A μGal MOEMS gravimeter designed with free-form anti-springs

Domain: earth_science
Published: 2025-02-20
Editorial outcome: accept

Abstract
Gravimeter measures gravitational acceleration, which is valuable for geophysical applications such as hazard forecasting and prospecting. Gravimeters have historically been large and expensive instruments. Micro-Electro-Mechanical-System gravimeters feature small size and low cost through scaling and integration, which may allow large-scale deployment. However, current Micro-Electro-Mechanical-System gravimeters face challenges in achieving ultra-high sensitivity under fabrication tolerance and limited size. Here, we demonstrate a μGal-level Micro-Opto-Electro-Mechanical-System gravimeter by combining a freeform anti-spring design and an optical readout. A multi-stage algorithmic design approach is proposed to achieve high acceleration sensitivity without making high-aspect ratio springs. An optical grating-based readout is integrated, offering pm-level displacement sensitivity. Measurements reveal that the chip-scale sensing unit achieves a resonant frequency of 1.71 Hz and acceleration-displacement sensitivity of over 95 μm/Gal with an etching aspect ratio of smaller than 400:30. The benchmark with a commercial gravimeter PET demonstrates a self-noise of 1.1 μGal Hz−1/2 at 0.5 Hz, sub-1 μGal Hz−1/2 at 0.45 Hz, and a drift rate down to 153 μGal/day. The high performance and small size of the Micro-Opto-Electro-Mechanical-System gravimeter suggest potential applications in industrial, defense, and geophysics.

Introduction
Constantly emerging applications, such as resource exploration, hazard forecasting, and geophysical research1–5 posed demanding needs for precise measurement of local gravitational acceleration. Compared with absolute gravimeters that measure the acceleration of a freely falling mass6,7, relative gravimeters measure the gravity variation via the displacement of a mass suspended by springs, thus incorporating relatively smaller size, lighter weight, and poorer stability8,9. However, conventional relative gravimeters, such as Scintrex CG6 and Microg LaCoste PET, are still large and expensive instruments. Micro-Electro-Mechanical-System (MEMS) gravimeter10–16 is an emerging field with potential applications in terms of their further miniaturization, mass production, and cost reduction. MEMS gravimeters with various strategies, such as lowering the spring constant10,11,13, enlarging the mass17, and applying highly sensitive readout18,19 have been published to achieve precise gravity variation measurement. However, MEMS gravimeters still face challenges compared with large-size competitors with μGal Hz-1/2-level sensitivity and corrected long-term stability of around 20 μGal/day. The first challenge is to achieve high acceleration sensitivity with an easily fabricated chip-scale structure. Achieving high sensitivity (or low resonant frequency) in a small size usually requires slender springs with a high aspect ratio and always brings the maximum stress dangerously close to the tolerable limit. Finding the optimal spring design requires a large parameter space and an efficient design approach. The second challenge is to integrate a readout with better displacement sensitivity without compromising the miniaturization and process compatibility. Although optical shadow sensors10,11, optical grating sensors20–22, and capacitive sensors23,24 with nm or sub-nm sensitivity have been subsequently introduced to MEMS gravimeters, the acceleration sensitivity could be pushed further via picometer- or femtometer-level displacement methodology25–30.

As an effective way to improve sensitivity, the anti-spring mechanism31,32 has been recently introduced into MEMS accelerometers33,34 and gravimeters10,11. For example, Middlemiss et al. developed a MEMS gravimeter with a resonant frequency of 2.3 Hz and sensitivity of 40 μGal Hz−1/2 based on an asymmetric arc-shaped anti-spring structure10,35. Tang et al. combined a pair of cosine-shaped negative stiffness springs with a pair of folded positive stiffness springs11, achieving a resonant frequency of 2.9 Hz, a sensitivity of 8 μGal Hz−1/2, and a dynamic range of 8 Gal. Boom et al. designed a MEMS accelerometer with a resonant frequency of 28.1 Hz based on arc-shaped anti-springs33. However, the limited degrees of freedom of these anti-spring mechanisms result in a small design parameter space, making it difficult to find a solution that balances ultra-high sensitivity, fabrication tolerance, and stress robustness. More specifically, the lowest reported resonant frequencies are always around 2~3 Hz, and the required etching aspect ratio is larger than 16, even exceeding 4010,36,37. The introduction of freeform geometries38–40 opens a path to address this issue owing to its larger design parameter space. Some researchers have explored the design of MEMS freeform structures. For example, Wang et al. used polynomial and Bézier curves to achieve global optimization of beams41,42, while the lack of local adjustability made these curves challenging to perform fine-tuning. Anti-springs undergo dramatic changes in stiffness during the loading process34. The optimal design of such freeform anti-springs (F-ASs) requires multiple iterations to tune the mechanical response. Therefore, although F-AS can provide a larger design parameter space, the vast parameter space makes it tough to find an optimal design in a single-stage optimization with conventional metaheuristic algorithms43–45. The local adjustability of freeform geometry and feasible algorithm are both necessary to decompose the expensive optimization problem46 of F-ASs.

In this paper, we demonstrate a Micro-Opto-Electro-Mechanical-System (MOEMS) gravimeter based on optimized locally adjustable F-ASs and a highly integrated optical grating-based readout. A multi-stage algorithmic approach is developed to spring design, providing a larger parameter space and feasible optimization. The first-stage optimization is performed to obtain a global design of the anti-spring geometry, and the second-stage optimization performs local fine-tuning of the geometry to meet multiple design requirements. A chip-scale spring-mass system is optimized to achieve high acceleration-displacement sensitivity over 95 μm/Gal (resonant frequency lower than 2 Hz) without making high-aspect ratio springs. The optical grating-based readout allows displacement measurement with 1.2 pm Hz−1/2 sensitivity and shows high integration and compatibility with the process. Designated manufacturing and packaging are implemented to minimize the impact of internal stress and external temperature and pressure, thereby enhancing long-term stability. The prototype of the MOEMS gravimeter is tested to have a noise level of 1.1 μGal Hz−1/2 at 0.5 Hz and a drift rate of 153 μGal/day. Co-site Earth tide and seismic measurements using the prototype and a commercial gravimeter are conducted to confirm the sensitivity and long-term stability. Such a small-size and high-performance MOEMS gravimeter could provide portable and precise measurements and platforms for geophysics and potential applications in prospecting and hazard forecasting. In addition, the algorithm-enabled multi-stage design approach of locally adjustable F-ASs opens a pathway toward the optimal design of complex structures with a large degree of freedom.

Results
Sensing unit of the MOEMS gravimeter with F-ASs and optical readout
Figure 1a illustrates the packaged sensing unit of the MOEMS gravimeter, including the grating, mechanical sensing unit, supporting layer, adhesive layer, and silicon substrate with patterned Pt resistors. The mechanical sensing unit converts the gravitational acceleration into the displacement of the proof mass suspended by springs. The displacement of the proof mass is measured by the optical grating-based readout, which comprises a pair of gold grid lines patterned on the fixed glass substrate and the proof mass. The supporting layer and adhesive layer are used to support and bond the device. The integrated Pt resistors are used to measure and control temperature.

The mechanical sensing unit is shown in Fig. 1b. It includes (I) the proof mass with metal grid lines at the surface center, (II) connecting rods, (III) F-ASs, and (IV) compression mechanisms. The sensitive direction is the y-axis, and the proof mass is connected to four F-ASs through two connecting rods. The four F-ASs have the same geometry and are symmetrically distributed on both sides of the proof mass to reduce cross-axis sensitivity in the x-direction. The other end of the F-AS is connected to the compression mechanism (Supplementary Fig. 17). This mechanism is used to actively compress the anti-spring with a certain distance to lower the stiffness along the y-axis. The working principle and semi-analytical calculation of the stiffness of the anti-springs, as well as its force-displacement relationship, can be found in Supplementary Note 1.

The F-ASs are constructed using B-spline curves due to their convex hull property and the local adjustment ability47, as shown in Fig. 1c. Control points P1, P2, …, Pn-1, Pn form a freeform curve, and the freeform curve is offset on both sides to form a closed area with connecting line segments. It is noted that the F-AS can be constructed by two separate B-spline curves, while the parameter space may be too large in actual trial designs for numerical computing-based optimization.

Multi-stage design of highly sensitive anti-springs with fabrication tolerance
This paper proposes a multi-stage algorithmic approach to design F-ASs to achieve high performance with constraints of stress and fabrication tolerance. The design flow is depicted in Fig. 2a, containing three main steps: (1) Initialization: Construct F-ASs and analytically predict the mechanical performance. (2) Multi-stage directional optimization: Conduct global optimization on the F-ASs, followed by local fine-tuning of the first-stage optimization results. (3) Evaluation: Evaluate the optimization results; perform local fine-tuning or global optimization if they do not meet the requirements; conduct a robustness analysis of the design if they meet the requirements.

Seven control points (P1~P7) were chosen to define the F-AS, considering the balance between the degree of freedom and computational expense. Table 1 lists ten variable structural parameters of the sensing unit and their upper and lower limits. The sensing unit has a total of 23 important structural parameters, which are listed in Supplementary Table 1. The thickness of the proof mass and springs was set to be the same as the thickness of a readily accessible wafer, taking advantage of the whole thickness of a wafer to construct a big mass. The freeform curve was a third-order quadratic B-spline curve. The control points P1 and P2 were overlapped and fixed to ensure a smooth connection between the anti-springs and the connecting rods. The x-coordinates of control points P6 and P7 were fixed to ensure that the length of the anti-springs in the x-axis direction remains unchanged. The coordinate ranges of the remaining control points were determined by the footprint of the unit. The lower limit of the spring width was determined by the etching aspect ratio.

Regarding the objective function of the optimization, we focused on the static response of the spring-mass system under nominal operating conditions because of the low-frequency characteristic of the gravitational acceleration10–12. Ultra-high acceleration-displacement sensitivity is the primary property of the sensing unit and serves as the first factor of the objective function. The linear range of acceleration-displacement response and the R-squared value within the range of 10 Gal are considered the second and third factors because the anti-spring structure usually exhibits highly nonlinear behavior34. The range of 10 Gal was selected considering the local gravitational acceleration variation at different locations48,49 and anti-spring-based competitors10,11. The optimization objective function for the F-AS is as follows:

FOM = f(S, G, R2) = 1 / (k1 × S + k2 × G + k3 × R2) (1)

where S represents the acceleration-displacement sensitivity, G represents the linear range, R2 is the R-squared value of the linear fit of the acceleration-displacement curve within the range of 10 Gal, and k1 to k3 are corresponding weight factors. Our purpose is to minimize the figure of merit (FOM). In the initialization step, we employed a semi-analytical method to predict the stiffness of the F-AS (Supplementary Note 1). In the multi-stage optimization, we used the Finite Element Method (FEM) to precisely evaluate the nonlinear performance of the F-AS (Methods).

For expensive optimization problems, it is beneficial to distribute the initial value of parameters covering the entire parameter space in the early stage of the optimization, and it is preferred to improve the convergence speed in the later stage. In this paper, both the first and second stages of optimization were conducted using the Adaptive Elite Learning Particle Swarm Optimization (AELPSO) algorithm, a self-built algorithm based on complementary sub-strategies for multimodal expensive problems (Methods).

The primary goal of the first-stage optimization is to pursue the three aforementioned objectives. The maximum stress value and spring width serve as soft constraints. The iteration number of the first-stage optimization was set to 200. Figure 2b illustrates the convergence graph, and the FOM can be observed to continue to decrease in the optimization. The global optimal results for the variable parameters are presented in Table 1. The geometry of the F-AS obtained by global optimization is shown in Fig. 2c, along with its compression and loading state. The simulation result and the fitting curve of the displacement to the applied acceleration response are depicted in black and red in the inset. The resonant frequency of the anti-spring was as low as 1.03 Hz, the acceleration-displacement sensitivity within the range of 10 Gal was 79.3501 μm/Gal, and the linearity was 0.9179. The sensitivity is lower than the frequency-calculated value by the relationship a = ω² × d because the sensitivity is the slope of the fitted straight line of the curve within a relatively large range, more likely an average slope. The sensing unit exhibited exceptionally high sensitivity and a moderate linearity that could be further improved. In addition, the maximum stress in the structure was 313.7 MPa, slightly exceeding the safe stress limit, which was set as 300 Mpa50,51.

The second-stage design was performed to adjust the anti-springs locally to address the issue of relatively low linearity and high local stress. The maximum stress and spring width are strictly constrained to be within 300 MPa and larger than 30 μm, respectively. As shown in Fig. 2c inset, the region with the maximum stress is located at the compression end of the F-AS. Adjusting control points P6 and P7, as illustrated in Fig. 2d, allows for the adjustment of local shape near the compression end without affecting the overall shape of the anti-spring based on the characteristics of the third-order quadratic B-spline curve. Table 1 list the upper and lower limits of the local fine-tuning parameters. The iteration number of the second-stage optimization was set to 50, considering the smaller number of parameters. The local fine-tuning results are presented in Table 1. The geometry of the optimized F-AS and its compression state are shown in Fig. 2e, where the insets depict the simulation result of the acceleration-displacement response. The resonant frequency of the anti-spring was 1.46 Hz, the mechanical sensitivity of the spring-mass system was 72.3056 μm/Gal, and the linearity was 0.9826. The maximum stress was reduced to 299.7 MPa, within the safe stress limit. It is noted that the linearity and maximum stress were modified without sacrificing much sensitivity and resonant frequency. The MEMS sensing unit still has one of the lowest reported resonant frequencies and the highest acceleration-displacement sensitivity. The modal analysis results of the optimized structure indicate that the cross-axis sensitivity is small (Supplementary Note 2).

Finally, a robustness analysis was conducted to confirm the feasibility. Supplementary Table 2 lists the parameters and parameter ranges of the sensing unit for the robustness analysis. We considered errors in both manufacturing processes and material properties. Errors arising from manufacturing processes include fillet radius, variations in the width of the anti-spring and the proof mass, and the compression distance. Errors arising from material properties include variations in Young’s modulus, density, and Poisson’s ratio. Single-parameter and multi-parameter robustness analysis results indicate that even in the worst case, the design can achieve a resonant frequency of 1.89 Hz, a sensitivity of 56.3262 μm/Gal, and linearity of 0.9762, respectively (single- and multi-parameter robustness analysis results see Supplementary Note 3). It demonstrates that the F-AS obtained by multi-stage design had excellent performance and fabrication tolerance. Note that the stress limit is safe enough to further reduce the width of the spring, and it is easy to obtain an F-AS-based structure with a resonant frequency below 1 Hz by setting a more tolerant constraint.

Experiment results
Displacement sensitivity measurement of the optical grating-based readout. The sensing unit of the MOEMS gravimeter was composed of the designated F-AS-based mechanical structure and an integrated optical grating cavity, as shown in Fig. 3a. The optical grating cavity allows pm-level displacement sensitivity and shows high integration. It comprises a pair of gold grid lines patterned on a fixed glass substrate and the proof mass, termed G1 and G2, respectively. Two gratings had the same period of 3 μm. A laser beam emitted from a source illuminated the optical cavity, generating a series of interferometric diffraction beams. For a certain cavity length, the light intensity of the first-order diffraction beam is related to the in-plane displacement of the proof mass along the y-axis but is insensitive to the x-axis displacement. The details can be found in Supplementary Note 4. The inset of Fig. 3a shows the simulation results of the light intensity as a function of the displacement. The intensity was detected by a photodetector, enabling the measurement of the displacement of the proof mass.

The displacement sensitivity of the readout was experimentally evaluated through the setup shown in Fig. 3b. A dissociated proof mass with grating G2 was bonded to a quartz plate, which was then bonded and fixed onto a piezoelectric ceramic (Model PA25LEW, Thorlabs Inc.). Grating G1 was mounted on a five-dimensional adjustment stage to facilitate the alignment of the cavity. Both gratings were mounted perpendicular to the ground, with their grid lines parallel to the x-axis direction. The expansion and contraction direction of the piezoelectric ceramic is the y-direction. First, the grating G2 was driven by a piezoelectric ceramic to test sensitivity and noise level. Figure 3c plots the experimental results of output voltage versus the input displacement, where each point corresponds to the average value of the static measurement result in 5 min. The response resembles the simulation results and has a sensitivity of 1.22 V/μm, with the standard deviations from 1 to 12 mV. The standard deviation error primarily arises from environmental vibrations and disturbances, which are transmitted through structural components to the cavity. At certain positions, specifically those with a larger slope, the response becomes noisier because a larger slope indicates higher sensitivity to the disturbances. The standard deviations correspond to acceleration errors from 8.2 to 98.3 μGal, combining the optical displacement sensitivity and the acceleration-displacement sensitivity of around 100 μm/Gal. The experimental result is not a perfectly linear response but rather resembles the typical response of such an optical cavity. Figure 3d shows the spectral density of the displacement measurement. It indicates that the noise level of the optical cavity is 1.2 pm Hz−1/2 at 1 Hz and below 5 pm Hz−1/2 within the frequency range of 0.5–1 Hz.

Acceleration sensitivity and resonant frequency measurement of the mechanical sensing unit. The fabricated sensing unit is shown in Fig. 4a. The shape of the proof mass was adjusted to avoid collisions with anti-springs. The adjustment did not change the center of the proof mass, nor did it affect the device’s modes and mechanical properties (Supplementary Note 5). In addition, we designed a sensing unit with single-convex freeform anti-springs (SF-ASs) using the same design approach to make a comparison with a smaller parameter space. The non-optimized structural parameters, the number of control points, the objective function, and design constraints remained unchanged for a fair comparison. The resonant frequency, sensitivity, and linearity of the sensing unit with SF-ASs were 2.08 Hz, 43.2631 μm/Gal, and 0.9822, respectively (optimization results see Supplementary Note 6). The fabricated structure is shown in Fig. 4b. This comparison demonstrates that the complete F-AS performs better than the SF-AS with a general layout.

Figure 4c illustrates the turntable experiment setup for the mechanical sensing unit. The mechanical sensing units were packaged using wafer-level manufacturing and bonding techniques (Supplementary Note 7). The manufacturing and packaging process is presented in Methods. The gravimeter prototype was fixed on the high-precision turntable (Model DP300, Dantsin-RPI Inc.). The sensitive axis of the gravimeter was parallel to the turntable, and the turntable surface was perpendicular to the ground. By rotating the turntable, the angle θ between the sensitive axis of the gravimeter and the gravitational acceleration g, pointing downward, could be finely changed. This variation in θ led to a change in the acceleration variation g cos θ along the sensitive axis. More specifically, a change of the angle of 0.0004° corresponds to the variation of the gravitational acceleration component applied to the sensitive axis of g*(sin(0.0004°) – 0) ≈ 7 mGal at the angle of 0° and g*(1 – cos(0 – 0.0004°)) ≈ 24 nGal at 90°. Herein, the turntable was first fast rotated around the z-axis to scan and then finely adjusted to locate the point with high sensitivity. As shown in Fig. 4d, the resonant frequency of the mechanical sensing units with complete F-ASs and SF-ASs were measured as 1.71 Hz and 2.45 Hz at the angle of 90°, close to the simulated results of 1.61 Hz and 2.77 Hz, considering the fabrication errors. The tiny deviation between the experiment and simulation is mainly due to the unobservable fabrication errors, as presented in Supplementary Fig. 18a. The resonant frequency could be lowered to 1 Hz or even below in consideration of the over etch, and it was realized in practice (see Supplementary Fig. 18b).

Subsequently, a fine scan was conducted within the dynamic range with a step size of 1.44 arcseconds. The system was paused at each angle for one minute to acquire the static response. Figure 4e presents the static response of the displacement versus the gravitational variation, where the red and blue colors represent the responses of sensing units based on the complete F-ASs and SF-ASs, respectively. The scattered points represent experimental data, and the solid lines represent the linear fitting. The time-domain data for some input acceleration points in one minute are shown in the insets. Within the last 30 s, the time-domain output curves at each acceleration point are highly stable after 30-s stabilization, with standard deviations from 4 nm to 14 nm, indicating virtually no drift. The fitted acceleration-displacement sensitivity of these two units is measured to be 95.3674 μm/Gal and 40.5394 μm/Gal, respectively, with linearity of 0.9985 and 0.9919. The linearity of 0.9985 demonstrates that the linearity is significantly enhanced via local fine-tuning. The experimental results indicate that the optimized F-AS-based structures exhibit low resonant frequency, high acceleration-displacement sensitivity of more than 95 μm/Gal, and reasonable linearity, which further demonstrate the superiority of multi-stage design.

Noise level measurement and comparison of gravimeters. The noise level of the MOEMS gravimeter was evaluated in a cave laboratory, where the MOEMS gravimeter and a reference commercial gravimeter, Microg LaCoste PET, were placed side by side, as shown in Fig. 5a. The reference gravitational acceleration obtained from the cave laboratory is 9.7944 m/s2. The temperature variation in the cave was within 100 mK per day, and the temperature variation was further decreased to within 5 mK by temperature control (Supplementary Note 8). The two instruments were positioned within 10 meters, detecting nearly identical gravitational signals. The output of the gravimeters had been recorded without vibration isolation or vacuum maintenance. Several teleseismic signals were recorded according to the benchmark with the reference gravimeter. Figure 5b shows the seismic signal generated by a magnitude 4.4 earthquake in Guangdong, China, on 4th November 2023. The P-waves and S-waves can be clearly distinguished. The acceleration power spectral density of the MOEMS gravimeter is shown in Fig. 5c. Data were recorded by a fast sampling rate of 10 Hz and a slow sampling rate of 0.1 Hz in order to observe the responses of Earth tremor peaks and tides. The noise floor of the prototype reaches down to 1.1 μGal Hz−1/2 at 0.5 Hz, and the ultimate noise floor can be smaller than 0.3 μGal Hz−1/2 at around 0.45 Hz; the diurnal and semi-diurnal components of the Earth tides are visible at 10−5 Hz; the Earth tremor peak can be observed at around 0.2 Hz and around 2-3 Hz. The noise bump around 0.1 Hz could be due to insufficiently precise temperature control. Finer temperature control and temperature compensation are expected to suppress the noise level in this frequency range. Supplementary Fig. 19 depicts the acceleration noise floor of the optical readout and the mechanism in the power spectral density plot. It demonstrates that, apart from the bump caused by thermal control, the optical readout noise is close to the self-noise floor, which can be considered the main limiting factor. Two pale red and deep red lines in Fig. 5d depict the data recorded by the MOEMS gravimeter from 11th to 17th June 2024 over five days, corresponding to the raw times series and data removing the linear drift. The prototype achieved reasonably good long-term stability, with a drift of 153 μGal/day. The drift is comparable to other MEMS gravimeters, which is around 100–300 μGal/day. According to the Allan deviation results shown in Fig. 5e, a bias instability of 6 μGal can be obtained with an integration time of 600 s. Comparing the data obtained from the reference gravimeter (in blue) shows a strong correlation coefficient of 0.9218, as shown in Fig. 5f, indicating that the MOEMS gravimeter has successfully observed Earth tides.

