Underwater High Precision Wireless Acoustic Positioning Algorithm Based on L-p Norm

Abstract

Underwater wireless acoustic positioning technology uses the geometric relationship between a target and a receiving array to determine the target’s position by measuring distances between the target and the array elements, that the receiving array is usually symmetry in space. It is an indirect measurement approach, so ranging errors can significantly impact positioning accuracy due to error transmission effects. To improve precision, a similarity-matched localization algorithm based on the L-p norm (LPM) is introduced. This algorithm constructs a distance vector model with environmental parameters and performs similarity analysis by computing the L-p norm of the distance vector and a reference copy vector for three-dimensional localization in the observation area. Unlike conventional methods, this technology directly matches distance vectors without coupling error transmission, thereby enhancing positioning accuracy even though it remains contingent upon ranging precision. To validate the algorithm’s efficacy, Monte Carlo simulations are employed to analyze the distribution patterns of positioning errors in both horizontal and three-dimensional spaces. The results show an improvement from a mean positioning error of 0.0475 m to 0.0250 m and a decrease in error standard deviation from 0.0240 m to 0.0092 m. The results indicate that LPM offers improved accuracy and robustness by circumventing traditional error transmission issues.

1. Introduction

Underwater wireless positioning technology, which determines the location of a sound source in underwater environments, is commonly applied in fields such as underwater acoustic communication, underwater navigation, underwater wireless sensor networks, marine archaeology, scientific research, and so on. The continuous development and improvement of underwater positioning technology will further drive advancements in these fields, providing more opportunities and possibilities for humanity to explore the resources of the ocean [1,2,3]. Underwater passive positioning technology comprises three main algorithms: Time Difference of Arrival (TDOA) [4], acoustic field analysis [5,6], and spatial array signal processing [7]. These algorithms exhibit varying applications and suitability within underwater environments as in

Table 1. Three main algorithms for underwater positioning technology.

Classification	Characteristic	Weaknesses
Time difference of arrival	Utilizing the time delay estimation information between array elements	Affected by the accuracy of time delay estimation and error transmission, sensitive to multipath effects and noise
Acoustic field analysis	Utilizing multipath effects caused by environmental information	sensitivity to the ocean environmental parameters
Spatial array signal processing	Enhancing positioning performance using calibrated designed spatial beams	Requires complex and precise array designs along with sophisticated signal processing algorithms.

.

The principle of time difference of arrival positioning algorithm involves distributing multiple receivers which can be also called receiving elements across different spatial locations and utilizing the geometric relationship between the target and the array to calculate the target’s position [8,9,10]. This method requires the use of techniques such as the Generalized Cross-Correlation (GCC) algorithm and Inter-Microphone Level Difference (IMLD) to estimate the distance from the target to the receiving elements. It constitutes an indirect measurement approach, susceptible to error transmission effects when ranging errors occur, resulting in lower positioning accuracy [11,12]. Theoretically, achieving three-dimensional positioning requires obtaining three sets of distances between the receiving elements and the target for calculation. Representative algorithms include the least squares method and hyperbolic intersection method [13,14,15]. While this algorithm’s principle is straightforward and easy to implement, it is sensitive to multipath effects and noise in underwater environments, so its positioning accuracy is limited [16].

The principle of acoustic field analysis algorithms involves utilizing the time delay and intensity differences caused by multipath effects, reflections, diffractions, etc., during the propagation of sound signals to calculate the position of a sound source [5,6]. Commonly used algorithms include Matched Field Processing (MFP) and Matched Mode Processing (MMP) [17,18]. These algorithms exhibit certain advantages in complex environments, as they can overcome the impact of multipath effects and environmental noise on sound source localization, providing higher positioning accuracy and stability. However, they are sensitive to the ocean’s environmental geology.

The principle of algorithms based on spatial array signal processing involves utilizing the spatial positioning relationships among sensors within an array and the phase differences of received signals to achieve signal separation and sound source localization [7]. Common signal processing algorithms include directional Conventional Beamformer (CBF) and super-resolution spectral estimation methods like Minimum Variance Distortionless Response beamformer (MVDR) mainly localize the sound source through weighting and phase adjustment of signals received by different hydrophones [19,20,21,22]. Array signal processing methods can effectively improve the signal-to-noise ratio and enhance the accuracy of sound source localization by appropriately designed array structures and suitable signal processing algorithms. Current research directions encompass optimizing array design, refining signal processing algorithms, and integrating machine learning and deep learning technologies to enhance the performance of sound source localization [23,24,25]. However, this approach requires complex and precise array designs along with sophisticated signal processing algorithms that also lead to high processing complexity. Meanwhile, significant synchronization mismatches between different array elements can greatly affect positioning accuracy.

