A Study on a Novel Inverted Ultra-Short Baseline Positioning System and Phase Difference Estimation

Abstract

Autonomous underwater vehicles (AUVs) are important tools for exploring and studying the ocean. To improve the stealthiness of AUV positioning, a new type of inverted ultra-short baseline (i-USBL) positioning system is proposed in this paper. The system uses a surface GPS buoy as a positioning base station to transmit positioning signals to the AUV, which receives the signals through the i-USBL and calculates its own coordinates. In this way, the power consumption of the AUV will be lower, improving its endurance. In addition, azimuth observation is one of the key observation quantities in ultra-short baseline positioning systems. Therefore, this paper derives two phase difference estimation algorithms for azimuth measurement, which are the Least Mean Square (LMS) algorithm and the Discrete Fourier Transform (DFT) algorithm. Simulation results show that when the signal-to-noise ratio (SNR) is less than or equal to 0 dB, the DFT algorithm has higher accuracy; when SNR is greater than or equal to 10 dB, LMS has slightly higher accuracy than DFT; however, DFT is more stable than LMS at all SNRs. When SNR is 20 dB, the maximum azimuth measurement error of LMS and DFT is 0.3301° and 0.2204°, respectively, which can meet the positioning accuracy requirements. The average running times of LMS and DFT are 0.085 s and 0.011 s, respectively, and LMS has a faster running speed. In summary, the LMS algorithm can be used for azimuth observation when the SNR is good, and DFT can be applied when the SNR is poor.

1. Introduction

Autonomous underwater vehicles (AUV) are important tools for studying the ocean [1]. Positioning technology is one of the key technologies for AUV underwater operations [2,3]. Electromagnetic waves undergo severe attenuation in water, so the sonic wave is usually used to transmit signals. The underwater acoustic positioning system is divided into long baseline, short baseline, and ultra-short baseline (USBL) based on the baseline length [4]. Among them, USBL has the smallest baseline length, which can be only a few centimeters, making it portable and independent, and it has thus become a research hotspot in the underwater technology field [5].

The conventional USBL is installed under the bottom of a ship. In order to obtain the navigation coordinates of underwater targets, the attitude of USBL also needs to be obtained, usually using attitude sensors [6]. However, it will be affected by large surface noise and ship vibration noise [7]. Therefore, Keith Vickery proposed that the development trend of underwater acoustic positioning systems would be to invert traditional acoustic positioning systems [8] and proposed a theoretical method for an inverted ultra-short baseline (i-USBL). Compared with the previous array installed on the mother ship, this method avoids the noise of the mother ship and can obtain a higher SNR, thus achieving a longer positioning distance [9]. It can meet the practical needs of multiple intelligent underwater vehicles for autonomous operations.

To obtain a higher SNR, the i-USBL system designed in this paper will invert the USBL and install it above the AUV. We provide positioning services for the underwater AUV by setting a GPS buoy on the sea surface. This positioning method can significantly improve the SNR when the USBL receives signals because the ship noise of the AUV is much smaller than that of the surface ship [10]. In addition, the AUV only receives acoustic signals from the GPS buoy and does not emit signals outward, which can greatly improve the concealment of the AUV.

While using USBL for positioning, it is necessary to estimate the phase difference between the two signals to calculate the azimuth. Then, the accuracy of phase difference estimation has a great impact on the positioning result [11,12].

Existing methods such as zero-crossing phase detection [13], digital correlation [14,15], etc., each have their own shortcomings in dynamic phase difference estimation. The zero-crossing phase detection method calculates the phase difference by the time when two signals cross the zero point. The calculation speed is fast, but the hardware cost is high. The digital correlation method estimates the phase difference by the correlation function of two signals, which has a strong noise suppression ability for random noise, but it is easily affected by harmonics, has large fluctuations, and requires synchronous sampling. Reference [16] uses adaptive phase tracking to track the signal phase changes and estimates the phase difference using the difference between the two estimated values of the phase tracker, but it is difficult to estimate dynamic phase difference. Reference [17] uses adaptive delay compensation for the input signal. The compensated signal is aligned with the reference signal, and the delay compensation factor is calculated and corrected under the least mean square error criterion. The phase difference is obtained from the delay compensation factor, but there is bias in the estimation due to the influence of noise. The spectral analysis of discrete Fourier transform (DFT) translates the signal from the time domain to the frequency domain and estimates the phase difference using the phase-frequency relationship. The least mean square (LMS) adaptive algorithm [18] does not require prior knowledge of the signal; it also has simple calculations and is easy to implement. It is widely used in the fields of adaptive filtering, frequency tracking, and delay estimation [19,20,21,22,23].

The difference between the new i-USBL positioning system proposed in this paper and the traditional USBL is that the USBL is inverted and installed above the AUV in this paper. The AUV calculates its own coordinates by receiving the positioning signal sent by the GPS buoy on the sea surface. Therefore, AUV does not need to transmit signals to the outside world. Passive positioning such as water surface can not only greatly reduce the power consumption of AUV but also make AUV more concealed.

The structure of the remaining sections of this paper is as follows. Section 2 provides a detailed explanation of the positioning principle proposed in this paper, as well as the positioning solution process. Section 3 introduces the applications of the LMS algorithm and the DFT algorithm in phase difference calculation. Section 4 presents simulation experiments conducted on the LMS algorithm and the DFT algorithm and compares the estimation accuracy of the two algorithms. Finally, Section 5 concludes this paper.

2. i-USBL Positioning Based on GPS Buoy

2.1. Coordinate System Definition

The conversion between coordinate systems is inseparable from the localization solution process. Therefore, the used coordinate systems are defined in this subsection.

Geodetic coordinate system O_g-X_gY_gZ_g (g-system): Let P be any point on the ellipsoidal plane of the Earth. Taking the Greenwich meridian as the prime meridian, the geodetic longitude L of P is defined as the angle between the meridian plane passing through P and the prime meridian plane, measured positively to the east and negatively to the west with respect to the prime meridian. The angle between the normal to the ellipsoidal plane passing through P and the equatorial plane is defined as the geodetic latitude B of P, with the equator serving as the reference circle and the north being positive and the south being negative. The geodetic height H is the distance from the ground point along the normal to the reference ellipsoid.

