A LSTM-RNN-Assisted Vector Tracking Loop for Signal Outage Bridging

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Introduction
Currently, users are capable of obtaining accurate positioning, navigation, and timing (PNT) information from a handheld or chip-scale Global Navigation Satellite System (GNSS) receiver [1][2][3]. Smartphones, smartwatches, or shared bicycles are all equipped with a chip-scale GNSS receiver for acquiring continuous PNT information. By its nature, the GNSS is a satellite-based navigation system, and the receivers receive the transmitted signals from the navigation satellites in space orbit covering the earth. A GNSS receiver usually consists of three major functional blocks (shown in Figure 1), which are signal acquisition, signal tracking, and navigation solution estimation [3][4][5]. The working flow of a typical GNSS receiver can be illustrated as follows: firstly, Radio Front (RF) receives the signal and then samples and downconverts the high-frequency navigation satellite signals for further processing in the later blocks; secondly, signal acquisition accomplishes coarse code and frequency estimation, and the results are converted to tracking loop component for initializing the signal tracking operation; thirdly, signal tracking is responsible for locking the signal and generating code and frequency measurements for navigation solution (position and navigation) determination in the final block. Due to the existence of the clock bias and drift, at least four satellites are necessary for the navigation solution estimation.
In a receiver, signal tracking refers to the code tracking and carrier signal tracking. Code tracking generates the pseudorange measurements for position estimation. Meanwhile, the carrier tracking generates the pseudorange rate measurements for velocity estimation. Commonly, a Least Square Method (LSM) or Kalman filter is employed for accomplishing the navigation solution estimation [3][4][5]. As aforementioned, in the above architecture, each channel of the receiver carries out the signal tracking loop individually, and the navigation solutions are not fed back to aid the signal tracking. This architecture is termed as Scalar Tracking Loop (STL), and this scheme has the following characteristics: each channel accomplishes tracking its corresponding satellite signal individually, no information exchanging happens among different channels, no information is fed back to the signal tracking or signal acquisition block, and the three consecutive blocks are comparatively independent. Basically, the STL ignores the inherent relationship between navigation solutions and each channel's signal tracking parameters (position information is determined with code tracking results, and the velocity is determined with the carrier tracking results). By its nature, all channels share common navigation solutions; this characteristic might be helpful for signal acquisition and tracking [1][2][3][4][5].
Different from the STL, Vector Tracking Loop (VTL) utilizes a center navigation filter (usually a Kalman filter) to process all the channels tracking together and generate navigation solutions. The architecture of the VTL is presented in Figure 2, navigation solutions are fed back to signal tracking for generating the local signal. Then, the local signal replica is mixed with the received signal for generating measurements for navigation solution updating. VTL is commonly composed of a Vector Delay Lock Loop (VDLL) and a Vector Frequency Lock Loop (VFLL). A VDLL was firstly proposed by Spilker, and it attracts serious con-cern from researchers since it has superior performance than STL in tracking weak signals [6]. Lately, Lashley et al. implemented a VFLL/VDLL and compared its performance with STL; the results showed that VTL performed 6.2 dB weak signal tracking threshold improvement with eleven satellites in view and 2.4 dB improvement with five satellites available [7][8][9][10]. In addition, Pany and Eissfeller found that VTL could track 10 dB-Hz weaker signals than STL.
Based on the VTL performance analysis, researchers were devoted to enhance the performance of the VTL in a signal challenging environment [11]. According to the past researches, the VTL encountered three major problems influencing the VTL performance. The first problem was the error propagation between channels, which was brought by the centralized processing of all the channels. The second problem was nonlinearity of the measurement equations; usually, the measurement equation was linearized and a Kalman filter was employed for processing it. The third problem was that signal outage or blockage negatively affected the VTL performance. For the first problem, Won and Eissfeller defined a geometry-related range accuracy coefficient, which could be employed for analyzing the interactions between different channels and how these channels shared the common information [12,13]. Channel interaction analysis could be a theoretical basis for designing an adaptive VTL for dealing with the condition that part of the signals was of low quality [12,13]. For the nonlinear problem, Chen designed an adaptive iterative extended Kalman (AIEKF) filter for VTL [12,13]; Xia designed an adaptive robust Cubature Kalman filter (AR-CKF) for enhancing the VTL navigation accuracy [14]. Apart from these methods, Grace proposed multivector tracking receiver architecture, and the positioning accuracy decreased with the collaborative positioning estimation of the multiple receivers [15]. Jiang employed a Chip Scale Atomic Clock (CSAC) to support VTL positioning, and the results showed that the CSAC could improve the VTL positioning accuracy [16]. The Dilution of Precision (DOP) values decreased with the CSAC as the local time reference [17].
Despite these efforts, a momentary signal blockage will inhibit VTL which generated continuously accurate navigation solutions. Limited work has been published on solving      [19]. Different from conventional Recurrent Neural Networks (RNN), Long Short-Term Memory-(LSTM-) RNN was designed with a unique "gate" structure which could make full use of the history information to build the fitting model [20][21][22]. LSTM-RNN performed better performance in the time-series data processing [20][21][22]. In this work, LSTM-RNN was employed to aid the navigation filter and bridge the signal outage. In the training stage, the output information from the discriminators and navigation filter would be adopted as the inputs of the LSTM-RNN. During signal outages, the LSTM-RNN was employed to predict innovative sequence in the navigation filter for aiding navigation solution generation. Performance evaluation for the LSTM-RNN-assisted VTL was carried out by comparing it with the conventional VTL without any assistance. The rest of this paper is organized as follows: Section 2 illustrates the basic architecture and equations of the VTL; the LSTM-RNN is introduced in Section 3, and the LSTM-RNN-assisted VTL is discussed; the LSTM-RNN-assisted VTL simulation and analysis are presented, and the results are discussed in Section 4. Conclusions are given in Section 5. Figure 3 presents the illustrative architecture of employed VTL; it has two essential components: signal tracking and navigation filter. In the VTL, firstly, the discriminators generate measurements for navigation filter, and then, the navigation filter employs the measurements to estimate the navigation solutions; secondly, the navigation solutions are fed back to generate local signal replica, which are mixed with the received signal in the correlators, and then generate the measurements again. The discriminators are placed after the correlators, and the information from the correlators is employed in the discriminators for measurement generation. Subsection 2.1 and Subsection 2.2 give the relationship between discriminators and the navigation filter and the related equations.