In general, the current challenges for underwater positioning technology are numerous. Research is primarily focused on addressing multipath effects and noise issues in underwater environments, optimizing sound source localization algorithms to meet the requirements of different underwater operational scenarios, enhancing positioning accuracy and stability, reducing hardware costs, etc. Additionally, integrating advanced acoustic sensor technology and signal processing methods to improve the performance and reliability of underwater acoustic positioning systems; exploring the application of machine learning and deep learning technologies in underwater acoustic positioning to enhance accuracy and real-time performance [26,27]. Especially, positioning accuracy is paramount in underwater acoustic positioning systems, it directly impacts the reliability and effectiveness of numerous underwater operations. High accuracy is essential for precise navigation, efficient resource mapping, and accurate data collection, which are critical for scientific research, environmental monitoring, and underwater construction. Therefore, enhancing positioning accuracy not only improves the performance and reliability of underwater systems but also ensures the success and safety of underwater missions. With continuous technological progress and innovation, underwater acoustic positioning technology will play a crucial role in a wider range of underwater application scenarios [28,29].

In the paper, assuming that time delay estimation and distance information have already been obtained, the purpose is to propose a high-precision positioning to optimize the error transmission process of TDOA, thereby improving the accuracy of TDOA algorithms.

The remainder of this paper is organized as follows. Section 2 describes the data model of the conventional TDOA localization algorithm. Section 3 defines and illustrates a similarity-matched localization algorithm based on the L-p norm. Section 4 compares the positioning performance of LPM and LS algorithms by applying the Monte Carlo analysis while varying environmental factors. Finally, Section 5 summarizes this work and presents its conclusions.

2. Underwater Localization Algorithm Using TDOA

When using the TDOA algorithm for positioning, the first step is to utilize time delay estimation to calculate distance information between the target and the array elements. Subsequently, the position is determined based on their geometric relationship. Therefore, this section first establishes a mathematical model of distance information using a three-element array as an example and then introduces the classical LS positioning algorithm.

2.1. Data Model

The underwater acoustic receiving array used by TDOA for localization usually is the three-element linear array, four-element planar array, or five-element cross-shaped array, and so on, the array geometry is typically some regular shape. To achieve three-dimensional localization in the water, let us assume that a circular array consisting of three elements is used. The array configuration and localization schematic are shown in the figure below. The center of the circular array serves as the origin of the Cartesian coordinate system, with the horizontal direction as the x-y plane and the vertical upward direction as the positive z-axis. The circular array is positioned in the x-y plane, with r representing the radius of the circular array. $A_1$, $A_2$, and $A_3$ denote the three elements, positioned at angles of 0°, 120°, and 240°, respectively. While in three dimensions, the coordinates of receiving elements are represented as $(x_i,y_i,z_i)$, $i=1,2,3$; $T(x,y,z)$ represents the underwater target to be localized with coordinates $(x,y,z)$; the distances from target to the elements of the circular array are denoted as $s_i$, $i=1,2,3$.

A schematic for time difference of arrival (TDOA) localization typically involves distance measurement and a localization algorithm. The distances between receivers and target source obtained using acoustic response include clock synchronization error, propagation delay variations, background noise and interference, decision thresholds, system errors caused by circuit delays, and so on. System errors can be eliminated through system calibration, but random errors cannot be eliminated. Let us assume that the obtained distance only includes random errors. Therefore, the distances obtained from the three receiving elements can be represented as:

$${\widehat{s}}_i=s_i+n_i,i=1,2,3$$

(1)

Here, ${\widehat{s}}_i$ represents the measured value of the distance from the target to the i-th array element, while $s_i$ represents the true value of this distance, and $n_i$ represents Gaussian white noise. The true value of the distance $s_i$ can be expressed as:

$$s_i=\sqrt{{(x-x_i)}^2+{(y-y_i)}^2+{(z-z_i)}^2}$$

(2)