Earth-centered, Earth-fixed coordinate system O_n-X_nY_nZ_n (e-system): Taking the center of the Earth’s ellipsoid as the origin, the X-axis points towards the intersection of the prime meridian and the equator. The Z-axis points towards the direction of the Earth’s north pole, with the north being positive. The Y-axis is perpendicular to the XOZ plane and forms a right-handed coordinate system.

Reference coordinate system O_r-X_rY_rZ_r (r-system): In this paper, the East-North-Up coordinate system is chosen as the reference coordinate system. A certain reference point on the sea serves as the acoustic center of the underwater transducer of the GPS float. The East, North, and Up directions are taken as the X, Y, and Z axes, respectively, forming a right-handed coordinate system.

USBL array coordinates system O_u-X_uY_uZ_u (u-system): As shown in Figure 1, the line connecting elements 1 and 3 serves as the X-axis, and the line connecting elements 2 and 4 serves as the Y-axis. The intersection of the X and Y axes is taken as the origin O_u of the coordinate system, which is also the acoustic center of the USBL array. The positive direction of the X-axis points towards element 1, and the positive direction of the Y-axis points towards element 2. The Z-axis is perpendicular to the X and Y axes, forming a right-handed coordinate system with a positive direction pointing upwards.

Vehicle coordinate system O_b-X_bY_bZ_b (b-system): This system is essentially the coordinate system of the gyroscope sensor, which is assumed to be aligned with the direction of motion of the underwater vehicle in this paper. The origin of the vehicle coordinate system is taken as the origin of the u-system, and the X, Y, and Z axes point towards the positive right, positive front, and positive up directions of the underwater vehicle, respectively, forming a right-handed coordinate system.

2.2. Coordinate Calculation

In coordinate transformation, three types of transformations are applied, including conversion from g-system to e-system, conversion from e-system to g-system, and conversion between right-handed rectangular coordinate systems. Therefore, the first three sections introduce coordinate transformations, followed by an analysis of the positioning solution in this paper.

2.2.1. Conversion from g-System to e-System

This paper does not provide a detailed derivation of the transformation. The conversion from g-system (B, L, H) to e-system (X, Y, Z) can be directly performed using the following equation [24]:

$$\left[\begin{array}{l}X\hfill \\ Y\hfill \\ Z\hfill \end{array}\right]=\left[\begin{array}{c}(N+H)\cos B\cos L\\ (N+H)\cos B\sin L\\ \left(N\left(1-e^2\right)+H\right)\sin B\end{array}\right].$$

(1)

In this equation, B, L, and H represent geodetic longitude, geodetic latitude, and geodetic height, respectively. $N=\frac{a}{\sqrt{1-e^2{\sin }^{2}B}}$ is the radius of the prime vertical circle, and $e^2$ is the square of the first eccentricity of the reference ellipsoid.

2.2.2. Conversion from e-System to g-System

To transform from the e-system (X, Y, Z) to g-system (B, L, H), the following formula can be used [25]:

$$\left\{\begin{array}{l}L=\arctan \left(\frac{Y}{X}\right)\hfill \\ B=\arctan \left(\frac{Z+{e^{\prime }}^{2}b{\sin }^3\theta }{\sqrt{X^2+Y^2}-e^{2}a{\cos }^3\theta }\right)\hfill \\ H=\frac{\sqrt{X^2+Y^2}}{\cos B}-N\hfill \end{array}\right.,$$

(2)

where $\theta =\arctan \left(\frac{Z\cdot a}{\sqrt{X^2+Y^2}\cdot b}\right)$, and ${e^{\prime }}^2$ represents the square of the second eccentricity.

To verify the correctness of the coordinate transformation, a MATLAB simulation is conducted as follows: first, the g-system (B, L, H) coordinates are transformed into the e-system (X, Y, Z) using Equation (1); then, the e-system (X, Y, Z) coordinates are transformed back to the g-system (B2, L2, H2) using Equation (2). The correctness of the coordinate transformation is evaluated by comparing the difference between (B, L, H) and (B2, L2, H2). The MATLAB simulation shows that when (B, L, H) = (117.1°, 39.1°, 1000 m), the error vector is [0, −3.6 × 10⁻¹⁰, −1.6 × 10⁻⁵], which indicates a very small error and confirms the correctness of the coordinate transformation.

2.2.3. Transformation among Right-Handed Coordinate Systems

The rotation transformation of the spatial rectangular coordinate system can be regarded as an extension of the transformation of the plane rectangular coordinate system. Not only does the rotation angle needs to be determined, but also the rotation axis. The following derivation presents the rotation formula among the spatial rectangular coordinate systems based on the basic rotation formula of the three axes.

According to the rotation formula among plane rectangular coordinate systems, the rotation angle $\alpha$ around the Z-axis can be obtained as follows:

$$\begin{array}{l}{x}^{\prime }=x\cos \alpha -y\sin \alpha \hfill \\ y^{\prime }=x\sin \alpha +y\cos \alpha \hfill \\ z^{\prime }=z\hfill \end{array}.$$

(3)

The rotation angle $\beta$ around the Y-axis:

$$\begin{array}{l}{z}^{\prime }=z\cos \beta -x\sin \beta \hfill \\ x^{\prime }=z\sin \beta +x\cos \beta \hfill \\ y^{\prime }=y\hfill \end{array}.$$

(4)

The rotation angle $\gamma$ around the X-axis:

$$\begin{array}{l}{y}^{\prime }=y\cos \gamma -z\sin \gamma \hfill \\ z^{\prime }=y\sin \gamma +z\cos \gamma \hfill \\ x^{\prime }=x\hfill \end{array}.$$

(5)

