Systems & Control Letters 57 (2008) 216–221www.elsevier.com/locate/sysconle

Bearings only multi-sensor maneuvering target tracking

Darko Mušicki∗

PO Box 1203, Doncaster East VIC 3109, Australia

Received 5 December 2006; received in revised form 2 August 2007; accepted 23 August 2007Available online 22 October 2007

Abstract

This paper presents a solution to target trajectory estimation when multiple asynchronous passive bearings only sensors with uncertainpositions are employed. Asynchronous target position triangulation is achieved. Gaussian mixture measurement presentation, together with atrack splitting algorithm allows space/time integration of the target position uncertainty with a simple algorithm. Gaussian mixture measurementpresentation incorporates sensor position uncertainty, as well as the spatial uncertainty brought by bearings only measurement. Each sensordetects the target emissions independently, and the measurements are incorporated into track as they arrive. Measurements by arbitrary numberof sensors can be incorporated, provided that the triangulation observability criterion is satisfied. The approach is verified by a single target,two sensors, two-dimensional surveillance simulation experiment.© 2007 Elsevier B.V. All rights reserved.

Keywords: Information fusion; Target tracking; Bearings only; Track splitting; Gaussian sum measurement representation; Asynchronous triangulation

1. Introduction

Target localization using bearings only measurements frommultiple sensors usually involves triangulation. Simultaneousmeasurements from multiple sensors locate the target in thebeam crossing [1]. The target emission must reach at leasttwo of the sensors concerned either via target antenna (wide)main beam, or via target antenna side lobe transmission.Any measurement received by only one sensor is discarded.A high quality network is necessary for real time targettracking.

A more effective approach is to integrate information fromevery bearings only measurement from each sensor indepen-dently as it arrives. This approach is termed “asynchronoustriangulation” in this paper. One does not need to check for si-multaneous measurements from multiple sensors and networkrequirements become less stringent. Measurements receivedby only one sensor are used and not discarded, increasingthe information content and improving the target tracking.

∗ Tel.: +61 4 11 347999.E-mail address: Darko.Musicki@gmail.com

URL: http://Darko.Musicki.googlepages.com.

0167-6911/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2007.08.008

Targets employing ESA (electronically steered array) radaremit energy in random direction at any given time, thus whenpassive tracking such objects using bearings-only measure-ments tracks must be updated at random intervals. This isthe problem that is solved in this paper and the author is notaware of any other publication which uses asynchronous tri-angulation to estimate target trajectory. Here the emphasis ison the single target trajectory estimation, no clutter situation isassumed.

The approach used here follows [8]. Non-Gaussian mea-surement pdf is presented as a Gaussian mixture. Track stateis presented as a set of components as in [11,7], where eachtrack component is track state estimate assuming one measure-ment component (element of the measurement Gaussian mix-ture presentation) per measurement to be correct. Two trackcomponents are different if they use different measurementcomponents from at least one measurement. The track compo-nent update process always uses one measurement componentwith a Gaussian pdf at one time, standard linear estimators(Kalman filter, IMM) can be used and there is no need to uti-lize non-linear estimation techniques (e.g. extended/unscentedKalman filters, particle filters). As the target can potentiallybe maneuvering, each component is one Interacting MultipleModel [7,4] (IMM) block. During the measurement update,

D. Mušicki / Systems & Control Letters 57 (2008) 216–221 217

each component of the measurement pdf presentation (oneGaussian pdf) is used to update each component of the trackstate pdf presentation creating a new track component for thenext measurement initiated track prediction/update cycle. Thisincreases the number of track components exponentially intime, and track component management control [3,10] mustbe used to limit the number of track components using thea posteriori probability of each track component.

This approach has some similarities to [9,5], where trackstate also consists of a number of components. However, thereare a number of differences. Both papers consider only singlesensor bearings—only tracking of non-maneuvering targets.In [9], a static bank of extended Kalman filters is initialized,one for each range interval, and then they are updated usingthe non-linear measurement. In [5], the first measurementpdf is presented by a Gaussian sum and is used to initializea static bank of Kalman filters. When updating using subse-quent measurements, each track component is updated usingonly corresponding measurement component. In both [9,5],the weights (probabilities) of each track component are up-dated recursively, using the nonlinear measurement. In thispaper, the track components are dynamic, created in eachmeasurement update, and their probabilities are updated usinglinear measurement component innovations. The system de-signer is offered a trade off between required computationalresources and track quality by choosing a number of retainedtrack components. This paper also accommodates uncertaintyin sensor positions. Measurement of detection time of arrivalis assumed known with high accuracy compared to the sen-sor motion, although that uncertainty can be included in amanner identical to incorporating uncertainty in the sensorposition.

