Abstract

A novel fifth-degree strong tracking cubature Kalman filter is put forward to improve the two-dimensional maneuvering target tracking accuracy. First, a new fifth-degree cubature rule, with only one point more than the theoretical lower bound, is used to approximate the intractable nonlinear Gaussian weighted integral in the nonlinear Kalman filtering framework, and a novel fifth-degree cubature Kalman filter is proposed. Then, the suboptimal fading factor is designed for the filter to adjust the filtering gain matrix online and force the residual sequences mutually orthogonal, thus improving the ability of the filter to track the mutation state, and the fifth-degree strong tracking cubature Kalman filter is derived. The suboptimal fading factor is calculated in a new method, which reduces the number of calculations for the cubature points from three times to twice without calculating the Jacobian matrix. The simulation results indicate that the proposed filter has the ability to track the maneuvering target and achieve higher target tracking accuracy and thus verifies the effectiveness of the proposed filter.

1. Introduction

For the last several decades, the target tracking has momentous applications in many fields, such as navigation guidance, military application, and sensor networks [1]. In target tracking problem, the process model is generally linear, while the measurement model, mainly including the measured range and bearing angle, is nonlinear [2, 3]. The essence of the target tracking is to use a series of measured ranges and bearing angle information to estimate the position and velocity of the target in real time; hence, it belongs to the nonlinear filtering problem, which has always been dealt with using the nonlinear Kalman filters [4].

In nonlinear Kalman filters, the extended Kalman filter (EKF) [5] is the most widely used one. In EKF, the nonlinear function is approximated using the first-order Taylor series expansion and, then, the standard Kalman filter is applied [6]. EKF is simple in principle; however, its filtering accuracy and stability may reduce for the strong nonlinear system, and it needs to calculate the Jacobian matrix, which is difficult to accomplish at times. Julier [7] adopts a set of sigma points to approximate the posterior probability density function (PDF) and proposes the unscented Kalman filter (UKF). UKF is a derivative-free filter and can achieve third-degree filtering accuracy [8, 9]. However, there exist some tunable parameters in the selection of sigma points, and the inappropriate selection may reduce the filtering accuracy. In addition, the weight on the center point may be negative for the high-dimensional system, which may degrade the numerical stability of the filter [10, 11]. Arasaratnam [12, 13] puts forward the cubature Kalman filter (CKF), where the intractable integral in nonlinear Kalman filter is decomposed into the spherical integral and the radial integral, which are approximated using different numerical integration rules. CKF contains a set of cubature points with equal weights; thus the numerical stability is guaranteed [14–16]. CKF can be regarded as a special case of UKF [17]; however, CKF gives the rigorous reason for the selection of parameters for the first time. In order to improve the accuracy of CKF, Jia [18] adopts the symmetric spherical interpolation and moment matching method to calculate the spherical integral and radial integral, respectively, and proposes the fifth-degree CKF. And then, Wang [19] employs the transformation group of the regular simplex instead of the symmetric spherical interpolation to derive the fifth-degree spherical simplex-radial CKF. These two filters improve the filtering accuracy effectively.

However, the conventional nonlinear Kalman filters cannot achieve the effective tracking of the maneuvering target. In order to improve the ability of the filter to fast track the mutation state, Zhou [20] proposes the strong tracking filter (STF) on the basis of EKF. The STF uses the suboptimal fading factor to realize the real time adjustment of the gain matrix to force the residual sequences mutually orthogonal [21, 22]; it has the ability to track the mutation state while inheriting the inherent defects of EKF. Correspondingly, the strong tracking UKF (STUKF) and strong tracking CKF (STCKF) are put forward in succession. There are two methods to calculate the suboptimal fading factor in these two types of filters. On the one hand, the fading factor is still calculated using the Jacobian matrix [23–25], which may bring some trouble in strong nonlinear systems. On the other hand, the fading factor is calculated using the equivalent expressions derived by the statistical regression method [26, 27], in which the predicted measurement covariance and the cross-covariance are contained. It results in the nonlinear transformation carried out for three times, which increases the amount of calculation while loses a certain precision.

