1 Introduction

Underwater vehicles, such as submarine, torpedoes and exploration ship, often need to locate and track targets during missions. Due to mission requirements, they usually work in passive mode that the observer can remain hidden. Only target angles are measured, which is called bearings-only target tracking [1]. In this case, the mapping between the measured angle and the target position is non-linear.

Several methods have been proposed for bearings-only target tracking, in which nonlinear filtering is the majority. For example, the extended Kalman filter (EKF) linearize the measurement equation using Taylor series expansion. It was mainly utilized in early researches on target tracking [2, 3], having the advantage of simplicity. However, the large estimation error of the first-order Taylor series approximation can make EKF suffer performance degradation [4]. Gordon et al. first proposed the use of particle filter algorithm in target tracking [5]. The particle filter algorithm has the advantage of not being affected by the nonlinearity in the model, but its random sampling leads to excessive calculations and problems such as particle degradation, limiting its practical application [6]. In contrast, deterministic sampling, which helps reduce the computational burden, has been used for unscented Kalman filter (UKF), cubature Kalman filter (CKF), central difference Kalman filter (CDKF), etc. [7,8,9]. And UKF is widely used in target tracking recently. Compared to EKF, the accuracy of UKF is significantly higher under the same conditions, reaching a second-order Taylor series approximation at least. However, if the dimension of the state vector is higher than three, its sample scheme may need some adjustments to overcome the propagation of a non-positive definite covariance matrix. It leads to an increase in estimation error [10].

Recent advances in deep learning have demonstrated its ability to extract the high-level features from raw sensory data. Temporal convolutional network (TCN) is a convolutional architecture designed for sequence signals [11]. It performs better than recurrent architectures like long-short term memory (LSTM) on many sequence model tasks such as Seq. MNIST and Music Nottingham [12]. Moreover, as the universal approximation theorem states, TCN is capable of approximating any continuous function to a reasonable accuracy. It means that the mapping between angle measurement and target dynamics can be fitting by TCN, although its non-linearity is existing.

Despite the above advantages, TCN is not suitable to be used directly in target tracking. Underwater targets maintain a uniform linear motion most of the time, which indicates the relationship between the target position and speed, as well as the relationship between the target positions at different times. But for a TCN, the estimation of the current target state only depends on the input observations, having nothing to do with other factors. Therefore, it is a natural idea to establish a state equation describing the target motion to help TCN utilize the relationship mentioned above, where the key is the identification of the equation parameters. Least squares method (LS) is widely used in parameter identification. It has good adaptability in linear and non-linear systems. However, a large amount of data is required to use it, which brings great computational burden. The recursive least squares method (RLS) proposed later overcomes the above-mentioned shortcomings in an iterative manner while maintaining accuracy.

In this paper, a TCN-based network combining RLS named TCN-RLS is proposed to improve the accuracy and stability of bearings-only target tracking. Main contributions are summarized as follows. 1) A TCN-based network is proposed to fit the mapping between angle measurement and target position. It is capable of estimating the target position with a higher accuracy. 2) The RLS is combined to further improve the accuracy and stability of target tracking, as well as estimate the target speed.

The rest of this paper is organized as follows. The dynamic model of bearings-only target tracking is review in Sect. 2. The TCN-RLS we proposed is illustrated in Sect. 3. In Sect. 4, simulation results for the performance of the TCN-RLS is shown and analyzed, followed by conclusions in Sect. 5.

2 Dynamic Model of Bearings-Only Target Tracking

Underwater targets maintain a constant speed most of the time, and only the case of vehicles moving on a two-dimensional plane is discussed in this paper. Thus the state of the target at time k can be expressed as

$$ {\varvec{X}}_{k} = \left[ {\begin{array}{*{20}c} {x_{k} } & {v_{x} } & {y_{k} } & {v_{y} } \\ \end{array} } \right] $$
(1)

where xk, yk represent target position, and vx, vy represent target velocity. The state equation of the system is defined by

$$ {\varvec{X}}(k + 1) = {\varvec{\varPhi}}{\varvec{X}}(k) + {\varvec{\omega}}(k) $$
(2)

where \({\varvec{\varPhi}}\) represents the state transition matrix, \({\varvec{\omega}}(k)\) represents the system noise with mean 0 and variance Q,

