MEMS three-axis accelerometer calibration method based on simulated annealing particle swarm optimization algorithm

Abstract

The application discloses a MEMS three-axis accelerometer calibration method based on a simulated annealing particle swarm optimization algorithm, adopts a centrifuge to calibrate a MEMS three-axis accelerometer, uses angular velocity generated by the centrifuge as specific force input of the MEMS three-axis accelerometer, applies the simulated annealing particle swarm optimization algorithm to a nonlinear model for solving to-be-calibrated parameters of the MEMS three-axis accelerometer, and identifies corresponding parameters; in the calibration process, firstly, six-position tumbling method is used to determine scale factors and zero offset values of the three-axis accelerometer, and initial values of particle swarm iteration are determined, so that the convergence speed and stability of the algorithm are improved. By rotating a slave shaft on which the accelerometer is installed by 180 degrees, the influence of centrifuge system error on parameter identification is reduced. The method effectively excites high-order coefficients of the MEMS three-axis accelerometer, significantly reduces the uncertainty of unknown parameters of the MEMS three-axis accelerometer, and greatly improves the calibration precision.

Description

Technical Field

[0001] This invention belongs to the technical field of MEMS inertial device parameter calibration, and mainly relates to a MEMS triaxial accelerometer calibration method based on simulated annealing particle swarm optimization algorithm. Background Technology

[0002] The development of Micro-Electro-Mechanical Systems (MEMS) technology has led to a trend towards miniaturization, high performance, and low cost of sensors. MEMS triaxial accelerometers, born from this context, are now widely used in inertial navigation, drones, and military weapons. Accelerometers based on MEMS technology are characterized by low cost, low power consumption, and miniaturization. However, their signal variations are also smaller than those of traditional accelerometers, making their accuracy highly susceptible to both internal systematic errors and external random errors.

[0003] MEMS triaxial accelerometers inherently possess a series of non-negligible systematic errors, including their own scaling factor, axis-to-axis coupling (i.e., non-orthogonality error), zero-bias error, and mounting errors between the package and the sensor. These errors can cause significant discrepancies between measurement results and actual data, thus affecting the final accuracy. Therefore, calibration of MEMS triaxial accelerometers is an important research direction in micro-inertial sensors.

[0004] Currently, while there is no unified method for calibrating MEMS triaxial accelerometers, they can be broadly categorized into two types. One type involves simultaneous triaxial calibration using a triaxial rate turntable within the acceleration vector domain. Based on Lotters' modulus calibration principle, regardless of the accelerometer's attitude within a gravitational field, the sum of its triaxial acceleration vectors is always equal to the vector of the local gravitational acceleration. By designing multi-position attitude methods, such as six-position, twelve-position, and twenty-four-position calibration methods, the triaxial acceleration output in these attitudes is statically measured, thereby calculating the triaxial accelerometer's zero bias, scaling factor, and first-order coupling coefficient. The other type is an on-site automatic calibration method that does not require precision equipment or a controlled environment. This involves rigidly connecting a simple handheld accelerometer measurement platform or motion capture system to the accelerometer to be calibrated. Similarly, by changing the attitude direction to capture the angular position of the MEMS triaxial accelerometer, the calibration is completed by iteratively solving nonlinear equations. However, with the maturity of MEMS triaxial accelerometer technology and the expansion of application scenarios, the accuracy and performance of MEMS triaxial accelerometer signals need to be further improved. Summary of the Invention

[0005] This invention addresses the need for further improvement in the accuracy and performance of MEMS triaxial accelerometer signals in existing technologies. It provides a MEMS triaxial accelerometer calibration method based on simulated annealing particle swarm optimization (PSO) algorithm. The method employs a centrifuge to calibrate the MEMS triaxial accelerometer, using the angular velocity generated by the centrifuge as the specific force input. The simulated annealing PSO algorithm is applied to the nonlinear model for solving the calibration parameters of the MEMS triaxial accelerometer, identifying the relevant parameters. During the calibration process, a six-position rolling method is used to determine the accelerometer's zero bias and scale factor confidence intervals. The centrifuge error is reduced by rotating the slave axis on which the accelerometer is mounted by 180 degrees. This invention effectively excites the higher-order coefficients of the MEMS triaxial accelerometer, significantly reduces the uncertainty of unknown parameters, and greatly improves calibration accuracy.

