

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 8 / 23

SLIDE 25

Sampling Vertices

Sample n non-overlapping single-robot positions where n > m m: number of robots n: number of samples

V

m = 3, n = 4

×3

Such V is called a pumped configuration

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 8 / 23

SLIDE 26

Sampling Vertices

Sample n non-overlapping single-robot positions where n > m m: number of robots n: number of samples

V

m = 3, n = 5

×3

Such V is called a pumped configuration

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 8 / 23

SLIDE 27

Sampling Vertices

Sample n non-overlapping single-robot positions where n > m m: number of robots n: number of samples

V

m = 3, n = 5

×3

Such V is called a pumped configuration

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 8 / 23

SLIDE 28

Sampling Vertices

Sample n non-overlapping single-robot positions where n > m m: number of robots n: number of samples

V

m = 3, n = 5

×3

Such V is called a pumped configuration

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 8 / 23

SLIDE 29

Sampling Vertices

Sample n non-overlapping single-robot positions where n > m m: number of robots n: number of samples

V

m = 3, n = 6

×3

Such V is called a pumped configuration

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 8 / 23

SLIDE 30

Sampling Vertices

Sample n non-overlapping single-robot positions where n > m m: number of robots n: number of samples

V

m = 3, n = 7

×3

Such V is called a pumped configuration

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 8 / 23

SLIDE 31

Connection

Construct a graph G = (V , E)

◮ Connect pairs of positions with a path ◮ Consider the rest of the positions as obstacles ◮ Add an edge if respective path does not collide with obstacles

G ×3

G is a geometrically-embedded graph

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 9 / 23

SLIDE 32

Connection

Construct a graph G = (V , E)

◮ Connect pairs of positions with a path ◮ Consider the rest of the positions as obstacles ◮ Add an edge if respective path does not collide with obstacles

G ×3

G is a geometrically-embedded graph

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 9 / 23

SLIDE 33

Connection

Construct a graph G = (V , E)

◮ Connect pairs of positions with a path ◮ Consider the rest of the positions as obstacles ◮ Add an edge if respective path does not collide with obstacles

G ×3

G is a geometrically-embedded graph

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 9 / 23

SLIDE 34

Connection

Construct a graph G = (V , E)

◮ Connect pairs of positions with a path ◮ Consider the rest of the positions as obstacles ◮ Add an edge if respective path does not collide with obstacles

G ×3

G is a geometrically-embedded graph

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 9 / 23

SLIDE 35

Connection

Construct a graph G = (V , E)

◮ Connect pairs of positions with a path ◮ Consider the rest of the positions as obstacles ◮ Add an edge if respective path does not collide with obstacles

G ×3

G is a geometrically-embedded graph

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 9 / 23

SLIDE 36

Connection

Construct a graph G = (V , E)

◮ Connect pairs of positions with a path ◮ Consider the rest of the positions as obstacles ◮ Add an edge if respective path does not collide with obstacles

G ×3

G is a geometrically-embedded graph

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 9 / 23

SLIDE 37

Properties of Graph

A set of m vertices represents a valid placement for the m robots

G ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 10 / 23

SLIDE 38

Properties of Graph

A set of m vertices represents a valid placement for the m robots

G ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 10 / 23

SLIDE 39

Properties of Graph

A set of m vertices represents a valid placement for the m robots

G ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 10 / 23

SLIDE 40

Properties of Graph

A set of m vertices represents a valid placement for the m robots Robots are allowed to move one at a time

G ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 10 / 23

SLIDE 41

Properties of Graph

A set of m vertices represents a valid placement for the m robots Robots are allowed to move one at a time

G ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 10 / 23

SLIDE 42

Properties of Graph

A set of m vertices represents a valid placement for the m robots Robots are allowed to move one at a time

G ×3

Such graph is called a pebble graph

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 10 / 23

SLIDE 43

Pebble Motion on a Graph

Variation of the pebble motion problem (Kornhauser ’84) Identical pebbles placed on vertices of graph Goal: move pebble from one placement to another

