Positioning method based on discrete return-to-zero neurodynamic model

Abstract

The invention relates to the technical field of underwater dynamic environment, in particular to a positioning method based on a discrete return-to-zero neurodynamics model, which comprises the following steps of: 1, analyzing a UASNs positioning problem existing in a continuous model, and establishing a continuous return-to-zero neurodynamics model; 2, a numerical difference method capable of being used for continuous system discretization is explored, and a proper numerical difference formula is selected; 3, deducing a discrete return-to-zero neurodynamic model which is easy to implement; and 4, solving a UASNs positioning problem through the discrete return-to-zero neurodynamics model to obtain a positioning result. According to the invention, UASNs positioning can be well carried out.

Description

Technical Field

[0001] The present invention relates to the technical field of underwater dynamic environments, and more particularly, to a positioning method based on a discrete reset neural dynamics model. Background Art

[0002] Underwater acoustic sensor networks (UASNs) have broad application prospects in the field of marine science due to their powerful underwater monitoring capabilities and flexibility in deployment. At the same time, related technical research in the field of UASNs has received extensive attention. For almost all underwater applications, providing accurate spatio-temporal data is a basic requirement for UASNs. Therefore, the research on node positioning technology has important scientific research value and practical significance. In the absence of global positioning system signals, underwater node positioning mainly uses the positioning parameter information between unknown nodes and their surrounding anchor nodes to establish equations to solve the positions of the unknown nodes. Based on whether ranging is relied on, UASNs node positioning methods can be divided into ranging-based positioning methods and ranging-free positioning methods. In recent years, various vehicles with mobile functions have gradually been applied to UASNs and participated in node positioning, which brings challenges to node positioning while improving network scalability. Moreover, different from land sensor nodes, nodes located underwater inevitably continue to produce dynamic displacements under the action of ocean currents. However, most traditional ranging-based or ranging-free positioning methods are static and do not consider the time-varying and dynamic characteristics of the positioning scenario, making it difficult to achieve accurate positioning. In addition, in practical applications, the acquisition of positioning parameter information is severely affected by environmental noise. In an open underwater environment, there are often noise interferences such as ships and marine organisms, which seriously reduce the positioning accuracy. Therefore, how to effectively suppress the influence of noise is another challenge faced by UASNs node positioning.

[0003] As an important branch of artificial intelligence, neural networks have gradually penetrated into the field of underwater acoustic communication technology. The reset neural dynamics method starts from the dynamic characteristics of neural networks, introduces a time parameter, and uses an error feedback mechanism to meet the requirements of real-time, online, fast, and accurate solution of time-varying and time-invariant problems. In addition, compared with other traditional neural dynamics methods (such as the traditional gradient neural dynamics method), the reset neural dynamics method has a faster convergence speed and higher solution accuracy while achieving real-time solution. Due to the above advantages, the reset neural dynamics method is also applied in distributed, routing-free, and ranging-free positioning scenarios in wireless sensor networks.

[0004] Neural networks are designed for underwater acoustic fields such as underwater acoustic detection, target recognition, and channel estimation, and the degree of integration is getting deeper and deeper. Correspondingly, the research depth and breadth of neural dynamics are also constantly expanding, and many directions of integration with industrial fields have emerged, such as fish image classification and recognition, robotic arm motion planning and control, and intelligent agent cooperative control. However, its application potential in underwater acoustic positioning technology still needs to be further explored. From the existing research results, it is undeniable that neural dynamics has great development potential and application prospects in the technical direction of UASNs node positioning. Using neural dynamics as a solution method to carry out node positioning has become a research trend in the UASNs field.

[0005] For dynamic problems in the form of A(t)x(t) = b(t), the discrete-time model performs online solving of dynamic problems based on current and / or past data. Essentially, the solution of dynamic problems is a future calculation problem involving unknown parameters. However, in actual solving, ι+1 and b ι+1 's information can only be obtained until time ι+1 t. This means that in the absence of future information ι+1 and b ι+1 , the key to the solution lies in using the existing information ι and b ι to replace the future information ι+1 and b ι+1 and completing the solution of x ι+1 within the time interval [ιτ, (ι + 1)τ).