[0006] To achieve the above objectives, the technical solution adopted by this invention is: a MEMS triaxial accelerometer calibration method based on simulated annealing particle swarm optimization algorithm, comprising at least the following steps:

[0007] S1: Determine the sources of systematic errors in the MEMS triaxial accelerometer and establish an input-output model of the MEMS triaxial accelerometer based on all errors; the systematic errors include at least zero-bias error, scaling factor error, cross-coupling error, and installation error, and the input-output model is specifically as follows:

[0008]

[0009] Among them, [A] x A y A z [pa] is the output measurement signal. x a y a z [S] represents the acceleration input. ij (i≠j=x,y,z) are cross-coupling terms, S i (i = x, y, z) are the scaling factors, S ii (i = x, y, z) is the second-order error term, [θ x θ y θ z [b] represents the installation error angle. x0 b y0 b z0 ] represents the zero bias error term, [e x e y e z [This represents random error;]

[0010] S2: Using the six-position rolling method, the MEMS triaxial accelerometer is placed successively at the ±1g position pointing to the x-axis, the ±1g position pointing to the y-axis, and the ±1g position pointing to the z-axis. After the accelerometer stabilizes, data is collected and processed to obtain the scaling factors and zero bias initial value intervals of the three axes.

[0011] S3: Install the MEMS triaxial accelerometer onto the centrifuge. Fix the MEMS triaxial accelerometer on the slave axis worktable. The centrifuge table rotates at a given angular velocity. After the centrifuge stabilizes, collect the triaxial signals of the MEMS triaxial accelerometer. After the data acquisition is completed, rotate the slave axis worktable 180 degrees around the slave axis of the centrifuge. The centrifuge table continues to rotate at a given angular velocity. After the centrifuge stabilizes, collect the triaxial signals of the MEMS triaxial accelerometer a second time.

[0012] S4: The simulated annealing particle swarm optimization algorithm is applied to solve the nonlinear model of the parameters to be calibrated in MEMS triaxial accelerometers. The parameters to be identified are mapped to the particle dimension, and the number of experimental samples represents the total number of particles. The particle swarm algorithm is used to update the velocity and position of the particles. The simulated annealing mechanism is introduced to accept the position of the particles with poor solutions. When a particle gets stuck in a local optimum, the particle is removed from the local extreme value region and continues to search for a better solution in the entire space. The search is iterated until the particle finally converges to the global optimum. At this time, the optimal solution position of the particle is the optimal value of the parameter to be calibrated in MEMS triaxial accelerometer.

[0013] As an improvement to the present invention, after step S2, the triaxial scale factor and initial zero bias of the MEMS triaxial accelerometer are specifically as follows:

[0014]

[0015] Among them, K i (i = X, Y, Z) represents the initial values ​​of the scale factors for each axis, b i (i = X, Y, Z) represents the initial zero bias values ​​for each axis, [A 1gi A -1gi (i = X, Y, Z) represents the collected values ​​at positions ±1g for each axis.

[0016] As an improvement to the present invention, after the two acquisitions in step S3, the triaxial signal output of the MEMS triaxial accelerometer is specifically as follows:

[0017] The results of the two x-axis acquisitions are processed as follows:

[0018]

[0019] The results of the two y-axis acquisitions are processed as follows:

[0020]

[0021] The results of the two z-axis experiments were processed as follows:

[0022]

[0023] in, (i = X, Y, Z) represents the processed measurement data for each axis, [a x ′a′ y a z [′] represents the first measurement data for each axis, [a x "a' y ′a z "" represents the second measurement data for each axis.

[0024] As another improvement of the present invention, step S4 specifically includes the following steps:

[0025] S41: For the identification error model: A a =K a U a +b a +e, K a Let A be the total transformation matrix. a For the measured value, U a For the ideal input value, b a Zero bias, e is random noise; parameter set (K) a ,b a The error model parameter set includes at least the scaling factor, zero offset, and cross-coupling coefficient; the objective function is constructed based on the principle of model calibration. Where N is the number of locations where experimental observations and data were collected;

[0026] S42: Initialize the parameters of the simulated annealing-based particle swarm optimization algorithm, wherein the scaling factor and zero bias term in the particle position are initialized to K in step S2. i (i = X, Y, Z) and b i (i = X, Y, Z), the particle velocity is initialized to 10%-20% of the parameter range, and the inertia weight ω is adaptively varied:

[0027]

[0028] In the formula ω m t is the maximum value of the inertia weight. m The maximum value for the iteration is set, and α is the coefficient of the weight change rate.