Other notable references in this field include [12,6]. In [12]use of multiple sensors in bearings only tracking is studied,where sensors are relatively close to each other (multiple sonararrays towed by the same ship), and receive the same sig-nal from the target. In [6] dynamic programming is used totrack maneuvering targets using bearings only measurements,where the track state space is discretized in two models, non-maneuvering and maneuvering.

In Section 2 the passive bearings only problem and targettrajectory models are defined. The measurement pdf presenta-tion as a Gaussian mixture is detailed in Section 3. Section 4details tracking of maneuvering targets with this approach.A simulation study presented in Section 5 shows the effec-tiveness of this method, followed by concluding remarks inSection 6.

2. The asynchronous triangulation target tracking problem

A two-dimensional surveillance problem is considered here,with the straightforward extensions for the three-dimensionalcase. Multiple sensors, stationary or moving, can detect andmeasure direction of electromagnetic, acoustic (or other) emis-sions from a target that needs to be localized. The targetmay be moving and maneuvering. It is assumed here that the

triangulation observability criterion:

• at least two sensors are not collocated with each other andalso neither is collocated with the target, and

• at least two sensors and the target are not collinear

is satisfied during some percentage of the tracking time. Thesecond criterion includes the first, both are given here for clarity.There are no computational singularities or divergences if theobservability criterion is not always satisfied.

Each sensor detects and measures the direction of target atrandom times, independent from one sensor to another. The po-sition of sensor s at the moment of target detection t is knownwith an error, assumed here to have Gaussian distribution withzero mean and covariance of Rs(s, t). The target trajectory fol-lows one of (finite) M predefined dynamic models. This modelalso assumes that target trajectory model changes happen onlyat the point of target detection, and is given by

xk = Fk−1(j)xk−1 + �k−1(j) (1)

where xk denotes the target trajectory state, k denotes the detec-tion number, j is a value of a random variable Jk−1 denoting thedynamic model which takes a discrete value 1, 2, . . . , M , andFk−1(j) is the value of the state propagation matrix betweenk − 1 and k. Process noise �k−1(j) is a zero mean and whiteGaussian noise sequence with covariance matrix Qk−1(j). Thedynamic model Jk evolves as a Markov Chain with given tran-sition probabilities, denoted by

�(j, �)�P {Jk = �|Jk−1 = j}, j, � ∈ [1, . . . , M]. (2)

Denote by yk measurement received at time k, and by yk theset of measurements up to and including yk , yk =⋃k

�=1y�. Thea priori and a posteriori probabilities of dynamic model aredenoted by

�k|k−1(j)�P {Jk = j |yk−1},

�k|k(j)�P {Jk = j |yk}. (3)

3. Measurement model

Each sensor detects the target at random times. The mea-surement is defined by the detecting sensor position, zs , andby the measurement azimuth angle, �m. Let the sensor posi-tion be known exactly. The target range uncertainty is definedby minimum and maximum range to target, denoted by Rminand Rmax, respectively. Denote by r and � the target positionin polar coordinate system centered at zs , by x and y the targetposition in Cartesian coordinate system, and by p(�|�m) thepdf of � given measurement. Cartesian pdf of target position is

p(x, y|�m) ={

0, r /∈ [Rmin, Rmax],2/(R2

max − R2min) · p�(�|�m), r ∈ [Rmin, Rmax]

(4)

218 D. Mušicki / Systems & Control Letters 57 (2008) 216–221

and is presented on the left-hand side of Fig. 2. Correspondingpolar target position pdf, given single measurement is given by

p(r, �|�m) ={

0, r /∈ [Rmin, Rmax],2r/(R2

max − R2min) · p(�|�m), r ∈ [Rmin, Rmax].