2.2. Conventional TDOA Localization Algorithm

In order to employ geometric positioning methods for target localization, Equation (2) is transformed into the following form:

$$\left\{\begin{array}{c}{s}_1^2={(x-x_1)}^2+{(y-y_1)}^2+{(z-z_1)}^2\\ s_2^2={(x-x_2)}^2+{(y-y_2)}^2+{(z-z_2)}^2\\ s_3^2={(x-x_3)}^2+{(y-y_3)}^2+{(z-z_3)}^2\end{array}\right.$$

(3)

In order to transform the system of three quadratic equations into a system of three linear equations, we subtract each pair of equations in Equation (3), resulting in the following Equation (4):

$$\left\{\begin{array}{c}(x_1-x_2)·x+({\mathrm{y}}_1-y_2)·y+(z_1-z_2)·z={\delta }_{12}\\ (x_1-x_3)·x+({\mathrm{y}}_1-y_3)·y+(z_1-z_3)·z={\delta }_{13}\\ (x_2-x_3)·x+({\mathrm{y}}_2-y_3)·y+(z_2-z_3)·z={\delta }_{23}\end{array}\right.$$

(4)

where:

$${\delta }_{ij}=\frac{(x_i^2+y_i^2+z_i^2)-(x_j^2+y_j^2+z_j^2)-s_i^2+s_j^2}{2}$$

(5)

At this point, the system of linear Equation (4) can be expressed in matrix form as following:

$$\mathbf{Ax}=\mathbf{b}$$

(6)

where:

$$\mathbf{A}=\left\{\begin{array}{ccc}(x_1-x_2)& (y_1-y_2)& (z_1-z_2)\\ (x_1-x_3)& (y_1-y_3)& (z_1-z_3)\\ (x_2-x_3)& (y_2-y_3)& (z_2-z_3)\end{array}\right.$$

$$\mathbf{x}={(x,y,z)}^T$$

(7)

$$\mathbf{b}={({\delta }_{12},{\delta }_{13},{\delta }_{23})}^T$$

(8)

Due to the fact that the receiving elements $A_1$, $A_2$, $A_3$ lie in the same horizontal plane, the third column of matrix A is zero. Therefore, matrix A is a rank-deficient matrix, and the linear equations can form an overdetermined system with rank deficiency. Consequently, it is not possible to directly calculate the three-dimensional coordinates of the target through Equation (5). More precisely, it is not possible to directly obtain the z-coordinate of the target through Equation (5). Therefore, when calculating the target position using Equation (5), the z-axis is ignored, which means:

$$\mathbf{A}=\left\{\begin{array}{ccc}(x_1-x_2)& (y_1-y_2)& \\ (x_1-x_3)& (y_1-y_3)& \\ (x_2-x_3)& (y_2-y_3)& \end{array}\right.$$

(9)

$$\mathbf{x}={(x,y)}^T$$

(10)

At this point, the linear Equation (5) becomes an overdetermined system with full column rank. It can be solved using the least squares algorithm to obtain the $(x,y)$ coordinates of the target:

$${\mathbf{x}}_{LS}={\left({\mathbf{A}}^T\mathbf{A}\right)}^{-1}{\mathbf{A}}^T\mathbf{b}$$

(11)

The result obtained from Equation (11) can be substituted into Equation (3) to determine the target’s z-coordinate. Since Equation (3) is a system of three quadratic equations, the z-coordinate may contain a spurious value that needs to be excluded based on the actual array deployment situation. Assuming the array deployment shown in Figure 1 is on the seabed, the true value of the z-coordinate is only positive. This algorithm is referred to as the Least Squares (LS) localization algorithm.

3. Similarity-Matched Localization Algorithm

As indicated by Equation (2) under the assumption that the positions of the array elements are fixed, the distance between a single receiving element and the target is a function of the target coordinates. Moreover, this distance is also influenced by environmental parameters, which can be represented as:

$$s_i=f_i(x,y,z,\psi )$$

(12)

where $\psi$ denotes a set of environmental parameters. The distances from different elements of the receiving array to the target form a distance-vector, expressed as:

$$\mathbf{s}(x,y,z)={(s_1\cdots {\mathrm{s}}_N)}^T$$

(13)

In this expression, $s_i$ as shown in Equation (12) represents the distance between the i-th element and the target; $\mathbf{s}$ is an $N\times 1$ column vector, where N is the number of array elements. When the true position of the target is $(x_t,y_t,z_t)$, the target’s distance vector can be represented as:

$${\mathbf{s}}_t(x_t,y_t,z_t)={(s_{t,1}\cdots s_{t,N})}^T$$

(14)

where $s_{t,i}=f_i(x_t,y_t,z_t,{\psi }_t)$.