Therefore, the rotation matrix between two spatial coordinate systems can be obtained as follows:

$$\left[\begin{array}{c}{X}^{\prime }\\ Y^{\prime }\\ Z^{\prime }\end{array}\right]=\left[\begin{array}{ccc}{\mathrm{c}}_1{\mathrm{c}}_2& {\mathrm{c}}_1{\mathrm{s}}_2{\mathrm{s}}_3-{\mathrm{s}}_1{\mathrm{c}}_3& {\mathrm{c}}_1{\mathrm{s}}_2{\mathrm{c}}_3+{\mathrm{s}}_1{\mathrm{s}}_3\\ {\mathrm{s}}_1{\mathrm{c}}_2& {\mathrm{s}}_1{\mathrm{s}}_2{\mathrm{s}}_3+{\mathrm{c}}_1{\mathrm{c}}_3& {\mathrm{s}}_1{\mathrm{s}}_2{\mathrm{c}}_3-{\mathrm{c}}_1{\mathrm{s}}_3\\ -{\mathrm{s}}_2& {\mathrm{c}}_2{\mathrm{s}}_3& {\mathrm{c}}_2{\mathrm{c}}_3\end{array}\right]\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right].$$

(6)

Equation (6) is a 3 × 3 rotation matrix $R$. The subscripts 1, 2, and 3 correspond to $\alpha$, $\beta$, and $\gamma$ respectively. $\mathrm{c}$ and $\mathrm{s}$ represent $\cos$ and $\sin$, where ${\mathrm{c}}_1$ represents $\cos \alpha$. The calculation of the rotation matrix $R$ depends on the chosen rotation method and the order of coordinate axis rotation. In this paper, the rotation mode is intrinsic rotations, and the order of coordinate axis rotation is z-axis ($\alpha$) → y-axis ($\beta$) → x-axis ($\gamma$).

2.2.4. Conversion from r-System to e-System

The transformation between the r-system and the e-system is essentially a transformation between right-handed coordinate systems. To convert from the r-system to the e-system, a rotation is first performed around the X-axis by an angle of $(90°-B)$, followed by a rotation around the Z-axis by an angle of $(90°+L)$, using the outer rotation convention. Here, L and B are the geodetic longitude and latitude of the origin of the reference coordinate system, respectively. After the rotation, the resulting coordinates differ from the Earth-centered, Earth-fixed coordinates via a translation, which is simply the addition of the e-system coordinates of the origin of the r-system. The e-system coordinates can be calculated using the equation in Section 2.2.1. Therefore, the transformation can be performed using the following equation:

$$\left[\begin{array}{l}{X}_e\hfill \\ Y_e\hfill \\ Z_e\hfill \end{array}\right]=\left[\begin{array}{ccc}-\sin L& -\sin B\cos L& \cos B\cos L\\ \cos L& -\sin B\sin L& \cos B\sin L\\ 0& \cos B& \sin B\end{array}\right]\cdot \left[\begin{array}{l}{X}_r\hfill \\ Y_r\hfill \\ Z_r\hfill \end{array}\right]+\left[\begin{array}{c}{X}_0\\ Y_0\\ Z_0\end{array}\right].$$

(7)

2.2.5. i-USBL Positioning Principle

Figure 2 shows the principle diagram of the i-USBL positioning system. Point A in Figure 2 represents the underwater transducers carried by the GPS buoy, while O_u denotes the acoustic center of the USBL transducer carried by the underwater vehicle. ${\theta }_x$, ${\theta }_y$, and ${\theta }_z$ represent the angles between the range R and the X_u, Y_u, and Z_u axes, respectively. By measuring the time difference of arrival between the first and third transducers and between the second and fourth transducers, as well as the range R, the coordinates of point A in the u-system can be calculated.

As shown in Figure 3. Since the slant range R is much greater than the distance d between elements 1 and 3, the received signals can be considered parallel to each other. Therefore, the relationship between the angle of the signal and the X_u axis of the u-system can be obtained:

$$\cos {\theta }_x=\frac{\lambda {\phi }_x}{2\pi d}$$

(8)

$$\cos {\theta }_y=\frac{\lambda {\phi }_y}{2\pi d},$$

(9)

where ${\phi }_x$ is the phase difference of the received signal between element 1 and element 3, and ${\phi }_y$ is the phase difference of the received signal between element 2 and element 4. $c$ is the speed of sound in seawater, and $d$ is the distance of the elements.

The relationship between the angle ${\theta }_z$ of the signal and the Z-axis of the u-system is given by:

$$\cos {\theta }_z=\sqrt{1-{\cos }^2{\theta }_x-{\cos }^2{\theta }_y}.$$

(10)

Hence, the coordinates of point A_u in the u-system can be expressed as:

$$A_u=\left[\begin{array}{l}{x}_u\hfill \\ y_u\hfill \\ z_u\hfill \end{array}\right]=\left[\begin{array}{c}R\cos {\theta }_x\\ R\cos {\theta }_y\\ R\cos {\theta }_z\end{array}\right].$$

(11)

The slant range R in Equation (11) can be obtained by measuring the one-way travel time of the signal. The signal contains the time of transmission $T$, and the time of arrival at the origin point O_u can be obtained by averaging the times of receiving the signal from the four elements, denoted as $To$. Thus, the slant range R can be calculated as $R=c(To-T)$.

In this paper, the rotation matrix $R_A^B$ is defined as the transformation matrix from coordinate A-system to coordinate B-system, e.g., $R_u^b$ is the rotation matrix from the u-system to the b-system. As shown in Figure 2, there exists a mounting angle error between the u-system and the b-system, which requires calibration [26]. The essence of the calibration is to obtain the rotation matrix $R_u^b$. Since there is no displacement between the two coordinate systems, the carrier coordinates of point A can be expressed as:

$$A_b=R_u^b{A}_u.$$

(12)

Assume that the reference coordinates of point A are denoted as $A_r$, and the reference coordinates of the USBL acoustic center O_u are denoted as $B_r$; then, the reference coordinates $B_r$ relative to $A_r$ can be expressed as:

$$T_r=A_r-R_b^r{A}_b=-R_b^r{A}_b.$$

(13)

Finally, the coordinates can be converted into geodetic coordinates based on the procedures outlined in Section 2.2.2 and Section 2.2.3.

The above process constitutes the complete solution process of the i-USBL positioning system. Throughout the entire process, only the surface GPS buoy and the underwater vehicle are required. The positioning calculation is performed on the underwater vehicle, enabling real-time and fast positioning of the underwater vehicle.