[0029] S43: Perform iterative optimization steps, dynamically adjust the inertia weight ω and learning factors c1 and c2 based on the number of iterations, update particle velocity and position according to the particle swarm, and calculate the fitness value of the new position:

[0030]

[0031] Perform the annealing operation:

[0032]

[0033] In the formula The fitness value represents the particle's new position. The fitness value represents the globally optimal fitness value;

[0034] If the fitness value is lower than the global fitness value, the new solution is better and is accepted; if the fitness value is higher than the global fitness value, the new solution is worse and is rejected with a certain probability. Accept new interpretations without immediately discarding them;

[0035] S44: After particle updates are complete, the global optimum is updated based on the latest particle swarm positions and attitudes, and the temperature T is adjusted according to T. t+1 =μT t The algorithm decays by setting the fitness function, target accuracy range, and number of iterations as convergence criteria. The algorithm terminates when the convergence criteria are met, and the output parameters are the MEMS triaxial accelerometer parameters to be identified (K). a ,b a ).

[0036] As another improvement of the present invention, in the parameter initialization of the simulated annealing particle swarm optimization algorithm in step S42, the number of particles N is given according to the experimental data, and the initial temperature T is set to T0 = 100E. initial (K), the annealing coefficient μ is set to 0.9~0.99, and the learning factors c1 and c2 are adaptively adjusted according to the number of iterations. In the early stage of the algorithm, c1>c2, focusing on individual search; in the later stage of the algorithm, c1<c2, focusing on group collaboration.

[0037]

[0038] In the formula c 1,max and c 2,max c is the maximum value of the learning factor. 1,min and c 2,min For the minimum learning factor, t m This is the maximum value set for iterations.

[0039] As a further improvement of the present invention, in the iterative optimization of step S43, the particle swarm optimization formula is specifically as follows:

[0040]

[0041] Wherein, at the t-th iteration, the velocity and position of the i-th particle are respectively and The optimal position found by the i-th particle during the iteration process is The average of the optimal values ​​of all individual particles, representing the optimal position searched by the swarm during the iteration process. It is the global optimal solution.

[0042] Compared with the prior art, the present invention has the following beneficial effects:

[0043] (1) Based on the characteristic that the centrifuge experimental platform can excite high-order parameters, this invention constructs a high-order input-output model and considers the system error of the centrifuge. It designs a scheme that can significantly reduce the system error and overcome the influence of the static working radius error and installation misalignment angle error of the centrifuge on the experimental data. At the same time, the initial values ​​of some parameters are determined by the six-position rolling method to ensure the accuracy of parameter identification, while reducing the search space, improving the convergence speed, and enhancing the stability of the particle swarm optimization algorithm based on simulated annealing.

[0044] (2) The particle swarm optimization algorithm based on simulated annealing proposed in this invention improves its ability to find the global optimum while ensuring convergence speed and iteration accuracy, overcoming the problem of premature convergence in traditional particle swarm optimization algorithms. It also improves the calibration accuracy of MEMS triaxial accelerometers. Attached Figure Description

[0045] Figure 1 This is a flowchart illustrating the calibration method of the present invention;

[0046] Figure 2 This is a schematic diagram showing the orientation of each input axis in the three-axis experiment of this invention;

[0047] Figure 3 This is a flowchart of the particle swarm optimization algorithm based on simulated annealing in step S4 of the method of the present invention;

[0048] Figure 4 This is a convergence curve of the simulated annealing particle swarm optimization algorithm without initial values, as presented in this invention.

[0049] Figure 5 This is a convergence curve with initial values ​​for the simulated annealing particle swarm optimization algorithm of this invention. Detailed Implementation

[0050] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0051] Example 1

[0052] This invention presents a centrifuge calibration method for MEMS triaxial accelerometers based on simulated annealing particle swarm optimization. Considering the systematic errors of the MEMS triaxial accelerometer and the experimental characteristics of the centrifuge, a high-order input-output nonlinear model of the MEMS triaxial accelerometer is established. Data is collected at ±1g positions along each axis of the MEMS triaxial accelerometer using a six-position rolling method to determine the confidence intervals of the scaling factor and zero bias for each axis. The angular velocity generated by the centrifuge is used as the specific force input for the MEMS triaxial accelerometer during calibration experiments. However, the systematic errors of the centrifuge can significantly interfere with the calibration experiments. Therefore, this invention designs an experimental method to reduce the systematic errors of the centrifuge by rotating the slave axis on which the accelerometer is mounted by 180 degrees to compensate for static working radius errors and misalignment angle errors. To address the tendency of the particle swarm optimization algorithm to get trapped in local optima, an annealing algorithm is introduced, allowing the particle swarm optimization algorithm to accept solutions with poor fitness with a certain probability, thus converging to the global optimum with probability 1.

