Optimizing MEMS-based navigation sensors for aerospace vehicles

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Introduction The use of microelectromechanical systems (MEMS) in space applications has shown potential to revolutionize future spacecraft systems, which is why careful attention to material selection is always required. Technology Readiness Levels (TRLs) are a method for estimating the maturity of MEMSbased devices offer miniaturization advantages [1]. MEMS technology offers miniaturization advantages but packaging and testing remain significant challenges that account for a major part of the final cost of MEMS devices [2]. MEMS-based devices offer a promising solution for navigation systems of autonomous aerospace vehicles. However, the performance of sensitive MEMS devices, such as magnetometers, can be significantly affected by the test environment (factors such as electrical activity and ferrous materials), which affects the magnetometer output [3]. Thus, thermal engineers and guidance, navigation, and control (GN&C) engineers have considered both the benefits and challenges when dealing with MEMS technology in space [4]. A potential approach suggested by researchers is topology optimization, which divides MEMS struc- tures into elements and assigns a design variable to each one to determine optimal material distribution [5]. On the other hand [6], emphasized the significance of MEMS features in achieving low mass and high reliability in aerospace systems, as the current industry standard for launching a satellite into low-Earth orbit (LEO) stands at roughly $5,000 per kilogram. MEMS navigation sensors have gained attention from aerospace engineers and a wide range of users due to their compact size, lightweight, and costeffectiveness [7]. These sensors have great potential for integration with space instruments and find widespread applications in aviation [8-10]. Their small dimensions and affordable manufacturing make them highly desirable for aerospace-related purposes, leading to a steady increase in global market demand [11]. In recent years, there has been increased interest in MEMS Inertial Measurement Units (IMUs) due to their small size and low cost. IMUs are used to obtain navigation data when GPS signalunavailable environments or electronic interference are present. However, a major disadvantage of IMUs has been the accumulated error when integrating them with navigation equations alone to find the position [12]. To improve the accuracy of navigation data, various IMU sensors, consisting of accelerometers and gyroscopes, are assembled together into a printed circuit board known as the IMU Cluster that calibration process is needed to optimize the systematic errors [13]. This review will examine topology optimiza- tion techniques as potential solutions for the design and material distribution optimization challenges encountered in MEMS navigation sensors. By addressing these advancements and challenges, this review aims to offer valuable insights to resear- chers, engineers, and industry professionals, foster- ing progress and innovation in MEMS-based navi- gation systems across various applications. 1. MEMS in Guidance, Navigation and Control MEMS technology involves micromachining silicon to create micron-scale structures such as cantilevers, free-standing bridges, membranes, and channels, which are then combined with micro- electronics fabrication methodology and technology to produce miniature systems that integrate electro- nics with sensors, transducers, and actuators. MEMS devices have potential applications in spacecraft GN&C systems for navigational functions. As shown in Table1 [4] the essential functional elements of a spacecraft GN&C system are sensors, processors, and actuators, and MEMS technology can be used to develop miniaturized sensing and control devices, including accelerometers, gyroscopes, star trackers, sun sensors, magnetometers, reaction wheels, and thrusters. MEMS technology for space can be categorized into various areas, such as inertial navigation, RF switches, and variable capacitors. However, reliability is a crucial concern for space hardware due to radiation, thermal cycling, thermal shocks, vibration, and mechanical shock [14; 15]. Table 1 Typical Spacecraft GN&C Attitude Sensing and Control Devices Attitude Sensing Devices Navigation Sensing Devices Attitude Control Actuation Devices Sun sensors Thrusters Earth sensors Magnetic torquers Horizon sensors IMU and IMMU* Antenna pointing gimbals Gyroscopes Gyroscopes Momentum wheels Accelerometers Accelerometers Reaction wheels Magnetometers Magnetometers Control moment gyros Fine guidance sensors Solar array drives N o t e. When magnetometers are combined to accelerometers and gyroscopes, IMUs are referred to as IMMUs. Alizadeh A., Saltykova O.A., Novinzadeh A.B. RUDN Journal of Engineering Research. 