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 11 / 23

SLIDE 44

Pebble Motion on a Graph

Variation of the pebble motion problem (Kornhauser ’84) Identical pebbles placed on vertices of graph Goal: move pebble from one placement to another

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 11 / 23

SLIDE 45

Pebble Motion on a Graph

Variation of the pebble motion problem (Kornhauser ’84) Identical pebbles placed on vertices of graph Goal: move pebble from one placement to another

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 11 / 23

SLIDE 46

Pebble Motion on a Graph

Variation of the pebble motion problem (Kornhauser ’84) Identical pebbles placed on vertices of graph Goal: move pebble from one placement to another

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 11 / 23

SLIDE 47

Pebble Motion on a Graph

Variation of the pebble motion problem (Kornhauser ’84) Identical pebbles placed on vertices of graph Goal: move pebble from one placement to another

Observation

Solution exists iff the number of pebbles within each connected component of the graph matches for the two placements

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 11 / 23

SLIDE 48

Several Pebbles Graphs are Typically Required

Some problems require more than one pebble graph

S T

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 12 / 23

SLIDE 49

The UPUMP Algorithm

1 Preprocess: 2 Query(S, T): Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 13 / 23

SLIDE 50

The UPUMP Algorithm

1 Preprocess: ◮ Sample pumped configurations V1, V2, . . . ◮ Generate respective pebble graphs G1, G2, . . . ◮ Connect pairs of graphs 2 Query(S, T): Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 13 / 23

SLIDE 51

The UPUMP Algorithm

1 Preprocess: ◮ Sample pumped configurations V1, V2, . . . ◮ Generate respective pebble graphs G1, G2, . . . ◮ Connect pairs of graphs 2 Query(S, T): ◮ Create the start and target pebble graphs GS and GT ◮ Connect to other pebble graphs ◮ Retrieve path Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 13 / 23

SLIDE 52

The UPUMP Algorithm

1 Preprocess: ◮ Sample pumped configurations V1, V2, . . . ◮ Generate respective pebble graphs G1, G2, . . . ◮ Connect pairs of graphs 2 Query(S, T): ◮ Create the start and target pebble graphs GS and GT ◮ Connect to other pebble graphs ◮ Retrieve path Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 13 / 23

SLIDE 53

Connecting Graphs

Given two pebble graphs G, G ′ find m non-colliding paths Each path moves robot from vertex in G to vertex in G ′ Robots move simultaneously

G G′

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 14 / 23

SLIDE 54

Connecting Graphs

Given two pebble graphs G, G ′ find m non-colliding paths Each path moves robot from vertex in G to vertex in G ′ Robots move simultaneously

G G′

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 14 / 23

SLIDE 55

Connecting Graphs

Given two pebble graphs G, G ′ find m non-colliding paths Each path moves robot from vertex in G to vertex in G ′ Robots move simultaneously

G G′

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 14 / 23

SLIDE 56

Connecting Graphs

Given two pebble graphs G, G ′ find m non-colliding paths Each path moves robot from vertex in G to vertex in G ′ Robots move simultaneously

G G′

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 14 / 23

SLIDE 57

Interference Graph

Construct a graph I = (Π, E)

G G′

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 58

Interference Graph

Construct a graph I = (Π, E)

V V ′

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 59

Interference Graph

Construct a graph I = (Π, E) Each vertex is a single-robot path from v ∈ V to v′ ∈ V ′

V V ′ I

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 60

Interference Graph

Construct a graph I = (Π, E) Each vertex is a single-robot path from v ∈ V to v′ ∈ V ′

V V ′ I

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 61

Interference Graph

Construct a graph I = (Π, E) Each vertex is a single-robot path from v ∈ V to v′ ∈ V ′

V V ′ I

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 62

Interference Graph

Construct a graph I = (Π, E) Each vertex is a single-robot path from v ∈ V to v′ ∈ V ′

V V ′ I

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 63

Interference Graph

Construct a graph I = (Π, E) Each vertex is a single-robot path from v ∈ V to v′ ∈ V ′