Table 2 compares the proposed MOEMS gravimeter and the reported MEMS and commercial gravimeters. Thanks to the multi-stage design of locally adjustable F-AS, the mechanical sensing unit achieves a resonant frequency of 1.71 Hz under nominal operating conditions, acceleration-displacement sensitivity of over 95 μm/Gal, and a dynamic range of over 10 Gal, which outperforms its previously reported counterparts. In addition, the highly integrated optical grating cavity enables the pm-level displacement measurement, further allowing the gravitational acceleration measurement with a noise level of 1.1 μGal Hz−1/2 at 0.5 Hz. The compact MOEMS gravimeter with ultra-high sensitivity, reasonable robustness, and high stability shows promise for portable and precise relative gravity measurement worldwide.

Methods
Finite element simulation
Besides using semi-analytical analysis to roughly predict the stiffness of the F-ASs, we used FEM to precisely simulate the nonlinear behavior of the F-ASs. The mechanical model was simplified as a 2D finite element model to reduce the computing costs because of the identical thickness of the die. More specifically, 3D simulation of the F-AS response requires a time of around two hours, and 2D simulation only requires less than one minute. The validity of the 2D FEM results was confirmed by the thorough comparison with the 3D FEM results (Supplementary Fig. 20). The 2D model was further simplified to half of the unit due to the symmetry. Thousands of elements and a mapped method were used to mesh the 2D model. The modulus tensor of single-crystal silicon was set according to reference 52. In addition, the density of single-crystal silicon was set to 2329 kg/m3. The boundary condition was that the displacement of the proof mass in the x-direction was zero due to structural simplification, a specified displacement was applied at the compression end of the F-ASs, and an inertial force was applied to the proof mass. The equilibrium equations for the F-ASs were established in the deformed state and updated with deformation. The solution was path-dependent, so the gravitational acceleration was increased gradually to achieve a slow growth in the load, ensuring good convergence.

Optimization algorithm
A self-built Particle Swarm Optimization algorithm53,54, AELPSO, was proposed to solve the expensive optimization problem. AELPSO employs four sub-swarms with complementary ‘CODA’ strategies that are adaptively fused via the adaptive elite learning (AEL) method to balance exploration and exploitation. The letters ‘C’, ‘O’, ‘D’, and ‘A’ stand for four learning exemplars generation strategies of Cross-, Ortho-, Dynamic-, All-, in which “C” and “A” use binomial crossing method and weighted average method to generate learning examples respectively, which excel in exploration, while “O” and “D” use updated orthogonal and dynamic (adaptive learning) strategies, which excel in exploitation. In the AEL method, better particles are selected to execute more strategies to escape local optima quickly; the number of particles executing all strategies is adaptively increased to prevent premature convergence from the exploratory to the exploitative state; finally, each time when the adaptive elite learning method is performed, a certain number of particles with the same subscript are selected from the population for elimination to avoid the risk of premature convergence and elimination of the best particles. Promising particles in AELPSO explore the search space in multiple ways and are more likely to find better solutions in a limited iteration number (typically 200) when faced with such an expensive optimization problem. A thorough benchmark with six state-of-the-art optimization algorithms was conducted, demonstrating the superiority of AELPSO in optimization performance and convergence speed (Supplementary Note 9).

