TBM cutter head rub-impact fault diagnosis method based on vibration frequency micro fluctuation

Abstract

The invention discloses a TBM cutterhead rub-impact fault diagnosis method based on vibration frequency micro fluctuation. The method comprises the following steps: collecting cutterhead spindle vibration signal data; obtaining each frequency component of the vibration signal according to the frequency spectrum component of the vibration signal; carrying out time-frequency analysis on the component signal by adopting a generalized synchronous compression transformation time-frequency analysis method to obtain a time-frequency distribution result of the component signal; according to a generalized synchronous compression transformation result of each component, extracting an instantaneous frequency ridge line of time-frequency distribution, removing a direct current component of the instantaneous frequency ridge line, calculating a frequency spectrum of the instantaneous frequency, and obtaining an oscillation frequency of the instantaneous frequency; judging whether periodic oscillation exists in the instantaneous frequency according to the frequency spectrum of each instantaneous frequency ridge line; if yes, judging whether the oscillation frequency is equal to the harmonic frequency of the vibration signal; and if the conditions are met, judging that the TBM cutterhead system has a rub-impact fault, and judging the rub-impact condition of the cutterhead according to the oscillation frequency. According to the invention, the accuracy and sensitivity of diagnosis are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of intelligent equipment status monitoring, and in particular relates to a TBM cutter head rubbing fault diagnosis method based on vibration frequency micro-fluctuation. Background Art

[0002] As an important construction equipment in modern tunnel engineering, the working efficiency and stability of the full-section rock tunnel boring machine (TBM) directly affect the construction progress and cost of the tunnel engineering. As one of the key components, the TBM cutterhead is in direct contact with the rock during the excavation process and is subjected to huge cutting forces and impact loads. Therefore, the working state of the cutterhead directly determines the overall performance of the TBM. During the TBM excavation process, cutterhead rubbing failure is a common and serious type of failure. Rubbing failure will not only lead to increased tool wear and reduced cutting efficiency, but may also cause a series of problems such as increased equipment vibration, increased noise, and increased energy consumption. In severe cases, it may even cause the equipment to shut down for maintenance, seriously affecting the construction progress and cost control.

[0003] At present, the main diagnostic methods for TBM cutter head friction faults include vibration spectrum analysis, acoustic emission monitoring, image processing, etc. These methods have their own advantages and disadvantages, but there are still some shortcomings in practical applications. For example, although the vibration signal spectrum analysis method can reflect the vibration state of the cutter head, when processing complex and changeable vibration signals, it may be affected by factors such as noise interference and signal attenuation, resulting in inaccurate diagnostic results. The acoustic emission monitoring method has high requirements for the collection and analysis of sound waves and is easily interfered by environmental noise. The image processing method relies on high-quality image data, and the computational complexity of the image processing algorithm is high.

[0004] Although the traditional time domain signal analysis method of TBM cutter head system rubbing fault can judge the system rubbing fault by observing the time domain waveform characteristics of the vibration signal, such as waveform distortion, clipping phenomenon, etc., in actual engineering, the vibration signal is often interfered by various noises. The traditional time domain analysis method is easily affected by waveform distortion when processing noisy signals, resulting in inaccurate diagnosis results. Using the spectrum characteristics of the vibration signal, including the power frequency (1X), multiple frequency (such as 2X, 3X, etc.) and low-frequency and high-frequency components, according to the rich spectrum characteristics that appear in the vibration signal, such as the appearance of subharmonics and high-order harmonics, it is an important feature to judge the rotor system rubbing fault. However, the spectrum information is often difficult to capture the subtle changes in the instantaneous frequency during the rubbing process of the TBM cutter head system, and is easily interfered by the spectrum characteristics of other fault types. Summary of the invention

[0005] Purpose of the invention: In order to more accurately diagnose TBM cutter head rubbing faults and improve the real-time and reliability of fault diagnosis, the present invention proposes a TBM cutter head rubbing fault diagnosis method based on vibration frequency micro-fluctuation. The method uses a high-precision non-stationary signal time-frequency analysis method to accurately extract the instantaneous frequency micro-fluctuation characteristics of the cutter head vibration signal, thereby realizing rapid and accurate diagnosis of cutter head rubbing faults.