2024;25(1):57-74 1.1. Current Types of MEMS Sensors in Aerospace Navigation 1.1.1. MEMS inertial sensors The use of Inertial Measurement Units (IMUs) has been integral to navigation and guidance systems in aerospace and military applications. Traditionally, IMUs consisted of three accelerometers and three gyroscopes to sense linear accelerations and angular velocities. However, advancements in Micro-ElectroMechanical Systems (MEMS) technology have resulted in the development of new low-power wireless transceiver-based applications, leading to smaller, more cost-effective IMUs. These IMUs are implemented as strap-down systems and offer advantages over traditional IMUs, including higher accuracy and lower power consumption. As can be seen in Figure 1 by measuring local angular velocity and linear acceleration, it is possible to determine the moving displacement or absolute position in the global inertial reference frame [16]. MEMS IMMU technology is a critical compo- nent in modern aerospace and defense systems, providing precise positioning, velocity, and orienta- tion data crucial for safe and reliable navigation. Furthermore, recent research has shown the poten- tial for MEMS IMUs to improve safety and mobility during space missions through the development of highly accurate gait detection algorithms for various positions of the human body [17-19]. This techno- logy could be integrated into astronaut clothing, providing real-time feedback on movement and position. 1.1.2. Performance Metrics To ensure cost-effectiveness, reliability, and safety, stable MEMS performance is necessary throughout its life cycle [20]. Achieving high sensitivity and accurate modeling of the sensors is crucial. The harsh environment of aerospace applications requires specialized designs and modeling techniques, such as micro-dampers and protective materials, to withstand conditions like vibrations, shocks, temperature gradients, and fluids. Additionally, compliance with export control regulations and aeronautics development standards and processes is essential, which may involve incorporating antiproliferation devices and technologies. Table 2 shows the necessary requirements for performance of some prevalent navigation sensors [21]. a b Figure 1. Global Positioning via Local Angular Velocity and Linear Acceleration Measurements. The inertial navigation basic: a - strap-down systems; b - position calculation S o u r c e : made by the Iniewski et al [16] Table 2 MEMS for Aerospace Navigations; Standards and Performance N o t e . The performance metrics of navigation sensors vary for each mission. Therefore, Table 2 specifies the minimum requirements for sensors when performing a simple mission in the aerospace field. To achieve detailed data about Characterization of Inertial Measurement Units under Environmental Stress Screening, refer to the studies of Capriglione et al [22]. For more study, Liu et al [23], and have referred to exact performance metrics of magnetometer. 2. Techniques for Optimizing MEMS-Based Navigation Sensors and Solutions 2.1. Fabrication and Production Optimization Fabrication optimizing methodologies are divided into two main parts: - selection of Smart Structure (Materials); - development of Mathematical Models for modeling and analysis purposes. 2.1.1. Smart Structure Smart technology has been extensively employed across science and engineering fields, offering immense potential for highly significant applications. It has successfully addressed challenges in aerospace and electronics through the utilization of innovative materials with electromechanical/magneto mechanical coupling capabilities. These materials have enabled the conversion of energy from one form to another, thereby facilitating the development of sensors and actuators from the same substances. A control mechanism integrated into the system has responded to sensor signals, determining the appropriate actions of the actuators. Researchers worldwide have devised methods to incorporate these components, introducing smartness into systems. Initially, this technology was applied in larger systems, but there has been a growing focus on miniaturization, particularly with the rise of microelectromechanical systems (MEMS), driven by the need for lightweight designs. The engineering of smart systems and MEMS has involved multidisciplinary research and has presented numerous technological challenges. As smart systems technology has expanded into various disciplines, there has been a crucial and timely need to consolidate technological advancements in specific areas, providing valuable insights for practicing researchers in science and engineering who are interested in potential applications of this technology [24; 25]. 