Manufacturing and packaging process
The manufacturing and packaging process is illustrated in Fig. 6. As shown in Fig. 6c, wafer 1 of grating G1 was made of a 400-μm-thick glass wafer with thermal expansion coefficients comparable to single-crystal silicon wafers, and the grating fabrication process on its surface was the same as the process used for the surface of the proof mass. The wafer 2 of mechanical sensing unit was based on a 400-μm-thick single-crystal silicon wafer with a crystal orientation of <100>, as illustrated in Fig. 6d. First, a 100-nm-thick layer of gold was deposited on the top surface of the silicon using magnetron sputtering. The gold layer was then patterned by a standard photolithography and peeling process, obtaining 3-μm-period grid lines. Next, a 10-μm-thick layer of AZ4620 photoresist was spin-coated on the top surface of the sample wafer and placed on a hot plate at 90°C for 90 s. The mask of the mechanical sensing unit’s was aligned and exposed using cross marks, followed by a positive development in TMAH for approximately 3 min, forming the patterned photoresist (Supplementary Fig. 21). Due to the large area to be etched, halo masking technology was used to design the mask (Supplementary Fig. 22). The mask was designed with a 200-μm-width halo, which was used to etch away the contour of the structure rather than all unwanted areas55. Paraffin was used as the mounting adhesive to affix the sample to a carrier wafer (with the photoresist side upward). The sample was then loaded into an SPTS dry-etching machine to run a deep reactive ion etching (DRIE) process for approximately 120 min until the wafer was etched through. The etched sample was soaked in acetone to remove the photoresist, and a plasma stripper was used to remove the residual photoresist. The sample was inverted in a dish, and alcohol and deionized water were used to detach the sample from the carrier silicon wafer. As shown in Fig. 6e, wafer 3 of the supporting layer was manufactured by deep wet etching of the thinned glass wafer with a relatively large thickness. Figure 6f shows that wafer 4 was made of a 400-μm-thick single-crystal silicon. Platinum resistors were first patterned on the backside of the silicon wafer. Then, a confinement trench was etched by the DRIE process. These four wafers were bonded by a low-stress bonding technique (Supplementary Note 7) sequentially to finish the packaging, as shown in Fig. 6g–i. This packaging offers reasonable shock resistance (Supplementary Note 10).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The work is interesting with its main contribution being in the form of the optimization algorithm (AELPSO) that helps in\ndesigning nonlinear structures - in this case the geometric anti-springs. However, I feel that the work lacked significant\ntechnical details, failed to provide a larger context to the application of this algorithm and at some points was not thorough in\nits analysis of results. Here, I list point by point some of my concerns and where some detailed analysis/information is\nmissing in its current form.\n1. The authors successfully show that they can improve the key metrics of the device through this algorithm. However, even\nafter their improvement over other designs that use geometric anti-springs, there are geometric anti-spring devices (Boom et\nal.) that have shown lower noise floors (1uGal/rt Hz) than this work. The authors brush of this work by saying their resonant\nfrequency is higher and their springs are simple design. However, the key metric in this case for a gravimeter is the\nnoise floor and Allan deviation rather than the resonant frequency or the complexity of the springs. In comparison to that work that\ndoes not use this optimization algorithm, it seems like simple design can also be beneficial for the end user. This reduces\nthe impact that the author’s work has on the community. There are even other MEMS devices that have shown better noise\nfloors eg Pike et al. than the current device. Can this algorithm be used to optimize these capacitive accelerometer structures\nas well? If the authors can show a larger application to this algorithm, then the paper will be much stronger. The complexity\nof the beams is not the crucial point rather the noise floor and the allan deviation are.\n2. The authors switch units a lot in the introduction between ng/ rtHz and ugal/ rt Hz. Please just use on set of units to avoid\nconfusing the reader.\n3. Line 177. The authors should list a priority on the parameters used in the objective function. They mention that the\nacceleration-displacement sensitivity is their most critical property and yet the first variable in equation 2 is the dynamic\nrange. It will be clearer of they listed the parameters that go into the objective function in a table and show the priority and\nwhether or not they are dependent on each other. For eg: high linearity could mean lower acceleration-displacement\nsensitivity.\n4. Line 259: Can the authors provide more explanation as to why the 5 minutes were needed to average the result for static\nmeasurement result? Is there drift? Other measurement artifacts?\n5. Line 272. The authors mention that the proof mass was adjusted to avoid collissions with the anti-springs. By doing so the\ncentre of mass has changed in the SF-AS design making it asymmetrical. Will that not cause tilting effects? Can the authors\nprovide more explanation on this in the text?\n6. Line 281. The authors mention a high-precision turntable was used. Can they provide make and model? Most turntables\nare very inaccurate and cannot provide enough accuracy on the tilt to calibrate/measure sensitivity using the tilt method.\n7. Line 295. Can the authors explain why the settling time of minutes is needed for measurement of each angle? Is it drift, or\nother measurement artifact or use of other equipments?\n8. Fig 4d. How do the authors account for the change of bandwidth across the dynamic range? The frequency changes from\n1.3Hz-2.6Hz between 0-5Gal. This means that the frequency components/noise will be much higher when measuring at\nhigher static acceleration.\n9. Fig 4e. One of the reason the FEA and the experimental results are not matched could be that in the experiment, the non\ntilting axis is not completely orthogonal to the gravity. This would result in some cross-coupling of gravity in the non-sensitive\naxis into the sensitive axis. This cross-coupling study must be done to better understand the performance of the\naccelerometer.\n10. Fig 5c Why are there two resonances in the plot? One at 1.3Hz and another at about 2/3Hz? Where is this coming from?\nIt needs to be discussed.\n11. Line 376. The authors mention that the computing cost is reduced significantly using this algorithm. However, there is no\ndirect comparison shown between the different computing costs. Please show the quantitative improvement of the\ncomputing costs if you are claiming such an improvement and how you would calculate such an improvement.\n12. Line 384. The authors mention that the acceleration needs to be gradually increased because of the path-dependency of\nthe solution. How does the accelerometer react to shocks? What this ever tried in an experiment?\n13. Fig6. Can the authors mention how they packaged the device since that has a significant part to play in high resolution\ngravimeters?\nIn my opinion, the authors have a good concept here but it is not fleshed out enough technically and neither is its impact full\nrealized in this work. I would like to suggest the authors take these comments and improve the paper before resubmitting it.", "reviewer_2": "The paper reports a new MOEMS gravimeter with a novel free-form anti-springs design. Authors claim that their approach\nallows their device to achieve both high-sensitivity as well as a larger dynamic range. They back-up their claims through: a)\nFEA simulations which show high-linearity over +/- 5 Gal acceleration pertubrations, b) experimental results which show a\nself-noise floor of 1.7 ng/rtHz for the MOEMS device.\nWhile the work of the authors is quite interesting and cool, and it is good to see new approaches to designing MEMS\nsensors being introduced since the publication by Middlemiss et al (2016), their motivation, and indeed of others who have\nrecently published in this new and developing field, unfortunately, seems to be a bit misplaced. To my understanding, the\nissues with the previously reported MEMS gravimeters are not related to their noise floor (Pike et al (2018) have already\nshown a noise floor of 0.25 ng/rtHz for the Imperial's MEMS seismometer), nor related to the dynamic range (Mustafazade et\nal 2020 have reported a dynamic range of +/1 g = +/1 980 Gal for Cambridge's Vibrating beam MEMS device). All MEMS\ngravimeters suffer from poor drift rates (Middlemiss et al and Prasad et al 2022 though demonstrated a drift of 100-300\nuGal/day for Glasgow's MEMS gravimeter which is comparable to CG-5), robustness concerns, and other stability issues.\nWhile it might seem that lowering the resonant frequency to sub-Hz is a pressing challenge, it is really not. It is well-known\nthat by tuning geomterical parameters (launch angle, length of the spring) of an anti-spring (Cella et all 2004), arbitrarily low\nresonant frequencies can be achieved but this comes at a cost. Low-resonant frequencies brings the maximum stress within\nthe springs dangerously close to the tolerable limits and/or increase the resonator's cross-axis sensitivity. These robustness\nissues mean that higher resonant frequency robust devices are more preferable. While this indeed will decrease the\nacceleration sensitivities, better, more sensitive read-outs (like the one proposed by the authors) should be used to make up\nfor the loss. So, in my opinion, while the proposed work is useful, it should not be pitched as a solution to a problem which in\nreality is not that critical.\nApart from the motivation issue, the authors need to provide a good context to their algorithmic approach to spring design.\nWhile they have cited Wang et al (2020, 2022) as the previous work on free-form spring design, use of genetic algorithms for\ndesigning compliant non-linear springs was really started by Jutte et al (2008) (Michigan) and others. While I agree their\nfocus was not to make MEMS devices, they did suggest that their approach could be used in MEMS. While I do not expect\nthe authors to cite each and every paper that has been since published, for a high-reputation journal like Nature Comms,\nI expect the authors to guide the readers to a few main previous works.\nThe experimental results and plots in the papers are a bit confusing and left me looking for details in the text. While I was\noften able to find the appropriate information, sometimes I did not. I have included more detailed comments on this later.\nIn summary, while the work is interesting, I am not entirely confident that it significantly advances and solves the persisting\nissues in the field. I encourage the authors to go through my comments and use them to improve and modify the manuscript.\nDeailed comments:\n1. (line#30) The opening sentence in the Abstract did not make a lot of sense, please rephrase\n2. (line#34) The sentence \"A multi-stage directional design scheme...\" in the Abstract is packing too much stuff in a single\nsentence and confusing, please simplify\n3. (line#37) I am not sure I have seen the use of the term \"mechanical sensitivity\" in the accelrometer/gravimeter community;\nplease stick with the standard terms or define any new terms properly\n4. (line#38) The displacement sensitivity of 4 pm/rtHz mentioned in the abstract is of the readout, not of the gravimeter;\nplease make this clear\n5. (line#40) The authors have switched between ng and uGals/Gals too many times; when talking about gravimeters avoid\nusing ng as it is not really a unit (though the accelerometer community prefers it). If the gravity itself is not the same\neverywhere, how can ng be constant?\n6. (line#49) Relative gravimeters, generally speaking, do not have better resolution that absolute gravimeters; please correct\n7. (line#63) Achieveing a resonant frequency belwo 1 Hz has not reamined elusive- there is no point of making a device that\nis difficult to handle\n8. (line#71) The authors switch between sensitivity and resolution; please avoid this and stick with one (preferably\nsensitivity)\n9. (line#72) Boom et al did not make a gravimeter\n10. (line#73) is having a simple geomtery an issue?\n11. (line#74) having more degrees of freedom is not advantageous if you can get the same result through a simpler design\n12. (line#80) Yes, the geometric anti-springs are designed so that they reach the lowest stiffness at 1g (nominal operating\ncondition). Doesn't the proposed F-AS also reach their minimum stiffness at 1g? How is this is not loading? How is F-AS\nthen different to a simpler geometric anti-spring?\n13. (line#81) I am not sure where in the ref#15 it has been mentioned that nonlinear stiffness tuning is experienced\n14. (line#83) what do the authors mean by non-stationary conditions? A few sentences ago the authors claimed that the\ngeometric anti-springs suffer from simple design but here they claim that these have complex morphology- please\nexplain/correct\n15. (line#90) pushed further\n16. (line#100) R-squared value of what? please explain\n17 (line#104) How many ng/rtHz?\n18 (line#118) The explanantion on how the anti-springs work is confusing. There is no sequence of events. Everything\nhappens seamlessly, not in discrete steps; please modify.\n19 (line#137) not immediately clear why convex nature is important? please elaborate\n20. (Fig1b) I did not understand the blue line. Do the authors mean that geomteric anti-springs behave as linear springs\nwhen they are not loaded?\n21. (line#173) As I have mentioned earlier, there are examples of MEMS gravimeters which have a much higher dynamic\nrange than 5 Gal. When the authors say \"... 5 Gal in conventional MEMS gravimeters\" they should back their comment with\ncitations.\n22. (line#210) I am confused about the \"mechanical sensitivity\" that the authors report. For a 1.46 Hz device, by the simple\na=w^2*x relationship, shouldn't the \"mechanical sensitivity\" be 118 um/Gal? why is it consistently smaller everywhere in the\nmanuscript?\n23. The authors should report different resonant modes for the device. It is not clear what the disadvantages of the design\nare.\n24. (line#256) It is not clear how G2 was being actuated using a piezo element? please clarify.\n25. (line#259) The authors did not report the standard devitation on their 5 minute measurements. An error of 0.4% might\nseem small but, the authors should report what this means in terms of acceleration error.\n26. (Fig 3b) The fitted line is not very linear for the lower displacements. Is this a limit of the readout?\n27. I am confused on what quasi-static mean here? Is using this term very necessary?\n28. (line#285) I did not understand how a change of 0.0004 deg in theta is equivalent to 7 mgal. Wouldn't delta_a = g*(1-\ncos(0-0.0004))?\n29. (line#295) how many minutes?\n30. (Fig 4) This plot is confusing. What do the authors mean by the working range? If the frequency is changing non-linearly\nin most of the working range how can it be the working range? In 'e', what is the reason behind the deviation of measured\npoints from the fitted straight line?\n31. (Fig#5b) Whenever presenting seismic data, please use real UTC time in the x-axis rather than relative time.\n31. (Fig#5c) If the authors are claiming that their device is a gravimeter, they should plot the asd down to 10^-5/10^-6\nseconds to capture the tidal signals\n32. The authors have not shown the raw, uncorrected data. The raw data in the supplementary seems to be drift corrected\nand averaged. What kind of drift do they see in their device?\n33. (line#380) What is the reasoning behind using the isotropic model for simulations?"}, "author_response": "Responseto Reviewers.\n“Title - Multi-stage directional design of locally adjustable freeform anti-springs: a pathway\ntowards an ng-level MOEMS gravimeter with 1 Hz resonant frequency and below” Manuscript\nNCOMMS-23-62597.\nWe would like to thank our Referees for their insightful, very detailed and useful comments.\nFollowing their suggestions and criticism, we clarified our interpretation, performed additional\nexperiments, and assessed theoretical and technical aspects in a more detailed discussion. A\ncentral component of this reply is our response to the comments involving long-term stability\nand robustness. We believe we made it clearer why the research reported here is an advance\nand achievement in the MEMS gravimeter, and we hope that the revised manuscript meets the\nhigh standards of Nature Communications. Below is our point-by-point response to our\nReviewer’scomments.\nReviewer 2 – general assessment: “The work is interesting with its main contribution being in the form\nof the optimization algorithm (AELPSO) that helps in designing nonlinear structures - in this case the\ngeometricanti-springs.However,Ifeelthattheworklackedsignificanttechnicaldetails,failedtoprovide\nalargercontexttotheapplicationofthisalgorithmandatsomepointswasnotthoroughinitsanalysisof\nresults. Here, I list point by point some of my concerns and where some detailed analysis/information is\nmissinginitscurrentform.”\nReviewer 1 – Response: We appreciate our reviewer’s remark on the importance and\ninterest of our work. We apologize for the previous omission of technical details. In the\nrevised manuscript and Supplementary Information we have appended (1) Technical details\nand analysis of the optimization algorithm (AELPSO) and the algorithm-enabled multi-stage\ndirectional design; (2) Other applications of AELPSO, including optimization problems of a\nMEMS disk resonator gyroscope and a membrane-based optical cavity for displacement\nsensing; (3) Content of improvements in noise floor and long-term stability. Our point-by-point\nresponse has been listed below.\nReviewer 1 - Comment #1: “The authors successfully show that they can improve the key metrics of the\ndevice through this algorithm. However, even after their improvement over other designs that use\ngeometricanti-springs,therearegeometricanti-springdevices(Boometal.)thathaveshownlowernoise\nfloors (1uGal/rt Hz) than this work. The authors brush of this work by saying their resonant frequency is\nhigher and their springs are simple design. However, the key metric in this case for a gravimeter is the\nnoise floor and Allan deviation rather than the resonant frequency or the complexity of the springs. In\ncomparison to that work that does not use this optimization algorithm, it seems like simple design can\nalsobe beneficial for the enduser.Thisreduces the impactthattheauthor’s workhas onthe community.\nThere are even other MEMS devices that have shown better noise floors eg Pike et al. than the current\ndevice. Can this algorithm be used to optimize these capacitive accelerometer structures as well? If the\nauthors can show a larger application to this algorithm, then the paper will be much stronger. The\ncomplexityofthebeamsisnotthecrucialpointratherthenoisefloorandtheallandeviationare.”\nReviewer 1 – Comment #1 – Response: Thanksfor your valuablecomment andsuggestion.\nWe fully agree with the reviewer that the key metric for a gravimeter is the noise floor and Allan\n2\ndeviation, or long-term stability. Our interpretationrests onthree pillars: (1) thenoise floor, (2) the\nlong-termstability,and(3)thealgorithmapplications.\n(1)Noisefloor\nIn fact, our primary design objective was to lower the resonant frequency through multi-stage\ndirectional design, thereby reducing the noise floor. As shown in Fig.R1 and Eq.R1 [IEEEJournal\nof solid-state circuits, 1996, 31(3): 366-375], lower resonant frequency means higher acceleration-\ndisplacement sensitivity and lower mechanical thermal noise, also known as Brownian noise.\nMechanical thermal noise is a component of the noise floor of the gravimeter, always the\nfundamental noise, so reducing mechanical thermal noise can theoretically lower the noise floor.\nHowever, it can be observed that mechanical thermal noise is related not only to the resonant\nfrequency but also to temperature, Q-factor, and mass. Notably, Boom et al. [2017 IEEE 30th\nInternationalConferenceonMicroElectroMechanicalSystems(MEMS).IEEE,2017: 33-36] and Pike et\nal. [2018 IEEE Micro Electro Mechanical Systems (MEMS). IEEE, 2018: 113-116] and Tang et al.\n[Microsystems & Nanoengineering, 2019, 5(1): 45] all put their device in a vacuum, leading to Q-\nfactors that are orders of magnitude greater than ours. In addition, the detector noise and data\nacquisition noise, including shot noise, Johnson noise, and quantizing noise, also affect the noise\nfloor.Theslightlyhighernoisefloorofourdevice,reportedintheoriginalmanuscript,comparedto\notherdevicesmightbeduetotheaforementionedreasons.\nFig.R1-Response (for Reviewer 1 – Comment #1). The noise model sources for the MOEMS\ngravimetershowingthetopologyanddomainoftheirinjectionintothesignalpath.[2018IEEEMicroElectro\nMechanicalSystems(MEMS).IEEE,2018:113-116]\n4k T\na  B 0 (R1)\nn mQ\nwhetherthenoisefloorcouldbelowered.ThephotodetectorwaschangedfromModel PA25LEW,\nThorlabs Inc. to Model S8745-01, Hamamatsu Inc. with a lower noise, and the data acquisition\nsystem was changed to a self-built AD7177-based circuit. Fig.R2a depicts the noise power\nspectral density curve of the 10-minute dark state data before and after replacing the\nphotodetectoranddataacquisitionsystem.It isevidentthat theoutputelectronic noiseinthedark\nstate can be lowered to smaller than 0.7 μV/Hz1/2 at 1 Hz, which improves by an order of\nmagnitude compared to before the replacement. By replacing the detector and data acquisition\nsystem, the noise level of the optical displacement can be lowered to 1.2 pm/Hz1/2, and the noise\nfloor of the gravimeter can be lowered down to 1.1 μGal/Hz1/2, as shown in Fig.R2b and c. It is\nimportant to note that our device was measured in an atmospheric environment, indicating that\nthere is still room for improvement in the overall noise floor of the device by vacuum packaging\nand testing. Nevertheless, the noise floor of our device is comparable to the best devices\navailable, and the non-vacuum packaging requires less processing complexity, which may be\nadvantageousinsomescenarios.\nminute dark state data before and after replacing the photodetector and data acquisition system. b Noise\nlevel of the optical measurement before and after modification. c Noise level of the acceleration\nmeasurementbeforeandaftermodification.\n(2)Long-termstability\nThe Allan deviation involves the long-term stability of the gravimeter. We implemented an\nintegrated solution for the gravimeter, including compact hermetic packaging, low-stress wafer-\nlevel manufacturing and bonding, and high-precision real-time temperature control. These\n5\nmeasures were undertaken to enhance its long-term stability, thereby reflected in the Allan\ndeviation.\nFig.R3-Response (for Reviewer 1 – Comment #1). a Schematic diagram and b Photograph of the\npackaged device. c-i Schematic diagram of the manufacturing and packaging process. c1 A glass wafer.\nc2 Metal patterning of the grating on the glass wafer. d1 A silicon wafer of the mechanical sensing unit.\nd2 Metal patterning of the grating on the silicon wafer. d3 Lithography patterning of the mechanical\nsensing unit. d4 Deep reactive ion beam etching of the mechanical sensing unit. d5 Release of the\nmechanical sensing unit. d6 Processed mechanical sensing unit. e1 A glass wafer. e2 Deep wet etching\nof theglasswafer. f1Asilicon waferof the substrate. f2Platinum patterningon thesiliconwafer. f3Deep\nreactive ion beam etching of the confinement trench. g Low-stress bonding of wafer 2 and wafer 4.\nh\nLow-stress bonding of wafer 3 and the bonded wafer 2+4. i Low-stress bonding of wafer 1 and the\nbondedwafer2+3+4.\nandthecorresponding processisillustratedinFig.R3ctoi. During theprocess, weemployedlow-\nstress wafer-level manufacturing and bonding techniques to reduce factors of structural stress,\ndeformation, hysteresis, and adhesive creep. These specifics will be further elaborated on in the\nfollowing.\nWe also fabricated platinum resistors on the substrate of the device using micro-manufacturing\ntechnology to achieve high-precision real-time temperature control. The distribution and\ndimensions of the resistors were optimized through FEA to ensure that the entire mechanical\nsensingunitmaintainsareasonabletemperaturegradientandstabilityduringtemperaturecontrol,\nas shown in Fig.R4a and b. The ambient temperature was set to 291.5 K (very close to the\ntemperature of our cave laboratory), and the target temperature was set to 294 K. In reality, our\ntemperature control is based on fuzzy PID [Micromachines, 2021, 12(12): 1511] with optimized\nparameters.Fig.R4cshowsthat itrealizedatemperaturestabilityofsmaller than3.3mK, whichis\nmeasured by the temperature measurement resistor. The inset shows the temperature data over\nmore than 8 hours of the measurement resistor and ambient temperature. It is evident that\ntemperaturecontrolcaneffectivelysuppresslong-termdriftsinambienttemperature.\nof the mechanical sensing unit and b the packaged device. c Temperature measured by the resistor, and\ntheambienttemperaturewasaround291.5K.Theinset presentsthemeasuredtemperatureover8hours\nasacomparisonwiththeambienttemperaturemeasuredbyathermistor.\nRegarding the manufacturing, we utilized a monolithic single-crystal silicon wafer to fabricate the\nmechanicalstructuretominimizeinternalstress.GlassWaferswiththermalexpansioncoefficients\ncomparable to single-crystal silicon (Scott AF 32) were used to accomplish wafer-level packaging\nto reduce interface stress. Conventional wafer bonding can easily result in stress at the bonding\ninterface due to applied pressure and high temperature. We developed a low-stress bonding\ntechnique,asshowninFig.R5a, toachievewafer-levellow-stresspackaging. Step1,we usedour\nself-developed automated dispensing system, as illustrated in Fig.R5b, to ensure the consistency\noftheadhesive;Step2,aclampingalignmentsystem,asshowninFig.R5e,wasusedtoalignthe\ntwo bonded wafers, see Fig.R5f and g; Step 3, the adhesive was cured by a specific stepwise\ncuring process. The bonding technique can be achieved by applying a certain small amount of\npressureatloworevenroomtemperatures.