From a Multi-robot Global Plan to Single-robot Actions

Bernd Br¨uggemann

, Elmar Langetepe

, Andreas Lenerz

and Dirk Schulz

Research Group “Umanned Systems”, Fraunhofer Institute for Communication, Information Processing and Ergonomics,

Neuenahrer Str 20, Wachtberg, Germany

Department of Computer Science I, University of Bonn, Bonn, Germany

Keywords:

Multi-robot Systems, Planning, Plan Execution, Coordination, Algorithmic Geometry.

Abstract:

Planning for and coordinating robots in a multi-robot system (MRS) is crucial for optimizing the performance

of the whole MRS. Thus a plan for the movement of the MRS must exist. If some centralized entity calculates

the plan, it may result in one plan for the whole group. Such a global plan, which shows how the MRS can

reach a goal state, has to be transformed into action guidelines for each robot. This task becomes harder if such

a global plan includes dependencies caused by necessary cooperation of the robots. In this paper we present

an approach to transform a global multi-robot plan into single-robot actions. We also provide a method to

determine how many robots are needed to fulﬁl the global plan while obeying some constraints. Here we

use plans generated by a coordinated navigation planner with spatial constraints, but the method could be

expanded to a more general class of plans built from a centralized entity.

1 INTRODUCTION

Currently, decentralized planning for multi-robot sys-

tems is known as the more ﬂexible and robust way of

coordinating robots. But often this results in a more

autonomous and a more ”free” behaviour of the sin-

gle robots (see, for example, (Alami et al., 1998)). As

the robots have to coordinate themselves and ﬁgure

out how to achieve the goal, their autonomous abil-

ities have to be strengthened. Additionally, the be-

haviour of each robot cannot be predicted in advance

very often. Thus, the operator has to accept that the

robot shows unpredictable behaviour to some extend

(for more information about decentralized MRS see,

for example, (Parker, 2008)).

In most cases such an autonomous decentralized

behaviour of the robots is desirable and beneﬁting,

especially if the robustness of such approaches is

needed. But there are certain situations imaginable

where the behaviour of the robot group should be as

predictable as possible. In such situations (like dan-

gerous environments where the danger cannot be de-

tected by the robots) a centralized coordination of the

robots may provide advantages.

Another precondition for an advantageous use of

MRS is a proper coordination of the robots. Such co-

ordination often leads to constraints the robots have

to obey. One of the most common constraints a MRS

has to fulﬁl is to keep up the communication between

the robots. This results in a MRS plan with spatial

and temporal constraints: the robots have to be at a

certain place on a certain time.

In this paper we present a method to determine

how many robots are needed to execute the con-

strained multi-robot global plan and how to transform

it into single actions for each robot. This multi-robot

plan does not only consist of a planned path for the

MRS but also of commands how to follow possible

constraints resulting from the robots’ coordination.

The resulting approach can automatically transform a

certain group of global plans into plans for each robot.

As there are numerous applications and algo-

rithms for centralized multi-robot control, there are

also several ways to deal with the transition of the

general MRS goal into the actions of the single robots.

In (Burgard et al., 2000) a multi-robot system has to

explore an unknown environment. In their approach a

central entity chooses target positions for each robot

by estimating the information gain of those positions.

In (Alami and da Costa Bothelho, 2002) a general

planning framework for a multi-robot system is pre-

sented. Here centralized and decentralized planning

algorithms are combined. The global plan is formu-

lated in a chain of subtasks which can be understood

by the corresponding robots. The tasks can be trans-

formed into robot actions on the robot as they are pre-

deﬁned behaviours. Sanchez and Latombe show that

some problems occur when transferring centralized

419

Brüeggemann B., Langetepe E., Lenerz A. and Schulz D..

From a Multi-robot Global Plan to Single-robot Actions.

DOI: 10.5220/0004016304190422

In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 419-422

ISBN: 978-989-8565-22-8

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

to decoupled planning. In (Sanchez and Latombe,

2002) they show that sometimes a solution cannot be

found because the decoupled planning process loses

the completeness. Although it is not expected that

the decentralized approach fails in general, you can

expect that the solutions will be slightly worse than

the centralized ones. Decoupled planning also often

needs some mechanisms to detect and to solve dead-

locks (like in (Ryan, 2010) or (Kaminka et al., 2010)).

