MEMS gyroscope reliability evaluation method based on rigidity and damping competitive degradation

Abstract

The invention discloses an MEMS gyroscope reliability evaluation method based on rigidity and damping competitive degradation, which comprises the following steps: decomposing a failure mode into a rigidity degradation sub-process and a damping degradation sub-process according to an MEMS gyroscope kinetic equation; wiener degradation models are established for the two degradation sub-processes respectively, a generalized linear predictor is introduced, and the coupling relation of the two degradation processes is quantified through a specific coefficient; joint estimation of a drift coefficient and a diffusion coefficient is realized by using a moment estimation method and a maximum likelihood estimation method, and a parameter estimation result is substituted into a competitive degradation reliability function, so that reliability evaluation of the high-performance MEMS gyroscope is realized.

Description

Technical Field

[0001] This invention belongs to the field of semiconductor anomaly detection and reliability analysis technology, and more specifically, relates to a high-performance MEMS gyroscope reliability analysis method based on stiffness and damping competition degradation. Background Technology

[0002] Currently, microelectromechanical systems (MEMS) gyroscopes are widely used in many key fields, such as spacecraft attitude estimation, robot motion control, and automated testing. Complex service environments are placing new and higher demands on their reliability assessment.

[0003] Existing research on MEMS reliability analysis and assessment is mainly divided into two categories: one is based on failure physics methods, which conducts research on the degradation mechanism at the micro level during the degradation process of MEMS products, establishes mathematical models, and conducts reliability analysis accordingly. This type of method can delve into the failure mechanism level, and the reliability analysis results can be linked to the failure mechanism. The other type is based on mathematical statistics, which conducts statistical analysis on the degradation process and life characteristics of MEMS products, and further conducts reliability assessment. This type of method can analyze reliability from the degradation process of macroscopic parameters, and the assessment process is easy to implement. Although the above two aspects of research have made great progress over the years, the following shortcomings still exist: (1) Degradation mechanism models are difficult to directly embed into statistical reliability analysis, while pure data-driven methods cannot reveal the root cause of failure, resulting in a long-term separation between degradation mechanism research and reliability assessment statistical models; (2) Failure events are often the result of the combined action of multiple failure mechanisms, and existing methods cannot describe the competition and promotion relationship between multiple failure mechanisms while revealing the failure mechanism. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide a reliability assessment method for MEMS gyroscopes based on the competitive degradation of stiffness and damping. By analyzing the dynamic model of the MEMS gyroscope, the two main failure mechanisms that cause failure are identified, and a model is established to describe the competitive and promoting relationship between the two failure mechanisms. Based on improving the interpretability of the model, the reliability assessment of high-performance MEMS gyroscopes can be achieved.

[0005] To achieve the above-mentioned objectives, this invention provides a reliability assessment method for MEMS gyroscopes based on the competing degradation of stiffness and damping, characterized by comprising the following steps:

[0006] (1) Failure mode decomposition based on the dynamic model of MEMS gyroscope;

[0007] (2) Establish stiffness degradation model and damping degradation model;

[0008] (3) Define the generalized linear predictor;

[0009] (4) Solve for the specificity coefficients λ1 and λ2 in the generalized linear predictor;

[0010] (5) Reliability analysis of stiffness and damping competition degradation;

[0011] (6) Estimate the parameters in the stiffness degradation model and damping degradation model of the linear drift Wiener process;

[0012] (7) Substitute the parameter estimation results of stiffness degradation and damping degradation into the reliability function of stiffness and damping competition degradation of MEMS gyroscope to obtain the prediction results of reliability function.

[0013] The objective of this invention is achieved as follows:

[0014] This invention presents a reliability assessment method for MEMS gyroscopes based on the competing degradation of stiffness and damping. According to the dynamic equations of the MEMS gyroscope, the failure mode is decomposed into stiffness degradation sub-processes and damping degradation sub-processes. Wiener degradation models are established for each of the two degradation sub-processes, and a generalized linear predictor is introduced. Specificity coefficients are used to quantify the coupling relationship between the two degradation processes. The moment estimation method and maximum likelihood estimation method are used to jointly estimate the drift coefficient and diffusion coefficient. The parameter estimation results are then incorporated into the competing degradation reliability function, thereby achieving reliability assessment of high-performance MEMS gyroscopes.

[0015] Meanwhile, the MEMS gyroscope reliability assessment method based on the competitive degradation of stiffness and damping in this invention also has the following beneficial effects:

[0016] (1) Based on improving the interpretability of the model, this invention achieves high-precision reliability assessment of high-performance MEMS gyroscopes under the stiffness and damping competition degradation mechanism, overcomes the limitation of traditional single degradation model that cannot reflect the interaction of failure mechanism, and avoids the "black box" modeling problem.