\nfabricatedtoserveasadhesivereservoirsforsiliconwafers(severalholesforglasswafers),which\nfacilitated stress homogenization during bonding. In addition, we selected a suitable adhesive\nthrough experiments with adhesive wafer samples and developed a specialized process to\nbalancethecuringtemperature,achievingoptimalperformancestability.\n9\ntechnique. b Photograph of the self-developed automated dispensing system. c Inverted pyramid arrays\nserving as adhesive reservoirs. d Photograph of the inverted pyramid arrays with a depth of around 30\nμm. e Photograph of the self-developed clamping alignment system. f Schematic diagram of the\nalignmentoftwowafers.gSchematicdiagramofthecontactoftwowafers.\nThe adhesive we selected was a self-developed composite adhesive. The curing agent is an\nimidazole compound, the filler is calcium carbonate, and the matrix is epoxy resin, with a mass\nratio of 20:20:60. To evaluate the mechanical properties of the adhesive, or relaxation modulus,\nwe used a specialized mold to fabricate adhesive specimens for measuring the relaxation\nmodulus,asshowninFig.R6a.Therelaxationmodulusparametersatdifferenttemperatureswere\ntestedusing a high-precision dynamic mechanical analyzer (DMA), as shown inFig.R6b. Thetest\nresultsatroomtemperatureareshowninFig.R6c.\nFig.R6-Response (for Reviewer 1 – Comment #1). a Photograph of the tested adhesive specimen. b\nPhotograph of the relaxation modulus test using DMA. c Measured relaxation modulus as a function of\ntime at room temperature. d Stepwise curing process containing five steps. e Photograph of the tested\nadhesivesiliconwaferanditsstressdistribution.\nboth curing temperature and degree of cure after multiple rounds of optimization. As shown in\nFig.R6d, it contains five steps. The first step is to raise the temperature to T , taking a time of t ;\n1 1\nthe second step is to maintain the temperature at T until time t ; the third step is to raise the\n1 2\ntemperature to T , at an end time of t ; the fourth step is to maintain the temperature at T until\n2 3 2\ntimet ;thelaststepisthenaturalcooltoroomtemperature,andthefinalendtimeist .Thetested\n4 5\nadhesive silicon wafer cured by this process had a stress distribution state, as shown inFig.R6e,\nwith a maximum stress of 0.02532Mpa, which proved that this process can achieve low-stress\ncuring.\ngravimeter and the reference gravimeter. The red line represents the MOEMS gravimeter, the blue line\nrepresents the reference gravimeter, PET, and the orange line represents the noise data of the MOEMS\ndevice. The red and the orange series were recorded at a sampling rate of 20 Hz. The green series\nrepresent data that were recorded at a sampling rate of 0.1 Hz. At 10-5 Hz, the diurnal and semi-diurnal\ncomponents of the Earth tides are visible; the Earth tremor peak can be observed at around 0.2 Hz; and\n13\nthe resonant peak of the device is located at around 1.6 Hz. b Raw data of the tide measurement with a\nsampling rate of 0.1 Hz. c Allan deviation of the data in b. d Raw data with the removal of the linear drift.\neAllan deviationof the series ind. f Comparisonof Earth tides measuredbythe MOEMSgravimeter and\nthe PET gravimeter. The red line is obtained from the MOEMS gravimeter, and the blue line is obtained\nfrom the reference gravimeter. The correlation coefficient between these two sets of data is 0.9218. g\nAllandeviationofthedataoftheMOEMSgravimeterinf.\nFig.R7 shows the noise floor, drift data, and Allan deviation results of our prototype MOEMS\ngravimeterafterimplementingallimprovements.Datawererecordedbyafastsamplingrateof20\nHz and a slow sampling rate of 0.1 Hz in order to observe the responses of Earth tremor peaks\nand tides. As shown in Fig.R7a, the noise floor of the device reaches down to 1.1 μGal/Hz-1/2 at\n0.5 Hz, and the ultimate noise floor of the device can be around 0.4 μGal/Hz-1/2 at 0.45 Hz; the\ndiurnal and semi-diurnal components of the Earth tides are visible at 10-5 Hz; the Earth tremor\npeak can be observed at around 0.2 Hz. According to the Allan deviation results, a bias instability\nof 6 μGal can be obtained with an integration time of 600 s. It is also observed that the device\nachieved reasonably good long-term stability, with a drift of 153 μGal/day. As a comparison, the\ndrift of our device is comparable to or even better than other MEMS gravimeters, which is around\n100-300 μGal/day [Nature, 2016, 531(7596): 614-617; Scientific reports, 2022, 12(1): 13091]. Overall,\nourdeviceachievesthehighestreportedacceleration-displacementsensitivity(orlowestresonant\nfrequency) on the chip scale and exhibits excellent noise floor and long-term stability. This\nperformance is attributed to our design methodology, as well as the packaging and measures\nmentionedearlier.\nPreviously, we regarded the long-term stability improvements as a relatively independent work,\nand the original manuscript focused on the innovation of the design and freeform anti-springs, so\nthispartwasnotextensivelyintroducedintheoriginalmanuscript.However,wegreatlyappreciate\thereviewersforraisingthese concerns.Wehaveaddedthe discussionof theabove contentinto\nboththerevisedmanuscriptandtheSupplementaryInformation.\n(3)Algorithmapplications\nThe proposed algorithm, AELPSO, is fit for multi-modal optimization problems. Herein, we added\ntechnical details and analyses regarding AELPSO and multi-stage directional design. We\nconducted a comprehensive comparison between AELPSO and common optimization algorithms\n(PSO/GA/DE) using the CEC test function sets, which were commonly used in the metaheuristic\nalgorithm field. Additionally, we selected two cases for application in other devices to validate the\n14\nuniversality of the algorithm, including a MEMS disk resonator gyroscope and a membrane-\nbased optical cavity for displacement sensing.\nIn the original manuscript and Supplementary Information, we have provided the framework and\npseudocode of AELPSO and its sub-strategies of ‘C’, ‘O’, ‘D’, and ‘A’. We will present the\nmathematicalmodelandconvergenceanalysisofthestrategiesinthefollowing.\nIn solving an optimization problem with a D-dimensional hyperspace, each particle i in the\npopulation has a velocityvector V =[v ,v ,v ,...,v ] anda position vector X =[x ,x ,x ,...,x ],\ni i1 i2 i3 iD i i1 i2 i3 iD\nwhere the subscripts i and D denote the index of the particle and the dimensions of the problem,\nrespectively. The personal best experience of each particle is recorded as p = [p , p , p , ..., p ]\ni i1 i2 i3 iD\nand then to generate the global optimum p . P andp serve as guidance for resetting the velocity\ng i g\nandpositionsofparticles.TheclassicallearningruleofPSOis:\nvt1 vt c r   p xt  c r   p xt  (R2)\ni,j i,j 1 1 i,j i,j 2 2 g i,j\nxt1xt vt1\ni,j i,j i,j\nwhere t is the generation number, ω is an inertia weight factor [IEEE International Conference on\nEvolutionary Computation, 1998: 69-73], c and c are the acceleration coefficients [Proceedings of\n1 2\nthe International Symposium on Micromachines and Human Science,1995:39-43], r and r are two\n1 2\nrandomvalues.\n‘C’) Cross- Strategy: The binomial crossover of the current particle’s personal best experience\nwith the personal best experiences of several randomly selected particles is set as the learning\nexemplar for the particle. Its update formula is shown in Eq.R4 and R5, where p and p are the\nr1 r2\npersonalbestexperiencesof tworandomlyselectedparticlesinthepopulation. Thepersonalbest\nexperience of the winning one (p ) through a tournament mechanism [IEEE Transactions on\nrw\nEvolutionary Computation, 2006, 10(3): 281-295] is crossed with the current particle’s personal best\nexperience to generate pc. It is noted that pc is only updated after a stagnation of p for S\ni\ngenerationsbecausefrequentpcupdatescanmakepopulationdiversitytoorichtoconverge.\nvt1 vt cr   pc xt  (R4)\ni,j 1 i,j i,j i,j i,j\npc p ||p ,p better  p ,p  (R5)\ni,j i,j rw,j rw,j r1 r2\n‘O’) Ortho- Strategy: The orthogonal combination of the current particle’s personal best\nexperience with the personal best experience in its neighborhood lb is set as the learning\ni\nexemplar for the particle. Its updated formula is shown in Eq.R6 and R7, which ⊕ represents an\northogonal combination [IEEE Transactions on Evolutionary Computation, 2011, 15(6): 832-847]. The\ntimecomplexity of the orthogonal process is O (MD2) (M and D represent the population size and\n15\nthe dimensions of problems, respectively), and frequent orthogonal operations will significantly\nincrease the algorithm’s time complexity. Therefore, po is only updated after a stagnation of p for\ni\nGgenerations.\nvt1 vt cr   po xt  (R6)\ni,j 2 i,j i,j i,j i,j\npo p lb (R7)\ni,j i i\n‘D’) Dynamic- Strategy: The minimum value among the personal best experiences of all particles\nand the current particle’s personal best experience are set as the learning exemplars for the\ncurrent particle, while the parameters are adjusted adaptively. Its update formula is shown in\nEq.R8, where ω , c , and c adaptively change with the variation of population diversity [IEEE\n3 1 2\nTransactionsonSystems,Man,andCybernetics:PartB,2009,39(5):1362-1381].\nvt1 vt c r   p xt  c r   p xt  (R8)\ni,j 3 i,j 1 i,j i,j i,j 2 i,j g i,j\nxt1xt vt1\ni,j i,j i,j\n‘A’) All- Strategy: The personal best experiences of each particle in its current neighborhood are\nset as the learning exemplars for the particle. Its update formula is shown in Eq.R9 and R10,\nwhere pa denotes the weighted average of the personal best experiences of all particles in p’s\ni\nneighborhood,Nisthenumberofparticlescontainedinthecurrentparticle’sneighborhood,andφ\nisarandomnumberuniformlydistributedbetween0and1.\nvt1 vt cr   pa xt  (R9)\ni,j 4 i,j i,j i,j i,j\n \npa \nN\n p /r ,r\nN\n (R10)\ni,j k1 k,j k,j i,j i,j k1 k,j\nConvergence analysis of the strategies: To better illustrate the convergence of our AELPSO\nalgorithm, we will analyze the convergence of the four update strategies mentioned earlier\nbased on knowledge from the analysis of discrete dynamic systems in modern control theory\n[Information Processing Letters, 2003, 85(6): 317-325]. Considering that we define final\nconvergence as the state when all dimensions of the particle's values become stable, we\ngenerallyconstruct ananalytical modelbased onthevariation patternof thejth dimensionof the\nith particle's value during the optimization process. Because the major difference between the\nfour strategies lies in their learning paradigm generation process and the model construction is\nsimilar, we present the construction process for the relatively complex D-strategy here. The\nvelocityupdateformula fortheD-strategy isshown asfollows:\nvt1 vt c r   p xt  c r   p xt  (R11)\ni,j i,j 1 1 i,j i,j 2 2 gbest i,j\nxt1xt vt1\ni,j i,j i,j\nexpectedvalue of 1/2.Then, substitutingEq.R11into Eq.R12,we obtainEq.R13:\nxt1  xt vt  c 1  p xt   c 2  p xt  (R13)\ni,j i,j i,j 2 i,j i,j 2 gbest i,j\nTherefore, proving the convergence of this strategy is equivalent to proving the convergence of\ntheconstructed dynamic system.Welet:\nc c\nc 1 2 (R14)\n2\nc c\np*  1 p  2 p (R15)\nc c i,j c c gbest\n1 2 1 2\nThenEq.R11will becomeEq.R16:\nvt1w*vt c*(p*xt )(R16)\ni,j i,j i,j\nFrom the above equation, it can be seen that the new gravitational coefficient c is equal to the\naverage of the particle's own gravitational coefficient c and the global particle's gravitational\n1\ncoefficient c . The attractor particle p* is also the weighted average of the local best and global\n2\nbest.\nThus, Eq.R17andR18 canberegarded asthedynamic equationsof alinear discretesystem：\nyt1  Ayt Bp*(R17)\nx t 1c  c\nyt  v i i , , j j t   ,A   c    ,B  c   (R18)\nIn dynamic system theory, A is called the state matrix, p* is called the input, and B is called the\ninput matrix.\ny*(t 1)  y*(t) (R19)\ny*p*,0 T (R20)\n \nThus, based on Eq.R19, we can derive Eq.R20. The convergence of the system means that the\nparticle's state will eventually reach the equilibrium state y*. According to modern control theory,\nthe eigenvalues of the state matrix A determine whether the system can reach an equilibrium\nstate. Weuse Eq.R21tocalculatetheeigenvalues:\n17\n2   c 1 0 (R21)\nWhen the absolute values of the eigenvalues are less than 1, the system is generally\nconsidered asymptotically stable, meaning the strategy will eventually converge. Based on this\ncondition,we can obtaintherangeofvalues fortheparametersω and c:\n01 ，c0 ，2c20(R22)\nThe parameter settings for the four strategies mentioned in this paper all satisfy Eq.R19,\ntherebyproving theconvergenceof thestrategiesdiscussedin thepaper.\nAs for the other three strategies, when the system composed of each strategy converges, the\nvalue ofy*is shown in Table R1.\nTableR1:ResultsofConvergenceAnalysisforEachSub-strategy\nStrategies y*\nC [pc i,j ,0]T\nO [po i,j ,0]T\nD [p*,0]T\nA [pa i,j ,0]T\nThe other three strategies prove convergence similarly to the D-strategy we mentioned above\nand will not be listed here. The above is the convergence analysis of the strategies in our\nalgorithm.\nAdaptive Elite Learning (AEL): In the updated formulas mentioned in the previous section, each\nstrategy uses the personal best experiences of particles to construct the learning exemplars.\nAlthough the CODA multi-strategies hybrid allows it to search more extensively in the parameter\nspace, the exemplars construction can only be carried out within the cognitive scope of one sub-\nstrategy, leading to an even worse optimization efficiency than the single strategy PSO variants.\nThe AEL method helps one sub-strategy understand information beyond its cognition. This\nmethod consists of three parts: first, let particles that perform a strategy and with good\nperformance perform other strategies besides this strategy in the next generation by fixed\nelimination and competition methods; second, use the current particle’s personal best experience\nto correct the exemplars constructed in all strategies; third, regulate the number of particles that\nperform the correction. By performing these operations, positive feedback can be formed to drive\nparticlestomovetobetterpositionscontinuously:\n1)Thefixed eliminationandcompetition methodsareappliedto makethe AELPSOmore suitable\nfor solving multimodal problems. The top-performing particles from all sub-swarmsare called elite\nparticles. The particles from fixed positions of each sub-swarm are called regular particles. Then,\nregular particles and elite particles will compete by fitness values. It is crucial to note that the\n18\ncompetingparticlesmustoriginatefromdifferentsub-swarms.Ifaregularparticle’sfitnessvalueis\nsuperior, it remains within its sub-swarm. However, if an elite particle prevails, it replaces the\nregular particle. Furthermore, the positions of regular particles are recorded. In each subsequent\napplication of the fixed elimination and competition methods, particles occupying these recorded\npositions are consistently selected as regular particles. This approach is implemented to prevent\nprematureconvergenceofthealgorithm.\n2) The top-performing personal best experiences are applied to form positive feedback. Following\nthe execution of the fixed elimination and competition methods, each sub-swarm is enriched with\nnewparticlesthatpossesssuperiorpersonalbestexperiences.Theseparticlesactivelyparticipate\nin the CODA multi-strategies, aiding the sub-swarms in generating improved learning exemplars.\nTheserefinedexemplarsguidetheparticlesineachsub-swarm,enablingthemtoconductamore\nefficientsearch.\n3) The nonlinear adaptive method is applied to regulate the number of replaced particles. In the\nfixed elimination and competition methods, the number of elite and regular particles is always\nequal,andtheirnumbersaredifferentinsub-swarmsofdifferentsizes.Thisnumberalsochanges\nwith the optimization process to mitigate the impact of the AEL method on population diversity.\nTheαisdefinedasfollows:\n2\n t \nt      (R23)\nmin max min I \nmax\nI is the maximum number of generations, t is the current number of generations, both are\nmax\nintegers, and the range of values for β and β is 0 to 1. According to Eq.R23, αt will increase\nmax min\nwith generations, and the rate of change will become faster and faster. The number of regular\nparticles is represented by αt*S , where S represents the size of the sub-swarm that performs\nk k\nstrategyk,butsincethecompetitionmethod,theactualnumbermustbelessthanαt*S .\nk\nWealsoappendedthepseudocodeofAELmethodbelow.\nAdaptive_Elite_Learning(t,X,V,P,αt,RINT,N,D,Pop).\n01:Pop={Pop ,Pop ,Pop ,Pop |Pop =X ∪V ∪P,1≤i≤4}\n1 2 3 4 i i i i\n02:RINT={r ,r ,r ,r |r ={r ,r ,r ,...,r |1≤j≤round(αt·N)}}\n1 2 3 4 i i,1 i,2 i,3 i,j i\n03:SortP inascendingorder,anddenotethesortedP asPs\ni i i\n04:SortXandVbyPs,anddonatesortedX,VasXs,Vs\ni\n05:Fork=1to4Do\n06: Fori=1toround(αt*N )Do\nk\n07: SelectanintegerfromRINT anddonateitasm\nk\n08: IfXs,Vs,Ps ∉Pop ANDPs <P\ni i i k i m\n09: X =(X -X )∪Xs\nk k m i\n10: V =(V -V )∪Vs\nk k m i\n11: P =(P -P )∪Ps\nk k m i\n12: EndIf\n19\n13: EndFor\n14:EndFor\n15:Outputresults:X,V,P.\nIn the original Supplementary Information, we used eight algorithms to benchmark and highlight\nthe superiority of the proposed AELPSO. However, we noticed that these algorithms were all\nvariantsof PSO,whichmaynotrepresentall metaheuristicoptimizationalgorithms. Therefore,we\ndeleted several PSO variants and added four other metaheuristic optimization algorithms, namely\nGA, PSO, GLPSO, and DCMA_EA, to conduct a more comprehensive comparison. GA is a\nclassic genetic algorithm [Statistical Computing, 1994, 4: 65-85]; GLPSO is an algorithm that\nintegratesPSOwithageneticalgorithm[IEEETransactionsonCybernetics,2015,46(10):2277-2290];\nDCMA_EA is an algorithm that integrates covariance matrix adaptation strategy and evolutionary\nalgorithm [IEEE Symposium on Differential Evolution (SDE), 2011:1-8]. The algorithms included in\nthe comparative analysis are DSPSO [Information Sciences, 2019, 480: 109-129], FMPSO [Swarm\nand Evolutionary Computation,2019, 44: 349-364], MFCPSO [IEEE Transactions on Fuzzy Systems,\n2023,31(7):2464-2478],GLPSO,GA,DCMA_EA,andtheproposedAELPSO.\nBefore conducting a comparative analysis of the algorithms in practical applications, we first\ntested their performance using test functions (can be regarded as simple optimization problems).\nInsteadofCEC2015,thistime, weusedtestfunctionsfromtheCEC2020andCEC2022datasets,\nandevaluatedthetestresultsbasedonspecificperformancemetrics.Thesemetricswerederived\nfromnon-parametrictestingusingtheMann-WhitneyU-test[AnnalsofMathematicalStatistics,1947,\n18(1): 50-60]. Let t be the result obtained by the jth algorithm on the kth test problem during the\nijk\nith run, where i=1, 2, …, m, and j=1, 2, …, n. The results of all n runs for the m algorithms on the\nkth problem are arranged in ascending (or descending) order. Then, scores are assigned\nsequentially from the best to the worst result, with the best result receiving a score of m*n, the\nnext best receiving m*n − 1, and so on. Fig.R8 shows the convergence plots of all algorithms for\neach test function. Table R3 lists the calculated metrics of all algorithms. It is evident that\nAELPSO outperforms the other algorithms in test functions based on the convergence plots and\nthesummarizedscores.\n20\n21\nforeachtestfunction,includingeightalgorithmsand62functions.\nTableR2Settingoftestproblems\nTestFunctions Dimensions PopulationSize\nCEC2020 5/10/15/20 40/50/80/100\nCEC2022 10/20 50/100\nTableR3Summaryoftestresults\nCEC2020Score CEC2022Score\nAlgorithms Rank Rank\n5D 10D 15D 20D 10D 20D\nDSPSO 76 77 74 76 2 73 64 4\nFMPSO 53 40 46 53 5 57 51 5\nMFCPSO 65 67 65 56 4 70 79 2\nGLPSO 38 24 26 21 7 36 36 6\nGA 30 28 33 40 6 25 41 7\nDCMA_EA 72 75 70 54 3 76 65 3\nAELPSO 89 91 96 90 1 80 92 1\nThen, we will demonstrate the effectiveness of AELPSO in two applications to validate the\nuniversality. One application is a structural optimization problem of a MEMS disk resonator\n23\ngyroscope(DRG), andthe other isa structural andmaterial optimization problem of a membrane-\nbased optical cavity for displacement sensing. The reason for choosing these two applications is\nthat the first problem is a typical expensive optimization problem, focusing on the optimization of\nMEMS resonator devices, whose operating principle is different from our F-AS-based flexural\nstructure. The second application is a relatively expensive optimization problem but in a vastly\ndifferentapplicationscenario.Webelievethesetwoapplicationsarerepresentative.\n(A)aMEMSdiskresonatorgyroscope\nThis application stems from the article [Microsystems & Nanoengineering, 2018, 4(1): 32]. It aims to\noptimizethespokelengthdistributionofaMEMSDRGtoachievealowerresonantfrequencyand\na higher thermoelastic quality factor (Q ). The structural parameters of the MEMS DRG are\nted\nshown in Fig.R9a, including the number of rings N, outer radius R, anchor radius r, ring width h,\nthickness T, and the length of the spokes in the ith layer L. Their values or ranges, and units are\ni\nlistedin Table R4. The variableparametersto be optimized are L (i= 1, 2, 3, …, N), and the total\ni\nlength of the spokes ∑L is constant, as shown in Fig.R9b. The remaining structural parameters\ni\nare fixed. In the article, the authors used the classical PSO combined with FEA to perform\nstructuraloptimizationonthis9-ringMEMSDRG.\nFig.R9-Response (for Reviewer 1 – Comment #1). a Schematic of a 9-ring MEMS DRG along with its\nstructuralparameters.bVariableparameterstobeoptimizedandtheirschematicdiagram.\nTableR4MEMSDRGstructuralparameters\nN Number ofrings 9 /\nR Outer radius 4 mm\nr Anchorradius 1.9 mm\nh Ringwidth 12 μm\nT Thickness 150 μm\nLengthofthespokesin\nL i(i=1,2,3…N) 0.2-0.6 mm\ntheithlayer\nTableR5MEMSDRGstructuralparameters\nResonantfrequency Resonantfrequency Q ted Q ted\nParameter\n(simulation) (inreference) (simulation) (inreference)\nEqual-width\n7.491kHz / 414k 392k\nDRG\nOptimized\n7.079kHz 7.08 kHz 489k 490k\nDRG\nWefirstusedFEAtoreproducetheresultsoftheequal-width9-ringMEMSDRGanditsoptimized\nSLD DRG, as listed in Table S5. It can be seen that our results were almost identical to those in\nthe article, indicating that our FEA simulation setup and results were accurate. Then, we used\nAELPSOtoreplacetheclassicalPSOfortheoptimizationprocessandcomparetheresults.\nIn the article, the classical PSO used 10 particles and 50 iterations. For a fair comparison, we\nchose 50 particles and 10 iterations for the proposed AELPSO in consideration of four sub-\nswarms. This ensures that the total number of particles and the function evaluations (FES = 500,\nmeansoptimization cost) arethe same. Boththe optimization iterations (or FES) in the articleand\nours were so small because this problem is very expensive. Each evaluation of the fitness value\nrequires completing a FEA-based calculation. For example, computing the resonant frequency of\nsuchastructureusingFEAtakesapproximately1minuteand21seconds,andcalculatingtheQ\nted\nrequiresevenmoretime,about33minutes.\nand Q ted of the DRG at each iteration in the classical PSO and AELPSO-based optimization progresses,\nwherein the dashed and solid lines are from reference [Microsystems&Nanoengineering,2018,4(1):32], and\nthereddotsarefromouroptimizationresults.\nThe convergence plots of the optimized resonant frequency and Q we obtained are shown in\nted\nFig.R10. It can be seen that for both resonant frequency and Q , the optimization driven by\nted\nAELPSO converges to better values (f = 7.077 kHz, Q = 490.639k, which were all better than\nted\nthe best values in the article f = 7.08 kHz, Q = 490k) by the eighth generation. Not only did it\nted\nconvergefasterthanthe optimizationprocessdescribedinthearticle, but theoptimalvalueswere\nalso superior. This clearly demonstrates the effectiveness of AELPSO in addressing such\nexpensiveoptimizationproblemsforMEMSdevicesbasedonFEA.\n(B)amembrane-basedopticalcavityfordisplacementsensing\nWe previously designed a displacement-sensing optical cavity based on membranes [Opto-\nElectronic Advances, 2023, 6: 220018]. At that time, we used a modified GA for optimization. The\noptical cavityiscomposedof mirror 1 (fixed mirror),an air layer, and mirror 2 (movablemirror), as\nshown in Fig.R11a. The displacement of mirror 2 changes the cavity length of the cavity and\ncauses corresponding changes in the optical intensity-displacement response curve. The mirrors\narecomposedofopticalthinfilmlayersandsubstrates.Giventhestablerefractiveindexandeasy\navailability of SiO wafers, we selected SiO as the substrate material for mirror 1. The material\n2 2\nandthicknessoftheopticalfilmandsubstratesignificantlyaffecttheopticalintensity-displacement\nresponse; thus, we consider these parameters as optimization variables, as shown in Fig.R11a.\ncontrast,asshownintheinsetofFig.R11b.Thefitnessfunctionisconstructedasfollows:\nSyemm*0.6（1 fit_value）*0.4, if fit_value0.95\nf  (R24)\nx, else\nwhereSyemmdenotesthesymmetryoftheresponse,fit_valueisafunctionofthecontrast,andxis\narelativelylargenumbertodiscardoutputsthatdonotmeetthecontrastrequirement.\n\nSyemm maxL,L \n1 2 2\nI I\nC  max min (R25)\nI I\nmax min\nC I\nfit_value  max\n2 2\noptical cavity along with its variable parameters. b Typical response of the light intensity versus the\ndisplacementofmirror2,thepaneldepictsthetargetresponseoftheoptimization.\nHerein, considering that this optimization problem is not as computationally expensive, we\ncompared not only the GA algorithm but also DSPSO, FMPSO, MFCPSO, GLPSO, GA, and\nDCMA_EA, totaling seven algorithms for comparison. The reason we did not apply this\ncomprehensive comparison in the first application is that the first optimization problem was too\ncomputationally expensive. Benchmarking all the algorithms would have exceeded the three-\nmonth deadline. The convergence plots of the fitness values for different algorithms are shown in\nFig.R12a. The convergence plots of the fitness values for the two sub-objectives, symmetry, and\ncontrast, are shown in Fig.R12b and c, respectively. Fig.R12d shows eight light intensity-\ndisplacement response curves. The red dashed line is obtained by reflecting the right half of the\ncurveabouttheblackdashedaxis,aimingtovisuallyillustratethesymmetryoftheresponsemore\nclearly. It can be observed that the optimization result of AELPSO is the best, with the best\nsymmetry and contrast. According to the convergence plots, AELPSO reaches optimal values\nsignificantly quicker than the other algorithms, demonstrating superior performance in both\nefficiencyandeffectiveness.