3. Phase Difference Estimation Algorithm

3.1. Least Mean Square Error Adaptive Phase Difference Estimation Algorithm

The essence of the LMS algorithm is to perform an orthogonal transformation on a signal to generate a new signal that matches another signal, and then adaptively update the weight coefficients based on the least mean square error to achieve an adaptive estimation of phase difference. The algorithm will be described in detail below.

From a certain perspective, the adaptive filtering algorithm can also be referred to as a performance surface search method. On the performance surface, it finds the optimal solution by continuously measuring whether the measurement points are close to the target value. Currently, one of the widely used surface functions is the mean square error (MSE) function, which is expressed as follows:

$$f\left(e\left(k\right)\right)=\zeta (k)\triangleq E\left\{|e(k){|}^2\right\}.$$

(14)

Equation (14) is designed to find the minimum value of the mean square error (MMSE) criterion function, which is used to derive the Wiener filter. The formula is given by:

$$\zeta =E\left\{d{\left(k\right)}^2\right\}+w^{T}Rw-2p^{T}w.$$

(15)

From this equation, it can be seen that the mean square error and the weight vector of the filter have a quadratic relationship. The introduction of the mean square error surface is to describe the mapping relationship of the function, and the quadratic function corresponding to the weight vector is a double parabolic surface. According to the MMSE criterion and the mean square error surface, at each time, the weight vector $w$ surface is updated along the projection direction of the steep descent of the mean square error, that is, the anti-gradient vector ζ of the objective function is iteratively updated. Since there is only one unique minimum value on the mean square error performance surface, as long as the convergence step is properly selected, no matter where the initial weight vector is, it can converge to a small point on the error surface or in its neighborhood. This method of solving the minimization problem along the opposite direction of the objective function gradient is generally called the fast descent method, and its expression is as follows:

$$w\left(k+1\right)=w\left(k\right)+\frac{1}{2}\mu (-{\widehat{\nabla }}_k),$$

(16)

where ${\widehat{\nabla }}_k$:

$$\begin{array}{ll}{\widehat{\nabla }}_k& =\frac{\partial }{\partial w_k}{\left|{\epsilon }_k\right|}^2\\ & =\frac{\partial }{\partial w_k}\left[{\left|d_k\right|}^2+w_k^H{x}_k^{*}{x}_k^T{w}_k-2\mathrm{Re}\left(d_k^{*}{x}_k^T{w}_k\right)\right]\\ & =2x_k^{*}{x}_k^T{w}_k-2d_k{X}_k^{*}\\ & =2x_k^{*}\left(x_k^T{w}_k-d_k\right)\\ & =-2{\epsilon }_k{x}_k^{*}\end{array}.$$

(17)

The complete expression of the least-mean-square (LMS) adaptive filtering algorithm based on the stochastic gradient algorithm is given as follows:

$$\left\{\begin{array}{c}y\left(k\right)\hspace{0.33em}=w^T\left(k\right)x(k)\\ \epsilon \left(k\right)=d\left(k\right)-y(k)\\ w\left(k+1\right)=w\left(k\right)+\mu \epsilon \left(k\right)x(k)\end{array}\right.,$$

(18)

where $\omega (k)$ represents the weight coefficient, $y(k)$ represents the calculated estimated signal, $x(k)$ represents the signal to be recalculated, $d(k)$ represents the desired signal, and $\epsilon (k)$ represents the error value between the desired signal and the estimated signal.

Suppose the sampled sequences of two identical sinusoidal signals with the same frequency are $s_1(k)$ and $s_2(k)$, then

$$\left\{\begin{array}{c}{s}_1(k)=A\sin \left(w_{0}k+{\theta }_1\right)+n_1(k)\\ s_2(k)=A\sin \left(w_{0}k+{\theta }_2\right)+n_2(k)\end{array}\right.\hspace{1em}k=1,2,\cdots ,N,$$

(19)

where $A$ represents the amplitude of the signal, ${\theta }_1$ and ${\theta }_2$ are the initial phases of the signal, and ${\omega }_0=2\pi f_0/f_s$ is the angular frequency, where $f_0$ and $f_s$ are the frequency of the sine wave signal and the signal sampling rate, respectively. $n_1(k)$ and $n_2(k)$ are Gaussian white noise, and N is the sampling length. Since there is only a phase difference between $s_1(k)$ and $s_2(k)$, according to trigonometry, one signal can be estimated through its orthogonality with another signal. Without loss of generality, let $s_2^{\prime }\left(k\right)=A\cos \left(w_{0}k+{\theta }_2\right)+n_2^{\prime }(k)$ be the orthogonality of $s_2(k)$, where $n_2^{\prime }(k)$ is the phase shift of noise $n_1(k)$. Then, the estimation of $s_1(k)$ and the estimation error can be obtained as follows:

$${\widehat{s}}_1(k)=w_s(k)s_2(k)+w_c(k)s_2^{\prime }(k)$$

(20)

$$\epsilon \left(k\right)=A\sin \left({\omega }_{0}k+{\theta }_1\right)-{\widehat{s}}_1(k)+n_3(k),$$

(21)

where $n_3\left(k\right)=n_1\left(k\right)-w_s(k)n_2\left(k\right)-w_c\left(k\right)n_2^{\prime }(k)$. The phase difference $\Delta \theta$ information is contained in the coefficients $w_s(k)$ and $w_c(k)$. The LMS algorithm is used to update $w_s(k)$ and $w_c(k)$ adaptively. When the mean square error $E[{\epsilon }^2(k)]\to \min$, the phase difference $\Delta \theta$ can be obtained by solving it. The design of the phase difference adaptive estimator is shown in Figure 4.

