3.1. The Design of the Algorithmic Model
The method based on phase ratio in [
20] neglected the influence of the spacing error of the array element. In this paper, we considered that there was a spacing error between the array elements, and that there were differences in the spacing errors between different groups of array elements, as seen in
Figure 4.
Since the truth-value of the spacing of array elements is unknown, and the spacing error of array elements is also difficult to calibrate accurately, this paper designed a USBL positioning calculation model based on the rotating array and reusing elements method, which is used to eliminate the influence of the spacing error of array elements on the USBL positioning system.
According to Equations (14)–(16) and (34), the purpose of this model is to equalize the spacing errors between the three elements groups, that is,
The spacing error between array elements is difficult to calibrate accurately. To make the three spacing errors completely equal, the most appropriate method is to use the elements reusing method. Thus, array elements 3, 4, 5, and 6 were removed, leaving only array elements 1 and 2, as shown in
Figure 5. When it is necessary to calculate the phase difference
and the phase difference
, the coordinate system is rotated so that array elements 1 and 2 on the coordinate axis are respectively rotated to the positions of elements 3–4 and 5–6, and the phase differences
and
are respectively calculated. In this process, only two array elements connected rigidly are used. In the whole positioning process, the spacing error between the two array elements is always
.
Figure 5 shows the USBL structure based on virtual six receiving elements array. The array was composed of array elements 1 and 2, which were connected rigidly. The positioning principle of the USBL acoustic receiving array was the same as that of the 6-element USBL receiving array shown in
Figure 4, and the error principle also was the same.
Figure 6 shows the rotating schematic diagram of the USBL array based on the rotating array and reusing elements.
Figure 6a is the schematic diagram of the
-frame rotating 90° clockwise around the
y-axis, so that array elements 1 and 2 of the original coordinate
x-axis can rotate to the position of array elements 5 and 6 on the
z-axis. Similarly,
Figure 6b is the schematic diagram of the
-frame rotating 90° clockwise around the
-axis, so that array elements 5 and 6 on the original
z-axis can rotate to the position of array elements 3 and 4 on the
y-axis.
Figure 6c is the schematic diagram of the USBL array
-frame rotating 90° clockwise around the z-axis, so that array elements 3 and 4 on the original y-axis can rotate back to the position of the array elements 1 and 3 on the
x-axis.
The calculation process is shown as follows:
Figure 7 depicts the flow chart of the USBL positioning calculation model based on the rotating array and reusing elements. First, the USBL positioning system acquires the depth information of the target
. Then, the USBL array obtains the phase difference
between array elements 1 and 2 by receiving the underwater acoustic signal from the target.
After acquiring the phase difference , the -frame is rotated around the y-axis to rotate the x-axis to the position of the z-axis, and the phase difference is measured. Similarly, the -frame is rotated from the z-axis around the x-axis to the position of the y-axis, and the phase difference is measured. After the three phase differences , , and are collected, the i-th USBL horizontal positioning calculation is performed. Then, the -frame is rotated around the z-axis and the y-axis is rotated back to the x-axis. At this time, if the positioning task is not finished, the program enters the next round of the target positioning solution, otherwise the positioning task is completed and ends this procedure.
In
Figure 6a,
where
is the phase difference between array elements 1 and 2;
is the spacing value between array elements 1 and 2, which contains the constant spacing error
; and
is the angle between the underwater acoustic signal line and the x-axis.
In
Figure 6b,
where
is the phase difference between array elements 5 and 6;
is the spacing value between array elements 5 and 6, which contains the constant spacing error
; and
is the angle between the underwater acoustic signal line and the z-axis.
