Vehicle Tracking Based on Fusion of Magnetometer and Accelerometer Sensor Measurements with Particle Filtering

Roland Hostettler, Petar M. Djuric´

This is a post-print of a paper published in IEEE Transactions on Vehicular Technology. When citing this work, you must always cite the original article:

R. Hostettler and P. M. Djuric,´ “Vehicle tracking based on fusion of magnetometer and accelerometer sensor measurements with particle filtering,” Vehicular Technology, IEEE Trans- actions on, vol. 64, no. 11, pp. 4917–4928, November 2015

DOI: 10.1109/TVT.2014.2382644

Copyright: c 2015 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. 1 Vehicle Tracking Based on Fusion of Magnetometer and Accelerometer Sensor Measurements with Particle Filtering Roland Hostettler, Member, IEEE, Petar M. Djuric,´ Fellow, IEEE

Abstract—In this article, we propose a method for vehicle Target tracking is often formulated as a state estimation tracking on roadways using measurements of magnetometers problem where the position of the target as a function of time is and accelerometers. The measurements are used to build a considered a random process [17]. The measurements obtained low-cost, low-complexity vehicle tracking sensor platform for highway traffic monitoring. First, the problem is formulated by from the sensors are described as a function of the states, for introducing the process model for the motion of the vehicle example, of the range and bearing of a target [18]. In many on the road and two measurement models, one for each of cases, these measurements are highly nonlinear functions of the sensors. Second, it is shown how the measurements of the states which, for their processing, often require the use the sensors can be fused using particle filtering. The standard of approximating methods, such as the extended Kalman or sampling importance resampling (SIR) particle filter is extended for processing of multi-rate sensor measurements and models particle filters. It has been shown that the latter is a suitable that employ unknown static parameters. The latter are treated approach in many different applications; see [2] for a thorough by Rao-Blackwellization. The performance of the method is overview. demonstrated by computer simulations. It is found that it is In this work, we address vehicle tracking by combining feasible to fuse the two sensors for vehicle tracking and that the magnetometer and accelerometer measurements in a single proposed multi-rate particle filter performs better than particle filters that process only measurements of one of the sensors. sensor unit that is mounted on the road surface as illustrated The main contribution of this paper is the novel approach of in [16]. Even though tracking using individual sensors has fusing the measurements of road-mounted magnetometers and been addressed before [14]-[16], the combination of the two accelerometers for vehicle tracking and traffic monitoring. sensors has not been considered yet. An obvious major ad- Index Terms—Particle filters, sensor fusion, vehicle tracking vantage is that the two different sensors come at low costs and that they complement each other in the sense that they I.INTRODUCTION measure completely different phenomena. Furthermore, the sensors can be integrated in small, battery-powered sensor Target tracking is of importance in many different applica- nodes and require less computational power than, for example, tions ranging from air traffic control [2] to tracking of mobile video-based systems. phone users within a cellular network [3]. Another important Magnetometers are sensors measuring the strength of the application is in intelligent transportation systems that allows Earth’s magnetic field at a given point. They are often used in for tracking of vehicles on roadways. This is of interest for compass applications, for example, in mobile phones. Metallic obtaining insightful information about the traffic which can be objects like vehicles cause local distortions, and they can used, for example for understanding traffic patterns such as be measured by a magnetometer and exploited for target congestions [4], [5] or for predicting/preventing accidents [6] tracking [19]. Commonly, magnetometers are vector sensors (we note that the prevention of accidents requires very high measuring the Cartesian components of the vector field. Ac- accuracy of tracking). The gathered information (aggregate or celerometers are also used in a number of applications. An individual) can be broadcast to individual vehicles equipped accelerometer attached to the road surface measures vibrations with corresponding receiver equipment [4]. in the road caused by dynamic loads of vehicles passing in Popular approaches for vehicle tracking itself are often its proximity. The features of the vibrations can be used to based on vision systems [7]-[10], radar [11], or a combina- estimate vehicle parameters. tion of such techniques [12]. Recently, solutions that employ When these sensors are used for tracking purposes, there are low-cost, low-complexity sensors such as microphones [10], several important issues that need to be addressed carefully. [13], magnetometers [14], [15], or accelerometers [16] have First, the sensor measurements are highly nonlinear functions emerged. of the states. Second, the measurements depend on a set Copyright (c) 2014 IEEE. Personal use of this material is permitted. of unknown, target- or material-specific parameters which, if However, permission to use this material for any other purposes must be included in the estimation problem, make the problem even obtained from the IEEE by sending a request to [email protected]. R. Hostettler is with the Department of Computer Science, Electrical and more challenging. Third, due to the different measurement Space Engineering, Lulea˚ University of Technology, Lulea,˚ Sweden (e-mail: principles, the sensors have different properties including [email protected]). different sampling rates and operating clocks that usually are P. M. Djuric´ is with the Department of Electrical and Computer Engi- neering, Stony Brook University, Stony Brook, NY 11794 USA (e-mail: not synchronized. [email protected]). The solutions to this type of problems require fusion of 2 information from the sensors and multi-rate processing. A Thus, the motion is represented by a simple one-dimensional methodology that is suitable for these problems is particle constant velocity motion model given by filtering. Some solutions for generically similar settings have   " T 2 # 1 Ts s been proposed in the literature. For example, a hybrid multi- xt = xt 1 + 2 ut, (1) rate Kalman-particle filter was proposed in [20]. There, it was 0 1 − Ts assumed that the measurements for the faster sensor depend | {z } | {z } ,A linearly on the state and hence, a Kalman filter was used ,B to update the particles when only the linear measurement where t = 1, 2,...,T is a discrete time index, Ts is the were available. The method was also applied to a tracking sampling time,  x problem where velocity and range measurements were avail- rt xt = x , (2) able. In [21], particle filtering with multi-rate sensors in a r˙t process industry context was considered. Similarly, [22] in- x x rt is the position on the road, and r˙t is the vehicle’s velocity. troduced multi-rate particle filtering in conjunction with robot The symbol ut can be interpreted as a random acceleration localization based on a set of different sensors. An unscented term that accounts for small changes in the vehicles velocity information filter for estimating the position of a vehicle fusing due to the drivers input, drag, and so on. It is a random variable the measurements from a camera, laser range finder, and GPS with a distribution given by receiver was proposed in [23]. Finally, there is a large body of 2 work on vehicle tracking using particle filtering and different p(ut) = (ut; 0, σu ), (3) N t types of sensors. For example, Yin et. al. propose to combine a where (u ; 0, σ2 ) signifies normal distribution of a scalar particle filter with the CamShift algorithm for vehicle tracking N t ut with mean zero and variance σ2 . using video in order to achieve scale-invariability and account ut The initial position of the vehicle is the point where the for disturbances such as occlusion and background clutter [24]. vehicle is in the proximity of the sensor and where the Another video-based method that is especially robust to partial measurements start being taken. It is thus given through the occlusion of the target was proposed in [25]. Finally, [26] also sensor range plus some uncertainty. Hence, it is modeled uses particle filtering for video-based vehicle tracking where according to particles are clustered by analyzing the motion coherence in p(x ) = (x ; µ ,C ), (4) order to form convex shapes of the tracked objects. 0 N 0 x0 x0

