Domain-Dependent View of Multiple Robots Path Planning Pavel Surynek Charles University, Prague Czech Republic.

Domain-Dependent ViewDomain-Dependent Viewof Multiple Robotsof Multiple RobotsPath PlanningPath Planning

Pavel SurynekPavel Surynek

Charles University, PragueCharles University, PragueCzech RepublicCzech Republic

Outline of the TalkOutline of the Talk

Problem definitionProblem definition Motivation by real-world problemsMotivation by real-world problems Motivation for the domain-dependent viewMotivation for the domain-dependent view

– why don’t use domain-independent approachwhy don’t use domain-independent approach Difficulty of the problemDifficulty of the problem Domain-dependent polynomial-time Domain-dependent polynomial-time

solving algorithmsolving algorithm Complexity analysisComplexity analysis

– polynomial-timepolynomial-time Final remarksFinal remarks

STAIRS 2008 Pavel Surynek

Multi-robot Path Multi-robot Path PlanningPlanning

Input:Input: Graph G=(V,E), V={vGraph G=(V,E), V={v11,v,v22,...,v,...,vnn} and a set } and a set of robots R={rof robots R={r11,r,r22,...,r,...,rμμ}, where }, where μμ<<nn– each roboteach robot is placed is placed in a vertexin a vertex (at most one robot in (at most one robot in

a vertex)a vertex)– a a robot can move into an unoccupiedrobot can move into an unoccupied vertex vertex

through an edge (no other robot is allowed to enter the through an edge (no other robot is allowed to enter the vertex)vertex)

– initial positionsinitial positions of robots ... simple function of robots ... simple function SS00: R: R→V→V

– goal positions goal positions of robots ... simple function of robots ... simple function SS++: R→V: R→V Task: Task: Find a sequence of allowed movesFind a sequence of allowed moves for for

robots such that all the robots reach their goal robots such that all the robots reach their goal positions starting from the given initial positionspositions starting from the given initial positions

STAIRS 2008 Pavel Surynek

(Ryan, 2007)

MotivationMotivation

Rearrangement of robots in tight spaceRearrangement of robots in tight space

Motion planning forMotion planning fora group of robotsa group of robots

Automated trafficAutomated trafficcontrolcontrol

STAIRS 2008 Pavel Surynek

Motivating experiments Motivating experiments (1)(1) Problems motivated by Problems motivated by heavy car trafficheavy car traffic

– ignoring traffic rulesignoring traffic rules

Several planners (SGPlan, SATPlan, LPG) were Several planners (SGPlan, SATPlan, LPG) were evaluated on these problemsevaluated on these problems

STAIRS 2008 Pavel Surynek

Graph Initial positions Goal positions

Motivating experiments Motivating experiments (2)(2)

STAIRS 2008 Pavel Surynek

ProblemNumber ofvertices/free

SolvablePlan length/optimal

SGPlan(seconds)

SATPlan(seconds)

LPG(seconds)

01 7/6 Yes 1/1 0.00 0.01 0.01

02 8/7 Yes 6/6 0.00 0.04 0.02

04 8/6 Yes 25/16 0.00 0.09 0.01

05 8/5 Yes 216/32 0.00 0.87 0.98

06 12/10 Yes 12/12 0.00 0.04 0.01

07 12/8 Yes 178/26 0.00 0.14 0.08

08 12/7 Yes 176/36 0.03 0.50 0.37

09 12/6 Yes 64/46 0.04 0.52 1.54

10 12/5 Yes 72/56 0.04 >120.0 3.46

11 12/4 Yes 112/60 0.05 >120.0 4.36

12 12/3 Yes 170/98 0.17 >120.0 5.25

13 13/4 Yes 154/112 0.61 >120.0 5.94

15 13/3 Yes N/A >120.0 >120.0 >120.0

16 12/10 Yes 10/10 0.00 0.04 0.02

17 12/8 Yes 30/24 0.02 0.16 0.03

18 12/4 Yes 124/N/A >120.0 >120.0 1.45

19 12/3 Yes 114/78 0.76 >120.0 6.23

20 12/2 Yes 208/120 0.33 >120.0 7.58

22 16/2 No N/A >120.0 >120.0 >120.0

23 14/2 No N/A >120.0 >120.0 >120.0

24 28/20 Yes 72/64 0.08 >120.0 0.11

Analysis of experimentsAnalysis of experiments

All the problems are small - graphs of about 15 All the problems are small - graphs of about 15 verticesvertices

SGPlan and SATPlan are trying to find SGPlan and SATPlan are trying to find shortest shortest possible solutionspossible solutions - this complicates the task - this complicates the task

SGPlan and SATPlan often SGPlan and SATPlan often do not solvedo not solve the the problem within the time problem within the time limit of 2 minuteslimit of 2 minutes

LPG searches for a LPG searches for a suboptimal solutionsuboptimal solution LPG performs as best however still not able to LPG performs as best however still not able to

solve several problems in the time solve several problems in the time limit of 2 limit of 2 minutesminutes

Conclusion: Domain-independent planners Conclusion: Domain-independent planners do not perform well on the problemdo not perform well on the problem

STAIRS 2008 Pavel Surynek

Difficult or easy?Difficult or easy?