\noptical cavity along with its variable parameters. b Typical response of the light intensity versus the\ndisplacementofmirror2,thepaneldepictsthetargetresponseoftheoptimization.\nIn conclusion, we really appreciate our reviewer’s remark on the crucial point of this work. We\nb\nWe\nbelieve that the discussion above makes a powerful case for the interpretation in terms of the\nnoise floor, long-termstability, andthegeneralizabilityof thealgorithm.\nChangesinthemanuscript.\n-Weessentially rewrotetheabstract, introduction,results, and methods.\nChangesintheSupplementaryInformation.\n-SupplementaryNote6.Low-stresswafer-levelmanufacturingandbondingtechniques.\n-SupplementaryNote7.High-precisionreal-timetemperaturecontrol.\n-SupplementaryNote8.Comparisonofconvergenceofmultiplealgorithms.\nugal/rtHz.Pleasejustuseonsetofunitstoavoidconfusingthereader.”\nReviewer 1 – Comment #2 – Response: We thank our referee for the comment. In the revised\nmanuscript, not only in the introduction but throughout the entire text, these two units have been\nunifiedtoμGal/Hz1/2.\nReviewer 1 - Comment #3: “Line 177.The authors shouldlista priorityon the parameters usedinthe\nobjective function. They mention that the acceleration-displacement sensitivity is their most critical\nproperty and yet the first variable in equation 2 is the dynamic range. It will be clearer of they listed the\nparametersthatgointotheobjectivefunctioninatableandshowthepriorityandwhetherornottheyare\ndependentoneachother.Foreg:highlinearitycouldmeanloweracceleration-displacementsensitivity.”\nReviewer 1 – Comment #3 – Response: We thank our referee for the constructive comment.\nIndeed, in the original manuscript, we did not list the objective function according to prioritywhen\npresenting the properties, which caused some ambiguity. We apologize for this. In the revised\nmanuscript, we have adjusted the order of the properties in the revised Eq.2. The order now is S,\nG, R2, which represents the priority of these properties. Additionally, in the revised\nSupplementary Information, we have added the parameters constituting the objective function\nalong with their weights, aslisted in Table R6. Herein, the weight of the linearityis large because\na highly linear curve usually has a R2 very close to 1. Typically, S ranges from 0 to 100 (μm/Gal),\nGrangesfrom0to10(Gal),and1000*(1−R2) isthesmallest,reflectingthepriority.\nTableR6Parametersoftheobjectivefunction\nAcceleration-\nLinearrange Linearity\ndisplacementsensitivity\nProperty (rangewithR2>0.99) (R-squaredvalue):\n(averageslope)\nUnit:Gal Unit:/\nUnit:μm/Gal\nSymbol S G R2\nWeight 1 1 1000*(1−R2)\nChangesinthemanuscript.\n-Line 215 “The optimization objective function for the F-AS is as follows: … where S represents\nthe acceleration-displacement sensitivity, G represents the linear range, and R2 is the R-\nsquaredvalue of thelinear fit of theacceleration-displacementcurve within therangeof ±5Gal.”\n-SupplementaryTable1-2.\nReviewer 1 - Comment #4: “Line 259: Can the authors provide more explanation as to why the 5\nminutes were needed to average the result for static measurement result? Is there drift? Other\nmeasurementartifacts?”\nReviewer 1 – Comment #4 – Response: We thank our referee for the constructive comment.\nRegarding the optical displacement measurement, the selection of 5 minutes was a result of\nconsidering factors including measurement duration, measurement stability, and power spectrum\nresolution. In the original manuscript, we averaged the 5-minute data in order to plot the voltage\nversusdisplacementcurve,asshownintheoriginalFig.3b.\nFig.R13-Response (for Reviewer 1 – Comment #4). a Static results of the output voltage versus the\ninput displacement, with the 5-minute time-domain data shown in the insets. b Noise performance of the\nopticaldisplacementmeasurementunit.\npoint are highly stable, with standard deviations from 1 to 12 mV, indicating virtually no drift. This\nis mainly due to the absence of a mechanical elastic mechanism to amplify the displacement,\nleading to good long-term stability. The noise power spectral density curve of the 5-minute data\nis shown in Fig.R13b. From the results, it can be observed that the power spectrum resolution\nreaches10-2 Hz,whichisdeemedsufficient forevaluatingtheopticaldisplacementmeasurement.\nIndeed, for each displacement point, we could choose a longer measurement time without\nsignificantly affecting the results because the time-domain response of the displacement\nmeasurement is relatively stable. However, this would obviously impact the overall testing\nduration.Therefore,aftercarefulconsideration,we decidedonameasurementtimeof5minutes.\nChangesinthemanuscript.\n-Revised Fig.3.\n-Line 323 “The time-domain data of each point are presented in the right panel. The response\nresembles the simulation results and performs a sensitivity of 1.28 V/µm, with the standard\ndeviations from 1 to 12 mV. The standard deviations correspond to acceleration errors from 8.2\nto 98.3 μGal, combining the optical displacement sensitivity and the acceleration-displacement\nsensitivity of around 100 μm/Gal. The experimental result is not a highly linear response but\nrather resembles the typical response of such an optical cavity. Fig. 3c shows the spectral\ndensity of the displacement measurement. It indicates that the noise level of the optical cavity is\n1.2pm Hz-1/2at 1Hz andbelow 5 pm Hz-1/2within thefrequencyrangeof 0.5to1 Hz.”\nReviewer 1 - Comment #5: “Line 272. The authors mention that the proof mass was adjusted to avoid\ncollissionswiththeanti-springs. By doingsothe centreof masshas changedinthe SF-ASdesignmaking\nit asymmetrical. Will that not cause tilting effects? Can the authors provide more explanation on this in\nthetext?”\nReviewer 1 – Comment #5 – Response: We appreciate the reviewer’s comment and\nsuggestion. We adjusted the outline of the proof mass in the xy-plane, as shown in Fig.R14a\n(The actual change in area is approximately 0.4%). The adjustment did not change the center of\nthe proof mass, nor did it affect the device's modes and mechanical properties. The modes (the\nfirst three modes) before and after the mass shape adjustment are shown in Fig.R14b (for\nstructurepresentedin theoriginalmanuscript).It canbeobservedthatthefrequencies ofthe first\nthreemodes are nearly identical. Thetiny differences are likely due tomesh errors. Furthermore,\n32\nit had almost no effect on its tilting effects. We also analyzed the cross-axis coupling of the\nstructure with and without the proof mass adjustment. Fig.R14c presents the cross-axis\nsensitivity in the x-direction of two structures. The cross-axis sensitivity is termed as\nsensitivity/sensitivity , and calculated from the displacement-acceleration curves in x and y-\nx y\ndirections.Itisevidentthatthecross-axissensitivityremainedunchanged.\nTherefore,itissafetoconcludethatthemassadjustmentdidnotcausetiltingeffects.\nFig.R14-Response(forReviewer1–Comment#5).aShapeofproof massbeforeandafteradjustment.\nb First three modes of the mechanical sensing unit before and after the proof mass adjustment. c Cross-\naxissensitivityinthex-directionoftwomechanicalsensingunits.\n-Line 342 “The adjustment did not change the center of the proof mass, nor did it affect the\ndevice'smodesandmechanical properties.(SupplementaryNote4).”\nChangesintheSupplementaryInformation.\n-Supplementary Note 4. Results of the mechanical sensing unit before and after the proof mass\nadjustment.\nReviewer 1 - Comment #6: “Line 281. The authors mention a high-precision turntable was used. Can\nthey provide make andmodel?Most turntables are very inaccurate and cannot provide enough accuracy\nonthetilttocalibrate/measuresensitivityusingthetiltmethod.”\nReviewer 1 – Comment #6 – Response: Weappreciatethereviewer’s comment. Weusedthe\nDP300 turntable from Dantsin-RPI Inc., as shown in Fig.R15a, which can achieve the highest\nangular positioning accuracy of 0.5''. As shown in the schematic Fig.R15b, this positioning\naccuracy is approximately equivalent to the calibrate sensitivity of delta_a = g*(1 – cos(5'')) ≈ 2.9\nnGal when the sensitive structure rotates to a position where the sensitive axis is parallel to the\ndirection of gravity. Even when the sensitive structure rotates to a position where the sensitive\naxis is perpendicular to the direction of gravity, the 0.5'' angular positioning accuracy is\napproximately equivalent to the calibrate sensitivity of delta_a = g*(sin(5'') – 0) ≈ 2.4 mGal.\nConsidering that our gravimeter primarily operates with the sensitive axis parallel to the direction\nof gravity, and the tilt method is only used to calibrate its displacement-acceleration sensitivity,\nwe can conclude that the turntable's angular positioning accuracy meets the calibrate/measure\nsensitivityrequirementsofthetiltmethod.\nand the mounted device. b Schematic of the relationship between the variation of acceleration and the\nangularpositioningaccuracy.\nChangesinthemanuscript.\n-Line 352 “The gravimeter was fixed on the high-precision turntable (Model DP300, Dantsin-RPI\nInc.).”\n-Line 357 “More specifically, a change of the angle of 0.0004° corresponds to the variation of\nthe gravitational acceleration component applied to the sensitive axis of g*(sin(0.0004°) – 0) ≈ 7\nmGalat theangle of 0° andg*(1–cos(0–0.0004°)) ≈ 24nGalat 90°.”\nReviewer 1 - Comment #7: “Line 295. Can the authors explain why the settling time of minutes is needed for measurement of each angle? Is it drift, or\nother measurement artifact or use of other equipments?”\nReviewer 1 – Comment #7 – Response: We appreciate the reviewer’s constructive comment.\nIn our turntable tilt experiments, called quasi-static measurement in the original manuscript, we\nselected to pause at each angle for 1 minute, more specifically 30 seconds for stabilization and\n30secondsformeasurement,bothconsideringmeasurementdurationandstabilitytime.\nRegarding the stability time, each rotation of the turntable requires approximately 30 seconds to\nstabilize, as depicted in Fig.R16b. Averaging the datawithin the 1-minute (excludingthe initial 30\nsecondsforturntablestabilization)wassolelyforplottingthedisplacement-accelerationresponse\nin the original Fig.4e. As shown in Fig.R16b, within the 30 s, the time-domain output curves at\neachacceleration point are highlystableafter stabilization, with standard deviations from 6nm to\n14nm,indicatingvirtuallynodrift.\nRegarding the measurement duration, the tilt method measurement requires a larger number of\npoints to achieve enough measure resolution of sensitivity in such a large range of displacement\nchange. For example, we measured 595 points in the range of 10 Gal, as shown in the original\nFig.4. The number of points is much larger than what we tested in the optical displacement\nmeasurement, refer toResponse #4. Thetestingtime perpoint also needstobecorrespondingly\nreduced; otherwise, the total duration would become unbearable. Therefore, the overall\nmeasurement time per point was selected as1 minute, significantly shorterthan 5 minutes. Even\nso, measuring the displacement-acceleration response of an F-AS-based structure in a range of\n10Galstillrequiresaroundtenhours.\nFig.R16-Response (for Reviewer 1 – Comment #7). a Results of the turntable tilt experiment, with red\nseries representing the results of the F-AS-based sensing units. The scatter represents the experimental\nresults, and the dashed and solid lines represent the simulation and linear fitting of the experimental\nresults.bTime-domaindataforsomeinput accelerationpointsinoneminute.\nIn the revised manuscript, we added the time-domain data plot corresponding to one point in the\nacceleration-displacement curve (see revised Fig. 4). Additionally, we included some\nexplanations in the main text on why 1 minute was chosen as the measurement time of each\nangleforthetiltmethod.\n-Revised Fig.4e.\n-Line368“Thesystem was pausedat eachangle forone minutetoacquirethestatic response.”\n-Line 372 “Within the 30 seconds, the time-domain output curves at each acceleration point are\nhighly stable after 30-second stabilization, with standard deviations from 6 nm to 14 nm,\nindicatingvirtually nodrift.”\nReviewer 1 - Comment #8: “Fig 4d. How do the authors account for the change of bandwidth across\nthe dynamic range? The frequency changes from 1.3Hz-2.6Hz between 0-5Gal. This means that the\nfrequencycomponents/noisewillbemuchhigherwhenmeasuringathigherstaticacceleration.”\nReviewer 1 – Comment #8 – Response: We appreciate the reviewer’s constructive comment\nand suggestion. Our interpretation restson threepillars: (1) the definitionof “dynamic range” and\n“mechanical sensitivity” in the original manuscript, (2) the frequency changes of the F-AS\nstructure,and(3)howwe enhancethe“dynamicrange”.\n(1)thedefinitionof“dynamicrange”and“mechanicalsensitivity” intheoriginalmanuscript\nFirst, we apologize for the use of imprecise definitions of “dynamic range” (or “working range”)\nand “mechanical sensitivity” in the original manuscript. In the original manuscript, we defined the\n“dynamic range” as the range where the R-squared value of the linear fit of the acceleration-\ndisplacementcurve isgreater than0.99. The “mechanical sensitivity” was definedasthe slopeof\nthelinearfitoftheacceleration-displacementcurvewithinthe5 Galrange.Thedefinitionsindeed\nhad some ambiguity. For example, some researchers defined the dynamic range as the range\nwith a variation in resonant frequency smaller than around 0.5 Hz [Nature, 2016, 531(7596): 614-\n617; Microsystems & Nanoengineering, 2019, 5(1): 45]. In the revised manuscript, we have\nendeavored to avoid using “dynamic range” and “mechanical sensitivity”, instead using “linear\nrange of acceleration-displacement response” and “acceleration-displacement sensitivity”. When\nthetermsmustbeused,wehaveclearlydefinedtheirfirstoccurrencestoavoidambiguity.\n(2)thefrequencychangesoftheF-ASstructure\nRegarding the frequency changes, we completely agree with the reviewers' comments that\nchanges in frequency can indeed affect noise and sensitivity. The reason our structure shows a\nfrequency change from 1.3 Hz to 2.6 Hz between 0-5 Gal is because we set the frequency\nminimum at the angle of 90°, as shown in original Fig.4d and Fig.R17a. This made the\nfrequency response like a V-shape curve and the acceleration-displacement response slightly\n37\nnonlinear, as shown in Fig.R17b. In the revised manuscript, we slightly adjusted the design by\nsecond-stage optimization to make the frequency minima located at the angle of approximately\n86.8° and 93.2°. Fig.R17c depicts the frequency response versus ∆a and angle from horizontal\nfor both simulation and experimental results. It indicates that this measure can suppress the\nfrequency change to within 0.7 Hz in the range of 5 Gal. The R-squared value of the linear fit of\nthe acceleration-displacement curve within the range of 5 Gal was improved to 0.9985, as\nshown in Fig.R17d. The expense is that the resonant frequency at the angle of 90° is slightly\nincreased to 1.6 Hz. However, due to the significant improvement in linearity within the 5 Gal\nrange, the acceleration-displacement sensitivity has not decreased but remained at a high value\nof 95.3674 μm/Gal. Overall, by regulating the angular position of the frequency minima, we can\neffectively reducethefrequencychangewithinthe5 Galrange.\nFig.R17-Response (for Reviewer 1 – Comment #8). a Experimental and simulation results of the V-\nshaped frequency versus the angle from horizontal and ∆a of the original F-AS. b Experimental and\nsimulation results of the W-shaped frequency versus the angle from horizontal and ∆a of the modified F-\nAS with expanded linear range. c Experimental and simulation results of the acceleration-displacement\nresponse of the original F-AS. d Experimental and simulation results of the acceleration-displacement\nresponseofthemodifiedF-AS.\n38\n(3)howwe enhancethe“dynamicrange”\nAs mentioned above, adjusting the angular position of the frequency minima can effectively\nenhance the linear range or linearity of the acceleration-displacement response, as illustrated in\nFig.R17. This is a measure that we have implemented in the revised manuscript. Another\nmeasure is to use electromagnetic force to maintain the structure in a force balance state at the\nworking point, such as an angular position of 90°, through feedback. This is also a part of our\nupcomingwork.\nAs shown in Fig.R18, we have designed and manufactured an electromagnetic force-driven\nfeedback system to achieve closed-loop control for the F-AS-based gravimeter. Of course, the\nfeedback mechanism is part of our future work and not a major concern of this paper. In the\nrevised manuscript, we have briefly mentioned how possible methods could enhance the\ngravimeter'sdynamicrange.\nFig.R18-Response (for Reviewer 1 – Comment #8). Schematic of the electromagnetic force-driven\nfeedback system, including magnetic conductors, permanent magnets, micromachined metal coils, and\nstructuralcomponents.\nIn conclusion, we really appreciate our reviewer’s remark on the frequency change across the\n“dynamic range”. Wehopetheabove responseadequatelyaddressesyour comments.\nChangesinthemanuscript.\n-Any “dynamic range” and “mechanical sensitivity” are replaced by “linear range of acceleration-\ndisplacementresponse” and “acceleration-displacement sensitivity”\n39\n-Revised Fig.4d.\n-Line 244 “The second-stage design was performed to adjust the anti-springs locally to address\ntheissue of relatively low linearity and highlocal stress. As shown in Fig. 2c inset, thefrequency\nof the structure obtained by global optimization versus operating angle, or acceleration, is a V-\nshaped curve with its frequency minimum located at the angle of 90°, making the linearity poor\nwithin ±5Gal.”\n-Line 252 “The second-stage optimization aims to transform the V-shaped frequency response\ninto a W-shaped response, locating frequency minima at around86.8° and 93.2° to enhancethe\nlinearity. In addition, the resonant frequency at 90° should be less than 1.9 Hz, and the\nmaximumstressis strictlyconstrained tobe within 300MPa.”\n-Line 374 “The fitted acceleration-displacement sensitivity of these two units is measured to be\n95.3674 µm/Gal and 40.5394 µm/Gal, respectively, with linearity of 0.9985 and 0.9919, all\npassing through the robustness analysis. The static experimental results indicate that the\noptimized F-AS-based structures exhibit unprecedentedly low resonant frequency, high\nacceleration-displacement sensitivity of more than 95 µm/Gal, and reasonable linearity, which\nfurtherdemonstratethesuperiorityof multi-stagedirectionaldesignand theoptimized F-AS.”\n…\nReviewer 1 - Comment #9: “Fig 4e. One of the reason the FEA and the experimental results are not\nmatched could be that in the experiment, the non tilting axis is not completely orthogonal to the gravity.\nThis would result in some cross-coupling of gravity in the non-sensitive axis into the sensitive axis. This\ncross-couplingstudymustbedonetobetterunderstandtheperformanceoftheaccelerometer.”\nReviewer 1 – Comment #9 – Response: We appreciate the reviewer’s constructive comment\nand suggestion. Westrongly agree that cross-axis coupling can introduceerrors in experimental\nresults, especially in tilt experiments where input from the non-sensitive axis always exists. In\nthe following, we will discuss the cross-coupling of the F-AS structure in detail. Herein, the y-\ndirectionisthesensitive axis of thestructure.\nFirstly, because the optical grating cavity-based displacement readout is insensitive to\ndisplacements in the x-direction, any displacement in the x-direction, whether it occurs or not,\nhaslittle impacton thereadout of thegravimeter. Thereasoncanbe illustratedin Fig.R19a. The\ngrating lines are parallel to the x-direction, and the laser spot is much smaller than the length of\nthe grating lines. The movement of the grating lines on the proof mass in the x-direction does\nnot affect the cavity output as long as the laser spot remains within the grating area. It is also\n40\ndemonstratedby theFDTD-based optical simulation, as shown in Fig.R19b. Therefore, it can be\nconsideredthatthecross-coupling of thegravimeter mainlyoriginates fromthe displacementsof\nthe proof mass in the y and z-directions caused by accelerations along the non-sensitive axes,\nincluding thex and z-directions.\nFig.R19-Response (for Reviewer 1 – Comment #9). a Schematic of the difference between the optical\ngrating cavity with and without an x-axis displacement of grating G2. b Simulation result of the light\nintensityversusthex-axisdisplacementofgratingG2.\nRegarding the cross-axis sensitivity for x-directional acceleration, we used FEA to analyze the\ndeformation of the F-AS structure with an x-directional acceleration when the structure is at the\nangle of 90° from the horizontal. Fig.R20b presents the displacement of the proof mass in both\nthe z-direction and y-direction when applying acceleration in the x-direction, represented by the\nred curve and blue curve. As shown in the red curve (almost invisible), applying acceleration in\nthe x-direction almost did not cause displacement of the proof mass in the z-direction due to the\nhighsymmetryin thez-direction. It is obvious that thedisplacement in the y-directionis also very\ntiny if the same vertical axis scale as in Fig.R20a is used. By comparing the curve with that in\nFig.R20a, we can derive the curve of cross-axis sensitivity for the x-directional acceleration, as\nshown in Fig.R20d. It can be seen that the cross-axis sensitivity due to x-directional cross-\ncoupling of gravity is smaller than 0.2% within the range of 200 Gal, which is an acceptable\nresult foranultra-sensitive structure.\nRegarding the cross-axis sensitivity for z-directional acceleration, we also present the results of\nthe input acceleration in the z-direction on the displacement of the proof mass, as shown in\nFig.R20c. Acceleration in the z-direction almost did not cause displacement of the proof mass in\nthe y-direction because of the symmetry, as demonstrated by the blue curve. The z-direction\ndisplacement versus the input acceleration in the z-direction is shown in the red curve in\nFig.R20c. The curve of cross-axis sensitivity for the z-directional acceleration can be obtained in\nFig.R20e. The cross-axis sensitivity due to z-directional cross-coupling of gravity is also smaller\n41\nthan 0.2% within the range of 200 Gal. This is primarily due to the significant difference in\nstiffness in the y and z-directions, resulting from the etching aspect ratio of 400:30. The cross-\naxis sensitivity for z-directional acceleration can be further reduced by increasing the etching\naspect ratio. In addition, the gravimeter is mounted on a turntable, with the turntable plane\nperpendicular to the ground, so the rotation of the turntable almost does not cause acceleration\nin thez-direction.\nFig.R20-Response (for Reviewer 1 – Comment #9). Simulation results of the proof mass displacement\nin the y and z-directions caused by accelerations along different axes a y-directional displacement of the\nproof mass as a function of the applied acceleration in the y-direction. b Displacement in the y and z-\n42\ndirectionscausedby thex-directionalacceleration. cDisplacementintheyandz-directions causedbythe\nz-directional acceleration. d Cross-axis sensitivity for the x-directional acceleration. e Cross-axis\nsensitivityforthez-directionalacceleration.\nFig.R21-Response (for Reviewer 1 – Comment #9). a-f Mode shapes and frequencies of the first six\nmodes of themechanical sensing unit at its frequencyminimaangles (86.8°and 93.2°). g-lMode shapes\nand frequencies of the first six modes of the mechanical sensing unit at the angle of 90° (nominal\noperatingcondition).\nFig.R21 shows the mode shapes and frequencies of the first six modes of the F-AS-based\nstructure at its frequency minima angles and at 90°. For the lowest frequency state, it can be\nseen that the first mode is the primary mode of motion in the y-direction, approximately at 1.20\nHz, and the second mode is the mode of motion in the z-direction, with a frequency of 18.90 Hz,\nmore than 15 times the difference from the first mode. The third and fourth modes are rotations\naround the y and x-directions, with frequencies of 34.00 Hz and 56.20 Hz, respectively. Of\ncourse, the rotations would bring about readout change of the optical cavity, but the large\ndifferencein frequencymeanstheresultingdeformation is actuallyvery small. The sixth modeis\nthe rotation around the z-direction, with a frequency of 83.11 Hz. For the nominal operating\ncondition, the first six resonant frequencies are 1.61 Hz, 21.35 Hz, 38.45 Hz, 64.67 Hz, 66.79\nHz, and94.56Hz, respectively. Theycorrespondtothemotion in they-direction,motion in thez-\ndirection, rotation aroundthe y-direction, motionin thex-direction, rotation aroundthe x-direction,\nand rotation around the z-direction. Let us perform a simple semi-quantitative calculation via\nfrequencies about the cross-axis sensitivity under nominal operating conditions. Because the\nfundamental frequency of the structure is very low (~1.6 Hz), the frequency closest to the\nsensing mode is more than 13.3 times different from it, corresponding to a sensitivity difference\nof morethan132 timesbythesimplea =ω2 ×d relationship.\nOverall, the proposed F-AS structure exhibits a reasonably small cross-coupling, which we\nconsider to be satisfactory for typical applications of gravimeters that are vertically mounted and\nhave a limited measurement range. The differences between the FEA and experimental results\nin Fig.4e may not only be attributed to the aforementioned impact of cross-coupling but also to\nthe near-zero stiffness effects and manufacturing errors of the F-AS structure. Nevertheless, the\ncorrelation between the FEA and experimental results reaches 0.9984, which we think is a\ncommendable value. We have included the above cross-coupling study in the revised\nSupplementaryInformation andadded somediscussionin therevised manuscript.\nChangesinthemanuscript.\n-Revised Fig.4f.\n-Line 264 “For the nominal operating condition, the first six resonant frequencies are 1.61 Hz,\n21.35 Hz, 38.45 Hz, 64.67 Hz, 66.79 Hz, and 94.56 Hz, respectively. They correspond to the\n44\nmotion in the y-direction, motion in the z-direction, rotation around the y-direction, motion in the\nx-direction, rotation around the x-direction, and rotation around the z-direction, as shown in Fig.