[0017] (2) This method can be embedded in the health detection system of MEMS gyroscope to realize real-time reliability updates and remaining life prediction, provide decision-making basis for system-level fault early warning, and reduce the risk of sudden failure. Attached Figure Description

[0018] Figure 1 This is a flowchart of the reliability assessment method for high-performance MEMS gyroscopes with stiffness and damping competition degradation according to the present invention.

[0019] Figure 2 This is the prediction result of the reliability function obtained in the embodiment. Detailed Implementation

[0020] The specific embodiments of the present invention will now be described with reference to the accompanying drawings to enable those skilled in the art to better understand the invention. It should be particularly noted that in the following description, detailed descriptions of known functions and designs that might obscure the main content of the invention will be omitted here.

[0021] Example

[0022] In this embodiment, as Figure 1 As shown, this invention provides a reliability assessment method for high-performance MEMS gyroscopes with competing degradation of stiffness and damping, comprising the following steps:

[0023] (1) Failure mode decomposition based on the dynamic model of MEMS gyroscope;

[0024] (1.1) A MEMS gyroscope can be abstracted as a "spring-damped-mass" system, and its dynamic equation is defined as follows:

[0025]

[0026] Where the x-axis is the driving direction, the y-axis is the detection direction, and m x and m y These are the equivalent mass of the system in the driving direction and the detection direction, respectively; c x and c y These are the equivalent damping coefficients of the system in the driving direction and the detection direction, respectively; k x and k y ω represents the equivalent stiffness of the system in the driving and detection directions; f0 represents the driving amplitude of the harmonic driving force; ω represents the resonant frequency; F(t) represents the Coriolis force; and t represents time.

[0027] (1.2) Perform a Laplace transform on the first equation in the dynamic equation and solve for the mechanical amplitude of the driving direction under open-loop operating conditions:

[0028]

[0029] in, The natural frequency in the driving direction; Damping ratio in the driving direction;

[0030] (1.3) Defining variables Solving for the extreme points of f(ω), the resonant frequency of the MEMS gyroscope is obtained as follows:

[0031]

[0032] (1.4) The expression for the damping ratio and the expression for natural frequency Substituting these values ​​into the resonant frequency expression, we can further rewrite the expression for the resonant frequency as follows:

[0033]

[0034] For ease of description in subsequent formulas, the equivalent mass m will be omitted in the following text. x stiffness k x and damping coefficient c x Using the subscript, we can simplify the expression for the resonant frequency as follows:

[0035]

[0036] Where c is the equivalent damping coefficient of the system in the driving direction after simplification, m is the equivalent mass of the system in the driving direction after simplification, and k is the equivalent stiffness of the system in the driving direction after simplification.

[0037] Therefore, the main reasons for the deviation of the resonant frequency ω from its normal value are the decrease in stiffness k and the increase in damping coefficient c. Cumulative stiffness damage has been confirmed to exist in MEMS silicon thin-film devices, and the main reason for this phenomenon is the change in the equivalent elastic modulus of the structural material caused by fatigue. Furthermore, for vacuum-packaged high-performance MEMS gyroscopes, gas leakage caused by packaging degradation directly leads to an increase in the damping coefficient. Therefore, a stochastic process model for stiffness degradation and damping degradation is first established below.

[0038] (2) Establish stiffness degradation model and damping degradation model;

[0039] (2.1) Define the stiffness degradation quantity k(t) as following a Wiener process with linear drift:

[0040] k(t) = μt + σ1B(t)

[0041] Where μ and σ1 are the drift coefficient and diffusion coefficient, respectively, representing the main trend and random walk of the degradation process; B(t) is the standard Brownian motion and satisfies

[0042] (2.2) Define the damping degradation quantity c(t) as following a Wiener process with half-saturation drift:

[0043]

[0044] Among them, α, β, and γ are drift term parameters, whose values ​​determine the main trend of the degradation process; σ2 is the diffusion coefficient, whose value represents the degree of random walk in the degradation process.

[0045] (3) Define the generalized linear predictor;

[0046] We construct a limit state function that comprehensively reflects the competing relationship between the two failure mechanisms using a generalized linear competition model; this is called a generalized linear predictor.

[0047]

[0048] Where, vector X = (k(t), c(t)); k th and c th λ1 and λ2 are the threshold values ​​for stiffness degradation and damping coefficient degradation, respectively; λ1 and λ2 are specificity coefficients used to represent the degree of influence of different covariates on the linear predictor output; k0 is the initial stiffness set; c0 is the initial damping coefficient set; in the above formula, we will (c(t)-c th ) / c0 and (k(t)-k th ) / k0 is called a covariate. Normalizing it can eliminate the problem of the two mechanisms being orders of magnitude apart, thus allowing for a more accurate capture of how the two failure mechanisms affect the output of the linear predictor.