(5)

The marginal distribution of the target radial distance, p(r|�m),from the sensor is not uniform, departing from [5]. As in [5],divide [Rmin, Rmax] in G components in such a manner that

Rmax(g)

Rmin(g)= �, g = 1 · · · G, (6)

� =(

Rmax

Rmin

)1/G

(7)

with Rmax(g) and Rmin(g) the maximum and minimum rangefor component g. Define dR(g) = Rmax(g) − Rmin(g) andRmean(g)=0.5(Rmax(g)+Rmin(g)) as the range difference andthe mean range of the component g. Assuming that azimuthmeasurement error has zero mean Gaussian distribution withcovariance �2

�, the measurement component g uncertainty pdfin Cartesian coordinates is approximated by a Gaussian pdfwith mean value

z(g) = zs + Rmean(g)

[cos(�m)

sin(�m)

](8)

and covariance

R0(g) =[

cos(�m) − sin(�m)

sin(�m) cos(�m)

]

×[(dR(g)/2)2 0

0 (Rmean(g)��)2

]

×[

cos(�m) sin(�m)

− sin(�m) cos(�m)

]. (9)

Fig. 1. Gaussian sum measurement representation.

Fig. 2. Left: True measurement pdf. Right: Gaussian mixture approximation.

Adding the sensor position error covariance, the final compo-nent g covariance is

R(g) = R0(g) + Rs(s, t). (10)

The probability �(g) that measurement component g containsthe target is proportional to the area covered by the measure-ment component,

�(g) =√|R(g)|∑Gh=1

√|R(h)| ≈ dR(g)Rmean(g)∑Gh=1dR(h)Rmean(h)

. (11)

Denote by z the two-dimensional Cartesian target position, itspdf given measurement �m is given by a Gaussian mixture

p(z) =G∑

g=1

�(g)N(z; z(g), R(g)), (12)

where Gaussian pdf of variable x, with mean m and covari-ance R is denoted by N(x; m, R). One Gaussian sum measure-ment model of five components is presented in Fig. 1, whereeach component is presented by a one-sigma ellipsis. A three-dimensional presentation of measurement pdf and associatedGaussian mixture measurement presentation pdf, Eq. (12), areshown in Fig. 2.

4. Target tracking

The track is modeled in the manner of track splitting [11,7]filter. The track, or target trajectory state estimate pdf, is aset of components. Each track component is a trajectory stateestimate obtained by applying one measurement componentper each measurement. As each measurement component has aGaussian pdf, standard linear estimators are used, in this paperInteracting Multiple Model (IMM) filter [7,4]. The pdf of thetrack at time k − 1 is a weighed sum of track component pdfs:

p(xk−1|yk−1) =Ck∑c=1

k−1(c)p(xk−1|c, yk−1), (13)

where each track component pdf is a weighed sum of IMMmodel pdfs

p(xk−1|c, yk−1) =M∑

j=1

�k−1|k−1(c, j)p(xk−1|c, j, yk−1), (14)

D. Mušicki / Systems & Control Letters 57 (2008) 216–221 219

p(xk−1|c, j, yk−1)

= N(xk−1; xk−1|k−1(c, j), Pk−1|k−1(c, j)),

Ck∑c=1

k−1(c) =M∑

j=1

�k−1|k−1(c, j) = 1 (15)

and where Ck denotes the number of track components, k−1(c)

denotes the posterior probability that measurement history oftrack component c is correct, �k−1|k−1(c, j) is the posteriorprobability that the target is following trajectory model j at timek−1 given that the track component c is a true track component.Track state estimate mean and covariance at time k − 1 andgiven that component c and trajectory model j are correct, aredenoted by xk−1|k−1(c, j) and Pk−1|k−1(c, j), respectively.

4.1. Track prediction

Measurements arrive at random times, the first step in trackupdate at time k is track state prediction from time k − 1 totime k. Values of Fk(j), Qk(j) and �j,� depend on the timebetween detections k and k −1. The details can be found in [1]and are not repeated here. Track prediction consists of IMMmixing and prediction operation for each track component cseparately. IMM mixing and prediction is a standard operation[1,7,4], which is not repeated here for reasons of space.

The outputs at time k of the IMM mixing and predictionoperations for track component c are

• a priori probability of model j, �k|k−1(c, j),• a priori track component state estimate mean and covariance,

xk|k−1(c, j) and Pk|k−1(c, j), respectively.

