Node Localization Using Mobile Robots in Delay Tolerant WSN

8/8/2019 Node Localization Using Mobile Robots in Delay Tolerant WSN

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Node Localization Using Mobile Robots inDelay-Tolerant Sensor NetworksPubudu N pathiranal, Nirupama ~ u l u s u ~ ,ndrey V Savkin3, Sanjay .Tha4, andThanh X ~ a n g ~

Deakin University, Victoria 3217, Australia pubuduedeakin . du . uPortland State University, Portland, OR 97207-0751, USA nbulusu@cs . dx . duUniversity of New South Wales, Sydney 2052,Australiaa. savkin@unsw . du . uUniversity of New South Wales, Sydney 2052,Australia s ha@ se . nsw . du . uPortland State University, Portland, OR 97207-0751, USA dangtx@cs . dx .edu

Summary. We present a novel scheme for node localization in a Delay-Tolerant Sensor Net-work (DTN). In a DTN, sensor devices are often organized in network clusters that may bemutually disconnected. Some mobile-robots may be used for data collection from the networkclusters. The key idea in our scheme is to use this robot to perform location estimation for thesensor nodes it passes by based on the signal strength of radio messages received from them.Thus we eliminate the processing constraints of static sensor nodes and the need for static ref-erence beacons. Our mathematical contribution is the use of a Robust Extended Kalman Filter(REKF) based state estimator to solve the localization. Compared to the standard extendedKalman filter, REKF is computationally efficient and also more robust, since it does not makeany assumptions about the measurement noise. Finally, we have implemented our localizationscheme on a hybrid sensor network testbed, and show that it can achieve node localizationaccuracy within lm in a large indoor setting.

1 IntroductionRecent years have witnessed a boom in sensor networks research [3,10] nd com-mercial activities [8]. This has been motivated by the wide range of potential appli-cations from environmental monitoring to condition-based maintenance of aircraft.Sensor networks are frequently envisioned to exist at large scale, and characterizedby extremely limited end-node power, memory and processing capability.The concept of a delay-tolerant sensor network (DTN) was first proposed byFall [13]. A DTN would typically be deployed to monitor an environment over along period of time, and characterized by non-interactive sensor data traffic. Sensorsare randomly scattered and organize into one or more clusters that may be discon-nected from each other. Each cluster has a cluster-head. Sensor information is typ-ically aggregated at the cluster heads, which tend to have more resources and areresponsible for communicating data to outside world. Wireless mobile robots (e.g.robomote [25] ) ,unmanned aerial vehicles can roam around the network to collect


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data from cluster heads, or to dynamically reprogram or reconfigure the sensors. Ex-amples of DTNs in existence are Sammi [5], Zebranet [12], and DataMules [24].

This chapter revisits the problem of node localization, i.e., estimating sensornode positions for a delay-tolerant sensor network. A DTN has several distinguishingcharacteristics which motivate alternate approaches to node localization than thosepreviously proposed. In a DTN, sensor nodes need neither be localized in real time,nor all at once. In this chapter, we propose a novel localization scheme for DTNs us-ing received signal strength (RSSI) measurements from each sensor device at a datagathering mobile-robot. Our contributions are threefold:a We motivate and propose a novel approach that allows onelmore mobile robots

to perform node localization in a DTN, eliminating the processing constraints ofsmall devices. Mobility can also be exploited to reduce localization errors andthe number of static reference location beacons required to uniquely localize asensor network.

a We develop a novel Robust Extended Kalman Filter (REKF) [18], based state es-timation algorithm for node localization in DTNs. Localization based on signalstrength measurements is solved by treating it as on-line estimation in a nonlineardynamic system (Section 3). Our model incorporates significant uncertainty andmeasurement errors and is computationally more efficient and robust in compar-ison to the extended Kalman filter implementation used to solve similar problemin cellular networks [15] [6] (simulations, section 4).

a We implement and validate our scheme on a novel hybrid sensor network testbedof motes, Stargates and Lego Mindstonn robots (section 5).

