Magnetometer Modeling and Validation for Tracking Metallic Targets

Niklas Wahlström and Fredrik Gustafsson

Linköping University Post Print

N.B.: When citing this work, cite the original article.

Niklas Wahlström and Fredrik Gustafsson, Magnetometer Modeling and Validation for Tracking Metallic Targets, 2014, IEEE TRANSACTIONS ON SIGNAL PROCESSING, (62), 3, 545-556. http://dx.doi.org/10.1109/TSP.2013.2274639

©2014 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. http://ieeexplore.ieee.org/ Postprint available at: Linköping University Electronic Press http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-88963

1 Magnetometer Modeling and Validation for Tracking Metallic Targets Niklas Wahlstrom,¨ Student Member, IEEE, and Fredrik Gustafsson, Fellow, IEEE

Abstract—With the electromagnetic theory as basis, we present The simplest far-field model for the metallic vehicle is a sensor model for three-axis magnetometers suitable for localiza- to approximate it as a moving magnetic dipole, which is tion and tracking as required in intelligent transportation systems parametrized with the magnetic dipole moment m that can and security applications. The model depends on a physical magnetic dipole model of the target and its relative position be interpreted as the magnetic signature of the vehicle. This to the sensor. Both point target and extended target models dipole model has previously been used for classification and are provided as well as a heading angle dependent model. The target tracking of ground targets in [4]–[16]. In a near-field suitability of magnetometers for tracking is analyzed in terms of scenario the signal structure is much more complex where local observability and the Cramer´ Rao lower bound as a function nonparametric methods have been shown to be successful. of the sensor positions in a two sensor scenario. The models are validated with real field test data taken from various road vehicles In [17], [18] an extensive investigation is presented, where which indicate excellent localization as well as identification of the measured signal itself has been interpreted as the mag- the magnetic target model suitable for target classification. These netic signature, which can be used for classification and re- sensor models can be combined with a standard motion model identification of the vehicle at a different location. and a standard nonlinear filter to track metallic objects in a Furthermore, several studies have been done exploring the magnetometer network. use of underwater magnetometers for tracking of vessels [19]– Index Terms—Magnetometers, tracking, magnetic dipole, ob- [21], where [21] has used the dipole model not only for servability localizing the vessel, but also for estimating the positions of the sensors. In [20] a Bayesian match-field approach has I.INTRODUCTION been used, which takes the attenuation of the seawater into RACKING and classification of targets are primary con- consideration. T cerns in automated surveillance and security systems. As in [4]–[16], [21], this work will be based on the dipole The tracking and classification information can be used for model. Similar to [4]–[7], [9]–[12] the theory will be applied statistical purposes, i.e. counting number of targets of a on experimental data. However, we will generalize some specific type and registration of their velocities and directions important assumptions that previously have been made: of arrival. In this work we will focus on metallic targets. • Prior work [5]–[16] is constrained to a single dipole point One way of sensing metallic objects is by making use of target model, which does not included the geometrical their magnetic properties. We know that metallic objects in- extend of the target. We suggest a model with multiple duce a magnetic field partly due to its permanently magnetized dipoles. This model was suggested by Wynn [4] without ferromagnetic content and partly due to the deflection of the being illustrated on simulated or real data, which will be earth magnetic field [1]. This induced magnetic field can be provided in this work. measured with distributed passive magnetometers [2], [3]. • Work in [4]–[10] is based on a linear motion assumptions, For moving metallic vehicles, the magnetometer measure- which simplifies the modeling of the dipole moment . ments will vary in time, which results in a time dependent m However, for other motions where the target orientation signal. Like in other common tracking application based on changes over time, the dependency on will be more radar, time difference of arrival and received signal strength, m complicated. One way of solving this is to marginalize this signal depends on the position, speed, orientation and the dipole moment as in [12]–[14]. However, this work target specific parameters. The difference between the appli- will explicitly describe the as a function of the target cations is summarized in the sensor model that relates the m orientation. quantities to the observations. The same motion models and filters can be used in all these application. However, we do not We will derive a general sensor model for extended mag- focus on suggesting particular motion models and filters since netic targets with target orientation dependent dipole moments these choices are standard in literature. Instead, we focus on observed in an arbitrary magnetometer network, where we the application specific fundamental questions of observability validate the models using statistical tools for a variety of and geometry for the sensor model. vehicles. Furthermore, in [16] the problem with lack of observability This work is supported by the Swedish Foundation for Strategic Research under the project Cooperative Localization. A preliminary version of this for a one-sensor scenario has been addressed. Our work will paper was presented at the Thirteenth International Conference on Information more precisely describe this unobservable manifold. Also the Fusion (FUSION), Edinburgh, U.K., 2010. optimal sensor deployment is analyzed in terms of local The authors are with the Department of Electrical Engineering, Division of Automatic Control, Linkoping¨ University, Linkoping,¨ SE 581-83, Sweden observability and the Cramer´ Rao Lower Bound (CRLB) as (e-mail: [email protected]; [email protected]) a function of the sensor positions. 2

The paper outline is as follows: Section II describes the B. Multiple Sensor Model sensor model and its extensions. In Section III a constant The sensor model (2) can easily be extended to handle velocity tracking model is presented and compared with other multiple sensors by introducing models from the literature based on the same assumption. In h (x ) = B + Jm(r θ )m (3a) Section IV observability and sensor deployment is discussed j k 0,j k − j k and in Section V the estimation methods are summarized. A where θj and B0,j are the position and the bias of the jth validation of the sensor model based on experimental data is sensor respectively. provided in Section VI. The paper closes with conclusions and The total sensor model is then given by proposals for future work. yj,k = hj(xk) + ej,k (3b) T II.THEORETICAL SENSOR MODEL T T T with xk = B0,1:J rk mk , (3c) In the statistical signal processing framework, a sensor where ej,k is the measurement noise of the jth sensor. model of a time-invariant system is given by C. Extended Target Model yk = h(xk) + ek, (1) The dipole model (2b) approximates the target with a single where y is the measurement, x is the state of the system and k k magnetic dipole and can as such be seen as a point target e is measurement noise, all at time instant kT , T being the k s s model. However, this is a fairly rough approximation since sample period. This section will present and discuss different vehicles do have a geometrical extension. By parameterizing sensor models suitable for target tracking of magnetic objects. the extended target with multiple dipoles, a more accurate model can be obtained. By that we get an extended target A. Single Sensor Point Target Model i model. Consider each dipole to be placed at ∆rk, relative to i Any magnetic target will induce a magnetic field, which the center of the target rk and having the dipole vector mk. can be measured by a magnetometer. By approximating the Then the point target model (2b) can be extended as target as a magnetic dipole [1], this field can be modeled as d m i i a magnetic dipole field. An expression of this field can be h(xk) = B0 + J (rk + ∆rk)mk (4) derived from Maxwell’s equations [22] (for a derivation see i=1 X for example [1]) and gives the following nonlinear model where d is the number of dipoles. The extended target model 2 in (4) reduces to a single target model (2b) if the relative µ0 3(rk mk)rk rk mk h(x ) = B + · − k k (2a) i k 0 5 positions ∆rk are much smaller than rk. The influence of the 4π rk m k k target extension can be analyzed by applying a Taylor series = B0 + J (rk)mk, (2b) m expansion of J (rk) in (2c) around rk. By truncating the Tay- with lor series after the second term, we reach the approximation