[0006] Technical solution: To achieve the purpose of the present invention, the technical solution adopted by the present invention is:

[0007] A TBM cutter head rubbing fault diagnosis method based on vibration frequency micro-fluctuation includes the following steps:

[0008] Step 1: Collect the vibration signal data of the cutter head spindle;

[0009] Step 2: According to the spectral components of the vibration signal, the signal is decomposed to obtain the frequency components of the cutter head spindle vibration signal;

[0010] Step 3: Use the generalized synchronous compression transform time-frequency analysis method to perform time-frequency analysis on the component signals decomposed in step 2 to obtain their time-frequency distribution results;

[0011] Step 4: According to the generalized synchronous compression transformation results of each component, the instantaneous frequency ridge of the time-frequency distribution is extracted, and after removing the DC component of the instantaneous frequency ridge, the spectrum of the instantaneous frequency is calculated to obtain the oscillation frequency of the instantaneous frequency;

[0012] Step 5: Determine whether the instantaneous frequency has periodic oscillation according to the frequency spectrum of each instantaneous frequency ridge line. If there is no periodic oscillation, it means that there is no friction fault; otherwise, execute step 6;

[0013] Step 6: Determine whether the TBM cutter head system has a friction fault, then determine whether the oscillation frequency is equal to the harmonic frequency of the vibration signal, and determine the condition of the cutter head friction based on the oscillation frequency.

[0014] Furthermore, the generalized synchronous compression transformation method in step 3 is as follows:

[0015] (1) First, the improved short-time Fourier transform of the vibration signal based on the Gaussian window function in different transformation forms is calculated. The calculation process is as follows:

[0016]

[0017] Where, T n (t,ω) is the time-frequency distribution result obtained by improving the short-time Fourier transform of the vibration signal based on the Gaussian window function in different transformation forms, t is the time variable, τ is the time shift variable, and x(τ) is the time domain vibration signal; is the time domain Gaussian window function, σ is the control parameter of the window function, n∈N " ;

[0018] For the N-order generalized synchrosqueezed transform, 2N-1 improved short-time Fourier transforms based on different forms of Gaussian window functions are calculated;

[0019] (2) Construct the generalized time-frequency matrix A, which is as follows:

[0020]

[0021] Where, the elements of the generalized time-frequency matrix A are the time-frequency distribution results obtained when n takes different values ​​in formula (1), n ​​= 1, 2, ..., 2N-1; T n =T n (t,ω);

[0022] (3) Solve the generalized instantaneous frequency explicit expression of the N-order instantaneous frequency frequency modulated signal based on the generalized time-frequency matrix A: The calculation formula is as follows:

[0023]

[0024] Where ω is the frequency domain independent variable, is the Gaussian window function parameter, To calculate the imaginary part of the function, A 1,n is the algebraic cofactor of the generalized time-frequency matrix A;

[0025] (4) Through multiple compression transformations, the time-frequency distribution energy of the short-time Fourier transform is rearranged at the generalized instantaneous frequency estimate to obtain the time-frequency distribution result; the generalized synchronous compression transformation algorithm is expressed as follows:

[0026]

[0027] In the formula, is the result of generalized synchronous compression transformation, T(t,η) is the time-frequency distribution result obtained by improving short-time Fourier transform of vibration signal based on Gaussian window function with different transformation forms, δ() is Dirac function, is the M-iteration result of the generalized instantaneous frequency estimation operator, and its N-order form is as follows:

[0028]

[0029] Furthermore, the ridge extraction method in step 4 is as follows:

[0030] The optimal path is designed by using the time-frequency ridge extraction method based on dynamic path optimization, combining the continuity of the time-frequency ridge and the maximum point of the time-frequency energy in the frequency direction;

[0031] The optimal path is constructed as follows:

[0032]

[0033] Where, {IF(1),…,IF(N)} is the extracted instantaneous frequency ridge, GST(m,n) is the M×N time-frequency distribution matrix, which represents the time-frequency extraction transformation result; m and n are discrete variables corresponding to the continuous variables t and ω, m = 1,…,M, n = 1,…,N; λ is the penalty factor used to ensure the continuity of the ridge;

[0034] IF(n) is the position taken in the nth column of the matrix GST(m,n). When columns 1 to N are traversed, the extracted ridge {IF(1),…,IF(N)} is formed, which is the optimal path.

