Distributed Optimisation of Communication Capacity forNetworked Robot Systems

Trung Dung Ngo

The More Than One Robotics Laboratory

Faculty of Science

University of Brunei Darussalam

ngo.trungdung@ubd.edu.bn

Abstract— A system of networked robots can be sent intounknown environments to explore and search desired objects.The advantage of such a systems is that mobile robots can auto-matically organize communication channels from one robot tothe other robot through an ad-hoc network of inter-connectingmobile robots. However, the challenge is how to maximizethe network link capacity of such channels for informationtransmission through the network.

This paper focuses on distributed optimization of commu-nication capacity of networked robot systems. The developedalgorithm guarantees that: (1) the connectivities of the mobilerobots are well maintained to establish the communicationlink through the ad-hoc network; (2) the network is scalableand the network link capacity is adaptable to the number ofnodes; (3) the network link capacity is improvable after thecommunication link is established; and (4) the computationalcomplexity of each robot converges to a constant over time.

Keywords— networked robots, distributed optimization, linkcapacity, max-flow min-cut

1. IntroductionResearch on networked robot systems have recently re-

ceived significant attentions due to its wide range of potential

applications. A network of mobile robots can be sent into

hazardous environments for exploration and rescue mission,

poison detection, and victim identification [3]. In compar-

ison, a system of networked robots have many advantages

over a wireless sensor network because of their mobility.

For example, mobile robots can navigate in cooperation,

they can find and replace the fault robots, they adapt to

conditions of environments, or they can carry different types

of sensor for coverage and data collection. However, it is

more challenging to deploy and manage a network of mobile

robots than a standard wireless sensor network because of

dynamic changes of network topology, e.g., relocation of

nodes may interrupt or break down the communication link.

In real-world applications, there are two primary steps

to apply a network of mobile robots in environments: de-ployment and network establishment, and network optimiza-tion. In the first step, we send networked robots into the

environment. The robots automatically communicate each

other to form a large network. Specifically, in many real-

world environments, the robots can only communicate each

other by line-of-sight manner because most radio commu-

nication channels are disturbed by the environmental sur-

rounding, e.g., metal shelves in the warehouse. Once the

ad-hoc network of mobile robots is formed and a robot

wishes to transmit its gathered information to an operator, a

communication link through inter-connected robots must be

established and maintained during communication time. In

the second step, capacity of the established communication

link should be maximized to accelerate the messages transfer,

which reduces a lot of risks e.g., fading or interruption.

In the scope of this paper, we slightly present the first step

by introducing the artificial potential force field that automat-

ically connects and maintains connectivities of mobile robots.

We mainly focus on the second step of how to maximize the

flow of information using graph theory in distributed manner.

1.1 Artificial Potential FieldThe artificial potential field was originally coined out for

single robot navigation [5] as the potential field represents

perception of the robot about obstacles in the environment.

The potential field is decomposed into attractive field and

repulsive field. The attractive field aims at directing the

robots to move towards a goal while the repulsive field

enables them to move away from the obstacles. The sum

of the attractive and repulsive fields is able to navigate the

robot in the environment.

The artificial potential field has been extended into various

ways and also developed for multi-robot systems. The poten-

tial force representing the local perception among robots, and

between robots and environments. A collection of artificial

potential functions used for potential forces in multiple

robotic systems is categorized into: linear function [8], [6],

quadratic form [4], [9], [10], and exponential expression[2],

[7]. In this paper, we utilize the potential force based on the

Artificial Physics as proposed in [4].

1.2 Graph Theory in Networked SystemsGraph theory is often used as a powerful tool to model

networked systems including networks of mobile robots.

Networked systems consist of a set of dynamical units that

interact through information exchange for its coordinated

operation. Using graph theory, operation of such systems can

be computed, controlled and even optimized w.r.t individual

and/or collective constraints.

To apply graph theory for networked robot systems, the

network problem is converted to a graph theoretic problem.

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For example, in [11], [12], [13], stable flocking of mo-

bile agents in both fixed, dynamic or switching topology

are described and proven by techniques of graph theory.

Formation control of multi-agent systems in [14], [15] is

also expressed in terms of the graph Laplacian. Distributed

coordination control of multi-agent systems in [16] used the

graph Laplacian for finding a non-linear feedback control

ensuring the connectedness of mobile agents. In [17], the

consensus problem of networked agents with capabilities of

switching topology and time delays is modeled by weighted

graph.

2. Problem FormulationWe define the optimization problem and identify perfor-

mance metrics in this section.

