Abstract

The paper investigates the problem of manoeuvring target tracking using angular measurements. The target dynamics is modelled by multiple switching models, while the measurements are collected asynchronously from possibly multiple moving platforms. The contribution of the paper is twofold. First a theoretical lower bound of the performance error is derived and analysed. Second, three tracking algorithms are proposed and compared to the theoretical bound. The proposed algorithms include: (i) the interactive multiple model (IMM) algorithm with extended Kalman filters, (ii) the IMM with unscented Kalman filters and (iii) the multi-mode particle filter.

Author Keywords: Target tracking; Passive ranging; Target motion analysis; Cramer–Rao bound; Nonlinear filtering; Manoeuvring target; Interactive multiple model; Particle filter

Abbreviations: , CA, constant acceleration; CV, constant velocity; CRLB, Cramer–Rao lower bound; EKF, extended Kalman filter; IMM, interactive multiple model; LOS, line of sight; MMPF, multi-mode particle filter; MMSE, minimum mean square error; pdf, probability density function; PF, particle filter; UKF, unscented Kalman filter; UT, unscented transform

α_k

index of dynamic model M_k; α_k{1,…,r}

E

expectation operator

F_k

transition matrix at time index k

h_jk

nonlinear measurement function (arctan)

H_k^j_k

Jacobian of h_jk

lth model history at time index k

j_k

sensor (platform) index; j_k{1,…,S}

J_k

Fisher information matrix at time index k

k

time index corresponding to time instant t_k

M_k

dynamic model in effect from t_k−1 to t_k; M_k{m₁,…,m_r}

n

dimension of the state vector

N

number of particles in the MMPF

p_ij

model transition probabilities

π_i

initial model probabilities

q_kⁿ

probability mass associated with nth particle n=1,…,N

Q_k

process noise covariance matrix

r

number of dynamic models

s_k^t

target state vector at time index k

s_k^j_k

observer j_k state vector at time index k

T_k

sampling interval

U_k

vector of dynamic model mismatch compensation at time index k

x_k

relative state vector at time index k

y_k

augmented state vector (in MMPF)

_k

set of particles at time index k, approximately distributed as the posterior density

z_k^j_k

measurement at time index k from sensor j_k

Z_k

collection of all sensor messages up to time index k.

Article Outline

Nomenclature

1. Introduction

2. Problem description

2.1. General formulation

2.2. Example: CV and CA models

3. Lower bound of performance

3.1. Derivation of the bound

3.2. Analysis of the bound

4. Tracking algorithms

4.1. IMM-EKF algorithm

4.2. IMM-UKF algorithm

4.3. MMPF

4.3.1. Prediction

4.3.2. Update

5. Numerical results

6. Conclusion

References

1. Introduction

The basic problem is sequential (on-line) estimation of target location, speed and heading from noise corrupted measurements of target line-of-sight (LOS) angle. The problem has a long history being of interest to various applications, such as for example airborne and underwater surveillance with passive sensors [4, 15 and 24]. It appeared in the literature under various names, some of them being passive ranging, target motion analysis and bearings-only tracking. If a single moving platform is collecting angular measurements, target state (position and velocity components) becomes observable only if the observer “outmanoevres” the target (i.e. observer motion is one derivative higher than that of the target and a component of this motion is perpendicular to the LOS) [4, 11 and 16]. A number of recursive algorithms for angle-only target tracking have been proposed in the literature, e.g. [1, 14, 22, 25 and 32].

Most of the research in the field of angle-only tracking, however, focused on the assumption that the target dynamics can be modelled by a constant velocity motion. To our best knowledge, only a few exceptions (with a manoeuvring target) are reported recently. An interactive multiple model (IMM) algorithm with two models is proposed in [5]. The first model for constant velocity (CV) target motion is based on the extended Kalman filter (EKF) in modified polar coordinates. The second model for manoeuvring target motion is an angle acceleration model. Le Cadre and Tremois [7] modelled the accelerating target using the CV model with process noise and developed a tracking filter in the framework of the hidden Markov model. Dufour and Mariton [10] used the IMMPDA filter (various configurations of the IMM) and assumed the collected measurements are from two synchronised static passive sensors.

In this paper we assume the target motion is modelled by multiple switching dynamic models. This means that the usual target state vector is appended with a discrete model (or regime) variable M{m₁,…,m_r}, whose transitions are modelled with a Markov chain. The angular measurements are collected by sensors on multiple possibly moving platforms. The platforms are assumed to be connected by a tactical data link capable of transmitting measurements as they occur (asynchronously) with a zero transmission delay. Furthermore, it is assumed that the probability of target detection is unity and there are no false alarms (thus ignoring the data association issues).