[0035] Beneficial effects: Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

[0036] The present invention can suppress the interference of noise to a certain extent and improve the reliability of the diagnosis result by decomposing the signal into various characteristic components and using a high-precision time-frequency analysis method to calculate the time-frequency distribution of the signal. The method of the present invention can extract the instantaneous frequency ridge of the fault signal, thereby revealing the time-frequency characteristics unique to the rubbing fault. Through the generalized synchronous compression transform time-frequency analysis method, the instantaneous frequency changes in the vibration signal, especially the tiny instantaneous frequency fluctuations, can be captured more accurately. This high-precision analysis makes it possible to detect rubbing faults early and identify subtle changes, thereby improving the accuracy and sensitivity of the diagnosis. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 It is a flow chart of the method proposed by the present invention.

[0038] Figure 2 This is a diagram of the Jeffcott rotor system.

[0039] Figure 3 It is the vibration response of the rotor system under rubbing fault conditions.

[0040] Figure 4 It is the time-frequency distribution result of the fundamental frequency component using different time-frequency analysis methods.

[0041] Figure 5 It is the signal analysis result of the fundamental frequency component.

[0042] Figure 6 It is the analysis process of the 2 times frequency component of the rubbing fault signal.

[0043] Figure 7 It is the analysis process of the 3 times frequency component of the rubbing fault signal.

[0044] Figure 8 It is the analysis process of the 4th frequency component of the rubbing fault signal. DETAILED DESCRIPTION

[0045] The technical solution of the present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0046] The present invention discloses a TBM cutter head rubbing fault diagnosis method based on vibration frequency micro-fluctuation, the process is as follows: Figure 1 As shown, the steps are as follows:

[0047] Firstly, the TBM cutter head rubbing fault signal is constructed.

[0048] In this embodiment, the Jeffcott rotor model is used to simulate the TBM cutter head rubbing fault to obtain the vibration response signal. The Jeffcott rotor system is as follows: Figure 2 shown.

[0049] Figure 2 (a) is a simplified diagram of the system. Figure 2 (b) is the dynamic analysis. According to the rotor system diagram, the center of the stator is set as the coordinate origin O, and the displacement of the rotor center O1 along the x and y directions of the coordinate axis is recorded as x(t) and y(t).

[0050] The excitation forces of the rotor system are the rotor unbalance force, gravity and friction. The dynamic equations of a single rotor along the x and y coordinate axes are as follows:

[0051]

[0052] In the formula, is the second-order derivative of x(t) and y(t), is the first-order derivative of x(t) and y(t), m is the concentrated mass of the rotor, c is the shaft damping, k is the stiffness coefficient, U is the rotor unbalance, ω is the rotation frequency, g is the gravitational acceleration, and F n (x,y) is the friction force along the x-axis, F o (x,y) is the friction force along the y-axis. The friction force is expressed as follows:

[0053]

[0054] In the formula, is the radial displacement of the rotor, k x is the stator stiffness, δ is the initial gap between the rotor and the stator, λ is the friction coefficient, ε(·) is the step function, and the expression is as follows:

[0055]

[0056] This shows that the rotor will rub only when the radial displacement r of the rotor is greater than or equal to the initial gap δ. Table 1 shows the parameters of the above rotor system. The above dynamic equations are solved by the Runge-Kutta method.

[0057] Table 1 Rotor system parameters

[0058]

[0059] By setting the initial gap between the rotor and the stator to δ=1e-4m and the rotation frequency to ω=100Hz, the friction failure of the TBM cutter rotor system is simulated.

[0060] Secondly, the signal is decomposed into components to obtain the various frequency components of the cutter head spindle vibration signal.

[0061] The vibration response time domain signal and its spectrum of the rotor system are as follows: Figure 3 As shown, Figure 3 (a) is the time domain waveform of the vibration signal. Figure 3 (b) in the figure is the spectrum of the rubbing signal. This shows that when the rotor system rubs, in addition to the 1st harmonic frequency (100Hz), there will also be 2nd harmonic frequency, 3rd harmonic frequency and other harmonic components in the spectrum. Filtering is performed according to each characteristic frequency component in the spectrum, and each component is analyzed separately.

