In the following content, we first introduced the framework of our proposed positioning method. We then describe the detailed function and specific implementation of each sub-module.
3.3. Signal Noise Reduction
Noise can cause significant disturbance to the underwater object’s positioning calculation. So, it is necessary to conduct noise reductions in the underwater acoustic signal to achieve a higher accuracy of the target’s positioning. However, the propagation of the acoustic signal in the water propagation is affected by the temperature, waves, and internal waves. It will make noise distribution random and irregular. We chose the least-square error rule as the best estimation criterion for the noise reduction of the signals received to reduce the noise interference and improve the positioning accuracy. According to this idea, the paper utilizes the Kalman filtering algorithm to process the obtained acoustic signals based on the minimum mean-square error estimation.
Kalman filtering estimates the system state from the sequences of uncertain observations using the predict–update cycle. First, the next system state and its uncertainty are predicted by an existing physical model and a statistical model, which describes any uncertain factors, including the process noise. This prediction is updated using the procedure observation and the difference between the prediction value and the observation value. Once the updated estimation is done, we can estimate a new predictive value.
The Kalman filtering algorithm requires only the system estimation data of the previous moment and the current measurement data in each operation; the algorithm is simple and easy to implement. It also has a wide range of applications in the field of underwater navigation. In this study, we utilize the Kalman filter algorithm to smooth the acoustic signal. The specific method we used is as follows.
In the underwater positioning array, shown in
Figure 3, the array elements receive acoustic signals from the underwater target. We sample the received signals and get a sequence
S = {
s1,
s2,
s3, …,
sk, …,
sn}, where
k is the
kth moment stamp,
n is the total number of sampling, and 1 ≤
k ≤
n.
To reduce the negative impact of noise, we utilize a Kalman filter to filter the signal {
} using Equations (2) and (3).
Here, X = {X1, X2, X3, …, Xk, …, Xn} is the state information of the system. H is the transition matrix (we set it to be a unit matrix), and G denotes the gain matrix; Wk−1 is the system noise at the moment k (we treat it as Gaussian white noise); {uk} presents the system observation information of the system; Xk denotes the state value of the system at moment k; and u = {u1, u2, u3, …, uk, …, un} is the observation signal of the system.
There are two procedures durin Kalman filtering: estimation and correction. In the estimation stage, we estimate the system state
at the moment
k by Equation (4), and the prior state estimate
is estimated recursively at the
kth moment by state
at moment
k−1, where there is no control input.
Equation (5) predicts the mean-squared error
of the system, and the prior covariance estimation
Pk−1 is obtained recursively at the
kth moment through the posterior covariance
at the previous time.
Equation (6) calculates the Kalman gain of the system, and the system Kalman gain
Kk is calculated at the
kth moment through the covariance estimation
at moment
k.
Equation (7) calculates the estimated value of the system state, and the system state value
is calculated at the
kth moment through the prior state estimation at the
kth moment.
We estimate the posterior error covariance
of the system by Equation (8). Here,
Kk is the system gain and
denotes the prior error covariance.
Finally, we can obtain the processing result
by Equation (9).
In these analyses, = {, , , …, , …, } is the state information of the system and is the state variable of the system at moment k. = {, , , …, , …, } is the prior state estimation of the system. is the prior estimation of the state variable at the kth moment, obtained from the state variable at moment k−1. A = 1 is the state transform coefficient acting on . H = 1 is the observation model matrix, which maps the real state space into observation space. = {, , , …, , …, } is a 1 × n prior estimation error covariance matrix; P = {P1, P2, P3, …, Pk, …, Pn} denotes a 1 × n posterior estimation error covariance matrix; Q = 0.1 represents the process noise covariance coefficient; R = 0.25 is the process noise covariance coefficient; the value of I is set to be 1; and K denotes the Kalman gain or mixing factor, calculated from the specific data. Its role is to minimize the posterior estimation error covariance. û = {û1, û2, û3, …, ûk, …, ûn} denotes the filtered value of the acoustic signal, and ûk denotes the filtered value of the acoustic signal at moment k.
The above process is repeated recursively to implement the filtering processing of the acoustic signal received. We transform the original signal u = {u1, u2, u3, …, uk, …, un} into a new signal û = {û1, û2, û3, …, ûk, …, ûn}. Similarly, we can get the processed signals ûx, ûy, and ûo from the x-axis array element, the y-axis array element, and the original array element, respectively. The filtering method adjusts the relevant parameters by an estimation error condition and realizes higher-quality signal filtering.