The contributions of this paper are as follows. First, we where µx0 is the mean sensor range, Cx0 the spread, and propose a multi-rate particle filtering method for vehicle 1 tracking fusing the measurements of two passive sensors, (x0; µx0 ,Cx0 ) = N/2 1/2 N (2π) Cx0 (5) a magnetometer and an accelerometer, where the measure- 1 | | > −1 (x0 µ ) C (x0 µ ) ments are in general asynchronous. This is a novel approach e− 2 − x0 x0 − x0 × and has not been considered before. Second, the unknown is the normal distribution of an N 1 random variable x0 with parameters in the sensor models are handled through Rao- × mean µx0 and covariance Cx0 . Blackwellization, simplifying the tracking problem and im- Finally, the position of the target is defined by the vector proving the performance. With our simulation results, we show  x that the proposed method can be used for implementing a low- rt y cost, high accuracy vehicle tracking sensor platform. rt = rt  (6) z The remainder of this paper is organized as follows. The rt problem is formulated in Section II and the particle filtering pointing from the sensor at the origin of the Cartesian coor- algorithm is introduced in Section III. How the particle filter z dinate system to the target. The position rt in the z-direction can be applied to the problem is shown in Section IV. Finally, z is assumed to be rt = 0, that is, it is assumed that the sensor simulation results are provided in Section V and concluding is in the same plane as the vehicle. Furthermore, the lateral remarks are given in Section VI. y position rt is assumed to be in the middle of the driving lane adjacent to the sensor and hence known. II.PROBLEM FORMULATION We formulate the problem by using a model describing the B. Magnetometer vehicle motion along the road and two measurement models, The magnetic mass of the vehicle causes small local mag- for the magnetometer and accelerometer, respectively. They netic distortions in the earth’s magnetic field in the vicinity of are described in turn below. the vehicle which can be measured using a magnetometer [15], [19]. It has been shown and verified [15] that the measured A. Motion Model magnetic field can be modeled by the static background field The vehicle is assumed to follow the course of the road, B0 plus a series of dipoles with locations rp and magnetic which is a very constrained environment. Furthermore, since moments mp as the time window where the vehicle is in the vicinity of the P 2 X 3(rp>mp)rp rp mp sensor is rather short (of magnitude 1 s ... 2 s), it can be safely − | | B = B0 + 5 . (7) assumed that the vehicle travels at a nearly constant speed. rp p=1 | | 3