V V ′ I

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 64

Interference Graph

Construct a graph I = (Π, E) Each vertex is a single-robot path from v ∈ V to v′ ∈ V ′ There is an edge between every two paths π, π′ ∈ Π that collide

V V ′ I

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 65

Interference Graph

Construct a graph I = (Π, E) Each vertex is a single-robot path from v ∈ V to v′ ∈ V ′ There is an edge between every two paths π, π′ ∈ Π that collide

V V ′ I

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 66

Interference Graph

Construct a graph I = (Π, E) Each vertex is a single-robot path from v ∈ V to v′ ∈ V ′ There is an edge between every two paths π, π′ ∈ Π that collide

V V ′ I

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 67

Interference Graph

Construct a graph I = (Π, E) Each vertex is a single-robot path from v ∈ V to v′ ∈ V ′ There is an edge between every two paths π, π′ ∈ Π that collide

V V ′ I

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 68

Interference Graph

Construct a graph I = (Π, E) Each vertex is a single-robot path from v ∈ V to v′ ∈ V ′ There is an edge between every two paths π, π′ ∈ Π that collide

Observation

m non-colliding paths ⇐ ⇒ Independent set of size m

V V ′ I

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 69

Interference Graph

Construct a graph I = (Π, E) Each vertex is a single-robot path from v ∈ V to v′ ∈ V ′ There is an edge between every two paths π, π′ ∈ Π that collide

Observation

m non-colliding paths ⇐ ⇒ Independent set of size m

V V ′ I

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 15 / 23

SLIDE 70

The UPUMP Algorithm

1 Preprocess: ◮ Sample pumped configurations V1, V2, . . . ◮ Generate respective pebble graphs G1, G2, . . . ◮ Connect pairs of graphs 2 Query(S, T): ◮ Create the start and target pebble graphs GS and GT ◮ Connect to other pebble graphs ◮ Retrieve path Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 16 / 23

SLIDE 71

KPUMP Algorithm

A k-color problem consists of k unlabeled problems For every unlabeled problem

◮ Sample a pumped configuration ◮ Generate geometric graph

Additional restrictions:

◮ Pumped configuration in different colors do not overlap ◮ Paths do no collide with stationary robots from other color

×3 ×2 ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 17 / 23

SLIDE 72

KPUMP Algorithm

A k-color problem consists of k unlabeled problems For every unlabeled problem

◮ Sample a pumped configuration ◮ Generate geometric graph

Additional restrictions:

◮ Pumped configuration in different colors do not overlap ◮ Paths do no collide with stationary robots from other color

×3 ×2 ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 17 / 23

SLIDE 73

KPUMP Algorithm

A k-color problem consists of k unlabeled problems For every unlabeled problem

◮ Sample a pumped configuration ◮ Generate geometric graph

Additional restrictions:

◮ Pumped configuration in different colors do not overlap ◮ Paths do no collide with stationary robots from other color

×3 ×2 ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 17 / 23

SLIDE 74

KPUMP Algorithm

A k-color problem consists of k unlabeled problems For every unlabeled problem

◮ Sample a pumped configuration ◮ Generate geometric graph

Additional restrictions:

◮ Pumped configuration in different colors do not overlap ◮ Paths do no collide with stationary robots from other color

×3 ×2 ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 17 / 23

SLIDE 75

KPUMP Algorithm

A k-color problem consists of k unlabeled problems For every unlabeled problem

◮ Sample a pumped configuration ◮ Generate geometric graph

Additional restrictions:

◮ Pumped configuration in different colors do not overlap ◮ Paths do no collide with stationary robots from other color

×3 ×2 ×3

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 17 / 23

SLIDE 76

Experimental Results

Implementation for disk robots and polygonal obstacles

◮ CGAL for local planner and collision detection

Algorithm was tested on a wide range of scenarios

◮ Varying number of robots and colors

Compared performance with

◮ OOPSMP’s implementation of PRM for fully-colored inputs ◮ Our implementation of PRM variant for k-color