m µ0 T 2 d J (rk) = (3rkrk rk I3), (2c) m m i i 4π r 5 − k k h(xk) B0 + J (rk) + ∇r J (rk) ∆r m (5) k ≈ k k k k i=1 where is a bias, is the position of the target relative to X r i B0 rk ,J (rk,∆rk) the sensor, is the magnetic dipole moment of the target mk d | {z } m m −1 r i i and ek (0, R) white Gaussian noise. The total sensor = B + J (r ) I + J (r ) J (r , ∆r ) m , ∼ N 0 k 3 k k k k model is then given by i=1 X where we have defined yk = h(xk) + ek (2d) ∂ T T T T ∇ f(x) = f (x) (6) with xk = B0 rk mk , (2e) x i1,...,in+1 i1,...,in { } ∂xin+1 The dipole model is an approximation of the induced magnetic and Jr( , ) is given by (30b). Further, it can be shown that field by considering the target to be a point source. This · · i approximation is valid if the target is far from the sensor m −1 r i 8 ∆rk 2 J (rk) J (rk, ∆rk) 2 k k (7) compared to its size, see Section II-C. k k ≤ 7 rk 2 k k The bias B0 will in theory be constant and equal to the i and with ∆rk 2 rk 2 we arrive at the point target model earth magnetic field. However, in practice it also includes the k k k k d magnetic distortion induced by other metallic objects in the h(x ) Jm(r ) mi = Jm(r )m , (8) environment, which we assume to be constant. Furthermore, k ≈ k k k k i=1 the bias B0 also includes a magnetic sensor bias, which X slightly depends on temperature. This temperature may vary where the dipole moment mk in point target model is related i over time due to changing weather condition and amount of to the dipole moments mk in the extended target models as daylight exposed to the sensor [17]. However, this drift is d i slowly varying and is assumed to be constant during the time mk = mk. (9) it takes for a vehicle to pass the sensor. i=1 X 3

Thus, the point target model is valid if the distance between magnetization due to the hard iron can be represented with the target and the sensor is much larger than the characteristic a magnetic dipole moment m0, which is independent of the length of the target, which has also been stated in [1]. external magnetic field and will thus always be constant in However, if this is not the case an extended target model might the reference frame of the vehicle. Since the sensors have the be considered. same reference frame as the world, which is not the same as the one of the vehicle, a transformation between these D. Uniformly Linear Array of Dipoles two reference frames has to be found. Generally, the heading Ψ, banking φ and inclination θ angles are used to define In this work, a row of dipoles will be analyzed. Since a the relative orientation of a vehicle with respect to world vehicle normally has its largest geometrical extension in the coordinates. Here, the world coordinates are aligned with the same direction as its course, a row of equidistantly distributed magnetic north, see Fig. 2. In this work, no banking and dipoles along the velocity vector vk will be considered (see inclination are considered since we assume that the road is flat, Fig. 1). The velocity vector vk will later be included in the i.e. it does neither bank nor incline. Furthermore, any slip is dynamic description in Section III. Furthermore, the size of the neglected. Consequently, the heading Ψk uniquely determines the orientation of the vehicle with respect to the sensors and m = mi k i k is defined as LP 4 mk y x Ψk = arctan2(vk, vk), (11) 3 mk 2 vk with arctan2 being the four quadrant arc-tangent. Now, with 1 mk mk 2 the rotation matrix ∆rk cos Ψk sin Ψk 0 rk θj − − Q(Ψk) = sin Ψk cos Ψk 0 , (12) Sensor j  0 0 1 the hard iron contribution of magnetic dipole moment can be Fig. 1: The vehicle parametrized with d = 4 dipoles aligned described as in a row in the direction of the vehicle course vk. All dipoles hard are equally distanced with the interval L/(d 1), i.e. a total mk = Q(Ψk)m0. (13) length of L. − On the other hand, the magnetization due to the soft iron is more complicated. For an isotropic medium (a spherical shell object is unknown, thus the distance between the first and the for example), the induced magnetic dipole moment m is last dipole L has to be regarded as a parameter. By combining soft parallel to the earth magnetic field B this with the multiple sensor model (3) we get 0 soft D d m = B0, (14) d m µ h (x ) = B + J (r θ + ∆ri )mi (10a) 0 j k 0,j k − j k k i=1 with D being a target characteristic scalar constant. X with However, in general the moment induction is a non-isotropic process and its description requires a symmetric 3 3-matrix i d + 1 L vk × ∆rk = i . (10b) P, which is known as the polarizability tensor [1]. In world − 2 d 1 vk T − k k coordinates this tensor reads Q(Ψk)PQ (Ψk) and we have The total sensor model is given by 1 msoft = Q(Ψ )PQT(Ψ )B . d k µ k k 0 yj,k = hj (xk) + ej,k (10c) 0 T T T 1:d T T Since a symmetric 3 3-matrix has 6 unknowns, this would with xk = (B0,1:J ) r v (m ) L . (10d) k k k heavily enlarge our state× space dimension (and presumably Note that if either the length is L = 0 or the number of dipoles lead to unobservability). Therefore, in this work we assume is d = 1, the model (10) will be equivalent with the point target that magnetic dipole moment is parallel to the earth magnetic model presented in (2). field. Thus we assume that the vehicles are isotropic, which might be a rough but workable assumption. E. Target Orientation Dependent Model Since magnetization is additive, the total magnetic dipole moment can be modeled as The magnetic dipole moment m will also depend on the k D orientation of the vehicle. To describe this dependency, the mk = Q(Ψk)m0 + B0. (15) source of the magnetization has to be decomposed into one µ0 hard iron contribution and one soft iron contribution as done In Fig. 2 a graphical representation of the model is presented. in [5], [13]. We know that metallic objects induce a magnetic field partly We have now decomposed the magnetic dipole moment into due to the ferromagnetic content (hard iron) and partly due two components, one representing the hard iron and one rep- to the deflection of the earth magnetic field (soft iron). The resenting the soft iron. The corresponding model parameters 4

Magnetic north III.CONSTANT VELOCITY TRACKING MODEL x B0 Ψk vk soft D The models presented in the previous section are not con- m = B0 y µ0 strained to any type of motion. However, in this work special mk interest will be given to the scenario when vehicles pass the z sensor with constant velocity, i.e. a straight line motion with constant speed. This prior knowledge enables us to formulate a static sensor model.

