Modeling Temperature Effects in a MEMS Ring Gyroscope: Toward Physics-Aware Drift Compensation

150 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 34, NO. 2, APRIL 2025 Modeling Temperature Effects in a MEMS Ring Gyroscope: Toward Physics-Aware Drift Compensation Mehran Hosseini-Pishrobat , Member, IEEE, and Erdinc Tatar , Member, IEEE Abstract— Temperature plays an indispensable role in the long-term performance of MEMS gyroscopes, and despite extensive studies in the literature, analytical treatment of temperature effects is still an open problem. This paper, to the best of our knowledge, is the first attempt to address this gap for ring gyroscopes. We start with a superposition principle that disentangles thermal displacement fields from the gyroscope’s nominal vibration. We set forth a geometrically nonlinear variational formulation to obtain the temperature-induced stiffness matrix. We conduct temperature tests on our 3.2 mm-diameter, 58 kHz ring gyroscopes equipped with 16 capacitive stress sensors. The experimental data validate our analytical modeling in the following key aspects: 1) The model accounts for not only changes in material properties but also a less explored factor, thermal stresses. Thanks to a strain interpolation module that leverages the measured stresses, the model predicts frequency variations consistently and captures hysteresis loops arising from residual stresses. Notably, we accurately estimate the deviation of the temperature coefficient of frequency (TCF) from the expected value −30 ppm/◦C (based on the widely known −60 ppm/◦C dependency of Young’s modulus of silicon). 2) The model is able to capture stiffness couplings in the orders of less than 0.1 N/m (in a 7 kN/m device) and closely predicts the quadrature error and its leakage into the in-phase channel. Additionally, the model incorporates temperature variations of mechanical scale factor, drive mode’s amplitude, damping coupling, and sense mode’s phase in terms of their contribution to the in-phase error. Based on these merits, our model serves as a building block toward drift compensation algorithms encompassing the underlying physics of the temperature effects. [2024-0163] Index Terms— Temperature effects, ring gyroscope, stress sensing, quadrature error, in-phase error. I. INTRODUCTION LONG-TERM performance deterioration is pervasive in MEMS gyroscopes, hindering their navigation-grade applicability. Temperature has a prominent role in this problem Received 20 September 2024; revised 13 December 2024; accepted 29 December 2024. Date of publication 15 January 2025; date of current version 4 April 2025. This work was supported by the European Union’s European Research Council (ERC) under Grant 101116162-0-drif-ERC-2023STG. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or ERC. Subject Editor V. Zega. (Corresponding author: Mehran Hosseini-Pishrobat.) Mehran Hosseini-Pishrobat is with the Department of Electrical and Electronics Engineering, Bilkent University, 06800 Ankara, Türkiye (e-mail: [email protected]). Erdinc Tatar is with the Department of Electrical and Electronics Engineering and the National Nanotechnology Research Center (UNAM), Bilkent University, 06800 Ankara, Türkiye (e-mail: [email protected]). Digital Object Identifier 10.1109/JMEMS.2024.3524796 due to its correlation with scale factor and bias drift [1],[2]. Therefore, understanding the effects of temperature is critical for improving the performance of these sensors beyond the current limits. On this basis, we performed several tests on our ring gyroscopes by exposing them to controlled temperature cycles. Several key observations emerged from these experiments: 1) According to the well-known ∼−60 ppm/◦C temperature dependency of silicon’s Young’s modulus [3], we expected to see a temperature coefficient of frequency (TCF) of ∼−30 ppm/◦C; however, we measured very different values such as −12 ppm/◦C and −14 ppm/◦C. 2) Over temperature cycles, hysteresis loops formed in the frequency-temperature plots, which point to some irreversible residual effects. 