[0053] This invention applies a simulated annealing particle swarm optimization algorithm to solve the nonlinear model of the parameters to be calibrated in a MEMS triaxial accelerometer, thereby identifying the corresponding parameters. For example... Figure 1 As shown, the specific steps include the following:

[0054] Step S1: Represent the systematic error of the MEMS triaxial capacitive accelerometer using a mathematical model, establish the error model of the MEMS triaxial accelerometer, determine the unknown parameters that need to be calibrated and solved, and finally confirm the complete high-order input-output model of the MEMS triaxial accelerometer.

[0055] The mathematical model of input and output includes the scaling factor S. i (i = x, y, z), zero offset [b] x0 b y0 b z0 Cross-coupling coefficient S ij (i≠j=x,y,z), installation error coefficient [θ] x θ y θ z Second-order nonlinear coefficient S ii (i = x, y, z), random error [e] x e y e z ].

[0056] Based on the subsequent centrifuge calibration experimental platform, it was determined that the input acceleration signal was a second-order signal, and the input-output model was expressed as follows:

[0057]

[0058] In the above model parameters, [A x A y Az [a] is the output measurement signal. x a y a z [S] represents the acceleration input. ij (i≠j=x,y,z) are cross-coupling terms, S i (i = x, y, z) are the scaling factors, S ii (i = x, y, z) is the second-order error term, [θ x θ y θ z [b] represents the installation error angle. x0 b y0 b z0 ] represents the zero bias error term, [e x e y e z [ ] represents random error, and the error model is represented as a matrix:

[0059] A a =K a U a +b a +e

[0060] In the formula K a Let A be the total transformation matrix. a For the measured value, U a For the ideal input value, b a The bias is zero, and e is random noise.

[0061] Step S2: Place the MEMS triaxial accelerometer sequentially at the ±1g positions pointing to the x-axis, y-axis, and z-axis. Based on the ideal model where all axes are uncoupled:

[0062]

[0063] The data collected at ±1g for each axis are as follows:

[0064] and

[0065] Among them, K i (i = X, Y, Z) represents the initial values ​​of the scale factors for each axis, b i (i = X, Y, Z) represents the initial zero bias values ​​for each axis, a i (i = X, Y, Z) represents the ideal acceleration input value, A i (i = X, Y, Z) represents the actual data collected. [A 1gi A -1gi(i = X, Y, Z) represents the acquired values ​​at ±1g positions on each axis. After the accelerometer stabilizes, the data is acquired and processed using the following formula:

[0066]

[0067] The processing yields the scaling factors and zero bias initial value intervals for the three axes.

[0068] Step S3: Install the MEMS triaxial accelerometer on the slave axis worktable of the centrifuge, generate centripetal acceleration input by setting the angular velocity, and perform a second experiment by rotating the accelerometer 180° around the slave axis after a single centrifugation experiment. After collecting the data from the two experiments, eliminate the static working radius error of the centrifuge and the installation misalignment angle error.

[0069] The inherent systematic errors of centrifuges, which cannot be ignored, originate from the static working radius and the misalignment angles of the three axes, which can significantly interfere with calibration experiments.