2 Related WorkIn this section, we review research most relevant to our work- i) delay-tolerantsensor networks, and (ii)sensor network localization.2.1 Delay-tolerant Sensor NetworksFall first proposed a Delay-tolerant Network architecture [13] for sensors deployedin mobile and extreme environments lacking an always-on infrastructure. These sen-sors are envisioned to monitor the environment over a long period of time. Herein,communication is based on an abstraction of message switching rather than packetswitching. The abstraction of moderate-length message (known as bundles) deliveryfor non-interactive traffic can provide benefits for network management because itallows the network path selection and scheduling functions to have a-priori knowl-edge about the size and performance requirements of requested data transfers.

DTNs are already being used in practice. DataMules [24] uses a Mule that peri-odically visits sensor devices and collects information from these devices, in effectproviding a message store-and-forward service, enabling low-power sensor nodes toconserve power. The Sammi Network [5] is a community of Sammi people, who are


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Node Localization Using Mobile Robots 85reindeer herders in Sweden who keep relocating their base. The Sammi communi-ties do not have a wired or wireless communication infrastructure. Their relocationis controlled by an yearly cycle which depends on the natural behavior of reindeer.In the Zebranet wildlife tracking system [12],wireless sensor nodes attached to ani-mals collect location data and opportunistically report their histories when they comewithin radio range of base stations. While previous research has focused on comrnu-nication abstractions, we are investigating the challenges and opportunities that arisefrom mobile data collecting elements in DTNs.

2.2 Sensor Network LocalizationLocalization is one of the most widely researched topics within the area of sensornetworks, and in robotics [2,7,19,23].Previous localization systems for sensor net-works [19,23]have been designed to simultaneously scale and continuously localizea large number of devices. To meet these requirements, devices usually computetheir own location from distance (or other measurements) made to nearby referencebeacons. However, localization requirements for these sensor networks are differentfrom delay-tolerant sensor networks.

In a DTN, nodes neither need to be localized concurrently nor continuously. Wetradeoff computational time for node localization for several other benefits. We canreduce the computational requirements for small sensor devices, by instead using oneor more mobile robots to compute the location of sensors. We can employ more so-phisticated algorithms since processing is performed by the robot rather than sensordevices. We can also reduce the number of static location reference beacons required,by exploiting the mobility of the robot.

Previous research has also investigated RSSI-based localization schemes [I]. Oneof the main drawbacks in RSSI-based localization schemes is the RSSI measurementnoise caused by short-scale and medium-scale fading. In our scheme, we reduce theimpact of fading, to make RSSI-based localization more viable. Because the robot-receiver is mobile, over a period of time, we can statistically eliminate the fadingnoise in RSSI measurements (not possible with static transmitter-receiver pair).

Previously, Kalman filters and Bayesian filters have been applied to the localiza-tion problem (mainly in the context of robotics and cellular networks). In this chapterwe propose using a Robust Extended Kalman Filter (REKF) s a state estimator inpredicting sensor locations. These robust state estimation ideas emerged from thework of Savkin and Petersen [22]. It not only provides satisfactory results [17], butalso eliminates the requirement of the knowledge of measurement noise in the more

For instance, Eren et a1 [4] estimate that to uniquely localize a sensor network of n nodesino(& logn ) )steps,O(&) reference location beacons are required using the iterativetrilateration scheme proposed in [23]. To localize with just 1 beacon, O(n) steps will berequired.In our scheme, assume that the mobile robot can localizeO(1ogn)sensors in each step(a single Filter computation). This is not an unreasonable assumption to make since logn 510, even for very large n. Using just 1mobile robot, node localization can be achieved inO(&) steps. To localize in O ( e ) teps, our scheme requiresO(/Zlflogn) steps.


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Node Localization Using Mobile Robots 87the Sensori in the same order, then xi (t) A x, (t) - xt t) . Furthermore, let u(t ) de-note the two dimensional driving/acceleration command of the mobile - obot fromthe respective accelerometer readings and w (t) denote the unknown two-dimensionaldrivinglacceleration command of the sensor if moving. Although it can be general-ized for moving sensor case, as most applications rely on stationary sensors, herewe consider the sensors as stationary and set w(t) = 0. This system can be repre-sented in graphical form in the form of an input(u(t)) and measurement (y) systemas in Figure 1. We omittedB2 as we only consider the case of stationary sensors. Thebasic idea in such a system is to estimate statex from measurement y. In the localiza-tion problem, as the sensor locations are unknown, we assume an arbitrary location(0,O). We show that this assumed state converges to the actual state and hence the un-known sensor location can be estimated (as the positionlstate of the mobile - obotis known) within the prescribed time frame.