2.1 Problem DefinitionA graph G(V,E) is given to describe a network of mobile

robots, where V is a set of mobile robots and E is the set of

communication link between them. V is categorized in three

sub-sets: the sources S - the robot wants to send messages,

the sinks T - the robot wants to receive message, and the

intermediates R - the robot acts as a hopper in the network,

where S and T must be non-empty.

In a network, each edge e(vi,v j) ∈ E is assigned a

nonnegative capacity c(vi,v j) ≥ 0. The network flow is a

communication channel that starts from a source s ∈ S, flows

through intermediates r ∈ R before being absorbed by a sink

t ∈ T . On each intermediate r, the inflow fi(r) is the total of

flows going into the node and the outflow fo(r) is the total

of flows going out the node. Thus, the flow f (vi,v j) must

satisfy the condition: f (vi,v j) ≤ c(vi,v j), for any r ∈ R, we

have fi(r) = fo(r), and for each flow between two nodes viand v j.

The value of a flow f , denoted val( f ), is the

total flow from the source s, denoted Out(s), to

the sink t, val( f ) = ∑e∈Out(s) f (s). Hereafter, we

define a maximum flow as a flow, f max, fulfilling

val( f )≤ val( f max),∀ f .

A partition of the network G is formed by Vs and Vt where

source s ∈ Vs and sink t ∈ Vt . A cut in the network is a set

of all edges that connect a vertex in Vs to a vertex in Vt .

The capacity of a cut is a sum of the capacity of edges in

the cut, denoted c(Vs,Vt). Hereafter, we define a minimum-cut, cmin(Vs,Vt) of the network as a cut with the minimum

capacity.

In an ad-hoc network of mobile robots, to ensure existence

of a flow starting from a source to a sink, the connectiv-

ities between the robots must be preserved. This task is

done by the artificial force field to keep the robots in the

desired distance. Once the network of mobile nodes has

been well established and maintained, finding a path with

maximum flow is required in order to maximize informa-

tion transmission from the source to the sink. In graph

theory, the Max-Flow Min-Cut theorem, so-called mincut[18] states that, for a given ad-hoc network, the value of

maximum flow is equal to the capacity of a minimum cut,

f max = cmin(Vs,Vt). Consequently, to find the maximum flow

of a network we can search for min-cut, which is also

recognized as the bottleneck of the network.

In a wireless network of mobile robots, the communication

capacity between two robots is inversely proportional to their

relative distance. Therefore, the communication capacity

is increased if the robots can move close to each other.

That is, the maximum flow of the network is increased by

controlling the robots moving to desired positions where

the total capacity of the minimum cut of the network is

improved.

The focus of this paper is to find min-cut, cmin(Vs,Vt) of

a network of mobile robots, G(V,E) between sources s ∈Vsand sinks t ∈ Vt in which each robot is only aware of its

nearest neighbours instead they know status of all robots in

the network as demonstrated in the centralized optimization

[1]. We also consider scalability and adaptability of the

network, and computational power of mobile robots when

the network is optimized.

2.2 Performance MetricsTo evaluate the developed optimization algorithm, we

define the following metrics.

• Improbability: The maximum flow should be improved

over time after the communication link between mobile

robots is established.

• Adaptability: Adding robots to the network should gain

the maximum flow. That is, the communication capacity

is increased if more intermediate routers are added.

• Convergence: The information flow between two robots

converges to a steady state solution in a certain scenario.

It means all the robots no longer need to consume

energy for their mobility when the network has been

maximized.

• Scalability: The computational complexity of a single

robot should not increase to infinity, even number of

robots added into the network is unlimitedly increased.

3. Distributed Optimisation of CommunicationCapacity

In this section, we describe the role of Artificial Physics

in networked robot systems before we present the distributed

optimization algorithm.

3.1 The role of Artificial PhysicsArtificial potential force field used widely to maintain

the relative positioning between robots in networked robot

systems. The robots can use either their sensing or commu-

nication capabilities to generate the potential force field that

forces them to keep their relative distance within the desired

distance . The Artificial Physics [4] is one kind of artificial

potential forces developed for connectivity preservation of

robot swarms. The Artificial Physics consists of attractive

forces pulling the robots closer when they are away each

other and repulsive forces pushing the robots away from

each other when they are close. There is a gap between

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attractive and repulsive forces, called neural force, in which

the robots act freely to maintain the networking linkage

without considering their positioning. The equation of the

artificial potential force on the robots can be seen in (1).