Vehicle rj,k A. Sensor Model for Constant Velocity

By assuming constant velocity vk Magnetometer hard mk = rk+1 = rk + Tsvk, Q(Ψk)m0 (17) Bdipole vk+1 = vk,

the heading Ψk will also be constant as well as the magnetic dipole moment mk according to (15). We could also consider Fig. 2: A stationary magnetometer measures the earth magnetic another parametrization of the target motion. However, if field B0 together with a magnetic dipole field Bdipole. The that motion model implies a time-varying heading Ψk, the magnetic dipole field is induced by a moving vehicle at magnetic dipole moment will no longer be constant and the position rj,k with velocity v0 and magnetic dipole moment target orientation dependent dipole model (15) has to be m = Q(Ψ )m + D B . The heading angle Ψ is defined considered. k k 0 µ0 0 k by the velocity direction. We can now define a static sensor model

yk = hk(x) + ek D and m0 will represent the real magnetic signature of the µ 3(r m)r r 2m target and will thus be constant even in a target maneuvering 0 k k k , (18) = B0 + · −5 k k + ek scenario. 4π rk k k where rk = r0 + kTsv0. F. General sensor model Here, the substitution The orientation dependent sensor model can now be com- rk = r0 + kTsv0 bined with the extended target model (10) which gives (19) d mk = m hd(x ) = B + Jm(r + ∆ri θ )mi (16a) j k 0,j k k − j k has been used in (2). Thus, now we have the parameter vector i=1 X T T T T T with x = B0 r0 v0 m , (20) i i i D mk = Q(Ψk)m0 + B0, (16b) where r0 is the initial position of the target, v0 its constant µ0 velocity and m its constant magnetic dipole moment. Thus, the 1 J model (18) has 12 parameters where 6 of them enter linearly B0 = B0,j, (16c) J in (18) (B0 and m) whereas the other 6 (r0 and v0) enter j=1 non-linearly. X y x Ψk = arctan2(vk, vk), (16d) In a similar manner also the extended target model pre- i d + 1 L vk sented in (10) can be reformulated under the constant velocity ∆rk = i . (16e) 1:d − 2 d 1 vk assumption. Here, all magnetic dipole moments m will be − k k constant. We have Here, the mean of all sensor biases B0,j has been taken as an estimate of the earth magnetic ﬁeld, which is needed for d d m i i describing the orientation of the soft iron contribution (14). h (x) = B0,j + J (rk + ∆r θj)m , (21a) j,k k − i=1 The total sensor model is then given by X d with yj,k = hj (xk) + ej,k (16f) T T T 1:d T 1:d T T i d + 1 L v0 with xk = (B0,1:J ) r v (m ) (D ) L ∆r = i and r = r + kT v . k k 0 k − 2 d 1 v k 0 s 0 − k 0k Note that if the heading Ψk is constant the magnetic (21b) i dipole moment mk will also be constant. Consequently, the i i The total sensor model is then given by parameters for the hard m0 and the soft iron D are not identiﬁable. Therefore, in such a scenario a parametrization of d yj,k = hj,k(x) + ej,k (21c) the dipole moment directly is more attractive solution, which T T T 1:d T T will be used in the next section. with x = (B0,1:J ) r0 v0 (m ) L . (21d) 5

B. Anderson function expansion for constant velocity C. Comparison The presented sensor model (18) describes the observed By comparing the two different models (18) and (22), the magnetometer signal if the target is moving with constant ve- following observations can be made locity. An equivalent model [4], [6]–[8] for the same problem • The Anderson function model (22) consists of 15 pa- setup can be formulated by modeling the signal itself. Direct rameters, whereas the dipole model (18) consists of only calculations [4] starting from (18) reveals that each of the three 12. Consequently, (22) is over-parametrized. The reason dimensions in the sensor response yk can be expressed as a is that the superposition of the Anderson functions in linear combination of three elementary functions. These func- (22) is non-linearly constrained and does not have all 9 tions are known as the Anderson functions [4]. By choosing degrees of freedom as given in (22). This can be seen the time origin at the closest point of approach (CPA) time by inspecting (25). Each of the unit vectors ˜r, v˜ and m˜ t CPA r CPA and normalizing it with the range CPA and the speed contribute with 2 degrees of freedom to the matrix C. By v = v , the resulting model is 0 k 0k taking the coefficient β and the constraint (˜r v˜) = 0 into account, C has in total only 6 independent components,· v0 or equivalently, the 9 coefficients in C have to obey 3 yk = B0 + C f (kTs tCPA) + ek, (22) · rCPA − (non-linear) constraints. These constraints are not present where the time dependency enters in the orthogonal Anderson in (22). Thus, (22) is a relaxation of (18). functions [4] • The dipole model (18) has 6 parameters entering non- T linearly in the model, whereas (22) has only 3. Since the f(t) = f0(t) f1(t) f2(t) (23a) relaxation reduces the non-linearity, (22) can be seen as n t a convexification of the more non-linear model (18). fn(t) = 5 , n = 0, 1, 2 (23b) (t2 + 1) 2 • The parametrization of (18) is directly related to the target and where C is a 3 3-matrix of coefficients. Consequently, trajectory (r0 and v0) and the magnetic signature of the × target m. This will make the multi sensor extension of (22) is linear in B0 and C, which includes 3 + 9 = 12 (18) trivial, see Section II-B. In contrast, a multi sensor parameters, and is nonlinear in the remaining 3 parameter v0, version of (22) has to rely on sub-optimal solutions where rCPA and tCPA, in total 15 parameters. each sensor estimates its own matrix Cˆ, which then are Both tCPA and rCPA are related to the parameters in (20) in the following way: The traveled distance between t = 0 and fused at a later stage, as done in [8]. In addition, (18) can easily be extended to tracking by t = tCPA is equal to the projection of r0 onto v0, see Fig. 3. − applying a motion model and a nonlinear filter, which has been done in [11]. This extension would for (22) be t =0 t = tCPA harder, since the shape of the Anderson functions relies v0 Target trajectory v0 on a linear motion of the target.