[0006] Usually, the ode45 solver in MATLAB is called to simulate and implement the circuit. However, the operation based on ode45 uses future time data in addition to the existing time data at each sampling interval. In addition, in the calculation of real numerical devices, the information of future time is not available and cannot be used at the current time. Therefore, the continuous-time reset neural dynamics model cannot be directly applied to numerical devices. Some scholars have proposed a discrete-time reset neural dynamics algorithm for solving time-varying linear equations. However, the sensitivity of this model to noise will lead to a decline or even failure in the solution performance. Some scholars have studied two discrete-time neural dynamics models for solving online time-varying complex-valued quadratic programming problems under linear constraints. The discrete neural dynamics model proposed by some scholars can effectively and accurately solve time-linear equations containing matrix inversion operations in the real domain.

[0007] Although some achievements have been made in the design and analysis of neurodynamics models for online solving of dynamic problems, for complex underwater dynamic environments, existing research still has some deficiencies and cannot be directly applied to UASNs node positioning, which is worthy of further improvement and optimization. For example, existing discrete-type work does not involve any solution methods with underwater noise suppression capabilities. Therefore, combining the time-varying characteristics of UASNs node positioning and challenges such as environmental noise, conducting research on UASNs positioning methods based on discrete-type reset neurodynamics models has certain scientific research value and practical significance. Summary of the Invention

[0008] The content of the present invention is to provide a positioning method based on a discrete-type reset neurodynamics model, which can better perform UASNs positioning.

[0009] A positioning method based on a discrete-type reset neurodynamics model according to the present invention includes the following steps:

[0010] 1. Analyze the UASNs positioning problems existing in the continuous model and establish a continuous reset neurodynamics model;

[0011] 2. Explore numerical difference methods that can be used for discretization of continuous systems and select appropriate numerical difference formulas;

[0012] 3. Derive an easily implementable discrete-type reset neurodynamics model;

[0013] 4. Solve the UASNs positioning problem through the discrete-type reset neurodynamics model to obtain the positioning result. Preferably, in step 1, the UASNs positioning problem is expressed as a dynamic matrix equation and described as:

[0014] S(t)q(t) = m(t)

[0015] In the formula, the coefficient matrix involves the coordinates of anchor nodes and the arrival angle or time difference of arrival of acoustic signals, where and p = m - 1; is a known vector, which also involves the coordinates of anchor nodes and the arrival angle or time difference of arrival of communication signals; is the vector to be solved, which involves the positions of unknown nodes;

[0016] The error function e(t) of the UASNs positioning problem is constructed in the following form:

[0017] e(t) = S(t)q(t) - m(t)

[0018] When e(t) is close to zero, the obtained solution q(t) is close to the theoretical solution q *(t); Therefore, the evolution formula of e(t) is designed as:

[0019]

[0020] where the parameters k1, k2 > 0 are used to adjust the convergence rate of which is the first-order time derivative of e(t), and τ represents the sampling interval; Next, expanding the above formula can obtain the continuous reset neural dynamics model:

[0021]

[0022] In the context of UASNs, S(t) is usually full column rank; Finally, the continuous reset neural dynamics model is expressed in the following form:

[0023]

[0024] where represents the inverse of the matrix S(t).

[0025] Preferably, in step two, the Euler forward difference formula is selected to discretize the continuous model, thereby obtaining the discrete reset neural dynamics model;

[0026] The Euler forward difference formula is:

[0027]

[0028] q ι 、q ι+1 represent vectors at times l and l + 1, represents the first-order time derivative of q ι .

[0029] Preferably, in step three, substituting the Euler forward difference formula into the continuous reset neural dynamics model, we get:

[0030]

[0031] where h1 = τk1, h2 = τ 2 k2; S ι represents the matrix S at time l, represents the inverse matrix of S ι at time l, m ι represents the vector m at time l, and g is the summation index variable;

[0032] Finally, for and using the Euler backward difference formula, the discrete reset neural dynamics model in the following form is obtained:

[0033]

[0034] Name the above model as the DIND model.