3) Quadrature and in-phase channels exhibited outputs in tandem with the temperature cycles, indicating nonuniform stress profiles that affect the stiffness symmetry of the gyroscope. Even with a continuous quadrature cancellation, the in-phase error persisted, which signifies variation sources other than quadrature leakage. The above observations point to thermal stresses as a major mechanism of temperature effects. Previous works [2],[4] have established the potential of stress calibration for drift compensation, and [5] developed an analytical model for the effects of mechanical stress. In practice, the source of stresses in MEMS gyroscopes is usually thermal rather than mechanical. Accordingly, we aim to expand such analytical modeling’s ambit to encompass the temperature effects and explain the experimental observations. This attempt will serve as a basis for understanding the physics behind the temperature dependency of long-term drift. Preliminary results of this study–for only frequency variations–have been presented in the conference paper [6]. A. Literature Review In [1], a drift compensation was developed for a high quality factor quadruple mass MEMS gyroscope, which prevented the need for a temperature sensor by taking advantage of the drive mode’s linear temperature dependency. In a comparable approach, the frequency-temperature characteristics of an on-chip integrated resonator were leveraged to calibrate a MEMS accelerometer [7]. For a NEMS-based gyroscope, it was shown in [8] that the drive and sense 1057-7157 © 2025 IEEE. All rights reserved, including rights for text and data mining, and training of artificial intelligence and similar technologies. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on April 16,2025 at 20:57:19 UTC from IEEE Xplore. Restrictions apply. HOSSEINI-PISHROBAT AND TATAR: MODELING TEMPERATURE EFFECTS IN A MEMS RING GYROSCOPE 151 Fig. 1. Ring gyroscope’s SEM (a), schematics (b), and arrangement of electrodes (c). modes’ TCF mismatch is responsible for the scale factor drift. The temperature dependency of drift for a tuning fork gyroscope was investigated in [9] based on temperature chamber tests, and material properties and electronics parameters variations were introduced as the main factors. For a JPL/Boeing Gyroscope, it was shown in [10] that drive/sense frequencies and quality factors have an exponential response–with a considerable time lag–to slow step-like temperature changes. A pole-zero cancellation controller was developed in [11] to guarantee above 90 Hz bandwidth for a dual-mass MEMS gyroscope against temperature-dependent frequency variations. A micro-oven was reported in [12] to thermally isolate an inertial measurement unit (IMU) and prevent temperature fluctuations using a PID controller. The bias drift behavior of a group of consumer-grade MEMS gyroscopes was studied in [13] over −40◦C to 85◦C, taking into account the temperature-dependency of quadrature and sense mode phase errors. Mechanical stresses were cited as the source of discrepancy between the experimental and predicted quadrature errors. Mode reversal has been proposed for temperature calibration of circular gyroscopes [14],[15]. In this method, the device’s operation periodically switches between the two wineglass modes to generate two different outputs whose combination cancels out the bias. In [16], recurrent neural networks with a blend of machine learning algorithms were used to model and compensate for the drift. Similarly, neural networks were utilized in [17] for temperature calibration to improve upon the conventional polynomial fitting. B. Relevance Analytical models for temperature effects in MEMS gyroscopes are absent in the literature. In that respect, the exigency of our work lies in addressing the following: •In the existing studies, the emphasis is often on the temperature dependency of the material properties, and the role and significance of thermal stresses have not been meticulously explored. •Current drift compensation methods focus on extracting patterns from input-output data (using, e.g., linear fitting or neural networks), where the physics underpinning the drift phenomenon is treated as a black box. The emphasis of such TABLE I RING GYROSCOPE’SPARAMETERS methods is, in effect, on correlation rather than causation. Our goal is to lay the groundwork for models that shed light on the governing physics and the root cause of the drift. C. Contribution Our gyroscope, shown in Fig. 1, has a double-ring structure designed to operate in the n=2 wineglass modes at ∼58 kHz. The device was fabricated by a silicon-on-glass (SOG) process from (111) silicon with wafer-level vacuum packaging [18], and its dimensions are given in Table I. Sixteen electrode pairs around the outer ring are used for differential drive, sense, quadrature cancellation, and frequency tuning. Moreover, we have 16 capacitive stress sensors (eight inner, eight outer, 45◦separated) to monitor external stresses across the substrate. We set out an analytical model for the effects of temperature on the stiffness distribution of this ring gyroscope. The departure point of our analysis is that we can delineate two mechanisms for such effects: 1) Change in material properties, especially the ∼−60 ppm/◦C dependency of Young’s modulus; 2) Thermal stresses that arise from 1) conflict between thermal deformations and structural constraints, 2) coefficient of thermal expansion (CTE) mismatches at the silicon-glass and die attachment interfaces. Our modeling adheres to the linear theory of thermoelasticity where strains are the linear sum of a thermal and a mechanical part [19]. The innovative features of our approach are as follows. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on April 16,2025 at 20:57:19 UTC from IEEE Xplore. Restrictions apply. 152 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 34, NO. 2, APRIL 2025 Fig. 2. Superposition of three loading states for modeling temperature effects. •We propose a superposition framework by distinguishing thermal components of the total displacement field from the nominal vibration of the gyroscope. Considering geometric nonlinearity, we present a potential function to calculate the stiffness matrix induced by such thermal displacements within a variational framework. •The on-chip stress sensors measure the mechanical strains that enable us to interpolate the substrate’s strain field in combination with thermal strains. We use this interpolation to calculate the displacements of the supporting beams and changes in the electrostatic gap. Furthermore, we show the mechanical strains contain residual effects over temperature cycles that account for frequency-temperature hysteresis. We demonstrate the effectiveness of our model through comparison with the experimental data. The model addresses the aforementioned experimental observations by successfully predicting the frequency variations, formation of frequency-temperature hysteresis loops, TCF, and quadrature/in-phase errors. Additionally, the model provides insight into the evolution of the gyroscope’s structural stiffness over temperature. This approach allows us to identify different types of stiffness based on how they manifest in the gyroscope’s performance. D. Organization Section II explains the solid mechanical basis of the modeling. Section III elaborates on the principles of stress sensing in connection with the analytical model. Section IV links the previous two sections to the gyroscope’s frequency variations and quadrature/in-phase errors. Section Vprovides the details of temperature tests and discusses the analyticalversus-experimental results. Section VI concludes the paper with a view on future direction. II. ANALYTICAL MODELING The mechanical structure of the gyroscope is composed of Ring#1 (outer ring), Ring#2 (inner ring), eight connecting beams between Rings#1 and 2, and eight supporting beams joining Ring#2 to the anchored inner structure (see Fig. 1-(b)). Our objective, in a nutshell, is to mathematically describe how this structure’s stiffness varies as temperature changes. Let T0be the initial stress-free temperature, and the gyroscope is subject to the thermal load 1T=T−T0. As the crux of our modeling, we decompose the displacement field of the gyroscope’s moving structure into the superposition of three distinct states (see Fig. 2): 1) State I. Thermal expansion (or contraction) of the moving structure while it is fixed at the supporting beams (i.e., homogeneous boundary conditions). The resulting stresses are due to conflicts between the thermal deformations of the beams and rings. 2) State II. The internal anchored structure undergoes thermal expansion (contraction) as well and pushes (pulls) the moving parts as a result (amounting to nonhomogeneous boundary conditions). The ensuing stresses are mostly exerted on Ring#2. 3) State III. Nominal n=2 wineglass vibration of the gyroscope in which, Ring#1 displacement field is governed by the mode shapes cos(2θ) (drive mode) and sin(2θ) (sense mode). Despite their shared thermal origin, we distinguish State I from State II since, for stiffness calculation, we are concerned with the gyroscope’s moving structure. State I describes the thermal effects endogenous to the moving structure, while State II accounts for the exogenous effects in the form of boundary loads at the supporting beams. States I and II take place in a slow time scale determined by the temperature cycles (e.g., in the order of minutes or hours) while State III occurs in a fast time scale defined by the operation frequency (in the order of µs). This slow-fast time scale separation indicates that States I and II can be conceived as the equilibrium displacement field of a quasistatic thermal loading. We then obtain the resulting stress field σ, which we can treat as a prestress in conjunction with the gyroscope’s nominal vibration. In a variational framework, we consider the modified potential function Udefined as [20] dU d−V=1 2EE2 n+σEn(1) (d−Vis the differential volume element) to describe the interaction between States I-III and to calculate the stiffness matrix Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on April 16,2025 at 20:57:19 UTC from IEEE Xplore. Restrictions apply. HOSSEINI-PISHROBAT AND TATAR: MODELING TEMPERATURE EFFECTS IN A MEMS RING GYROSCOPE 153 [kσ i j ] ∈ R2×2induced by σ: dkσ i j d−V=∂2 ∂Qi∂QjdU d−V−dU d−Vσ=0Q1,Q2=0 .