In this paper, a novel fifth-degree strong tracking cubature Kalman filter (5-STCKF) is proposed to further improve the two-dimensional maneuvering target tracking accuracy. A new fifth-degree cubature rule is utilized to approximate the intractable nonlinear Gaussian weighted integral, and a novel fifth-degree cubature Kalman filter is put forward. To improve the ability of the filter to track the mutation state, the suboptimal fading factor is designed to adjust the filtering gain matrix online and force the residual sequences mutually orthogonal and the 5-STCKF is derived. The suboptimal fading factor is calculated in a new method, which reduces the number of calculations for the cubature points from three times to twice without calculating the Jacobian matrix. The proposed 5-STCKF has the ability to track the mutation state and can achieve better performance compared with the 5-CKF and 5-SSRCKF in maneuvering target tracking applications. In addition, the number of cubature points needed in the 5-STCKF is less than those in the 5-CKF and 5-SSRCKF. The simulation results verify the validity of the filter.

The rest of this paper is organized as follows: the novel fifth-degree cubature Kalman filter is proposed in Section 2, the novel fifth-degree strong tracking cubature Kalman filter is derived in Section 3, the simulation results and analysis are presented in Section 4, and the conclusion is given in Section 5.

2. A Novel Fifth-Degree Cubature Kalman Filter

The following discrete time nonlinear system is considered:where denotes the state vector at time , represents the measurement vector, and and denote the known nonlinear process and measurement functions, respectively. is the zero mean Gaussian white process noise, with the covariance being , and is the zero mean Gaussian white measurement noise, with the covariance being .

Given the measurements , the minimum mean square error (MMSE) estimate of the state with the estimate error covariance is given below:where denotes the posterior PDF.

For nonlinear Kalman filters, the posterior PDF is assumed to be Gaussian distribution and the following five integrals in the filtering framework are required to be calculated [12]:where denotes the Gaussian distribution with mean and covariance , and represent the prior state estimate and estimate error covariance, respectively, is the predicted measurement, and and denote the measurement covariance and cross-covariance, respectively.

It can be seen from (4) to (8) that the key problem in nonlinear Kalman filters is to calculate the intractable integral in the form of , where is arbitrary nonlinear function. In general, it is hard to achieve the analytical solution of , and the numerical approximation should be considered. For this, the integral is taken into account first.

There already exist some cubature rules, including the third-degree spherical-radial cubature rule and fifth-degree spherical-radial cubature rule, to calculate , and it is proved that the fifth-degree cubature rule can achieve higher accuracy than the third-degree one. For more details, please refer to [12, 18].

In this paper, through linear transformation, a novel fifth-degree cubature rule for calculating the integral is given below.

It is pointed out in [28] that can be calculated approximately using the following fifth-degree cubature rule:where perm means all distinct permutations, and represent the binomial coefficients that equal to and , respectively, and the parameters , , and satisfy the following equations. For more details, please refer to [28, 29].

However, (9) is not suitable for calculating in its current form. In order to simplify (9), we define as an n-order identity matrix and the matrix subscript is utilized to denote the ith column and, then, (9) can be written in the following form:

Further, we define the following variables:

Then, (13) is transformed into the following simplified form:

It can be proved that has the following equivalent form [12]:such that can be approximated using the cubature rule below:where the cubature points and corresponding weights are given below and the specific values of A, B, and C are given in [29].where and represent the matrices constituted by and , respectively.

Take as an example, which will be used in the simulation section; the parameters in (19) are listed in Table 1.

Remark 1. The number of points needed in the fifth-degree cubature rule proposed in [18] is , and that in the spherical simplex cubature rule proposed in [19] is . However, the number of points needed in the proposed cubature rule (19) is , which is less than those of the aforementioned two filters and only one more than the theoretical lower bound of the fifth-degree rules, that is, [29].

Remark 2. The cubature rule (19) can be applied in the condition that , and for the conventional two-dimensional maneuvering target tracking problem, the maximum state dimension is seven (including two position variables, two velocity variables, two acceleration variables, and an optional turn rate); hence, the proposed cubature rule is suitable in the application of maneuvering target tracking.

Based on the cubature rule and the points and weights, the novel fifth-degree cubature Kalman filter is proposed in the nonlinear Kalman framework as follows.

Step 1 (filter initialization). Take the initial values and into consideration.
Cycle , and complete the following steps.