The model order P depends on the magnetic extension of varying due to seasonal influences). wa,t is measurement noise the vehicle as well as the distance between the vehicle and with the sensor. It has been shown [15] that a low model order p(w ) = (w ; 0, σ2), (13) a,t N a,t a of P = 1 ... 3 is sufficient for small- to mid-sized vehicles passing the sensor at a reasonable distance. For simplicity, we The location of the two axles is described by the two vectors assume a single dipole (P = 1) here, however, the extension pointing at the center of each axle given by to higher orders is straightforward. Furthermore, the static l/2 l/2 background field B0 can be subtracted which finally yields r1,t = rt +  0  and r2,t = rt  0  , (14) the measurement model 0 − 0 2 3(rt>m)rt rt m and l is the wheelbase which is approximately 2.5 m for ym,t = −5 | | +wm,t, (8) rt passenger cars and is hence assumed known [32]. | | {z| } The bandwidth of the road surface waves depends mainly on ,hm(xt) the material parameters of the road which in turn can depend where y is a 3 1 vector containing the vector components on location and climate. It has been indicated that a bandwidth m,t × of the magnetic field, wm,t is the measurement noise with a of 1 kHz should be sufficient for year-round operation [28] distribution given by and hence, the sampling rate for the accelerometer is chosen to be 2 kHz. This is again supported by the sensor under p(w ) = (w ; 0, σ2 I ), (9) m,t N m,t m 3 consideration [33]. m is the so called magnetic dipole moment, a generally unknown target specific parameter, and I is the 3 3 identity III.PARTICLE FILTERING 3 × matrix. Particle filters are well-suited for state estimation of non- The bandwidth of the signal measured by the magnetometer linear, dynamic systems such as the one introduced in Sec- depends on a number of different factors such as the speed of tion II [34], [35]. Here, we first describe the particle filter as the passing vehicles and magnetic moment and is typically a state estimator as known from the literature [34], [35] and lower than 100 Hz. Hence, here, the magnetometer is sampled then we show a way of extending the filter for using it in at a sampling rate of 200 Hz. In practice, this sampling rate multi-rate sensor scenarios. is supported by the sensors being evaluated for a prototype system [27]. A. Sampling Importance Resampling Particle Filter Assume that the dynamic system of interest is described by C. Accelerometer xt = f(xt 1, ut) (15a) Similarly, the interaction between the vehicle and the road − y = h(x , w ), (15b) causes road surface waves that can be measured by an ac- t t t celerometer. It has been shown that the wave propagation where t = 1,...,T , x is a d 1 state vector, y a d 1 t x × t y × can be described as waves spreading circularly from the measurement vector, ut and wt are process and measurement source [28] with transfer function noises, respectively, and the initial state is x p(x ), with 0 ∼ 0 p(x ) being the prior distribution of x . It is well known G(ω) = A(ω) (1)( k(ω) r ) 0 0 0 (10) that a closed form solution for the filtering density p(x y ) H − | | t| 1:t where A(ω) is a material parameter and k(ω) the complex only exists for special cases of (15), for example when the wavenumber. (1)(x) is the Hankel function of the first kind probability density functions are Gaussian and the system is H0 of order zero [29]. linear. Furthermore, a common approach in tracking is to model For other cases, one has to resort to approximating method- the source as a random process [30], which is also used here. ologies. One of them is particle filtering [34], [35], which is Hence, the input is described by the random variable also the approach that we adopt in this work. With particle filtering, we approximate the distributions of interests with v (0, 1). (11) particles, x(m), and their associated weights, w(m), where m t ∼ N t t is the index of a particle and its weight. Each particle can be Together with a narrow-band approximation and describing interpreted as a candidate value of the state of the system while each axle of a vehicle as an individual source, the measurement its corresponding weight can be interpreted as the probability model for a two-axled vehicle becomes (see [31] for a detailed of that value. Finally, note that the weights themselves sum derivation) up to one.  (1) (1)  A particle filtering method has three steps, (a) particle y = κi (iη r ) + (iη r ) v + w , (12) a,t H0 | 1,t| H0 | 2,t| t a,t generation (propagation), (b) weight computation, and (c) | {z } resampling. The last step is necessary to avoid degeneracy ,ha(xt) of the discrete representation of the target distribution. With where i = √ 1, κ = A and η = Im(k) are material reampling, the particles with low weights are removed and − | | parameters, which in general are unknown (they may be slowly the ones with large weights are replicated. The removal and 4 replication is conducted in random fashion according to the B. Multi-rate SIR Particle Filter weights of the particles. Now, instead of having a single sensor assume that the There are many different particle filtering methods available system is measured by L different sensors at L different in the literature but the original one goes under the name sampling rates fs,l for l = 1,...,L. Hence, we can not sequential importance resampling (SIR) [36], and it is the use Algorithm 1 directly and have to extend it in order to method we use in this paper. For completeness, the SIR accommodate the different sampling rates. particle filtering algorithm as it was introduced in [36] is First, assume that the ratios summarized in Algorithm 1. fs,max Rl = (18) Algorithm 1 (SIR Particle Filter). fs,l (m) (m) 1) Set t 0 and initialize x p(x ), w = 1/M are integers where f is the highest sampling frequency. ← 0 ∼ 0 0 s,max for m = 1,...,M. (This assumption is made for ease of presentation. Particle 2) Set t t + 1; continue if t T , otherwise terminate. filters can handle measurements from sensors acquired with ← ≤ 3) Propagate the particles, any relationship between the sampling frequencies [37].) The measured signals can then be represented by (m)  (m) xt p xt xt 1 . ∼ | − y1,R1t = h1(xR1t, wR1t),

4) Calculate the non-normalized weights y2,R2t = h2(xR2t, wR2t), . (19)   . w˜(m) = w(m)p y x(m) . t t t| t yL,RLt = hL(xRLt, wRLt),

where Rlt denotes that the sample of the lth sensor is 5) Normalize the weights measured only at time instants Rlt (relative to the fastest (m) sensor which has samples at t = 1, 2,...,T ). Furthermore, (m) w˜t w = . a reasonable assumption is that the measurement noises wRlt t PM (m) m=1 w˜t and wRkt are uncorrelated, that is,

1  2 E wRltwRkt = 0 for l = k. (20) PM  (m) − { } 6 6) If w < MT resample with re- m=1 t Without loss of generality, assume that we are given the (m) (m) placement such that P (xt = xt ) = wt and set measurements of two sensors (L = 2) with R1 = 1 and R2 = (m) 2 t wt = 1/M. . Hence, at odd, both sensors provide new measurements 7) Roughen the particles according to (16)-(17). whereas at t even, only sensor 1 provides new data. Then, at 8) Return to step 2. any even time instant t1, the posterior can be expressed as

p(xt1 y1,1:t1 , y2,1:2:t1 1) In this algorithm M is the size of the particle set and MT | − (21) p(y1,t1 xt1 ) p(xt1 y1,1:t1 1, y2,1:2:t1 1), is a positive number less than M. In step 6 we decide if ∝ | | − − resampling should take place or not. If the “effective sample where stands for “proportional to,” size” of the particle set (expression on the left of the inequality ∝  > in step 6) is less than the preselected value MT , resampling yl,1:Rl:t = yl,1 yl,Rl+1 . . . yl,t , (22) is performed. Finally, step 7 is a standard trick to mitigate and sample impoverishment [36]. It improves sample diversity by p(y1,t xt ) (23) adding an additional jitter to the particles which reduces the 1 | 1 effects of the dependency introduced by the resampling step. is the likelihood. The roughening is performed by adding an independent jitter On the other hand, for t1 + 1 we have new measurements (m) c to each particle xt where from both sensors; hence