APF(vi,v j) =G∗m(vi)∗m(v j)

r2(1)

where the robots are denoted as vi and v j, their relative

distance is denoted as r, and the gravitational constant that

can be chosen at initialization is denoted as G.

We developed two kinds of Artificial Physics with two

different weight factors for different functionalities of the

mobile robots in the network. The robots acting as mobile

hopper are assigned normal mass while the robots working

as detectors finding a source or as receivers receiving infor-

mation are assigned with heavier mass, giving them higher

priority to guarantee network connectivities as illustrated in

Fig 1.

In the ad-hoc network, a robot is capable of communicat-

ing with its immediate neighbors if they are mutually within

their communication range. In other words, their communi-

cation capacity is proportional to their relative distance. In

fact, the communication range between two robots is usually

longer than their sensing range, thus the robots can always

communicate each other if they are within the desired relative

distance.

In a directed graph G(V,E), only edges E(vi,v j) that

are within the relative distance r between two robots i and

j are taken into consideration for the communication link

establishment. Such links are also used for link capacity

optimization. The other communication links out of this

range is considered as noise which is cancelled out by the

robots themselves.

(a) Base Stations (b) Robots

Fig. 1. Artificial potential forces

3.2 dLinkMind - Distributed Optimization AlgorithmIn our previous work [1], it has been proven that the

centralized algorithm can optimize the communication link

capacity, but it can not overcome two specific problems:

computational complexity and scalability. The computational

cost of the centralized algorithm is dramatically increased

according to the number of robots, n, added in the network,

O((|V |.|E|2)n). Moreover, the network with the centralized

algorithm is not scalable because the robots must have

knowledge of the entire network to find the bottleneck.

The crucial difference between centralized and distributed

manners is that all robots are supposed to get all information

(a) R1 measures the flow through itsdirect connections in mirror of thecapacity

(b) The immediate neighbours areasked for their connection directionsand corressponding flows

(c) All obtained flow capacities arecompared to find out the most satu-rated connection

(d) A new direction vector is gener-ated

Fig. 2. The local min-cut identification method

from the other robots in the network with the centralized

algorithm while they only need to get information from its

immediate neighbours in the distributed manner.

If the Edmond-Karp algorithm [18] is directly imple-

mented in the distributed scheme, it requires that all the

robots would have had all the other robot positions. However,

in the real world, the robots do not have the processing power

and communication bandwidth to retrieve the positions and

information of the other robots in the entire network. In other

words, if the number of deployed robots increases to infinity,

it is a giant demand of the data transmission protocol and

bandwidth, the processing power, and data repository.

In the distributed scheme, the robots can not estimate min-

cuts because the virtual flow can not be utilized. Instead

they must use their perception to estimate where the min-cut

might be located. Because the robots are capable of finding

saturated links by measuring information flow throughout

them, they decide which direction they move towards to

get off the saturation. However, the information flow based

direction may be different from the Artificial Physics guided

direction if there exist neighbouring robots not contributing

to the network flow.

If no flow goes through the network, the robots maintain-

ing the distance with the other immediate neighbours are

only under constraints of the Artificial Physics. However, it

could be better if the robots out of flows still attempt to im-

prove their connectivity in-between, which might indirectly

improve the network flow.

If there exists a flow throughout the network, it is im-

portant for the robots to determine where the possible min-

cut is located in. Because the min-cut is the most saturated

connection of the robot, it has to communicate with their

immediate neighbours to measure the information flow and

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corresponding flow capacity in the flow graph. Based on

this graph and flow measurement, the robot identifies which

communication link is the most stressed. For example, as

illustrated in Fig 2, the robot R1 would ask its direct

neighbours to find a saturated link.

As a robot can communicate with its immediate neigh-

bours to know about its link capacity and flows, we can

use such two elements to estimate which link is saturated

( a good candidate to be the min-cut). We propose a new

method, named Stress Factor, to calculate how stressed a

robot is. The function is based on differential of the link

capacity and the information flow throughout it as stated in

the equation ( 2).

SF(vi) =

⎧⎨⎩

∑v j∈N c(vi,v j)−∑v j∈N f (vi,v j)

∑v j∈N f (vi,v j)if f (vi,v j) �= 0

∞ else

(2)

We define two switching operation modes of the robots:

connectivity optimization and link capacity optimization. If

no flow is detected, the robots work in the connectivity

optimization mode, otherwise they operate in the flow opti-

mization mode. Note that two sub-modes, saturated link and

non-saturated link are embedded in the flow optimization.

The machine states of distributed optimization algorithm can

be seen in Figure 3.