(2) Here, Enis the Green-Lagrange strain [21] associated with the nominal n=2 vibration of the gyroscope (that is, State III) and Q1and Q2are the displacements of the cos(2θ) and sin(2θ) modes, respectively. In Equation (1), it is essential to Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on April 16,2025 at 20:57:19 UTC from IEEE Xplore. Restrictions apply. 154 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 34, NO. 2, APRIL 2025 Fig. 3. Temperature variations of silicon and glass CTEs. use the Green-Lagrange strain to capture all the second-order terms that produce stiffness (refer to the discussion of geometric nonlinearity in [5]). Following the aforementioned superposition principle and Equation (2), the total stiffness matrix of the gyroscope K=Km+KT+KS−Ke−Kβ,(3) comprises the following parts: •Km= [km i j ] ∈ R2×2is the nominal mechanical stiffness matrix of the wineglass mode shapes in the absence of thermal stresses (State III). If the value of this matrix at the initial temperature T0is Km,0, then it should be scaled by the variations of Young’s modulus, Km=E(T) E(T0)Km,0. •KT= [kT i j ] ∈ R2×2is the thermal stiffness matrix resulting from State I of thermal loading, capturing the stiffness produced by thermal deformations of the moving structure while no external boundary conditions are applied. •KS= [kS i j ] ∈ R2×2is the stress stiffness matrix corresponding to State II of thermal loading; this stiffness matrix reflects the effects of stresses that expansion (contraction) of the inner structure imposes on the moving parts. •Ke= [ke i j ] ∈ R2×2is the stiffness matrix associated with the linear electrostatic forces. •Kβ= [kβ i j ] ∈ R2×2is the equivalent stiffness matrix of the Duffing-type nonlinear electrostatic forces. Note that Keand Kβappear with the negative sign in (3) because of their electrostatic origin. The calculation details of Kmand Kecan be found in [22] and [23], respectively. We approximate Kβusing He’s variational method, which has been shown to provide good accuracy in predicting resonance frequencies of oscillators with polynomial nonlinearities [24]. Our approach to compute KTand KSconsists of 1) calculating the displacement fields of States I and II using the Ritz method [25]; 2) obtaining the stress field σin Equation (1) based on the strain-displacement relation and the constitutive law; 3) determining the induced stiffness using Equations (2). An exhaustive treatment of this procedure based on variational principles of solid mechanics is reported in [26], and we here only quote the final results for the stiffness matrices in Box 1. It is important to note that the equations for KTand KSrepresent an extensional-type stiffness, which is proportional to E A/Ras opposed to bending-type having the E I/R3factor. These two stiffness originate from the interaction of the extensional stresses across the rings and the bending motions of the n=2 mode shapes and could be hardening or softening depending on the sign of the stresses. As a result, to capture this type of stiffness in the modeling, the assumption of centerline inextensibility–ubiquitous in the ring gyroscope literature–should be revoked. A. Temperature-Dependent Material Properties Based on the widely used approximation dE/EdT≈ −60 ppm/◦C[3], we consider the following model for the Young’s modulus: E(T)≈170 1−60 ×10−6(T−20)[GPa],(4) where Tis in ◦C. According to the CTE measurements in [27] and [28] for silicon and borosilicate glass, respectively, we use the following polynomial approximations valid for T∈ [20◦C,100◦C]: Si: α(T)≈2.3332 +9.2618 ×10−3T −1.8901 ×10−5T2[ppm/◦C]; (5) Substrate: ˜α(T)≈3.1137 +1.5215 ×10−3T −1.0524 ×10−5T2+1.3980 ×10−8T3[ppm/◦C].