Modeling Discriminator Output and Navigation Solutions.
The relationship between position error and the code phase error is described in Figure 4. In this figure, the position error and the code phase error can be modelled using Equation (1): where the φ code,k is the code phase error, ΔX k+1 is the position error vector, the position error refers to the motion between k th position X k and ðk + 1Þ th position X k+1 , and its projection on the line-of-sight vector is corresponding to the code phase increment. In fact, the code phase error is also affected by the local clock bias, and the detailed relationship between the user's position and code phase error is listed as where the c is the light speed, t b,k+1 is the clock bias at ðk + 1Þ th epoch, and ω k+1 is the code phase noise. Specifically, the cos ðθÞ is determined by the relative position of the receiver and the user. Apart from the relationship between position error and code phase error, the velocity and carrier frequency error have an analogous connection. Equations (4) and (5) where the Δf k+1 refers to the carrier frequency error at ðk + 1Þ th epoch, t d,k+1 is the local time drift, and the η k+1 is the noise. V k+1 is the velocity vector at ðk + 1Þ th epoch; V k is the estimated velocity vector at k th epoch.

Navigation Filter
Model. As aforementioned, subsection 2.1 illustrates how to utilize the code phase and frequency error estimate of the position and velocity increment. Based on the above analysis, this subsection will give the model of the navigation filter. The navigation filter is usually a Kalman filter, which consists of a state equation and measurement equation.
In the state equation, the position errors, velocity errors, clock bias, and clock drift are the state variables, and the equation is given as where the F k,k+1 is the state transfer matrix and the W is the state noise. More details of the state equation could be found in our previously published paper [3].
Based on the analysis of the relationship between the state variables and the measurements, thus, the measurement equation could be formed as where the Z is the measurements which is composed of the outputs from discriminators; specifically, the code discriminators produce code increments and the carrier frequency discriminators offer frequency errors between epochs. H k is the matrix describing the relationship between the state variables and the measurement information; Λ k is the measurement noise matrix. In addition, more information about how to calculate the above matrix or measurements is revealed in our previous papers [3].

