1. Introduction
Accurate and robust underwater target tracking and advanced parameter estimation are becoming increasingly important research areas in marine science. Underwater surveillance is a core component of national defense and economic development [
1,
2]. Good estimates of the motion parameters of an underwater uncooperative target not only can be used to obtain useful prior information about a target but also can provide support for further military or economic actions. Therefore, developing more advanced underwater uncooperative target tracking techniques to more robustly and accurately track the target is of great value.
Two kinds of tracking mechanisms can be applied to underwater target tracking scenarios, namely, active tracking [
3,
4] and passive tracking [
5,
6,
7]. By utilizing the active sonar systems mounted on facility platforms (usually ship-borne or shore-based platforms), the active tracking system can track an underwater target with high precision in terms of both angle information and range information. However, because the active tracking system depends on the active sonar system, which needs to emit acoustic signals, the tracking system has a high power consumption and is hard to conceal. In addition, due to its implementation on fixed shore-based facility platforms and exposed ship platforms, the active tracking system lacks mobility and the capacity for concealment. However, flexibility and concealment are key considerations for applications in national defense. Therefore, passive tracking systems, which accomplish tracking only via the angle information obtained from the passive radiated noise created by the underwater uncooperative target, have been examined by several researchers [
8,
9,
10]. In the specific underwater tracking scenario in which an underwater target operates at a depth that is far less than the distance between the target and the passive tracking system, passive underwater tracking can be modeled as a 2-D bearing-only tracking (BOT) problem, which is subject to a large number of technical challenges.
According to References [
11,
12], the underwater 2-D BOT problem is characterized by the inherent challenges in low observability and high nonlinearity. As a result, the tracking system is unobservable when only one fixed observer works during the passive tracking procedure. To ensure that the tracking process is observable and thus to satisfy the requirements of robust estimation, the single observer must maneuver with significantly more agility than that of the target; if this cannot be achieved, more observers are needed [
13,
14]. Usually, deploying more observers not only saves energy but also allows for concealment. Therefore, the deployment of more than one observer during the target tracking procedure can resolve the issue of low observability.
Due to the high nonlinearity of the bearing-only measurements, a 2-D BOT is difficult in the context of underwater target tracking [
15]. To address this problem, a few nonlinear tracking techniques have been proposed for passive underwater target tracking scenarios. Among the techniques that ensure reliable tracking of an uncooperative underwater target, nonlinear Bayesian estimating techniques are the most popular due to their robust and accurate performance. By assuming that the disturbance caused by the environment has a Gaussian stochastic process, significant research accomplishments have been achieved by various researchers [
16,
17,
18,
19]. An extended Kalman filter (EKF)-based underwater target tracking algorithm was designed by Reference [
16] to perform suboptimal estimations of an underwater target. Considering the linearization error introduced by the first-order Taylor series expansion, a group of determined sigma points was utilized by Reference [
17] to linearize the nonlinear measurement equations in target tracking and thus to theoretically enhance the tracking accuracy of the second-order Taylor series expansion. Similarly, to enhance tracking accuracy, more complex tracking algorithms based on the cubature Kalman filter (CKF) and sigma-point Kalman filter with interpolation were proposed by References [
18,
19], respectively. Unlike the abovementioned studies, which utilized linearization techniques or determined sigma points to linearize the nonlinear system models, a particle filter (PF)-based estimating scheme using arbitrary “particles” to reform the nonlinear tracking system and to track the target was proposed by Reference [
20]. By utilizing the Monte Carlo method to generate particles with an arbitrary distribution, the PF-based tracking algorithms can accurately estimate the distribution of the uncertainties of the tracked states. By generating many particles during the tracking procedure, the PF-based tracking algorithms can theoretically achieve accurate tracking results in any nonlinear tracking systems with any distributions of the uncertainties. However, the PF-based tracking algorithms have a significant computational burden, particularly when the system is highly nonlinear. As a result, the applications of PF-based tracking techniques are limited.