In contrast to global planners which are planning

directly on robot actions our approach introduces an

intermediate step between global planning and local

action execution. This enables the use of global plans

which are better understandable for human operators.

The remainder of the paper is as follows: ﬁrst,

in section 2, constrained global plans as we are us-

ing them for multi-robot systems are deﬁned and mo-

tivated. As such global plans do not directly re-

port about the resources needed to execute them, we

analyse the question how many robots are needed.

This results in an algorithm which not only gives the

needed number of robots but also a visiting sequence

for the target positions. Section 3 shows the resulting

approach to compute a plan for each robot. We close

the paper with our conclusions and some words about

future work.

2 GLOBAL PLANS

As stated in the introduction, we have to coordinate

the robots to take advantage of the MRS. So in a MRS

the robots can, for example, serve as communication

relays. Thus, to keep up communication to robots far

away from the operator additional robots have to be

placed. The need of robots as relay station can be

viewed in a more general way to allow additional con-

straints besides communication. A centralized plan-

ner for such constrained navigation problems can be

found in (Br¨uggemann and Schulz, 2010). Such an

application leads to a different kind of global plan,

the constrained multi-robot global plan. Here the po-

sition of each robot is chosen so the constraint is not

violated. Each position is either a target position for

one robot or a relay position a robot has to take over.

As such relay positions assure that the constraint is

not violated, a robot is not allowed to move any more

as soon as it is placed there.

As the constraint should not be violated during the

execution of the plan, the order of arrival at the posi-

tions is crucial. Additionally, there are situations in

which the constraint is obeyed between the relay po-

sitions but not at the path in between. In such cases

additional temporary relays are needed. Such tem-

porary relay robots are needed during the movement

of other robots, but when all robots have reached the

next relay position, the temporary relay robots are al-

lowed to move away. An example of such an con-

strained multi-robot global plan can be seen in ﬁgure

Figure 1: An example of a constrained multi-robot global

plan. The red marker indicates the starting position. The

green markers are relay positions. These are necessary to

keep up the communication between the starting point and

the target positions (yellow markers). The blue marker is

a temporary relay position. To reach the rightmost target

position a robot has to be placed there. If it is taken, a new

task can be assigned to the robot on the temporary relay

position.

3 TRANSFER GLOBAL PLANS

TO LOCAL ACTIONS

To transform the constrained global multi-robot plan

into local robot actions we have to determine at ﬁrst

how many robots are needed to fulﬁl the plan without

violating the constraint. As normally the number of

robots is limited within a MRS, we try to minimize

the number of robots needed to execute the plan. This

turns out to be a difﬁcult problem due to the tempo-

rary relay robots. They are needed to get one or more

robots over certain paths of the plan but then can be

used somewhere else. Hence, if there are temporary

relay robots needed ,there are situations in which they

raise the total amount of needed robots and other sit-

uations in which they do not.

To get a more formal view of the problem we re-

formulate the global plan into so called navigation

trees. Here the relay positions are the nodes. Two

nodes are connected if there is a path between them

in the global plan. Each edge in the navigation tree

has a weight equal to the number of temporary relay

positions on the corresponding path. A transforma-

tion of a global plan into a navigation tree can be seen

in ﬁgure 2.

With the help of the navigation tree, the problem

ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics

420

Figure 2: Transformation of a global plan into a navigation

tree.

of ﬁnding the lowest number of robots needed for the

global plan can be reformulated as multi-agent graph

traversal problem: Find the lowest number of agents

which are needed to visit every node of the navigation

tree while obeying the following rules:

• If an agent visits an unvisited node, that agent has

to stay there.

• To pass an edge the number of agents must be at

least

– equal to the edge weight if the following node

is already visited, and

– equal to 1 plus the edge weight if the following

node is not yet visited.

A sequence S which obeys the rules above and needs

n robots provides a base for the translation of global

maps into local actions. Additionally, we want to min-

imize n. This results in the following algorithms:

Algorithm 1 computes the number of robots

needed to visit every node in the sub-tree of v

and

return to v

. The dominating edges are those edges

which have a higher weight than any other edge in

their sub-tree. A cluster is the sub-tree of a dominat-

ing edge. Algorithm 2 computes the minimal number

of robots needed to traverse the navigation tree. The

traversal of the robots can be expressed as a sequence

of leafs visited. Unfortunately, trying out every pos-

sible sequence would result in exponential computing

time. However, it can be shown that all leafs, except

one leaf the robot group stops at, have to be visited

in the descending order of their corresponding domi-

nating edges’ weights. Thus, algorithm 2 counts how

many robots are needed for each leaf taken as the leaf

the robots have to stop at. This not only gives the

minimal number of robots needed to execute the glo-

bal plan but also the visiting order of all target posi-

tions.