p(xt +1 y1,1:t +1, y2,1:2:t +1) c (0, Σ). (16) 1 | 1 1 ∼ N p(y1,t1+1, y2,t1+1 xt1+1) p(xt1+1 y1,1:t1 , y2,1:2:t1 1). ∝ | | − Σ is a diagonal covariance matrix with elements (24) Furthermore, given the independence (20) of y1,t1+1 and 1/dx 2 y2,t +1, the likelihood can be factorized as [Σ]i = (KEM − ) (17) 1

p(y1,t1+1, y2,t1+1 xt1+1) =p(y1,t1+1 xt1+1) where K is a tuning parameter and E is the difference | | (25) p(y x ), between the largest and smallest values of all particles in × 2,t1+1| t1+1 direction i. By choosing the covariance as given in (17), the which now directly leads to the multi-rate particle filter as standard deviation is normalized such that the roughening follows. is proportional (with proportionality K) to the distance E First, note that the particle weights are calculated using (see [36] for details). p(y x ) in step 4 of Algorithm 1. Hence, in the above case, t| t 5 we would simply use the likelihood (23) for calculating the IV. PROPOSED METHOD particle weights when t is even and (25) for t odd. Given the problem formulation in Section II and the particle The generalization to an arbitrary number of sensors is filtering algorithms in Section III, it is now shown how the straightforward. At any given time t, sensor l delivers a latter is applied for vehicle tracking. In the following, there are new measurement if the remainder of the integer division of two subsections. In the first, we show how to apply the particle t 1 and R is 0 (note that all the sensors start acquiring − l filter to the problem as formulated directly, assuming known measurements at t = 1). That is, if parameters. In the second, we present how the problem can t 1 be dealt with when the parameters of the model are unknown. t 1 Rl − = 0, (26) − − Rl A. Known Parameters the lth sensor will produce a measurement at t. Then, we can (m) (m) Assuming that all the parameters in the measurement mod- define the non-normalized weight w˜t,l for the particle xt from the measurement of sensor l as els (8) and (12) are known, it is easy to apply the particle filter. The system is completely defined by (1), (8), and (12) (  (m) j t 1 k p yl,t x , if t = Rl − + 1 (m) t Rl and hence, the predictive density and likelihoods are given by w˜t,l = | (27) 1, otherwise 2
p(xt+1 xt) = xt+1; Axt, σut BB> , (31a) | N 2
(m) p(ym,t xt) = ym,t; hm(xt), σ I3 , (31b) and the overall non-normalized weight w˜t becomes m | N 2 2
L p(ya,t xt) = ya,t; 0, ha(xt) + σa , (31c) (m) (m) Y (m) | N w˜t = wt 1 w˜t,l . (28) and thus, applying the particle filter is straightforward. − l=1 Modifying step 4 in Algorithm 1 with (27)-(28) finally leads B. Unknown Parameters to the multi-rate particle filter as summarized in Algorithm 2. A more realistic scenario dictates that neither the parameters Algorithm 2 (Multi-rate SIR Particle Filter). of the measurement function nor the noise characteristics are (m) (m) known. Taking this into account, a straightforward solution 1) Set t 0 and initialize x0 p(x0), w0 = 1/M for m←= 1,...,M. ∼ would be to also include these parameters in the estimation 2) Set t t + 1; continue if t T , otherwise terminate. problem. However, these parameters are often not of interest ← ≤ 3) Propagate the particles, and what is more, they would unnecessarily also increase the dimension of the problem. Instead, the nuisance parameters (m)  (m) xt p xt xt 1 . can be modeled as unknowns with a prior distribution p(θ). ∼ | − This has the advantage that one is then able to use Rao- 4) Calculate the non-normalized weights Blackwellization (also called marginalization) in order to a) For each l = 1,...,L, calculate the individual integrate out the unknown parameters analytically [38]. Rao- (m) particle weights w˜t,l according to (27). Blackwellization can be thought of as weighted averaging over b) Calculate the total non-normalized particle weight all the possible values of the parameters, where the probability (m) w˜t according to (28). density function of the parameters is the weighting function. 5) Normalize the weights The advantages of this approach are that the dimension of spaces where particles have to be generated is reduced and that w˜(m) w(m) = t . the method does not have to deal with constant parameters, t PM (m) m=1 w˜t which in particle filtering require special treatment. All this entails more accurate and robust tracking. The implementation  2 1 PM  (m) − of the Rao-Blackwellization requires use of hyperparameters 6) If m=1 wt < MT resample with re- for the priors of the unknown parameters. It turns out that that (m) (m) placement such that P (xt = xt ) = wt and set the methods are quite robust against the assumed values of (m) wt = 1/M. these parameters. The likelihoods quickly overtake the priors 7) Roughen the particles according to (16)-(17). defined by these hyperparameters and the posterior accurately 8) Return to step 2. points to parts of its support where the true values of the unknown parameters might be. Next, we explain how we When the measurements of the different sensors are fully implement Rao-Blackwellization on both, the magnetometer asynchronous, they are processed as soon as they arrive. For and accelerometer measurements. example, let the latest time instant with a measurement be t1 1) Magnetometer: The 3 1 magnetic moment m in (8) × and the next measurement be obtained at t2 > t1. Then one is generally unknown. By assigning a prior distribution to m, generates the particles for t2 by it can be marginalized out so that the measurement becomes x(m) p(x x(m)) (29) independent of m. First, note that hm(xt) can be rewritten as t2 ∼ t2 | t1 3(r m)r r 2m 3r r r 2I and their non-normalized weights are computed by t> t t t t> t 3 hm(xt) = −5 | | = −5 | | m. (32) rt rt (m) (m) (m) | | | | {z| } w˜t2 = wt1 p(yt2 xt2 ). (30) | ,Hm 6