[0049] Therefore, the generalized linear predictor is a state function with respect to time. While describing the degradation state of MEMS, it integrates both stiffness and damping processes, making the reliability assessment model based on it more interpretable.

[0050] In this embodiment, the initial stiffness coefficient k0, the initial damping coefficient c0, and the stiffness degradation threshold k are... th and damping degradation threshold c th The values ​​of are shown in Table 1.

[0051] Table 1 shows the values ​​of the initial parameters and degradation threshold in this embodiment;

[0052] Parameters in this embodiment <![CDATA[k0]]> <![CDATA[c0]]> <![CDATA[k th ]]> <![CDATA[c th ]]> Value 54.86 N / m <![CDATA[9.523×10 -5 ]]> 52N / m <![CDATA[16×10 - 5]]>

[0053] (4) Solve for the specificity coefficients λ1 and λ2;

[0054] (4.1) Let r = c / (mω) x First, solve for the resonant frequency. The sensitivity to the stiffness coefficient k(t) and damping coefficient c(t) is denoted as follows: and The expression is as follows:

[0055]

[0056] (4.2) Define the specificity coefficients of the generalized linear predictor by normalization to solve the problem of the initial values ​​of stiffness coefficient and damping coefficient spanning orders of magnitude, and then calculate the specificity coefficients λ1 and λ2.

[0057]

[0058] In this embodiment, the equivalent mass in the driving direction is m = 4.99 × 10⁻⁶. -7 kg, and the specificity coefficients λ1 and λ2 calculated from this are shown in Table 2. The negative sign of λ2 in the results indicates that the increase in damping is the main reason for the resonant frequency shift.

[0059] Table 2 shows the specificity coefficients λ1 and λ2 calculated in the examples;

[0060] Specificity coefficient <![CDATA[λ1]]> <![CDATA[λ2]]> Value 0.54 -0.46

[0061] (5) Reliability analysis of stiffness and damping competition degradation;

[0062] (5.1) Define the reliability R(t) as the probability that the system does not experience an excessive resonant frequency shift at time t, which is the probability that the generalized linear predictor g(X,t) > 0:

[0063]

[0064] Where P(·) represents calculating the probability;

[0065] (5.2) Substitute the stiffness degradation model k(t) and the damping degradation model c(t) into the expression for the reliability R(t), and define the variable Z(t) = ak(t) + bc(t), where the variable... The reliability function for the competing degradation of stiffness and damping in a MEMS gyroscope is expressed as:

[0066]

[0067] Where, μ Z Z(t) is the mean function of Z(t), and Φ(·) is the cumulative distribution function of the standard normal distribution.

[0068] (6) Estimate the parameters in the stiffness degradation model and damping degradation model of the linear drift Wiener process;

[0069] (6.1) Define the observation time points of the observation data as t1 < t2 < ... < t n Where n is the number of observation time points;

[0070] (6.2) Estimate the parameters μ and σ1 in the stiffness degradation model of the linear drift Wiener process;

[0071] (6.2.1) Define the time increment Δt j =t j -t j-1j = 2, 3, ..., n, define the stiffness degradation increment Δk(j) = k(j) - k(j-1), j = 2, 3, ..., n;

[0072] (6.2.2) The estimated values ​​of parameters μ and σ1 are obtained using the first second moment method:

[0073]

[0074] The parameter estimation results of the stiffness degradation process in this embodiment are shown in Table 3:

[0075] Table 3 shows the parameter estimation results of the stiffness degradation process in this embodiment;

[0076]

[0077] (6.3) Estimate the parameters α, β, γ and σ2 in the damping degradation model of the half-saturated drift Wiener process. Due to the high nonlinearity of the half-saturated model, this invention uses maximum likelihood estimation and solves the parameter estimation results when the likelihood function is maximized using the interior point method and the quasi-Newton method.