[0035] The beneficial effects of the present invention are as follows:

[0036] (1) The present invention proposes a positioning method based on a discrete reset neural dynamics model, which focuses on solving the problem that continuous models are difficult to implement on numerical devices, improving the practicability of the method. At the same time, it also takes into account the suppression of underwater noise;

[0037] (2) Compared with existing classical methods, the model proposed in the present invention has superiority in both solution accuracy and noise suppression. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 is a flowchart of a positioning method based on a discrete reset neural dynamics model;

[0039] FIG. 2(a) is a trajectory diagram of q(t) generated by the DIND model in an ideal zero-disturbance environment;

[0040] FIG. 2(b) is a trajectory diagram of q(t) generated by the PSO algorithm in an ideal zero-disturbance environment;

[0041] FIG. 2(c) is a trajectory diagram of q(t) generated by the NI algorithm in an ideal zero-disturbance environment;

[0042] FIG. 2(d) is a trajectory diagram of q(t) generated by the MZDL1 model in an ideal zero-disturbance environment;

[0043] FIG. 2(e) is a diagram of the algorithm convergence in an ideal zero-disturbance environment;

[0044] FIG. 2(f) is an X, Y coordinate error diagram in an ideal zero-disturbance environment;

[0045] FIG. 2(g) is a positioning error diagram in an ideal zero-disturbance environment;

[0046] FIG. 2(h) is a diagram of the number of iterations corresponding to different initial errors in an ideal zero-disturbance environment;

[0047] FIG. 3(a) is in a constant disturbance environment, a trajectory diagram of q(t) generated by the DIND model;

[0048] FIG. 3(b) is in a constant disturbance environment, a trajectory diagram of q(t) generated by the PSO algorithm;

[0049] FIG. 3(c) is in a constant disturbance The q(t) trajectory diagram generated by the NI algorithm in the

[0050] Figure 3(d) shows that under a constant perturbation environment, the q(t) trajectory diagram generated by the MZDL1 model;

[0051] Figure 3(e) shows that under a constant perturbation environment, the diagram of the algorithm convergence situation;

[0052] Figure 3(f) shows that under a constant perturbation environment, the X and Y coordinate error diagram;

[0053] Figure 3(g) shows that under a constant perturbation environment, the positioning error diagram;

[0054] Figure 3(h) shows that under a constant perturbation environment, the schematic diagram of the number of iterations corresponding to different initial errors;

[0055] Figure 4(a) shows that under a bounded random perturbation environment, the q(t) trajectory diagram generated by the DIND model;

[0056] Figure 4(b) shows that under a bounded random perturbation environment, the q(t) trajectory diagram generated by the PSO algorithm;

[0057] Figure 4(c) shows that under a bounded random perturbation environment, the q(t) trajectory diagram generated by the NI algorithm;

[0058] Figure 4(d) shows that under a bounded random perturbation environment, the q(t) trajectory diagram generated by the MZDL1 model;

[0059] Figure 4(e) shows that under a bounded random perturbation environment, the diagram of the algorithm convergence situation;

[0060] Figure 4(f) shows that under a bounded random perturbation environment, the X and Y coordinate error diagram;

[0061] Figure 4(g) shows that under a bounded random perturbation environment, the positioning error diagram;

[0062] Figure 4(h) shows that under a bounded random perturbation environment, the schematic diagram of the number of iterations corresponding to different initial errors. Detailed implementation manner

[0063] To further understand the content of the present invention, the present invention will be described in detail in combination with the accompanying drawings and embodiments. It should be understood that the embodiments are only for explaining the present invention and not for limiting it.

[0064] Embodiment

[0065] As Figure 1 shown, this embodiment provides a positioning method based on a discrete reset neural dynamics model, which is characterized in that it includes the following steps:

[0066] 1. Analyze the UASNs positioning problem existing in the continuous model and establish a continuous reset neural dynamics model;

[0067] 2. Explore numerical difference methods that can be used for discretizing the continuous system and select appropriate numerical difference formulas;

[0068] 3. Derive a discrete reset neural dynamics model that is easy to implement;

[0069] 4. Solve the UASNs positioning problem through the discrete reset neural dynamics model to obtain the positioning result.

[0070] In step 1, the UASNs positioning problem is expressed as a dynamic matrix equation and described as:

[0071] S(t)q(t) = m(t)

[0072] In the formula, the coefficient matrix involves the coordinates of the anchor nodes and the arrival angle or time difference of arrival of the acoustic wave signals, where and p = m - 1; is a known vector, which also involves the coordinates of the anchor nodes and the arrival angle or time difference of arrival of the communication signals; is the vector to be solved, which involves the position of the unknown node;

[0073] The error function e(t) of the UASNs positioning problem is constructed in the following form:

[0074] e(t) = S(t)q(t) - m(t)

[0075] When e(t) is close to zero, the obtained solution q(t) is close to the theoretical solution q * (t); therefore, the evolution formula of e(t) is designed as:

[0076]

[0077] where the parameters k1, k2 > 0 are used to adjust the convergence rate of is the first-order time derivative of e(t), and τ represents the sampling interval; Next, expanding the above equation can obtain the continuous reset neural dynamics model:

[0078]

[0079] In the context of UASNs, S(t) is usually full column rank; Finally, the continuous reset neural dynamics model is expressed in the following form:

[0080]

[0081] where, represents the inverse of the matrix S(t).