2.1.2. Mathematical Modeling In mathematical modeling, the governing differential equation of a system is essential. There are two methods to achieve this. The first method involves isolating a small block from the continuum system, analyzing the 3-D state of stress on the block, and writing the equilibrium equation to obtain the governing equation. Approximations in lower dimensions can be derived from the 3-D equations by converting stresses into stress resultants. This method, known as the Theory of Elasticity, involves dealing with tensors and vectors. An alternative approach is the energy method, where minimizing the energy functional yields the desired governing equations and their associated boundary conditions. This method is widely used in discrete modeling techniques. The Finite Element Method (FEM) is extensively used for analyzing smart structures, but it may become computationally prohibitive for certain scenarios such as high-frequency loads or Structural Health Monitoring (SHM) in composites. In such cases, wave-based techniques are employed [26; 27]. Alizadeh A., Saltykova O.A., Novinzadeh A.B. RUDN Journal of Engineering Research. 2024;25(1):57-74 2.2. Review of Navigation Sensors Fabrication Optimizing Methodologies and Mechanisms In order to enhance the structural optimization of micro-electromechanical devices and select materials that effectively increase optimization parameters such as construction costs, production efficiency, and performance, it is imperative to establish a fundamental framework that serves as a basis for relevant research and provides general principles. Consequently, Varadan et al. [26], have addressed these fundamental principles and their design calculations. However, unlike the compilation of materials by Nithtianov et al. [28], specific optimization solutions have not been discussed. Instead, their work serves as a comprehensive reference for reviewing general principles and the fundamental concept of utilizing smart materials alongside their mathematical modeling. Ananthasuresh’s research provides a compre- hensive overview of MEMS design concepts, focusing on system-level synthesis methodology for modifying sensor structure topology. The study emphasizes automating design processes for fixed MEMS transducer topologies, involving identifying design variables, establishing constraints (as shown in Table 3), and formulating a mixed-integer nonlinear optimization problem. The optimization prob- lems are solved through a sequence of linearized sub-problems using sensitivity information. These large-scale problems involve thousands to millions of design variables and relatively few constraints. Clearly in Figure 2, the geometrical parameters and other details of Table 3 are shown. Various methods, such as Optimality Criteria, Sequential Linear Programming, and the Method of Moving Asymptotes (MMA), are employed to solve these optimization problems, with MMA being popular for advanced topology optimization. Quadratic programming methods are not used due to their high computational cost. The solutions to these problems are mesh-dependent and non-unique, but schemes can be implemented to ensure well-posed problems, such as modifying sensitivities within a fixed radius of the element [29] ’(.)*+- , $ ∑;9<$ 839:9 ’(’/)*+-4 ,. (1) = ’/0 /0 ∑5467 234 Where the mesh-independent convolution operator (weight factor) 839 is written as: 839 ==>9; -@ABC(D,A) EA∈G│@ABC(D,A)≤=>9;J,D=1,…,G. (2) In this expression, the operator @ABC (D,A) is defined as the distance between the center of the element D and the center of an element A. The convolution operator 839 is zero outside the filter area. The convolution operator for element A is seen to decay linearly with the distance from element D. It is worthwhile noting that the sensitivity converges to the original sensitivity when the filter radius =>9; approaches zero and that all sensitivities will be equal (resulting in an even distribution of material) when =>9; approaches infinity [29]. Table 3 a b c d Figure 2. Geometrical parameters and dimensions of the micro resonator elements (sensors): a - shuttle mass; b - folded-flexure; c - comb drive with N movable ‘rotor’ fingers; d - close-up view of comb fingers S o u r c e : made by the Ananthasuresh [29] As part of the investigation into the proposed constraints in design and synthesis, Kläui [30] conducted a comprehensive exploration of geo- metrically confined domain walls, employing a range of magnetometry and imaging techniques. The research encompassed the examination of spin structures, phase diagrams, thermal excitations, stray fields, and magnetic dipolar coupling. How- ever, to enhance the scientific rigor and reliability of the findings, further details regarding the metho- dological aspects, including experimental setups, sample preparation, and measurement protocols, are required. Additionally, a more comprehensive data analysis would be beneficial. Compliant mechanisms overcome challenges faced by traditional rigid-body mechanisms, such as backlash, wear, and increased part count. They utilize single-piece flexible structures for force and motion transmission, resulting in dimensions and cost savings in MEMS fabrication. According to the studies conducted by Shuib et al. [31], there are two common approaches for designing compliant mechanisms that include the kinematics-based approach, which represents compliant segments as rigid links with added torsional springs, and the structural optimization-based of approach, which focuses on determining the topology, shape, and size of the compliant mechanism through numerical methods like