A method for compensating fiber optic gyroscope vibration errors based on VMD and FPA-WT

Abstract

The present invention discloses a fiber optic gyroscope vibration error compensation method based on VMD and FPA-WT, which belongs to the field of fiber optic gyroscope testing. The method combines the advantages of variational mode decomposition (VMD) and wavelet transform (WT), decomposes the fiber optic gyroscope vibration signal through VMD, effectively avoids the endpoint effect and modal aliasing phenomenon that appear in empirical mode decomposition (EMD), and realizes the separation of effective signal and zero-bias drift error signal; the decomposed effective signal is filtered and denoised through wavelet transform. In wavelet transform, the selection of initial parameters has a very important influence on its denoising effect, so the flower pollination algorithm (FPA) is used to select the optimal parameters of wavelet transform. The present invention combines the advantages of VMD and FPA-WT algorithms, reduces the vibration error of the fiber optic gyroscope, and improves the performance of the fiber optic gyroscope in a vibration environment.

Description

Technical Field

[0001] The present invention relates to the field of fiber optic gyroscope testing, and in particular to a fiber optic gyroscope vibration error compensation method based on VMD and FPA-WT. Technical Background

[0002] Fiber optic gyroscopes (FOGs) use the Sagnac effect to provide angular velocity information about their carriers. They offer advantages such as solid-state operation, high reliability, and adaptability to diverse application requirements. They play a vital role in aerospace, autonomous driving, robotics, and other fields, garnering widespread attention. However, under stress, the optoelectronic components within FOGs can generate significant errors. This is primarily due to vibration-induced additional birefringence in the fiber, resulting in nonreciprocal phase errors caused by changes in the fiber's refractive index. This problem can be addressed through hardware optimization or algorithmic compensation. Hardware optimization increases the difficulty of mass production of FOGs and is less effective in practical applications. Algorithmic compensation, however, requires no additional components, offers greater flexibility and lower costs, making it a research hotspot. The empirical mode decomposition algorithm decomposes signals based on the inherent timescale characteristics of the data without requiring predefined basis functions. It is theoretically applicable to any signal decomposition type, particularly for nonlinear and nonstationary signals, and can be applied to FOG vibration error compensation. However, in the presence of excessive noise, modal aliasing and endpoint effects can occur during the empirical mode decomposition process, leading to unstable or even negative decomposition frequencies. Therefore, a method is needed to overcome the shortcomings of EMD decomposition and effectively compensate for the vibration error of the fiber optic gyroscope. Summary of the Invention

[0003] The purpose of the present invention is to provide a fiber optic gyroscope vibration error compensation method based on VMD and FPA-WT, which is used to compensate for the errors generated by the optoelectronic devices inside the fiber optic gyroscope under a vibration environment and improve the accuracy of the fiber optic gyroscope in a dynamic environment.

[0004] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0005] A method for compensating fiber optic gyroscope vibration errors based on VMD and FPA-WT includes the following steps:

[0006] Step 1: Decompose the FOG output signal using the VMD algorithm: Establish a variational constraint model and solve the bandwidth of each modal function component IMF, calculate the IMF center frequency, and achieve adaptive decomposition of the signal;

[0007] Step 2: Obtain the optimal parameters of wavelet transform through FPA algorithm;

[0008] Step 3: Use wavelet transform to filter each level of IMF components decomposed by VMD algorithm;

[0009] Step 4: Use inverse wavelet transform to reconstruct the FOG output signal to obtain the compensated FOG output.

[0010] Furthermore, the step 1 includes:

[0011] 1.1) Perform a Hilbert transform on the FOG output signal x(t) to obtain the analytical form of each modal function component IMF, and then obtain the marginal spectrum:

[0012]

[0013] Among them, u k (t) is the kth eigenmode function; δ(t) is the impulse function; j is a complex unit; t is the time;

[0014] 1.2) The construction frequency is the exponential term of the center frequency of each eigenmode function The corresponding eigenmode function is modulated to the baseband using multiplication of the exponential function:

[0015]

[0016] Among them, ω k is the center frequency of the kth eigenmode function;

[0017] 1.3) By calculating the square of the L2 norm of the first-order derivative of the demodulated signal shown in formula (2), the bandwidth of the corresponding intrinsic mode function is estimated, and the constrained variation problem in the VMD algorithm is expressed as:

