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Location Based Power Control for Energy Critical Sensors in aDisconnected Network

Pubudu N. Pathirana

Abstract— This paper provides a location based powercontrol strategy for disconnected sensory nodes deployedfor long term service. Power conservation is of importanceparticularly when sensors communicate with a mobile robotused for data collection. The proposed algorithm usesestimations from a Robust Extended Kalman Filter (REKF)with RSSI measurements, in implementing a sigmoid func-tion based power control algorithm which essentially ap-proaches a desired power emission trajectory based oncarrier-to-interference ratios(CIR) to ensure interference-less reception. The more realistic modelling we use incor-porates physical dynamics between the mobile robot andthe sensors together with the wireless propagation param-eters between the transmitter and receiver to formulate asophisticated and effective power control strategy for theexclusive usage of energy critical disconnected nodes in asensory network increasing their life span.�� Power control, Location tracking, mobil-

ity modelling, Kalman Filter, Ad hoc networks.

I. INTRODUCTION

Recent years have witnessed a boom in sensor net-work research [1] and commercial activities[2]. This hasbeen motivated by the wide range of potential applica-tions from environmental monitoring to condition-basedmaintenance of aircraft. Sensor networks are frequentlyenvisioned to exist at large scale, and characterized byextremely limited end-node power, memory and process-ing capability. In wireless sensory networks, the sensornodes are usually deployed in an adhoc pattern in adisconnected network with individual nodes given thetask of transmitting sensory data to a mobile robotic host.The primary and the widely researched problem in theadhoc sensor networks is the localization of the sensorynodes[3], [4], [5]. In [6], a novel localization schemeusing received signal strength (RSSI) measurements wasproposed from each sensor device at a data gatheringmobile-robot. DataMules[7] use a Mule that periodicallyvisits sensor devices and collects information from thesedevices, in effect providing a message store-and-forwardservice, enabling low-power sensor nodes to conservepower. Sensors are randomly scattered and organize intoone or more clusters that may be disconnected from eachother. Each cluster has a cluster-head. Sensor informationis typically aggregated at the cluster heads, which tend tohave more resources and are responsible for communicat-ing data to the outside world. These sensors are inherentlyfaced with power conservation issues due to limitedbattery power available hampering durability and theeffectiveness of the network. Therefore, conservation ofsensory power by means of controlling the transmission

This work was supported Australian Research Council(ARC)P.N. Pathirana is with the school of Engineering and

Technology, Deakin University, Victoria 3217, [email protected]

power is of paramount importance. Further, the powercontrol is intended to provide each sensor an acceptableconnection by eliminating interference and guaranteeingQuality of Service(QoS). The transmitter power controlin cellular systems has attracted much attention duringrecent times in the context of achieving a desired carrierto interference ratio at the receiver. The key objectivesof power control in a mobile communication point isto achieve power saving for the mobile terminal as wellas eliminating unnecessary interferences[8], determiningcapacity and the quality of service[9]. The cochannelinterference caused by frequency reuse is the single mostrestraining factor on the systems capacity[10]. A unifi-cation of convergence results for cellular radio systemsemploying iterative power control methods are given in[8]. Power control has shown to increase the call carryingcapacity for both channelized systems[11] and singlechannel CDMA systems[8][12]. Both synchronous andasynchronous schemes introduced have demonstrated toconverge iteratively[13][8]. Most of these work has beenin the context of mobile terminals and haven’t consideredthe physical separation of the transmitter and the receiver.In this paper, we are concerned with the power saving ofthe sensors stationary in the network. A mobile robot isused to collect data from the dispersed sensors. Similar tothe developments in the standard mobile communicationapproach, we expect to achieve a desired CIR for thecommunication of the sensors with the mobile robot.Further, we also introduce a new approach exploiting thedependance of transmission power over the distance oftransmission in our power control strategy. The optimalpower level that the transmitter needs to transmit dependson the relative distance between the transmitter and thereceiver which in turn relates to the transmission powervia Raleigh fading of the communication channel. There-fore, the receiver can conserve power by transmittingat a power level just enough for minimum interferencereception. In this application, we extend these ideasof minimizing channel interference(particularly in theframe work of Fixed Assignment[8][14]) when multiplenodes transmit data to a single base station and moreimportantly we do so while transmitting at the minimumpossible power with the intention of saving the sensorenergy. Our power control strategy can readily be used inconjunction with the node localization algorithms(i.e[6]or the Non-DTN types). This algorithm can be usedin centralized implementation or in a fully distributedfashion. If the sensory nodes have sufficient comput-ing power or can afford dedicated hardware, then thenodes can implement a local estimator, or else, theestimator can be implemented at the mobile robot andthe location estimates need to be transmitted over the