topology optimization. While compliant mechanisms offer benefits, their design and analysis still pose challenges, including the lack of formal synthesis methods and the complexity of determining force-deflection relationships and optimizing design variables that further research is needed to enhance their effectiveness. The silicon accelerometer utilizes piezore- sistive techniques to convert mechanical motion into an electrical signal. It consists of a silicon base, double cantilever beams, and piezo resistors. Kal and Das [32], explain well this accelerometer achieves a range of ± 13 g with low off-axis acceleration, high resolution, and linearity. this design specifically suggested for aircraft motion sensing in avionics. they stated, the quartz doubleended tuning fork (DETF) accelerometer is qualified for space applications. In completing this research process, Liu et al. [33], focused on a robust optimum design of shape and size for an accelerometer fabricated by silicon micromachining technology is proposed to minimize the effect of Alizadeh A., Saltykova O.A., Novinzadeh A.B. RUDN Journal of Engineering Research. 2024;25(1):57-74 variations from micro fabrication without a prelimi- nary assumption on the probabilistic distributions. The sensitivity analysis technology is employed to reduce design space and to find the key parameters that have greatest influence on the accelerometer. Then, the constraint conditions and objective functions for robust optimization and the corresponding mathe- matical model are presented. The unit stiffness sensitivity of the spring beams should be equal to the unit mass sensitivity of the detecting mass: │ │ Max , Min : 1 Max . . For this purpose, the unit stiffness sensitivity of the spring beams should be equal to the unit mass sensitivity of the detecting mass. where ζ is a random variable; k is the stiffness of the spring beam; m is the mass the detecting mass; is the initial value of the robust constraints. Max , Min are Maximum and minimum fabrication errors influence on the performance of the accelerometer and Max , is objective function. This formulation is practically applicable since no statistical information on the uncertainties is required during the process of the robust optimal analysis in advance. Considering that the magni- tude of fabrication errors and uncertainties in an accelerometer structure are comparatively large, the present robust optimal design method can be valuable for practical accelerometer design. The optimization problem is solved by the Multiple-island Genetic Algorithm is employed to solve the optimization problem, and the results are shown in Table 4. A design methodology for optimizing MEMS impedance matching networks based on the uniformity of the Smith chart coverage is presented by Domingue et al. [34], that approach is validated through a comparison between traditional coplanar waveguide (CPW) designs and improved designs using a slow-wave (SW) structure. The proposed reconfigurable impedance matching network based on distributed MEMS transmission line (DMTL) coupled with the SW structure achieves a 25 % reduction in physical length compared to traditional DMTL. In this regard, a better view of the results of this study are shown in Figure 3. Table 4 Results of Robust Optimization Description Symbols Initial design Optimum design Design variables (mm) 8 8.774876 5 4.485502 1.5 1.567798 0.8 0.847795 0.4 0.379551 1.5 1.592313 1storder natural frequency (Hz) 25.1349 22.489963 2storder natural frequency (Hz) 40.1076 39.126760 Frequency difference (Hz) f 15 16.636797 Sensitivity of detecting capacitance k 2.5947e12 3.16742e12 Robust constraints 1.3056e13 1.0914e13 A team of researchers [35], proposed a metho- dology for determining true chaos in distributed mechanical systems. It focuses on a beam structure with contact interaction, modeling nonlinearity using the Cantor model. The problem is formulated using partial differential equations and solved as an ordinary differential equation system using finite differences and Runge - Kutta methods. The analysis includes nonlinear dynamics methods and the qualitative theory of differential equations. The study confirms the existence of chaotic behavior, synchronization of oscillations, and the convergence of results through signal analysis as shown in Figure 4. Parameter values and methods for reliability and validity are established. The findings emphasize the with contact interaction reveals reduced chaotizaimportance of considering nonlinearities, and a tion through increased equations and regularization comparison between linear and nonlinear problems of oscillations. Figure 3. Measured return loss performance and Measured power transfer performance: From left-side: Measured return loss performance over the gamma plane for the fabricated designs (in decibels): a - CPW design; b - SW design. From right-side: Measured power transfer performance over the gamma plane for the fabricated designs (in decibels): a - CPW design; b - SW design S o u r c e : made by the Domingue et al. [34] t t a b Figure 