Fig. 1. Location estimation system

3.2 RSSI Measurement modelIn wireless networks, the distance between two communicating entities is observableusing the forward link RSSI (received signal strength indication) of the receiver.Measured in decibels at the mobile-robot for our case, RSSI can be modelled asa two folds effect: due to path loss and due to shadow fading [15]. Fast fading isneglected assuming that a low-pass filter is used to attenuate Rayleigh or Ricianfade. Denoting the ith ensor as Sensori (figure 2), the RSSI from the Sensori, pi(t )can be formulated as [27]


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Fig. 2. Network Geometry

where poi is a constant determined by transmitted power, wavelength, and antennagain of the mobile - obot. E is a slope index (typically 2 for highways and 4for microcells in the city), and vi ( t ) s the logarithm of the shadowing component,which is considered as an uncertainty in the measurement. di ( t ) epresents the dis-tance between the mobile - obot and Sensori, which can be further expressed interms of the position of the i th sensor with respect to the location mobile-robot i.e.,(xi(t) , i( t))

di(t)= (xi t )2 yi ( t )2 ) (3 )For sensors within a network cluster, we uses measurements at a single mobile-robotas opposed to multiple ones [15]. The observation vector

is sampled progressively as the mobile - obot moves in the coverage area. Themeasurement equation for the measurements made by the mobile - obot for the nnumber of sensors are in the form of,where v(t)= [vl t ) . v, (t) ]' ith

We provide a brief intuitive explanation of REKF here (see Appendix for detailedtheoretical background). We use the state space model (a representation of the dy-namic system consisting of the mobile - obot and the n sensors using a set of


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Node Localization Using Mobile Robots 89differential equations derived from simple kinematic equations). Our dynamic sys-tem considers two noise inputs: (i) measurement noise (this is standard with anymeasurement), v in y = C(x) + v and (ii) w - acceleration is also considered noiseas it is unknown. In this application, the initial condition errors are quite significantas no knowledge is available regarding the sensor locations. This issue is directly ad-dressed as the proposing algorithm is inherently robust against estimation errors ofthe initial condition (see equation 17 in the appendix). The two noise inputs and theinitial estimation errors have to satisfy IQC equation (equation 17 in the appendix).If there exists a solution to the Ricati equation (see section A.2 in the appendix), thenthe IQC presented in a suitable form by the equation 17 is satisfied and the states canbe estimated from the measurements using equation 19 which is a Robust version ofthe Extended Kalman Filter (REKF) .

In the application of REKF in a delay-tolerant network, the ith system (mobile-robot and the Sensori) during a corresponding time interval is represented by thenonlinear uncertain system in (12) together with the following Integral QuadraticConstraint(1QC) (from equation 17) :

Here Qi)O, Ri)O and Ni)O with i E { 1 , 2 , 3 ) are the weighting matrices for eachsystem i. The initial state(xo) is the estimated state of respective systems at startup.It is essentially derived from the terminal state of the previous system together withother data available in the network (i.e robot position and speed) to be used as theinitial state for the next system taking over the tracking. With an uncertainty relation-ship of the form of (7), the inherent measurement noise (see equation 5), unknownmobile robot acceleration and the uncertainty in the initial condition are consideredas bounded deterministic uncertain inputs. In particular, the measurement equationwith the standard norm bounded uncertainty can be written as (equation 5)

where 161 5 < with


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90 Pubudu et a13.3 Robust versus Optimal State EstimationREKF tends to increase the robustness of the state estimation process and reduce thechance that a small deviation from the Gaussian process in the system noise causesa significant negative impact on the solution. However, we lose optimality and oursolution will be just sub-optimal. To explain the connection between REKF and thestandard extended Kalman Filter, consider the system 12 with

where Ko(x, ) s some bounded function, and v)O is a parameter. Then, the REKFestimateZ(t) or the system 12,8,17 defined by 19,20 converges to ZO(t)s u tendsto 0. Here ZO(t)s the extended Kalman state estimate for the system 12 with theGaussian noise [w(t)' v(t)' ] satisfying

The parameter u in 8 describes the uncertainty in the system and measurement noise.For small v, our robust state estimate becomes close to the Kalman state estimatewith Gaussian noise. For larger v, we achieve more robustness but less optimality.We show via simulation that for larger uncertainty (which is quite realistic) our robustfilter still performs well whereas the standard extended Kalman estimate diverges.