• A saturated link found: the robot moves towards the

direction of the connectivity.

• No saturated link found: the robot moves towards the

direction where the flow is the best, which are the

highest possible place of a min-cut.

• No flow detected: the robot switches the state machine

from the flow optimization to the connectivity optimiza-

tion.

(a) The flows and capacities (b) Force vectors and their stress fac-tors

Fig. 4. An example of decentralized algorithm

We illustrate the concept of Stress Factor in the simulation

as seen in Fig 4. The robot with the minimal stress factor

is the most stressed, where the robot should move towards

in order to increase the link capacity and release the stress

concurrently. If the robot is not affected by any flow, the

link still has the maximum communication capacity, which

is usually much higher than the stress factor.

When the robots communicate in-between, the Stress

Factor of a robot, which are frequently updated by the

current flow information, can indicate whether it has to

cooperate with its neighbouring robots for improvement of

communication link capacity. In Fig 4, all edges in 4(a)

represent their flow capacity at the initial state. There are two

min-cuts associated to two base stations so we expect that

two robots are driven towards the base stations to improve

the link capacity. In 4(b), the Stress Factors of each robot is

calculated. The robots obtaining null from the base stations

generate a force vector towards the base stations. The Stress

Factor of the robot located in the top is an infinity, meaning

that it must move towards one of its neighbours.

The algorithm for each robot is based on neighbouring

states. If the saturated links are found in the first loop, the

second loop is skipped. If no saturated links is discovered,

the Stress Factor of the current robot is calculated and used

in the second loop for searching the smallest Stress Factor

of the neighbours. Once the lowest stress factor is found,

the node creates a force vector in the direction towards the

lowest Stress Factor.

Algorithm 1 dLinkMind - Distributed Optimization Algo-

rithm1: StressFactor := 0

2: for each BaseStation do3: Saturation := neighbour.capacity−neighbour. f low4: if Saturation = 0 then5: add neighbour.vector to mincuts6: end if7: StressFactor := StressFactor +

Satuation/neigbour. f low8: end for9: if mincuts = 0 then

10: MinStressFactor := StressFactor11: for each BaseStation do12: f actor := requestStressFactor(neighbour)13: if f actor ≥ MinStressFactor then14: add neighbour.vector to mincuts15: MinStressFactor := f actor16: end if17: end for18: end if19: return mincuts

3.3 Experiments and Results

We have developed series of experiments to examine the

dLinkMind algorithm. The developed algorithm is evaluated

according to four criteria proposed in performance metrics

as seen in 2.2. Similar to the centralized algorithm [1], we

have set-up experiments of 3 and 4 base stations, and of 10

to 20 robots placed randomly in the scenario. Based on the

results of experiments, we conclude that:

• Scalability: Since each robot only communicates with

its immediate neighbours, the computational complexity

on each robot does not depend on the size of the

network. Hence, the network is scalable.

• Improvability: the average maximum flow is improved

over time in all experiments. However, variances of

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Fig. 3. The state machine of distributed optimization

(a) 3 base stations and 10 robots in 100 tests (b) 3 base stations and 15 robots in 100 tests

(c) 3 base stations and 20 robots in 100 tests (d) 4 base stations and 15 robots in 86 tests

Fig. 5. Average maximum flow of four experiment setups in time scale: (a) 3 base stations and 10 robots, 100 tests;(b) 3 base stations and 15 robots,100 tests, (c) 3 base stations and 20 robots, 100 tests; (d) 4 base stations and 15 robots, 86 tests

communication capacity are relatively higher than the

centralized algorithm, meaning that the reliability of

the distributed algorithm is seemingly lower than the

centralized algorithm.

• Convergence: the average maximum flow in all four

cases seemingly converges to a constant over time.

We have also conducted experiments of 10 randomly

generated scenario of 9 to 21 robots and executed in 10000

simulation steps to calculate the average maximum flow. We

affirm that:

• Adaptabilitiy: the average maximum flow does increase

when we added more robots used as mobile routers

into the scenario. This affirms that we can increase

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Fig. 6. Average max-flow measured after 10000 simulations steps with participation of 9 to 21 robots

the number of robots to increase the communication

efficiency of the network.

4. CONCLUSIONS

We have described the distributed optimization of com-

munication link capacity for networked robots systems. The

distributed optimization algorithms - dLinkMind - is a result

of integrating the artificial potential force and graph theory

based information flow optimization. We have proven that

three criteria, improvability, adaptability and convergence are

satisfied with the distributed algorithm as in section 3.3.