It can be derived from Equation (30) that
Since array elements 5 and 6 are obtained by rotating array elements 1 and 2 of the x-axis on the USBL acoustic array around the y-axis to the z-axis, and the two elements are rigidly connected before and after the rotation, it can be seen that
. Thus, Equation (38) can be written as
The calculation of the
y-axis coordinate value of the target
in the USBL
-frame is performed as follows. First, the USBL matrix
-frame is rotated around the z-axis, so that array elements 1 and 2 on the x-axis are rotated to the positions of array elements 3 and 4 on the y-axis, and the phase difference
is measured. Then, the
-frame is rotated around the x-axis, so that array elements 3 and 4 on the y-axis are rotated to the positions of array elements 5 and 6 on the z-axis, and the phase difference
is measured. Finally, the y-axis coordinate value of the target
in the
-frame is calculated according to the depth value
of the target
and the phase difference values
and
. The USBL acoustic array structure and its rotation sequence are shown in
Figure 8.In order to save the positioning calculation time and adapt to the dynamic positioning environment, the horizontal position of the target
T can be solved in the order shown in
Figure 7. After the USBL acoustic positioning system completes the x-axis positioning calculation, the phase difference
of array elements 3 and 4 on the y-axis can be obtained immediately. As shown in
Figure 6b, after acquiring the phase difference
of array elements 5 and 6 on the z-axis, the
-frame rotates around x-axis, and rotates array elements 5 and 6 on the z-axis to the position of elements 3 and 4 on the y-axis to measure the phase difference
. Then, combined with the target depth value
, complete the positioning calculation of the target (y-axis direction) in the
-frame.
where
is the depth value of the target and
is the depth value of the USBL acoustic array. Equations (39)–(41) are the basic equations for the USBL positioning calculation based on rotating array and reusing elements method. This positioning method is suitable for USBL positioning calculations where the target depth
is known.
It can be seen from Equation (39) that the positioning accuracy of the USBL is only related to the depth value of the target and the error of the two phase differences and . The positioning accuracy is independent of the spacing of the array elements . Therefore, this model can completely eliminate the influence of the spacing error on the positioning accuracy of the USBL.
3.2. Error Analysis
In this section, the error source analysis of the USBL positioning model based on rotating array and reusing elements is performed. Apply the complete differential to Equation (39), and the positioning error (x-axis) of the target in the
-frame can be obtained.
Under the condition that each error term is independent of each other, the mean square error of the target in the x-axis of the
-frame is
Contrasting Equation (43) and Equation (18), it can be found that when compared with the USBL positioning calculation method based on the slant range and azimuth method, the proposed USBL positioning calculation method based on the rotating array and reusing elements method eliminates the horizontal positioning error caused by the signal wavelength error and the USBL element spacing error . The slant range measurement error is replaced by the depth measurement error . In the case of long distance, the depth measurement error is much smaller than the slant measurement error.
In the case of high-precision underwater positioning, the USBL positioning model based on the rotating and array element reuse should consider the USBL horizontal positioning error caused by the rotary angle error of the rotating device. During the process of the rotary device rotation, it is difficult to rotate the rotary device at exactly 90°. When the line connecting the two array elements after rotation does not completely coincide with the target coordinate axis, the distance between the two array elements projected onto the coordinate axis will be less than the spacing value of the actual array elements. As shown in
Figure 6a, the acoustic array coordinate system U is rotated about the
-axis, rotating the array element 1–2 on the
-axis to the positions of the array elements 5–6 on the
-axis. However, in the actual rotation process, when the
-axis and the
-axis do not completely coincide, there is a small error angle
between them, as shown in
Figure 9, where the projection of the array elements 1–2 on the
-axis are array elements
–
. The distance between them is defined as the actual array spacing
.
The space error
of the array elements caused by the rotation error of the rotary device is
In Equation (47), is the rotation angle error factor, where it can be seen that the rotation angle error of the USBL rotating device will produce the spacing error of the array elements, and the array element spacing error is one of the important parameters affecting the positioning accuracy of the USBL. The influence of the rotation angle error on the positioning accuracy of the USBL will be discussed by the numerical simulation method.
It can be seen from
Table 5 that a 1° rotation angle error can produce a positioning error of about 0.15‰ in the horizontal distance in the rotating and array element reuse model. That is, a rotation error of 1° can cause about a 1.5 m horizontal positioning error in the horizontal positioning distance of 10,000 m. Usually, the rotation error will be less than 0.1°, and the horizontal rotation position error of 10,000 m at this level will cause a horizontal positioning error of 0.015 m, which is a very small error value. This error can be neglected even in high-precision, full-depth positioning operations. It can be seen that the rotation angle error of the USBL rotating device was less than 0.1°, which has little effect on the horizontal positioning accuracy in the USBL positioning model based on the rotary and array reusing elements method.