(6) As plotted in Fig. 3, the silicon’s CTE has a 22% variation when the temperature rises from 20 ◦C to 100 ◦C while this value for the substrate is %1. III. STRESS SENSING A total of 16 stress sensors measure thermal stresses across the gyroscope (Fig. 4-(a)). Eight sensors are placed in the internal suspension region, and the rest are located outside the outer electrodes. This configuration helps us capture the stress effects on the supporting beams and electrostatic gap. Each stress sensor has an unbalanced bridge-type mechanical structure that converts and amplifies x-displacements in the y-direction (red lines in Fig. 4-(b)), which then is picked up by AC-modulated differential capacitance reading [4],[29]. It is important to note that the stress sensors do not respond to pure unconstrained thermal expansion (e.g., in a matched CTEs scenario). The FEM simulations visualized in Fig. 4-(b) demonstrate that pure thermal load (1T=50◦C) generates a negligible displacement gradient across the stress sensor’s interdigitated fingers and has no appreciable effect on their gaps. On the contrary, the pure mechanical load εxx = +42 µm/m (1x=50 nm applied to the underneath anchors) produces a significantly more pronounced displacement gradient that the stress sensor can resolve and pick up. In our gyroscope, these mechanical strains primarily result from the CTE mismatches of the different materials, including PCB, solder, ceramic package, and die-attach epoxy. The latter is particularly important due to its large CTE (∼140 ppm/◦C) and propensity for developing residual stresses. On this basis, the output ˜εi j (t)of the stress sensor located at polar coordinates (˜ Ri,˜ θj)is given by ˜εi j (t)=∂˜ur ∂r(r, θ, t)(˜ Ri,˜ θj)− ˜εT(t), (7) Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on April 16,2025 at 20:57:19 UTC from IEEE Xplore. Restrictions apply. HOSSEINI-PISHROBAT AND TATAR: MODELING TEMPERATURE EFFECTS IN A MEMS RING GYROSCOPE 155 Fig. 4. Stress sensors’ distribution (a); FEM results for y-displacement (nm) under pure thermal and mechanical loads (b) (working principle is explained in red lines). Fig. 5. Interpolation of substrate’s strain and displacement fields; ˜εTand εTare the thermal strains at the substrate and silicon layer, respectively. where ˜urand ˜εTare the radial displacement and thermal strain in the substrate level. We then proceed by the interpolation algorithm explained in [5] to obtain the radial displacement (see Fig. 5for visualization): ˜ur(r, θ, t)≈Zr 0 φ⊤(ϱ, θ)a∗(t)dϱ+ ˜εT(t)r,(8) where a∗is the optimal strain interpolation coefficient with the vector of Fourier-type basis functions φ⊤(ϱ, θ): a∗(t):=arg min a 1 2X i,j˜εi j (t)−φ⊤(˜ Ri,˜ θj)a2 , φ(r, θ) := [1,r]⊤⊗ [1,cos(θ), sin(θ), . . . , cos(Nhθ), sin(Nhθ)]⊤.(9) Here, ⊗stands for the tensor product, and Nhis the selected number of harmonics. This interpolation process enables us to calculate the displacements 1i(t)of supporting beams and variations of electrostatic gaps at the silicon layer. As illustrated in Fig. 5, these calculated displacements comprise two parts: 1) The displacement imposed by the substrate via anchors. We obtain this part based on the substrate’s interpolated displacement field (8) and glass CTE. 2) The displacement due to the silicon’s thermal expansion/contraction, calculated based on the silicon CTE and the distance from anchors to the points of interest (bases of supporting beams and edges of electrodes). Electronic noise is the dominant factor in the performance of the stress sensors given that they function at 10 kHz, which is considerably below their ∼300 kHz mechanical resonance. We found the resolution of these sensors to be 5.3 nano-Strain, which proved to be sufficient in capturing the thermal stresses during the experiments. IV. TEMPERATURE EFFECTS ON GYROSCOPE’S PERFORMANCE The stiffness matrix in Equation (3) is the key for quantifying temperature effects on the gyroscope’s frequencies and output errors. The matrix Kcan be conceived as the perturbation of the initial stiffness K0at the reference temperature T0: K0=Km,0−Ke,0−Kβ,07→ K=K0+δK, δK=E(T) E(T0)−1Km,0+KT+KS−(Ke−Ke,0) −(Kβ−Kβ,0)(10) where Km,0,Ke,0, and Kβ,0are the initial nominal, electrostatic, and Duffing stiffness matrices, respectively, and δKis the perturbing stiffness. Based on (10), a perturbation analysis can be carried out for the variations of eigenvalues (frequencies) and eigenvectors (mode shapes’ orientation). We refer to [5],[26] for the mathematical details of such an analysis and only cite the pertinent results here. Assume that initially (that is, corresponding to K0) the gyroscope has the mode shapes cos(2θ−ψ1,0)and sin(2θ−ψ1,0)with the frequencies ω1and ω2and modal masses m∗ 1and m∗ 2, respectively. The rotation angles ψ1,0, ψ2,0are due to fabrication imperfections and can be determined from the initial quadrature error (see Box 2 for an illustration). Under the effects of temperature, these modal characteristics will be perturbed such that ωi,07→ ωi,0+δωi and ψi,07→ ψi,0+δψi: δωi≈δk∗ ii 2m∗ iωi,0 , δψi≈δk∗ 12 m∗ i(ω2 1,0−ω2 2,0),i=1,2,(11) where δk∗ i j are the components of the stiffness perturbation δKas observed from the (initial) rotated mode shapes: δK∗=T⊤δK T,T=cos(ψ1,0)−sin(ψ2,0) sin(ψ1,0)cos(ψ2,0).