4. Comparison of beam signals. Beam signals, with and without geometric nonlinearity: a - Beam 1; b - Beam 2 S o u r c e : made by the Krysko et al. [35] The recent research of Nguyen, Saltykova and Krysko [36], focused on analyzing the nonlinear dynamics of MEMS cantilever beams using mathematical models and the finite element method with the ANSYS software package. The researchers investigated the dependence of vibration charac- teristics on geometric parameters and considered spatial vibrations in a 3D space. The study presented results such as time signals, phase portraits, wavelet spectra, and Fourier spectra. The advantage of using the finite element method is its ability to capture complex geometries and non- linear behavior accurately. However, the explicit integration method, specifically the Euler method, employed for solving the problem has limitations in terms of stability and accuracy, especially for longterm simulations. The research highlights the importance of considering spatial vibrations and provides insights into the effects of beam geometry on vibration amplitudes and frequencies. Overall, the research contributes to the understanding of MEMS cantilever beam dynamics but could benefit from more advanced numerical integration tech- niques to improve accuracy and stability in longduration simulations. The main purpose of the investigated methodologies for fabrication on the surface of the smart material or the desired semiconductor is to check the possibility of reducing the dimensions of the desired sensor IC, and the proposed calculations and modeling are more focused on the manufacturing accuracy. Scientists [28], mentioned this issue in the review of the presented equations and modeling that mechanical sensitivity in the event of unmatched modes is significantly decreased since the quality factor of the sense mode is no longer fully exploited. In particular, for the range mrq,, <∆j<jR the mechanical sensitivity for the unmatched frequency condition can be expressed as: ∆tu │wxyz{|}~ =2t!dd, ="O(C)#$RR = ∆Ω =4 %P&&,, pdm S())m,*+q,, /0{ .12*+3456(m q)T) . As a rule of thumb, the higher the required bandwidth, the lower the mechanical sensitivity that can be obtained from a MEMS gyroscope. On the other hand other studies explain the electrome- chanical systems of accelerometers and provide insights into their operation and interface circuitry. Shaeffer et al. [37], discussed the challenges of achieving DC accuracy, minimizing drift, and addressing thermal variations for accelerometers. Finally, they mentioned upon vibratory rate gyroscopes, which measure angular rate of rotation based on detecting Coriolis acceleration. Table 5 has referred more typical methodologies to optimizing MEMS fabrication. Table 5 Optimization Methodologies Based on Challenges and Typical Techniques Selecting Challenge Main Solution or techniques Advantages/Disadvantages Ref Overall, the challenge lies in developing a CMOS compatible integrated MEMS process that enables the integration of process for fabricating a differential Deep Reactive Ion Etching (DRIE) MEMS and CMOS technologies, while ensur- [38] capacitance-based sensor on a SOI process ing proper electrical isolation, maintaining (Silicon-On-Insulator) wafer mechanical The sandwiching technique has several advantages over spin coating and selfTo find an effective method for de- planarization. It ensures thickness uniformity, positing thick layers of SU-8 photo- 1) Spin Coating eliminates surface irregularities, and allows resist, particularly for applications 2) Self-Planarization for efficient production of thick and uniform [38] such as microfluidics and polymeric 3) Sandwiching coatings. The technique is simple, fast, and membranes fabrication using litho- achieves the desired thickness in a single graphy step, overcoming some limitations of other coating methods To optimize the sensitivity of microcantilever sensors by incorporating stress - concentration - regions (SCR) and modeling their effects Integration of stress concentration regions (SCR) on a microcantilever sensor to improve its performance Advantage of the technique used is that it enhances the performance of the microcantilever sensor by increasing - the dif- ferential surface stress through the incorporation of SCR holes. However, the disad- [38] vantage is that designs with multiple SCR holes may not provide significant improvements in performance while adding complexity and cost. Challenge Characterization of residual stresses in microelectromechanical systems (MEMS) Integrating gyroscope and accelerometer sensors into a compact package while maintaining mechanical stability and electrical compatibility with existing systems To develop and optimize a IMU for use in the Lobster-Eye X-ray Satellite to achieve high precision and stability in measuring angular rates while considering the specific requirements and constraints of satellite applications Improving performance parameters, and optimizing design features to enhance the precision and robustness of MEMS vibrating gyroscopes for applications in harsh environments Main Solution or techniques Measuring deformation or deflection and applying mathematical models to extract stress values and material properties 1) Heavily doped silicon sensor mechanisms and a MEMS foundry at Honeywell 2) Use of a vibration isolator in the IMU design, leveraging the design of the vibration isolator 1) Frequency Stabilization and Frequency Split Mitigation. 