[0018]

[0019] Among them, {u k}={u1,u2,…,u K} represents the set of all eigenmode functions,

[0020] {ω k}={ω1,ω2,…,ω K} represents the set of center frequencies of each eigenmode function; ||||2 represents the L2 norm;

[0021] 1.4) To find the optimal solution to the constrained variational problem, we introduce the augmented Lagrangian function L into the VMD algorithm. By using the quadratic penalty factor α and the Lagrangian multiplication operator λ(t), we transform the constrained variational problem into an unconstrained variational problem. The augmented Lagrangian function L is expressed as:

[0022]

[0023] Where α represents the penalty factor; λ(t) represents the Lagrange multiplier; ω k is the center frequency of the kth intrinsic mode function; <.> represents the vector inner product;

[0024] 1.5) Solve the augmented Lagrangian function L and finally obtain K IMF components to achieve adaptive decomposition of the signal.

[0025] Furthermore, the step 1.5) is specifically as follows:

[0026] The VMD algorithm uses the multiplication operator alternating direction method to continuously update λ ite+1 The value of , find the minimum point of L;

[0027] For u k Update issue, Expressed as:

[0028]

[0029] Equation (5) is transformed into the frequency domain using isometric transformation under the L2 norm:

[0030]

[0031] in, represents the frequency domain expression of the kth intrinsic mode function calculated at the iteth iteration; ω is the angular frequency; x(ω) represents the frequency domain form of the fiber optic gyroscope output signal x(t); is the conjugate form of x(ω); λ(ω) is the frequency domain form of λ(t), is the conjugate form of λ(ω); E represents the space that is integrable up to the second derivative and square integrable; j is the imaginary unit;

[0032] Let the first term of formula (6) be ω = ω - ω k ,but:

[0033]

[0034] Using the Hermitian symmetric matrix, Equation (7) is transformed into the form of non-negative frequency integral:

[0035]

[0036] That is, the solution to the quadratic optimization problem is expressed as:

[0037]

[0038] Similarly, we get The expression is:

[0039]

[0040] Furthermore, the step 2 includes:

[0041] 2.1) Initialize the parameters of the FPA algorithm: the number of flower populations n, the transition probability p, the maximum number of iterations N, and the random number rand that follows a uniform (0, 1) distribution.

[0042] 2.2) Calculate the fitness of each solution and find the optimal solution g in the current flower population by comparison * And its fitness value f(g * );

[0043] 2.3) Generate a random number rand. If the transition probability p ≥ rand condition is met, perform a global search, update the solution according to formula (11), and perform out-of-bounds processing;

[0044]

[0045] in, are the specific pollen positions of the i-th pollen individual after the t+1th and tth iteration updates respectively; g * represents the optimal pollen position in the current pollen population during initialization or after multiple iterations; l is a control parameter, representing a D-dimensional pollination intensity vector, each latitude of which is a random number obeying the levy distribution, and the calculation formula is:

[0046]

[0047] Where Γ(h) is the standard gamma function; h is the independent variable defined on (-∞, +∞), which is generally 1.5 in this algorithm; s is the step size, which can randomly transfer pollen from one flower to another, and is defined as follows:

[0048]

[0049] Among them, U and V are random numbers that obey the Gaussian distribution, and the Gaussian distribution is as follows:

[0050]

[0051] 2.4) If the transition probability p < rand condition holds, perform a local search, update the solution according to formula (15), and perform out-of-bounds processing:

[0052]

[0053] Among them, ∈ is a random number that obeys uniform distribution on [0,1]; Indicates the specific pollen positions of any two pollen individuals in the pollen population at the tth iteration update; j and k represent different pollen individuals;

[0054] 2.5) Calculate the fitness value corresponding to the new solution obtained in step 2.3) or 2.4). If the fitness value of the new solution is better than Then replace the current solution and the current fitness value with the new solution and the fitness value corresponding to the new solution respectively, otherwise keep the current solution and the current fitness value;

[0055] 2.6) If the fitness value corresponding to the new solution is better than the global optimal value, that is, f(X new )<f(g * ) then update the global optimal solution and the global optimal value;

[0056] 2.7) Determine whether the iteration termination condition is met: If it is met, output the optimal value and optimal solution. The optimal pollen individual g * That is, the optimal parameters of wavelet transform; otherwise, return to step 2.3).