[0078] (6.3.1) Construct the joint probability density f(c(t1), c(t2), ..., c(t) at different observation times. n ))for:

[0079]

[0080] Among them, the damping degradation increment Δc(t) j It follows the following distribution:

[0081]

[0082] (6.3.2) Construct the negative log-likelihood function as follows:

[0083]

[0084] Where θ=(α,β,γ,σ2) are the model parameters to be estimated;

[0085] (6.3.3) According to the physical meaning in the damped degradation model, the parameters must satisfy the constraint γ>0, σ c Since the likelihood estimate is greater than 0, we transform the maximum likelihood estimation into the following constrained optimization problem:

[0086]

[0087] stγ>0,σ2>0

[0088] (6.3.4) Introducing the obstacle function φ(θ)=-og(γ)-log(σ2), the constrained optimization problem is transformed into an unconstrained optimization problem. The unconstrained objective function is then:

[0089]

[0090] Where z is the obstacle parameter, used to control the weight of the obstacle function φ(θ);

[0091] (6.3.5) Set the initial value of the obstacle parameter, where the initial value z > ε; set the adjustment step Δz of the obstacle parameter; set the maximum number of iterations n. max and the maximum number of iterations m max ;

[0092] (6.3.6) Calculate the gradient

[0093] (6.3.7) Determine whether the following conditions are met: Or the current iteration count has reached the maximum iteration count n. max If the condition is met, proceed to step (6.3.10); otherwise, proceed to step (6.3.8).

[0094] (6.3.8) Using the BFGS quasi-Newton method to approximate the inverse H of the Hessian matrix -1 And determine the search direction as The step size η is determined using the Armijo condition, and the parameters are updated. The Armijo condition here assumes that the new function value must be far enough below the tangent at the current point. Using the Armijo condition, we can quickly select a safe and efficient step size, which is a commonly used method for determining the descent step size in iterative algorithms such as gradient descent, Newton, and quasi-Newton.

[0095] (6.3.9) Increment the iteration count of the inner loop by 1, and then return to step (6.3.6);

[0096] (6.3.10) Determine whether the following conditions are met: z < ε or the current iteration round reaches the maximum iteration round m. max If the condition is met, proceed to step (6.3.12); otherwise, proceed to step (6.3.11).

[0097] (6.3.11) Update obstacle parameters Then increment the iteration number of the outer loop by 1, and return to step (6.3.6);

[0098] (6.3.12) Output the current parameter estimation results.

[0099] The parameter estimation results of the damping degradation process in this embodiment are shown in Table 4:

[0100] Table 4 shows the parameter estimation results of the damping degradation process in this embodiment.

[0101]

[0102] (7) Substitute the parameter estimation results of stiffness degradation and damping degradation into the reliability function of the MEMS gyroscope stiffness and damping competing degradation to obtain the prediction result of the reliability function:

[0103]

[0104] The reliability function of the high-performance MEMS gyroscope with competing degradation of stiffness and damping, finally calculated in this embodiment, is as follows: Figure 2 As shown, we also used circles to mark the time when the reliability dropped to 90%, indicating that the operating status should be monitored at about 24 months, at which point the possibility of MEMS device resonant frequency shift failure is relatively high.

[0105] The above results demonstrate that the reliability analysis method for high-performance MEMS gyroscopes with competing degradation of stiffness and damping proposed in this invention not only enables accurate prediction of the reliability function but also further explains the competing relationship between the two failure mechanisms of stiffness degradation and damping degradation.

[0106] Although the illustrative specific embodiments of the present invention have been described above to enable those skilled in the art to understand the invention, it should be understood that the invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the invention as defined and determined by the appended claims, and all inventions utilizing the concept of the present invention are protected.