Assuming there is a target at any position in the observation space, and the position is ${\mathbf{s}}_c(x,y,z)$. The distance vector of the target at that location can be calculated using Equation (13), termed the replica vector ${\mathbf{s}}_c(x,y,z)$, represented as:

$${\mathbf{s}}_c(x,y,z)={[s_{c,1}(x,y,z)\cdots s_{c,N}(x,y,z)]}^T$$

(15)

Here, $s_{c,i}=f_i(x,y,z,\psi )$, the values of which are related to environmental parameters and can be determined using acoustic field computation software like Kraken. In the absence of environmental parameter effects, this value can be derived from Equation (2).

To ascertain the spatial location of the target, the following solution model can be constructed as follows:

$$\underset{(x,y,z)}{max}S\left({\mathbf{s}}_t(x_t,y_t,z_t),{\mathbf{s}}_c(x,y,z)\right)$$

(16)

where $S(·)$ is a similarity function between two vectors. The mathematical implication of this model is to find a spatial position that maximizes the similarity between two vectors, thereby determining the coordinates of the target.

Thus, it is necessary to establish a similarity function suitable for three-dimensional spatial localization. Let us define the similarity function S as the negation of the L-p norm of the difference between the distance vectors:

$$S({\mathbf{s}}_t,{\mathbf{s}}_c)=-{∥{\mathbf{s}}_t-{\mathbf{s}}_c∥}_p$$

(17)

And let:

$$d({\mathbf{s}}_t,{\mathbf{s}}_c)={∥{\mathbf{s}}_t-{\mathbf{s}}_c∥}_p=\sqrt[p]{\sum _{i=1}^N{\left|s_{t,i}-s_{c,i}\right|}^p},1\le p<\infty$$

(18)

Here, $d({\mathbf{s}}_t,{\mathbf{s}}_c)$ represents the Minkowski distance between the distance vectors ${\mathbf{s}}_t$ and ${\mathbf{s}}_c$. When $p=1$, this distance denotes the Manhattan distance or the city block metric. When $p=2$, it represents the Euclidean distance. A similarity function value of zero indicates that the two distance vectors are most similar. Consequently, the localization model expressed in Equation (14) can be reformulated as:

$$\underset{(x,y,z)}{min}d({\mathbf{s}}_t,{\mathbf{s}}_c)$$

(19)

That is

$$\underset{(x,y,z)}{min}\sqrt[p]{\sum _{i=1}^N{\left|s_{t,i}-s_{c,i}\right|}^p},\mathrm{for}1\le p<\infty$$

(20)

Since the similarity function utilized in the localization model is the L-p norm, this algorithm can be termed the L-p norm-matched localization (LPM) algorithm for simplification. The localization error can be calculated by the following expression:

$$err=\sqrt{{(x-x_t)}^2+{(y-y_t)}^2+{(z-z_t)}^2}$$

(21)

where $(x,y,z)$ are the coordinates of the target as computed by the algorithm.

4. Numerical Simulations and Analysis

To compare and analyze the localization performance of the LPM algorithm, the LS algorithm is used as a reference in simulations. However, the LS algorithm lacks the ability to adapt to changes in environmental parameters. Therefore, in the simulations, the influence of environmental parameters such as sound speed gradients on ranging is ignored. Additionally, we assume the deployment configuration depicted in Figure 1, with the array having a radius of 5 m, so $A_1$ is at (5.0, 0, 0), $A_2$ is at (−2.5, 4.33, 0), $A_3$ is at (−2.5, −4.33, 0). The target is only above the array. Meanwhile, in the simulations assume ranging error follows a Gaussian distribution with zero mean and variance of 0.03 m, and the number of Monte Carlo iterations set to 100.

In this section, positioning simulations in the x-y two-dimensional plane are conducted, and the positioning results in the two-dimensional space are shown. Subsequently, positioning simulations in the three-dimensional plane are performed to analyze the positioning accuracy.