Considering the linearization errors of nonlinear tracking algorithms and the significant computational load of PF-based tracking techniques, novel tracking mechanisms must be developed. In particular, new approaches are required for the scenario of bearing-only underwater uncooperative target tracking, in which the measurements are strongly nonlinear, but the kinematics is linear when the target operates in constant velocity (CV) mode. Rather than linearizing the system model to address the linearization error, a novel recursive tracking scheme, named the two-step filter, was developed by References [
21,
22]. By setting a nonlinear projection of the states to be tracked and the nonlinear measurements into a new extended space, the nonlinear measurement model can be transformed into a linear model in the first-step extended state estimation [
21]. Then, by taking the nonlinear projection as a function of the original states, and the estimation bias from the first step, the original states can be extracted from the extended state via a recursive Newton iteration technique. The two-step filter has a superior performance when applied to nonlinear measurements because it eliminates the nonlinearities in the measurement update procedure, particularly when the measurements are abundant but the process model is simple [
22]. In addition, based on the prototype designed by References [
21,
22], some modifications of the two-step filter have been made by several researchers. Reference [
23] modified the time update procedure of the traditional two-step filter to ensure its robustness. This work was extended by Reference [
24], in which a more comprehensive and complete time update method for the traditional two-step filter was proposed. Reference [
25] proposed an adaptive two-step filter based on the modified Sage-Husa technique in combination with the multiple model technique to estimate the measurement noise online and thus to achieve higher tracking accuracy. In addition to theoretical studies, References [
26,
27] implemented the two-step filtering technique in applications of satellite attitude estimation and aerial target tracking. The studies verified the performance of the two-step filter and demonstrated its superior characteristics as an alternative solution to the various Bayesian nonlinear filters (i.e., EKF, UKF, PF, etc.). However, few studies can be found in which this approach has been implemented in underwater uncooperative target tracking despite the fact that the CV and nonlinear bearing-only measurement models represent a perfect tracking prototype for the two-step filter. In addition, modifications implemented by former researchers are not perfect in the underwater target tracking scenario because the uncertain underwater environment not only influences the measurements but also generates uncertainties in the model kinematics. Furthermore, the measurement update procedure has been paid little attention in previous research. As a result, few modifications have been made, leading to an incomplete theory of two-step tracking.
This study aimed to robustly and accurately implement passive tracking of an underwater uncooperative target under the conditions of uncertain underwater disturbances while taking the advantages and shortcomings of existing tracking techniques into consideration. For this purpose, a novel adaptive two-step tracking algorithm is proposed. Inspired by the accomplishments of References [
23,
25] in the research area of aerospace engineering, this study adopted a modified Sage-Husa technique for online estimation of both the process noise and measurement noise to mitigate the influences caused by underwater uncertain disturbances. In addition, considering the specific scenario of underwater uncooperative target tracking in which measurements are limited, a negative matrix modification method in conjunction with a regularized Newton-Gauss iteration technique is proposed for both the time update and the measurement update procedures to ensure greater robustness and accuracy of the adaptive two-step filter. The main contributions of this study are summarized as follows:
First, a modified Sage-Husa online noise estimator was developed to simultaneously estimate the uncertain process noise and measurement noise during the underwater uncooperative target tracking procedure;
Second, a negative matrix modification method was utilized in the first step of the ATSF to ensure the time update process was steady. In addition, a regularized Newton-Gauss iteration technique was used in the measurement updating procedure to increase the robustness of the numerical recursion operation;
Finally, an adaptive two-step filter (ATSF) that combines an online noise estimator and a robust numerical recursive technique was used to robustly and accurately track the underwater uncooperative target.
The remainder of this paper is organized as follows. In
Section 2, the problem mentioned above is formulated by introducing the kinematics of the underwater uncooperative target and the bearing-only measurement model. In
Section 3, the principle of nonlinear least-squares estimation and the scheme of the traditional two-step filter for passive underwater uncooperative target tracking are reviewed. The modified Sage-Husa online noise estimator, the negative matrix modification method, the regularized Newton-Gauss iteration technique, and the adaptive two-step filter (ATSF) for underwater uncooperative target tracking are proposed in
Section 4.
Section 5 and
Section 6 present the comprehensive simulation results and a discussion, respectively, to verify the effectiveness of the designed algorithm. Finally, in
Section 7, the conclusions are drawn.
4. Adaptive Two-Step Bearing-Only Underwater Uncooperative Target Tracking
From Algorithm 1 proposed in the former section, two main drawbacks exist in the traditional two-step filter. First, the traditional two-step filter assumes the process noise, and the measurement noise remain unchanged and are known as prior information during the first-step estimation. However, in a real bearing-only underwater uncooperative target tracking scenario, the process noise and the measurement noise are unknown and uncertain. As a result, online noise estimating techniques should be considered to increase the robustness and accuracy of the tracking procedure. In addition, both the first-step and second-step iteration processes have their own drawbacks. As described in References [
23,
24], Equation (14) in the time-update procedure of the first-step estimation sometimes becomes negative. To address this problem, References [
23,
24] provide several numerical and analytical solutions to avoid the negative time-updated extended covariance matrix. Then, in the second step, the original state
is extracted recursively from the estimated extended state
by the Newton-Gauss method. However, the iteration procedure sometimes diverges because, during the bearing-only underwater uncooperative target tracking procedure, the Hessian matrix becomes ill-conditioned.
Considering the uncertainties in the tracking process, various kinds of adaptive tracking techniques have been developed to deal time different uncertain situations. References [
29,
30] considered the irregular sampling time during the measuring procedure, several robust parameter estimating algorithms are proposed. If the target is a far slow-moving object, the measurement intervals can be regarded as regular and adaptive tracking techniques dealing with the uncertain noise can be adopted to enhance the tracking accuracy. Inspired by the former research of References [
23,
24,
25,
27] and considering the application simplicity and the estimation performance, an adaptive two-step bearing-only underwater uncooperative target tracking filter (ATSF) is proposed in this paper. The proposed ATSF consists of three main parts, namely, online noise estimation, first-step negative matrix correction, and the second-step regularized Newton-Gauss iteration. The details are described in this section.