The Hilbert transform is applied to $s_2(k)$ to obtain the analytic signal, in which the real and imaginary parts are $s_2(k)$ and $s_2^{\prime }(k)$, respectively. By applying Equation (20) to calculate ${\widehat{s}}_1(k)$, and using the least mean square algorithm to adaptively update $w_s$ and $w_c$, the phase difference can be obtained:

$$\Delta \theta ={\theta }_1-{\theta }_2=\arctan \left(w_c/w_s\right).$$

(22)

The adaptive updating of $w_s$ and $w_c$ can be achieved by the LMS algorithm, and the specific formula is given as follows:

$$\left\{\begin{array}{l}{w}_s(k+1)=w_s(k)-\frac{1}{2}\mu \cdot \frac{\partial {\epsilon }^2(k)}{\partial w_s}=w_s(k)+\mu \epsilon (k)s_2(k)\hfill \\ w_c(k+1)=w_c(k)-\frac{1}{2}\mu \cdot \frac{\partial {\epsilon }^2(k)}{\partial w_c}=w_c(k)+\mu \epsilon (k)s_2^{\prime }(k)\hfill \end{array}\right.,$$

(23)

where $\mu$ is the step size of a single iteration.

3.2. Discrete Fourier Transform Solution Phase Difference

The phase difference of two measured sinusoidal signals with the same frequency can be calculated using the discrete Fourier transform (DFT). Specifically, the two measured sinusoidal signals are first sampled to obtain two discrete sequences, and then the DFT is applied to each sequence to obtain their spectra. The maximum amplitude value of each spectrum is located, and the corresponding spectral lines are found. The initial phase values of the two sequences are then calculated separately using the real and imaginary parts of the values of the corresponding spectral lines. Finally, the phase difference between the two measured signals can be obtained by subtracting the two initial phase values. The specific derivation process is shown as follows.

Suppose there are two sinusoidal signals:

$$Y_1(t)=\sin (2\pi ft+{\theta }_1)$$

(24)

$$Y_2(t)=\sin (2\pi ft+{\theta }_2),$$

(25)

where $t$ represents the time sequence, $f$ is the frequency of the sine wave, ${\theta }_1$ and ${\theta }_2$ are the initial phases of $Y_1(t)$ and $Y_2(t)$, respectively.

Under ideal sampling conditions (i.e., whole-cycle sampling), the frequency of the sine wave exactly falls on the N equal divisions of the sampling frequency, that is, $f=m{f}_s/N$, where m takes any positive integer value in 0, 1, 2, …, N/2.

After frequency extraction of the above $Y_1(t)$ and $Y_2(t)$, two discrete-time sequences can be obtained, namely:

$$Y_1(n)=\sin (2\pi fn{T}_s+{\theta }_1)=\sin (2\pi nf/f_s+{\theta }_1)=\sin (2\pi mn/N+{\theta }_1)$$

(26)

$$Y_2(n)=\sin (2\pi mn/N+{\theta }_2).$$

(27)

The N-point discrete Fourier transform (DFT) operation of $Y_1(t)$ is defined as follows:

$$\begin{array}{ll}{Y}_1(k)& =DFT[Y_1(n)]\\ & ={\displaystyle \sum _{k=0}^{N-1}{Y}_1(n)W_N^{kn}}\\ & ={\displaystyle \sum _{k=0}^{N-1}\sin (\frac{2\pi mn}{N})}{e}^{-j\frac{2\pi kn}{N}}\\ & =\frac{j}{2}{e}^{j{\theta }_1}{\displaystyle \sum _{k=0}^{N-1}\left\{e^{-j2\pi (m+k)n/N}-e^{j2\pi (m-k)n/N}\right\}}\end{array},$$

(28)

where k = 0, 1, …, N − 1.

According to the orthogonality of complex exponential periodic sequences, it can be observed that $Y_1(t)=0$ when $k\ne m$. $Y_1(t)\ne 0$ only when $k=m$:

$$Y_1(k)=-\frac{1}{2}Nj\cdot e^{j{\theta }_1}.$$

(29)

Therefore, for the sine sequence $Y_1(k)$, its initial phase is given by

$${\theta }_1=\arctan \frac{I_m\left[Y_1(m)\right]}{R_e\left[Y_1(m)\right]},$$

(30)

and for the sine sequence $Y_2(k)$, its initial phase is given by:

$${\theta }_2=\arctan \frac{I_m\left[Y_2(m)\right]}{R_e\left[Y_2(m)\right]}.$$

(31)

Hence, the phase difference $\Delta \theta$ between the loop $Y_1(t)$ and $Y_2(t)$ can be expressed as:

$$\Delta \theta ={\theta }_2-{\theta }_1=\arctan \frac{I_m\left[Y_1(m)\right]}{R_e\left[Y_1(m)\right]}-\arctan \frac{I_m\left[Y_2(m)\right]}{R_e\left[Y_2(m)\right]}.$$

(32)

The above deduction shows that when $k=m$, the spectral values of $Y_1(t)$ and $Y_2(t)$ reach their maximum values. According to Equations (30) and (31), the initial phase of the two tested sine signals can be calculated, and the phase difference $\Delta \theta$ between them can be obtained by subtracting the two initial phase values using Equation (32). Moreover, when frequency stability and noise are not considered, the phase values detected by the DFT method based on ideal sampling are very accurate. In this case, the calculated signal phase is correct.

4. Simulation

In the positioning process of the i-USBL system, the estimation of the azimuth is the key observation, which is achieved by measuring the phase difference between different signals. Therefore, a high-precision phase estimation algorithm is required. To intuitively see the impact of phase difference on practical applications, it is necessary to convert the phase difference into the corresponding equivalent distance difference (EDD) $\Delta d$ in seawater, which is done using the following equation:

$$\Delta d=\frac{\Delta \theta }{2\pi f_0}c,$$

(33)

where $\Delta \theta$ is the estimated value of phase difference in radians, $f_0$ is the center frequency of the estimated sinusoidal signal, and $c$ is the sound velocity in water, and the value in simulation is 1500 m/s.

To evaluate the precision and accuracy of the method, this paper performs statistical analysis on the errors of 1000 simulation experiments, calculating the mean error (ME), standard deviation (STD), and root-mean-square error (RMSE) of each method:

$$\mathrm{ME}=\frac{1}{N}{\displaystyle \sum _{i=1}^N{x}_i-x}$$

(34)

$$\mathrm{STD}=\sqrt{\frac{1}{N-1}{\displaystyle \sum _{i=1}^N{\left(x_i-\overline{x}\right)}^2}}$$

(35)

$$\mathrm{RMSE}=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(x_i-x\right)}^2}},$$

(36)

where $x_i$ represents the estimated equivalent range difference in the ith simulation experiment under the same experimental conditions, $x$ represents the true value, $N$ represents the number of Monte Carlo simulations, which is 1000 in this section, and $\overline{x}$ represents the mean value of all simulation results.