$${\varvec{\varPhi}}= \left[ {\begin{array}{*{20}c} 1 & {\Delta t} & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & {\Delta t} \\ 0 & 0 & 0 & 1 \\ \end{array} } \right] $$
(3)
$$ {\varvec{Q}} = \delta_{q}^{2} \left[ {\begin{array}{*{20}c} {\Delta t^{3} /3} & {\Delta t^{3} /2} & 0 & 0 \\ {\Delta t^{3} /2} & {\Delta t} & 0 & 0 \\ 0 & 0 & {\Delta t^{3} /3} & {\Delta t^{3} /2} \\ 0 & 0 & {\Delta t^{3} /2} & {\Delta t} \\ \end{array} } \right] $$
(4)

where \(\delta_{q}^{2}\) represents the process noise intensity of the system, and \(\Delta t\) represents the interval of sampling.

The measurement equation of the system can be expressed as

$$ {\varvec{Z}}_{k} = f({\varvec{X}}_{k} ) + {\varvec{v}}_{k} $$
(5)

where vk represents the measurement noise with mean 0 and variance R. And the observation function can be defined by

$$ f({\varvec{X}}_{{\varvec{k}}} ) = \arctan [(x_{k} - x_{k}^{(o)} )/(y_{k} - y_{k}^{(o)} )] $$
(6)

where \(x_{k}^{(o)}\) and \(y_{k}^{(o)}\) are the observer position at time k.

3 TCN-Based Network Combining RLS

3.1 TCN-Based Network

A TCN-based network is presented to fit the mapping between the observation sequence and target position. As shown in Fig. 1, the sequence composed of the last 20 target angles measured by the observer is the input, represented as Zk-19, …, Zk-1, Zk, and the target position at the current moment \(\left[ {\begin{array}{*{20}c} {\hat{x}_{k} } & {\hat{y}_{k} } \\ \end{array} } \right]\) is output accordingly.

Unlike traditional convolutional neural networks such as one-dimensional convolution, causal convolutions are used in TCN. It means the output at time k has nothing to do with elements later than time k in the previous layer.

Fig. 1.
figure 1

Proposed TCN-based network.

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In addition, dilated convolutions that achieve an exponentially large receptive are employed in TCN. For the proposed network, the dilated convolution operation F(·) on elements s is expressed as

$$ F(s) = ({\varvec{X}}*_{d} f)(s) = \sum\limits_{i = 0}^{k - 1} {f(i) \cdot {\varvec{X}}_{s - d \cdot i} } $$
(7)

where X represents the input of TCN, f is a filter, \(*_{d}\) presents the dilated convolution operation, d is the dilation factor. k represents the size of filter, and s-d·i indicates the direction of the past.

3.2 Recursive Least Squares

A state equation describing the target motion based on TCN is represented as

$$ {\varvec{Y}}(k) ={\varvec{\varPhi}}(k){\varvec{\theta}} + {\varvec{e}}(k) $$
(8)
$$ {\varvec{Y}}(k) = \left[ {\begin{array}{*{20}c} {\hat{x}_{k} } & {\hat{y}_{k} } \\ \end{array} } \right] $$
(9)
$${\varvec{\varPhi}}(k) = \left[ {\begin{array}{*{20}c} k & 1 \\ \end{array} } \right] $$
(10)
$$ {\varvec{\theta}} = \left[ {\begin{array}{*{20}c} {\hat{v}_{x} } & {\hat{v}_{y} } \\ {\hat{x}_{0} } & {\hat{y}_{0} } \\ \end{array} } \right] $$
(11)

where \(\hat{x}_{k}\) and \(\hat{y}_{k}\) in \({\varvec{\varPhi}}(k)\) are estimated by TCN, \({\varvec{\theta}}\) is the estimated parameter, and e(k) is the error vector. The RLS is used to update parameter in an iterative manner as shown below, then target velocity at time k can be estimated.

$$ {\varvec{K}}(k) = {\varvec{P}}(k){\varvec{\varPhi}}(k) $$
(12)
$$ {\varvec{P}}(k) = {\varvec{P}}(k - 1) - \frac{{{\varvec{P}}(k - 1){\varvec{\varPhi}}(k){\varvec{\varPhi}}^{{\varvec{T}}} (k){\varvec{P}}(k - 1)}}{{1 +{\varvec{\varPhi}}^{{\varvec{T}}} (k){\varvec{P}}(k - 1){\varvec{\varPhi}}(k)}} $$
(13)
$$ \hat{\user2{\theta }}(k) = \hat{\user2{\theta }}(k - 1) + {\varvec{K}}(k)[[\begin{array}{*{20}c} {\hat{x}_{k} } & {\hat{y}_{k} ] -{\varvec{\varPhi}}(k)} \\ \end{array} \hat{\user2{\theta }}(k - 1)] $$
(14)

where \(\hat{\user2{\theta }}(0)\) is selected based on experience, P(0) can be taken as E, the identity matrix.

