A MEMS accelerometer nonlinear error compensation method

Abstract

The application discloses a MEMS accelerometer nonlinear error compensation method, comprising the following steps: establishing a MEMS accelerometer temperature nonlinear error fractional order model, and performing discretization processing on the model to obtain an accelerometer output iteration equation; performing parameter identification on the fractional order model by using a cuckoo algorithm according to the accelerometer output to obtain a MEMS accelerometer nonlinear error model; obtaining an accelerometer system function according to the fractional order model, and designing a nonlinear error correction circuit on the basis, subtracting the correction circuit output from the MEMS accelerometer output to obtain the MEMS accelerometer output after the nonlinear error correction, so that the nonlinear error of the accelerometer can be effectively reduced, complex mathematical operations are avoided, and the calculation time delay is reduced.

Description

Technical Field

[0001] This invention belongs to the field of MEMS accelerometers, and particularly relates to a method for nonlinear error compensation of MEMS accelerometers. Background Technology

[0002] MEMS accelerometers are widely used in inertial navigation, military, autonomous driving, consumer electronics, robotics, and other fields. Accelerometer accuracy directly affects the accuracy and performance of control systems. Improving accelerometer accuracy is an urgent requirement for the development of aerospace, defense, industry, transportation, and medical fields. The principle is as follows... Figure 1 This diagram illustrates an accelerometer that detects the linear displacement of a sensitive mass block under acceleration. This accelerometer can be simplified to a sensitive mass block, a spring, and a damping structure. According to Newton's second law, F=ma, the acceleration force can be obtained through the mass block. Based on this, a displacement equation for the mass block is established according to Hooke's law and damping, and the acceleration is calculated by measuring the mass block's displacement. Furthermore, analyzing the error generation mechanism, establishing an error model, and studying error compensation measures to improve measurement accuracy are the relentless pursuits of many scholars.

[0003] Hooke's Law states that within the elastic limit, the elastic force of an ideal spring is directly proportional to its length. In reality, neither ideal elastic bodies nor ideal viscous bodies exist in nature; viscoelastic bodies, falling between these two extremes, are widely found. Viscoelastic bodies exhibit relaxation and creep phenomena during deformation, exhibiting global correlation and memory effects. Time-independent integer-order differential equations can only describe instantaneous changes and are insufficient for describing physical phenomena with global correlation and memory characteristics. Fractional differentials are a generalization of integer-order differentials, and research shows that fractional-order models can more accurately describe physical phenomena with global correlation and memory characteristics. A representative fractional-order Kelvin viscoelastic model is as follows:

[0004]

[0005] Where E is the elastic modulus of the material, ε is the material strain, η is the viscosity coefficient, and α is the fractional derivative order. Existing research results show that the fractional viscoelastic model can describe the system characteristics more accurately with fewer parameters; the fractional model can describe the mechanical properties of viscoelastic materials over a wider frequency range.

[0006] Since fractional-order models can more accurately describe the mechanical properties of viscoelastic materials, such as Figure 1The displacement-accelerometer relationship shown should also conform to a fractional-order mathematical model. Integer-order models can only approximate the actual system, and their modeling accuracy will inevitably be compromised. Fractional-order models are a generalization of integer-order models; any complex integer-order model can be simplified using a fractional-order model. Based on the above analysis, using a fractional-order model to study the input-output relationship of a MEMS accelerometer can more accurately reflect the system characteristics and achieve higher modeling accuracy. Therefore, using a fractional-order model to study nonlinear error compensation in MEMS accelerometers is of greater significance. Summary of the Invention

[0007] Objective: To address the complex nonlinear error mechanisms and difficulties in correction of MEMS accelerometers, this invention proposes establishing a fractional-order mathematical model and identifying its parameters to obtain the accelerometer system function. Based on this system function, a nonlinear error correction circuit is designed to achieve real-time correction of nonlinear errors. The method proposed in this invention is more accurate, helps reduce nonlinear errors, and avoids complex mathematical calculations, thus improving real-time processing performance.