Second, it is assumed that the prior of m conditioned on where kκ is a positive hyperparameter, we marginalize with 2 σm is given by respect to κ and obtain Z 2 2 2 ∞ 2 p(m σm) = (m; µm, σmKm), (33) p(y x , η, v , σ ) = p(y x , κ, η, v , σ ) | N a,t| t t a a,t| t t a −∞ which is the conjugate prior for (31b) and Km is a 3 3 2 × p(κ σa, vt)dκ matrix. Then, the marginalized distribution becomes Z × | ∞ 2
Z = ya,t; κHavt, σa (42) 2 ∞ 2 2 N p(y x , σ ) = p(y x , σ , m)p(m σ ) dm −∞ m,t t m m,t t m m  2  | | | kκσa Z −∞ κ; µκ; dκ ∞ × N v2 = y ; H m, σ2
t m,t m m (34) 2 2
N = ya,t; µκHavt, σ (1 + kκHa ) . −∞ N a m; µ , σ2 K
dm × N m m m Next, the unknown measurement noise variance can be 2
= y ; H µ , σ (I + H K H>) . N m,t m m m 3 m m m marginalized by assuming again an inverse Gamma prior, The last equality in (34) is due to a well-known property of 2 2 p(σa) = IG(σa; αa, βa). (43) Gaussian random variables [39]. 2 Marginalization then yields a t-distribution, similar to (37), The second unknown is the noise variance σm. The inverse Gamma distribution defined as that is, α Z ∞ β α 1 β 2 2 2 IG(x; α, β) = x− − e− x (35) p(ya,t xt, η, vt) = p(ya,t xt, η, vt, σa)p(σa)dσa Γ(α) | | −∞ (44)  β (1 + k H2) is known to be the conjugate prior for the unknown variance = t y ; 2α , µ H v , a κ a . a,t a κ a t α of a Gaussian random variable [38]. We assign it as a prior to a 2 σm, that is, Furthermore, it is given that vt (0, 1). Unfortunately, 2 2 ∼ N p(σm) = IG(σm; αm, βm). (36) marginalization with respect to vt cannot be used directly in (44). However, p(ya,t xt, η, vt) can be approximated by a 2 | Upon marginalizing σm (see Appendix), we obtain Gaussian distribution as

R ∞ 2 2 2 2
p(ym,t xt) = p(yt xt, σm)p(σm)dσm p(ya,t xt, η, vt) ya,t; µκHavt, β(1 + kκHa ) , (45) | −∞ | | ≈ N (37)  β (I +H K H> )  and then, marginalization yields = t y ; 2α ,H µ , m 3 m m m , m,t m m m αm Z ∞ p(y x , η) p(y x , η, v )p(v )dv where t(x; ν, µ, C) is the multivariate t-distribution with ν a,t| t ≈ a,t| t t t t degrees of freedom given by Z −∞ ∞ a 2
= yt ; µκHavt, β(1 + kκHa ) (46) ν+n
N Γ 2 −∞ t(x; ν, µ, C) = n/2 n/2 1/2 (v ; 0, 1)dv π ν Γ(ν/2) C × N t t | | ν+n (38) 2 2 2
 1  2 = ya,t; 0, β(1 + kκHa ) + µκHa . (x µ)>C− (x µ) − N 1 + − − , × ν The last unknown parameter η appears as an argument to the Hankel function (1)(x) in (12). It is thus impossible to and where x is of dimension n 1, µ is the mean of x, and H0 × analytically marginalize with respect to it and hence, it has C is an n n positive definite scale matrix. × to be taken care of in a different way. One straightforward 2) Accelerometer: Similar to the magnetometer, the ac- approach is to augment the state vector with the unknown celerometer measurement model depends on a set of unknown parameter as parameters, see (12). Starting from x  x˜ = t (47) t η p(y x , κ, η, v , σ2) = (y ; h (x )v , σ2), (39) a,t| t t a N a,t a t t a which leads to the augmented process which follows directly from (12), we note that    T 2  1 Ts 0 s  (1) (1)  2 ha(xt)vt = κ i 0 (iη r1,t ) + 0 (iη r2,t ) vt (40) x˜t = 0 1 0 x˜t 1 + Ts  ut (48) H | | H | | − | {z } 0 0 1 0 ,Ha | {z } | {z } ˜ ,A B˜ is linear in κ. After assigning a conjugate prior conditioned , 2 with prior on σa and vt, i.e., 2 p(η) = (η; µη, σ ) (49)  2  N η 2 kκσa p(κ σa, vt) = κ; µκ; 2 , (41) which is used to propose samples at t = 1. | N vt 7

3) Summary: Finally, the system where the unknown 2 parameters have been addressed is defined by (37), (46), 1 and (48). This leads to the predictive density and likelihoods µT / defined by 0 x ym,t y   m,t y ˜ 2 ˜ ˜ y 1 m,t p(˜xt+1 x˜t) = x˜t+1; Ax˜t, σut BB> , (50a) z | N − ym,t  β (I + H K H ) 2 p(y x˜ ) = t y ; 2α ,H µ , m 3 m m m> , − 0 200 400 600 800 1,000 m,t| t m,t m m m α m t (50b) 2 2 2
(a) p(ya,t x˜t) ya,t; 0, βa(1 + kκHa ) + µκHa , (50c) | ≈ N 2 10− where αm, βm, Km, βa, kκ, and µκ are hyperparameters. × 2 2 V. NUMERICAL ILLUSTRATION m/s