[0082] In step two, the Euler forward difference formula is selected to discretize the continuous model, thereby obtaining the discrete reset neural dynamics model;

[0083] The Euler forward difference formula is:

[0084]

[0085] q ι 、q ι+1 represent the vectors at times l and l + 1, represents the first-order time derivative of q ι .

[0086] In step three, substituting the Euler forward difference formula into the continuous reset neural dynamics model, we get:

[0087]

[0088] In the formula, h1 = τk1, h2 = τ 2 k2; S ι represents the matrix S at time l, represents the inverse matrix of S ι at time l, m ι represents the vector m at time l, and g is the summation index variable;

[0089] Finally, for and using the Euler backward difference formula, the discrete reset neural dynamics model in the following form is obtained:

[0090]

[0091] The above model is named the DIND model.

[0092] Comparison and verification

[0093] In this embodiment, the DIND model, two classical algorithms, namely the NI algorithm and the PSO algorithm, and the MZDL1 model are used to solve the UASNs positioning problem and compared to verify the superiority of the DIND model.

[0094] Consider a two-dimensional UASNs scenario of 20m * 20m, where the positions of the unknown nodes and six anchor nodes are fixed. The above four are all used to solve the node positioning problem, and simulation experiments are constructed under three perturbation environments: 1) zero perturbation; 2) constant perturbation; 3) random bounded perturbation. The parameters involved in the experiment are the sampling interval τ = 0.01s, h1 = 0.1, h2 = 0.5; the learning factors c1 = c2 = 2, the inertia factors κ1 = 1, κ2 = 6, a = 0.2, b = 5. The comparison and analysis are mainly carried out from aspects such as the q(t) trajectory, the algorithm convergence situation, the positioning error ||q(t) - q * (t)||2, and the X and Y coordinate errors and so on. Among them, the functions used to measure the algorithm convergence situation are defined as the following two. The DIND model, the NI algorithm, and the MZDL1 model adopt the steady-state error ||e(t)||2, while the PSO algorithm adopts the fitness function.

[0095] For case 1), as Figures 2(a)-2(d) shown, the orange circles represent the anchor nodes, the purple triangles represent the unknown nodes, and the black dots show the q(t) trajectories, that is, the process solutions, of different algorithms during the solution process. The red stars represent the positions of the unknown nodes obtained when the algorithms reach the convergence state. It can be intuitively observed that the red stars in the three subgraphs all cover the purple triangles, indicating that the DIND model, the PSO algorithm, the NI algorithm, and the MZDL1 model can all calculate the positions of the unknown nodes with relatively low errors. Among them, the DIND model and the NI algorithm produce fewer process solutions, indicating that these two models can obtain the positions of the unknown nodes after fewer iteration times. Due to the characteristics of the PSO algorithm itself, the process solutions it produces are more scattered and may jump from an intermediate solution close to the unknown node to a direction away from the true position, resulting in a waste of solution resources. In contrast, although the MZDL1 model produces more process solutions, it quickly jumps from a relatively far initial solution to an intermediate solution close to the unknown node and gradually approaches the unknown node. As for the algorithm convergence situation, as shown in Fig. 2(e), due to the different definitions of the functions, the initial values of the PSO algorithm and the other three algorithms on the vertical axis are different, but all four algorithms finally reach the convergence state. The convergence situations of the four algorithms for the positioning errors of the X and Y components of the unknown node coordinates are reflected in Fig. 2(f). For the positioning error index, it can be observed from Fig. 2(g) that the NI algorithm and the DIND model have better performance, with a faster convergence speed than the PSO and MZDL1 models and a lower positioning error.

[0096] In addition, from the perspective of the number of iterations, the above four algorithms are compared and analyzed in this embodiment, as shown in Fig. 2(h). In the face of different initial errors, both the DIND model and the NI algorithm can obtain the positions of unknown nodes with fewer iterations than the MZDL1 model and the PSO algorithm. Due to the setting of the static UASNs positioning scenario, as a numerical algorithm, the NI algorithm shows more prominent advantages. As the initial error increases, the number of iterations of the DIND model does not show an obvious increasing trend and remains at about 20 times. Due to its own characteristics, the number of iterations of the PSO algorithm shows large fluctuations and an increasing phenomenon.