4 SimulationsTo examine the performance of the Robust Extended Kalman Filter for a sensornetwork, we simulate a mobile-robot equipped with a radio transceiver moving in thesensor coverage area. We assume the network knows the acceleration of the mobile-robot via GPS and accelerometer readings but has no information about the sensors.We simulate two scenarios- arge sensors and small sensors.4.1 Large SensorsIn Scenario I, we simulate large sensors scattered over a wide area. We expect sen-sors to be low cost and equipped with modest transmitters. To model this, we use aslow sampling rate of 2 see per sample. We simulate a mobile robot and four sen-sors. The algorithm can be scaled to as many sensors as required by appropriatelyincreasing the number of mobile robots. Simulation parameters are listed in Table 1.The mobile-robot measures the forward link signal from four sensors and estimatesthe state of the system from an arbitrary initial estimate (zero). Figure 3 shows howthe estimated sensor location from an initial position converges to the actual sensorpositions within the simulation time. Figure 4 shows the distance variation in X andY directions separately for each sensor as well as the predicted distances approach-ing the actual distances. Figure 6 shows that the extended Kalman filter cannot beused with large uncertain instances as it diverges.


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Node Localization Using Mobile Robots 91'arameter

T I 5mins I Simulation time

QR

Base stationtransmission powerdiag{10,10,10,1, Weighting on

70,25,2,5, the Initial100,1,20,1, viscosity40,40,20,1F

Value

Table 1. Simulation parameters for Scenario I.

Comments

diag{2,2,11,1,1,371, 21

diag(2 x l o 3 , 1.7 x l o3 , }10 x lo3.51 x lo3

AmazTSxi 0)X: (0)X: (0)X: (0)X, (0)

4.2 Small Sensors

Weighting on theuncertainty in thevehicle driving command

Weighting on themeasurement noise

In Scenario 11, we use sensors with much lesser signal strength (600mW) as in [ l11with higher sampling rate as in commercially available systems. To demonstrate scal-ability, we increase both the time scale and the number of sensors in this scenario.The simulation parameters are given in Table 4. The arbitrary acceleration of themobile robot is taken as u( t) = A [-3 sin(0.2t) + 41 0.9 cos(0.05t) +421.In the dynamical system simulation, we choose the functions given in Table 4for the arbitrary mobile-robot acceleration (u), with qbl, and 4 2 being uniform ran-dom distributions in the interval [0 O.lAma,]. We consider uniformly distributedmeasurement noise in the interval [0 0.01 1 y (t)111 with y (t) being the noise freemeasurement with = 0.05. The equation for the state estimation and the corre-sponding Riccati Differential equation obtained from equation 19 and 20 are:

50m/sPa2s

[SoOm 25oOm0 01'[25OOm 500m 0 01'[100om9000m 0 01'[2500m 125OOm0 01'

[2500m -25oOm20ms-I 10ms-'1'

where

weighting on u(t)Sampling nternal

lst ensor initial state2nd sensor initial state3Cdensor initial state4th sensor initial state

mobile - obot initialstate


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Assurn4iWpoaiS(00)-2000 I I I I I3WO -m -1WO 0 1WO m

X direction (m)

Fig. 3. Location estimation trajectories converging to the actual sensor locations.

Fig. 4. Estimation convergence for sensors 1 , 2 , 3 and 4.

P b ) = C W )as shown in equation 6. Also here

xo is x(O), the relative initial dynamic state of the system. In the second scenario, tensensors with lesser signal strength are used with the mobile robot. Figure 5(a) plots


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Node L ocalization Using Mobile Rob ots 93

Fig. 5. (a) Estimation path for each sensor (Scenario 2). (b) Percentage error and he absoluteerror from the initial estimation.