The paper studies an error performance bound for the described problem, derived under the assumption that the switching model history is known and that process noise is zero. The bound is a conservative lower bound, useful in predicting the performance for various target–observer trajectories, data link bandwidth constraints, sampling intervals and sensor accuracies. The bound is also important in assessing the condition of observability for a given scenario. Three algorithms for target tracking in the described environment are also proposed in the paper and compared to the theoretical bound. Two of the algorithms are based on the IMM scheme to deal with the switching target dynamics. Filtering is done in one algorithm using the EKFs, while in the other this was done using the recently proposed unscented Kalman filters (UKFs) [18]. The third tracking algorithm, referred to as the multi-mode particle filter (MMPF), uses the sequential Monte-Carlo estimation method [9] for both switching dynamic mode estimation and filtering. The presentation in the paper will be in the context of airborne platforms, although the methodology is completely general.

The paper is organised as follows. Section 2 presents the mathematical formulation of the problem. Section 3 describes the derivation of the performance bound and its implications. The proposed tracking algorithms are presented in Section 4. Section 5 is devoted to the performance comparison and numerical results. The conclusions of the study with future research directions are given in Section 6.

2. Problem description

2.1. General formulation

Consider a two-dimensional (2D) target–observer encounter. Target motion obeys one of a finite number of dynamic models at a time. Typically for non-manoeuvring (quiescent) periods, the CV model is appropriate. For manoeuvring intervals the choice is between the coordinate uncoupled models and coordinate turn models [21]. In this paper we consider only the linear dynamic models which include all polynomial models (constant acceleration (CA), constant jerk (CJ), etc.) with or without process noise.

Suppose that target dynamics can switch (jump) from one model to the other during the tracking period. Let us denote the model in effect during the interval of time from t_k to t_k+1 as M_k+1{m₁,…,m_r}, where r is the number of possible models. Target dynamics now can be described as

The angular measurements are collected asynchronously by multiple observers at non-uniform time intervals T_k=t_k+1−t_k. Let z_k^j_k denote an angular measurement, with subscript k (as before) corresponding to the time when the measurement was recorded (t_k) and superscript j_k{1,2,…,S} denoting the observer (sensor) which supplied the measurement (S is the number of observers). As an illustration, the time axis with a possible set of collected measurements with S=3 observers is shown in Fig. 1. The measured angles are referenced clockwise positive to the y axis and are given by

(4)

The state vector of observer j_k, , is assumed to be known exactly at time t_k (information supplied by an on-board inertial navigation system and possibly aided with a GPS). The observer state vector is defined to match the components and size of the target state vector, so that we can introduce the relative state vector

The S observers (platforms) are connected by a tactical data link. Assume that each observer attempts to broadcast a message which consists of: time stamp t_k, the ownship state s_k^j_k and the measurement z_k^j_k collected at time instant t_k. Due to the bandwidth constraints, however, not all of the messages are transmitted, so that the tracking filter on platform j receives all the local measurements and only occasional external messages via the data link. Time delays in the transmission are assumed to be zero.²

The sequential angle-only tracking problem can be defined as follows: given a set of sensor messages: Z_k={(t_i,s_i^j_i,z_i^j_i)} from i=1,…,k, and the target motion model described by ((1) and (2)), estimate the state vector s_k^t or x_k (they are related by (5)) at time t_k.

2.2. Example: CV and CA models

Suppose the target motion is described by two models (r=2):

Model m₁: CV model for non-manoeuvring target flight segments. The target state is given by , the ownship state is . Transition matrix is

(7)

(8)

(9)

Model m₂: CA model for manoeuvring target flight segments. The target state is given by , the ownship state is . Transition matrix is

(10)

(11)

(12)

3. Lower bound of performance

The optimal Bayesian solution to the problem formulated in the previous section would require to construct the posterior density p(x_k|Z_k) where x_k is the relative state, and Z_k was defined as the collection of multi-platform messages up to time t_k. The optimal solution, unfortunately, cannot be derived analytically, even for r=1 (single dynamic model) case, because the measurement equation (3) is non-linear. A number of approximate solutions have been proposed in the literature for r=1 (CV) case, and they range from the EKF based solutions (in the Cartesian or modified polar coordinates) to the particle filters [1, 2, 14, 22, 25 and 32]. For r>1, the situation is even more difficult, as the required density p(x_k|Z_k) becomes a mixture density with the number of components growing exponentially with time [3]. In the absence of the optimal analytic solution, one has to resort to approximate solutions. However, in dealing with approximations, it would be beneficial to have a lower bound of performance which could predict the best achievable performance even before running the algorithms, and would help in the assessment of the level of approximation introduced by a particular algorithm. A derivation of such a bound is the subject of this section. The bound will be derived under the assumption that process noise v_k is zero and the model history is known.