(12) Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on April 16,2025 at 20:57:19 UTC from IEEE Xplore. Restrictions apply. 156 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 34, NO. 2, APRIL 2025 The variations of frequencies, mode shape orientation, and quality factor changes over temperature give rise to errors in the gyroscope’s quadrature and in-phase channels. We have provided the detailed expressions in Box 2 for a mode-mismatched gyroscope. We have assumed that the drive mode is excited into resonance by the AC force F(t)=Fdcos(ω1t)(using PLL and AGC loops), and the frequency split is much larger than the bandwidth. According to the equations in Box 2, the following remarks are in order: •Temperature-dependent variations of mechanical scale factor (SF, see Box 2 for the definition), drive force/displacement (Fd,Ad), and sense mode’s phase (β2) cause variations of the overall scale factor. •The misalignment between the nominal drive-sense axes and the actual mode shapes (the angles ψ1,2) leads to a projection of the mismatch between the modes onto the gyroscope’s output (represented by the difference between the Aand Bterms). This misalignment results in Type I and II errors, and, according to (11), depends on stiffness coupling. •Type I error is the main component of the quadrature, which can be considered as the equivalent rate of the stiffness coupling (since k12 ∝ |1ω|sin(2ψ1)). The demodulation angle ϕdem is used to compensate for the nonideal phase angle of the sense mode. Type I quadrature error can leak into the in-phase due to this phase error (compare the cos(ϕdem)and sin(ϕdem)components). Since quadrature is often orders of magnitude larger, this type of leakage could produce significant in-phase errors. Type II mainly affects the in-phase because the bandwidth is smaller than the frequency split. We note that scale factor variation is present in Type I and II through drive force/displacement amplitude changes. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on April 16,2025 at 20:57:19 UTC from IEEE Xplore. Restrictions apply. HOSSEINI-PISHROBAT AND TATAR: MODELING TEMPERATURE EFFECTS IN A MEMS RING GYROSCOPE 157 Fig. 6. Flowchart of modeling temperature effects. Fig. 7. Temperature test setup; the temperature sensor (PTAT) and the gyroscope front-ends are beneath the daughter board; the aluminum lid protects the wire bonds. •Type III error arises from a direct in-phase term, d, which we assume originates from the damping coupling between the two modes. In that respect, not only scale factor variations, but also temperature fluctuations of d impact this type of error. In effect, Type III represents temperature-prone components of the in-phase that would persist even after eliminating Type I and II with a quadrature control loop. Fig. 8. Temperature profiles and drive mode’s bandwidth variations (initial value: 1.8 Hz) during Tests#1 and 2. •Type IV reflects the effect of scale factor variations on extracting the rate. The flowchart in Fig. 6illustrates the algorithmic procedure of the model from experimental input data to the gyroscope’s performance outputs. V. RESULTS AND DISCUSSION A. Gyroscope Specifications We will report two datasets–labeled as Tests#1 and 2–from two different devices. For the device in Test#1, we measured the frequencies and quality factors of the drive (cos(2θ)) Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on April 16,2025 at 20:57:19 UTC from IEEE Xplore. Restrictions apply. 158 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 34, NO. 2, APRIL 2025 Fig. 9. Stress sensors’ measured data versus temperature for Test#1 (i: inner, o: outer, and numbers indicate the angular position). Fig. 10. Drive mode’s frequency variations over time and temperature; experimental versus analytical predictions. and sense (sin(2θ)) modes as 57.637 kHz, 31.93 ×103and 57.723 kHz, 30.13 ×103, respectively. The initial uncompensated quadrature error is -3225◦/s, which amounts to a mode shape rotation angle of −3.3◦. Details of extracting the mode shape rotation angle from the frequency response can be found in [22]. After correcting for electrostatic softening and the Duffing effect, we obtained the initial mechanical frequency of the drive mode as 58.045 kHz. Similar specifications with a slight variance hold for the device in Test#2. B. Experimental Setup Fig. 7shows the setup used for the temperature experiments. The daughter board contains the MEMS die with an on-PCB heater underneath it. The gyroscope’s front-ends Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on April 16,2025 at 20:57:19 UTC from IEEE Xplore. Restrictions apply.