[0057] Furthermore, the step 3 includes:

[0058] 3.1) Select the wavelet basis function according to the IMF waveform; obtain the optimal parameters of the wavelet transform according to the FPA algorithm in step 2, and set the number of decomposition layers, scale parameters, and translation parameters of the wavelet transform;

[0059] 3.2) Perform wavelet decomposition on the IMF component, and the decomposition equation is:

[0060]

[0061] Among them, ω j,k represents the kth wavelet coefficient after the jth wavelet decomposition; ψ a,b represents the wavelet function, ψ * (.) is ψ a,b (t) is the conjugate; f represents the signal to be decomposed; a is the scale parameter, which changes the bandwidth of the filter and determines the frequency information in the wavelet transform; b is the translation parameter, which determines the time domain information in the wavelet transform;

[0062] Discrete wavelet function ψ a,b Expressed as:

[0063]

[0064] Among them, q represents the support domain of the wavelet function, and k represents the adjustment coefficient;

[0065] 3.3) The wavelet coefficients containing noise coefficients are filtered by the soft threshold function to complete signal denoising; the threshold function is:

[0066]

[0067] Where, k and a0 represent adjustment coefficients, which are always positive; sgn(.) represents the step function; Represents the wavelet estimation coefficient after threshold processing; T represents the set threshold, Where N represents the length of the wavelet transform signal of this layer, σ represents the standard deviation of noise, and j is the number of decomposition layers.

[0068] Furthermore, the output formula of the compensated fiber optic gyroscope is:

[0069]

[0070] Where X(t) represents the FOG signal after noise reduction and inverse wavelet transform; C represents a signal-independent constant; ω j,k represents the kth wavelet coefficient after the jth wavelet decomposition; ψ a,b (t) represents the discrete wavelet function.

[0071] Compared with the prior art, the present invention has the following beneficial effects:

[0072] The present invention combines the advantages of the VMD algorithm and the FPA-WT algorithm. The VMD algorithm is used in the vibration signal compensation of the fiber optic gyroscope, which effectively avoids the endpoint effect and modal aliasing phenomenon existing in the traditional empirical mode decomposition method, and realizes the separation of the effective signal and the error signal of the fiber optic gyroscope under vibration. The wavelet transform has an adjustable sampling step size for different frequencies in the time domain, has better time-frequency window characteristics than the short-time Fourier transform, and can better remove the noise existing in the IMF component. At the same time, the FPA algorithm is used in the present invention to select the optimal parameters of the wavelet transform, thereby improving the noise reduction effect of the wavelet transform. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 This is a flow chart of a method for compensating fiber optic gyroscope vibration errors based on VMD and FPA-WT, shown in an embodiment of the present invention;

[0074] Figure 2 is a diagram showing the VMD decomposition results of the fiber optic gyroscope output under random vibration in this embodiment;

[0075] Figure 3 3 is a comparison chart of the compensation effects of different algorithms on random vibration of the fiber optic gyroscope in this embodiment. DETAILED DESCRIPTION

[0076] The technical solution of the present invention is further described in detail below with reference to the accompanying drawings.

[0077] The present invention provides a fiber optic gyroscope (FOG) vibration error compensation method based on variational mode decomposition (VMD) and a flower pollination algorithm (FPA)-wavelet transform (WT). The variational mode decomposition method determines the number of modal decompositions based on the actual conditions of the FOG output signal, adaptively matches the optimal center frequency and limited bandwidth of each mode, and effectively decomposes the given signal, overcoming the endpoint effect and modal aliasing problems existing in EMD, and effectively compensating for FOG vibration errors. In a vibrating environment, the higher frequency bands of the FOG output signal contain a large amount of noise. Therefore, the present invention addresses this issue by using a wavelet transform to perform noise reduction on the VMD-separated signal. The noise reduction effect of the wavelet transform depends on the selection of its parameters. To address this issue, the present invention uses a flower pollination algorithm to determine the optimal parameters for FOG vibration error noise reduction.