Moreover, this distributed algorithm is scalable because the

computational power of each robot does not depend on the

size of the network since the robots only communicate with

its immediate neighbours to obtain their relative positions

and communication capacity in-between.

We believe that the distributed algorithm is more advan-

tageous than the centralized algorithm in the real world

applications. This distributed algorithm is also simple to

be implemented on the real robots than the centralized

algorithms.

ACKNOWLEDGEMENTS

The authors acknowledge contribution of the student group

09gr870 working at the More-Than-One Robotics laboratory,

Department of Electronic Systems, Aalborg in the Spring

2009. The work presented in this paper is a part of their

project.

References[1] Ngo T.D. LinkMind: Link Optimization in Swarming Mobile Sensor

Networks. Sensors, 2011; 11(8):8180-8202[2] Horward, A.; Mataric, M.; Sukatme, G. Mobile Sensor Network De-

ployment Using Potential Fields: A Distributed, Scalable Solution tothe Area Coverage Problem. In Proceedings of the Sixth InternationalSymposium on Distributed Autonomous Robotics Systems, Fukuoka,Japan, 25–27 June 2002; pp. 229-208.

[3] Ogren, P.; Fiorelli, E.; Leonard, E.N. Cooperative control of mobilesensor networks: Adaptive gradient climbing in a distributed environ-ment. IEEE Trans. Autom. Control 2004, 49, 1292-1302.

[4] Spears, W.; Spears, D.; Hamann, J.; Heil, R. Distributed, Physics-based Control of Swarms of Vehicles. Autonomous Robot. 2004, 17,137-162.

[5] Khatib, O. Real-time obstacle avoidance for manipulators and mobilerobots. Int. J. Rob. Res. 1986, 5, 90-99.Journal article

[6] Reif, J.; Wang, H. Social potential fields: A distributed behavioralcontrol for autonomous robots. Rob. Autonomous Syst. 1999, 27, 171-194.

[7] Mikkelsen, B.S.; Jespersen, R.; Ngo, T.D. Probabilistic Commu-nication based Potential Force for Robot Formations: A PracticalApproach. In Proceedings of the 10th International Symposium onDistributed Autonomous Robotics Systems, Lausanne, Switzerland, 1–3 November 2010; pp. 229-208.

[8] Elkaim, G.; Siegel, M. A Lightweight Control Methodology forFormation Control of Vehicle Swarms. In Proceedings of the 16thIFAC World Congress, Prague, Czech Republic, 4–8 July 2005.

[9] Ge, S.S.; Cui, Y.J. New potential functions for mobile robot pathplanning. IEEE Trans. Rob. Autom. 2000, 16, 615-620.

[10] Kim, H.D.; Shin, S.; Wang, O.H. Decentralized control of autonomousswarm systems, using artificial potential functions: Analytical designguidelines. Int. J. Intell. Rob. Syst. 2006, 45, 369-394.

[11] Olfati-Saber, R. Flocking for Multi-agent Dynamic Systems: Algo-rithms and Theory. IEEE Trans. Autom. Control 2006, 51, 401-420.

[12] Tanner, G.H.; Jadbabai, A.; Pappas, J.G. Stable Flocking of MobileAgents, Part I: Fixed Topology. In Proceedings of the 42nd IEEEConference on Decision and Control, Maui, HI, USA, 12 December2003; pp. 2010-2015.

[13] Tanner, G.H.; Jadbabai, A.; Pappas, J.G. Stable Flocking of MobileAgents, Part II: Dynamic Topology. In Proceedings of the 42nd IEEEConference on Decision and Control, Maui, HI, USA, 12 December2003; pp. 2016-2021.s

[14] Desai, P.J. A Graph theoretic approach for modeling mobile robotteam formations. J. Rob. Syst. 2002, 19, 511-525.

[15] Dong, W.; Guo, Y. Formation control of nonholonomic mobile robotsusing graph theoretical methods. Lect. Notes Econ. Math. Syst. 2007,588, 369-386.

[16] Ji, M.; Egerstedt, M. Distributed coordination control of multi-agentsystems while preserving connectedness. IEEE Trans. Rob. 2007, 23,693-703.

[17] Olfati-Saber, R.; Murray, M.R. Consensus problems in networks ofagents with switching topology and time-delays. IEEE Trans. Autom.Control 2004, 49, 1520-1533.

[18] Ford, L.R.; Fulkerson, D.R. Maximal flow through a network. Can. J.Math. 1956, 8, 399-404.

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