4.1. Analysis of x-y Plane Localization Error

In this section, assuming that the target’s z-axis coordinate is known, there is no need to locate the z-coordinate. The true coordinates of the target in the x-y plane are (3.5, 4.0) in meters. In the simulations conducted in this section, the observation space in the x-y plane is defined as a circular area with a radius of 6 m centered around the array’s center. When using the LPM algorithm for localization, it is necessary to traverse the entire localization space with a stepping precision of 0.05 m.

The positioning simulation without errors is depicted in Figure 2. The center of the diamond represents the target’s true position, and the center of the plus sign indicates the localization result obtained using the LPM algorithm. From this figure, it can be observed that the LPM algorithm can accurately localize the target within the x-y plane.

When the positioning error is a Gaussian error with zero mean and a variance of 0.03 m, Figure 3 presents the localization results of the target’s x-coordinate for comparing the performance of the LPM algorithm and the conventional LS positioning algorithm. It can be observed that both algorithms can accurately localize the target, but the LS localization curve exhibits slightly greater fluctuations compared to LPM. This is reflected in Table 1, where the mean x-coordinate values obtained by both algorithms are equal at 3.5 m. However, the standard deviation of the x-coordinate obtained by the LS algorithm is 0.0047 m larger than that of LPM. The graphical results align with the numerical findings presented in the table.

Figure 4 presents the localization results of both algorithms for the target’s y-coordinate. From the figure, it is evident that the mean values of the localization results for both algorithms are around 4.0 m, indicating the correct localization of the y-coordinate by both methods. Additionally, the graph shows that the fluctuation in the y-coordinate localization curve for LPM is smaller than that of the LS algorithm, highlighting the superiority of LPM over LS in terms of y-coordinate localization.

Figure 5 illustrates the comparison of localization errors between the two algorithms when the same error exists as in Figure 3. The overall trend of the curves in the graph indicates that the error variation range of the LPM algorithm is smaller than that of the LS algorithm, demonstrating that LPM provides relatively stable localization results. Moreover, it can be observed from the graph that the maximum localization error of the LPM algorithm does not exceed 0.04 m, whereas the maximum error of the LS algorithm exceeds 0.1 m. This difference arises because the LS algorithm requires distance information for its calculations, leading to an error propagation effect. In contrast, the LPM algorithm directly matches the distance information, eliminating the impact of error propagation and thus improving positioning accuracy.

Table 2. Comparisons of localization results for LPM and LS at (3.5, 4.0).

	LS	LPM	$\left|\mathit{LS}-\mathit{LPM}\right|$
Mean (x)/m	3.5000	3.5000	0
Std (x)/m	0.0303	0.0256	0.0047
Mean (y)/m	4.0000	4.0000	0
Std (y)/m	0.0420	0.0299	0.0121
Mean (Error)/m	0.0457	0.0250	0.0207
Std (Error)/m	0.0240	0.0092	0.0148

provides a quantitative comparison of the localization results obtained by the LS and LPM algorithms for the target at location (3.5, 4.0). The comparison primarily focuses on the aspects of mean and standard deviation, where the mean is used to assess the accuracy of the algorithm’s localization while the standard deviation reflects the fluctuation characteristics of the localization results. From Table 2 it can be observed that both the mean and standard deviation of localization errors are smaller for LPM compared to LS. The standard deviation of the LPM algorithm is smaller than that of the LS algorithm, indicating that the dispersion of the Monte Carlo simulation results for the LPM algorithm is lower. Therefore, the results shown in Table 2 indicate that compared to the LS algorithm, the positioning results of the LPM algorithm are more accurate, stable, and reliable.

Assuming the observation area is a circular region with a radius of 6 m, the layout of the three elements array is shown in Figure 1. Within this area, we perform a traversal to simulate the positioning errors throughout the entire region. Figure 6 depicts the error plane of the LS algorithm in the observation area with star-shaped markers indicating the positions of three elements. From Figure 6, it can be observed that the positioning accuracy is highest near the center of the array, while it is lowest at the locations near the array elements. The highest localization accuracy is observed at the center position (0,0) of the x-y plane.

Figure 7 is a contour map of Figure 6, so it also displays the localization errors for the LS algorithm within the observation area. From the figure, it can be observed that the highest localization accuracy is achieved at the center position, and as the distance from the array’s center increases, the localization accuracy decreases. The positioning error near the center position is approximately 2.5 cm, while at the edge positions, the positioning error is around 5 cm.