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0-7803-9701-0/06/$20.00 ©2006 IEEE

network for the sensor’s use. Our model incorporatessignificant uncertainty and measurement errors and iscomputationally more efficient and robust in comparisonto the extended Kalman filter implementation used tosolve similar problem in cellular networks [15] [16]. Insection II we introduce the sensor - mobile robot dynamicmodel together with the measurement model based onRaleigh fading for RSSI measurement data. Section IIIpresents power control algorithms derived in the contextof an ad-hoc senor network while the simulation resultsare given in section IV.

II. SENSOR - MOBILE BASE STATION DYNAMIC

MODEL

We use the terminology Car for a mobile agent usedfor data collection in the sensory network (other mobilevehicles or robots fitted with a base station would fit intothe same category). The dynamic model for the �� sensor,�� and the mobile user to be used in this approach canbe given in two-dimensional Cartesian coordinates as[17],[18]:

�� (II.1)

��

��

�

��

��

�

��

�

� �

��

��

�� (II.2)

The dynamic state vector �� X�� X�� Y�� Y��

��, where X�� and

Y�� represent the position of the mobile agent withrespect to the �� sensor at time �, and their first-order derivatives �X�� and �Y�� represent the relativespeed along the X and Y directions. In other words,if x� ��

� representsthe absolute state (position and velocity in the Xand Y directions respectively) of the mobile user,and x��

��

��

��denotes the

absolute state of the �� sensor1 in the same order, then�� x� �� x��. Furthermore, let �� denotethe two dimensional driving and acceleration commandsof the �� from the respective accelerometer readings.Now we can extend the dynamic model for the

number of sensors randomly dispersed in a disconnectednetwork. The complete state of the system would be� �

��

��and the state space system

�� (II.3)

where � � �� and � �

��

A. Measurement model

As in cellular systems, the distance between the mobileand a known base station is practically observable. Suchinformation is inherent in the forward link RSSI (receivedsignal strength indication) of a reachable base station.Measured in decibels at the mobile station(robot) and

1Notice that x�� has zero �� and �� elements as the sensors are

immobile

also at each sensory node, RSSI can be modelled ashaving two components: one from path loss and one fromshadow fading[15]. Fast fading is neglected, assumingthat a low-pass filter is used to attenuate Rayleigh orRician fade. Denoting the �th sensor as �� (FigureV.1), the RSSI from ��, � received by the mobilerobot, can be formulated as[19]

� � �� (II.4)

where �� is the transition power of the �� sensorwhich is also determined by constants such as the wave-length, and antenna gain of ��. �� is the estimatedstate by the �� sensor and we expect our estimator toperform such that �� . Notice that �� is a functionof ��, as our proposed techniques’ transition power isalso dependent on the distance from the transmitter to thereceiver. � is a slope index (typically two for highwaysand four for microcells in the city), and �� is the loga-rithm of the shadowing component, which is consideredas an uncertainty in the measurement. �� represents thedistance between the mobile robot and ��, whichcan be further expressed in terms of the relative positionof the mobile robot and the �� sensor, i.e., �X�� Y�� as��

�X�

� � Y�

�

��. Similarly, the RSSI measurement

at the �� sensor :

�� (II.5)

where, �� is the constant transition power of the mobilerobot.

III. POWER CONTROL

As in [14], suppose a wireless system where N sensorsshare the same channel at a given instance. We canassume that the signal of mobile � will be receivedcorrectly if the CIR at base � is not less than a givenvalue �� . i.e

��

��

�� (III.1)

with denoting the number of sensory nodes in thenetwork sharing the same channel at a given instance,�� is the receiver noise at the base station �.

��

��

� �

��

�� ...

��

��

��

��

� � �

��

(III.2)

and the solution for III.2 in the equality gives the

��

� (III.3)

where �� and �� . If the transmissionpower of sensors are given by a � � �

� , where �� , then we use

654 2006 IEEE International Conference on Industrial Informatics

�� f��

��

��

��

��

�(III.4)

where, � � � �� and��

��

is time derivativeof �� 2. Here f � �

� � �� i.e f��

��

��

� . Also, f�a� � � ifa � � and thus the equilibrium point is the approximatedoptimal time function of the transition power. �� issigmoid function and satisfies the condition �� if � � �. Due to the fact that mobile transmission powerin practical systems cannot be arbitrarily large, when� � �denote the H� norm (magnitude) we impose the constraintand define

� � �� (III.5)

where, �� is the mobile user maximum transmissionpower. We state the following proposition which is anextension to the one in [14].