\n2f. Because the fundamental frequency of the structure is very low (~1.6 Hz), the frequency\nclosest to the sensing mode is more than 13.3 times different from it, corresponding to a\nsensitivitydifferenceof morethan 132 timesbythesimplea = ω2 × d relationship. Therefore,the\ncross-axis sensitivity of the structure can be considered small, which is satisfactory for typical\napplicationsof gravimetersthat arevertically mountedand have alimitedmeasurement range.”\nReviewer 1 - Comment #10: “Fig 5c Why are there two resonances in the plot? One at 1.3Hz and\nanotheratabout2/3Hz?Whereisthiscomingfrom?Itneedstobediscussed.”\nReviewer 1 – Comment #10 – Response: Weappreciatethereviewer’sconstructivecomment.\nAnother resonance is the earth tremor peak. The earth tremor peaks are always at 0.2-0.4 Hz\nand 2-4 Hz [Journal of Volcanology and Geothermal Research, 2003, 119(1-4): 161-187; Journal of\nGeophysical Research: Solid Earth, 2003,108, 2506-2520; 2023IEEE 36th International Conference on\nMicro Electro Mechanical Systems (MEMS). IEEE, 2023: 841-844], with an amplitude of 10 to 100\nμGal. The resonance can be clearly observed by the MOEMS gravimeter, indicating the\nsensitivity of our device. We have added an explanation about the second resonance in the\nrevised manuscript.\nChangesinthemanuscript.\n-Line 410“theEarthtremor peakcanbe observedat around0.2Hz andaround2-3Hz”.\nReviewer 1 - Comment #11: “Line 376. The authors mention that the computing cost is reduced\nsignificantly using this algorithm. However, there is no direct comparison shown between the different\ncomputing costs. Please show the quantitative improvement of the computing costs if you are claiming\nsuchanimprovementandhowyouwouldcalculatesuchanimprovement.”\nReviewer 1 – Comment #11 – Response: We thank the reviewer’s constructive comment and\nsuggestion. More specifically, 3D simulation of the F-AS response requires a time of one hour\nand 52 minutes, and 2D simulation only requires 55 seconds. The time difference reaches over\ntwo ordersof magnitude.Therefore,we supposethatit issafetoconclude thestatement.\nChangesinthemanuscript.\n-Line 470 “More specifically, 3D simulation of the F-AS response requires a time of one hour\nand52 minutes,and 2D simulationonlyrequires55 seconds.”\ngraduallyincreasedbecauseofthepath-dependencyofthesolution.Howdoestheaccelerometerreactto\nshocks?Whatthisevertriedinanexperiment?”\nReviewer 1 – Comment #12 – Response: We thank the reviewer’s constructive comment and\nsuggestion. In the original manuscript, our mention of “the acceleration needs to be gradually\nincreased” referred to the simulation process, as depicted in Fig.R22. It is necessary to\nincrementally increase the load acceleration (or force) using small load steps to ensure the\nsimulation converges and outputs accurate results. This does not mean that acceleration must\nbe gradually increased during actual measurements. We apologize for the imprecise wording in\nthe original manuscript. This issue has been corrected in the revised manuscript, where we\nhave clarifiedthat thestatement pertainsspecifically tothesimulation.\nFig.R22-Response(forReviewer1–Comment#12).SimulationprocessoftheF-AS.\nRegarding the shock resistance, we have taken into consideration the ability to protect it from\nshocks by designing the limit mechanisms and packaging. Under the constraints of the\n46\npackaging and limit mechanisms, the proof mass has a motion space of x: ±200 μm, y: +5700\nμm, -1450 μm, z: +100 μm, -200 μm. Fig.R23a, c, e, and g show the simulated response and\nmaximumstress of thepackagedstructureunder accelerations of5 gin thex, y,and z-directions.\nHerein, we simulated the effect of acceleration in only one direction for the x and z axes\nbecauseof itshighsymmetryin thesetwo axes. It canbe observedthat due tothelimitingeffect,\nthe maximum stress of the structure remains 343.273 MPa when subjected to 5 g acceleration\nin all three directions, indicating that the structure can withstand at least 5 g of load. When the\napplied acceleration is increased to 100 g, the structural stress, due to the limiting effect,\nremains at a maximum of 473.715 MPa, not exceeding the allowable stress of silicon, as shown\nin Fig.R23f. In addition, the gravimeter generally does not experience particularly large shocks\nduring transportation and use, and is usually protected by vibration isolation measures.\nTherefore,it canbe consideredthatthestructurecan copewithusual shocks.\n47\nstress of the packaged structure under accelerations of +5 g in the x-direction, +5 g and -5 g in the y-\ndirection, and +5 g in the z-direction. b, d, f, h Simulated response and maximum stress of the packaged\nstructureunderaccelerations of +100ginthe x-direction, +100gand -100ginthey-direction, and+100g\ninthez-direction.\nIn fact, we also conducted some semi-quantitative validation experiments. For instance, by\napplying an acceleration of more than ±1 g to the packaged structure through inversion, we\nobserved that the structure continued to work without any damage, as demonstrated in\nSupplementaryVideo 2.\nWe have included the above discussion regarding shock resistance in the Supplementary\nInformation.\nChangesintheSupplementaryInformation.\n-SupplementaryNote9. Shock resistanceanalysis.\n-SupplementaryVideo 2.\nReviewer 1 - Comment #13: “Fig6. Can the authors mention how they packaged the device since that\nhasasignificantparttoplayinhighresolutiongravimeters?”\nReviewer 1 – Comment #13 – Response: We thank the reviewer’s constructive comment. A\ndetailed description of the packaging has already been provided in Response #1. The\nschematic of the compact packaging and the corresponding process flow is shown in Fig.R3. In\nthe revised manuscript, we have included a brief introduction of packaging and related\ndiscussions, with the remaining content placed in the Supplementary Information. We also\nreplacedtheoriginal Fig.6withthefollowing Fig.R24.\npackaging process. a1 A glass wafer. a2 Metal patterning of the grating on the glass wafer. b1 A silicon\nwafer of the mechanical sensing unit. b2 Metal patterning of the grating on the silicon wafer. b3\nLithography patterning of the mechanical sensing unit. b4 Deep reactive ion beam etching of the\nmechanical sensing unit. b5 Release of the mechanical sensing unit. b6 Processed mechanical sensing\nunit. c1 A glass wafer. c2 Deep wet etching of the glass wafer. d1 A silicon wafer of the substrate. d2\nPlatinum patterning on the silicon wafer. d3 Deep reactive ion beam etching of the confinement trench. e\nLow-stressbonding of wafer2and wafer 4. fLow-stress bonding of wafer3andthe bonded wafer 2+4. g\nLow-stressbondingofwafer1andthebondedwafer2+3+4.\nChangesinthemanuscript.\n-Revised Fig.6.\n-Line 500-523 “The manufacturing and packaging process is illustrated in Fig. 6…. These four\nwafers were bonded by a low-stress bonding technique (Supplementary Note 6) sequentially to\nfinishthepackaging, asshown inFig. 6g-i.”\nChangesintheSupplementaryInformation.\n-SupplementaryNote6. Low-stresswafer-level manufacturingandbonding techniques.\n-SupplementaryNote7. High-precision real-timetemperaturecontrol.\nReviewer 2 – general assessment: The paper reports a new MOEMS gravimeter with a novel free-form\nanti-springs design. Authors claim that their approach allows their device to achieve both high-sensitivity\nas well as a larger dynamic range. They back-up their claims through: a) FEA simulations which show\nhigh-linearity over +/- 5 Gal acceleration pertubrations, b) experimental results which show a self-noise\nfloorof1.7ng/rtHzfortheMOEMSdevice.\nWhile the work of the authors is quite interesting and cool, and it is good to see new approaches to\ndesigning MEMS sensors being introduced since the publication by Middlemiss et al (2016), their\nmotivation, and indeed of others who have recently published in this new and developing field,\nunfortunately, seems to be a bit misplaced. To my understanding, the issues with the previously reported\nMEMSgravimetersarenotrelatedtotheirnoisefloor(Pikeetal(2018)havealready shownanoisefloor\nof 0.25 ng/rtHz for the Imperial's MEMS seismometer), nor related to the dynamic range (Mustafazade et\nal 2020 have reported a dynamic range of +/1 g = +/1 980 Gal for Cambridge's Vibrating beam MEMS\ndevice).AllMEMSgravimeterssufferfrompoordriftrates(MiddlemissetalandPrasadetal2022though\ndemonstratedadriftof100-300uGal/dayforGlasgow'sMEMSgravimeterwhichiscomparabletoCG-5),\nrobustness concerns, and other stability issues. While it might seem that lowering the resonant frequency\nto sub-Hz is a pressing challenge, it is really not. It is well-known that by tuning geomterical parameters\n(launch angle, length of the spring) of an anti-spring (Cella et all 2004), arbitrarily low resonant\nfrequenciescanbeachievedbutthiscomesatacost.Low-resonantfrequenciesbringsthemaximumstress\nwithin the springs dangerously close to the tolerable limits and/or increase the resonator's cross-axis\nsensitivity. These robustness issues mean that higher resonant frequency robust devices are more\npreferable. While this indeed will decrease the acceleration sensitivities, better, more sensitive read-outs\n(liketheoneproposedbythe authors)shouldbeusedtomakeupfor theloss.So,inmyopinion,whilethe\nproposed work is useful, it should not be pitched as a solution to a problem which in reality is not that\ncritical.\nReviewer 2 – Comment #0 – Response 1: We appreciate our reviewer’s remark on the\ninterest of our work. We fully agree with your insightful comment that MEMS gravimeters suffer\nfrom poor drift rates, robustness, and other stability issues. How to balance ultra-high sensitivity\nwith long-term stability and robustness is currently a major concern for the community. In fact,\nwe have undertaken significant efforts to enhance its long-term stability and robustness,\nalthough these improvements were not emphasized in the original manuscript. We apologize for\n51\nthe original manuscript focusing solely on introducing the F-AS structure and multi-stage\ndirectional design methodology. In the following, our interpretation rests on three pillars: (1) the\nlong-term stability, (2) the balance between sensitivity and robustness, and (3) the cross-axis\nsensitivity. Finally, we obtained a drift rate of 153 μGal/day, a noise level of 1.1 μGal/Hz1/2,\nacceleration-displacement sensitivity of over 95 μm/Gal, a maximum stress smaller than 300\nMPa, andcross-axis sensitivitysmaller than0.2%.\n(1)How weenhancelong-term stability\nWeimplementedanintegratedsolutionforthegravimeter,includingcompacthermeticpackaging,\nlow-stress wafer-level manufacturing and bonding, and high-precision real-time temperature\ncontrol. These measures were undertaken to minimize the impact of internal stress and external\ntemperature and pressure, thereby enhancing long-term stability. The stability was reflected in\nthereasonablylowdriftrateandAllandeviation.\nThe schematic diagram and the manufactured photograph of the compact package are shown in\nFig.R25a and b, and the corresponding process is illustrated in Fig.R25c. During the\nmanufacturing process, we employed low-stress wafer-level manufacturing and bonding\ntechniques to reduce factors of structural stress, deformation, hysteresis, and adhesive creep.\nThesespecificswillbefurtherelaboratedoninthefollowing.Wealsofabricatedplatinumresistors\non the substrate of the device using micro-manufacturing technology to achieve high-precision\nreal-timetemperaturecontrol.\npackaged device. c-i Schematic diagram of the manufacturing and packaging process. c1 A glass wafer.\nc2 Metal patterning of the grating on the glass wafer. d1 A silicon wafer of the mechanical sensing unit.\nd2 Metal patterning of the grating on the silicon wafer. d3 Lithography patterning of the mechanical\nsensing unit. d4 Deep reactive ion beam etching of the mechanical sensing unit. d5 Release of the\nmechanical sensing unit. d6 Processed mechanical sensing unit. e1 A glass wafer. e2 Deep wet etching\nof theglasswafer. f1Asilicon waferof the substrate. f2Platinum patterningon thesiliconwafer. f3Deep\nreactive ion beam etching of the confinement trench. g Low-stress bonding of wafer 2 and wafer 4.\nh\nLow-stress bonding of wafer 3 and the bonded wafer 2+4. i Low-stress bonding of wafer 1 and the\nbondedwafer2+3+4.\nThe distribution and dimensions of the resistors were optimized through FEA to ensure that the\nentire mechanical sensing unit maintains a reasonable temperature gradient and stability during\ntemperature control, as shown in Fig.R26a and b. The ambient temperature was set to 291.5 K\n53\n(very close to the temperature of our cave laboratory), and the target temperature was set to 294\nK. Inreality, ourtemperaturecontrolisbasedonfuzzyPID[Micromachines,2021,12(12):1511] with\noptimized parameters. Fig.R26c shows that it realized a temperature stability of smaller than 3.3\nmK, which is measured by the temperature measurement resistor. The inset shows the\ntemperature data over more than 8 hours of the measurement resistor and ambient temperature.\nIt is evident that temperature control can effectively suppress long-term drifts in ambient\ntemperature.\nFig.R26-Response (forReviewer2 – Comment #1). aSimulation results of the temperature distribution\nof the mechanical sensing unit and b the packaged device. c Temperature measured by the resistor, and\ntheambienttemperaturewasaround291.5K.Theinsetpresentsthemeasuredtemperatureover8hours,\nasacomparisonwiththeambienttemperaturemeasuredbyathermistor.\nRegarding the manufacturing, we utilized a monolithic single-crystal silicon wafer to fabricate the\nmechanicalstructuretominimizeinternalstress.GlassWaferswiththermalexpansioncoefficients\ncomparable to single-crystal silicon (Scott AF 32) were used to accomplish wafer-level packaging\nto reduce interface stress. Conventional wafer bonding can easily result in stress at the bonding\n54\ninterface due to applied pressure and high temperature. We developed a low-stress bonding\ntechnique, as shown in Fig.R27a, to achieve wafer-level low-stress packaging. Step 1, we used\nour self-developed automated dispensing system, as illustrated in Fig.R27b, to ensure the\nconsistency of the adhesive; Step 2, a clamping alignment system, as shown in Fig.R27e, was\nused to align the two bonded wafers, see Fig.R27f and g; Step 3, the adhesive was cured by a\nspecific stepwise curing process. The bonding technique can be achieved by applying a certain\nsmallamountofpressureatloworevenroomtemperatures.\nAs shown in Fig.R27c and d, several inverted pyramid arrays with a depth of around 30 μm were\nfabricatedtoserveasadhesivereservoirsforsiliconwafers(severalholesforglasswafers),which\nfacilitated stress homogenization during bonding. In addition, we selected a suitable adhesive\nthrough experiments with adhesive wafer samples and developed a specialized process to\nbalancethecuringtemperature,achievingoptimalperformancestability.\n55\ntechnique. b Photograph of the self-developed automated dispensing system. c Inverted pyramid arrays\nserving as adhesive reservoirs. d Photograph of the inverted pyramid arrays with a depth of around 30\nμm. e Photograph of the self-developed clamping alignment system. f Schematic diagram of the\nalignmentoftwowafers.gSchematicdiagramofthecontactoftwowafers.\nThe adhesive we selected was a self-developed composite adhesive. The curing agent is an\nimidazole compound, the filler is calcium carbonate, and the matrix is epoxy resin, with a mass\nratio of 20:20:60. To evaluate the mechanical properties of the adhesive, or relaxation modulus,\nwe used a specialized mold to fabricate adhesive specimens for measuring the relaxation\nmodulus, as shown in Fig.R28a. The relaxation modulus parameters at different temperatures\nwere tested using a high-precision dynamic mechanical analyzer (DMA), as shown in Fig.R28b.\nThetestresultsatroomtemperatureareshowninFig.R28c.\nFig.R28-Response (for Reviewer 2 – Comment #1). a Photograph of the tested adhesive specimen. b\nPhotograph of the relaxation modulus test using DMA. c Measured relaxation modulus as a function of\n57\ntime at room temperature. d Stepwise curing process containing five steps. e Photograph of the tested\nadhesivesiliconwaferanditsstressdistribution.\nRegarding the adhesive curing process, we developed a stepwise curing process that considered\nboth curing temperature and degree of cure after multiple rounds of optimization. As shown in\nFig.R28d, it contains five steps. The first step is to raise the temperature to T , taking a time of t ;\n1 1\nthe second step is to maintain the temperature at T until time t ; the third step is to raise the\n1 2\ntemperature to T , at an end time of t ; the fourth step is to maintain the temperature at T until\n2 3 2\ntimet ;thelaststepisthenaturalcooltoroomtemperature,andthefinalendtimeist .Thetested\n4 5\nadhesivesiliconwafercuredbythisprocesshadastressdistributionstate, asshowninFig.R28e,\nwith a maximum stress of 0.02532Mpa, which proved that this process can achieve low-stress\ncuring.\ngravimeter and the reference gravimeter. The red line represents the MOEMS gravimeter, the blue line\nrepresents the reference gravimeter, PET, and the orange line represents the noise data of the MOEMS\ndevice. The red and the orange series were recorded at a sampling rate of 20 Hz. The green series\nrepresent data that were recorded at a sampling rate of 0.1 Hz. At 10-5 Hz, the diurnal and semi-diurnal\ncomponents of the Earth tides are visible; the Earth tremor peak can be observed at around 0.2 Hz; and\n59\nthe resonant peak of the device is located at around 1.6 Hz. b Raw data of the tide measurement with a\nsampling rate of 0.1 Hz. c Allan deviation of the data in b. d Raw data with the removal of the linear drift.\neAllan deviationof the series ind. f Comparisonof Earth tides measuredbythe MOEMSgravimeter and\nthe PET gravimeter. The red line is obtained from the MOEMS gravimeter, and the blue line is obtained\nfrom the reference gravimeter. The correlation coefficient between these two sets of data is 0.9218. g\nAllandeviationofthedataoftheMOEMSgravimeterinf.\nFig.R29 shows the noise floor, drift data, and Allan deviation results of our prototype MOEMS\ngravimeterafterimplementingallimprovements.Datawererecordedbyafastsamplingrateof20\nHz and a slow sampling rate of 0.1 Hz in order to observe the responses of Earth tremor peaks\nand tides. As shown in Fig.R29a, the noise floor of the device reaches down to 1.1 μGal/Hz1/2 at\n0.5 Hz, and the ultimate noise floor of the device can be around 0.4 μGal/Hz1/2; the diurnal and\nsemi-diurnal components of the Earth tides are visible at 10-5 Hz; the Earth tremor peak can be\nobservedataround0.2Hz.AccordingtotheAllandeviationresults,abiasinstabilityof6μGalcan\nbe obtained with an integration time of 600 s. It is also observed that the device achieved\nreasonably good long-term stability, with a drift of 153 μGal/day. As a comparison, the drift of our\ndevice is comparable to or even better than other MEMS gravimeters, which is around 100-300\nμGal/day [Nature,2016,531(7596):614-617;Scientificreports,2022,12(1):13091]. Overall, our device\nachievesthehighestreportedacceleration-displacementsensitivity(orlowestresonantfrequency)\non the chip scale and exhibits excellent noise floor and long-term stability. This performance is\nattributedtoourdesignmethodology,aswellasthepackagingandmeasuresmentionedearlier.\nThis part was not extensively introduced in the original manuscript because we previously\nregarded the long-term stability improvements as a relatively independent work, and the original\nmanuscript focused on the innovation of the design and freeform anti-springs. However, we\ngreatly appreciate the reviewers for raising these concerns. We have added the discussion of the\nabovecontentintoboththerevisedmanuscriptandtheSupplementaryInformation.\n(2)How webalancethesensitivityand robustnessviamulti-stage directional design\nSimilarly, we fully agree with the reviewer's comment regarding the conflict between resonant\nfrequency/sensitivity and maximum stress/robustness. Indeed, tuning geometrical parameters\ncan arbitrarily lower the resonant frequency, but doing so typically increases stress and cross-\naxis crosstalk, thereby reducing robustness. However, our multi-directional design was, to some\nextent,aimedataddressingthisissue.\nincludingthetargetsandconstraintsoftwostages.\nAs shown in Fig.R30, our multi-stage directional design maintained the balance between\nsensitivity and robustness through three points. First, it leveraged the free-form geometry to\nachieve a large design parameter space with a relatively small number of control points (or\nvariable parameters). A larger parameter space implies a higher possibility of finding solutions\nthat balance sensitivity and stress robustness. Secondly, we used the local adjustability of B-\nspline-basedfree-formgeometrytodecomposethecomplexandexpensiveoptimizationproblem.\nobjective, herein is resonant frequency/sensitivity, by relaxing constraints on secondary\nobjectives such as stress. The second stage involves local adjustments to ensure the overall\nstructural performance meets the stress robustness constraints without significantly reducing the\nfirst-stage performance. The primary and secondary objectives and constraints can be flexible,\nand the total number of stages is not limited to two. Local adjustability is indispensable because\ntheprimaryobjectivesof thefirst-stageoptimizationwouldundergosignificant changeswithoutit.\nThirdly, we utilized AELPSO to drive the multi-directional design process. This ensures that the\nentiredesignachievesanacceptableconvergence speedandresult. Ourmulti-directional design\nprovides a methodologyto expand the designparameter space and reducethe complexity of the\noptimizationproblem,finallymakingitfeasibletorunandconverge.\nWe have demonstrated that the multi-directional design helps the structure meet three\nrequirements in the original manuscript: ultra-low resonant frequency of around 1 Hz, maximum\nstress of less than 300 MPa, and a reasonable linear range of acceleration-displacement\nresponse. In the revised manuscript, we have taken the reviewer's suggestion by appropriately\nincreasing the resonant frequency of the F-AS structure at the angle of 90° from horizontal and\nenhancing the linearity of the acceleration-displacement response. The difference is that we\nadjustedthesecondaryobjectivestothefrequencyminimalocatingattheanglesofaround86.8°\nand93.2°. Then,amodifiedF-ASstructurewithafrequencyresponseshown inFig.R31bandan\nacceleration-displacement response shown in Fig.R31d was obtained. It is evident that the\nlinearity of the acceleration-displacement response was enlarged, and the maximum stress was\nstillsmallerthan300MPa. Thecompromiseisaslightlyreducedresonant frequencyattheangle\nof 90°. Even so, the reduced resonant frequency of 1.61 Hz was still the reported lowest\nresonantfrequencyforon-chipstructures.\nshaped frequency versus the angle from horizontal and ∆a of the original F-AS. b Experimental and\nsimulation results of the W-shaped frequency versus the angle from horizontal and ∆a of the modified F-\nAS with expanded linear range. c Experimental and simulation results of the acceleration-displacement\nresponse of the original F-AS. d Experimental and simulation results of the acceleration-displacement\nresponseofthemodifiedF-AS.\n(3)Cross-axis sensitivity\nIn the following, we will discuss the cross-coupling of the F-AS structure in detail. Herein, the y-\ndirectionisthesensitive axis of thestructure.\nFirstly, because the optical grating cavity-based displacement readout is insensitive to\ndisplacements in the x-direction, any displacement in the x-direction, whether it occurs or not,\nhas little impacton thereadout of thegravimeter. The reason can be illustrated in Fig.R32a. The\ngrating lines are parallel to the x-direction, and the laser spot is much smaller than the length of\nthe grating lines. The movement of the grating lines on the proof mass in the x-direction does\nnot affect the cavity output as long as the laser spot remains within the grating area. It is also\n63\ndemonstrated by the FDTD-based optical simulation, as shown in Fig.R32b. Therefore, the\ncross-coupling of the gravimeter mainly originates from the displacements of the proof mass in\nthe y and z-directions caused by accelerations along the non-sensitive axes, including the x and\nz-directions.\nFig.R32-Response (for Reviewer 2 – Comment #1). a Schematic of the difference between the optical\ngrating cavity with and without an x-axis displacement of grating G2. b Simulation result of the light\nintensityversusthex-axisdisplacementofgratingG2.\nRegarding the cross-axis sensitivity for x-directional acceleration, we used FEA to analyze the\ndeformation of the F-AS structure with an x-directional acceleration when the structure is at the\nangle of 90° from the horizontal (nominal operation condition). Fig.R33b presents the\ndisplacement of the proof mass in both the z-direction and y-direction when applying\nacceleration in thex-direction, represented bythe redcurve and blue curve. As shown in thered\ncurve (almost invisible), applying acceleration in the x-direction almost did not cause\ndisplacementof theproof massin thez-direction duetothe highsymmetryin thez-direction. Itis\nobviousthat thedisplacement inthe y-directionis alsovery tiny if thesamevertical axisscale as\nin Fig.R33a is used. By comparing the curve with that in Fig.R33a, we can derive the curve of\ncross-axis sensitivity for x-directional acceleration, as shown in Fig.R33d. It can be seenthat the\ncross-axis sensitivity due to x-directional cross-coupling of gravity is smaller than 0.2% within\ntherangeof 200Gal, whichis anacceptableresult foran ultra-sensitive structure.\nRegarding the cross-axis sensitivity for z-directional acceleration, we also present the results of\nthe input acceleration in the z-direction on the displacement of the proof mass, as shown in\nFig.R33c. Acceleration in the z-direction almost did not cause displacement of the proof mass in\nthe y-direction because of the symmetry, as demonstrated by the blue curve. The z-direction\ndisplacement versus the input acceleration in the z-direction is shown in the red curve in\nFig.R33c. The curve of cross-axis sensitivity for z-directional acceleration can be obtained in\nFig.R33e. The cross-axis sensitivity due to z-directional cross-coupling of gravity is also smaller\n64\nthan 0.2% within the range of 200 Gal. This is primarily due to the significant difference in\nstiffness in the y and z-directions, resulting from the etching aspect ratio of 400:30. The cross-\naxis sensitivity for z-directional acceleration can be further reduced by increasing the etching\naspect ratio. In addition, the gravimeter is mounted on a turntable, with the turntable plane\nperpendicular to the ground, so the rotation of the turntable almost does not cause acceleration\nin thez-direction.