[0062] Then, the generalized synchronous compression transform time-frequency analysis method is used to perform time-frequency analysis on the decomposed component signals to obtain their time-frequency distribution results. The generalized synchronous compression transform method is as follows:

[0063] (1) First, the improved short-time Fourier transform of the vibration signal based on the Gaussian window function in different transformation forms is calculated. The calculation process is as follows:

[0064]

[0065] Where, T n (t,ψ) is the time-frequency distribution result obtained by improving the short-time Fourier transform of the vibration signal based on the Gaussian window function in different transformation forms, t is the time variable, τ is the time shift variable, and x(τ) is the time domain vibration signal; is the time domain Gaussian window function, σ is the control parameter of the window function, n∈N " .

[0066] For the N-th order generalized synchrosqueezed transform, 2N-1 improved short-time Fourier transforms based on different forms of Gaussian window functions are calculated.

[0067] (2) Construct a generalized time-frequency matrix A to solve the generalized instantaneous frequency explicit expression of the N-order instantaneous frequency frequency modulated signal. The generalized time-frequency matrix is ​​constructed as follows:

[0068]

[0069] Where, the elements of the generalized time-frequency matrix A are the time-frequency distribution results obtained when n takes different values ​​in formula (1), n ​​= 1, 2, ..., 2N-1; T n =T n (t,ω).

[0070] (3) Solve the generalized instantaneous frequency explicit expression of the N-order instantaneous frequency frequency modulated signal based on the generalized time-frequency matrix A: The calculation formula is as follows:

[0071]

[0072] Where ω is the frequency domain independent variable, is the Gaussian window function parameter, To calculate the imaginary part of the function, A 1,n is the algebraic cofactor of the generalized time-frequency matrix A.

[0073] (4) Through multiple compression transformations, the time-frequency distribution energy of the short-time Fourier transform is rearranged at the generalized instantaneous frequency estimate to obtain the time-frequency distribution result; the generalized synchronous compression transformation algorithm is expressed as follows:

[0074]

[0075] In the formula, is the result of generalized synchronous compression transformation, T(t,η) is the time-frequency distribution result obtained by improving short-time Fourier transform of vibration signal based on Gaussian window function with different transformation forms, δ() is Dirac function, is the M-iteration result of the generalized instantaneous frequency estimation operator, and its N-order form is as follows:

[0076]

[0077] Next, according to the generalized synchronous compression transformation results of each component, the instantaneous frequency ridge of the time-frequency distribution is extracted, and after removing the DC component of the instantaneous frequency ridge, the spectrum of the instantaneous frequency is calculated to obtain the oscillation frequency of the instantaneous frequency.

[0078] The ridge extraction method is as follows:

[0079] The optimal path is designed by using the time-frequency ridge extraction method based on dynamic path optimization, combining the continuity of the time-frequency ridges and the maximum point of the time-frequency energy in the frequency direction.

[0080] The optimal path is constructed as follows:

[0081]

[0082] In the formula, {IF(1),…,IF(N)} is the extracted instantaneous frequency ridge, GST(m,n) is the M×N time-frequency distribution matrix, which represents the time-frequency extraction transformation result; m and n are discrete variables corresponding to the continuous variables t and ω, m = 1,…,M, n = 1,…,N; λ is the penalty factor used to ensure the continuity of the ridge. In formula (7), n takes a certain value to represent a certain step in the path.

[0083] IF(n) is the position taken in the nth column of the matrix GST(m,n). When columns 1 to N are traversed, the extracted ridge {IF(1),…,IF(N)} is formed, which is the optimal path.

[0084] Finally, the frequency spectrum of each instantaneous frequency ridge is used to determine whether there is periodic oscillation in the instantaneous frequency. If there is no periodic oscillation, it means that there is no rubbing fault. Otherwise, it is determined that a rubbing fault has occurred in the TBM cutter head system, and then it is determined whether the oscillation frequency is equal to the harmonic frequency of the vibration signal. The cutter head rubbing condition is determined based on the oscillation frequency.