Proposition 1. Equation III.4 converges to��

��

�� in the time interval �� ,

starting from an arbitrary initial power vector� �� if the following are satisfied:

(i) The system is feasible(i.e., �� )(ii) f is chosen as the sigmoid function.

(iii) The following inequalities are satisfied.

a) � ��

b) � � ��

��

�� forsome � �

Proof: Taking � � � � ��

�� and

then we can directly use the uniqueness theorem [20]to prove that � � � � � � � and hence ��

�� (as in [14]). When �� is the global

Lipschitz constant of the sigmoid function, applying themean value theorem for the sigmoid function, we canstate the inequality

�f�� f��

�f�� f��

with �� which can be made < 1 Bychoosing � � . See [14]

Proposition 2. The algorithm has a quadratic conver-gence in the neighborhood of ��

�� .

Proof: Applying Forward Euler method to obtainthe difference equation

�� f ��

��

�� (III.6)

Similar to the proposition 1, and using [14], ��

� and, as �� , ��

�� quadrati-

cally.The measurement equation of a dynamic system can

be written in the following form:

�� (III.7)

2Notice that��

��

�is a function of � due to the structure

of the state space formulation

This more general nonlinear equation is for the receiversignal strength measuring two body dynamic system andis valid for the general case of mobile sensor mobile basestation case. where ��

� with

��

��

��

X�� Y��

��

...� � � � ��

�X � �

� � Y � ��

...��

�X��

� � Y��

��

(III.8)for � number of sensors with � � � is given by equationIII.4. In the application of REKF to the sensory network,the �� system (Mobil robot and the �� sensor) during thetime interval is represented by the standard nonlinear un-certain system together the Integral Quadratic Constraintin [21] as in [6]

IV. IMPLEMENTATION AND SIMULATIONS

To examine the performance of the Robust ExtendedKalman Filter based power control algorithm imple-mented on a Ad Hoc sensor network, simple simulationswere carried out for four typical sensors used for datauploading to a mobile covering the interested area. Thenetwork is assumed to have location and accelerationinformation for the mobile robot via GPS and accelerom-eter readings, while no such information is available withrespect to the power critical sensors. We simulated thetwo estimation processes by :

(i) Measurement from the sensors for distributed stateestimation with power control and localization in-corporated.

(ii) The centralized estimation from the mobile robot.

The simulated service area contains a mobile robot andfour typical sensors and can obviously be scaled for asmany sensors required.

A. Distributed Implementation

For the state estimation by sensors, the sensors donot have access to the acceleration commands of themobile robot. Therefore, � � � in II.1 is considered as anuncertainty input(as � in [17], [18]). For the distributedimplementation, the corresponding Riccati differentialequation obtained from REKF[21] together with equationII.4/II.5 are as follows:

��

�� !� ��

� � � ��

�"� � ��

�� #$�� #��

� �� !�

� ��

�"� � ��

�� %�� (IV.1)

B. Centralized Implementation

For the state estimation by the mobile robot, thecorresponding Riccati differential equation obtained from

2006 IEEE International Conference on Industrial Informatics 655

from REKF[21] together with equation II.4/II.5 are asfollows:

��

��

��

�� (IV.2)��

� ��

��

��

��

��

��

��

�� ...

��

��

with

��

��

As shown in equation III.8. Here, �� Finding ��, is an integral part in this approach.Using the generalized inverse definition for vectors andtaking � � �� for the following definitions :

��

��f ��

��

��

��

��

��

(IV.3)

��

��

� ��

��

��

��

�� ...

�� ...

��

��

� ��

(IV.4)

with �� corresponding to the mobile robot and the ��

sensor dynamic state.