Claims

1. A MEMS gyroscope reliability assessment method based on stiffness and damping competing degradation, characterized in that, Comprising the following steps: (1) Failure mode decomposition based on the dynamic model of MEMS gyroscope; (1.1) Abstract the MEMS gyroscope as a "spring-damper-mass" system, and define its dynamic equation as: where x-axis direction is the driving direction, y-axis direction is the detection direction, m x and m y are the equivalent mass of the system in the driving direction and the detection direction, respectively; c x and c y are the equivalent damping coefficient of the system in the driving direction and the detection direction, respectively; k x and k y are the equivalent stiffness of the system in the driving direction and the detection direction, respectively; f0is the driving amplitude of the simple harmonic driving force, ω is the resonance frequency; F(t) is the Coriolis force, and t represents time; (1.2) Perform Laplace transform on the first equation in the dynamic equation, and solve the mechanical amplitude in the driving direction under open-loop working condition: wherein is the natural frequency of the drive direction; is the damping ratio of the drive direction; (1.3), define variable Solving the extreme point of f(ω), the resonant frequency of MEMS gyroscope is obtained as: (1.4) the expression for the damping ratio and the expression for the natural frequency into the expression for the resonance frequency, so that the expression for the resonance frequency is further written as: For the convenience of subsequent formula description, the expression of resonance frequency is simplified as: Wherein, c is the equivalent damping coefficient of the system in the driving direction after simplification, m is the equivalent mass of the system in the driving direction after simplification; k is the equivalent stiffness of the system in the driving direction after simplification; (2) Establishing stiffness degradation model and damping degradation model; (2.1) Define the stiffness degradation amount k(t) to obey the Wiener process with linear drift: k(t) = μt + σ1B(t) where μ and σ1are the drift and diffusion coefficients, respectively, B(t) is a standard Brownian motion, and satisfies (2.2) Define the damping degradation amount c(t) to obey the Wiener process with semi-saturation drift: Wherein, α, β, γ are drift term parameters, and σ2 is the diffusion coefficient; (3) Define the generalized linear predictor g(X, t); where the vector X = (k(t), c(t)); k th and c th are the stiffness degradation amount threshold value and the damping coefficient degradation amount threshold value, respectively; λ1 and λ2 are specificity coefficients to represent the degree of influence of different covariates on the output of the linear predictor; k0 is the set initial stiffness; c0 is the set initial damping coefficient; (4) Solving the specificity coefficients λ1 and λ2; (4.1) define variable r = c / (mω x ); (4.2) Solving the resonance frequency The sensitivities with respect to the stiffness coefficient k(t) and the damping coefficient c(t) are denoted by and The expressions are as follows: (4.3) according to the sensitivity and calculate the specificity coefficients λ1and λ2; (5) Reliability analysis of stiffness and damping competing degradation; (5.1) Define the reliability R(t) as the probability that the system does not occur resonance frequency deviation beyond the limit at time t, that is, the probability that the generalized linear predictor g(X, t) > 0: Wherein, P(·) represents the probability; (5.2) Substitute the stiffness degradation model k(t) and the damping degradation model c(t) into the expression of reliability R(t), and define a variable Z(t) = ak(t) + bc(t), where the variable The reliability function of the MEMS gyroscope with competitive degradation of stiffness and damping is represented as where μ Z (t) is the mean function of Z(t), and Φ(·) is the cumulative distribution function of the standard normal distribution. (6) Estimating the model parameters defined in step (2); (6.1) defining observation time points of the observation data as t1 < t2 <... < tn n where n is the number of observation time points; (6.2) Estimating the parameters μ and σ1 in the linear drift Wiener process stiffness degradation amount model; (6.2.1), define time increment Δt j = t j -t j-1 , j = 2, 3,..., n, define stiffness degradation increment Δk(j) = k(j) - k(j-1), j = 2, 3,..., n; (6.2.2) The estimated values of parameters μ and σ1 are obtained by using the first-order second-moment method as: (6.3) Estimating the parameters α, β, γ and σ2 in the semi-saturation drift Wiener process damping degradation amount model; (6.3.1), the joint probability density f(c(t1), c(t2),..., c(t n )) is given by: where the increment of the damping degradation amount, Δc(t j ) is subject to the following distribution: (6.3.2) Construct the negative log-likelihood function as: Wherein, θ = (α, β, γ, σ2) is the model parameter to be estimated; (6.3.3) Transform the maximum likelihood estimation into the following constrained optimization problem: s.t. γ > 0, σ2 > 0 (6.3.4) Introduce the barrier function φ(θ) = -log(θ) - log(σ2), and transform the constrained optimization problem into an unconstrained optimization problem, then the unconstrained objective function is: Wherein, z is the barrier parameter, which is used to control the weight of the barrier function φ(θ); (6.3.5) setting an initial value of the barrier parameter, the initial value z > ε, setting an adjustment step Δz of the barrier parameter; setting a maximum number of iterations n max and a maximum iteration round m max ; (6.3.6), computing gradients (6.3.7), judge whether the following condition is satisfied: or the current iteration number reaches the maximum iteration number n max If the condition is satisfied, go to step (6.3.10) ; otherwise, go to step (6.3.8) ; (6.3.8), the inverse H of the Hessian matrix is approximated by the BFGS quasi-Newton method -1 and the search direction is determined as The step size η is determined by the Armijo condition, and the parameters are updated (6.3.9) Add 1 to the iteration number of the inner loop, and then return to step (6.3.6); (6.3.10), determine whether the following condition is satisfied: z < ε or the current iteration reaches the maximum iteration m max If the condition is satisfied, go to step (6.3.12); otherwise, go to step (6.3.11). (6.3.11), updating the barrier parameters The iteration round of the outer loop is then incremented by one and the process returns to step (6.3.6). (6.3.12), outputting the result of the current parameter estimation (7) Substitute the parameter estimation results of stiffness degradation and damping degradation into the reliability function of MEMS gyroscope stiffness and damping competing degradation, and obtain the prediction results of the reliability function:

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