As shown in

Table 1. Basic experimental parameters.

Physical Quantity	Value
Signal sampling rate	192 kHz
Sound velocity in water	1500 m/s
Element spacing d	0.5 m

, the basic experimental parameters in this section are set accordingly.

4.1. LMS Phase Difference Estimation Experiment

In this section, MATLAB is used to verify the LMS phase difference calculation. Monte Carlo simulations are conducted to estimate the phase difference using the LMS algorithm 1000 times under the same parameter conditions. In the simulation, the frequency of the sine signal is set to 30 kHz, and signals with phase differences of −60°, −45°, −30°, 30°, 45°, and 60° are estimated for each of the two signals. To consider the errors under different SNRs, noise with the following SNRs is added to the two signals: −30 dB, −20 dB, −10 dB, 0 dB, 10 dB, 20 dB, 30 dB, 40 dB, 50 dB, and 60 dB.

As shown in

Table 2. The ME (10⁻⁶ m) of EDD estimated by LMS in different SNRs when estimating different EDDs.

SNR	$\mathbf{\Delta }\mathit{\theta }$
	−60°	−45°	−30°	30°	45°	60°
−30 dB	−22,020.4716	−19,513.5927	−16,605.3267	−7911.2563	−5809.8387	−4377.2037
−20 dB	−20,221.8419	−17,930.1896	−15,061.2322	−7663.7161	−5858.5711	−3800.4058
−10 dB	−1396.8126	−369.4449	42.2249	−6.9370	−70.5884	59.8000
0 dB	0.6834	3.9036	−0.2880	−2.2440	−2.7233	5.1642
10 dB	0.2451	0.3378	1.6742	−1.7330	−0.3106	−1.1913
20 dB	−0.2872	0.2270	0.0065	0.0060	−0.0437	−0.1105
30 dB	−0.0598	0.0068	−0.1171	−0.1001	−0.0618	−0.0056
40 dB	−0.0474	−0.0681	−0.0582	−0.0553	−0.0230	−0.0260
50 dB	−0.0427	−0.0422	−0.0473	0.0001	−0.0043	0.0224
60 dB	−0.0426	−0.0476	−0.0430	−0.0117	−0.0001	0.0139

, the ME of LMS algorithm in estimating equivalent range difference under different experimental settings is presented. It can be seen that when the SNR is negative, the value of the ME is very large, with six of them reaching the centimeter level, indicating very low accuracy. When the SNR is greater than or equal to 30 dB, the ME is generally maintained at the order of magnitude of ${10}^{-8}$, with only one case reaching the order of magnitude of ${10}^{-10}$, indicating that the accuracy of LMS algorithm remains almost unchanged when the SNR is greater than or equal to 30 dB.

Table 3. The STD (10⁻⁶ m) of EDD estimated by LMS in different SNRs when estimating different EDDs.

SNR	$\mathbf{\Delta }\mathit{\theta }$
	−60°	−45°	−30°	30°	45°	60°
−30 dB	13,818.3860	14,472.6070	14,004.6620	14,547.1930	13,985.9700	14,416.3330
−20 dB	14,994.4710	14,838.9680	15,052.2990	14,063.1170	13,845.3150	13,808.4440
−10 dB	7465.8710	3719.9010	2587.6040	2155.4610	2092.4350	2187.2590
0 dB	139.0160	139.5840	136.1900	142.3900	135.7840	140.1170
10 dB	27.0330	27.1680	27.9970	26.9830	27.3990	26.9530
20 dB	8.2020	8.3290	8.0870	8.1470	8.3230	7.8240
30 dB	2.5650	2.5710	2.5730	2.6580	2.5460	2.6260
40 dB	0.7830	0.8530	0.8480	0.8010	0.8340	0.8300
50 dB	0.2690	0.2620	0.2570	0.2530	0.2530	0.2670
60 dB	0.0820	0.0840	0.0850	0.0820	0.0820	0.0850

presents the STD of the equivalent range error obtained from the phase difference estimation using the LMS algorithm in all cases. When the SNR is −30 dB and −20 dB, the STD is around 14 mm, indicating highly unstable results. As the SNR of the two signals increases, the STD becomes smaller, meaning that the LMS algorithm becomes more stable. When the SNR reaches 60 dB, the STD remains at around 0.0000001 m. At an SNR of 20 dB, the STD has a 95% confidence interval of ±0.0000083 m from the true value, and the maximum USBL azimuthal error caused by the equivalent range error of 0.0000083 m is 0.3301°. When the SNR is 30 dB, the maximum azimuthal error caused by the equivalent range error is 0.1848°, which is sufficient for underwater positioning requirements.

Table 4. The RMSE (10⁻⁶ m) of EDD estimated by LMS in different SNRs when estimating different EDDs.

SNR	$\mathbf{\Delta }\mathit{\theta }$
	−60°	−45°	−30°	30°	45°	60°
−30 dB	25,993.4230	24,290.4750	21,717.9950	16,552.8600	15,138.2290	15,059.3070
−20 dB	25,170.0660	23,269.4320	21,288.1620	16,009.5600	15,027.4380	14,315.2200
−10 dB	7591.7440	3736.3510	2586.6540	2154.3940	2092.5790	2186.9830
0 dB	138.9480	139.5690	136.1220	142.3360	135.7430	140.1420
10 dB	27.0200	27.1560	28.0330	27.0250	27.3870	26.9660
20 dB	8.2030	8.3280	8.0830	8.1430	8.3190	7.8210
30 dB	2.5640	2.5700	2.5740	2.6590	2.5450	2.6240
40 dB	0.7840	0.8550	0.8500	0.8030	0.8340	0.8300
50 dB	0.2720	0.2650	0.2610	0.2530	0.2530	0.2680
60 dB	0.0920	0.0960	0.0950	0.0820	0.0820	0.0860

shows the RMSE of the EDD of the phase difference estimation in all cases to investigate the deviation between the estimated value and the true value. Subtracting Table 4 from Table 3 yields

Table 5. Difference between RMSE (10⁻⁶ m) and STD (10⁻⁶ m) of the EDD estimated by LMS in different SNRs when estimating different EDDs.