The target position at time k is finally estimated as

$$ \left[ {\begin{array}{*{20}c} {\hat{x}_{k} } & {\hat{y}_{k} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} k & 1 \\ \end{array} } \right]\hat{\user2{\theta }}(k) $$
(15)

The pseudocode of the TCN-RLS is given hereinafter.

figure a

4 Simulation

4.1 Initial Conditions

The observer takes a zigzag movement, with initial state of [0 m 0 m/s 100 m 10 m/s]. The initial state of the target is set randomly in a certain range to verify the generalization of the algorithm, with x0, y0 ∈ [800 m 1200 m], and vx, vy ∈ [5 m/s 10 m/s] (see Fig. 2).

Target angle β is measured by observer, which contains white noise with a mean value of 0 and a standard deviation of 1°. A total of 80 iterations of observation is taken, with a sampling interval of 10 s.

Fig. 2.
figure 2

An instance of the tracking process.

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4.2 Simulation Results

10000 samples that meet the initial conditions are generated to train the TCN, with the ratio of the training set to the test set being 9:1. 50 samples randomly selected test set are used to compare the tracking performance of TCN-RLS with EKF and UKF. The initial estimation of the traditional algorithm is set to be [1000 m 7.5 m/s 1000 m 7.5 m/s], which is the expected value of the initial target state. Since TCN does not provide its estimation until 20th iteration, only the 20th to 80th iterations of simulation results are counted. The root mean square error (RMSE) is utilized to represent the tracking error, which can be expressed as

$$ RMSE_{position} = \sqrt {\frac{1}{k}\sum\limits_{i = 1}^{k} {[(x_{i} - \hat{x}_{i} )^{2} + (y_{i} - \hat{y}_{i} )^{2} ]} } $$
(16)
$$ RMSE_{velocity} = \sqrt {\frac{1}{k}\sum\limits_{i = 1}^{k} {[(v_{x} - \hat{v}_{xi} )^{2} + (v_{y} - \hat{v}_{yi} )^{2} ]} } $$
(17)

The estimation errors for an instance of tracking with TCN-RLS, EKF, and UKF are shown in Figs. 3 and 4. EKF has the largest estimation error due to the excessive linearization error, while UKF maintains a relatively small estimation error. Compared to these traditional methods, the TCN-RLS has the smallest estimation error during the entire tracking process. After 50 Monte Carlo experiments, the average RMSEposition of EKF, UKF, and TCN-RLS are calculated to be 654.5, 360.1, and 217.4, respectively, and RMSEvelocity are 1.8, 1.3, and 1.2, respectively, which supports the analysis mentioned above.

The estimation of the current target state provided by TCN only depends on the input observations, which may cause large fluctuations in the estimation sequence. And RLS is utilized to establish a state equation describing the relationship between the target position and speed, as well as the relationship between the target positions at different times.

Fig. 3.
figure 3

Comparison of the RMSEposition for TCN-RLS, EKF and UKF.

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Fig. 4.
figure 4

Comparison of the RMSEvelocity for TCN-RLS, EKF and UKF.

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As shown in Fig. 5, the target position sequence estimated by TCN-RLS forms almost a straight line,with points remains basically the same distance in it. While the target position sequence estimated by TCN fluctuates greatly as predicted. Compared with the single TCN, the target position sequence estimated by TCN-RLS is more stable and close to the true value.

Fig. 5.
figure 5

Estimated target position of the algorithm before and after adding RLS in one instance.

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5 Conclusions

In this paper, an improved TCN-based network combining RLS is presented for bearings-only target tracking. A TCN-based network is first proposed to fit the mapping between the observation sequence and target position, and the RLS is utilized to improve the stability and accuracy during target tracking. Simulation results show that the TCN-RLS has the smallest estimation error during the entire tracking process, compared to traditional EKF and UKF.