[0008] Technical Solution: To achieve the objectives of this invention, the technical solution adopted is: a MEMS accelerometer nonlinear error compensation method, comprising the following steps:

[0009] Step 1: Establish a fractional-order model of the MEMS accelerometer based on the fractional-order model of viscoelastic materials and the principle of MEMS accelerometers;

[0010] Step 2: Discretize the fractional-order model and obtain the MEMS accelerometer acceleration iteration equation based on the properties of fractional-order differential equations;

[0011] Step 3: Place the MEMS accelerometer on the centrifuge test bench, collect the MEMS accelerometer output according to the set acceleration change, and record the acceleration and corresponding accelerometer output at each moment;

[0012] Step 4: Optimize the undetermined parameters of the model based on the data collected in Step 3;

[0013] Step 5: Establish the MEMS acceleration input-output equation, and design a nonlinear compensation circuit based on the input-output equation to realize nonlinear error compensation of the MEMS accelerometer.

[0014] Furthermore, step 1 establishes the fractional-order model of the MEMS accelerometer as follows:

[0015]

[0016] Where t represents time, a(t) represents acceleration that changes with time, y(t) is the output of the MEMS accelerometer, and k, y0 are undetermined constants; The α-th order Caputo fractional differential of y(t), 0 < α < 1, is defined as:

[0017]

[0018] Optimal estimation of unknown parameters k, y0, and α is performed using discrete sampled data.

[0019] Furthermore, step 2 yields the MEMS accelerometer acceleration iterative equation:

[0020]

[0021] Where t i This indicates the time of the i-th sampling, and the sampling interval h = t. i -t i-1 ,a(t i ) for t i The acceleration at time t-ih, y(t-ih) represents the accelerometer output at time t-ih, and Γ(·) is the gamma function. express The roundness.

[0022] Furthermore, in step 4, the cuckoo algorithm is used to identify the undetermined parameters k, y0, and α, as follows:

[0023] Step 4.1: Initialization;

[0024] 1) Generate M bird nest locations for the identification parameters S{k,y0,α}:

[0025] S i,j =S j,min +r0(S j,max -S j,min (4)

[0026] Among them, S i,j This represents the position of the i-th nest in the j-th generation, where i = 1, 2, …, M, j = 1, 2, 3, S. j,min ,S j,max This represents the lower and upper bounds of the j-th dimension variable; r0∈[0,1] is a random number;

[0027] 2) According to formula (3), using parameter S i,j {k,y0,α} and accelerometer output y(t) i ) Calculate the estimated value of the accelerometer output

[0028] 3) Define the objective function as the root mean square error between the measured and estimated acceleration values:

[0029]

[0030] From the j-th generation M-th objective function Fit(S) i,j In the next generation, the smallest function is selected as the Fit(S) value, and the nest position S{k,y0,α} corresponding to the Fit(S) value is retained.

[0031] Step 4.2: Global Search;

[0032] 1) Update the bird's nest position according to the following formula:

[0033]

[0034] Where ξ represents the step size, Represents the dot product of vectors; This indicates the location of the Bird's Nest before the update. Le'vy(λ) represents the new solution under global search, i.e., the updated bird's nest position; Le'vy(λ) represents a random walk with a step size following a Le'vy distribution.

[0035] 2) Utilize the bird's nest positions before and after the update respectively and Corresponding parameters and Calculate the acceleration estimate according to formula (3);

[0036] 3) Calculate the parameters using formula (5) respectively. and Objective function of the obtained acceleration estimate and

[0037] 4) Select the objective function according to the following formula. and The location of the bird's nest corresponding to a small or medium objective function value is taken as the optimal bird's nest location. renew:

[0038]

[0039] Step 4.3: Local search;

[0040] 1) Create new nesting locations:

[0041]

[0042] in, express A new solution for local search, namely, the new location of the bird's nest. This represents two random nest locations in the current population, where r and θ are random numbers uniformly distributed in the interval [0,1], and H(·) represents the Heaviside function;

[0043] 2) Utilizing the location of the bird's nest and Corresponding parameters and The acceleration estimate is obtained according to formula (3), and the estimated acceleration and the measured acceleration are substituted into formula (5) to calculate the parameters. and objective function and

[0044] 3) Select the objective function according to the following formula. and The location of the bird's nest corresponding to a small or medium objective function value is taken as the optimal bird's nest location. renew:

[0045]

[0046] The updated As the optimal position judge If the objective function value is less than a threshold, the iteration ends, and the global optimal solution P is obtained. * That is, the optimal position, and the parameter P corresponding to the optimal position. * {k,y0,α} are used as the final identification parameters; otherwise, proceed to step 4.2 to begin the next generation global search.