C. Discussion of results

The simulation of a mobile robot used for a data collec-tion in a sensory ad hoc network formed in a �� kmsuburban area used four sensors for the purpose of clarityand simplicity in the demonstration. Clearly, the systemcan be scaled up to as many sensors as necessary as in[6]. The locations of the typical sensors and the arbitrarypath of the mobile robot is shown in figure V.2. Wechose realistic values for the simulation parameters (i.e.,velocities and acceleration of the vehicle) to depict a realapplication. The mobile robot and the sensors measurethe forward link signal as in the GSM system, for thepurpose state estimation(locations) in order to implementthe power control strategy. As the proposed algorithmscan be either centralized or decentralized, the figures 5.3,5.4, 5.5 and 5.6 indicates the sensor power dissipation to-gether with the ideal power dissipations. For each sensor,the estimation error(mean squared) for power dissipationwhen implemented as decentralized and centralized is

compared in 5.7, 5.8, 5.9 and 5.10, respectively. It isevident from these, in terms of achieving the desiredpower dissipation levels, our algorithm performs veryclose to the ideal behavior and can be used in either formdepending on the network requirements and limitations.The percentage power saving for each sensor is shownin figure 5.11 against the cases when sensors used witha fixed constant transmission power. Here the minimumvalue 300 Watts is used as it is the required minimumpower to achieve interference less communication for thesimulated case. Evidently, the proposed power controlstrategy improves the power conservation significantlywith respect to the constant power dissipation by sensorynodes. As the RSSI based measurements are subjectedto higher noise levels and the inherent uncertainty inthe initial state(particularly the location of sensors), wepropose using a REKF as opposed to EKF. Estimationerror comparison with the extended Kalman filter and theREKF in terms of mean squared error is shown in figure5.12 and shows the efficiency in using this algorithm ascompared with the standard extended Kalman filter.

V. CONCLUSION

We have provided a power control scheme for an adhoc sensory network consisting power critical sensors anda mobile robot used for data collection. To the best of ourknowledge there have been no other existing studies onpower control of such networks. Energy conversation isan issue of paramount importance as these sensors aretypically used for a long period of time with limited bat-tery power. The desire was to develop an effective, robustand easily implementable algorithm with less burden onthe system resources without compromising the networkissues that exist inherently. One fundamental problem inthese networks is the localization of the sensors. Thepower control strategy we propose is designed to be im-plemented in conjunction with the localization algorithm.Emerging from recent theoretical developments, REKFscan successfully be used in the power conservation of thesensory nodes in the network while the localizations arebeing performed. This is due to the specific structure inthe joint systems model consisting sensors and the mobilerobot that simplify overall power control strategy. Further,this algorithm can be implemented centrally(at the mobilerobot) or in a decentralized fashion. The decentralizedimplementation considerably reduces the network trafficas the system’s state is not transmitted to sensors. In thedecentralized implementation, the sensors estimate thedynamic state with respect to them and the accuracy instate estimation for both scenarios are significantly close.

REFERENCES

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[2] Crossbow Technologies Incorporated. http://www.xbow.com.[3] N. Priyantha, A. Chakraborty, and H. Balakrishnan. The cricket

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[4] A. Savvides, C. Han, and MB. Srivastava. Dynamic fine-grainedlocalization in ad-hoc networks of sensors. In Proceedings of ACMMobiCom 2001, pages 166–179, Rome, Italy, July 2001. ACM.


[5] Jeffrey Hightower and Gaetano Borriello. Location systems forubiquitous computing. IEEE Computer, 34(8):57–66, August2001.

[6] P.N.Pathirana, N.Bulusu, A.V.Savkin, and S.Jha. Node localizationusing mobile robots in delay-tolerant sensor networks. IEEETransactions on Mobile Computing, 4(3):285–296, 2005.

[7] R. Shah, S. Roy, S. Jain, and W. Burnette. Datamules: Modelinga three-tier architecture for sparse sensor networks. In Elsevier adhoc networks journal, volume 1, pages 215–233. Elsevier, 2003.

[8] D.R.Yates. A framework for uplink power control in cellular radiosystems. IEEE Journal on Selected Areas in Communications,13(7):1341–1347, September 1995.

[9] C.W.Sung and W.S.Wong. The convergence of an asyncronousecooperative algorithm for distributed power control in cellularsystems. IEEE Transactions on Vehiclular Technology, 48(2):563–570, March 1999.

[10] J.Zander. Distributed cochannel interference control in cellularradio systems. IEEE Transactions on Vehiclular Technology,41(3):305–311, August 1992.

[11] J.Zander. perfromance of optimum transmitter power controlin cellular radio systems. IEEE Transactions on VehiclularTechnology, 41(1):57–62, February 1992.

[12] A.M. Viterbi and A.Viterbi. Erlang capacity of a power-controlledcdma system. IEEE Journal of Selected Areas in Communication,11(1):892, August 1993.