2) Anti-Interference Design of External Environment. 3) Self-Calibratio In summary, the techniques and solutions in this research involve understanding the fundamentals of vibrating gyroscopes, addressing mode-matching challenges, exploring different gyroscopic designs like gimbal and multi-axis gyroscopes, and finding ways to minimize mode mismatch during microfabrication processes Advantages/Disadvantages Ref Relies on the Stoney formula, which is based on simplified assumptions and may [39] lead to modeling errors in certain cases Vibration isolator helps attenuate external vibrations at critical sensor resonant frequencies, allowing the IMU to maintain full [40] performance even under harsh tactical conditions with high vibration levels Overall, the technique and solution offer significant advantages in terms of performance, size, and versatility, making the MEMS IMU a valuable choice for highprecision navigation applications. However, [41] certain trade-offs and limitations (such as: Limitations in Absolute Accuracy & Sacrifice in Range and Bandwidth) should be considered based on the specific requirements of the application 1) Mode Mismatch Challenges. 3) 2) Sensitivity to External Factors. Limited Bandwidth. [42] 4) Complex Design and Calibration Ending of the Table 5 N o t e . These are classical methods for optimizing Micro-Electro-Mechanical sensors and navigation micro devices. Additional sources provide ample opportunities to explore and extract further examples for analysis [43]. 3. Modern Fabrication Methodologies In contemporary design practices, there is a prominent inclination among scientists to leverage various levels of artificial intelligence (AI) with the aim of diminishing calculation errors and curtailing production costs associated with micro-electrome- chanical devices [44]. This trend has consequently facilitated the amalgamation of conventional optimization techniques with cutting-edge technologies such as machine learning, neural networks, and related methodologies [45; 46]. 3.1. Unveiling Modern Methodologies for MEMS Navigation Sensor Fabrication Fontanella et al [47], conducted research to optimize the calibration of an IMU by addressing bias drift error caused by temperature variations in MEMS gyroscopes. They proposed using BackPropagation (BP) Neural Networks as a solution to improve calibration accuracy and reduce residual errors compared to the polynomial fitting method. The study included an analytical model for bias, a description of the standard calibration procedure, and a comparison of flight attitude angles calculated using both methods. In this context, the Least Mean Squares (LMS) curve fitting method is employed for constructing a temperature model of the sensor's zero-bias. By using a polynomial of order m to approximate the relationship between experimental data, the following expression is derived: 79 = 89 - ∑>9 d k9:99. Here, :9 represents temperature, 89 represents the corresponding gyro output, and 79 represents the error between the gyro output and the value calculated using the regression equation (where ‘i’ ranges from 1 to n, the number of samples of static IMU data). Employing LMS theory, the aim is to minimize the square of 79 to determine the optimal coefficients k9: φ(kd, k$, … , k>) = ∑;9<$ 79m ⟶ min. After thorough evaluation, the efficacy of the neural network calibration technique was conclusively demonstrated. Finally, the useful effect of neural network on beam signals is shown in Figure 5. a b Figure 5. Comparison of beam signals: a - Structure of the BP Neural Network adopted for modelling; b - Compensated bias, obtained using the BP Neural Network calibration method (z-axis gyroscope of Axitude AX1) S o u r c e: made by the Fontanella et al. [47] Modeling the random drift of MEMS gyroscopes is an important research area because it directly contributes to improving the accuracy of MEMS gyroscopes. For this purpose, in line with Fontanella et al’s research [47], the key contribution of Xing et al.’s research [48], is in reconstructing MEMS gyroscope random drift data using PSR method and subsequent analysis using both BPANN and CPSO-LSSVM methods. It is important. In this study investigated the non-linear and nonconstant random drift characteristics of MEMS gy roscopes. The proposed approach in this paper consists of using a wavelet filter to reduce the noise in the original data of MEMS gyroscopes, followed by reconstruction of the random drift data using phase space reconstruction (PSR). The reconstructed data are then used to build a model using a least squares support vector machine (LSSVM), with the model parameters optimized through chaotic particle