[0070] Let the static working radius be R, the static radius error be ΔR, and the misalignment angle errors of the three axes of the tooling installation be [θ]. I θ O θ P The centrifuge operates at an angular velocity of ω. Taking the x-axis as the centripetal acceleration input as an example, the ideal acceleration input for the three axes is:

[0071]

[0072] Taking into account errors, the actual working input would be:

[0073]

[0074] It can be seen that the force input of each axis is mixed with several errors. Although the magnitude is small, under high g input, even small errors can be excited into non-negligible interference input terms. Consider rotating the MEMS triaxial accelerometer around the slave axis stage to the back, that is, after a single centrifugation experiment, rotating the accelerometer's attitude 180° around the axis where the slave axis stage is located. The centrifuge's working angular velocity is still ω. At this time, the static working radius changes from R+ΔR to R-ΔR. The working input at this time is:

[0075]

[0076] The triaxial acceleration ratio input before attitude rotation and the triaxial acceleration ratio input after attitude rotation are processed as follows:

[0077]

[0078] Simplified MEMS triaxial accelerometer input:

[0079]

[0080] Considering θ i Both ΔR and cosθ are small quantities. i ≈1,,sinθ i ≈θ i The product of smaller quantities is a higher-order smaller quantity that can be neglected, i.e., ΔR·sinθ in the formula. P ≈0,ΔR·sinθ O ≈0,sinθ I ·sinθ P ≈0, then

[0081]

[0082] At this point, the single-axis acceleration input to each axis can be considered as eliminating the static working radius error and misalignment angle error of the centrifuge.

[0083] Therefore, a MEMS triaxial accelerometer was installed on the slave axis stage of a centrifuge. Initially, the x-axis signal was used as the centripetal acceleration force input, and the centrifuge motor started operating at a given angular velocity ω (the centripetal acceleration input must be at least greater than g). Once the centrifuge stabilized, data acquisition of the triaxial output signals began. After data acquisition, the slave axis stage was rotated 180° around the centrifuge's slave axis, and the negative x-axis signal was used as the centripetal acceleration force input. The centrifuge motor continued operating at the given angular velocity ω, and data acquisition began after stabilization. The same operation was repeated for the y-axis and z-axis centripetal acceleration signals to obtain the corresponding experimental data. Figure 2 This is a schematic diagram showing the orientation of the input axis and other axes in this embodiment. From left to right, the orientation is the positive installation direction with the x, y, and z axes as the input axes.

[0084] For the two experiments on the x-axis, the experimental results were processed as follows:

[0085]

[0086] For the two experiments on the y-axis, the experimental results were processed as follows:

[0087]

[0088] For the two experiments along the z-axis, the experimental results were processed as follows:

[0089]

[0090] For each acceleration input on each axis, there is a corresponding acceleration output that is processed.

[0091] Step S4: Globally optimize the experimental data based on the Simulated Annealing Particle Swarm Optimization (SAPSO) algorithm. By dynamically adjusting the inertia weight, learning factor, and annealing temperature, the error model parameters are calculated. Specifically, this includes: mapping the parameters to be identified to the particle dimension vector in the particle swarm; mapping the number of samples to the total number of particles; constructing a fitness function as the optimization objective to minimize the sum of squared residuals between the measured values ​​and the model predictions; iteratively updating the particle positions until the convergence criterion is met by combining the velocity update formula of the particle swarm algorithm and the probability acceptance mechanism of simulated annealing; and outputting the global optimal solution as the calibration parameters of the MEMS triaxial accelerometer.

[0092] Particle Swarm Optimization (PSO) is a swarm-based stochastic optimization algorithm that treats each individual particle as a massless, volumeless particle in the search space. Each particle is given a velocity and its state is adjusted based on its current best position and the best position of the entire swarm, gradually moving towards the optimal region. Its main steps include initializing the particle swarm, updating the individual's best position and the global best position, updating particle velocity and position, determining convergence conditions, and returning to the iteration. However, it suffers from drawbacks such as being prone to getting trapped in local optima, premature convergence, and insufficient optimization accuracy. Simulated Annealing (SA) is derived from the thermodynamic process of solid cooling, where the energy of an object reaches its minimum value as the temperature decreases. Based on the Metropolis criterion, it can accept not only the optimal solution but also, with the probability of temperature changes, worse solutions during the temperature decrease process, which helps in searching for the global optimum and reduces the risk of getting trapped in local optima. However, excessively large datasets and excessively high initial temperatures lead to slow convergence.