KPUMP outperforms PRM

◮ When the number of robots ≥ 3 Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 18 / 23

SLIDE 77

Selected Scenario

Unlabeled (23.2s) 2-Color (20.3s) 4-Color (32.9s) Fully-Colored: Decoupled (213.7s) Fully-Colored: Coupled (1.9s) Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 19 / 23

SLIDE 78

Selected Scenario

Unlabeled (23.2s) 2-Color (20.3s) 4-Color (32.9s) Fully-Colored: Decoupled (213.7s) Fully-Colored: Coupled (1.9s) Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 19 / 23

SLIDE 79

Selected Scenario

Unlabeled (23.2s) 2-Color (20.3s) 4-Color (32.9s) Fully-Colored: Decoupled (213.7s) Fully-Colored: Coupled (1.9s) Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 19 / 23

SLIDE 80

Selected Scenario

Unlabeled (23.2s) 2-Color (20.3s) 4-Color (32.9s) Fully-Colored: Decoupled (213.7s) Fully-Colored: Coupled (1.9s) Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 19 / 23

SLIDE 81

Selected Scenario

Unlabeled (23.2s) 2-Color (20.3s) 4-Color (32.9s) Fully-Colored: Decoupled (213.7s) Fully-Colored: Coupled (1.9s) Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 19 / 23

SLIDE 82

Selected Scenario

Unlabeled (23.2s) 2-Color (20.3s) 4-Color (32.9s) Fully-Colored: Decoupled (213.7s) Fully-Colored: Coupled (1.9s) Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 19 / 23

SLIDE 83

Comparison with PRM: k-Color

23.2s 20.3s 32.9s PRM failed after 10min

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 20 / 23

SLIDE 84

Comparison with PRM: Fully-Colored

1.9s 213.7s 51s PRM failed after 10min (7 robots)

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 21 / 23

SLIDE 85

Conclusion

Descried the new k-Color problem

Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 22 / 23

SLIDE 86

Conclusion

Descried the new k-Color problem New technique

◮ Sampling-based ◮ Tailor-made for multi-robots

Main building blocks

◮ Pumped configurations ◮ Pebble graphs Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 22 / 23

SLIDE 87

Conclusion

Descried the new k-Color problem New technique

◮ Sampling-based ◮ Tailor-made for multi-robots

Main building blocks

◮ Pumped configurations ◮ Pebble graphs

Simple to implement

◮ Collision detector ◮ Local planner

Algorithm avoids operations in high dimensions

◮ Only interactions between pairs of robots are considered Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 22 / 23

SLIDE 88

Conclusion

Descried the new k-Color problem New technique

◮ Sampling-based ◮ Tailor-made for multi-robots

Main building blocks

◮ Pumped configurations ◮ Pebble graphs

Simple to implement

◮ Collision detector ◮ Local planner

Algorithm avoids operations in high dimensions

◮ Only interactions between pairs of robots are considered

Fast

◮ Applicable to many robots ◮ Works well even for the fully-colored case Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 22 / 23

SLIDE 89

Future Work

1 Probabilistic completeness ◮ Seems possible with a few modifications (?) ◮ Work in progress on a variant of KPUMP that is complete Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 23 / 23

SLIDE 90

Future Work

1 Probabilistic completeness ◮ Seems possible with a few modifications (?) ◮ Work in progress on a variant of KPUMP that is complete 2 Path optimization ◮ Connect positions in pebble graphs with RRT ◮ Reduce length of paths induced by pebble problems Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 23 / 23

SLIDE 91

Future Work

1 Probabilistic completeness ◮ Seems possible with a few modifications (?) ◮ Work in progress on a variant of KPUMP that is complete 2 Path optimization ◮ Connect positions in pebble graphs with RRT ◮ Reduce length of paths induced by pebble problems 3 Hardness of Unlabeled and k-Color ◮ Unlabeled “warehouse problem” is trivial! Kiril Solovey (TAU) k-Color Motion Planning WAFR, 2012 23 / 23