(a) (b)Fig. 6. (a) Using standard Kalman filter as the s tate estimator. (b) Divergence of state estima-tion when using standard Kalrnan filter.

the estimated trajectory approaching each respective sensor location from an initialestimate of each sensor location of (0,O).Figure 5(b) plots the percentage error inlocalizing each sensor with respect to the initial estimation error.


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94 Pubudu et a15 Implementationand Experimental ResultsWe have implemented our REKF-based localization scheme to verify its computa-tional efficiency and estimation accuracy in a real environment. We now describe thisimplementation and report on preliminary experimental results.5.1 Hybrid Sensor PlatformOur hybrid platform consists of three devices- i) motes (ii) Stargates and (iii)Lego Mindstorm robots. They differ in processing, memory, battery, and mobilitycapabilities; and also in their operating systems software.

MotesMotes [16], shown in Figure 7, are our resource impoverished devices and run theTinyOS event-driven operating system. The Mica2 mote sensors deployed in our ex-periment use the CClOOO radio from ChipCon, which provides an analog RSSI mea-surement that can be connected to an analog to digital converter (ADC) to producedigital signals. These RSSI measurements can be used for localization.

Stargate [26], shown in Figure 7, is a resource-rich node that provides more capabil-ities than the MICA motes. It is a powerful Linux based single board computer withIntel 400MHz X-Scale processor (PXA255), Compact Flash, PCMCIA, Ethernet,USB Host, 64 MB SDRAM and an additional interface to communicate with a mote.We use a Stargate as the computational substrate for the mobile-robot. The Stargateruns the Robust Extended Kalman Filter which is implemented in Java. RSSI read-ings are measured for the mote interfaced with the Stargate, communicating withother sensors.

Lego MindStorm: Mobility

PlatformMICA2Stargate

We use the popular Lego MindStorm [14] platform, shown in Figure 7 to emulatea mobile robot. It is a programmable, non-maneuvring robot that constructed byconnecting small Legos together. A Stargate is mounted onto the Lego. An Infra Red

Table 2. Stargate and MICA2 Comparison

CoreAtmega l28L4Intel PXA255 Xscale

ProcessorRISCRISC

Data Path8 bits32 bits

MHz4MHz400MH z

Memory128KB program flash, 4 KB data32 MB flash, 64 MB SDRAM


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Node Localization Using Mobile Robots 95tower is used to program the RCX box, which controls the Lego MindStorm. Atthe core of the RCX is a Hitachi H8 microcontroller with 32K external RAM. Themicrocontroller is used to control three motors, three sensors, and an infrared serialcommunications port. Both the driver and firmware accept and execute commandsfrom the PC through the IR communications port. To calculate the velocity, the LegoMindStorm is programmed to move at a constant speed (selected from eight poweroptions) in a straight line.

(a) Mica2 mote (b) Stargate (c) Lego Mindstorm RobotFig. 7. Hardware Components of the Hybrid Testbed.

Fig. 8. Indoor sensor distribution and robot navigation topology.

5.2 Experimental ResultsWe placed the motes in the topology shown in Figure 8. and programmed the LegoMindstorm robot to move in a straight line. We measured the distance travelled andtime elapsed to accurately deduce the velocity of a mobile robot (sampling rate =0.3s). The values of the weighting matrices are tuned with simulations using themodelled system parameters.


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Computational EfficiencyTo characterize computational efficiency, we measured the REKF computation time(for 10 sensors) as a function of the number of RSSI samples for the following cases:

Java implementation (Sun's JVM) on Pentium IV 3GHz machineMatlab implementation on Pentium IV 3GHz machineJava implementation, with Open-Wonka on Stargate 400MHz machineThe computation time in milliseconds is shown in Table 3. The performance of Mat-

lab and Java running on Pentium IV 3 GHz does not differ much (in the same or-der). However for Stargate, the performance degrades significantly with the num-ber of samples. The Stargate has very limited memory (64 MB SDRAM ) whichmakes Open-Wonka's garbage collector inefficient. To improve the performance onthe Stargate, we can (i) break down the computation on Stargate to a smaller subsetsof samples, or (ii) implementREKF in C instead of Java.