In this section we demonstrate the performance of the / 0 proposed tracking method by computer simulations using the a,t y 2 models introduced in Section II. We considered five different − scenarios. First, the multi-rate particle filter was run with 0 200 400 600 800 1,000 known parameters in order to show that the approach is t feasible. Second, the marginalized particle filter was simu- lated and its performance analyzed. Third, two consecutively (b) passing vehicles, one fast and one slower, were simulated. Fig. 1. Simulated measurement signals. (a) Magnetometer signal and (b) Fourth, we analyze the sensitivity of the proposed method to accelerometer signal. hyperparameters. Fifth, we compared the method of fusing the sensors using particle filtering to particle filters for the A. Known Parameters individual sensors only, a particle filter fusing both sensors First, we show how the multi-rate particle filter performs but not using multi-rate processing, and the standard unscented when all the model parameters are known, that is, by using the Kalman filter (UKF) [40]. In each setup, a total of 100 Monte model summarized in (31). Samples of simulated measurement Carlo (MC) simulations were executed to evaluate the average signals of the magnetometer and accelerometer are displayed performance. in Fig. 1. We obtained all the results with the assumption that The hyperparameters for the magnetometer were set as the vehicle started at rx = 5 m with r˙x = 20 m/s. 0 − 0 1 In the case of known parameters, we expected good per- µm = 1 , and Km = I3, formance because there were no uncertainties about them. 1 Figure 2 shows the mean error of the estimated states from the 100 simulations (solid line). We see that both states con- and the parameters of the magnetometer measurement noise verge quickly from the initial offset (due to the initialization) were toward zero. Further, the 2σ-bounds (marked with dotted 2 5 2 σm,0 = 1 10− , αm = 6, and βm = σm,0(αm 1). lines) indicate that even the uncertainty about the estimated × − states converges quickly as the vehicle approaches and passes Furthermore, for the constanst κ and η of the accelerometer, the sensor. Finally, it is also worth pointing out the initial the hyperparameters were behavior of the filter (up to t 200). While the position ≈ µκ = 100, and kκ = 1, error converged almost immediately, the speed estimation error converged somewhat more slowly as the target approached the and sensor which is due to the fact that the speed is not observed 2 µη = 4, and ση = 1, through the measurement directly. respectively. For the accelerometer measurement noise, we used B. Unknown Parameters 2 8 2 In Fig. 3, we show the results for the same scenario as in σ = 1 10− , α = 6, and β = σ (α 1). a,0 × a a a,0 a − the previous section, however in this case the parameters are The choice of these parameters yields typical signal- and noise assumed unknown. The results were obtained by using the levels encountered in commercially available sensors. model (50). The mean of the state estimation error in Fig. 3 In all the cases, the initial distribution for the particles was suggests that even in this case, both the position and speed chosen as Gaussian with mean and covariance given by error converged towards zero for the multi-rate particle filter. However, note that the uncertainty (dotted line) is larger as  10 10 0  µ = and C = . compared to the the case where the parameters were known. x0 −15 x0 0 100 This can be attributed to the increased uncertainty in the In all the simulations, we used M = 1,000 particles. likelihood caused by the marginalization. 8

m 0.4 2 /

} 0.2 1 x t µT r

0 / x − 0 ym,t x t 0.2 y m,t ˆ r y − y m,t { 1 z

E 0.4 − y − m,t 0 200 400 600 800 1,000 2 − 0 500 1,000 1,500 2,000 2,500 3,000 t t (a) (a)

2 4 10− m/s × / 2

} 2 2 x t ˙ r 0 m/s − 2 / 0 x t ˙ ˆ r − a,t { 4 y E − 2 0 200 400 600 800 1,000 − t 0 500 1,000 1,500 2,000 2,500 3,000 (b) t Fig. 2. Mean estimation error with known parameters. (a) Position and (b) (b) speed. Fig. 4. Simulated measurement signals for two consecutive vehicles. (a) Magnetometer signal and (b) accelerometer signal.

m 0.4 /

} 0.2 1 x t m r 0 / 0.5 − } x t x t 0.2 r ˆ r −

{ 0 −

E 0.4 x − t ˆ r 0.5

0 200 400 600 800 1,000 { − t E 1 − 0 500 1,000 1,500 2,000 2,500 3,000 (a) t (a) 4 m/s

/ 2 10 } x t m/s ˙ r 0 / 5 − 2 } x t x t ˙ ˙ ˆ r − r 0 { 4 − E − x t 5 ˙ ˆ

0 200 400 600 800 1,000 r

{ −

t E 10 − 0 500 1,000 1,500 2,000 2,500 3,000 (b) t Fig. 3. Mean estimation error with unknown parameters. (a) Position and (b) speed. (b) Fig. 5. Mean estimation error for two vehicles following each other. (a) Position and (b) speed. C. Two Vehicles Fig. 4 illustrates the measurement signal for two vehicles passing by the sensors one after another. The first vehicle Fig. 5 (solid lines) together with the 2σ-bounds (dotted lines). x passed the sensor with an initial speed of r˙0 = 20 m/s while First, note how the particle filter was only activated whenever the second vehicle passed with r˙x = 10 m/s. The two vehicles a vehicle was present. Here this was the case from t 100 to 0 ≈ started at rx = 5 m. The particle filter was initialized t 900 for the first vehicle and from t 1,500 to t 2,600 0 − ≈ ≈ ≈ whenever a vehicle was detected in front of the sensor using for the second vehicle. For both vehicles, the position error of the magnetometer (vehicle detection using magnetometers the multi-rate filter converged to zero very quickly (Fig. 5a), has been shown to be reliable and rather simple, see, for whereas the speed error converged more slowly (Fig. 5b), as example [41]). Again, 100 MC simulations were run. was the case in the experiment with a single vehicle. This is The results of the mean estimation error are presented in especially pronounced for the second, slower vehicle. 9

TABLE I 103

PARAMETERSFORTHEHYPERPARAMETERSENSITIVITYANALYSIS. m Fusion

/ Acc. ) x Case m µm Km t Mag.

r 0  T  T (ˆ 10 a) 1 1 1 1 1 1 I3  T  T 5 b) 1 1 1 1 1 1 1 × 10 I3 T T     RMSE c) 2 2 2 1 1 1 I3 3  T  T 5 10− d) 2 2 2 1 1 1 1 × 10 I3 0 200 400 600 800 1,000 t 1 (a) a)

m 0.5 b)