[0085] In this embodiment, the 1-fold frequency component signal is analyzed. Figure 4 is the time-frequency distribution result of this component obtained by different time-frequency analysis methods. Figure 4 (a) in the figure is the time-frequency distribution result of short-time Fourier transform. Since the time-frequency resolution is too low, it cannot accurately describe the time-frequency characteristics of the component. Figure 4 (b) is the time-frequency distribution result of synchronous compression transformation. Figure 4 (c) is the time-frequency distribution result of the synchronous extraction transform. In (b) and (c), the synchronous compression transform and the synchronous extraction transform can only represent that the 1-fold frequency of the signal is 100 Hz, but they cannot provide more information, and they cannot use their time-frequency distribution to obtain the oscillation of the frequency component. Figure 4 (d) in the figure is the time-frequency distribution result of the generalized synchronous compression transform. Due to the use of a higher-order signal model construction algorithm, the generalized synchronous compression transform can better characterize the oscillation of the frequency components in the time-frequency distribution.

[0086] In order to accurately obtain the oscillation frequency of the instantaneous frequency, Figure 5 Given from Figure 4 (d) The instantaneous frequency ridges and their spectrum extracted from the time-frequency distribution of the generalized synchronization extraction transform. Figure 5 (a) in the figure is the instantaneous frequency ridge extraction result. Figure 5(b) in the figure is the spectrum of the instantaneous frequency ridge. According to the spectrum of the instantaneous frequency ridge, it can be found that the instantaneous frequency has strong oscillations, which indicates that a rubbing fault has occurred in the rotor system. At the same time, the fluctuation frequency of the instantaneous frequency also reflects the frequency of occurrence of rubbing. In this simulation signal, the frequency of occurrence of rubbing fault is equal to the fundamental frequency, indicating that a rubbing occurs between the rotor and the stator once per rotation.

[0087] The same analysis method is used to analyze the 2nd, 3rd and 4th frequency characteristic components. Figure 6 This is the analysis process of the 2-fold frequency component. Figure 6 (a) in the figure is the time-frequency distribution result calculated by generalized synchrosqueezing transform, which has higher time-frequency resolution and can clearly observe the weak time-frequency fluctuations of the 2-fold frequency component. Figure 6 (b) in the figure is the instantaneous frequency ridge extraction result, which is the ridge extraction result of the high-order synchronous compression transform time-frequency distribution, and can more clearly depict the fluctuation of the 2nd frequency. In order to more clearly characterize the fluctuation characteristics of the 2nd frequency instantaneous frequency, the spectrum analysis of the instantaneous frequency ridge without the DC component is performed, as shown in Figure 1. Figure 6 As shown in (c) in the figure, it can be found that the fluctuation frequency of the instantaneous frequency of the 2nd frequency is also 100Hz, which is consistent with the analysis result of the 1st frequency.

[0088] Figure 7 and Figure 8 This is the analysis process of the 3rd frequency component and the 4th frequency component. Figure 7 (a) is the time-frequency distribution result of the generalized synchrocompression transform of the 3-fold frequency component. Figure 7 (b) is the instantaneous frequency ridge extraction result of the 3-fold frequency component. Figure 7 (c) in the figure is the spectrum diagram of the instantaneous frequency ridge of the 3rd frequency component. Figure 8 (a) is the time-frequency distribution result of the generalized synchronous compression transform of the 4-fold frequency component. Figure 8 (b) is the instantaneous frequency ridge extraction result of the 4th frequency component. Figure 8 (c) in the figure is the spectrum diagram of the instantaneous frequency ridge of the 4th frequency component. The resolution of the time-frequency distribution result of the generalized synchronous compression transform is higher, and the weak time-frequency fluctuations of the characteristic frequency multiplication component can be observed more clearly. By extracting the ridge of the time-frequency distribution of the high-order synchronous compression transform, the fluctuation of the characteristic frequency multiplication can be more clearly portrayed. Spectral analysis of the instantaneous frequency ridge with the DC component removed can more clearly characterize the fluctuation characteristics of the instantaneous frequency of the characteristic frequency multiplication, and the fluctuation of the instantaneous frequency is also 100Hz. This shows that the instantaneous frequency of each characteristic component in the signal has produced weak fluctuations due to the friction fault, and the fluctuation frequency is consistent, all 100Hz. The fluctuation frequency is equal to 1 times the rotation frequency, which proves that the friction fault occurs once per rotation.