[0078] The algorithm flow chart is as follows Figure 1 As shown, the main steps are:

[0079] Step 1: Decompose the FOG output signal using VMD algorithm;

[0080] In this embodiment, the VMD algorithm calculation steps in step 1 are:

[0081] 1.1) Perform a Hilbert transform on the FOG output signal x(t) to obtain the analytical form corresponding to each modal function component (Intrinsic Mode Function, IMF), and then obtain the marginal spectrum:

[0082]

[0083] Among them, u k (t) is the kth eigenmode function; δ(t) is the impulse function; j is a complex unit; and t is the time.

[0084] 1.2) The construction frequency is the exponential term of the center frequency of each eigenmode function The corresponding eigenmode function is modulated to the baseband using multiplication of the exponential function:

[0085]

[0086] Among them, ω k is the center frequency of the kth eigenmode function.

[0087] 1.3) By calculating the square of the L2 norm of the first-order derivative of the demodulated signal shown in equation (2), the bandwidth of the corresponding intrinsic mode function is estimated. At this point, the constrained variation problem in the VMD algorithm can be expressed as:

[0088]

[0089] Among them, {u k}={u1,u2,…,u K} represents the set of all eigenmode functions,

[0090] {ω k}={ω1,ω2,…,ω K} represents the set of center frequencies of each eigenmode function; ||||2 represents the L2 norm.

[0091] 1.4) To obtain the optimal solution to the constrained variational problem, the VMD algorithm introduces the augmented Lagrangian function L, and uses the quadratic penalty factor α and the Lagrangian multiplication operator λ(t) to transform the constrained variational problem into an unconstrained variational problem.

[0092] The augmented Lagrangian expression of formula (3) is as follows:

[0093]

[0094] Among them, α represents the penalty factor, which can ensure the accuracy of signal reconstruction; λ(t) represents the Lagrange multiplier, which can strengthen the constraint; ω k is the angular frequency of the kth intrinsic mode function, that is, the center frequency of the kth intrinsic mode function; <.> represents the vector inner product.

[0095] 1.5) In order to further solve the above variational problem, the VMD algorithm adopts the Alternate Direction Method of Multipliers (ADMM) by continuously updating λ ite+1 The value of , find the minimum point of L.

[0096] For u k Update issue, It can be expressed as:

[0097]

[0098] The Parseval / Plancherel Fourier isometric transform is used to transform Equation (5) into the frequency domain under the L2-norm:

[0099]

[0100] in, represents the frequency domain expression of the kth intrinsic mode function calculated at the iteth iteration; ω is the angular frequency; x(ω) represents the frequency domain form of the fiber optic gyroscope output signal x(t); is the conjugate form of x(ω), and the superscript ^ indicates the conjugate form; λ(ω) is the frequency domain form of λ(t); E represents the space that is integrable up to the second-order derivative and square integrable; j is the imaginary unit.

[0101] Let the first term of formula (6) be ω = ω - ω k ,but:

[0102]

[0103] Using the Hermitian symmetric matrix, Equation (7) is transformed into the form of non-negative frequency integral:

[0104]

[0105] That is, the solution to the quadratic optimization problem can be expressed as:

[0106]

[0107] In the same way, we can get The expression is:

[0108]

[0109] To sum up, the specific implementation process of the VMD algorithm is:

[0110] ① λ 1 , m is initialized to 0;

[0111] ② Let ite=ite+1;

[0112] ③ Update u using equations (9) and (10) k 、ω k ;

[0113] ④ Let k = k + 1, and repeat ③ until k = K, where K is the number of IMF components;

[0114] ⑤Use Update λ;

[0115] ⑥ Repeat ②~⑤ until the constraints are met

[0116] Through the VMD algorithm, K IMF components can be finally obtained to achieve adaptive decomposition of the signal.

[0117] Step 2: Obtain the optimal parameters of wavelet transform through FPA;

[0118] In this embodiment, the FPA calculation steps in step 2 are:

[0119] 2.1) Initialize the parameters of the FPA algorithm: the number of flower populations n, the transition probability p, the maximum number of iterations N, and the random number rand that follows a uniform (0, 1) distribution.

[0120] 2.2) Calculate the fitness of each solution and find the optimal solution g in the current flower population by comparison * And its fitness value f(g * ).

[0121] 2.3) Generate a random number rand. If the transition probability p ≥ rand is met, perform a global search, update the solution according to formula (11), and perform out-of-bounds processing.