3.1. Derivation of the bound

The error covariance matrix C_k of an unbiased estimator of the state vector is bounded from below as follows:

(13)

(14)

J_k=E{[_xklog p(x_k,Z_k)][_xklog p(x_k,Z_k)]′}

(15)

r=1 case. For the case of a single dynamic model (r=1) the joint density function p(x_k,Z_k) is a multivariate Gaussian density, and the information matrix J_k can be calculated using the following recursion [29]:

(17)

H_k^j_k=[_xk[h_k^j_k(x_k)]′]′.

(18)

(19)

r>1 case. Next we consider the case of multiple switching models (r>1). Note that at time t_k the number of possible sequences of models, or model histories, is L=r^k. Let us denote one (ℓth) such a history as [3]

(21)

(22)

Suppose now that the correct model history is known. Then if and zero otherwise, leading to the simplification of (22):

(23)

(25)

The proposed error performance bound, which assumes that model history is known, is a conservative bound. This can be shown as follows. From ((13) and (23)) it follows that

(26)

(27)

C_kC_k^*[J_k^*]⁻¹,

Initial information matrix. For estimation (filtering) purposes, the ignorance about the initial state vector x₀ is usually modelled with a Gaussian density. Let the covariance matrix of this density be P₀. Then recursion (24) can be initialised as follows:

(29)

(30)

3.2. Analysis of the bound

Consider a multiobserver-target encounter shown in Fig. 2(a). Observer 1 is a primary surveillance platform and we analyse the performance bound for its tracking filter. This platform flies in a circle (coordinated turn motion [3]) of radius 4.25 km and at the speed of 900 km/h. Its acceleration is 1.5g, where g=9.81 m/s² is the acceleration due to gravity. Its angular measurements are fairly accurate, with the error standard deviation of σ₁=0.2°. Observer 2 is a supporting static platform with the error standard deviation of angular measurements σ₂=1°. The target is flying with a CV motion (no process noise) in the first 60 s. Then in the next 30 s, it is making a turn with the CA (no process noise) of 1.3g. In the last segment of its trajectory, the target switches back to the CV motion. Fig. 2(b) shows the target speed and acceleration as a function of time.

Since observer 1 is performing a coordinated turn (its motion has an infinite number of higher order polynomial derivatives), it is guaranteed to “outmanoeuvre” the target moving with the CV–CA–CV motion. Thus the error performance bound is supposed to converge even for an autonomous operation of observer 1 (when the data link is OFF). For the case when the data link is turned ON, we will in this particular example, assume that the first message from observer 2 is received 35.4 s after the beginning of the observation period. Subsequent messages from observer 2 then arrive every 16 s. The sampling interval of observer 1 is 2 s in all cases.

For the calculation of the initial covariance matrix P₀ in ((29) and (30)) the following parameters are used: σ_r=25 km, σ_v=300 m/s and σ_a=g.

The lower bounds of mean-square error for unbiased estimation of target position and velocity along x and y axes, are shown in Fig. 3. As explained earlier, the bounds were calculated using (24), (28) and (15). The performance curves in Fig. 3 correspond to the square-root of the expression in (15). The dashed lines in Fig. 3 show the performance for the case when the data-link is OFF (autonomous observer 1). Initially the bounds have a large value, reflecting a limited amount of prior knowledge (with uncertainty P₀). As the measurements become available, the bound decreases until the time instant of 60 s, when the target motion switches to the CA model. During the interval from 60 to 90 s, the bounds actually rise, but had this interval been prolonged, the bounds would have eventually started to decrease (since the observer “outmanoeuvres” the target). The solid lines in Fig. 3 correspond to the case when the data link is ON. Note how in this case the bounds drop sharply at the time instants where the data from observer 2 become available (these time instants are indicated by vertical dotted lines).

4. Tracking algorithms

This section describes three recursive estimators (filters) designed for tracking a manoeuvring target using asynchronous multi-platform angle only measurements. All of them are based on approximations, and our goal will be to compare their performance with the theoretical CRLBs discussed earlier. Note that, unlike the theoretical CRLB computation, the tracking filters make no assumption about target state being deterministic and they have no prior knowledge of model history. As before we will consider r=2 case (two switching models), m₁ being the CV model and m₂ being the CA model.

4.1. IMM-EKF algorithm

For a CV target (r=1), the EKF in the Cartesian coordinates is one of the first algorithms proposed for angle-only tracking [3 and 4]. It is based on a linearisation of the nonlinear measurement equation (3) using the first-order Taylor series expansion of arctan function. The implicit assumption made in the development of the EKF is that the posterior density p(x_k|Z_k) is Gaussian. Following the same approach, we have developed the EKF in the Cartesian coordinates for the CA target model.³ The EKF equations can be found in [3 and 4], but for completeness are repeated here. The prediction step of the EKF, for both CV and CA dynamics, is given by

(34)

(36)

(37)

(38)