Step 2 (time update). The posterior state estimate and estimate error covariance are used instead of and in (20) to calculate the cubature points , which are propagated using the nonlinear process function to obtain the following points :

The prior state estimate and the estimate error covariance are calculated below:where is given in (21).

Step 3 (measurement update). The prior state estimate and estimate error covariance are used instead of and in (20) to calculate the cubature points , which are propagated using the nonlinear measurement function to obtain the following points :

The predicted measurement , the predicted measurement covariance and the cross-covariance are calculated as follows, respectively:

The Kalman filtering gain , the posterior state estimate , and the posterior estimate error covariance are calculated, respectively.

3. A Novel Fifth-Degree Strong Tracking Cubature Kalman Filter

The maneuvering target tracking problem can be regarded as a mutation state tracking problem. It can be seen from (30) that once the state has an abrupt change, the residual may increase thereupon, and if remains minimum as it tends to be as the filter is stable, the proposed fifth-degree cubature Kalman filter may lose the ability to track the mutation state. Therefore, we should adjust online to satisfy the following two criterions:

The first criterion ensures the optimal filter, and the second criterion, which is called the orthogonality principle [20], plays a key role in tracking the mutation state. It has been proved that the residual series in Kalman filter are mutually orthogonal, which can be regarded as metrics to evaluate the performance of the filter. If we adjust online to force the criterion (b) established once the state takes an abrupt change, the filter has the ability to track the mutation state then and is named strong tracking filter.

It has been proved that (33) has the following expression:where and denote the Jacobian matrix and represents the residual covariance.

It can be seen from (34) that, in order to force , the following must hold:

By using the statistical linear regression method, the following can be achieved:

Combined with (37), (35) is transformed into the following form:

Then, by substituting (29) into (38), we have

On account of which is a nonzero matrix, (39) turns into

In order to adjust online, the time-varying suboptimal fading factor is introduced into the prior estimate error covariance and (24) is modified as follows:where is the time-varying suboptimal fading factor.

Similarly, and are modified according to (36) and (37) as below:

By substituting (42) into (40), we obtain that

Define and , and (44) turns into

By calculating the trace of both sides of (45), we achieve the following suboptimal fading factor:

The residual covariance can be estimated as follows:where denotes the forgetting factor and is generally set to be .

In practical applications, is often modified as , where denotes the softening factor to smooth the state estimate.

Thus, the novel fifth-degree strong tracking cubature Kalman filter is derived by substituting (42), (43) instead of (27), and (28) and (41) into (31), and the calculation process is listed in Figure 1.

Figure 1 

The calculation process of the proposed 5-STCKF.

Remark 3. In the previous nonlinear strong tracking filters, and are contained in the calculation of [26, 27] and results in the cubature points being calculated three times in a filtering cycle, which may reduce the filtering accuracy due to the loss of the higher order moment information. However, in the proposed 5-STCKF, and are calculated using a different method and the cubature points are calculated only twice in a filtering cycle as the conventional CKF, which may achieve more accuracy estimate and better computational efficiency.

Remark 4. There is no need to calculate the Jacobian matrix; thus, the proposed 5-STCKF is a derivative-free filter.

4. Simulation Results and Analysis

In this section, a maneuvering target tracking simulation is taken to test the performance of the proposed filter

4.1. Maneuvering Target Tracking Models

The dynamic model of the two-dimensional maneuvering target is given below:where denotes the target state, represents the control input, and is the zero mean Gaussian process noise. , , and denote the state transformation matrix, the control input matrix, and the noise input matrix, respectively, which are given below:where is the time interval.

For the tracking system, the target state and the control input are unknown; in this case, the target state and the control input can be used to form the augmented state vector , and (48) is modified as follows:where and denote the augmented state transformation matrix and noise input matrix, respectively, which are given below:

The measurement model is given as follows:where denotes the four-quadrant inverse tangent function and is the zero mean white Gaussian measurement noise.

4.2. Simulation Results and Analysis

In this simulation, the initial location of the target is and the initial velocity of the target is . The location of the radar is . The total simulation time is 400s, and the target takes a high maneuvering with the acceleration given below. The trajectory of the target and the location of the radar is shown in Figure 2.