Kalman Filter Implementation in VTL.
As the navigation filter model illustrated in the above subsection, the following equations will give the updating and the details of the Kalman filter in VTL. Commonly, a typical Kalman filter contains a prediction procedure and state updating procedure [3]. The state prediction is as The updating is as follows: where the ΔallX − k+1 is the predicted state vector using the previous state and state transfer matrix; P − k+1 is the variance matrix for the predicted state vector; K k+1 is the gain matrix, which defines the updating weight between the predictions and the new measurements. In addition, the matrix Q and R represent the state matrix and observation noise covariance matrix individually, and the specifications are given as Equations (13)-(15).
where the function Ef⋅g denotes the expectation calculation.

LSTM-RNN-Assisted VTL
LSTM-RNN is a popular variant of basic RNN; firstly, we introduce the components and structure of it [16]. How the LSTM-RNN works will be illustrated and explained in this section. Secondly, the implementation of LSTM-RNN in VTL will be introduced, and the details will be given.

LSTM-RNN.
A classic LSTM-RNN is composed of three "gates," which are "input gate," "output gate," and "forget gate" [16]. These gates have different functions, and they work together for accomplishing the task. Figure 5 gives the structure of the LSTM-RNN unit. As presented, the first gate is the "input gate." The input gate regulates the input data and the update of the state vector. The following two formulas are given to describe the procedure.
where the Wei i and Wei C are the weights, and the b i and b C are the bias; these parameters will be determined in the training procedure. As listed in Figure 5, the new candidate cell stateC t is constructed by the tanh function, then,C t is multiplied with the vector i t , and the multiplication results are added to the cell state for updating.

International Journal of Aerospace Engineering
The second is the "forget gate," which is responsible for regulating the previous information. This operation is usually carried out using a sigmoid function. Assuming the input vector is I t , then the function is given as where, in Equation (18), σð⋅Þ is the sigmoid function, Wei f is the updating weights vector, and b f is the bias vector. The output of this gate f t is a vector that contains numbers between 0 and 1, which represents the keeping degree of the corresponding values. The last gate is the output gate; the output gate is employed for regulating the outputs. As listed in Figure 5, two functions (a sigmoid function and a tanh function) are included in this gate to carry out the operation. This sigmoid function is listed in Equation (19), and the tanh function is listed in Equation (20).
Finally, the cell state output is 3.2. LSTM-RNN-Assisted VTL Implementation. The designed LSTM-RNN-assisted VTL is illustrated in Figure 6. The LSTM-RNN is integrated with the navigation filter; the past innovative sequences are the input to the LSRM-RNN to assist the VTL. Figure 5 is the just a LSTM-RNN unit, and Figure 7 is the LSTM-RNN from time t − 1 to t + 1.
The information exchanging is also illustrated in Figure 7. The innovative sequence of the VTL navigation filter is employed as the inputs of the LSTM-RNN, and the innovation is written as While the signal is normal, the LSTM-RNN is trained for determining some parameters, and then, the LSTM-RNN will output the predicted information to aid the VTL during a signal outage. As listed in Figure 6, each channel has a unique LSTM-RNN which obtains the information from the navigation filter. Considering the computation load and

Simulation and Results
Section 2 introduces the basic formulas and the details of the proposed method. In this section, a simulation will be carried out using a flexible Matlab software receiver. The LSTM-RNN was implemented in Matlab and incorporated with the VTL. An intermediate frequency (IF) data was collected from a hardware signal simulator. Through the GUI of the signal simulator, each satellite could be blocked or opened during the simulation. This section is divided into two subsections: (1) firstly, the setting up of this simulation is given in detail, including the trajectory details, the VTL working flow, and LSTM-RNN setting parameters; (2) secondly, the positioning results and the related analysis are given, including the position error comparison with a common VTL during signal outages. (3) SVM and RBFNN were employed to compare with the LSTM-RNN in terms of the positioning errors. Figure 8 shows the 3D dynamic trajectory. The numbers presented in the figure represent different flying task or motion; the specific meanings of all the numbers are listed in Table 1. There are seven different flying tasks in the trajectory. Then, the trajectory file is input to a GNSS signal hardware simulator; an IF data collector is employed for collecting the GNSS signal. The sampling frequency is 10 MHz, and the IF frequency is 3.02 MHz.