[13] D.P. Bertsekas and J.N. Tsitsiklis. Paralle and DistributedComputation. Prentice Hall, New Jercy, 1989.

[14] Z.Uykan and H.N.Koivo. Sigmoid-basis nonlinear power-controlalgorithms for mobile radio systems. IEEE Transactions onVehicular Technilogy, 53(1):265–271, January 2004.

[15] T. Liu, P. Bahl, and I. Chlamtac. Mobility modelling, locationtracking, and trajectory prediction in wireless ATM networks.IEEE Journal on Selected Areas in Communications, 16(6):922–936, August 1998.

[16] M. Hellebrandt and R. Mathar. Location tracking of mobilesin cellular radio networks. IEEE Transactions on VehicularTechnology, 48(5):1558–1562, September 1999.

[17] A.V. Savkin, P.N. Pathirana, and F.A. Faruqi. The problem of pre-cision missile guidance: LQR and H� control frameworks. IEEETransactions on Aerospace and Electronic Systems, 39(3):901–910, July 2003.

[18] P.N.Pathirana, A.V.Savkin, and S.Jha. Location estimation andtrajectory prediction for cellular networks with mobile base sta-tions. IEEE Transactions on Vehicular Technology, 53(6):1903–1913, November 2004.

[19] H.H. Xia. An analytical model for predicting path loss in urbanand suburban environments. In PIRMC, 1996.

[20] M.Vidyasagar. Nonlinear Systems Analysis. Prentice Hall, NewJercy, 1993.

[21] I.R. Petersen and A.V. Savkin. Robust Kalman Filtering forSignals and Systems with Large Uncertainities. Birkhauser,Boston, 1999.

Sensor 1

Sensor 2Sensor 3

Sensor n

Mobile Robot

Fig. V.1. Network geometry

0 0.5 1 1.5 2 2.5

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

X − direction

Y −

direct

ion

Sensor 1

Sensor 2

Sensor 3

Sensor 4

Fig. V.2. Location of Sensors and the mobile vehicle

trajectory

0 10 20 30 40 50 60160

180

200

220

240

260

280

300

320

340

Time (minutes)

Pow

er (d

B)

Desired power (popt) Actual transmitted power (p)

Fig. V.3. Transmission power for sensor 1

0 10 20 30 40 50 60160

180

200

220

240

260

280

300

Time (minutes)

Pow

er (d

B)

Desired power(popt)Actual power (p)


0 10 20 30 40 50 60160

180

200

220

240

260

280

300

320

340

360

Time (minutes)

Pow

er (p

)

Desired power (popt)Actual power (p)


2006 IEEE International Conference on Industrial Informatics 657

0 10 20 30 40 50 60160

180

200

220

240

260

280

300

320

340

360

Time (minutes)

Pow

er (d

B)

Desired power (popt)Actual power (p)

Fig. V.6. Transmission power for sensors 4

0 10 20 30 40 50 60−4

−3

−2

−1

0

1

2

3

4

Time (minutes)

Per

cent

age

erro

r in

pow

er d

issi

patio

n :1

00(p

−p0p

t )/p

Centralized power controlDecentralized power control

Fig. V.7. sensors 1

0 10 20 30 40 50 60−4

−3

−2

−1

0

1

2

3

4

Time (minutes)

Per

cent

age

erro

r in

pow

er d

issi

patio

n : 1

00(p

−pop

t )/p


Fig. V.8. sensors 2

0 10 20 30 40 50 60−4

−3

−2

−1

0

1

2

3

4

Time (minutes)

Per

cent

age

erro

r in

pow

er d

issi

patio

n


Fig. V.9. sensors 3

0 10 20 30 40 50 60−4

−3

−2

−1

0

1

2

3

4

Time (minutes)

Per

cent

age

erro

r in

pow

er d

issi

patio

n


Fig. V.10. sensors 4

300 320 340 360 380 400 420 440 460 480 5005

10

15

20

25

30

35

40

45

50

Constant transmition power (Watts)

Per

cent

age

pow

er s

avin

g

Sensor 1Sensor 2Sensor 3Sensor 4

Fig. V.11. Power saving for the simulated case

0 10 20 30 40 50 600

20

40

60

80

100

120

140

160

Time (minutes)

MS

E(E

KF)

−MS

E(R

EK

F)

Estimation my sensors(distributed)Centralized estimation by mobile robot

Fig. V.12. EKF and REKF comparison


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