[0047] Furthermore, substituting the obtained optimal parameter estimates of k, y0, α into the fractional-order model of the MEMS accelerometer, and based on the Laplace transform property of fractional-order differential equations, the Laplace transform is obtained:

[0048]

[0049] Where s represents the Laplace operator, and Y(s) and A(s) represent the Laplace transforms of y(t) and a(t), respectively;

[0050] Treating the MEMS accelerometer as a system, with acceleration a(t) as the system input and accelerometer output y(t) as the system output, the Laplace relation of the accelerometer input and output is obtained:

[0051]

[0052] Furthermore, the method for designing a nonlinear compensation circuit is as follows:

[0053] The MEMS accelerometer output y(t) is used as the input to the compensation circuit. The output z(t) of the compensation circuit is the output after nonlinear error compensation of the MEMS accelerometer. The Laplace transform of z(t) is Z(s). The system function of this nonlinear compensation circuit is h(s). The relationship between A(s) and Z(s) is:

[0054]

[0055] According to the properties of linear time-invariant systems, A(s) and Z(s) satisfy the following conditions:

[0056]

[0057] Where k2 is a constant representing the amplitude gain, and the constant τ represents the time delay of the output relative to the input; the designed system function is: h(s) = k2e -sτ s α Based on h(s), a circuit was designed to implement nonlinear error compensation for the MEMS accelerometer.

[0058] Furthermore, the circuit is designed based on the system function h(s), as follows:

[0059] Based on the value of α, the frequency equivalence relation is used to analyze s. α Rationalize the system and substitute it into the system function expression; decompose the updated system function into n cascaded subsystems h. m (s), 1≤m≤n;

[0060]

[0061] Where a m ,b m ,c m To utilize the frequency equivalence relation for s α The constant of the expression obtained by rationalization;

[0062] According to h m (s) Design the corresponding circuit, where the MEMS accelerometer output y(t) is used as the input to h1(s), and h1(s), h2(s), ..., h n (s) Module circuits are cascaded sequentially, h n The output of the (s) module circuit is the output after compensation for the nonlinear error of the MEMS accelerometer.

[0063] Beneficial effects: Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

[0064] 1. This invention uses a fractional-order model to establish the nonlinear error of MEMS accelerometers, which theoretically can approximate the actual model with arbitrary precision, resulting in more accurate modeling.

[0065] 2. This invention uses fractional derivatives to establish a nonlinear error model, which is more in line with the mechanical principles of MEMS accelerometers and can more accurately reflect the actual physical process.

[0066] 3. This invention utilizes the established fractional-order model to design a compensation circuit. The output of the compensation circuit is the output after nonlinear error compensation, which avoids complex calculations and can obtain measurement results in real time, making the measurement more accurate and faster. Attached Figure Description

[0067] Figure 1 This is a schematic diagram of a MEMS accelerometer.

[0068] Figure 2 Block diagram for implementing nonlinear error in MEMS accelerometers;

[0069] Figure 3 This is a block diagram for nonlinear error compensation in MEMS accelerometers.

[0070] Figure 4 According to h i (s) Design the corresponding circuit schematic. Detailed Implementation

[0071] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0072] This invention provides a method for nonlinear error compensation in MEMS accelerometers. A fractional-order mathematical model is established, and the input-output system function is obtained based on the fractional-order model. A nonlinear error compensation circuit is designed based on the inverse function of the system function, thereby achieving real-time compensation of the nonlinear error. The implementation block diagram is shown below. Figure 2 As shown, the MEMS accelerometer nonlinear error compensation method comprises two parts: fractional-order modeling of the nonlinear error and equivalent model compensation. Specific implementation includes:

[0073] Step 1: Establish a fractional-order model of the MEMS accelerometer based on the fractional-order model of viscoelastic materials and the principle of MEMS accelerometers; the model is as follows:

[0074]

[0075] Where t represents time, a(t) represents acceleration that changes with time, y(t) is the output of the MEMS accelerometer, and k, y0 are undetermined constants; The α-th order Caputo fractional differential of y(t), 0 < α < 1, is defined as:

[0076]

[0077] The core of fractional modeling is to use discrete sampled data to make optimal estimates of unknown parameters k, y0 and α.