[0097] Table 1 Comparison of positioning errors generated by algorithms in the face of constant perturbations of different sizes (unit: m)

[0098] Disturbance value DIND PSO NI MZDL1 1 2.35 2.80 2.39 0.15 2 0.52 3.08 0.52 0.73 3 0.04 4.20 0.09 0.87 4 0.40 5.27 0.59 0.96 5 0.20 6.32 1.05 1.03 6 0.22 7.36 1.49 1.08

[0099] Table 2 Comparison of positioning errors generated by algorithms starting from different initial values (unit: m)

[0100]

[0101]

[0102] For case 2), from Figures 3(a)-3(d) It is observed that the positions of the unknown nodes calculated by the DIND model are closer to the true positions. In contrast, the positions calculated by the NI algorithm, the PSO algorithm, and the MZDL1 model have a large deviation from the true positions. As for the positioning error, as Figures 3(e)-3(g) shown, the DIND model has the highest solution accuracy and the positioning error is lower than the other three algorithms. In addition, to further highlight the advantages of the DIND model, by adjusting the size of the constant perturbation value, the DIND model is compared with the other three algorithms in terms of positioning error. The summary results are shown in Table 1 and Fig. 3(h). As the constant perturbation increases, the positioning error of the PSO algorithm gradually increases and is significantly higher than that of other algorithms. Due to the adoption of the integral feedback mechanism and the combination of the Kalman filtering technology, the DIND model is less affected by the perturbation and demonstrates the optimal anti-perturbation performance.

[0103] The filtering process of the Kalman filtering technology for the measurement information is as follows:

[0104] Range-based positioning methods rely heavily on accurate distance measurements (i.e., AoA, TDoA measurements). However, due to the complexity of the underwater environment, the measurement information obtained from nature is often contaminated by environmental noise and is not pure. If not taken seriously and processed, the deviation of the measurement information will lead to serious positioning errors. Therefore, the Kalman filter algorithm is used to filter the measurement data contaminated by noise. The Kalman filter is one of the most important and commonly used filters. This filter describes a recursive solution to the problem of discrete data linear filtering, incorporating past measurement estimation errors into new measurement errors to estimate future errors. Therefore, this filter produces relatively accurate estimates based on inaccurate and uncertain measurements. The two core parts of the Kalman filter algorithm are update and prediction. Based on the previous state estimate (i.e., the measurement estimate at the previous moment) and the previous estimation uncertainty, the Kalman gain is calculated and the current state estimate and uncertainty are provided. Its mathematical form is described as follows:

[0105]

[0107] K g = p g,g-1 H T (Hp g,g-1 H T + R) -1

[0108] where the superscript T represents the transpose operation; g (g = 1, 2,...) represents the index of iteration; the vector represents the state estimate, which refers to the distance measurement information here; represents the estimation uncertainty; is the Kalman gain; represents the state transition matrix; is the covariance matrix of the process noise; is the covariance matrix of the measurement noise. Subsequently, the prediction process pushes the current state estimate and uncertainty p g,g-1 to the next state. Its mathematical form is described as follows:

[0109]

[0110] p g,g =(I - K g H)p g,g-1

[0111] where the vector is the measurement value, which refers to the unprocessed measurement information α1(t),..., α m (t) or ΔT 21(t),...,ΔT m1 (t); α1(t),...,α m (t) represents the angles of signal arrival from the unknown node to the 1st,..., mth anchor nodes in the AoA positioning method, ΔT 21 (t),...,ΔT m1 (t) represents the time differences of the signal propagating from the unknown node to the 2nd,.., mth and the 1st anchor nodes respectively in the TDoA positioning method, denotes the identity matrix. Even if the initial values and p 0,0 are not precise, through the above iterative process, the Kalman filtering algorithm can effectively filter out noise and converge to the true distance measurement information or