[0093] Combining the advantages of the two algorithms mentioned above—particle swarm optimization (PSO) with its fast iteration speed, relatively simple structure, and easily adjustable parameters, and simulated annealing (SUS) with its high search accuracy and greater convergence to the global optimum—this invention employs a method for solving nonlinear equations based on the simulated annealed particle swarm optimization (SA-PSO) algorithm. The parameters to be identified are mapped to particle dimensions in space, with the number of experimental samples representing the total number of particles. The particle swarm optimization algorithm is then used to update the particle velocity and position. A simulated annealing mechanism is introduced, accepting poor solution positions probabilistically. When a particle gets trapped in a local optimum, it is removed from the local extreme region and can continue searching for a better solution in the entire space. As the number of iterations increases, the particles eventually converge to the global optimum. The optimal solution position of the particle is the optimal value of the parameter to be calibrated in the MEMS triaxial accelerometer.

[0094] The experimental data were processed using a particle swarm optimization algorithm based on simulated annealing to identify parameters. In a D-dimensional space, the total number of particles is N. At the t-th iteration, the velocity and position of the i-th particle are respectively... and The optimal position found by the i-th particle during the iteration process is This is called the individual optimum. It refers to the optimal position searched by the population during the iteration process. This is called the global optimum. In the next iteration, the particle's velocity and position in the search space at the next moment are determined based on the particle's own and the group's experience, as well as the inertia weight ω. To reduce the probability of a particle getting trapped in a local optimum, the average of all individual particle optima is used. The individual optimal value replaces that of a single particle. The improved particle swarm optimization iterative update formula is as follows:

[0095]

[0096] The detailed implementation steps of this step are shown in the flowchart below. Figure 3 As shown:

[0097] First, determine the parameter set (K) of the identification error model. a ,b a This includes the scaling factor, zero-point offset, first-order inter-axis coupling coefficient, second-order nonlinear coefficient, and second-order nonlinear coupling coefficient of the MEMS triaxial accelerometer. A fitness function is constructed based on the principle of mode calibration. Where N is the number of locations observed and collected in the experiment, and the parameters obtained when the fitness function reaches its minimum value (i.e., the optimal value) are the accelerometer calibration parameters. The position vector of each particle corresponds to (K... a ,b a ), and define the parameter range for each parameter according to its corresponding physical meaning.

[0098] Next, the SA-PSO algorithm parameters are initialized: the number of particles N is given based on the experimental data, and the learning factors c1 and c2 are adaptively adjusted according to the number of iterations. A linear change is designed where c1 > c2 in the early stages of the algorithm, emphasizing individual search; and c1 < c2 in the later stages, emphasizing group cooperation.

[0099]

[0100] In the formula c 1,max and c 2,max c is the maximum value of the learning factor. 1,min and c 2,min For the minimum learning factor, t m This is the maximum value set for iterations.

[0101] The initial temperature T can be set according to the initial fitness, denoted as T0 = 100E. initial (K), annealing coefficient μ is generally

[0102] The annealing coefficient is set to 0.9–0.99. A larger coefficient results in a stronger global search capability but a slower convergence speed, and vice versa. The inertia weight determines the degree to which each particle retains its own motion state in each iteration during velocity updates; its value is adaptively varied.

[0103]

[0104] In the formula ω m t is the maximum value of the inertia weight. m The maximum iteration value is defined by α, where α is the coefficient for the rate of change of weights. The scaling factor and zero bias terms in the initial dimension of the particle are given by an initial value range, and subsequent iterations must not exceed this range; otherwise, the iteration will terminate immediately. The remaining identification parameters are randomized to ensure coverage of the global solution space, and the particle velocity is initialized to approximately 10%-20% of the parameter range to avoid premature convergence.

[0105] Then, an iterative optimization step is performed, dynamically adjusting the inertia weight ω and learning factors c1 and c2 based on the number of iterations. Afterward, the particle velocity and position are updated according to the particle swarm optimization formula, and the fitness value at the new position is calculated.

[0106]

[0107] Perform the annealing operation:

[0108]

[0109] In the formula The fitness value represents the particle's new position. This represents the globally optimal fitness value. If the fitness value is lower than the global fitness value, the new solution is better and should be accepted. If the fitness value is higher than the global fitness value, the new solution is worse and should be rejected with probability. The new solution is accepted, but not immediately discarded. After the particle swarm is updated, the global optimum is updated based on the latest particle swarm positions and attitudes, and the temperature T is determined according to T. t+1 =μT t Decay is defined by setting the algorithm's termination condition based on a given final convergence criterion: the standard deviation of parameter variation reaches the required accuracy, and the maximum number of iterations is set.