The pdf of predicted target trajectory state is a weighed sum ofcomponent pdfs:

p(xk|yk−1) =Ck∑c=1

k−1(c)p(xk|c, yk−1), (16)

p(xk|c, yk−1) =M∑

j=1

�k|k−1(c, j)p(xk|c, j, yk−1), (17)

p(xk|c, j, yk−1) = N(xk; xk|k−1(c, j), Pk|k−1(c, j)), (18)

Ck∑c=1

k−1(c) =M∑

j=1

�k|k−1(c, j) = 1. (19)

The prior pdf of measurement zk in Cartesian coordinates isgiven by

pk�p(zk|yk−1) =G∑

g=1

�k(g)pk(g), (20)

pk(g)�p(zk|g, yk−1) =Ck∑c=1

k−1(c)pk(c, g), (21)

pk(c, g)�p(zk|c, g, yk−1)

=M∑

j=1

�k|k−1(c, j)pk(c, g, j), (22)

pk(c, g, j)�p(zk|c, j, g, yk−1)

=N(zk(g); zk(c, j), Sk(c, g, j)), (23)

zk(c, j) = Hxk|k−1(c, j),

Sk(c, g, j) = HP k|k−1(c, j)HT + Rk(g). (24)

where H is the standard linear system measurement matrix.

4.2. Track update

Each pair {existing track component, measurement compo-nent} generates a new component for the next scan, as derivedin [8]. The probability of new component c+ = {c, g} is ob-tained by applying the Bayes formula and is given by

k(c+) = k−1(c)�k(g)pk(c, g)

pk

. (25)

State estimate of new component c+ is defined by xk|k(c+, j)

and Pk|k(c+, j) and �k|k(c+, j), j =1 · · · M . A posteriori prob-ability of trajectory model j of new track component c+ isgiven by

�k|k(c+, j) = pk(c, g, j)�k|k−1(c, j)∑M�=1 pk(c, g, �)�k|k−1(c, �)

= pk(c, g, j)�k|k−1(c, j)

pk(c, g). (26)

Mean xk|k(c+, j) and covariance Pk|k(c+, j) of a posterioristate estimate of model j of new component c+ are obtained byapplying Kalman filter state update on measurement (compo-nent g) with mean zk(g) and covariance Rk(g) and a priori stateestimate of model j of old component c with mean xk|k−1(c, j)

and covariance Pk|k−1(c, j). They are standard Kalman filterformulae [1]. Track state estimate pdf after track update isgiven by Eq. (13), with k − 1 replaced by k and Ck replacedby Ck+1 = Ck · Gk . Track structure remains with a differentnumber of components.

4.3. Component control

The number of track components created at scan k and en-tering scan k + 1 equals

Ck+1 = Ck · G. (27)

The number of track components therefore grows exponen-tially, and their number must be controlled. A number of tech-niques [3,2] exists to control the number of track components,from pruning the track components with low probability k(c),to track component subtree pruning, to sophisticated merging[10,13]. Track component control is not a part of the algorithmitself, however it is a necessity in any practical implementation.

220 D. Mušicki / Systems & Control Letters 57 (2008) 216–221

It is important to remember that after track component controlexercise the constraint

Ck+1∑c=1

k(c) = 1 (28)

must continue to hold. If some track components are removeddue to a form of pruning, then the probabilities of the remainingtrack components must be corrected so that the constraint ofEq. (28) holds.

Computational complexity is proportional to the number oftrack component updates. In scan k the computational complex-ity is theoretically proportional to the computational complex-ity of Ck ·G IMM filters. This usually implies one to two orderof magnitude higher computational complexity than the IMMfilter with the same measurement and track state estimation di-mension, and the same IMM models. Careful practical imple-mentation techniques, including component selection (gating)[3] as well as separation of component probability calculation(25) and component pruning from track component update, candramatically reduce this complexity to that of several IMMfilters.

4.4. Track output

The preferred form of track output consists of the mean valueand associated error covariance matrix of track trajectory esti-mate denoted by xk|k and Pk|k , respectively. They are given by

xk|k =Ck+1∑c=1

k(c)

M∑j=1

�k|k(c, j)xk|k(c, j)

Pk|k =Ck+1∑c=1

k(c)

M∑j=1

�k|k(c, j)(Pk|k(c, j)

+ xk|k(c, j)xk|k(c, j)T) − xk|kxTk|k . (29)

These values are output-only and are not used in subsequentrecursions.