[0112] For case 3), from Figures 4(a)-4(d) it is observed that, compared with the NI algorithm, PSO algorithm and MZDL1 model, the position of the unknown node calculated by the DIND model is closer to the true position. Regarding the positioning error, as Figures 4(e)-4(g) shown, the DIND model has the highest solution accuracy, the positioning error quickly reaches the convergence state and is lower than the other three algorithms. In addition, to further highlight the advantages of the DIND model, by adjusting the magnitude of the initial error, the DIND model is compared with the other three algorithms in terms of positioning error. The summary results are shown in Table 2 and Figure 4(h). As the initial error increases, the positioning errors of the NI algorithm, PSO algorithm and MZDL1 model gradually increase, while the positioning error generated by the DIND model slightly increases but is significantly lower than the other three algorithms. This benefits from the fact that the model integrates the Kalman filtering technology and the integral feedback mechanism, greatly reducing the influence brought by random bounded perturbations and demonstrating excellent robustness. The above simulation results show that the proposed DIND model has superiority and effectiveness in terms of accuracy and robustness when solving the UASNs positioning problem.

[0113] In view of the fact that the continuous model is difficult to implement on numerical devices, a numerical difference method that can discretize the continuous model for solving dynamic problems is explored. Taking the practicality of the positioning method as the starting point, the idea of discretization is introduced to optimize the model, and a positioning method based on the discrete reset neural dynamics model is proposed. Specifically, the model is discretized through the Euler forward difference formula to obtain its implementation form in the discrete time domain. The integral feedback mechanism and the Kalman filtering algorithm are continued to further ensure the robustness of the positioning method against environmental noise and internal perturbations. Strict theoretical analysis and computer simulations verify the effectiveness and feasibility of this method for solving the UASNs positioning problem under various perturbations.

[0114] The above has schematically described the present invention and its implementation manners. This description is not restrictive. What is shown in the drawings is only one of the implementation manners of the present invention, and the actual structure is not limited thereto. Therefore, if those of ordinary skill in the art are inspired by this and, without departing from the gist of the present invention, design similar structural manners and embodiments to this technical solution without creative efforts, they shall fall within the protection scope of the present invention.

Claims

1. A positioning method based on a discrete zeroing neural dynamics model, characterized in that: The following steps are involved: First, analyze the UASNs localization problem existing in the continuous model and establish a continuous zeroing neural dynamics model; 2. Explore the numerical difference method that can be used for discretization of continuous systems and select the appropriate numerical difference formula; 3. Derive a discrete zeroing neural dynamics model that is easy to implement; 4. Solve the UASNs positioning problem through the discrete zeroing neural dynamics model and obtain the positioning result.

2. The positioning method based on the discrete zeroing neural dynamics model according to claim 1 is characterized in that: In step 1, the UASNs positioning problem is expressed as a dynamic matrix equation and described as: S(t)q(t)=m(t) In the formula, the coefficient matrix Involving the coordinates of the anchor node and the arrival angle or arrival time difference of the sound wave signal, where And p = m-1; is a known vector, similarly involving the coordinates of the anchor node and the arrival angle or arrival time difference of the communication signal; is the vector to be solved, involving the position of the unknown node; The error function e(t) of the UASNs positioning problem is constructed as follows: e(t)=S(t)q(t)-m(t) When e(t) is close to zero, the solution q(t) is close to the theoretical solution q * (t); therefore, the evolution formula of e(t) is designed as: Among them, parameters k1, k2> 0 are used to adjust The convergence rate of is the first-order time derivative of e(t), and τ represents the sampling interval; Next, the above formula is expanded to obtain the continuous zeroing neural dynamics model: In the context of UASNs, S(t) is usually full rank; finally, the continuous zeroing neural dynamics model is expressed as follows: in, represents the inverse of the matrix S(t).

3. The positioning method based on the discrete zeroing neural dynamics model according to claim 2 is characterized in that: In step 2, the Euler forward difference formula is selected to discretize the continuous model, thereby obtaining a discrete zeroing neural dynamics model; Euler forward difference formula is: q ι ,q ι+1 represents the vector at time l and l+1, Indicates q ι The first time derivative of .

4. The positioning method based on the discrete zeroing neural dynamics model according to claim 3 is characterized in that: In step 3, by substituting the Euler forward difference formula into the continuous zeroing neural dynamics model, we get: Where h1 = τk1, h2 = τ 2 k2;S ι represents the matrix S at time l, Indicates time S ι The inverse matrix, m ι represents the vector m at time l, and g is the summation index variable; Finally, and Using Euler's backward difference formula, we get the following discrete zeroing neural dynamics model: The above model is named DIND model.

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