r0 rCPA = r0 + tCPAv0 IV. MAGNETOMETER POTENTIALFOR LOCALIZATION (r v )/ v AND TRACKING − 0 · 0 k 0k In this section limitations and possibilities of the magne- Fig. 3: The CPA time t and CPA range r can be expressed CPA CPA tometer will be discussed in terms of observability and the by the initial position and the velocity . r0 v0 Cramer´ Rao lower bound as a function of the placement of a second sensor. This gives the CPA time and CPA range Without loss of generality the bias B0 is assumed to be (r0 v0) (r0 v0) known and the discussion will be based on the unbiased static tCPA = · v0 = · (24a) T T T T − v0 k k −(v0 v0) sensor model in (18) with the state x = r0 v0 m k k . · to make the analysis cleaner. We will therefore consider the rCPA = rCPA = r0 + tCPA v0 . (24b) k k k · k model Furthermore, also the coefficients in C are related to (20) µ 3(r m)r r 2m h (x) = 0 k · k − k kk , in that k 4π r 5 (27) k kk C = β c1 c2 c3 , (25a) where rk = r0 + kTsv0. with A. Single Sensor Observability c1 = 3(˜r m˜ )˜r m˜ , (25b) · − To analyze observability, a local analysis can be performed. c2 = 3(˜r m˜ )v˜ 3(v˜ m˜ )˜r, (25c) · − · Consider the Fisher information matrix (FIM), which under c3 = 3(m˜ v˜)v˜ m˜ , (25d) Gaussian assumptions of the sensor noise e (0, R) reads · − k ∼ N where the following normalization has been used N T µ0 m rCPA v0 m 0 0 −1 0 β = , ˜r = , v˜ = , m˜ = . (26) (x ) , ∇hk (x )R ∇hk(x ), (28) 4π r3 r v m I CPA CPA 0 kX=1 6 where x0 are the true parameters. Any zero eigenvalues of only be regarded as the tangent of the unobservable one- this matrix makes it singular, which indicates a lack of local dimensional manifold at this point. Denote this manifold 0 T observability at x . The unobservable subspace is spanned by x˜(u) = ˜r0(u) v˜0(u) m˜ (u) , where u is a scalar param- the eigenvectors corresponding to the zero eigenvalues, also eter. For each u there will be a λ(u) such that known as the kernel of the matrix. Indeed, it can be shown ˜r (u) ˜r (u) that the kernel of (x0) is at least of dimension one. d d 0 0 I x˜(u) = v˜0(u) = λ(u) v˜0(u) . (33) du du Proposition 1. Consider the sensor model (27) and its infor- m˜ (u) 3m˜ (u) mation matrix (28). Then By choosing the parametrization  u = λ(u)−1, we get the r0 r0 following unobservable manifold (x0) v = 0, for all x0 = v 9. (29) 0 0 R ur I · 3m m ∈ 0 x˜(u) = uv0 . (34)       T T T T u3m Proof: Define x¯ = r0 v0 3m and apply x¯ to the Jacobian It is instructive to substitute x˜(u) into (18) and conclude that r0 h1:k(x˜(u)) is independent of the parameter u, which means 0 r r m ∇hk(x )x¯ = J (rk, m) kTsJ (rk, m)J (rk) v0 that all points on this manifold will give the same output yk. ·   3 3m For example, multiplying r0 and v0 with 2, and m with 2 = 8 r r m will result in the same output yk. = J (rk, m)r0 + kTsJ (rk, m)v0 + 3J (rk)m r m The result is physically quite intuitive since the magnetome- = J (rk, m)rk + 3J (rk)m. ter does not measure any absolute distances and the system can m r be arbitrarily scaled without changing the measured output. where J (rk) and J (rk, m) are the Jacobians of hk(x) with Thus, a small vehicle driving slowly close to the sensor will respect to m and rk and respectively give rise to the same signal as a large vehicle driving fast far 1 Jm(r ) = (3r rT r 2I ) (30a) from the sensor. Furthermore, the cubic scaling of the magnetic k r 5 k k − k kk 3 k kk dipole moment m is reasonable since it is related to the volume r 3 T of the vehicle. J (rk, m) = (rk m)I3 + rkm r 5 · The conducted observability analysis also gives insight k kk (r m) to the comparison between the dipole model (18) and the + mrT 5 k · r rT . (30b) k − (r r ) k k Anderson function model (22): k · k Further, we can show that • The sensor model (22) can easily be re-parametrized excluding the unobservable manifold (34) by introducing r m J (rk, m)rk = 3J (rk)m (31) the scale parameter α = v0/rCPA. This parametrization − has also been used in [4], [6]–[8]. An equivalent re- and from this we get parametrization in (18) would inevitably increase the ∇h (x0)x¯ = 3Jm(r )m + 3Jm(r )m = 0 k, model complexity. k − k k ∀ • By substituting (34) into (24a) it can be stated that tCPA which leads to is independent of u and consequently observable. This N means that tCPA can be uniquely determined from only (x0)x¯ = ∇hT(x0)R−1 ∇h (x0)x¯ = 0. I k k one vector magnetometer and does not necessarily need k=1 X =0 be aided with other sensors as done in [6]–[8] to achieve observability for that parameter. Furthermore, notice that | {z } Thus, the kernel is at least given by the following one- only one axis of the vector magnetometer is sufficient dimensional subspace: to achieve observability for tCPA. This can be seen by decomposing its signal into the three Anderson functions 0 T T T T Ker (x ) λ r0 v0 3m λ R . (32) (23b), which each of them contains the information of I ⊇ | ∈ t . In most cases the kernel dimension is exactly equal to CPA one, meaning that (32) is fulfilled with equality. For ex- T B. Single Sensor Observability with Prior Information ample, for the following parameters r0 = 3 1 0 , T T − The lack of observability can be solved by using prior v0 = 1 0 0 and m = 1 1 1 , the kernel of the information matrix indeed has dimension one. However, knowledge about the target trajectory. This situation has been studied in [6] and [7], which includes the possibility to use if m is orthogonal to both r0 and v0 the kernel will be T information about the range at CPA, for example by knowing larger. For example, for the parameter r0 = 3 1 0 , T T − the geometry of the road on which the vehicle is traveling. By v0 = 1 0 0 and m = 0 0 1 the kernel of the using (24) this gives the extra scalar measurement information matrix has dimension two. Due to the non-linearity, the expression of the unobservable 0 (r0 v0) yCPA = hCPA(x ) = rCPA = r0 · v0 (35) 0 k k − (v v ) · subspace (32) is only valid at the point x and can therefore 0 0 ·

7 and the corresponding information matrix can be computed as Furthermore, under the assumption that the measurement noise of different sensors are independent, the information (x0) = ∇hT (x0)α−1∇h (x0), (36) ICPA CPA CPA matrices for all sensors are additive, which gives where α 0 since the information in (24b) comes with zero J → 0 0 uncertainty. Furthermore, since this prior knowledge is inde- J (x , θ1:J ) = (xj ) (40) I I j=1 pendent of the sensor information, the information matrices X are additive, which gives By again using Proposition 1 we have