Figure 8 depicts the distribution of localization errors in the observation area for the LPM algorithm, presented as a contour map. The pattern of error distribution in Figure 8 is similar to Figure 6, with the overall localization errors being smaller than those in Figure 6.

Figure 9 is a contour map of Figure 8. Compared to Figure 7, the LPM algorithm exhibits smaller localization errors. The minimum positioning error near the center position is approximately 1.2 cm, while the maximum positioning error at the edge positions is around 2.2 cm.

4.2. Analysis of Three-Dimensional Spatial Localization Errors

Assuming the observation space extends from the x-y plane to three-dimensional space, where the scale of the x-y plane remains a circular area with a radius of 6 m, and the scale along the z-axis ranges from 0 m to 6 m, creating a cylindrical cuboid observation space. In the simulation, horizontal slices are taken at intervals of 0.1 m along the z-axis (height), and each slice undergoes traversal for localization simulation, thereby achieving traversal simulation in three-dimensional space. The array deployment and error conditions remain unchanged during the simulation.

When the height of the x-y plane is 6 m. Figure 10 and Figure 11 are contour maps of localization errors for the LS and LPM algorithms, respectively. Comparing Figure 7 and Figure 10, as well as Figure 9 and Figure 11, it can be observed that when the slice height is the same, the LPM algorithm outperforms the LS algorithm. Additionally, when the slice height is 6 m, both the LS and LPM algorithms exhibit overall localization errors greater than those at a height of 0 m.

The results of localization simulations for other slices are similar to those depicted in Figure 6, Figure 7, Figure 8 and Figure 9. Figure 12 illustrates the average error variation curves for x-y plane slices at different heights. Figure 6 represents the average of 10 measurements, while the value at each point on the curve in Figure 12 is the average of 100 Monte Carlo simulations.

In order to observe the relationship between positioning error and target height, during the traversal of three-dimensional space, the average positioning error is defined by the following equation:

$$f\left(z\right)=\frac{1}{N}\sum _{x,y}err(x,y,z)$$

(22)

Here, N represents the number of traversals in the x-y plane, and x, y, z represent the coordinates of the target location, and $err(x,y,z)$ represents the positioning error at the location $(x,y,z)$.

Figure 12 shows the average error curve at different heights using Equation (22), from this figure, it is evident that within the range of 0 to 6 m, the localization error of the LPM algorithm is smaller than that of the LS algorithm. When z is 6 m, the localization accuracy of LPM is approximately 3 cm higher than that of LS. Furthermore, when the target moves away from the plane where the base array is located, the localization errors of both LPM and LS gradually increase, but the rate at which the LPM error increases is slower than that of LS.

5. Conclusions

In the field of underwater target localization, especially in long baseline systems or precise positioning within smaller scales, geometric calculations based on distances remain the main technique for localization. However, this technique is susceptible to error propagation patterns and environmental factors such as ocean currents and sound speed gradients, leading to lower localization accuracy. To enhance the precision of passive acoustic localization, this paper proposes an L-p norm matched localization method, which utilizes matching localization techniques for three-dimensional localization based on completed distance measurements. Simulation experimental results indicate the following: Firstly, near the array’s center position, both localization algorithms are more accurate, with localization accuracy gradually decreasing as the target moves away from the array’s center. Secondly, On the plane where the receiving array is located, the localization accuracy is higher, while it gradually decreases as the localization plane moves away from the array plane.

Overall, the LPM algorithm demonstrates a higher localization accuracy. In all the simulation experiments, the worst localization accuracy of the LS algorithm is about 5 cm, while that of the LPM algorithm is 2.2 cm, indicating a significant improvement in localization accuracy. However, a drawback of the LPM algorithm is its slower computational speed due to the need to traverse the distance database of the observation area during localization. Nonetheless, the introduction of a database brings another significant advantage to the LPM algorithm. In the scenario where environmental parameters are accurately known, the algorithm can incorporate the influence of environmental factors into the construction of the distance database. This helps eliminate the impact of environmental parameters on localization, achieving functionalities that are challenging to accomplish with geometric positioning alone.

The noise model in this paper currently only considers Gaussian white noise; the analysis of the influence of a more complex noise model is necessary. Additionally, the validation work in this paper is currently limited to Monte Carlo simulations, and subsequent experimental verification of the work will be conducted.