3.4. Positioning Computing
3.4.1. Positioning Principle
Firstly, we introduce the traditional principle of phase difference positioning. For the single-frequency CW signal, the phase information is the most commonly used for the ultra-short baseline positioning system. The phase difference between the received signals is measured to solve the positioning problem. We utilize different frequencies to distinguish the target response signals in multi-target situations. The following describes the principle of target location using the phase difference between single-frequency CW signals.
We performed underwater positioning, as shown in
Figure 4. In the specific coordinate system, we need to determine the coordinate (
x,
y,
z) of target
S. We deployed two orthogonal linear arrays on the
x-axis and
y-axis, respectively, and the array center is the origin of the coordinates.
The target radius is OS, and its direction cosine is
Here,
represents the radius OS and the
x-axis angle;
denotes the radius OS and y-axis angle;
R is the target slant distance; and S’ is the projection of S on the “
xo
y” plane, and the angle
θ between it and the
x-axis is the target horizontal azimuth. We can obtain the angle
θ:
Here, r is the target horizontal slant distance and z is the target depth. We can determine it through a depth measurement sensor.
Equations (10)–(15) are the basic formulas for positioning calculation. The above equations can work out the target position parameters. Considering the propagation of the acoustic signals between the element array and object, and the propagation differences between the elements, we can get the equations approximately.
Here, is the phase difference of the received signal from the adjacent array element on the x-axis; represents the wavelength of the acoustic signal; and denotes the phase difference of the received signal from the adjacent array element on the y-axis.
With Equations (10) and (11), we can obtain Equations (18) and (19). We determine the coordinate (
x,
y,
z) of the target relative to the USBL’s position.
In the formulas above, R = c × ∆t/2, c is the acoustic velocity in water and ∆t is the time difference from the transmission to the receiver. So, the distance R is equal to c × ∆t/2, so the actual measurements are , , c, and ∆t.
3.4.2. Ambiguity Problem Solution for the Phase Difference
During the actual measurement, the acoustic velocity c is the same. The error ∆t is negligible, and the accuracy of the slope distance R estimated by the response ranging method is high enough. The estimation accuracy of R is called the vertical estimation accuracy. The estimation accuracy of x and y is called the horizontal coordinate estimation accuracy. It follows from Equations (17)–(19) that they mainly depend on the measurement accuracy of the phase differences and .
After the noise reduction in
Section 3.3, we analyze the signals
ûx,
ûy, and
ûo to obtain the phase difference between the signals. Then we plug the phase differences
ϕ and
ψ into Equations (18) and (19) to get the coordinates (
x,
y) of the target; the values of
x and
y are then plugged into Equation (12). We calculate the depth
z of the target to achieve the positioning calculation.
However, there exist ambiguity issues of a phase difference. We solve this problem as follows. According to Equations (18) and (19), considering the distance between Element 1 and Element 3, and the distance between Element 2 and Element 3 on the
x-axis, Equations (20) and (21) are obtained, respectively.
Here, and are the projections of the distance R on the x-axis and in the Array Element 1 direction, respectively.
In this research work, ϕ13 is the phase difference between Element 1 and Element 3. There is no phase ambiguity. So, we can utilize it to confirm there is phase ambiguity in ϕ32 and ϕ34 for a high positioning accuracy or not. The signal delay differences between all azimuths received by Array Element 1 and Array Element 3 are both equal to τ13 and less than T/2. So, the phase difference ϕ23 is equal to 8ϕ13. When |ϕ13| is greater than 22.5°, ϕ23 will be located in the multi-valued interval, and the difference between the measured value ϕ23 and the true value is one or several cycles; this as long as ϕ23 is added or subtracted by an integer multiple of 360° and compared with 8-fold of ϕ13. If the phase difference is less than one cycle, we can determine that the ϕ23 is correct. This step will solve the phase difference multi-valued fuzzy of a large array.
Next, we consider the positioning accuracy of the new array. ϕij and ψij have the same phase difference measurement accuracy; i.e., ∆ϕ13 = ∆ϕ23 = ∆ϕ when the background noise of the array elements is independent.
We regard Equation (20) as the positioning formula on the
x-axis of the traditional USBL array. We treat Equation (21) as the improved positioning formula on the
x-axis of the USBL array. Without considering the measurement errors of the slope
R and the sound speed c, we differentiate both sides of Equations (20) and (21) respectively to get the influence factor of
xd and
xL:
Since L = 8 d, the array element with spacing L can improve the positioning accuracy 8-fold from Equations (20) and (21). The array element with spacing L can solve the positioning accuracy problem to achieve a high-precision phase measurement positioning based on the array element with a spacing d, solving the multi-valued phase difference measurement blurring.
From the illustration above, we analyzed the computation complexity of the proposed Kalman filtering-based underwater positioning method. Suppose the number of positioning points is M, the sample point number of the acoustic signal is n. The computational complexity is O(Mn).