0 0 0 0 T T T T +CPA(x ) = (x ) + CPA(x ). (37) Ker (x ) λ (r0 θj) v0 3m λ R . (41) I I I I j ⊇ − | ∈ If the condition in (32) is fulfilled with equality, we only Assume that (41) is fulfilled with equality for two different 0 T T T T sensors at θ and θ . Then the intersection of their kernels need to require that CPA(x ) r0 v0 3m = 0 for 1 2 0 I · 6 will be empty if θ = θ . Under these assumptions we will +CPA(x ) to have full rank. This is indeed true if r0 is not 1 2 I then reach observability6 for a system with two sensors. parallel to v0. Proposition 2. Consider the sensor model (35) and its information matrix (36). Then D. Comparison We can now compare two extensions from the single sensor r0 setup 0 CPA(x ) v0 = 0 (38) • Setup 1) One sensor with prior knowledge of CPA range I · 3m • Setup 2) Two sensors   if and only if r0 is parallel to v0. Both scenarios resolve the observability issue reported for T the single sensor setup except for some cases with special Proof: Define T T T and apply to the x¯ = r0 v0 3m x¯ geometry. However, Setup 2 requires that both coordinate Jacobian systems of the two sensors are aligned with each other and r also that their relative position is known in order not to violate T T 0 0 r (r0·v0)r ∇h CPA CPA the sensor model (39). This issue is less critical in Setup 1 CPA(x )x¯ = kr k v ·v kr k 01×3 v0 CPA ( 0 0) CPA ·   h i 3m since any misalignment of the single sensor will not change   the range to the target and will thus not violate the sensor Since (r v ) = 0 we have CPA · 0 model. Furthermore, in Setup 2 the two sensors need to be synchronized. 0 (rCPA r0) ∇hCPA(x )x¯ = · r k CPAk (r r )(v v ) (r v )(r v ) E. Cramer´ Rao Lower Bound = 0 · 0 0 · 0 − 0 · 0 0 · 0 0 (v0 v0) rCPA ≥ So far we have only compared the models concerning their · k k observability properties. However, in order to find a good and is only fulfilled with equality if and only if r is parallel 0 sensor deployment we want to quantify the best achievable to v . This leads to 0 estimation performance as a function of the sensor positions. N This analysis is justified with the CRLB [23], which states that 0 T 0 −1 0 CPA(x )x¯ = ∇h (x )R ∇hCPA(x )x¯ = 0 any unbiased estimate must have a covariance matrix larger I CPA kX=1 than or equal to the inverse of the FIM if and only if r0 is parallel to v0. Cov(xˆ) (x0)−1 0. (42) Consequently, we do not achieve observability if r0 is − I parallel to v0, which corresponds to rCPA = 0. However, The design goal in the sensor deployment is then to make the this situation can be considered as unphysical, since it would inverse of the information matrix as small as possible in order correspond to a scenario where the sensor is on the target to minimize the covariance of the estimate. We do this analysis trajectory. for the partitions of the state space corresponding to r0, v0 and m separately. We notice that for a positive semidefinite matrix any diagonal sub matrix is also positive semidefinite. C. Multi Sensor Observability By following [23] and using that property on (42) we get The lack of observability can also be solved with a second − [Cov(xˆ) (x0) 1] 0 (43) sensor. To handle multiple sensors, the sensor model has to − I ii be slightly expanded. Let the jth sensor be positioned at θj. and consequently The target parameter relative to the jth sensor will then be 0 −1 T T T T Cov(xî) = [Cov(xˆ)]ii [ (x ) ]ii (44) xj = (r0 θj) v m and the total sensor model is I − 0 given by where i corresponds to a partition of the state vector. The design goal in the sensor deployment is then to minimize any y = h (x ) + e , for all j J. (39) k,j k j k,j ∈ of the three criteria 8

In this framework, also the covariance of the estimate xˆ can be estimated. The covariance can be approximated as [26] 0 −1 , 0 −1 , 0 −1 [ (x ) ]r0r0 2 [ (x ) ]v0v0 2 [ (x ) ]mm 2 I I I −1 (45) N 2 d T −1 d depending on if we are interested in a good estimation per- Cov(xˆ) ∇h (xˆ) R ∇h (xˆ) . (47) ≈  j,k j j,k  j=1 formance for the position r0, velocity v0 or magnetic dipole kX=1 X   moment m. In order to examine where a second magnetometer which can be derived from a second order approximation of should be placed, these three criteria can be computed for d the cost function (46b), where we use that ∇V (x) x=xˆ = 0 different positions θ2 of the second sensor. In [10] similar at optimum. | criteria has been analyzed. For most values x0 the conclusion is that you should deploy a two sensor system symmetrically at both sides of the target trajectory if a good estimation C. Model Validation performance for the position r0 and magnetic dipole moment The number of dipoles d can for this model class be m is of interest. On the other hand, for the velocity v0 it considered as a model order parameter, which will affect the is better to place both sensors at the same side of the target statistical properties of the cost function. Under the assumption trajectory and as separate as possible, which is intuitive. In that both the measurement equation hj,k(x) and the noise the experiment presented in this work, the first of these two ej,k (0, Rj) are correctly modeled, the cost function deployments has been considered. V d ∼will N be χ2 distributed, n and n being the di- (xˆ) Nny −nx y x mension of each measurement and the state space respectively. V. ESTIMATION For the extended target model in (16) this will be As already mentioned, in this work special interest will be given to scenarios where vehicles pass the sensors with con- ny = dim(y1:J,k) = 3J (48) T stant velocity. Therefore, in this section the estimation methods n = dim( (B )T rT vT (m1:d)T L ) to be used with the constant velocity models presented in x 0,1:J 0 0 Section III will be given. These methods will later be used = 3J + 3 + 3 + 3d + 1 = 3J + 7 + 3d. (49) in the sensor model validation presented in Section VI. Since the mean of χ2 is equal to Nn n , the Nny −nx y x normalized cost function − A. Sensor noise covariance d d V¯ (xˆ) = V (xˆ)/(Nny nx) First, the distribution of the sensor noise ej,k will be − estimated for each of the magnetometers. According to [24], = V d(xˆ)/(3NJ 3J 7 3d) (50) the noise of the magnetometer [25] is white Gaussian, i.e. − − − is considered and V¯ d(xˆ) 1 would indicate a correct its samples are i.i.d. and normally distributed with zero mean. ≈ Such a stochastic process is uniquely defined by its covariance model. Therefore, the normalized cost function can be used for matrix R. This matrix can be estimated from a measurement choosing an appropriate model order. However, in this work sequence without any moving targets by using the sample no explicit model order selection rule will be proposed. covariance N D. State and parameter estimation 1 T Rj = (y y¯ )(y y¯ ) , where N 1 j,k − j j,k − j By using the maneuvering target model presented in Section − k=1 X II-E, the constant velocity assumption can be relaxed. By 1 N y¯ = y . combining that sensor model with an appropriate motion j N j,k k=1 model, a non-linear filter can be applied to estimate all states X included in the model. Related results have been reported in Here Rj represents the covariance of the jth sensor. [11] where a (nearly) constant velocity model has been applied and the estimation has been performed with an Extended B. Parameter estimation using constant velocity model Kalman Filter using data for vehicles passing a three-way The parameter x introduced in the models (3) and (16) can intersection. be estimated using weighted least square by minimizing the cost function VI.SENSOR MODEL VALIDATION xˆ = arg min V d(x) (46a) x In order to validate the proposed sensor models, real exper- N J imental data has been collected with magnetometers contained V d(x) = (y hd (x))TR−1(y hd (x)). j,k − j,k j j,k − j,k in commercial-grade measurement units [25]. In this section, j=1 kX=1 X the experimental setup will be described and the results of (46b) the validation will be given using the estimation methodology Note that under the Gaussian assumption of the measurement presented in Section V. In accordance with the results in noise ej,k, (46) will be both a minimum variance and a Section IV-A and IV-C, all measurements have been done with maximum likelihood estimate. two sensors in order to reach observability. 9

A. Experimental Setup The two magnetometers have been placed close to a straight road (see Fig. 4). In accordance with the discussion in Section IV-E they have been deployed symmetrically at each side of the road. In the experimental setup a relative distance of 9.0 m between the sensors has been used. As reference data a video (a) Vehicle 1 (b) Vehicle 2 (c) Vehicle 3 y Sensor 2 θ2 =[0 4.5 0]

m Road r θ k − 2 z x

rk v0 Vehicle r θ (d) Vehicle 4 (e) Vehicle 5 (f) Vehicle 6 k − 1 Sensor 1 θ = [0 4.5 0] Fig. 5: The vehicles used in the sensor model validation. 1 −

Video camera TABLE I: The values of the normalized cost function V 1(xˆ)/(Nn n ) for the point target model d = 1. N is y − x Fig. 4: Sensor setup for the sensor model validation. rk is a number of samples, ny the dimension of each measurement vector from the origin to the vehicle, v0 is the velocity of the and nx the state dimension. vehicle, m is the magnetic moment of the vehicle and θ1 and 1 θ2 are the positions of the two sensors. Vehicle V (xˆ)/(Nny − nx) N ny nx 1 360 70 6 15 2 142 54 6 15 camera has been used recording all objects within the sensor 3 10 32 6 15 range area. Data from six vehicles (Fig. 5) passing the sensors 4 14 44 6 15 5 51 27 6 15 are used for the sensor model validation in Section VI-C and 6 17 22 6 15 VI-D.