\nFig.R33-Response (for Reviewer 2 – Comment #1). Simulation results of the proof mass displacement\nin the y and z-directions caused by accelerations along different axes a y-directional displacement of the\nproof mass as a function of the applied acceleration in the y-direction. b Displacement in the y and z-\n65\ndirectionscausedby thex-directionalacceleration. cDisplacementintheyandz-directions causedbythe\nz-directional acceleration. d Cross-axis sensitivity for the x-directional acceleration. e Cross-axis\nsensitivityforthez-directionalacceleration.\nFig.R34-Response (for Reviewer 2 – Comment #1). a-f Mode shapes and frequencies of the first six\nmodes of themechanical sensing unit at its frequencyminimaangles (86.8°and 93.2°). g-lMode shapes\nand frequencies of the first six modes of the mechanical sensing unit at the angle of 90° (nominal\noperatingcondition).\nFig.R34 shows the mode shapes and frequencies of the first six modes of the F-AS-based\nstructure at its frequency minima angles and at 90°. For the lowest frequency state, it can be\nseen that the first mode is the primary mode of motion in the y-direction, approximately at 1.20\nHz, and the second mode is the mode of motion in the z-direction, with a frequency of 18.90 Hz,\nmore than 15 times the difference from the first mode. The third and fourth modes are rotations\naround the y and x-directions, with frequencies of 34.00 and 56.20 Hz, respectively. Of course,\nthe rotations would bring about readout change of the optical cavity, but the large difference in\nfrequency means the resulting deformation is actually very small. The sixth mode is the rotation\naround the z-direction, with a frequency of 83.11 Hz. For the nominal operating condition, the\nfirst six resonant frequencies are 1.61 Hz, 21.35 Hz, 38.45 Hz, 64.67 Hz, 66.79 Hz, and 94.56\nHz, respectively. They correspond to the motion in the y-direction, motion in the z-direction,\nrotation around the y-direction, motion in the x-direction, rotation around the x-direction, and\nrotationaroundthez-direction.\nLet us perform a simple semi-quantitative calculation via frequencies about the cross-axis\nsensitivity under nominal operating conditions. Because the fundamental frequency of the\nstructure is very low (~1.6 Hz), the frequency closest to the sensing mode is more than 13.3\ntimes different from it, corresponding to a sensitivity difference of more than 132 times by the\nsimple a = ω2 × d relationship. Therefore, the cross-axis sensitivity of the structure can be\nconsidered small, which is satisfactory for typical applications of gravimeters that are vertically\nmountedandhave alimitedmeasurement range.\nIn summary, we fully agree with the reviewer's comment on the importance of stability and\nrobustness for a MEMS gravimeter, which is also the motivation of our work. In the original\nmanuscript, we focused on the free-form geometry and multi-stage directional design and,\ntherefore, did not emphasize the drift rates, stability, and robustness issues. We deeply\napologize for that. In the revised manuscript, we clearly outlined the bottleneck issues in MEMS\ngravimeters and the motivation for this work. We have included a brief description of packaging\nand long-term stability improvement measures. All the aforementioned discussions were\nappended in the Supplementary Information. Besides, the narrative of the original manuscript\nhas been significantly revised. We have focused on how the multi-stage directional design can\nbalanceultra-highsensitivityandstressrobustnesswhile maintaininghighprocessfeasibility.\nChangesinthemanuscript.\n-Weessentially rewrotetheabstract, introduction,results, and methods.\nChangesintheSupplementaryInformation.\n-SupplementaryNote6.Low-stresswafer-levelmanufacturingandbondingtechniques.\n-SupplementaryNote7.High-precisionreal-timetemperaturecontrol.\ngeneral assessment #2: Apart from the motivation issue, the authors need to provide a good context to\ntheiralgorithmicapproachtospringdesign.WhiletheyhavecitedWangetal(2020,2022)astheprevious\nworkonfree-formspringdesign,useofgeneticalgorithmsfordesigningcompliantnon-linearspringswas\nreally started by Jutte et al (2008) (Michigan) and others. While I agree their focus was not to make\nMEMS devices, they did suggest that their approach could be used in MEMS. While I do not expect the\nauthors to cite each and every paper that has been since published, for a high-reputation journal like\nNatureComms,Iexpecttheauthorstoguidethereaderstoafewmainpreviousworks.\nReviewer 2 – Comment #0 – Response 2: We appreciate the reviewer’s constructive\nsuggestions.Inthe revised manuscript,we have not only citedthe work of Juttle et al. [Journalof\nmechanical design, 2008, 130(8): 081403] but also the works of Foli et al. [International journal of\nheat and mass transfer, 2006, 49(5-6): 1090-1099] and Melnyk et al. [Proceedings of MIXDES 2013.\nIEEE, 2013: 352-354] to guide readers to previous research involving the use of metaheuristic\nalgorithmsfordesigningcompliant non-linear springsand MEMS structures.\nChangesinthemanuscript.\n-Line 88 “Previous researchers have already combined metaheuristic algorithms into the design\nof MEMS devices.”\n-reference\ngeneral assessment #3: The experimental results and plots in the papers are a bit confusing and left me\nlooking for details in the text. While I was often able to find the appropriate information, sometimes I did\nnot.Ihaveincludedmoredetailedcommentsonthislater.\nIn summary, while the work is interesting, I am not entirely confident that it significantly advances and\nsolves the persistingissues in the field. I encourage the authors to go through my comments and use them\ntoimproveandmodifythemanuscript.\nReviewer 2 – Comment #0 – Response 3: We appreciate the reviewer’s constructive\ncomments. In the revised manuscript, we have thoroughly revised the presentation of the\n68\nexperimentalresultsand plots,including but not limitedtothefollowing point-to-point responses.\nWe hope we have made it clearer and that the revised manuscript meets the standards of\nNatureCommunications.\nReviewer 2 - Comment #1: “(line#30) The openingsentenceintheAbstractdidnotmakealotofsense,\npleaserephrase”\nReviewer 2 – Comment #1 – Response: We appreciate the reviewer's suggestion. We have\nthoroughly revised the abstract. The opening sentence was rephrased as “MEMS gravimeters,\nhaving found success in applications, still face challenges in maintaining ultra-high sensitivity\nandgoodstability androbustnessin termsof their limitedvolume.”\nReviewer 2 - Comment #2: “(line#34) The sentence “A multi-stage directional designscheme...” in the\nAbstractispackingtoomuchstuffinasinglesentenceandconfusing,pleasesimplify”\nReviewer 2 – Comment #2 – Response: We appreciate the reviewer's suggestion. The\nsentence has been simplified to “A multi-stage directional design scheme is proposed to\nbalancethetrade-offbetween thesensitivity andtherobustnessof theF-AS.”\nReviewer 2 - Comment #3: “(line#37) I am not sure I have seen the use of the term “mechanical\nsensitivity”intheaccelrometer/gravimetercommunity;pleasestickwiththestandardtermsordefineany\nnewtermsproperly”\nReviewer 2 – Comment #3 – Response: We appreciate the reviewer's precious comment.\nWe apologize for the use of the imprecise definition of “mechanical sensitivity” in the original\nmanuscript. More precisely, it should be defined as “acceleration-displacement sensitivity”,\nwhich refers to the displacement of the proof mass to each Gal change in acceleration. The unit\nis μm/Gal. Because the acceleration-displacement curve was not perfectly linear, we used the\nlinear fit of the acceleration-displacement curve within the 10 Gal range to represent\n“acceleration-displacement sensitivity”. In the revised manuscript, we have endeavored to avoid\nusing “mechanical sensitivity”, instead using “acceleration-displacement sensitivity”. When the\nterm must be used, we have provided our clear definition upon their first occurrences to avoid\nambiguity.\nabstractisofthereadout,notofthegravimeter;pleasemakethisclear”\nReviewer 2 – Comment #4 – Response: Thanks for our referee’s comment. Indeed, the\ndisplacement sensitivity is of the readout. The abstract has been thoroughly rephrased. The\nsentence was revised as “It demonstrates a self-noise of 1.1 μGal Hz-1/2 at 0.5 Hz and Earth\ntides measurement ability with the help of the readout achieving a displacement sensitivity of\n1.2pm Hz-1/2.”\nReviewer 2 - Comment #5: “(line#40) TheauthorshaveswitchedbetweennganduGals/Gals toomany\ntimes; when talking about gravimeters avoid using ng as it is not really a unit (though the accelerometer\ncommunityprefersit).Ifthegravityitselfisnotthesameeverywhere,howcanngbeconstant?”\nReviewer 2 – Comment #5 – Response: Thanks for our referee’s comment. The reviewer is\nright that these two are different units. In the revised manuscript, the unit has been unified to\nμGals/Gals.\nReviewer 2 - Comment #6: “(line#49) Relative gravimeters, generally speaking, do not have better\nresolutionthatabsolutegravimeters;pleasecorrect”\nReviewer 2 – Comment #6 – Response: Thanksfor our referee’scomment. Wehave deleted\nthe“thebetter resolution” description.\nReviewer 2 - Comment #7: “Achieveing a resonant frequency belwo 1 Hz has not reamined elusive-\nthereisnopointofmakingadevicethatisdifficulttohandle”\nReviewer 2 – Comment #7 – Response: Weappreciate thereviewer'sconstructive comment.\nWe fully agree with the reviewer that it is unnecessary to pursue a structure that is difficult to\nhandle. We have significantly revised this section to state that achieving chip-scale ultra-low\nfrequencies, such as 1 Hz, and simultaneously maintaining robustness is very challenging. For\nexample, low resonant frequencies always bring the maximum stress within the springs\ndangerouslyclosetothetolerablelimits.\nAs discussed in Comment #0 – Response 1, the multi-directional design was, to some extent,\naimed at addressing this issue. In addition, the designed F-AS-based structure was not difficult\n70\nto manufacture and handle. The designed structure only required an etching aspect ratio of\n400:30 (with a beam width of 30 μm) for manufacturing. This requirement is less demanding\ncompared to the designs by Middlemiss et al. and Liu et al. [Scientific reports, 2022, 12(1): 13091;\nMicrosystems & Nanoengineering, 2019, 5(1): 45; IEEE Transactions on Industrial Electronics, 2023,\n71(6): 6418-6426], which were larger than 20:1. The lenient manufacturing process, combined\nwith compact packaging, can achieve good stability and cross-axis sensitivity, as well as some\nresistance to shocks (see Supplementary Video 2). It also demonstrates that our structure is not\nthatdifficult tohandle.\nChangesinthemanuscript.\n-Line 64 “Regarding acceleration-displacement sensitivity, the biggest problem is designing a\nchip-scale structure with an ultra-low resonant frequency and simultaneously maintaining\nrobustness and stability, which is very challenging due to the size restrictions of the die and the\nmass. For example, low resonant frequencies always bring the maximum stress within the\nspringsdangerouslyclosetothetolerablelimits.”\nChangesintheSupplementaryInformation.\n-SupplementaryVideo2\nReviewer 2 - Comment #8: “(line#71) The authors switch between sensitivity and resolution; please\navoidthisandstickwithone(preferablysensitivity)”\nReviewer 2 – Comment #8 – Response: Thanks for our referee’s comment. We have unified\nthesetwotermsas “sensitivity” in therevised manuscript.\nReviewer2- Comment #9: “(line#72)Boometaldidnotmakeagravimeter”\nReviewer 2 – Comment #9 – Response: Thank you for the reviewer's comment. We\napologize for the lack of precision in our wording. In the revised manuscript, we have changed\nthedescription to“Boom et al. designeda MEMS accelerometer …”.\nReviewer2- Comment #10: “(line#73)ishavingasimplegeometeryanissue?”\nReviewer 2 – Comment #10 – Response: We appreciate the reviewer's constructive\ncomment. Oure previous statement was somewhat imprecise and inappropriate. Simple\n71\ngeometry and limited degrees of freedom are not an issue. The accurate statement should be:\nthe simple geometry and limited degrees of freedom result in a small design parameter space,\nmaking it difficult to find a solution that balances sensitivity and stress robustness, as well as a\nreasonable linear range. More specifically, the parameter space for anti-spring mechanisms is\ncomplex due to their nonlinear responses. Simple geometries, such as arc beams, may have\ndifficulty achieving results that balance multiple performances even when using the same\noptimization algorithm. We take the arc-AS, SF-AS, and F-AS as the illustration. Fig.R35a\ndepicts the convergence plot of the global optimization for three geometries in 200 iterations.\nFig.R35b to d show their convergence plots of sensitivity, R2, and the maximum stress,\nrespectively. The plots demonstrate that the proposed F-AS has the largest design parameter\nspace and performs the best fitness value. Although the linearity and stress of the F-AS are\nslightly inferior compared tothe other two, the significantly superior sensitivity and fitness clearly\nindicatethat F-AScanfinda better andmore balancedsolutionwithin thesameiterations.\nFig.R35-Response (for Reviewer 2 – Comment #10). a Convergence plot of the fitness value of the\nglobal optimization for three geometries. b Convergence plot of the sensitivity of the global optimization\n72\nforthree geometries. cConvergence plot of the linearity of the global optimization for three geometries. d\nConvergenceplotoftheMaximumstressoftheglobaloptimizationforthreegeometries.\nIn the revised manuscript, we have significantly revised the introduction and removed this\nsentence, changing it to “ However, the simple geometry and limited degrees of freedom of\ntheseanti-spring mechanisms result in a small design parameter space, makingit difficult tofind\nasolutionthat balancessensitivityand stressrobustness, aswell asareasonablelinear range.”\nChangesinthemanuscript.\n-Line 78 “However, the simple geometry and limited degrees of freedom of these anti-spring\nmechanisms result in a small design parameter space, making it difficult to find a solution that\nbalancesultra-highsensitivity, stressrobustness,and areasonablelinear range.”\nReviewer 2 - Comment #11: “(line#74) havingmore degrees offreedomis notadvantageous if youcan\ngetthesameresultthroughasimplerdesign”\nReviewer 2 – Comment #11 – Response: We appreciate the reviewer's constructive\ncomment. We fully agree with the reviewer that more degrees of freedom are unnecessary if\ncomparable performance can be achieved with asimpler design. However, as mentioned above,\nfor an anti-spring mechanism, low degrees of freedom results in a small design parameter\nspace and relatively poor adjustability, making it difficult to find a solution to balance multiple\nobjectives, such as ultra-high sensitivity and robustness. Fig.R35 demonstrates that arc-shaped\nand SF-AS beams were unable to obtain comparable performance even with the same\noptimization algorithm at the same number of iterations. These optimization results have similar\nmaximum stress,but theF-ASstructure performstheoptimalsensitivity/resonant frequency.\nFor\nstructures like arc beams to achieve sensitivity comparable to that of the F-AS structure, the\nbeam width would need to be reduced, leading to increased maximum stress and decreased\nrobustness. Of course, it might be possible to find a simpler design with comparable\nperformance given more optimization time, but this would inevitably increase the overall\noptimization time and computational cost. This is why we opted for a freeform geometry beam\nwithhigher degreesof freedom.\nWe apologize for the unclear statement. In the revised manuscript, we have significantly revised\nthe introduction and removed this sentence, changing it to “The simple geometry and limited\n73\ndegreesof freedom result in a small designparameter space, making it difficult tofind a solution\nthatbalancessensitivityand stressrobustness,aswell asareasonable linear range.”\nChangesinthemanuscript.\n-Line 78 “However, the simple geometry and limited degrees of freedom of these anti-spring\nmechanisms result in a small design parameter space, making it difficult to find a solution that\nbalancesultra-highsensitivity, stressrobustness,and areasonablelinear range.”\nReviewer 2 - Comment #12: “(line#80) Yes, the geometric anti-springs are designed sothat they reach\nthe lowest stiffness at 1g (nominal operating condition). Doesn't the proposed F-AS also reach their\nminimum stiffness at 1g? How is this is not loading? How is F-AS then different to a simpler geometric\nanti-spring?”\nReviewer 2 – Comment #12 – Response: We appreciate the reviewer's constructive\ncomment. Indeed, our F-AS structure was also designed to reach the lowest stiffness under\nnominal operating conditions and required loading. From this perspective, the F-AS structure is\nno different from simpler geometric anti-springs. Therefore, thedescription has been removed in\ntherevised manuscript.\nAdditionally, we would like to mention that in the revised manuscript, the F-AS was adjusted by\nthe second-stage optimization to make the lowest stiffness slightly deviate from the nominal\ncondition. As shown in Fig.R36a and b, the adjustment helps to avoid the V-shaped frequency\nresponse, which is the reason behind the limited linear range of acceleration-displacement\nresponse(previously defined as the “dynamic range” or “working range”). Instead, the frequency\nminima were located at the angle of approximately 86.8° and 93.2°, making the frequency\nresponse like a W-shaped curve. This measure can suppress the frequency change to within\n0.8 Hz in therange of 5 Gal and expand the linear range. The R-squaredvalueofthelinear fit of\nthe measured acceleration-displacement curve within the range of 5 Gal improved to 0.9985, as\nshown in Fig.R37d. The expense was that the resonant frequency at the angle of 90° is slightly\nincreased to 1.61 Hz. Even so, the acceleration-displacement sensitivity did not decrease but\nremained at a high value of 95.3674 μm/Gal due to the significant improvement in linearity\nwithin the5 Galrange.\nshaped frequency versus the angle from horizontal and ∆a of the original F-AS. b Experimental and\nsimulation results of the W-shaped frequency versus the angle from horizontal and ∆a of the modified F-\nAS with expanded linear range. c Experimental and simulation results of the acceleration-displacement\nresponse of the original F-AS. d Experimental and simulation results of the acceleration-displacement\nresponseofthemodifiedF-AS.\nOverall, we apologize for the potentially misleading description, and have deleted it in the\nrevised manuscript.\nChangesinthemanuscript.\n-Line 86 “However, anti-springs usually require pre-loading and experience nonlinear stiffness\ntuning during work” to “However, anti-springs usually experience nonlinear stiffness tuning\nduringwork.”\nReviewer 2 - Comment #13: “(line#81) I am not sure where in the ref#15 it has been mentioned that\nnonlinearstiffnesstuningisexperienced”\nreplacedthereferencewith amore feasible one [Microsystems&Nanoengineering,2024,10(1):43].\nReviewer 2 - Comment #14: “(line#83) what do the authors mean by non-stationary conditions? A few\nsentencesagotheauthorsclaimedthatthegeometricanti-springssufferfromsimpledesignbutherethey\nclaimthatthesehavecomplexmorphology-pleaseexplain/correct”\nReviewer 2 – Comment #14 – Response: We thank the reviewer's comment. We apologize\nfor the imprecise description in the original manuscript. The “non-stationary conditions” referred\nto the near-zero stiffness condition of the anti-springs in terms of their nonlinear buckling\nbehavior, as illustrated in state 3 in the revised Fig.1a4. The deformed slender beams, despite\ntheir initially simple shape, will exhibit a complex morphology, or more accurately the\ndeformation, in the low stiffness region in the force-displacement plot. We also provided a\ndetailed explanation of how theanti-springs work in the revised Fig.1 and response toComment\n#18. Indeed, the “non-stationary conditions” or “complex morphology” were inaccurate\nstatements and were not the key points as to why anti-springs require more degrees of freedom\nandlocal adjustability.\nIn the revised manuscript, we have significantly revised the introduction and corrected the\ndescription.\nChangesinthemanuscript.\n-Line 86 “However, the simple geometry and limited degrees of freedom of these anti-spring\nmechanisms result in a small design parameter space, making it difficult to find a solution that\nbalancesultra-highsensitivity, stressrobustness,and areasonablelinear range.”\n-Line 91 “Thus, the local adjustability of free-form geometry is also required to decompose the\nexpensive optimizationproblem of F-ASs, makingit feasibletorunand converge.”\nReviewer2- Comment #15: “(line#90)pushedfurther”\nReviewer 2 – Comment #15 – Response: We thank the reviewer's comment. It has been\ncorrectedas “pushedfurther”.\nReviewer2- Comment #16: “(line#100)R-squaredvalueofwhat?pleaseexplain”\nvalue here refers to the R-squared value of the linear fit of the acceleration-displacement curve\nwithin arangeof 10Gal.\nIntherevisedmanuscript, wehave addedadetailed explanation oftheterm R-squared value.\nChangesinthemanuscript.\n-Line 86 “and R2 is the R-squared value of the linear fit of the acceleration-displacement curve\nwithin therangeof ±5Gal.”\nReviewer2- Comment #17: “(line#104)Howmanyng/rtHz?”\nReviewer 2 – Comment #17 – Response: Wethank the reviewer's comment. It should be 1.7\nμGal/Hz1/2 in the original manuscript. In the revised manuscript, it should be 1.1 μGal/Hz1/2. We\nhave included thespecificnumberin therevised manuscript.\nReviewer 2 - Comment #18: “(line#118) The explanation on how the anti-springs work is confusing.\nThereisnosequenceofevents.Everythinghappensseamlessly,notindiscretesteps;pleasemodify.”\nReviewer 2 – Comment #18 – Response: We appreciate the reviewer's constructive\ncomment. We apologize for the confusing explanation in the original manuscript. Herein, we\nredrawFig.1 andrewrite theworkingprinciple of theanti-springstepbystep.\nIn a force equilibrium state, the greater the compression on a slender beam, the lower its lateral\nstiffness. In axial compression experiments on slender beams, deflection occurs when the\ncompression is much smaller than the theoretical critical load value, though the increase in\ndeflection is very slow. As the compression approaches the theoretical critical load, the\ndeflectionincreasesrapidly. The compressedanti-springis basedonthisprinciple.\nFig.R38a State 0: The initial configuration is shown in Fig.R37a. Two identically shaped slender\nbeams are symmetrically arranged and connected to a proof mass, with the outer ends\nconnected to a compression mechanism. For clarity, the proof mass is not shown in the figure.\nTheinitial x-directional lengthof thespringstructureis l.\nFig.R37bState1: Let us definethepositive y-direction asdownward. The uncompressedbeams\ndo not deform when the input force F is zero, corresponding to a displacement of zero. If a\ncompression distance of d in the x-direction is applied to two beams, the proof mass does not\nc\ndisplace in the x-direction due to the symmetry of the structure. To maintain equilibrium, the\nproof mass will displace by −d in the y-direction, with a compressing force F (denoted by the\n0 c\nblackbidirectional arrow) on thebeam.\nFig.R37c State 2: When a y-direction force is applied to the proof mass and gradually increases,\nthe proof mass gradually moves downward. When the force reaches F , the displacements of\n1\nthe proof mass relative to the initial position for the uncompressed and compressed structures\nare d ′ and −d , respectively. In both spring structures, the slender beams are further\n1 1\ncompressed, reducing the y-directional stiffness of the structures. Because the axial\ncompression force F of the compressed beams is greater than that in the uncompressed\nc\nbeams, the y-directional stiffness of the compressed beams is lower. As illustrated in the figure,\nthe black bidirectional arrow of the compressed beams is longer than that in the uncompressed\nbeams, leading to a greater slope or lower stiffness shown in the force-displacement\nrelationship.\nFig.R37d State 3: When the y-direction force reaches F ’ and F , respectively, the beams in\n2 2\nboth spring structures achieve maximum compression with a proof mass displacement of d ,\n2\nresultingin thelowest stiffnessin theforce-displacement relationship. As illustratedin thefigure,\nthe black bidirectional arrows (axial compression force) of the two structures are both the\nlongest, leading to the maximum slopes in their force-displacement relationships. However, the\naxial compression force of the compressed beams is greater and closer to the critical load\ncompared to that in the uncompressed beams due to the initial compression. Consequently, the\nminimum stiffness of the compressed structure is much lower than that of the uncompressed\nstructure.Ourproposedanti-springsoperate in thisstate.\nFig.R37e State 4: As the force continues to increase, the compression in the beams of both\nspring structures will decrease and may even transition to tension, resulting in a gradual\nincreasein stiffness.\nworking principle. a State 0 corresponds to the Initial configuration of the anti-spring structure. b State 1\nof the compressed and uncompressed structures without applied y-directional force. c State 2 of the\ncompressedand uncompressed structures with ay-directional forceof F 1. The stiffness of both structures\nbegins to decrease in this state. d State 3 of the compressed and uncompressed structures with y-\ndirectional forces of F 2 ’ and F 2, and the displacement of the proof masses are both d 2. This state\ncorresponds to the lowest stiffness state. e State 4 of the compressed and uncompressed structures with\nafurtherincreasedy-directionalforceofF 3.Thestiffnessofbothstructuresbeginstoincreaseinthisstate.\nAll the above explanations have been included in the revised manuscript, and Fig.1 has been\nreplaced.\nChangesinthemanuscript.