[0078] Step 2: Discretize the fractional-order model and obtain the MEMS accelerometer acceleration iteration equation based on the properties of fractional-order differential equations;

[0079]

[0080] Where t i This indicates the time of the i-th sampling, and the sampling interval h = t. i -t i-1 ,a(t i ) for t i The acceleration at time t-ih, y(t-ih) represents the accelerometer output at time t-ih, and Γ(i) is the gamma function. express The roundness.

[0081] Step 3: Place the MEMS accelerometer on the centrifuge test bench, collect the MEMS accelerometer output according to the set acceleration change, and record the acceleration and corresponding accelerometer output at each moment.

[0082] Step 4: Based on the data collected in Step 3, use the Cuckoo Algorithm to optimally estimate the undetermined parameters k, y0, and α of the model; the method is as follows:

[0083] Step 4.1: Initialization;

[0084] 1) Generate M bird nest locations for the identification parameters S{k,y0,α}:

[0085] S i,j =S j,min +r0(S j,max -S j,min (4)

[0086] Among them, S i,j This represents the position of the i-th nest in the j-th generation, where i = 1, 2, …, M, j = 1, 2, 3, S. j,min ,S j,max This represents the lower and upper bounds of the j-th dimension variable; r0∈[0,1] is a random number;

[0087] 2) According to formula (3), using parameter S i,j {k,y0,α} and accelerometer output y(t) i ) Calculate the estimated value of the accelerometer output

[0088] 3) Define the objective function as the root mean square error between the measured and estimated acceleration values:

[0089]

[0090] From the j-th generation M-th objective function Fit(S) i,j In the next generation, the smallest function is selected as the Fit(S) value, and the nest position S{k,y0,α} corresponding to the Fit(S) value is retained.

[0091] Step 4.2: Global Search;

[0092] 1) Update the bird's nest position according to the following formula:

[0093]

[0094] Where ξ represents the step size, and ξ is taken as 1 / 10 of the scale of the corresponding variable. Represents the dot product of vectors; This indicates the location of the Bird's Nest before the update. Le'vy(λ) represents the new solution under global search, i.e., the updated bird's nest position; Le'vy(λ) represents a random walk with a step size following a Le'vy distribution.

[0095] 2) Utilize the bird's nest positions before and after the update respectively and Corresponding parameters and Calculate the acceleration estimate according to formula (3);

[0096] 3) Calculate the parameters using formula (5) respectively. and Objective function of the obtained acceleration estimate and

[0097] 4) Select the objective function according to the following formula. and The location of the bird's nest corresponding to a small or medium objective function value is taken as the optimal bird's nest location. renew:

[0098]

[0099] Step 4.3: Local search;

[0100] 1) Create new nesting locations:

[0101]

[0102] in, express A new solution for local search, namely, the new location of the bird's nest. This represents two random nest locations in the current population, where r and θ are random numbers uniformly distributed in the interval [0,1], and H(·) represents the Heaviside function;

[0103] 2) Utilizing the location of the bird's nest and Corresponding parameters and The acceleration estimate is obtained according to formula (3), and the estimated acceleration and the measured acceleration are substituted into formula (5) to calculate the parameters. and objective function and

[0104] 3) Select the objective function according to the following formula. and The location of the bird's nest corresponding to a small or medium objective function value is taken as the optimal bird's nest location. renew:

[0105]

[0106] The updated As the optimal position judge If the objective function value is less than a threshold, the iteration ends, and the global optimal solution P is obtained. * That is, the optimal position, and the parameter P corresponding to the optimal position. * {k,y0,α} are used as the final identification parameters; otherwise, proceed to step 4.2 to begin the next generation global search.

[0107] Step 5: Establish the MEMS acceleration input-output equation, and design a nonlinear compensation circuit based on the input-output equation to realize nonlinear error compensation of the MEMS accelerometer.