B. Sensor Noise Covariance in Fig. 5 is presented. By this we can conclude that larger Without any targets, each magnetometer measures a station- vehicles generally produce a worse fit with the measured data ary magnetic field together with sensor noise. This stationary than smaller vehicles do. This is intuitively clear since the magnetic field mainly consists of the earth magnetic field point target model assumes that the target has no geometrical but also of induced magnetic fields from other stationary extension, which is a more rough approximation for large ferromagnetic objects in the environment. We do not need vehicles than for small vehicles. However, for all vehicles, to assume that the magnetic field in the environment is the normalized cost function is much larger than 1. homogeneous, however we do assume it to be stationary. Finally, by simulating the model with the estimated pa- By using data from a time window without any moving rameter xˆ we get an estimate of the measured quantity targets present, the sensor noise covariance can be estimated yˆj,k = hj,k(xˆ), which can be compared with the measured using the approach described in Section V-A. In this manner, magnetic field yj,k. Note that the residual yj,k yˆj,k Rj is the two symmetric 3 3 -matrices R and R for the two exactly what is being minimized in the NLSk framework.− k In × 1 2 sensors in use are estimated to be Fig. 6a the estimated x-component of the magnetic field for x sensor 1, yˆj=1,k, is compared with its measured equivalence x 0.1303 0.0073 0.0114 yj=1,k. We can see that the main character of the signal has −15 − − R1 = 10 0.0073 0.1112 0.0117 , (51a) been caught. However, there is potential to get a better fit. −0.0114 0.0117 0.1558  − 0.1500 0.0205 0.0215  D. Extended Target Model Validation −15 R2 = 10 0.0205 0.1937 0.0310 , (51b) In order to improve the model fit, the extended target model 0.0215 0.0310 0.1483 (16) with d > 1 will considered. 1) Model Order Selection and Model Validation: For an where the unit of the measurement is Tesla. extended target model, an arbitrary model order can be cho- sen which here is the number of dipoles d. In Fig. 7, the C. Point Target Model Validation normalized cost function (50) is displayed as a function of the Since the road is straight, the constant velocity assumption model order d. for all vehicles applies. We start with modeling the vehicles For most vehicles, the normalized cost function rapidly as a single dipole. Since we have a scenario with multiple decreases with higher model order d. For the smaller vehicles sensors, the sensor model presented in II-B will be considered. it can be seen that the normalized cost function converges In Table I the normalized cost function for the vehicles towards 1, which corresponds to a correct model. 10

−5 x 10 Magnetic field x−direction, sensor 1, Vehicle 4, Point Target Model TABLE II: The normalized cost function for the extended 1.8 1.78 target model d = 3 and the estimated length between the ﬁrst

1.76 and the last dipole with the corresponding standard deviation. Measured 1.74 NLS 3 1.72 Nr V (xˆ)/(Nny − nx) N ny nx Length [m] 1.7 1 302.6 70 6 22 9.09 (0.03) Magnetic field [T] 1.68 2 76.7 54 6 22 11.25 (0.04)

1.66 3 1.6 32 6 22 3.95 (0.36) 4 1.5 44 6 22 3.56 (0.15) 1.64 0 0.5 1 1.5 2 2.5 3 3.5 5 1.5 27 6 22 4.98 (0.25) Time [s] 6 1.3 22 6 22 2.07 (0.24) (a) The point target model with one dipole

−5 x 10 Magnetic field x−direction, sensor 1, Vehicle 4, Grid of 3 dipoles 1.78 2) Length Estimation: Extended target models also produce 1.76 new states, where length of the row of dipoles has a direct 1.74 Measured NLS physical interpretation. The estimate of this length is also 1.72 presented in Table II and do correspond to the size of the 1.7 vehicles given in Fig. 5. Notice that this length does not Magnetic field [T]

1.68 exactly correspond to the actual length of the vehicle since it is based on an approximation of the target using a finite 1.66 0 0.5 1 1.5 2 2.5 3 3.5 Time [s] number of dipoles. However, a long dipole row should indicate (b) The extended target model with a row of three dipoles a long vehicle and vise versa, which then might be used for classifying vehicles of different sizes. Fig. 6: The measured magnetic field yx in the x-direction j=1,k 3) Initial state and Velocity Estimation: Furthermore, the together with the NLS-estimated value hx (xˆ) for vehicle j=1,k extended target model produces reasonable estimates of the 4 with a 90 % confidence interval. initial position ˆr0, Table III, and the velocity vˆ0, Table IV. All target trajectories are almost parallel to the x-axis in The normalized cost function for Vehicle 1−6 and different model orders accordance with the experimental setup. Furthermore, Vehicle 1, 3 and 4 are estimated to head in positive x-direction (right) Vehicle 1 (see Table IV), which is correct according to Fig. 5. ) ˆ x Vehicle 2 ( 2 d 10 ¯

V Vehicle 3 Vehicle 4 TABLE III: The estimated initial position ˆr0 in Cartesian Vehicle 5 Vehicle 6 coordinates (xyz) for the extended target model d = 3 with standard deviations in parenthesis. Also reference data for the three Cartesian coordinates is given. For example, for

1 10 Vehicle 1 the x-component of ˆr0 is 12.47 m with a standard deviation of 0.04 m and according− to the reference data this parameter should be negative. Value of the normalized cost function

Nr Estimated initial position ˆr0 [m] Ref. data 100 1 -12.47, 0.13, 4.36 (0.04, 0.02, 0.04) -, 0, +

1 2 3 4 2 13.68, -0.21, 0.32 (0.06, 0.03, 0.05) +, 0, + Number of dipoles (d) 3 -10.49, -1.47, 0.57 (0.27, 0.09, 0.09) -, 0, + ¯ d 4 -8.73, -2.00, 0.25 (0.14, 0.08, 0.07) -, 0, + Fig. 7: The normalized cost function V (xˆ) as a function of 5 11.43, -1.47, 0.59 (0.20, 0.10, 0.10) +, 0, + the model order d of the extended target model. 6 8.78, 0.93, 0.42 (0.22, 0.14, 0.09) +, 0, +