[0122]

[0123] in, are the specific pollen positions of the i-th pollen individual after the t+1th and tth iteration updates respectively; g * represents the optimal pollen position in the current pollen population during initialization or after multiple iterations; l is a control parameter, representing a D-dimensional pollination intensity vector, each latitude of which is a random number obeying the levy distribution, and the calculation formula is:

[0124]

[0125] Where Γ(h) is the standard gamma function; h is the independent variable defined on (-∞, +∞), which is generally 1.5 in this algorithm; s is the step size, which can randomly transfer pollen from one flower to another, and is defined as follows:

[0126]

[0127] Among them, U and V are random numbers that obey the Gaussian distribution, and the Gaussian distribution is as follows:

[0128]

[0129] 2.4) If the transition probability p < rand condition holds, perform a local search, update the solution according to equation (15), and perform out-of-bounds processing.

[0130]

[0131] Among them, ∈ is a random number that obeys uniform distribution on [0,1]; represents the specific pollen positions of any two pollen individuals in the pollen population at the tth iteration update; j and k represent different pollen individuals.

[0132] 2.5) Calculate the fitness value corresponding to the new solution obtained in step 2.3) or 2.4). If the fitness value of the new solution is better than The current solution and the current fitness value are replaced by the new solution and the fitness value corresponding to the new solution respectively, otherwise the current solution and the current fitness value are retained.

[0133] 2.6) If the fitness value corresponding to the new solution is better than the global optimal value, that is, f(X new )<f(g * ) then update the global optimal solution and global optimal value.

[0134] 2.7) Determine whether the iteration termination condition is met: If it is met, output the optimal value and optimal solution. The optimal pollen individual g * That is, the optimal parameters of wavelet transform; otherwise, return to step 2.3).

[0135] Step 3: Use wavelet transform to filter each level of IMF components obtained by VMD decomposition;

[0136] In this embodiment, the wavelet transform calculation steps in step 3 are:

[0137] 3.1) Based on the IMF waveform, select the wavelet basis function that can approximate the waveform step by step; obtain the optimal parameters of the wavelet transform according to the FPA algorithm in step 2, and set the number of decomposition levels, scale parameters, translation parameters, etc. of the wavelet transform.

[0138] 3.2) Perform wavelet decomposition on the IMF component, and the decomposition equation is:

[0139]

[0140] Among them, ω j,k represents the kth wavelet coefficient after the jth wavelet decomposition; ψ a,b represents the wavelet function, ψ * (.) is the conjugate of the wavelet function; f represents the signal to be decomposed; a is the scale parameter, which changes the bandwidth of the filter and determines the frequency information in the wavelet transform; b is the translation parameter, which determines the time domain information in the wavelet transform.

[0141] Discrete wavelet function ψ a,b It can be expressed as:

[0142]

[0143] Among them, q represents the support domain of the wavelet function, and k represents the adjustment coefficient.

[0144] 3.3) The wavelet coefficients containing noise coefficients are filtered through the soft threshold function to complete the signal denoising. The threshold function is:

[0145]

[0146] Where, k and a0 represent adjustment coefficients, which are always positive; sgn(.) represents the step function; Represents the wavelet estimation coefficient after threshold processing; T represents the set threshold, Where N represents the length of the wavelet transform signal of this layer, σ represents the standard deviation of noise, and j is the number of decomposition layers.

[0147] Furthermore, the output formula of the compensated fiber optic gyroscope is:

[0148]

[0149] Where X(t) represents the FOG signal after noise reduction and inverse wavelet transform; C represents a signal-independent constant; ω j,k represents the kth wavelet coefficient after the jth wavelet decomposition; ψ a,b (t) represents the discrete wavelet function.

[0150] In order to further verify the implementation effect of the present invention, in this embodiment, the fiber optic gyroscope is first placed on a vibration table so that the gyroscope axis is parallel to the vibration direction, and random vibration is applied to it to obtain a random vibration signal of the gyroscope.

[0151] Secondly, the VMD method is used to decompose the fiber optic gyroscope output to obtain various IMF components with different center frequencies, such as Figure 2 shown.

[0152] Third, the optimal wavelet transform parameters are obtained through the FPA algorithm, and the decomposed IMF components are denoised using wavelet transform.