Numberof Samples

Estimation AccuracyTo evaluate estimation accuracy, we report on an experiment with four sensors, sincethe visualization of estimation convergence is clearer with a smaller number of sen-sors. The four sensors are positioned at (6.1,6)m, (12.2,13.6)m, (18.3,21.2)m and(24.4,13.6)m. The mobile vehicle(robot) s initially located at (0,15.2)m and moveswith a velocity of 2.52mlmin. For our indoor implementation, with the collecteddata and modelling the RSSI, we use poi = 160mw and E = 3(equation2).Figure 9 shows the localization of the four sensors with approximate error oflm. The results are very close to the real positions. The Lego Mindstorm is a nonmaneuvering robot (constant velocity). By incorporating vehicle maneuver furtherimprovements to the localization error can obviously be made.

I'00016000 27 56Table 3. Computation Times for REKF Implementations

Pentium-N (Java)(seconds)

6 Algorithmic Improvements and Future WorkExperimental evaluation revealed the importance of finding the right parameters forthe weighting matrices. Our next step is to implement automatic tuning of these pa-rameters through machine learning techniques and enhance the localization scheme

Pentium-N (MATLAB)(seconds)

STARGATE(Java:(seconds)


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Node Localization Using Mobile Robots 97

45% - r am sm 0 5% raa rso rn mXdmmm(m)

Fig. 9. (a) Sensor localization, and (b) localization error for the real system.

significantly. The algorithm stores all sensor states during the experiment periodin order to validate the convergence of estimations. To improve computational ef-ficiency, in a production system, where only the final estimation is needed, only twostates for each sensor need to be stored (instead of nearly 500).

In general, the use of mobile robots in Delay-Tolerant Sensor Networks opensup a number of other interesting research possibilities. Once sensor localization hasbeen performed, it is possible to create a topological map (or a path profile) that canbe optimized so that the data collecting mobile-robot can follow this path for datacollection in the shortest possible time, or to meet storage and power requirements.Mobile robots could act as relays between disconnected portions of the network,thereby forming a relay network. The trajectory of a mobile robot can be dynamicallyrecalculated so that a mobile robot can slow down whenever it needs to download alot of information. We intend to explore these aspects in future work.

7 ConclusionsIn this chapter, we have provided a scheme for node localization using mobile robotsin a delay-tolerant sensor network (DTN). To the best of our knowledge no otherstudy has been done for such a network. DTNs are commonly deployed in long-termenvironmental monitoring applications. In a DTN, node localization need not hap-pen in real time. Using one or more mobile robots to compute the location of sensorsallows us to tradeoff computational time for node localization for several other ben-efits. First, we can eliminate processing constraints for small sensor devices. We canemploy more sophisticated algorithms since processing is performed by the robotrather than sensor devices. Second, we can reduce the number of static location ref-erence beacons required, by exploiting mobility of the robot. Third, it makes RSSI-based localization more viable. Because the robot-receiver is mobile, over a periodof time, we can statistically eliminate the fading noise in RSSI measurements.


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PoiN

Q

Value Comments

l o - 8 , l o - , l o - 8 , l o - 8 , the Initial5,5,5,5, viscositysolution

Weightingon he

uncertaintyin the userdriving

command

Weightingon the

measurementnoise

l O m i n s I Simulation time5 0 m / s - ' weighting on u(t)TlS tensor initial state2"d sensor initial state

[-4500m1 500m 0 01' 3'" sensor initial state

[-2500m -3500m 0 01' I 6 t h sensor initial state[-1500m -3000m 01' I 7 t h sensor initial stateI-1OM)m 22.50111 01' I gt sensor initial stateI-1700m l 000m O 01' 1 g th sensorinitial state13000m 900 m 0 01' 1 l o t h sensor initial state

2500m 2500m 2 m s- I 10ms-'1' 1 m o b i l e - o b o t initial stateTable 4. Simulation parameters for Scenario11.