With the model order d = 3, the equivalent plot to Fig. 6a TABLE IV: The estimated velocity vˆ0 in Cartesian coordinates can be found in Fig. 6b. From this it can be concluded that an (xyz) for the extended target model d = 3 with standard almost perfect fit has been achieved. This model order will be deviations in parenthesis. Also reference data is given. used for a more thorough analysis and all results are presented in Tables II, III, IV and V for each vehicle. Nr Estimated velocity vˆ0 [m/s] Ref. data In Table II the normalized cost function for the vehicles 1 6.41, -0.27, -1.54 (0.02, 0.01, 0.02) +, 0, 0 2 -5.98, 0.20, 0.28 (0.03, 0.01, 0.02) -, 0, 0 is given. We have seen that this quantity should equal one 3 7.37, 0.33, 0.21 (0.16, 0.06, 0.06) +, 0, 0 for correct models. By comparing Table II with Table I, we 4 5.34, 0.28, 0.20 (0.08, 0.05, 0.04) +, 0, 0 see that the small vehicles 3–6 have taken huge steps in this 5 -9.74, 0.34, 0.02 (0.15, 0.09, 0.08) -, 0, 0 6 -11.56, 1.18, 0.03 (0.25, 0.19, 0.11) -, 0, 0 direction, whereas the large vehicles 1–2 still have a long way to go. This faces the real difficulties finding feasible models for large vehicles using models with a reasonable state space 4) Height Estimation: The z-component of the initial posi- dimension. tion ˆr0 has the interpretation of being the height of the car’s 11

metallic center over the road plane. According to Table III

300 300 this estimate is reasonable, being positive for all vehicles and mˆ (Ψ) Q Ψ · m signiﬁcantly larger for Vehicle 1. 200 200 ( ) ˆ 0

5) Dipole Estimation: Finally, by making use of the su- Dˆ Dˆ B0 B µ0 0 perposition principle (9), the total magnetic dipole moment 100 100 µ0

0 0 of the vehicles can be calculated by summarizing all dipoles. South−North [J/T] South−North [J/T] This estimate is given in Table V. However, for these values y y −100 −100 Ψ = 95◦ Ψ = 275◦ we do not have any reference data more than that large mˆ x x mˆ (Ψ) z z Q Ψ · mˆ should correspond to large vehicles. −200 ( ) 0 −200

−200 −100 0 100 200 300 −200 −100 0 100 200 300 West−East [J/T] West−East [J/T] TABLE V: The values of the estimated total magnetic moment (a) Vehicle 5 heading north. (b) Vehicle 5 heading south. mˆ = d mˆ i in Cartesian coordinates (xyz) for the extended i=1 target model d = 3 with standard deviations in parenthesis. 300 300 AlsoP reference data for the three Cartesian coordinates is 200 200 given. 100 Dˆ 100 Dˆ mˆ (Ψ) B0 B0 µ0 µ0 mˆ (Ψ) 2 Nr Estimated magnetic dipole moment mˆ [Am ] kmˆ k Rf. 0 0 South−North [J/T] South−North [J/T] Q(Ψ) · mˆ 0 Q Ψ · m 1 25, -819, -1528 (10, 5, 10) 1734 Large ( ) ˆ 0 y y 2 305, 62, -1872 (12, 5, 12) 1897 Large −100 −100 Ψ = 30◦ Ψ = 210◦ 3 -22, -5, -580 (8, 5, 13) 581 Small x x z z 4 -129, -71, -430 (8, 5, 12) 454 Small −200 −200

−200 −100 0 100 200 300 −200 −100 0 100 200 300 5 -142, 142, -438 (8, 7, 13) 482 Small West−East [J/T] West−East [J/T] 6 195, -62, -265 (10, 7, 19) 335 Small (c) Vehicle 5 heading north-east. (d) Vehicle 5 heading south-west. Fig. 8: The estimated magnetic dipole moments mˆ (Ψ) (the crosses) for different heading angles Ψ given with a 90% con- E. Validation of Direction Dependent Target Model fidence interval (the small ellipses) together with the simulated The proposal of decomposing the magnetic dipole moment mˆ (Ψ) using the estimated parameter. The magnetic dipole m into two components as described in (15) cannot be moment can successfully be divided into two components, validated from one scenario with constant heading since the one being parallel to the earth magnetic field, and one being two components cannot be resolved. As stated in [5], different oriented in the reference frame of the vehicle. observations of the same target at multiple headings are needed to allow the induced and permanent dipole moments to be The estimated parameter θˆ = mˆ 0 are given in Table VI resolved. In this validation, Vehicle 5, Fig. 5e, has been used Dˆ in controlled experiments heading at different speeds in four together with their standard deviations √P . ii different directions, 2-4 times in each direction. For each scenario the corresponding mˆ (Ψ) has been es- TABLE VI: The estimated hard iron dipole moment ˆr0 in ˆ timated with the row of dipoles extended target model with Cartesian coordinates (xyz) and estimated soft iron scalar D model order d = 3, which in Section VI-D has been proved with standard deviations in parenthesis. to be a good model. Now we define 2 3 Estimated hard iron dipole moment mˆ 0 [Am ] Dˆ [m ] mˆ (Ψ1) Q(Ψ1) B0/µ0 -203, 124, -267 (2.1, 1.7, 4.9) 1.054 (0.011) mˆ (Ψ2) Q(Ψ2) B0/µ0     . . . . m In order to validate (15), predictions of the magnetic dipole Y =   , ΦT =   , θ = 0 , mˆ (Ψi) Q(Ψi) B0/µ0  D moment m for new Ψ can be made by using mˆ and Dˆ     0  .   .   .   .  Dˆ     mˆ (Ψ) = Q(Ψ)mˆ + B . (52) mˆ (Ψ ) Q(Ψ ) B /µ  0 0  n   n 0 0 µ0     The predictions can be compared with the previously estimated Cov mˆ (Ψ1) 0 . mˆ (Ψi). In Fig. 8 the excellent fit for the x- and y-component R =  ..  , of mˆ (Ψi) and mˆ (Ψi) is displayed.   By this it can be verified that the magnetic dipole moment m  0 Cov mˆ (Ψn)    can be decomposed into two components: one being parallel to where iare the different scenarios performed at heading angle the earth magnetic field with the scalar multiplier D/µ0, and Ψ . The weighted least squares estimate of θ can be found by i one being oriented in the reference frame of the vehicle m0 minimizing the residual Y ΦTθˆ , i.e. by solving k − kR where the parameters m0 and D are vehicle specific. Note that − T − (ΦR 1Φ )θˆ = ΦR 1Y m0 and D cannot be found only from a single measurement since mˆ (Ψ) contains three components whereas m0 and D in with the corresponding covariance matrix total contains four. In other words, the target needs to make P = (ΦR−1ΦT)−1. maneuvers in order to excite its complete magnetic signature. 12