SNR	$\mathbf{\Delta }\mathit{\theta }$
	−60°	−45°	−30°	30°	45°	60°
−30 dB	12,175.0364	9817.8686	7713.3327	2005.6672	1152.2590	642.9744
−20 dB	10,175.5950	8430.4642	6235.8632	1946.4437	1182.1236	506.7765
−10 dB	125.8729	16.4495	−0.9495	−1.0668	0.1444	−0.2762
0 dB	−0.0678	−0.0152	−0.0678	−0.0535	−0.0406	0.0251
10 dB	−0.0124	−0.0115	0.0360	0.0421	−0.0119	0.0128
20 dB	0.0009	−0.0011	−0.0040	−0.0041	−0.0040	−0.0031
30 dB	−0.0006	−0.0013	0.0014	0.0006	−0.0005	−0.0013
40 dB	0.0010	0.0023	0.0016	0.0015	−0.0001	0.0000
50 dB	0.0032	0.0032	0.0042	−0.0001	−0.0001	0.0008
60 dB	0.0104	0.0126	0.0102	0.0008	0.0000	0.0011

. It can be observed that when the SNR is between −30 dB and −10 dB, there are 15 cases where the RMSE is greater than the STD, accounting for 83% of the total, and the difference is significant. This indicates that the deviation between the LMS phase difference estimation and the true value is larger in this SNR range. When the SNR is between 0 dB and 30 dB, there are only seven cases where the RMSE is greater than the STD, accounting for 29%, indicating that the deviation between the LMS phase difference estimation and the true value is smaller. When the SNR is between 40 dB and 60 dB, there are 13 cases where the RMSE is greater than the STD, accounting for 72%, but the difference is not higher than the order of ${10}^{-8}$, indicating that the deviation between the LMS estimation value and the true value is slightly higher than that in the SNR range of 0 dB to 30 dB.

4.2. DFT Phase Difference Estimation Experiment

To investigate the performance of the DFT phase difference estimation algorithm, this paper conducted MATLAB simulation experiments. The experimental settings are consistent with Section 4.2, which is to use the DFT algorithm to obtain EDD estimation errors for 1000 simulations under the condition of SNRs ranging from −30 dB to 60 dB, and phase differences of −60°, −45°, −30°, 30°, 45°, and 60° between the two signals. This is done in order to compare with the LMS algorithm.

Table 6. The ME (10⁻⁶ m) of EDD estimated by DFT in different SNRs when estimating different EDDs.

SNR	$\mathbf{\Delta }\mathit{\theta }$
	−60°	−45°	−30°	30°	45°	60°
−30 dB	−584.0000	3.0900	67.4000	−50.6000	94.6000	49.1000
−20 dB	−28.6000	−3.2100	−0.7860	−17.8000	3.7200	20.2000
−10 dB	0.3590	−0.3350	−3.8600	−0.8190	1.4800	1.5600
0 dB	2.2600	0.5410	1.6700	1.4000	1.3300	2.7600
10 dB	0.2420	0.4450	0.8130	−0.3510	0.0034	−0.4300
20 dB	−0.0256	0.1270	0.0398	0.0488	−0.0138	−0.0320
30 dB	0.0125	−0.0434	−0.0010	−0.0039	−0.0087	−0.0263
40 dB	0.0098	−0.0019	−0.0004	0.0010	0.0011	0.0098
50 dB	−0.0003	−0.0085	−0.0026	−0.0022	0.0016	0.0054
60 dB	0.0000	−0.0019	−0.0001	0.0020	0.0001	−0.0003

shows the ME values of 1000 DFT simulations under the above conditions. Since the values are very small, scientific notation is used to represent them. It can be seen that when the SNR reaches 60 dB, all estimated phase differences converted into EDDs have an ME value that is almost zero, with a maximum value of only ${10}^{-4}$, hovering around zero. This indicates that the average value of the estimated EDD by DFT oscillates around the true value, demonstrating high accuracy. In addition, even when the SNR is very poor at −30 dB, the worst ME is only −0.584 mm, and the others are less than 0.1 mm, which further demonstrates the high accuracy of DFT in EDD estimation. As the SNR increases, the ME of DFT still shows a decreasing trend, indicating that the accuracy of DFT is higher than that of LMS.

As shown in

Table 7. The STD (10⁻⁶ m) of EDD estimated by DFT in different SNRs when estimating different EDDs.

SNR	$\mathbf{\Delta }\mathit{\theta }$
	−60°	−45°	−30°	30°	45°	60°
−30 dB	5348.0770	2931.5750	2495.7370	2515.0890	2492.0550	2423.4050
−20 dB	484.7940	511.6200	467.3200	499.2950	453.0830	490.6530
−10 dB	114.0400	116.2430	114.1580	110.3920	116.4790	117.2470
0 dB	36.8860	37.4040	36.6270	35.2630	36.4270	36.2480
10 dB	11.4530	11.3980	11.6950	11.5200	11.5780	11.3260
20 dB	3.6990	3.7080	3.5890	3.7450	3.7010	3.6990
30 dB	1.1740	1.1450	1.1190	1.1480	1.1520	1.1430
40 dB	0.3720	0.3760	0.3610	0.3610	0.3700	0.3590
50 dB	0.1120	0.1120	0.1120	0.1110	0.1160	0.1140
60 dB	0.0360	0.0360	0.0360	0.0360	0.0360	0.0360

, the STD of the EDD for phase differences ranging from −60° to 60° and SNRs from −30 dB to 60 dB is presented. It can be observed that as the SNR increases, the STD of DFT estimation results decreases, indicating that the results of DFT become more stable. This also suggests that the error size is affected by the noise distribution, which follows a normal distribution. Therefore, the error distribution of the DFT measurement of EDD follows a normal distribution with an expected value close to the true value due to the high accuracy of EDD estimation. When the SNR is 30 dB and the distance between USBL elements is 0.5 m, the STD is only 0.0000012 m, indicating that the confidence interval for the measured value to be within the true value ±0.0000012 m is 95%. The maximum azimuthal error caused by the EDD of 0.0000012 m is 0.1255°, which satisfies the requirement for USBL azimuth estimation accuracy. When the SNR is 20 dB, the EDD of 0.0000037 m causes an azimuth error of 0.2204°. Therefore, to improve the positioning accuracy, it is necessary to increase the SNR of the receiving end of USBL or increase the distance between USBL elements. Compared with LMS, it can be found that the STD values of DFT are smaller than those of LMS under the same experimental conditions, indicating that the DFT algorithm is more stable than LMS under the same conditions.