We proposed applying a Robust Extended Kalman Filter based state estimatorfor node localization. It is computationally more efficient and robust to measurementnoise compared to the more commonly used extended Kalrnan filter irnplementa-tion. Real experiments in a large indoor area show that the localization accuracyis approximately lm. This compares favorably with previously proposed RSSI lo-


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Node Localization Using Mobile Robots 99calization schemes in an indoor setting [I] (accuracies within 3m), as well as withfiner-grained acoustic time-of-flight localization schem es [1 9,2 3] (accuracies vary10cm - 25cm). Now that we have validated our ideas through simulation, implemen-tation and experiment, we are working on further localization schem es refinementsand on other mobile-robot uses in delay-tolerant sensor networks.

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AppendixA Set-Value state estimation with a non-linear signal modelWe consider a nonlinear uncertain system of the form

as a general form of the system given by equation 1with measurement equation inthe form of equation5, and defined on the finite time interval [0,s] .Here, x( t ) E Wndenotes the state of the system y(t) E W1 is the measured output and ~ ( t )WQ isthe uncertainty output. The uncertainty inputs are w(t ) E WP and v( t ) E W1. Also,u( t )E Wm is the known control input. We assume that all of the functions appearingin (12) are with continuous and bounded partial derivatives. Additionally, we assumethat K ( x ,u ) s bounded. This was assumed to simplify the mathematical derivationsand can be removed in practice [18] [9]. The matrixB2 s assumed to be independentof x, and is of full rank.


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Node Localization Using Mobile Robots 101The uncertainty in the system is defined by the following nonlinear integral con-straint [18,20]

where d 2 0 is a positive real number. Here, a,L1 and L2 are bounded non-negative functions with continuous partial derivatives satisfying growth conditionsof the type

where I I - I I is the euclidian norm with P ) O , and+ = @, L1,Lz.Uncertainty inputsw -),v ( . satisfying this condition are called admissible uncertainties. We considerthe problem of characterizing the set of all possible states X, f the system (12)attime s > 0 which are consistent with a given control input u0 . ) and a given outputpath y o ( - ) .e., x E X, if and only if there exists admissible uncertainties suchthat if uO(t )s the control input and x( .) and y ( - ) re resulting trajectories, thenx(s)= x and y(t) = yO(t) ,or all 0 I t I s.A .l The State EstimatorThe state estimation set X, s characterized in terms of level sets of the solutionV ( x , ) of the PDE

a-V+ r n -~w- {V,V. (A (x , O)+B ~ W )at-Li (w , o - C ( x ) )+ L2 ( K ( x , O) ) = 0

V ( . O ) = @. (15)The PDE ( 15) can be viewed as a filter, taking observations u0( t )yo( t ) 0 I t I sand producing the set X, s a output. The state of this filter is the function V ( - s ) ;thus V is an information state for the state estimation problem.Theorem 1.Assume the uncertain system (1 2) , (1 3 ) satisjies the assum ptions givenabove . Then the corresponding set of possible states is given by

X,= { x ER" : V ( x , s ) d ) , (16)where V ( x , ) s the unique viscosity so lution of 15) in C (Bn [0, ])Pmofi see [l8].


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102 Pubudu et a1A.2 A Robust Extended Kalman FilterHere we consider an approximation to the PDE (15) which leads to a Kalman filterlike characterization of the set X,. Petersen and Savkin in [18] presented this asa Extended Kalman filter version of the solution to the Set Value State Estimationproblem for a linear plant with the uncertainty described by an Integral QuadraticConstraint (IQC). This IQC is also presented as a special case of equation 13. Weconsider uncertain system described by (12) and an integral quadratic constraint ofthe form

where N)O,Q)Oand R)O.For the system (12), (17), the PDE (15) can be written as

Considering a function 2(t)defined as 2(t) A argminx V ( x , ) , nd the followingequations (19),(20) and (21), define our approximate solution to the PDE (18) :

X ( t ) s defined as the solution to the Riccati Differential Equation (RDE)


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Node Localization Using Mobile Robots 103The function V(x,)was approximated by a function of the form

Hence, it follows from Theorem 1 that an approximate formula for the set X, s givenby

This amounts to the so called Robust Extended Kalman Filter(REKF) general-ization presented in [18].

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