[0153] Fourth, the signal is reconstructed using inverse wavelet transform to obtain the compensated fiber optic gyroscope output.

[0154] Finally, the performance of the proposed method and wavelet transform in vibration error compensation is compared. Figure 3 As shown; and using the angular random walk (N) and bias instability (B) in the allan variance as evaluation indicators, the method of the present invention is compared with the ensemble empirical mode decomposition (EEMD), and the results are shown in Table 1.

[0155] Table 1 Allan variance compensation results under random vibration

[0156]

[0157]

[0158] The experimental results shown in Table 1 show that the angular random walk and bias instability of the FOG signal processed by VMD and FPA-WT algorithms are significantly reduced compared to the original FOG signal and the FOG signal decomposed by EEMD. This demonstrates that the present invention can effectively compensate for FOG vibration errors and improve the performance of FOGs in vibration environments.

[0159] The technical content and technical features of the present invention have been disclosed as above. However, those skilled in the art may still make various substitutions and modifications based on the teachings and disclosures of the present invention without departing from the spirit of the present invention. Therefore, the scope of protection of the present invention should not be limited to the contents disclosed in the embodiments, but should include various substitutions and modifications that do not depart from the present invention and be covered by the claims of the present invention.

Claims

1. A method for compensating fiber optic gyroscope vibration errors based on VMD and FPA-WT, characterized in that: The following steps are involved: Step 1: Decompose the FOG output signal using the VMD algorithm: Establish a variational constraint model and solve the bandwidth of each modal function component IMF, calculate the IMF center frequency, and achieve adaptive decomposition of the signal; Step 2: Obtain the optimal parameters of wavelet transform through FPA algorithm; Step 3: Use wavelet transform to filter each level of IMF components decomposed by VMD algorithm; Step 4: Use inverse wavelet transform to reconstruct the FOG output signal to obtain the compensated FOG output.

2. The fiber optic gyroscope vibration error compensation method based on VMD and FPA-WT according to claim 1, characterized in that: The step 1 comprises: 1.1) Perform a Hilbert transform on the FOG output signal x(t) to obtain the analytical form of each modal function component IMF, and then obtain the marginal spectrum: Among them, u k (t) is the kth eigenmode function; δ(t) is the impulse function; j is a complex unit; t is the time; 1.2) The construction frequency is the exponential term of the center frequency of each eigenmode function The corresponding eigenmode function is modulated to the baseband using multiplication of the exponential function: Among them, ω k is the center frequency of the kth eigenmode function; 1.3) By calculating the square of the L2 norm of the first-order derivative of the demodulated signal shown in formula (2), the bandwidth of the corresponding intrinsic mode function is estimated, and the constrained variation problem in the VMD algorithm is expressed as: Among them, {u k }={u1,u2,…,u K } represents the set of all eigenmode functions, {ω k }={ω1,ω2,…,ω K } represents the set of center frequencies of each eigenmode function; || ||2 represents the L2 norm; 1.4) To find the optimal solution to the constrained variational problem, we introduce the augmented Lagrangian function L into the VMD algorithm and use the quadratic penalty factor α and the Lagrangian multiplication operator λ(t) to transform the constrained variational problem into an unconstrained variational problem. The augmented Lagrangian function L is expressed as: Where α represents the penalty factor; λ(t) represents the Lagrange multiplier; ω k is the center frequency of the kth intrinsic mode function; <.> represents the vector inner product; 1.5) Solve the augmented Lagrangian function L and finally obtain K IMF components to achieve adaptive decomposition of the signal.

3. The fiber optic gyroscope vibration error compensation method based on VMD and FPA-WT according to claim 2, characterized in that: The step 1.5) is specifically as follows: The VMD algorithm uses the multiplication operator alternating direction method to continuously update λ ite+1 The value of , find the minimum point of L; For u k Update issue, Expressed as: Equation (5) is transformed into the frequency domain using isometric transformation under the L2 norm: in, represents the frequency domain expression of the kth intrinsic mode function calculated at the iteth iteration; ω is the angular frequency; x(ω) represents the frequency domain form of the fiber optic gyroscope output signal x(t); is the conjugate form of x(ω); λ(ω) is the frequency domain form of λ(t), is the conjugate form of λ(ω); E represents the space that is integrable up to the second derivative and square integrable; j is the imaginary unit; Let the first term of formula (6) be ω = ω - ω k ,but: Using the Hermitian symmetric matrix, Equation (7) is transformed into the form of non-negative frequency integral: That is, the solution to the quadratic optimization problem is expressed as: Similarly, we get The expression is:

4. The fiber optic gyroscope vibration error compensation method based on VMD and FPA-WT according to claim 1, characterized in that: The step 2 includes: 2.1) Initialize the parameters of the FPA algorithm: the number of flower populations n, the transition probability p, the maximum number of iterations N, and the random number rand that follows a uniform (0, 1) distribution. 2.2) Calculate the fitness of each solution and find the optimal solution g in the current flower population by comparison * And its fitness value f(g * ); 2.3) Generate a random number rand. If the transition probability p ≥ rand condition is met, perform a global search, update the solution according to formula (11), and perform out-of-bounds processing; in, are the specific pollen positions of the i-th pollen individual after the t+1th and tth iteration updates respectively; g * represents the optimal pollen position in the current pollen population during initialization or after multiple iterations; l is a control parameter, representing a D-dimensional pollination intensity vector, each latitude of which is a random number obeying the levy distribution, and the calculation formula is: Where Γ(h) is the standard gamma function; h is the independent variable defined on (-∞, +∞) and is set to 1.5; λ is the Lagrange multiplication operator; and s is the step size, which randomly transfers pollen from one flower to another, defined as follows: Among them, U and V are random numbers that obey the Gaussian distribution, and the Gaussian distribution is as follows: 2.4) If the transition probability p < rand condition holds, perform a local search, update the solution according to formula (15), and perform out-of-bounds processing: Among them, ∈ is a random number that obeys uniform distribution on [0,1]; Indicates the specific pollen positions of any two pollen individuals in the pollen population at the tth iteration update; j and k represent different pollen individuals; 2.5) Calculate the fitness value corresponding to the new solution obtained in step 2.3) or 2.4). If the fitness value of the new solution is Then replace the current solution and the current fitness value with the new solution and the fitness value corresponding to the new solution respectively, otherwise keep the current solution and the current fitness value; 2.6) If the fitness value corresponding to the new solution is better than the global optimal value, that is, f(X new )<f(g * ) then update the global optimal solution and the global optimal value; 2.7) Determine whether the iteration termination condition is met: If it is met, output the optimal value and optimal solution. The optimal solution g * That is, the optimal parameters of wavelet transform; otherwise, return to step 2.3).

5. The fiber optic gyroscope vibration error compensation method based on VMD and FPA-WT according to claim 1, characterized in that: The step 3 includes: 3.1) Select the wavelet basis function according to the IMF waveform; obtain the optimal parameters of the wavelet transform according to the FPA algorithm in step 2, and set the number of decomposition layers, scale parameters, and translation parameters of the wavelet transform; 3.2) Perform wavelet decomposition on the IMF component, and the decomposition equation is: Among them, ω j,k represents the kth wavelet coefficient after the jth wavelet decomposition; ψ a,b represents the wavelet function, ψ * (.) is ψ a,b (t) is the conjugate; f represents the signal to be decomposed; a is the scale parameter, which changes the bandwidth of the filter and determines the frequency information in the wavelet transform; b is the translation parameter, which determines the time domain information in the wavelet transform; q represents the support domain of the wavelet function; Discrete wavelet function ψ a,b Expressed as: Wherein, k represents the adjustment coefficient; 3.3) The wavelet coefficients containing noise coefficients are filtered by the soft threshold function to complete signal denoising; the threshold function is: Where, k and a0 represent adjustment coefficients, which are always positive; sgn(.) represents the step function; Represents the wavelet estimation coefficient after threshold processing; T represents the set threshold, Where N represents the length of the wavelet transform signal of this layer, σ represents the standard deviation of noise, and j is the number of decomposition layers.

6. The fiber optic gyroscope vibration error compensation method based on VMD and FPA-WT according to claim 1, characterized in that: The output formula of the compensated fiber optic gyroscope is: Where X(t) represents the FOG signal after noise reduction and inverse wavelet transform; C represents a signal-independent constant; ω j,k represents the kth wavelet coefficient after the jth wavelet decomposition; ψ a,b (t) represents the discrete wavelet function.

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