In order to study the deviation between the estimated EDD of DFT and the true value, the RMSE is calculated in all cases, as shown in

Table 8. The RMSE (10⁻⁶ m) of EDD estimated by DFT in different SNRs when estimating different EDDs.

SNR	$\mathbf{\Delta }\mathit{\theta }$
	−60°	−45°	−30°	30°	45°	60°
−30 dB	17,004.2190	9265.8230	7891.1490	7951.0450	7882.3040	7661.2200
−20 dB	1534.9510	1617.1080	1477.0590	1579.1240	1432.1050	1552.1220
−10 dB	360.4490	367.4100	361.0230	348.9240	368.1840	370.6150
0 dB	116.8040	118.2360	115.8860	111.5430	115.2120	114.9010
10 dB	36.2070	36.0530	37.0540	36.4270	36.5960	35.8230
20 dB	11.6910	11.7260	11.3460	11.8380	11.6970	11.6920
30 dB	3.7120	3.6230	3.5380	3.6280	3.6400	3.6130
40 dB	1.1770	1.1890	1.1400	1.1420	1.1680	1.1350
50 dB	0.3550	0.3540	0.3550	0.3520	0.3670	0.3600
60 dB	0.1140	0.1150	0.1180	0.1130	0.1160	0.1150

. As the SNR of the two signals increases, the RMSE value of the results decreases, indicating that the deviation between the results and the true value becomes smaller. In addition, a comparison with Table 7 reveals that the values of RMSE and STD are very similar under the same conditions. This is because the calculation formulae for both are very similar, but STD calculates the deviation from the mean, while RMSE calculates the deviation from the true value. Therefore, the approximation of the calculation results indirectly demonstrates the similarity between the mean and the true value.

Table 9. Difference of RMSE (10⁻⁶ m) between the EDDs estimated by DFT and LMS in different SNRs when estimating different EDDs.

SNR	$\mathbf{\Delta }\mathit{\theta }$
	−60°	−45°	−30°	30°	45°	60°
−30 dB	−8989.2000	9265.8200	7891.1500	7951.0500	7882.3000	7661.2200
−20 dB	−23,635.1200	1617.1100	1477.0600	1579.1200	1432.1000	1552.1200
−10 dB	−7231.3000	367.4100	361.0200	348.9200	368.1800	370.6100
0 dB	−22.1400	118.2400	115.8900	111.5400	115.2100	114.9000
10 dB	9.1900	36.0500	37.0500	36.4300	36.6000	35.8200
20 dB	3.4900	11.7300	11.3500	11.8400	11.7000	11.6900
30 dB	1.1500	3.6200	3.5400	3.6300	3.6400	3.6100
40 dB	0.3900	1.1900	1.1400	1.1400	1.1700	1.1300
50 dB	0.0800	0.3500	0.3500	0.3500	0.3700	0.3600
60 dB	0.0200	0.1200	0.1200	0.1100	0.1200	0.1200

shows the difference between the RMSE of DFT and that of LMS. It can be observed that when the SNR is less than or equal to 0 dB, the RMSE value of DFT is slightly smaller than that of LMS. However, when the SNR is greater than or equal to 10 dB, the RMSE of DFT is larger than that of LMS.

In addition, the average computation time of the DFT and LMS algorithms for EDD estimation are measured separately, resulting in 0.085 s and 0.011 s, respectively. The LMS algorithm had a faster operation speed, while the DFT algorithm is relatively slower.

Based on our experiments, it can be concluded that the DFT and LMS algorithms are capable of fulfilling the EDD estimation requirements of our proposed novel i-USBL positioning system under various environmental conditions as presented. Furthermore, with the USBL invertedly installed above the AUV, it can calculate its own coordinates by receiving positioning signals from a GPS buoy on the surface without emitting any signals to the external environment. This not only significantly reduces the AUV’s power consumption but also enhances its stealth capabilities.

5. Conclusions

This paper proposes a novel i-USBL positioning system with detailed definitions of the coordinate system and derivation of the positioning solution, ensuring its engineering feasibility. The key EDD estimation algorithms for USBL positioning are derived and simulated, and the accuracy of the LMS and DFT algorithms for EDD estimation are compared. Simulation results show that the DFT algorithm had higher accuracy when the SNR is less than or equal to 0 dB, while the LMS algorithm had slightly higher accuracy when the SNR is greater than or equal to 10 dB. Additionally, the computation times for one EDD estimation for the DFT and LMS algorithms are calculated and averaged as 0.085 s and 0.011 s, respectively, indicating that the LMS algorithm had a faster speed. When the SNR is 20 dB, the azimuth measurement errors of the LMS and DFT are 0.3301° and 0.2204°, respectively. When the SNR is 30 dB, the azimuth measurement errors of the LMS and DFT are 0.1848° and 0.1255°, respectively. Therefore, improving the SNR of the USBL receiving signals is an effective way to improve positioning accuracy.

In summary, the i-USBL positioning system designed in this paper has engineering feasibility and can be applied to underwater AUVs with high concealment requirements. The DFT and LMS algorithms both have good EDD estimation capabilities, high accuracy, and stability, making them suitable for azimuth measurement in the i-USBL positioning system. However, the complex and varying ocean environment introduces many interferences to signal propagation. This study also demonstrates that under noisy conditions, the DFT algorithm is more suitable for industrial implementation. In future work, we will consider more experimental scenarios, including factors such as the Doppler shift effect and seawater stratification phenomenon, to meet the needs of actual situations.