[0108] Substituting the obtained optimal parameter estimates of k, y0, and α into the fractional-order model of the MEMS accelerometer, and based on the Laplace transform property of fractional-order differential equations, the Laplace transform is obtained:

[0109]

[0110] Where s represents the Laplace operator, and Y(s) and A(s) represent the Laplace transforms of y(t) and a(t), respectively;

[0111] Treating the MEMS accelerometer as a system, with acceleration a(t) as the system input and accelerometer output y(t) as the system output, the Laplace relation of the accelerometer input and output is obtained:

[0112]

[0113] The method for designing a nonlinear compensation circuit is as follows:

[0114] The MEMS accelerometer output y(t) is used as the input to the compensation circuit. The output z(t) of the compensation circuit is the output after nonlinear error compensation of the MEMS accelerometer. The Laplace transform of z(t) is Z(s). The system function of this nonlinear compensation circuit is h(s). The relationship between A(s) and Z(s) is:

[0115]

[0116] According to the properties of linear time-invariant systems, A(s) and Z(s) satisfy the following conditions:

[0117]

[0118] Where k2 is a constant representing the amplitude gain, and the constant τ represents the time delay of the output relative to the input; the designed system function is: h(s) = k2e -sτ s α Based on h(s), a circuit was designed to implement nonlinear error compensation for the MEMS accelerometer.

[0119] Design the circuit based on the system function h(s), as follows:

[0120] Based on the value of α, the frequency equivalence relation is used to analyze s. α Rationalize the system and substitute it into the system function expression; decompose the updated system function into n cascaded subsystems h. m (s), 1≤m≤n;

[0121]

[0122] Where a m ,b m ,c m To utilize the frequency equivalence relation for s α The constant of the expression obtained by rationalization;

[0123] According to h m (s) Design the corresponding circuit, where the MEMS accelerometer output y(t) is used as the input to h1(s), and h1(s), h2(s), ..., h n (s) Module circuits are cascaded sequentially, h n The output of the (s) module circuit is the output after compensation for the nonlinear error of the MEMS accelerometer.

[0124] In the embodiments, the rationalized correspondences are as follows:

[0125]

[0126]

[0127]

[0128]

[0129] If α = 0.3, then:

[0130]

[0131] Design a nonlinear error compensation circuit based on h(s). The circuit design method is illustrated using k2 = 39.8107 and τ = 0 as an example. When k2 = 39.8107 and τ = 0,

[0132]

[0133] The above system function is decomposed into several subsystems:

[0134]

[0135] The block diagram for MEMS accelerometer nonlinear error compensation is shown below. Figure 3 As shown, the MEMS accelerometer output y(t) is used as the input to the compensation circuit, and the compensation circuit output z(t) is the output of the MEMS accelerometer after nonlinear error compensation.

[0136] According to h i Design the corresponding circuit for (s)(i=1,2,3,4). i (s)(i=1,2,3,4) can be represented as the following expression:

[0137]

[0138] Design based on this expression, as follows: Figure 4 The circuit shown. The components satisfy the following conditions:

[0139] R i14 =R i24 =R i23 =R i =R o =1Ω.

[0140] The MEMS accelerometer output y(t) is used as the input of h1(s), and the module circuits h1(s), h2(s), h3(s), and h4(s) are cascaded in sequence. The output of the h4(s) module circuit is the output of the MEMS accelerometer after nonlinear error compensation.

Claims

1. A method for compensating nonlinear errors in a MEMS accelerometer, characterized in that: Includes the following steps: Step 1: Establish a fractional-order model of the MEMS accelerometer based on the fractional-order model of viscoelastic materials and the principle of MEMS accelerometers; Step 2: Discretize the fractional-order model and obtain the MEMS accelerometer acceleration iteration equation based on the properties of fractional-order differential equations; Step 3: Place the MEMS accelerometer on the centrifuge test bench, collect the MEMS accelerometer output according to the set acceleration change, and record the acceleration and corresponding accelerometer output at each moment; Step 4: Optimize the undetermined parameters of the model based on the data collected in Step 3; Step 5: Establish the MEMS acceleration input-output equation, and design a nonlinear compensation circuit based on the input-output equation to realize nonlinear error compensation of the MEMS accelerometer.

2. The MEMS accelerometer nonlinear error compensation method according to claim 1, characterized in that: Step 1 establishes the fractional-order model of the MEMS accelerometer as follows: （1） in Indicates time, This represents acceleration that changes over time. For MEMS accelerometer output, , These are undetermined constants; express of Caputo type fractional differential, Its definition is: （2） Using discrete sampled data to analyze unknown parameters , and Perform the optimal estimate.