/ Fusion c) m/s 2 x t Acc. / 10 r 0 d) ) Mag. x t − ˙ ˆ r x t 0.5 ( ˆ r − 0 1 10 − 0 200 400 600 800 1,000 RMSE t 0 200 400 600 800 1,000 (a) t (b) 10 a) Fig. 7. Comparison of the multi-rate particle filter and individual particle filters for each sensor. RMSE for (a) the position and (b) the speed. m/s 5 b)

/ c) x t

˙ 0 r d)

− 5 E. Comparison to Other Filters x t ˙ ˆ r − 10 In order to further illustrate the performance of the proposed − 0 200 400 600 800 1,000 method, it is compared to individual particle filters for each of t the sensors first. Again, a single passage of a vehicle and 100 (b) Monte Carlo simulations using the setup mentioned above are considered. As a measure of comparison, the root mean square Fig. 6. Examples of the proposed method using different hyperparameters. (a) Position and (b) speed. error (RMSE) of the individual states over the 100 simulations was chosen. Figure 7 illustrates the obtained results. It can be seen D. Sensitivity to Hyperparameters that the filter fusing both measurements (solid line) has the lowest RMSE for both states, the position (Fig. 7a) and speed The proposed method heavily relies on Rao- (Fig. 7b). The filter only using the magnetometer (dashed line) Blackwellization, which involves analytical integration performs as well as the fusion filter initially but does not attain of some of the unknown parameters. Rao-Blackwellization as low errors once the vehicle is in front of the sensor around requires the use of prior distributions of the parameters to t 500. Finally, the filter only using the accelerometer (dotted ≈ be integrated, and these priors are in turn defined by their line) performs considerably worse than both other filters. own parameters (hyperparameters). Hence, in this section, the As mentioned above, popular alternatives to state estimation sensitivity of the method to the hyperparameters introduced in nonlinear systems using particle filtering includes different through the priors is illustrated. For this purpose, a single non-linear Kalman filters such as the extended Kalman filter vehicle passage with varying true- and hyperparameters is (EKF) or the UKF [40]. It is generally believed, that the considered. The combinations of the different parameters are UKF is superior to the EKF and hence, the particle filter is given in Table I. compared to the UKF here. For comparison, we employed The results are shown in Fig. 6. If the location hyperpa- an UKF that uses multi-rate fusion of the data just as the rameter µm is close to the true value of m, the performance proposed particle filter. This is achieved by calculating the time does not vary much depending on the prior variance-related and measurement updates by simply using all the available parameter Km, see cases a) and b). However, it can be seen measurements for any given t. Additionally, the proposed that choosing µm significantly different from the true value of method is also compared to a particle filter fusing the two m can lead to deteriorated performance, see case c). This effect sensors where both sensors are sampled at the same sampling can be mitigated by instead increasing the hyperparameter Km frequency. and thereby increasing the prior variance (case d), see (33)). The results of this comparison are shown in Fig. 8. Even We note that an increased prior variance entails less reliance though the UKF (dashed) initially converges, it has difficulties on the prior and more on the likelihood, which is why the to actually track the vehicle which is reflected in increased results of case d) are better than the ones of case c). RMSE around t 250 for both, the position and speed. The ≈ 10

102 accordingly. This can, for example, be achieved by gathering

m MR-PF enough measurement data and building the appropriate prior

/ PF ) 0 and using empirical Bayes methods, see [42]. x t 10 UKF r

(ˆ In summary, it is apparent that tracking should be done by using both sensors. It is important to point out that the 2 10− multi-rate particle filter does not require equal sampling rates RMSE or synchronized sampling by the two sensors. These sensors 0 200 400 600 800 1,000 can provide samples according to their sampling rates. This, t in general, allows for conservation of energy and reduction of (a) necessary computations. It is also worth mentioning scenarios where the proposed 102 method has difficulties to accurately track vehicles. This MR-PF m/s PF mostly happens when the single-target assumption is violated. /

) 0 UKF Typical examples when this occurs are vehicles crossing in x

t 10 ˙ ˆ r

( front of the sensor or vehicles following each other very

2 closely. In order to accommodate such scenarios, the target 10− model has to be extended to account for the additional vehicle. RMSE 0 200 400 600 800 1,000 Finally, note that the simulations are conducted using the models introduced in Section II. Since there is no model t mismatch between the developed estimator and the simulation (b) setup, the numerical illustrations do not provide inaccuracies Fig. 8. Comparison of fusion using the multi-rate particle filter, a regular due to any mismatches between reality and the assumed particle filter, and an unscented Kalman filter. RMSE for (a) the position and (b) the speed. models. We would like to point out that the model for the magnetometer measurements has been verified (see, for ex- ample, [15]) whereas the one for accelerometer measurements time at which this happens coincides with the time where the is still under way. If it turns out that the accelerometer model vehicle enters the range of the sensor (see Fig. 1). Closer should be significantly different, the particle filtering method inspection of the results revealed that the UKF is not able proposed in this paper can be straightforwardly modified. to run on the accelerometer measurements only (in between the magnetometer sampling instants) which eventually leads to divergence. By contrast, both particle filters are able to do VI.CONCLUSION it, and therefore, they clearly outperform the UKF. Hence, the In this paper we addressed the problem of vehicle tracking additional complexity of the particle filter results in a gain in based on multi-rate particle filtering that fuses measurements performance. Comparing the multi-rate particle filter to the from two different sensors. The sensors are an accelerometer regular particle filter (dotted), we can see that the regular and a magnetometer and they operate with different sampling particle filter performs better. This is not surprising since it rates. The measurement models include unknown parameters, can make use of data which is unavailable to the multi-rate and we handle them by using Rao-Blackwellization. The particle filter. results indicate that the proposed approach is feasible. As expected, better results for the state estimates were obtained F. Discussion when the multi-rate particle filter used the measurements of both sensors as opposed to using measurements of each sensor The above simulations show that the multi-rate particle filter individually. Furthermore, the performance of the filter when performed well. It successfully tracks the vehicle with low the parameters of the models were assumed unknown was very uncertainty. As expected, it yielded best results because it pro- close to that of the filter that knew the correct values of the cesses information from the two sensors in comparison to the parameters. particle filters that rely on single sensors only. The reason this is not a surprising result is because the statistical information from the sensors is additive, that is, more measurements lead APPENDIX to more accurate estimates. Furthermore, the simulations indicate that even though some In order to find the marginalized distribution for p(ym,t xt), 2 | parameters of the model are unknown, they can be treated by where σm has been eliminated, we start from the likeli- Rao-Blackwellization. The results suggest that the error for hood (34) given by the proposed multi-rate algorithm was of the same magnitude 2 2
p(y x , σ ) = y ; H µ , σ (I + H K H>) , as in the case where the parameters were assumed known. m,t| t m N m,t m m m 3 m m m It was also shown that the choice of the hyperparameters and the prior distribution affect the method in the sense that poorly chosen parameters can lead to deteriorated performance. In practice, one often αm βm 2 βm 2 αm 1 2
− − − σm has insight in the system and can calibrate the parameters p(σm) = σm e . Γ(αm) 11