Setting Up.
Before operating the VTL, some initial information is necessary, for instance, position, velocity, signal tracking parameters, and ephemeris. Table 2 lists the acquisition results including the satellites RPN, Doppler, and code offset, which are essential for initializing the VTL. As revealed in Table 2, eight satellites are included in the simulation. Particularly, GPS is included in this simulation for representing the GNSS. In aspects of the employed LSTM-RNN, considering     2  5  5579  5530  3  13  1441  4628  4  26  3024  8266  5  7  4311  3688  6  8  3272  7368  7  25  -2288  8989  8  2  1688  79   6 International Journal of Aerospace Engineering the computation load, the LSTM-RNN is a single-layer structure, the learning rate is 0.01, the training epoch is 100, and the batch size is 30.

LSTM-RNN VTL Results and Analysis.
Firstly, with the proper settings, only two satellites are included in the VTL to estimate the navigation solutions, which aims to evaluate the performance of the LSTM-RNN VTL under the condition that part of the signals is blocked at the beginning of the trajectory. As listed in Table 2, eight satellites are acquired, and we randomly select two satellites (PRN 04 and PRN 08) in this simulation. Figure 9 presents the positioning errors including the comparison between VTL and LSTM-RNN-assisted VTL. As shown in Figure 8 Moreover, Figure 10 reveals the position error comparison for signal outages without any available satellites. Figure 10(a) presents the latitude error; Figures 9(b) and 9(c) plot the longitude and altitude errors. In Figure 10, the LSTM-RNN VTL is compared with a common VTL with available satellites. As plotted in Figure 9, while there are only two satellites employed in VTL, the positioning errors diverge quickly and over 500 m. Therefore, we selected the VTL with enough satellites as the reference. Figure 10 presents the positioning errors with three different signal outages, and each outage continues approximately 20 seconds. In aspects of the latitude errors ( Figure 10(a)), with the LSTM-RNN assistance, the errors still grow during the signal outage. For the outage 1, the maximum errors reach to 15 m; the outage 2 and outage 3 reach to approximately 23 m. In aspects of the longitude errors ( Figure 10 In this simulation, the signal outages all last for 20 seconds, and the results reveal that most of the positioning errors are below 20 m, which is a somewhat satisfying result. As presented in these figures, we can find that the errors of  The implementation of the SVM and RBFNN heavily references the literature [18,19]. The testing trajectory is the same as that presented in Figure 8. The comparison is composed of two parts: outage with two available satellites and outages with no available satellites.
Firstly, the positioning results of the long-time outage with two available satellites are presented in Figure 11. The yellow lines represent the results from the SVM, and the green lines represent that from the RBF-NN. As presented, the LSTM-RNN performs the smallest positioning errors during the testing time. The RBF-NN also performs smaller positioning errors than the SVM.
Secondly, the positioning error comparison results between the three methods during the signal outages are presented in Figure 12. The green lines represented the positioning errors from SVM, the red lines presented the positioning errors from LSTM-RNN, and the blue lines represent that from RBFNN. While the signal is normal, the system worked under the integration model; therefore, only blue lines are visible in Figure 12 (excluding the signal outages). It is obvious that the LSTM-RNN suppresses the positioning errors effectively and obtains the best performance compared with SVM and RBFNN.

Conclusions
In this paper, a LSTM-VTL was investigated for improving the VTL navigation accuracy during signal outages. A dynamic trajectory with signal blockages was employed to evaluate the performance of the proposed LSTM-RNNassisted VTL. With the LSTM-RNN aiding, the position errors were reduced, which supported the conclusion that LSTM-RNN was effective for improving the VTL during signal outages. However, we thought there were still some limitations as follows: (1) The LSTM-RNN should be trained before its deployment in the VTL; the training was time-consuming,     International Journal of Aerospace Engineering which might increase the computation load of the processor (2) As presented in Figure 12, position errors of the LSTM-RNN-aided VTL still diverge over time during a signal outage; longer time signal outage will bring about larger positioning errors In the future, we think the following work is worth further investigation: (1) Better or more approximate neural networks are beneficial for improving the performance of the VTL during a signal outage. It is interesting to investigate some other variants of RNN in this application, for instance, Gated Recurrent Unit (GRU) (2) A filed test is necessary for evaluating the performance of the proposed method, for instance, city canyon, and it is worthwhile for improving positioning accuracy for VTL in signal challenging environments (3) For a gourd vehicle or UAV, there are other sensors equipped, and it is of significance to investigate the integration of VTL with these sensors for more robust positioning performance

Data Availability
The data and code used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare no conflicts of the paper publication.