Is the problem difficult or easy (Is the problem difficult or easy (NP-completeNP-complete or or polynomial-time solvablepolynomial-time solvable)?)?

Answer:Answer: If at least two vertices are unoccupied If at least two vertices are unoccupied the problem is polynomial-time solvablethe problem is polynomial-time solvable

Outline Outline of the algorithm:of the algorithm:– decompose the graphdecompose the graph into bi-connected into bi-connected

components (result is a tree where vertices are components (result is a tree where vertices are represented by bi-connected components)represented by bi-connected components)

– move robotsmove robots into their goal bi-connected into their goal bi-connected components or decide that it is not possiblecomponents or decide that it is not possible

– solve the problemsolve the problem within the within the individual bi-individual bi-connected componentsconnected components

STAIRS 2008 Pavel Surynek

Algorithm:Algorithm: decomposition decomposition into bi-connected into bi-connected componentscomponents DecompositionDecomposition of the graph G=(V,E) into of the graph G=(V,E) into

bi-connected componentsbi-connected components– an undirected G=(V,E) graph is bi-connected if |V|an undirected G=(V,E) graph is bi-connected if |V|

≥3 and for every v≥3 and for every vV H=(V-{v},E’) is connected (E’ V H=(V-{v},E’) is connected (E’ is E restricted on V-{v})is E restricted on V-{v})

Result is a Result is a treetree of singleton vertices and of singleton vertices andbi-connected components ... denoted bi-connected components ... denoted BB11,B,B22,...,B,...,Bkk

– (B(Bii is a vertex or a bi-connected component) is a vertex or a bi-connected component)

STAIRS 2008 Pavel Surynek

B1

B2

B3

B4

B5

B6

B6

Algorithm:Algorithm: moving robots moving robots into goal bi-connected into goal bi-connected components (1)components (1) Observation:Observation: Arbitrary vertex in the graph Arbitrary vertex in the graph

can be made unoccupiedcan be made unoccupied

Observation:Observation: A robot canA robot canbe moved to arbitrary vertexbe moved to arbitrary vertexwithin the bi-connectedwithin the bi-connectedcomponentcomponent

STAIRS 2008 Pavel Surynek

G=(V,E)r1

r2

r3 r4

G=(V,E)r4

r3

r2r1

“Shift” robots along a path connecting unoccupied vertex and vertex to be freed

r2

r4

r5

r1

r3

r6 r7

r8

r9r12

r10r11

r13

r14 r15

r16

r17

C1

C2

C3

C4

Algorithm:Algorithm: moving robots moving robots into goal bi-connected into goal bi-connected components (2)components (2) Observation:Observation: To move a robot from one To move a robot from one

bi-component to another bi-component we bi-component to another bi-component we need to free a path between the componentsneed to free a path between the components

If the path cannot be freed the problem is If the path cannot be freed the problem is unsolvableunsolvable

STAIRS 2008 Pavel Surynek

bi-connectedcomponent

bi-connectedcomponent

Bj Bi

r1

Algorithm:Algorithm: solving problem solving problem within bi-connected within bi-connected componentcomponent Observation:Observation:

Bi-connected componentBi-connected componentcan be constructed fromcan be constructed froma cycle by adding loopsa cycle by adding loops– We inductively placeWe inductively place

robots into loopsrobots into loopsstarting with the laststarting with the lastloop and proceedingloop and proceedingto the original cycleto the original cycle

– By placing robots in aBy placing robots in aloop we obtain theloop we obtain thesmaller problemsmaller problem

– For the last cycle we needFor the last cycle we needtwo unoccupied verticestwo unoccupied vertices

STAIRS 2008 Pavel Surynek

r3 r4 r5

r1

r2

2-connectedremainder

r6

? ? ?

? ??

?

?

2-connectedremainder

r1

? ? ?

? ?r1

?

?

2-connectedremainder

r2

? ? ?

Complexity analysisComplexity analysis

DecompositionDecomposition of the graph into of the graph into bi-bi-connectedconnected components: components:

O(|V|+|E|)O(|V|+|E|) Moving robotsMoving robots into goal bi-connected into goal bi-connected

components:components:O(|V|O(|V|33))

Solving problem in the individualSolving problem in the individualbi-connectedbi-connected component: component:

O(|V| O(|V|33)) The algorithm requires O(|V|The algorithm requires O(|V|33) steps in total) steps in total

STAIRS 2008 Pavel Surynek

Conclusion and remarksConclusion and remarks

We proposed a We proposed a polynomial-time solving polynomial-time solving algorithmalgorithm for multiple robots path planning for multiple robots path planning problem in the case where there are at least problem in the case where there are at least two unoccupied verticestwo unoccupied vertices

The work on the specialized The work on the specialized domain-domain-dependent algorithmdependent algorithm is partially motivated is partially motivated by inefficiency of domain-independent by inefficiency of domain-independent planners on the problemplanners on the problem

Currently we are working on the Currently we are working on the generalizationgeneralization of the approach - some of the approach - some theoretical results are already known (Wilson, theoretical results are already known (Wilson, 1973)1973)

STAIRS 2008 Pavel Surynek