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Ing. degree in electrical and communication engi- [27] Honeywell, 1, 2 and 3 Axis Magnetic Sensors neering from Bern University of Applied Sciences, HMC1051/HMC1052/HMC1053, datasheet. Switzerland in 2007, and the M.Sc. degree in electri- [28] R. Hostettler, M. Lundberg Nordenvaad, and W. Birk, “The pavement cal engineering and Ph.D. degree in automatic con- as a waveguide: Modeling, system identification, and parameter estima- trol from Lulea˚ University of Technology, Sweden tion,” Instrumentation and Measurement, IEEE Transactions on, vol. 63, in 2009 and 2014, respectively. no. 8, pp. 2052–2063, 2014. He is currently working as a research associate [29] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. at Lulea˚ University of Technology, Sweden. His Washington, DC, USA: National Bureau of Standards, 1964. research interests include statistical signal process- [30] H. L. Van Trees, K. L. Bell, and Z. Tian, Detection, Estimation, and ing with applications to target tracking, intelligent Modulation Theory: Part 1: Detection, Estimation, and Filtering Theory, transportation systems, and sensor networks. 2nd ed. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2013. [31] R. Hostettler, “Traffic monitoring using road side sensors: Modeling and estimation,” Ph.D. Thesis, Lulea˚ University of Technology, June 2014. [32] The International Council on Clean Transportation, “European Vehicle Market Statistics,” 2013. Petar Djuric´ (S’86–M’90–SM’99–F’06) received [33] ST Microelectronics, LIS344ALH MEMS Inertial Sensor, datasheet. the B.S. and M.S. degrees in electrical engineering [34] M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial from the University of Belgrade, Yugoslavia, in 1981 on particle filters for online nonlinear/non-gaussian bayesian tracking,” and 1986, respectively, and the Ph.D. degree in Signal Processing, IEEE Transactions on , vol. 50, no. 2, pp. 174–188, electrical engineering from the University of Rhode February 2002. Island, USA, in 1990. [35] P. M. Djuric,´ J. H. Kotecha, J. Zhang, Y. Huang, T. Ghirmai, M. F. He is currently a Professor with the Department of Bugallo, and J. M´ıguez, “Particle filtering,” IEEE Signal Processing Electrical and Computer Engineering, Stony Brook Magazine, vol. 20, no. 5, pp. 19–38, 2003. University, Stony Brook, NY, USA, where he joined [36] N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to in 1990. From 1981 to 1986, he was a Research nonlinear/non-Gaussian Bayesian state estimation,” IEE Proceedings-F, Associate with the Vincaˇ Institute of Nuclear Sci- vol. 140, no. 2, pp. 107–113, 1993. ences, Belgrade. His research has been in the area of signal and information [37] L. Geng, M. F. Bugallo, A. Athalye, and P. M. Djuric,´ “Indoor tracking processing with primary interests in the theory of signal modeling, detection, with RFID systems,” Journal of Selected Topics in Signal Processing, and estimation; Monte Carlo-based methods; signal processing over networks; vol. 8, no. 1, pp. 96–105, 2014. and applications of the theory in a wide range of disciplines. He has been [38] A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and invited to lecture at many universities in the United States and overseas. D. B. Rubin, Bayesian Data Analysis. CRC Press, 2013. Prof. Djuric´ was a recipient of the IEEE Signal Processing Magazine Best [39] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Paper Award in 2007 and the EURASIP Technical Achievement Award in Theory. Prentice Hall, 1993. 2012. In 2008, he was the Chair of Excellence of Universidad Carlos III [40] E. A. Wan and R. Van Der Merwe, “The unscented Kalman filter de Madrid-Banco de Santander. From 2008 to 2009, he was a Distinguished for nonlinear estimation,” in Adaptive Systems for Signal Processing, Lecturer of the IEEE Signal Processing Society. He has been on numerous Communications, and Control Symposium 2000. AS-SPCC. The IEEE committees of the IEEE Signal Processing Society and of many professional 2000, 2000, pp. 153–158. conferences and workshops. He is the Editor-in-Chief of the new journal [41] J. Ding, S.-Y. Cheung, C.-W. Tan, and P. Varaiya, “Signal processing IEEE Transactions on Signal and Information Processing over Networks. Prof. of sensor node data for vehicle detection,” in Intelligent Transportation Djuric´ is a Fellow of the IEEE.