3. The MEMS accelerometer nonlinear error compensation method according to claim 2, characterized in that: Step 2 yields the MEMS accelerometer acceleration iteration equation: （3） in Indicates the first Sampling time, sampling interval , for acceleration at any moment express Accelerometer output at any given time For gamma function, express The roundness.

4. The MEMS accelerometer nonlinear error compensation method according to claim 3, characterized in that: In step 4, the Cuckoo algorithm is used to identify the parameters to be determined. , and The method is as follows: Step 4.1: Initialization; 1) For the identification parameter S{ , , Generate M bird nest locations: （4） in, This indicates the position of the i-th bird's nest in the j-th generation. , , This represents the lower and upper bounds of the j-th dimension variable; It is a random number; 2) According to formula (3), using the parameters { , , } and accelerometer output Calculate the estimated value of the accelerometer output ; 3) Define the objective function as the root mean square error between the measured and estimated acceleration values: （5） From the Mth objective function of the jth generation Choose the smallest function as Value, will The value corresponds to the location of the bird's nest, S{ , , }Preserved for the next generation; Step 4.2: Global search; 1) Update the location of the bird's nest according to the following formula: （6） in Indicates step size, Represents the dot product of vectors; This indicates the location of the Bird's Nest before the update. This represents the new solution under the global search, i.e., the updated location of the Bird's Nest; Indicates step size follows Random walk of the distribution; 2) Utilize the bird's nest locations before and after the update respectively and Corresponding parameters { , , }and { , , According to formula (3), the acceleration estimate is calculated; 3) Calculate the parameters using formula (5) respectively. and Objective function of the obtained acceleration estimate and ; 4) Select the objective function according to the following formula. and The location of the bird's nest corresponding to a small or medium objective function value is taken as the optimal bird's nest location. renew: （7） Step 4.3: Local search; 1) Create new nesting locations: （8） in, express A new solution for local search, namely, the new location of the bird's nest. This represents two random nest locations in the current population. and A random number uniformly distributed within the interval [0,1]. This represents the Heaviside function; 2) Utilizing the location of the bird's nest and Corresponding parameters { , , }and { , , According to formula (3), the acceleration estimate is obtained, and the estimated acceleration and the measured acceleration are substituted into formula (5) to calculate the parameters. and objective function and ; 3) Select the objective function according to the following formula. and The location of the bird's nest corresponding to a small or medium objective function value is taken as the optimal bird's nest location. renew: (9) The updated As the optimal position ,judge If the objective function value is less than a threshold, the iteration ends and the global optimal solution is obtained. That is, the optimal position, and the parameters corresponding to the optimal position. { , , } is used as the final identification parameter; otherwise, proceed to step 4.2 to begin the next generation global search.

5. The MEMS accelerometer nonlinear error compensation method according to claim 3, characterized in that: The result , , Substituting the optimal parameter estimates into the fractional-order model of the MEMS accelerometer, and based on the Laplace transform property of the fractional-order differential equation, the Laplace transform is obtained: ， in Represents the Laplace operator. , They represent , The Laplace transform of; Treating MEMS accelerometers as a system, acceleration As a system input, the accelerometer output As the system output, the Laplace relation of the accelerometer input and output is obtained: 。 6. The MEMS accelerometer nonlinear error compensation method according to claim 5, characterized in that: The method for designing a nonlinear compensation circuit is as follows: MEMS accelerometer output As the input of the compensation circuit, the output of the compensation circuit is... This is the output after compensation for the nonlinear error of the MEMS accelerometer. The Laplace transform is ; The system function of the nonlinear compensation circuit is ; and The relationship is: ， According to the properties of linear time-invariant systems, and The following conditions must be met: ， in The constant represents the amplitude gain. This represents the time delay of the output relative to the input; the system function is designed as follows: and according to Design a circuit to implement nonlinear error compensation for MEMS accelerometers.

7. The MEMS accelerometer nonlinear error compensation method according to claim 6, characterized in that: According to system functions The circuit design is as follows: according to The value of is determined by using the frequency equivalence relation. Rationalize the system and substitute it into the system function expression; decompose the updated system function into n cascaded subsystems. ; ， in To utilize frequency equivalence relations to The constant of the expression obtained by rationalization; according to Design the corresponding circuit for the MEMS accelerometer output. As Input, and , ,…, The module circuits are cascaded in sequence. The output of the module circuit is the output after compensation for the nonlinear error of the MEMS accelerometer.

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