VII.CONCLUSIONS [13] L. Merlat and P. Naz, “Magnetic localization and identification of vehicles,” in Unattended Ground Sensor Technologies and Applications In this paper, we have addressed the problem of localizing V, vol. 5090. SPIE, 2003, pp. 174–185. metallic targets using magnetometers. We have presented a [14] A. S. Edelstein, “Magnetic tracking methods and systems,” U.S. Patent general sensor model based on the magnetic dipole model. 6 675 123, Jan. 6, 2004. [15] M. Birsan, “Non-linear Kalman filters for tracking a magnetic dipole,” in This model has been extended by introducing a model con- Proc. Int. Conf. on Marine Electromagnetics, London, UK, Mar. 2004. sisting of a series of dipoles, which also contains the length of [16] ——, “Unscented particle filter for tracking a magnetic dipole target,” the vehicle in an extended state vector. Also a new model with in Proc. MTS/IEE OCEANS, Washington, DC, Sep. 2005. [17] S. Y. Cheung and P. Varaiya, “Traffic surveillance by wireless sensor heading angle dependent dipole moments has been proposed, networks: Final report,” Institute of Transportation Studies, University which has been accomplished by decomposing the influence of California, Berkeley, Tech. Rep., Jan. 2007. of soft and hard iron. The sensor model allows the target to [18] S. Y. Cheung, S. Coleri, B. Dundar, S. Ganesh, C. W. Tan, and P. Varaiya, “Traffic measurement and vehicle classification with single magnetic be observed in an arbitrary magnetometer network. sensor,” J. Transportation Research Board, vol. 1917, pp. 173–181, We have also provided a detailed observability analysis of 2005. the model and compared it to other models based on the so [19] Z. A. Daya, D. L. Hutt, and T. C. Richards, “Maritime electromagnetism and DRDC signature management research,” Defence R&D Canada, called Anderson functions in the closest related papers. We Tech. Rep., Dec. 2005. have also provided measures for analyzing the optimal sensor [20] M. Birsan, “Electromagnetic source localization in shallow waters using deployment in a magnetometer sensor network using criteria Bayesian matched-field inversion,” Inverse Problems, vol. 22, no. 1, pp. 43–53, 2006. based on the Cramer´ Rao lower bound [21] J. Callmer, M. Skoglund, and F. Gustafsson, “Silent localization of un- Further, our model has been validated with a selection of derwater sensors using magnetometers,” EURASIP Journal on Advances dedicated field experiments. It has been found that the natural in Signal Processing, 2010. [22] J. C. Maxwell, “A dynamical theory of the electromagnetic field,” single dipole model cannot be validated for vehicles close Philosophical Trans. of the Royal Society of London, vol. 155, pp. 459– to the sensor when a commercial-grade sensor is used in its 513, 1865. field of coverage. It has been found that this extended target [23] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Upper Saddle River, NJ: Prentice-Hall, 1993. model works excellently on small vehicles. However, for large [24] D. Tornqvist,¨ “Statistical fault detection with applications to IMU vehicles close to the sensor challenging modeling work still disturbances,” Lic. thesis, Dept. Elect. Eng., Linkoping¨ Univ., Linkoping,¨ remains. Finally, target orientation dependent dipole moments Sweden, 2006. [25] MTi and MTx User Manual and Technical Documentation, Xsens are identifiable only if the vehicle is observed in different Technologies B.V., Enschede, The Netherlands, 2005. orientations. Therefore, this model has been validated using [26] F. Gustafsson, Statistical Sensor Fusion, 1st ed. Lund, Sweden: the same vehicle in controlled experiments heading in different Studentlitteratur, 2010. directions with good results. In addition, these experiments also verify that the dipole vectors are reproducible for a certain vehicle. Niklas Wahlstrm (SM’11) received the M.Sc. degree in applied physics and electrical engineering in REFERENCES 2010 and the Tech. Lic. degree in automatic control in 2013, both from Linkping University, Sweden. [1] J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1999. Since 2010 he is working towards the Ph.D. [2] J. Lenz, “A review of magnetic sensors,” Proc. IEEE, vol. 78, pp. 973– degree at the Division of Automatic Control, De- 989, Jun. 1990. partment of Electrical Engineering, Linkping Uni- [3] J. Lenz and S. Edelstein, “Magnetic sensors and their applications,” versity. His research interests include sensor fusion, IEEE Sensors J., vol. 6, no. 3, pp. 631–649, Jun. 2006. localization and mapping, especially using magnetic [4] W. M. Wynn, “Detection, localization, and characterization of static sensors. magnetic dipole sources,” in Detection and Identification of Visually Obscured Targets. Philadelphia, PA: Taylor and Francis, 1999, pp. 337–374. [5] J. W. Casalegno, “All-weather vehicle classification using magnetometer Fredrik Gustafsson is professor in Sensor In- arrays,” in Proc. SPIE Unattended Ground, Sea, and Air Sensor Tech- formatics at Department of Electrical Engineering, nologies and Applications IV, vol. 4743, 2002, pp. 205–212. Linkoping¨ University, since 2005. He received the [6] R. J. Kozick and B. M. Sadler, “Joint processing of vector-magnetic and M.Sc. degree in electrical engineering 1988 and the acoustic-sensor data,” in Proc. SPIE Unattended Ground, Sea, and Air Ph.D. degree in Automatic Control, 1992, both from Sensor Technologies and Applications IX, vol. 6562, 2007. Linkoping University. During 1992-1999 he held [7] ——, “Classification via information-theoretic fusion of vector-magnetic various positions in automatic control, and 1999- and acoustic sensor data,” in Proc. Int. Conf. on Acoustics, Speech and 2005 he had a professorship in Communication Sys- Signal Processing, vol. 2, Apr. 2007, pp. II–953 –II–956. tems. His research interests are in stochastic signal [8] ——, “Algorithms for tracking with an array of magnetic sensors,” in processing, adaptive filtering and change detection, Proc. Sensor Array and Multichannel Signal Processing Workshop, Jul. with applications to communication, vehicular, air- 2008, pp. 423 –427. borne, and audio systems. He is a co-founder of the companies NIRA [9] C. T. Christou and G. M. Jacyna, “Vehicle detection and localization Dynamics (automotive safety systems), Softube (audio effects) and SenionLab using unattended ground magnetometer sensors,” in Proc. 13th Int. Conf. (indoor positioning systems). on Information Fusion, Edinburgh, UK, Jul. 2010. He was an associate editor for IEEE Transactions of Signal Process- [10] N. Wahlstrom,¨ J. Callmer, and F. Gustafsson, “Magnetometers for ing 2000-2006 and is currently associate editor for IEEE Transactions on tracking metallic targets,” in Proc. 13th Int. Conf. on Information Fusion, Aerospace and Electronic Systems and EURASIP Journal on Applied Signal Edinburgh, UK, Jul. 2010. Processing. He was awarded the Arnberg prize by the Royal Swedish [11] ——, “Single target tracking using vector magnetometers,” in Proc. Academy of Science (KVA) 2004, elected member of the Royal Academy of Int. Conf. on Acoustics, Speech and Signal Processing, Prague, Czech Engineering Sciences (IVA) 2007, elevated to IEEE Fellow 2011 and awarded Republic, May 2011, pp. 4332–4335. the Harry Rowe Mimno Award 2011 for the tutorial “Particle Filter Theory [12] M. Rakijas, “Magnetic object tracking based on direct observation of and Practice with Positioning Applications”, which was published in the AESS magnetic sensor measurements,” U.S. Patent 6 269 324, Jul. 31, 2001. Magazine in July 2010.