4.5. Track initialization

Track is initialized by the first measurement, using the pro-cedure outlined in [5], extended here to accommodate an IMMfilter bank per track component. Each component of the firstmeasurement translates into one component of the track. EachIMM model of the component has equal probability,

�1|1(c, j) = 1

M, c = 1 · · · G, j = 1 · · · M . (30)

Position components (x and y) of x1|1(c, j) and P1|1(c, j) areequal to z1(c) and R1(c), respectively. Velocity and accelera-tion components of x1|1(c, j) are equal to zero. Velocity andacceleration components of P1|1(c, j) are equal to V 2

maxI2/3and A2

maxI2/3, respectively, where Vmax and Amax denote maxi-mum target speed and acceleration, respectively, and I2 denotesthe identity matrix of dimension two.

5. Simulation study

Each simulation run simulates 320 s of one target passivetracking by two bearings only sensors. The target follows atrajectory shown in Fig. 3, where the trajectory is displayedevery 4 s. Target motion consists of 8 segments, 40 s each:

(1) uniform motion with constant velocity of 9 m/s,(2) exponential acceleration motion, with acceleration a=v0

exp(t), where v0 is velocity at the start of the segment, tdenotes time since segment start, and = 0.015 s−1,

(3) exponential deceleration, with acceleration a=v0 exp(t),where v0 is velocity at the start of the segment, t denotestime since segment start and = −0.015 s−1,

(4) right turn with the angular velocity of �/36 rad/s,(5) exponential acceleration with = 0.015 s−1,(6) exponential deceleration with = −0.015 s−1,(7) left turn with angular velocity �/36 rad/s, and(8) uniform motion.

The IMM filter consists of four models of target motion:

(1) Uniform motion: target moves on a straight line with con-stant velocity.

(2) Acceleration: target moves with constant acceleration.(3) Left coordinated turn with constant angular velocity

� = �/30 rad/s.(4) Right coordinated turn with constant angular velocity �=

�/30 rad/s.

Initial [x, y] positions of sensors are [−10,000 m −10,000 m] and [0 − 10,000 m]. Both sensors move uniformlywith speed of 30 m/s in the x direction. Both sensors detect tar-get randomly and independently, with exponential distributionof time between detections with the mean of 0.5 s. Therefore,the number of target detections from each sensor follows aPoisson distribution. Angle measurement rms errors are 1◦ forboth sensors. Sensor position errors with independent zeromean Gaussian distribution and covariance matrices of 100I2are simulated.

Experiments “M3T20” and “M5T10” model measurementsby three and five component Gaussian mixtures, respectively,and retain 20 and 10 track components, respectively, withRmin = 200 m and Rmax = 20,000 m. After each track update,simple track component control is used to retain only trackcomponents with highest probabilities. More sophisticated

0 200 400 600 800 1000 1200 1400 1600

0

100

200

300

400

Fig. 3. Target trajectory.

D. Mušicki / Systems & Control Letters 57 (2008) 216–221 221

50 100 150 200 250 300

60

80

100

120

140

160

time

rms e

rror

M3T20

M5T10

Fig. 4. RMS estimation error.

means of component control may yield lower estimationerrors or faster execution. Each simulation run is repeated500 times. RMS estimation error over time is presented onFig. 4. Notice that higher number of measurement compo-nents (“M5T10”) converges somewhat faster, and higher num-ber of track components (“M3T20”) has somewhat smallerfinal and peak errors. To put the results in perspective, the“beam widths” at the position of the target are 245 m and175 m, respectively, and if both sensors were to detect simulta-neously, the “beams” would cross at an angle of 45◦. Increasingthe angle (up to 90◦) and/or using additional sensors woulddecrease track estimation errors further.

6. Conclusions

This paper presents a way to track a maneuvering tar-get using multiple asynchronous passive angle only sensorswith uncertain positions. By using Gaussian mixture presen-tation of both measurements and track estimates, need fornon-linear estimation using for example extended Kalman fil-ters, unscented Kalman filters or particle filters is eliminated.IMM estimation integrates seamlessly in this scheme, provid-ing computationally efficient estimation of target maneuvers.

As the number of track components grows exponentially intime, a component management scheme needs to be imple-mented. A trade off between the number of track componentsand the track quality can be used in practice to tune the systemdesign.

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