[0110] Figure 4 The objective function E(K) is obtained by establishing the SA-PSO algorithm with the x-axis as the input direction. t The convergence plot, the experimental data after step S3 corresponding to the particle swarm optimization sample, the initial values ​​are random, and the SA-PSO algorithm is iterated in step S4, with the termination condition set as follows: the number of iterations is 200, and the standard deviation of each parameter is less than 10. -2The graph shows that although the algorithm gets stuck in a local optimum during iteration, the annealing mechanism allows the objective function to escape the local optimum and continue searching for the global optimum. It can be seen that the fitness function converges to 10 after approximately 120 iterations. -3 . Figure 5 The objective function E(K) obtained by establishing the SA-PSO algorithm with the x-axis as the input direction is used. t The convergence graph, after step S2, shows the scale factor and zero-biased initial value intervals corresponding to the initial values ​​of some dimensions of the particle swarm, with the initial values ​​of other dimensions being random. The experimental data after step S3 corresponds to the particle swarm samples. The SA-PSO algorithm is iterated in step S4, with the termination condition being... Figure 4 The conditions are consistent. It can be seen that, under the selected initial value conditions, the fitness function converges to 10 after approximately 40 iterations. -3 .

[0111] Figure 4 and Figure 5 A comparison reveals that although the final convergence values ​​of the objective function are roughly the same, Figure 5 The convergence speed and stability under the given conditions are significantly better than those under the given conditions. Figure 4 This improves the algorithm's speed and avoids wasting resources searching in an unreasonable solution space. Since all parameters have converged to the target optimal value, the algorithm model can be considered stable, and the identification results are reliable. Stopping the computation and outputting the solved parameters are the MEMS triaxial accelerometer parameters (K) to be identified. a ,b a The generalization ability of the model can be verified using test data that was not used in the training, and the effect of parameter identification can be determined by calculating the residual RMS value.

[0112] It should be noted that the above content merely illustrates the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. For those skilled in the art, various improvements and modifications can be made without departing from the principle of the present invention, and all such improvements and modifications fall within the scope of protection of the claims of the present invention.

Claims

1. A calibration method for MEMS triaxial accelerometers based on simulated annealing particle swarm optimization algorithm, characterized in that: A centrifuge was used to calibrate the MEMS triaxial accelerometer. The angular velocity generated by the centrifuge was used as the specific force input of the MEMS triaxial accelerometer. The simulated annealing particle swarm optimization algorithm was applied to the nonlinear model to solve the parameters to be calibrated of the MEMS triaxial accelerometer, and the corresponding parameters were identified. The calibration process involves determining the accelerometer zero bias and the confidence range of the scaling factor using a six-position rolling method, and reducing centrifuge error by rotating the driven shaft on which the accelerometer is mounted by 180 degrees; it includes at least the following steps: S1: Determine the sources of systematic errors in the MEMS triaxial accelerometer and establish an input-output model of the MEMS triaxial accelerometer based on all errors; the systematic errors include at least zero-bias error, scaling factor error, cross-coupling error, and installation error, and the input-output model is specifically as follows: ； in, To output the measurement signal, For acceleration input, ( ) represents a cross-coupling term. ) represents the scaling factor. ) represents the second-order error term. For installation error angle, For the zero bias error term, This is random error; S2: Using the six-position rolling method, the MEMS triaxial accelerometer is placed successively at the ±1g position of the x-axis, the ±1g position of the y-axis, and the ±1g position of the z-axis. Data is collected after the accelerometer stabilizes to obtain the scaling factor and zero bias initial value range of the three axes. S3: Install the MEMS triaxial accelerometer onto the centrifuge. Fix the MEMS triaxial accelerometer on the slave axis worktable. The centrifuge table rotates at a given angular velocity. After the centrifuge stabilizes, collect the triaxial signals of the MEMS triaxial accelerometer. After the data acquisition is completed, rotate the slave axis worktable 180 degrees around the slave axis of the centrifuge. The centrifuge table continues to rotate at a given angular velocity. After the centrifuge stabilizes, collect the triaxial signals of the MEMS triaxial accelerometer a second time. S4: The simulated annealing particle swarm optimization algorithm is applied to solve the nonlinear model of the parameters to be calibrated in MEMS triaxial accelerometers. The parameters to be identified are mapped to the particle dimension, and the number of experimental samples represents the total number of particles. The particle swarm algorithm is used to update the velocity and position of the particles. The simulated annealing mechanism is introduced to accept the position of the particles with poor solutions. When a particle gets stuck in a local optimum, the particle jumps out of the local extreme value region and continues to search for a better solution in the entire space. The search is iterated until the particle finally converges to the global optimum. At this time, the optimal solution position of the particle is the optimal value of the parameter to be calibrated in MEMS triaxial accelerometer.