4.1. Modified Sage-Husa Online Noise Estimation
The Sage–Husa online noise estimator was first introduced by Reference [
31] for linear system noise estimation. For the nonlinear tracking system depicted by Equations (1) and (3), the first and second momentum of the uncertain noise at tracking time
can be estimated online using the nonlinear Sage–Husa estimator as follows:
where
and
are the first-order momentum of the process noise and measurement noise, respectively;
is the transfer matrix from tracking time
to
; and
and
are represented as follows:
From Equations (24) to (29), it can be found that the classic nonlinear Sage–Husa online noise estimator must utilize all of the smooth values of the state within a certain tracking period; as a result, the total process is hard to compute. According to Reference [
20], the recursive suboptimal representation of the nonlinear Sage–Husa online noise estimator can be represented as follows:
where
is the filter gain by a designed tracking algorithm,
is the covariance of the estimated state at time k, and
is the innovation represented as the following equation:
The representation of is dependent on different nonlinear tracking algorithms.
For the uncertain noise, the most recent measurement should be given more attentions than the historical data. Therefore, the fading factor
at tracking time
is introduced to Equations (30) to (33) as follows:
with
It can be found in Equation (39) that, when the index is close to 1, the noise calculated by Equations (35) to (38) focuses on the measurements from the total tracking period. On the contrary, if is close to 0, the estimated noise is more focused on the current time innovation. Consequently, by tuning the value of index , the fading factor changes dynamically to affect the online noise estimation performance. By combining Equations (35) to (39), we obtain the modified Sage–Husa online noise estimator to address the uncertain characteristic of the process and measurement noise during the tracking procedure.
4.2. First-Step Negative Matrix Correction
In the first-step time update procedure of the traditional two-step filter described as Equation (15), the matrix subtraction operation sometimes results in an ill-conditioned
, as discovered by References [
23,
24]. Thus, to ensure the positive characteristic of the computed
, a small positive scalar
is introduced to modify the original, as proposed by Reference [
23] as the following:
Although References [
24,
26] utilized a more comprehensive analytical expression to calculate the extended state and its covariance matrix in the first-step time updating step, the continuous differential computation introduces a significant amount of computational complexity. Therefore, we utilize the regularization technique to make
positive.
4.3. Second-Step Regularized Newton-Gauss Iteration
From Equation (21), the Hessian matrix is inverted during the iteration process. However, this inversion diverges when the values of components of the Hessian matrix
are nearly zero, particularly when the target is uncooperative and few measurements can be obtained. Therefore, to ensure the robustness of tracking, a regularization technique is adopted. The implementation of the regularization process in Equation (21) can be represented as the following equation:
where
is the coefficient of the regularization process and the other matrixes have the same definitions as previously mentioned in
Section 3.
From Equation (41), by setting a specific small value , the inversion computation can avoid the ill-conditioned situation and thus ensures a stable second-step iteration while extracting the original state from the extended state .
4.4. Adaptive Two-Step Filter for Bearing-Only Underwater Uncooperative Target Tracking
From Equations (37), (38), (40), and (41), in conjunction with the traditional two-step filter described in
Section 3, the adaptive two-step filter (ATSF) for passive underwater uncooperative target tracking can be summarized as Algorithm 2, as follows:
Algorithm 2: Adaptive two-step filter for bearing-only underwater uncooperative tracking. |
- 1:
Considering the system model depicted by Equations (1) and (3), initialize the state and the covariance matrix ; - 2:
Set up the initial extended sate and extended covariance matrix according to Equation (8); - 3:
Estimate the process noise and measurement noise online by Equations (37) and (38); - 4:
Estimate the extended state and the extended covariance matrix at the th tracking step by the first-step estimation depicted by Equations (12) to (14), Equation (40), and Equations (16) to (18); - 5:
Extract the original state from the extended state by the second-step estimation depicted by Equations (41), (22), and (23);
- 6:
Calculate the iteration difference , and check whether is less than the preset threshold for stopping the iteration. If is larger than the preset threshold, the iteration for the second step continues; otherwise stop the iteration of the second step;
- 7:
Time propagation to run the whole algorithm at tracking time . Then, jump to step (3).
|
Compared to Algorithm 1 proposed in
Section 3, it is clear that the modified step (3) in Algorithm 2 enables the online noise estimation process to increase the accuracy of the tracking procedure. In addition, the modified steps (4) and (5) in Algorithm 2 ensure that the two-step filter has a robust estimation performance. Hence, the proposed ATSF can increase the robustness and accuracy of the passive underwater uncooperative target tracking process.