swarm optimization (CPSO). The CPSO-LSSVM method effectively reduces the standard deviation of random drift through compensation, as shown in Table 6, and shows superior prediction accuracy compared to BP-ANN, as shown by statistical indices such as MAE, RMSE and ARE are shown for the test data set. Table 6 The statistical analysis of BP-ANN and CPSO-LSSVM. MAE Group Ⅰ Group Ⅱ Group Ⅲ Model MAE,°/S RMSE, °/S ARE BP-ANN 0.0421 0.0554 11.10% CPSO-LSSVM 0.0099 0.0263 8.86% Before compensation,°/S 0.00354 0.00412 0.00328 After compensation, °/S 0.00065 0.00072 0.00053 Note. MAE - mean absolute error; RMSE - root mean square error; ARE - average relative error. Upon examining Table 5, a notable reduction in the standard deviation of the random drift was evident following compensation. These findings provide additional evidence to support the effectiveness and credibility of the CPSO-LSSVM method. Consequently, this approach proves to be a viable and gratifying means of constructing the model for MEMS gyroscope random drift. The set of these researches, despite all the advantages, did not provide an answer for the performance of the proposed solution at higher resonant frequencies and expanding its frequency range. Therefore, Pertin et al [49] have focused on optimizing the design of a conical piezoelectric MEMS vibration energy harvester using artificial intelligence techniques, in order to increase the harvester’s performance at higher resonant frequencies and expand its frequency range. FEM simulations generated datasets used to train an artificial neural network (ANN) for optimization algorithms. Two training methods of Levenberg - Marquardt (LM) [50; 51] and Scaled Conjugate Gradient (SCG) [52; 53] and two optimization methods of GAM (Goal Attainment Method) [54] and Genetic Algorithm (GA) [55-59] were compared. The results showed that the genetic algorithm with ANN trained by SCG provided the best solution. The optimized structure achieved over five times more power below 200 Hz and a wider frequency range. The proposed harvester is suitable for low-frequency energy harvesting and can be applied to similar structures. The approach allows for further improvements and investigations using different algorithms for ANN training or optimization (In order to learn more about the structure of the methods mentioned in this paragraph, refer to the given references). Different training methods and approaches yield distinct artificial neural network (ANN) fitting functions, which are evaluated using two key metrics: mean squared error (MSE) and regression coefficient (R). The MSE measures the dissimilarity between the desired target ‘t’ and the ANN's generated output ‘a’, with lower values indicating better performance. The regression coefficient R, also known as the coefficient of correlation, gauges the correlation between the target and the ANN's output, aiming for a value closer to one for superior performance where N is the number of examples in a dataset, and C9̅ is the arithmetic mean of the target values. ̅ . Regression coefficients above 95.152 % were obtained using the Levenberg-Marquardt (LM) training algorithm, considered satisfactory. The correlation between targets and ANN outputs is not automatically computed for the two outputs, but separate assessments are conducted to ascertain the ANN’s capability to achieve favorable results for new data. The LM training method outperformed the SCG training algorithm in terms of MSE, particularly in predicting the frequency and power of the first resonant mode. The LM training method yielded an MSE an order of magnitude lower than the SCG training method’s MSE for predicting the normalized frequency of the resonant mode (3.8398×10-4). The ANN fitting functions obtained using the LM training algorithm exhibited regression coeffi- cients above 95.152 %, indicating good performance as shown in Figure 6. However, since the ANN generates output as a two-column vector, the correlation between the targets and the ANN outputs is not automatically calculated. Separate analysis was conducted to assess the ANN's capability to achieve satisfactory results for new data. On the other hand, applying the SCG training algorithm yielded even better regression coefficients, surpassing 95.468 %. The LM training method outperformed other methods in terms of mean squared error (MSE), particularly in predicting the frequency and power of the first resonant mode. The MSE for the LM method was 3.8398×10-4, significantly lower than that of the SCG method. Although a perfect fit to the training data does not guarantee a reliable predictive model, the ANN was further tested using unseen data. 3.2. Integrating Optimizer Techniques for Managing Computational Complexity The challenge of researchers is finding optimal, suboptimal solutions and reducing computing costs for MEMS design and developing a bionic CAD system [60]. On the other hand meeting modern MEMS element design requirements requires complex multi-physics analysis, accurately describing the project structure. Furthermore, new knowledge accumulation