Claims

1. A TBM cutter head rubbing fault diagnosis method based on vibration frequency micro-fluctuation, characterized in that: The steps include: Step 1: Collect the vibration signal data of the cutter head spindle; Step 2: According to the spectral components of the vibration signal, the signal is decomposed to obtain the frequency components of the cutter head spindle vibration signal; Step 3: Use the generalized synchronous compression transform time-frequency analysis method to perform time-frequency analysis on the component signals decomposed in step 2 to obtain their time-frequency distribution results; Step 4: According to the generalized synchronous compression transformation results of each component, the instantaneous frequency ridge of the time-frequency distribution is extracted, and after removing the DC component of the instantaneous frequency ridge, the spectrum of the instantaneous frequency is calculated to obtain the oscillation frequency of the instantaneous frequency; Step 5: Determine whether the instantaneous frequency has periodic oscillation according to the frequency spectrum of each instantaneous frequency ridge line. If there is no periodic oscillation, it means that there is no friction fault; otherwise, execute step 6; Step 6: Determine whether the TBM cutter head system has a friction fault, then determine whether the oscillation frequency is equal to the harmonic frequency of the vibration signal, and determine the condition of the cutter head friction based on the oscillation frequency.

2. A TBM cutter head rubbing fault diagnosis method based on vibration frequency micro-fluctuation according to claim 1, characterized in that: The generalized synchronous compression transformation method in step 3 is as follows: (1) First, the improved short-time Fourier transform of the vibration signal based on the Gaussian window function in different transformation forms is calculated. The calculation process is as follows: Where, T n (t,ω) is the time-frequency distribution result obtained by improving the short-time Fourier transform of the vibration signal based on the Gaussian window function in different transformation forms, t is the time variable, τ is the time shift variable, and x(τ) is the time domain vibration signal; is the time domain Gaussian window function, σ is the control parameter of the window function, n∈N " ; For the N-order generalized synchrosqueezed transform, 2N-1 improved short-time Fourier transforms based on different forms of Gaussian window functions are calculated, n = 1, 2, ..., 2N-1; (2) construct a generalized time-frequency matrix A, and solve the generalized instantaneous frequency explicit expression of the N-order instantaneous frequency frequency modulated signal based on the generalized time-frequency matrix A; (3) Through multiple compression transformations, the time-frequency distribution energy of the short-time Fourier transform is rearranged at the generalized instantaneous frequency estimate to obtain the time-frequency distribution result.

3. The TBM cutter head rubbing fault diagnosis method based on vibration frequency micro-fluctuation according to claim 2 is characterized in that: Construct the generalized time-frequency matrix A in the following form: Where, the elements of the generalized time-frequency matrix A are the time-frequency distribution results obtained when n takes different values ​​in formula (1), n ​​= 1, 2, ..., 2N-1; T n =T n (t,ω); Solve the generalized instantaneous frequency explicit expression of N-order instantaneous frequency frequency modulated signal based on the generalized time-frequency matrix A The calculation formula is as follows: Where ω is the frequency domain independent variable, is the Gaussian window function parameter, To calculate the imaginary part of the function, A 1,n is the algebraic cofactor of the generalized time-frequency matrix A.

4. The TBM cutter head rubbing fault diagnosis method based on vibration frequency micro-fluctuation according to claim 3 is characterized in that: The calculation formula of generalized synchronous compression transformation is as follows: In the formula, is the result of generalized synchronous compression transformation, T(t,η) is the time-frequency distribution result obtained by improving short-time Fourier transform of vibration signal based on Gaussian window function with different transformation forms, δ() is Dirac function, is the M-iteration result of the generalized instantaneous frequency estimation operator, and its N-order form is as follows:

5. The TBM cutter head rubbing fault diagnosis method based on vibration frequency micro-fluctuation according to claim 4 is characterized in that: The ridge extraction method in step 4 is as follows: The optimal path is designed by using the time-frequency ridge extraction method based on dynamic path optimization, combining the continuity of the time-frequency ridge and the maximum point of the time-frequency energy in the frequency direction; The optimal path is constructed as follows: Where, {IF(1),…,IF(N)} is the extracted instantaneous frequency ridge, GST(m,n) is the M×N time-frequency distribution matrix, which represents the time-frequency extraction transformation result; m and n are discrete variables corresponding to the continuous variables t and ω, m = 1,…,M, n = 1,…,N; λ is the penalty factor used to ensure the continuity of the ridge; IF(n) is the position taken in the nth column of the matrix GST(m,n). When columns 1 to N are traversed, the extracted ridge {IF(1),…,IF(N)} is formed, which is the optimal path.

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