2. The MEMS triaxial accelerometer calibration method based on simulated annealing particle swarm optimization algorithm according to claim 1, characterized in that: In step S2, the six-position rolling method is applied. After the accelerometer stabilizes, data is collected. The triaxial signal output of the MEMS triaxial accelerometer is as follows: ； in, The initial values ​​of the scale factors for each axis are represented. Represents the initial zero bias value for each axis. This represents the collected values ​​at ±1g positions for each axis.

3. The MEMS triaxial accelerometer calibration method based on simulated annealing particle swarm optimization algorithm according to claim 1, characterized in that: After the two data acquisitions in step S3, the triaxial signal output of the MEMS triaxial accelerometer is as follows: The results of the two x-axis acquisitions are processed as follows: ； The results of the two y-axis acquisitions are processed as follows: ； The results of the two z-axis experiments were processed as follows: ； in, For the processed measurement data of each axis, These are the initial measurement data for each axis. These are the second measurement data for each axis.

4. The MEMS triaxial accelerometer calibration method based on simulated annealing particle swarm optimization algorithm according to claim 2, characterized in that; Step S4 specifically includes the following steps: S41: Determine the parameter set in the identification error model, wherein the identification error model is specifically: ； in, The total transformation matrix is... For measured values, For ideal input values, Zero bias, It is random noise; The error model parameter set includes at least the scaling factor, zero offset, first-order cross-coupling coefficient, and second-order cross-coupling coefficient; the objective function is constructed based on the principle of model calibration. ； Where N is the number of locations where experimental observations and data were collected; S42: Initialize the parameters of the simulated annealing-based particle swarm optimization algorithm, where the scaling factor and zero-bias term in the particle position are initialized as follows: and The particle velocity is initialized to 10%-20% of the parameter range, and the learning factor... Adaptive adjustment based on the number of iterations; inertia weight Adaptive changes: ； In the formula This represents the maximum value of the inertia weight. The maximum value of the set iterations, This is the coefficient representing the rate of change of the weight. S43: Perform iterative optimization steps, dynamically adjusting the inertia weight based on the number of iterations. Learning factor , Based on the particle swarm's updated particle velocity and position, calculate the fitness value for the new position: ； Perform the annealing operation: ； In the formula The fitness value represents the particle's new position. The fitness value represents the globally optimal fitness value; If the fitness value is lower than the global fitness value, the new solution is better and should be accepted; if the fitness value is higher than the global fitness value, the new solution is worse and should be rejected based on probability. Accept new interpretations without immediately discarding them; S44: After particle updates are complete, the global optimum is updated based on the latest particle swarm positions and attitudes, and the temperature T is adjusted accordingly. The algorithm is terminated when the convergence criteria are met. The parameters obtained from the solution are the MEMS triaxial accelerometer parameters to be identified. The fitness function, target accuracy range, and number of iterations are set as convergence criteria. .

5. The MEMS triaxial accelerometer calibration method based on simulated annealing particle swarm optimization algorithm as described in claim 4, characterized in that: In step S42, during parameter initialization based on the simulated annealing particle swarm optimization algorithm, the particle number N is given according to the experimentally acquired data, and the initial temperature T is set to... Annealing coefficient The learning factor is set to 0.9~0.

99. Adaptive adjustments are made based on the number of iterations in the early stages of the algorithm. Focusing on individual search, later stages of the algorithm Emphasis on group collaboration: ； ； In the formula and The maximum value of the learning factor. and The minimum value of the learning factor. This is the maximum value set for iterations.

6. The MEMS triaxial accelerometer calibration method based on simulated annealing particle swarm optimization algorithm as described in claim 5, characterized in that: In the iterative optimization of step S43, the particle swarm optimization formula is specifically as follows: ； Wherein, at the t-th iteration, the velocity and position of the i-th particle are respectively and The optimal position found by the i-th particle during the iteration process is , The average of the optimal values ​​of all individual particles, representing the optimal position searched by the swarm during the iteration process. It is the global optimal solution.

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