and synthesis of sub-optimal solutions are also crucial. Koryagin et al’s proposed approach [61], offers a promising direction for bionic CAD development, balancing computational complexity and competitive device solutions. Cognitive adaptation and the use of cognitive knowledge banks are proposed to handle the complexity of MEMS design that goal is to find optimal and suboptimal solutions by utilizing cognitive knowledge banks and adaptive methods. Thus, there is a complex mathematical model of the designed system shown below: D=CDq ,{EOF/t}:{E@k/G},{EHI/J}→{ESl/8}L, where: k is a mathematical model, A & B is a set of input, output variables of the model, C is a semantic network (Figure 7), J is a semantic network synthesis operator, M is a computing mechanisms complex, N E is a structured system pattern. a b a b c d c d Figure 6. Regression diagrams for the ANN trained by applying Levenberg - Marquardt algorithm & Scaled Conjugate gradient Algorithm: a - training dataset; b - validation dataset; c - dataset used for testing; d - dataset with all three aforementioned groups of data S o u r c e: made by the Pertin et al. [49] Figure 7. Example of the semantic wave: B - home domain, Ƃ - home domain, E - terminal nodes, U - intermediate vertex S o u r c e : made by the Koryagin et al. [61] Partial differential equations (PDEs) describe MEMS element dynamics. Simulating them is complex due to multiple physical processes. SAM technology enables optimization in MEMS design cycles. The bionic CAD system’s core architecture includes models, preprocessor, bionic search, and postprocessor blocks. It has allowed parallelization, search management, and solution generation. Validation involves comparing calculation results with published data and experiments. The adaptive MEMS design system based on SAM tech reduces computational costs and simulation errors. Computational intelligence is widely used in MEMS design, but bionic systems with AI and cognitive technologies are more promising com- pared to traditional methods. 4. Discussion To overcome the challenges associated with optimizing MEMS-based navigation solutions, a comprehensive approach must be adopted. This includes meticulous material selection, sophisti- cated mathematical modeling techniques, and rigorous calibration procedures. The integration of AI methods can certainly enhance the opti- mization process, but it should be acknowledged that the foundation for effective optimization lies in addressing fundamental issues. The optimization of functional errors in MEMS sensors within electronic circuits has long been a challenge for users. Leveraging the Kalman filter and artificial intelligence has significantly enhanced the accuracy and precision of navigation mission outputs. However, researchers are actively seeking more optimal and integrated solutions to further optimize these sensors at the user level. These endeavors serve as a promising basis for future research in the field [62; 63]. Conclusion In the conclusion of this paper several key results and insights can be summarized. These conclusions reflect the findings and contributions: 1. Optimizing MEMS navigation sensors for aerospace vehicles poses challenges in material selection and structural complexities. 2. Integration of sensors into an IC and efficient mathematical modeling are crucial for performance. While AI can optimize sensor data, ensuring seamless integration and compatibility between AI algorithms and the sensor hardware can be challenging. 3. Calibration during startup and mitigating functional errors by users are essential. 4. MEMS-based sensors may have limitations in terms of accuracy and precision, especially in demanding navigation applications. Addressing these issues could involve improving the sensor’s calibration and error correction mechanisms. 5. Future research should refine material selection, advance mathematical models, and explore novel calibration techniques. Enhancing sensor performance and reliability in aerospace requires a multidimensional approach and focus on fundamental challenges.

About the authors

Ali Alizadeh

RUDN University; K.N. Toosi University of Technology

Author for correspondence.
Email: ali.rim.alizadeh@gmail.com
ORCID iD: 0009-0006-0673-1893
SPIN-code: 1755-9674

M.S Student of Control in Technical Systems-Space Engineering of the Department of Mechanics and Control Processes, Academy of Engineering, RUDN University; M.S Student of Space Engineering, Faculty of Aerospace Engineering, K.N. Toosi University of Technology

Moscow, Russia; Tehran, Iran

Olga A. Saltykova

RUDN University

Email: saltykova-oa@rudn.ru
ORCID iD: 0000-0002-3880-6662
SPIN-code: 3969-6707

Ph.D. of Physico-mathematical Sciences, Associate Professor of the Department of Mechanics and Control Processes, Academy of Engineering

Moscow, Russia

Alireza B. Novinzadeh

K.N. Toosi University of Technology

Email: novinzadeh@kntu.ac.ir
Ph.D. of Space Engineering, Associate Professor and Head of the Department of Space Engineering, Faculty of Aerospace Engineering Tehran, Iran

References