With this visiting order and the following notation

we are able to transform the global plan into local

robot actions:

• → A : Go to position A

• A

′

: Wait at position A until the robot which stays

at A arrives

• ||x, y: Wait until the robots with number x and y

have reached the next relay node

Algorithm 1: Count number of robots when returning to

node v

1: procedure TRAVERSE1(node v

)

2: Be v

the node the group of agents is at the

moment

3: for every node v

′

6= v

on the path π =

. . . v

4: if v

′

is not visited before then

5: mark v

′

as visited

6: change m (total robots) and u (robot

available) corresponding to the need of v

′

7: end if

8: end for

9: Let cluster C be the ﬁrst unvisited cluster in

10: if u < # nodes in C+ weight of e

then

11: m = m + (# nodes in C+ weight of e

−u)

12: u = # nodes in C+ weight of e

13: end if

14: mark the dominating edge e

15: u = u - # nodes in C+ weight of e

16: end procedure

Algorithm 2: Find minimal number of robots needed and

corresponding visiting sequence.

1: procedure TRAVERSE2(node v

)

2: for all clusters C

in v

3: mark C

as visited

4: TRAVERSE1(v

)

5: mark C

as unvisited

6: TRAVERSE2(v)

7: if k

opt

> counter then

8: k

opt

= counter

9: end if

10: mark all clusters as unvisited

11: end for

12: end procedure

The two wait commands are necessary to ensure

that no robot violates the constraint given by the con-

strained global map as well as to gather enough robots

to pass the edges. At ﬁrst we assume that we move all

robots as one group. Then we present an optimization

of the robot actions which results in a parallel execu-

tion of those actions.

To ﬁnd an optimal visiting sequence, we build the

action guidelineof each robot step by step. Each robot

gets its own plan. The path is split into sub-paths

consisting only of positions and temporary nodes.

Whenever a robot receives the command ”move to X”

”→ X

′

” is added to the action guideline. Please notice

that we always add the command ”wait until robot for

that node is there (’)”. This guarantees that the con-

FromaMulti-robotGlobalPlantoSingle-robotActions

421

Action guidelines:

0 :→ A

1 :→ A

′

→ B

2 :→ A

′

→ B

′

→ tR1||5, 6, 7, 8 → tR2

′

→ tR3

′

→ D

3 :→ A

′

→ B

′

→ tR1

′

→ tR2||5, 6, 7, 8, 2 → tR2

′

→ D

′

→ tR4||4, 5, 6, 7, 8 → v

l,2

4 :→ A

′

→ B

′

→ tR1

′

→ tR2

′

→ tR3||5, 6, 7, 8, 2, 3→ D

′

→ tR4

′

→ v

′

l,2

→ tR4||5, 6, 7, 8→ D

′

→ v

l,1

5 :→ A

′

→ B

′

→ tR1

′

→ tR2

′

→ tR3

′

→ D

′

→ tR4

′

→ v

l,2

→ tR4

′

→ D

′

→ v

′

l,1

→ D

′

→ tR3||8 → tR2

′

→ tR1

′

→ B

′

→ tR5||6, 7, 8 → v

l,3

6 :→ A

′

→ B

′

→ tR1

′

→ tR2

′

→ tR3

′

→ D

′

→ tR4

′

→ v

′

l,2

→ tR4

′

→ D

′

→ v

′

l,1

→ D

′

→ tR3

′

→ tR2||8, 5 → tR1

′

→ B

′

→ tR5

′

→ v

′

l,3

→ tR5||7, 8 → B

′

→ A

′

→ tR6||7, 8 → C

7 :→ A

′

→ B

′

→ tR1

′

→ tR2

′

→ tR3

′

→ D

′

→ tR4

′

→ v

′

l,2

→ tR4

′

→ D

′

→ v

′

l,1

→ D

′

→ tR3

′

→ tR2

′

→ tR1||8, 5, 6 → B

′

→ tR5

′

→ v

′

l,3

→ tR5

′

→ B

′

→ A

′

→ tR1

′

→ C

′

→ tR7||8 → E

8 :→ A

′

→ B

′

→ tR1

′

→ tR2

′

→ tR3

′

→ D

′

→ tR4

′

→ v

′

l,2

→ tR4

′

→ D

′

→ v

′

l,1

→ D

′

→ tR3

′

→ tR2

′

→ tR1

′

→ B

′

→ tR5

′

→ v

′

l,3

→ tR5

′

→ B

′

→ A

′

→ tR6

′

→ C

′

→ tR7

′

→ E

′

→ v

l,4

Figure 3: Resulting action guidelines from the navigation tree in ﬁgure 4.

0 1

3 1 1

v v

l,1

l,2

l,3

l,4

B C

D E

Figure 4: Example navigation tree.

straint is never violated. It also does no harm because,

if the robot for that node is already there, then this

command has no effect.

We also have to deal with temporary relay posi-

tions, which we name tR

. While crossing edges with

a weight, a deﬁned number of robots is needed. They

are placed on thetR

nodes and have have to stay there

until the whole group reaches the next non-tR

node.

After that the temporary robots can also go forward.

The robot farthest away from the position of the group

has to move ﬁrst, then the second farthest, and so on.

Up to this point we have moved all robots from

the beginning in one group. In most cases this is

not necessary. Thus, we propose an optimizing step

which can be done after calculating the action guide-

lines for the robot group. This optimization uses the

fact that a robot which passes a node v

in the di-

rection of the leafs and back without getting at least

one ”||” command in between is not necessary. So all

the commands between two appearances of v

can be

removed. However, this remove operation has some

side effects: As the waiting commands ”||” are always

related to the whole robot group, a robot whose action

guideline was partly removed has to be removed from

the waiting lists of the other robots. A resulting plan

transformation can be seen in ﬁgure 3.

4 CONCLUSIONS

In this paper we address the problem of transforming

a constrained global multi-robot plan into plans for

each single robot in that multi-robot system. There-

fore, we introduce navigation trees to determine the

minimal number or robots for a general global plan.

This results not only in the minimal number of robots

but also in a visiting sequence for the target positions.

Although the combined approaches, building a global

plan with our planning approach together with the

automatic translation to robot action guidelines are

tested on real robots (on a MRS with up to 6 robots)

an exhaustive evaluation of the performance is nec-

essary. Especially the online optimization via inter-

changing guidelines is a focus of future work.

REFERENCES

Alami, R. and da Costa Bothelho, S. (2002). Plan-based

multi-robot cooperation. In Beetz, M., Hertzberg, J.,

Ghallab, M., and Pollack, M., editors, Advances in

Plan-Based Control of Robotic Agents, volume 2466

of Lecture Notes in Computer Science, pages 65–95.

Springer Berlin / Heidelberg. 10.1007/3-540-37724-

7 1.

Alami, R., Fleury, S., Herrb, M., Ingrand, F., and Robert, F.

(1998). Multi-robot cooperation in the martha project.

Robotics Automation Magazine, IEEE, 5(1):36–47.

Br¨uggemann, B. and Schulz, D. (2010). Coordinated navi-

gation of multi-robot systems with binary constraints.

In 2010 IEEE/RSJ International Conference on Intel-

ligent Robots and Systems. IEEE Computer Society.

Burgard, W., Moors, M., Fox, D., Simmons, R., and Thrun,

S. (2000). Collaborative multi-robot exploration. vol-

ume 1, pages 476–481 vol.1.

Kaminka, G., Erusalimchik, D., and Kraus, S. (2010).

Adaptive multi-robot coordination: A game-theoretic

perspective. In Robotics and Automation (ICRA),

2010 IEEE International Conference on, pages 328–

334. IEEE.

Parker, L. E. (2008). Distributed intelligence: Overview

of the ﬁeld and its application in multi-robot systems.

Journal of Physical Agents, 2(1).

Ryan, M. (2010). Constraint-based multi-robot path plan-

ning. In IEEE International Conference on Robotics

and Automation.

Sanchez, G. and Latombe, J.-C. (2002). Using a prm

planner to compare centralized and decoupled plan-

ning for multi-robot systems. In Robotics and Au-

tomation, 2002. Proceedings. ICRA ’02. IEEE Inter-

national Conference on, volume 2, pages 2112–2119

vol.2.

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