\n-Revised Fig.1.\n-Line 123 to 154 “In a force equilibrium state, the greater the compression on a slender beam,\nthe lower its lateral stiffness. In axial compression experiments on slender beams, deflection\noccurswhen thecompression is muchsmaller thanthetheoretical critical load value, thoughthe\nincrease in deflection is very slow. As the compression approaches the theoretical critical load,\nthe deflection increases rapidly. The compressed anti-spring is based on this principle. The\n79\nschematic of the anti-spring structure is shown in Fig. 1a, and Fig. 1b illustrates the force-\ndisplacementrelationship of theanti-springstructure.\n…\nState 4: As the force continues to increase, the compression in the beams of both spring\nstructures will decrease and may even transition to tension, resulting in a gradual increase in\nstiffness.”\nReviewer 2 - Comment #19: “(line#137) not immediately clear why convex nature is important?please\nelaborate”\nReviewer 2 – Comment #19 – Response: We appreciate the reviewer's constructive\ncomment. As illustrated in Fig.R38, convex nature refers to a curve being enclosed within the\nconvex hull formed by its control polygon. In structural design, using curves with convex nature\noffersseveralbenefits:\nStability and Predictability: Curves with convex nature remain within the convex hull of their\ncontrol points, ensuring that their shape and position are stable and predictable. This is crucial\nforstructural designasit ensuresthecurve doesnot deviatefrom theexpectedrange.\nEase of Optimization: During the optimization process, convex curves are easier to handle since\ntheir shape and position are confined by the control points. This prevents the curve from\nstrayingoutsidethedesignspace,allowing formore effective parameter optimization.\nFig.R38-Response (for Reviewer 2 – Comment #19). Demonstrations of freeform curves without and\nwith convex nature. a The Ferguson curve without convex nature. b The B-spline curve with convex\nnature.\nReviewer 2 - Comment #20: “(Fig1b) I did not understand the blue line. Do the authors mean that\ngeomtericanti-springsbehaveaslinearspringswhentheyarenotloaded?”\nReviewer 2 – Comment #20 – Response: We appreciate the reviewer's constructive\ncomment. Weapologize for the misleading illustration in the original manuscript. As illustrated in\nFig.R37 and revised Fig.1, unloaded anti-springs also behave in a nonlinear force-displacement\nrelationship. However, the response appears approximately linear because its slope is much\nsmaller or its stiffness is much higher than that of the loaded anti-spring. Herein, we use the\noptimized F-AS presented in the manuscript as an example to demonstrate its acceleration-\ndisplacement relationships with and without compression. Fig.R39 shows the simulation results\nof the acceleration-displacement relationship for these two states. The red and blue lines\nrepresent the responses of the F-AS with and without compression, respectively. Actually, the\nF-AS without compression is not a linear spring, but it behaves as a linear spring when we put\nthese two plots together because the sensitivity is far smaller than that of the F-AS with\ncompression.\nFig.R39-Response (for Reviewer 2 – Comment #20). Simulation results of the acceleration-\ndisplacement relationship of the optimized F-AS for these two states. The blue line represents the\nresponseoftheuncompressedF-AS,andtheredlinerepresentstheresponseofthecompressedF-AS.\nChangesinthemanuscript.\n-Revised Fig.1b.\ngravimeters which have a much higher dynamic range than 5 Gal. When the authors say “... 5 Gal in\nconventionalMEMSgravimeters”theyshouldbacktheircommentwithcitations.”\nReviewer 2 – Comment #21 – Response: We appreciate the reviewer's constructive\ncomment. Indeed, as noted by the reviewer, some MEMS gravimeters, such as the vibrating\nbeam-based devices proposed by Mustafazade et al. [Scientific Reports, 2020, 10(1): 10415] and\nthose closed-loop schemes [IEEE Transactions on Industrial Electronics, 2023, 71(6): 6418-6426;\nIEEE Transactions on Instrumentation and Measurement, 2024, 73: 9504610], can achieve a dynamic\nrange far exceeding 5 Gal. However, the “dynamic range” of anti-springs was always small.\nHerein, we apologize for using the imprecise definition of “dynamic range” in the original\nmanuscript. In the original manuscript, the “dynamic range” was defined as the range where the\nR-squared value of the linear fit of the acceleration-displacement curve is greater than 0.99. It\nwas truly a weak and imprecise definition, resulting in some ambiguity. We have avoided using\n“dynamic range” insteadof “linear rangeof acceleration-displacementresponse” or “linearrange”\nin the revised manuscript. The acceleration-displacement sensitivity (previous “mechanical\nsensitivity”) and linear range are conflicting parameters in the near-zero stiffness region of anti-\nsprings, as depicted in Fig.R38d. For an anti-spring-based structure with acceleration-\ndisplacementsensitivitygreater than 40μm/Gal [Nature,2016,531(7596):614-617;Scientificreports,\n2022,12(1):13091;Microsystems & Nanoengineering,2019, 5(1): 45], maintaining a linear range of 5\nGalisnoteasy.\nIn the revised manuscript, we have corrected the sentence. In addition, we included Middlemiss\netal.andTangetal.’sworkascitationstobackthecomment.\nChangesinthemanuscript.\n-Line 209 “As a comparison, the linear range of acceleration-displacement response is often\nwithin 5Galin conventional anti-spring-basedMEMS gravimeters1,15.”\nReviewer 2 - Comment #22: “(line#210) I am confused about the “mechanical sensitivity” that the\nauthors report. For a 1.46 Hz device, by the simple a=w^2*x relationship, shouldn't the “mechanical\nsensitivity”be118um/Gal?whyisitconsistentlysmallereverywhereinthemanuscript?”\nReviewer 2 – Comment #22 – Response: We appreciate the reviewer's insightful comment.\nThe reviewer is right that the relationship holds for a typical second-order resonant system.\nFig.4e in the original manuscript), the maximum slope of the acceleration-displacement curve\ncorresponds to the sensitivity of around 126 μm/Gal, close to the calculated 118 μm/Gal. The\nmaximum slope is obtained by calculating the displacement difference within small intervals of\n0.1 Gal in the lowest stiffness region. The reason the “mechanical sensitivity” is consistently\nlower than the calculated value is due to the misleading definition of “mechanical sensitivity”\nused in the original manuscript. As previously mentioned, the “mechanical sensitivity” in the\noriginal manuscript denoted the slope of the fitted straight line of the acceleration-displacement\ncurve within 10 Gal. It was more likely an average slope/sensitivity within the range. The\naverage slope was necessarily smaller than the slope corresponding to the lowest resonant\nfrequency, which is themaximum slope.\nThe acceleration-displacement relationship of the linear range-adjusted structure (the measured\nresonant frequency at the angle of 90° is ~1.6 Hz) is shown in Fig.R40b. Although the\nacceleration-displacement curve still exhibits some nonlinearity, the average sensitivity in 10\nGal is 95.3674 μm/Gal, which is very close to the frequency-calculated sensitivity of ~98 μm/Gal\nfrom 1.6Hz. Itindirectlyproves theeffectivenessof thelinear rangeadjustment.\nFig.R40-Response (for Reviewer 2 – Comment #22). a Experimental and simulation results of the\nacceleration-displacement response of the original F-AS. b Experimental and simulation results of the\nacceleration-displacementresponseofthemodifiedF-AS.\nIn the revised manuscript, we have changed the terms “mechanical sensitivity” and “dynamic\nrange” to “acceleration-displacement sensitivity” and “linear range of acceleration-displacement\nresponse”, respectively, and clarified their definitions. We also explained, at the first mention of\nthesevalues, why theydiffer from thosecalculated usinga=ω²×d.\nChangesinthemanuscript.\n83\n-Any “dynamic range” and “mechanical sensitivity” are replaced by “linear range of acceleration-\ndisplacementresponse” and “acceleration-displacement sensitivity”\n-Revised Fig.4d.\n-Line 237 “The sensitivity is lower than the frequency-calculated value by the relationship a = ω²\n× d because the sensitivity is the slope of the fitted straight line of the curve within a relatively\nlarge range, more likely an average slope. It is necessarily smaller than the largest slope\ncorrespondingtothefrequency-calculatedsensitivity.”\nReviewer 2 - Comment #23: “The authors should report different resonant modes for the device. It is\nnotclearwhatthedisadvantagesofthedesignare.”\nReviewer 2 – Comment #23 – Response: We appreciate the reviewer's constructive\ncomment. Refer to Comment #0 – Response 1, Fig.R41 shows the mode shapes and\nfrequencies of the first six modes of the F-AS structure at its frequency minima angles and at\n90° (nominal operating condition). For the lowest frequency state, the first six resonant\nfrequencies are 1.20 Hz, 19.90 Hz, 34.00 Hz, 56.20 Hz, 56.81 Hz, and 83.11 Hz, respectively.\nThey correspond to the motion in the y-direction, motion in the z-direction, rotation around the y-\ndirection, rotation around the x-direction, motion in the x-direction, and rotation around the z-\ndirection. For the nominal operating condition, the first six resonant frequencies are 1.61 Hz,\n21.35 Hz, 38.45 Hz, 64.67 Hz, 66.79 Hz, and 94.56 Hz, respectively. They correspond to the\nmotion in the y-direction, motion in the z-direction, rotation around the y-direction, motion in the\nx-direction, rotation around the x-direction, and rotation around the z-direction. Although the\nabsolute value of the modal spacing is not large, for a structure with a fundamental frequency of\naround1.6Hz, therelativemodalspacing canbeconsidered large.\nFrom the modes and the modal spacing, the cross-axis sensitivity can be roughly distinguished.\nLet us perform a simple semi-quantitative calculation via frequencies about the cross-axis\nsensitivity of the nominal operating condition. Because the fundamental frequency of the\nstructure is very low (~1.6 Hz), the frequency closest to the sensing mode is more than 13.3\ntimes different from it, corresponding to a sensitivity difference of more than 132 times by the\nsimple a = ω2 × d relationship. Therefore, the cross-axis sensitivity of the structure can be\nconsideredsmall.\nmodesofthemechanicalsensingunit.\nIn the revised manuscript, we have added different resonant modes in the revised Fig.2. In\naddition, thediscussionabout themodesandthemodalspacing were includedin thetext.\nChangesinthemanuscript.\n-Revised Fig.4f.\n-Line 264 “For the nominal operating condition, the first six resonant frequencies are 1.61 Hz,\n21.35 Hz, 38.45 Hz, 64.67 Hz, 66.79 Hz, and 94.56 Hz, respectively. They correspond to the\n85\nmotion in the y-direction, motion in the z-direction, rotation around the y-direction, motion in the\nx-direction, rotation around the x-direction, and rotation around the z-direction, as shown in Fig.\n2f. Because the fundamental frequency of the structure is very low (~1.6 Hz), the frequency\nclosest to the sensing mode is more than 13.3 times different from it, corresponding to a\nsensitivitydifferenceof morethan 132 timesbythesimplea = ω2 × d relationship. Therefore,the\ncross-axis sensitivity of the structure can be considered small, which is satisfactory for typical\napplicationsof gravimetersthat arevertically mountedand have alimitedmeasurement range.”\nReviewer 2 - Comment #24: “(line#256) It is not clear how G2 was being actuated using a piezo\nelement?pleaseclarify.”\nReviewer 2 – Comment #24 – Response: We appreciate the reviewer's comment. As shown\nin Fig.R42, a dissociated proof mass with grating G2 was bonded to a quartz plate, which was\nthen bonded and fixed onto a piezoelectric ceramic (Model PA25LEW, Thorlabs Inc.). Grating\nG1was mounted on a five-dimensional adjustment stageto facilitate the alignment of the cavity.\nBoth gratings were mounted perpendicular to the ground, with their grid lines parallel to the x-\ndirection. The expansion and contraction direction of the piezoelectric ceramic is the y-direction.\nDuring the optical displacement measurement experiment, the piezoelectric ceramic, when\nenergized, drives G2toproducedisplacementin they-direction.\nFig.R42-Response (for Reviewer 2 – Comment #24). Photograph and schematic of the optical\ndisplacementmeasurementsetupbasedonapiezoelectricceramic.\nmeasurement setup in the revised Fig.3. In addition, the clarification of how the piezoelectric\nceramicdrives thegrating hasalso beenincluded.\nChangesinthemanuscript.\n-Revised Fig.3b.\n-Line 315 “The optical displacement measurement setup is shown in Fig. 3b. A dissociated\nproof mass with grating G2 was bonded to a quartz plate, which was then bonded and fixed\nonto a piezoelectric ceramic (Model PA25LEW, Thorlabs Inc.). Grating G1 was mounted on a\nfive-dimensional adjustment stage to facilitate the alignment of the cavity. Both gratings were\nmounted perpendicular to the ground, with their grid lines parallel to the x-direction. The\nexpansion and contractiondirectionof thepiezoelectricceramicis they-direction.”\nReviewer 2 - Comment #25: “(line#259) The authors did not report the standard deviation on their 5\nminute measurements.Anerror of 0.4%mightseemsmallbut,the authorsshouldreport whatthis means\\intermsofaccelerationerror.”\nReviewer 2 – Comment #25 – Response: We appreciate the reviewer's constructive\ncomment. Wehave addedthe time-domain dataof the static optical displacement measurement\nfor each point of displacement, along with their standard deviations, as shown in Fig.R43 and\ntherevised Fig.3.\nAs shown in Fig.R43, within the 5-minute, the time-domain output curves at each displacement\npoint are highly stable, with standard deviations from 1 to 12 mV. The standard deviations\ncorrespond to acceleration errors from 8.2 to 98.3 μGal, combining the optical displacement\nsensitivityofaround1.28V/μmandtheacceleration-displacementsensitivityofaround95μm/Gal.\nThe calculated acceleration error of 8.2 μGal is also comparable to the Allan deviation presented\nintherevisedmanuscript.\ninputdisplacement,withthe5-minutetime-domaindatashownintheinsets.\nChangesinthemanuscript.\n-Revised Fig.3.\n-Line 323 “The time-domain data of each point are presented in the right panel. The response\nresembles the simulation results and performs a sensitivity of 1.28 V/µm, with the standard\ndeviations from 1 to 12 mV. The standard deviations correspond to acceleration errors from 8.2\nto 98.3 μGal, combining the optical displacement sensitivity and the acceleration-displacement\nsensitivity of around 100 μm/Gal. The experimental result is not a highly linear response but\nrather resembles the typical response of such an optical cavity. Fig. 3c shows the spectral\ndensity of the displacement measurement. It indicates that the noise level of the optical cavity is\n1.2pm Hz-1/2at 1Hz andbelow 5 pm Hz-1/2within thefrequencyrangeof 0.5to1 Hz.”\nReviewer 2 - Comment #26: “(Fig 3b) The fitted line is not very linear for the lower displacements. Is\nthisalimitofthereadout?”\nReviewer 2 – Comment #26 – Response: We appreciate the reviewer's constructive\ncomment. The reviewer is right that the optical displacement response is not very linear. The\nlight intensity-displacement curve obtained by simulation is a periodic response, with the\nresponse curve for one period shown in Fig.R44a. It is a sine-like response (but not a perfect\nsine curve), and the response is related to the cavity spacing. The experimental result, as\nshown in Fig.R44b, was not a highly linear response but rather resembled the simulation results.\nThis indeed could be a limit of the readout due to the non-linear response and the small\n88\nmonotonic interval. However, it is a typical response of an optical cavity, and the repeatability of\nthis response is high. The repeatability means that the nonlinear response can be readily\ncalibrated, and the measurement has a reasonable accuracy of sub-nanometer. The small\nmonotonicinterval canbe expanded through fringecountingor phase unwrapping [Appliedoptics,\n1991, 30(28): 4046-4055; Applied optics, 1993, 32(17): 3047-3052], which is a common practice in\nthefield of optical measurement andhasalso beenutilized in our work.\nFig.R44-Response (for Reviewer 2 – Comment #26). a light intensity-displacement curve obtained by\nFDTD simulation for one period. b Exeripmental results of the output voltage versus the input\ndisplacement.\nThisdiscussion hasbeen addedin therevisedmanuscript and Supplementary Information.\nChangesinthemanuscript.\n-Revised Fig.3.\n-Line 323 “The response resembles the simulation results and performs a sensitivity of 1.28\nV/µm, withthestandarddeviationsfrom 1 to12 mV.”\nReviewer 2 - Comment #27: “I am confused on what quasi-static mean here? Is using this term very\nnecessary?”\nReviewer 2 – Comment #27 – Response: We appreciate the reviewer's constructive\ncomment. Thequasi-static measurement referstodriving thesensing unit along thez-axis using\na turntable, pausing for 1 minute at each angle to read the data of 1 minute. This aimed to test\nthe static response of the sensing unit. We fully agree with the reviewer that the term “quasi-\nstatic” is not necessary and may cause some ambiguity. We have deleted this term, instead\nusing “measurement” rather than“quasi-staticmeasurement” in therevised manuscript.\nReviewer 2 - Comment #28: “(line#285) I did not understand how a change of 0.0004 deg in theta is\nequivalentto7mgal.Wouldn'tdelta_a=g*(1-cos(0-0.0004))?”\nReviewer 2 – Comment #28 – Response: We appreciate the reviewer's constructive\ncomment. As illustrated in Fig.R45, an increase of the angle of 0.0004° at the angle of 0° from\nhorizontal corresponds to the change of the gravitational acceleration component applied to the\nsensitive axis of 7 mGal (delta_a = g*(sin(0.0004°) – 0) ≈ 7 mGal), which is the coarsest change.\nWhile at the angle of 90° from horizontal, it should be delta_a= g*(1-cos(0-0.0004°)) = 24.37nGal,\nwhich is the finest change. We apologize for the imprecise statement in the original\nmanuscript and have added a detailed explanation of the equivalent conversion at two angles.\nFig.R45-Response (for Reviewer 2 – Comment #28). Schematic of the relationship between the\nvariationofaccelerationandtheangularchange.\nChangesinthemanuscript.\n-Line 357 “More specifically, a change of the angle of 0.0004° corresponds to the variation of\nthe gravitational acceleration component applied to the sensitive axis of g*(sin(0.0004°) – 0) ≈ 7\nmGalat theangle of 0° andg*(1–cos(0–0.0004°)) ≈ 24nGalat 90°.”\nReviewer 2 – Comment #29 – Response: We appreciate the reviewer's comment. It was 1\nminute for each angle. We also appended the time-domain data of 1 minute in the revised\nFig.4e.\nWe selected to pause at each angle for 1 minute, more specifically 30 seconds for stabilization\nand 30 seconds for measurement, both considering measurement duration and stability time.\nRegarding the stability time, each rotation of the turntable requires approximately 30 seconds to\nstabilize, as depicted in Fig.R46b. Averaging the datawithin the 1-minute (excludingthe initial 30\nsecondsforturntablestabilization)wassolelyforplottingthedisplacement-accelerationresponse\nin the original Fig.4e. As shown in Fig.R46b, within the 30 s, the time-domain output curves at\neachacceleration point are highlystableafter stabilization, with standard deviations from 6nm to\n14nm,indicatingvirtuallynodrift.\nRegarding the measurement duration, the tilt method measurement requires a larger number of\npoints to achieve enough measure resolution of sensitivity in such a large range of displacement\nchange. For example, we measured 595 points in the range of 10 Gal, as shown in the original\nFig.4. The number of points is much larger than what we tested in the optical displacement\nmeasurement. The testing time per point also needs to be correspondingly reduced; otherwise,\nthe total duration would become unbearable. Therefore, the overall measurement time per point\nwas selected as 1 minute, significantly shorter than 5 minutes. Even so, measuring the\ndisplacement-accelerationresponseofanF-AS-basedstructureinarangeof10Galstillrequires\naroundtenhours.\nseries representing the results of the F-AS-based sensing units. The scatter represents the experimental\nresults, and the dashed and solid lines represent the simulation and linear fitting of the experimental\nresults.bTime-domaindataforsomeinput accelerationpointsinoneminute.\nIn the revised manuscript, we added the time-domain data plot corresponding to one point in the\nacceleration-displacement curve (see revised Fig. 4). Additionally, we included some\nexplanations in the main text on why 1 minute was chosen as the measurement time of each\nangleforthetiltmethod.\nChangesinthemanuscript.\n-Revised Fig.4e.\n-Line368“Thesystem was pausedat eachangle forone minutetoacquirethestatic response.”\n-Line 372 “Within the 30 seconds, the time-domain output curves at each acceleration point are\nhighly stable after 30-second stabilization, with standard deviations from 6 nm to 14 nm,\nindicatingvirtually nodrift.”\nReviewer 2 - Comment #30: “(Fig 4) This plot is confusing. What do the authors mean by the working\nrange? If the frequency is changingnon-linearly in most of the working range how can it be the working\nrange?In'e',whatisthereasonbehindthedeviationofmeasuredpointsfromthefittedstraightline?”\ncomment. Our interpretation rests on two pillars: (1) the definition of “working range” and (2) the\nreasonbehind thedeviation.\n(1)thedefinitionof “workingrange”\nFirst, we apologize for the use of the imprecise definition of “working range” in the original\nmanuscript. As mentioned above, we defined the “working range” or “dynamic range” as the\nrange where the R-squared value of the linear fit of the acceleration-displacement curve is\ngreater than 0.99. It was truly a weak and imprecise definition, resulting in some ambiguity.\nSome other researchers defined the dynamic range as the range with a variation in resonant\nfrequency of smaller than around 0.5 Hz [Nature, 2016, 531(7596): 614-617; Microsystems &\nNanoengineering, 2019, 5(1): 45]. In the revised manuscript, we have endeavored to avoid\n“working range” and “dynamic range”, instead using “linear range of acceleration-displacement\nresponse”. When the terms must be used, we have provided our clear definition upon their first\noccurrences toavoid ambiguity.\n(2)thereasonbehind thedeviation of measuredpointsfrom thefittedstraightline\nAs depicted in Fig.R37 and revised Fig.1b, the acceleration-displacement response of an anti-\nspring-based structure should be a nonlinear curve. For a F-AS-based structure with a V-\nshaped frequency response, as presented in the first submission, the nonlinearity is somehow\nsignificant. Although the R-squared value of the linear fit in the range of 10 Gal was 0.9908,\nthere were still some deviations between the measured points, also the simulation results, and\nthe fitted straight line. As mentioned above, we have expanded the linear range by local\nadjustmentto makethe frequencyminima located at the angleof approximately 86.8° and 93.2°.\nFig.R47 depicts the acceleration-frequency and acceleration-displacement responses before\nand after adjustment. The R-squared value of the linear fit in the range of 10 Gal was improved\nfrom 0.9908 to 0.9985. It is evident that the linearity is significantly enhanced, with a smaller\ndeviationof measuredpointsfrom thefittedstraightline.\nshaped frequency versus the angle from horizontal and ∆a of the original F-AS. b Experimental and\nsimulation results of the W-shaped frequency versus the angle from horizontal and ∆a of the modified F-\nAS with expanded linear range. c Experimental and simulation results of the acceleration-displacement\nresponse of the original F-AS. d Experimental and simulation results of the acceleration-displacement\nresponseofthemodifiedF-AS.\nIn the revised manuscript, we have added the discussion about the deviation of measured\npointsfrom thefittedstraight line.\nChangesinthemanuscript.\n-Revised Fig.4.\n-Line 374 “The fitted acceleration-displacement sensitivity of these two units is measured to be\n95.3674 µm/Gal and 40.5394 µm/Gal, respectively, with linearity of 0.9985 and 0.9919, all\npassing through the robustness analysis. The linearity of 0.9985 demonstrates that the linearity\nis significantlyenhancedvia adjustment.”\nthex-axisratherthanrelativetime.”\nReviewer 2 – Comment #31 – Response: We appreciate the reviewer's constructive\ncomment. In the revised manuscript, the time axis coordinates have all been changed from\nrelative timetoreal UTCtimewhen presentingseismicdata.\nChangesinthemanuscript.\n-Revised Fig.5.\nReviewer 2- Comment #32: “(Fig#5c)Iftheauthorsareclaimingthattheirdeviceisagravimeter,they\nshouldplottheasddownto10^-5/10^-6secondstocapturethetidalsignals”\nReviewer 2 – Comment #32 – Response: We appreciate the reviewer's constructive\ncomment. We have provided the amplitude spectral density plot of the post-regression data in\nFig.R29a and revised Fig.5c. The green line represents data that were recorded at a sampling\nrateof 0.1Hz. The Earthtidesarevisible at thebandof 10-5 Hz.\nChangesinthemanuscript.\n-Revised Fig.5c.\nReviewer 2 - Comment #33: “The authors have not shown the raw, uncorrected data. The raw data in\nthe supplementary seems to be drift corrected and averaged. What kind of drift do they see in their\ndevice?”\ncomment. In the revised manuscript, we have provided the raw data in revised Fig.5d. It\nbehaved at a drift rate of around 153 μGal/day. The drift is approximately linear. Although the\ndrift is a bit larger than that of Glasgow's MEMS gravimeter, the magnitude is already\ncomparable. In the future, its stability might be further improved through vacuum packaging and\nhigher precision temperature control. The drift data were not extensively introduced in the\noriginal manuscript because we previously regarded the long-term stability improvements as a\nrelatively independent work, and the original manuscript focused on the innovation of the design\nand freeform anti-springs. However, we greatly appreciate the reviewers for raising these\nconcerns. As previously mentioned, we have substantially appended the relevant content in the\nrevised manuscript.\nChangesinthemanuscript.\n-Revised Fig.5d.\nReviewer 2 - Comment #34: “(line#380) What is the reasoning behind using the isotropic model for\nsimulations?”\nReviewer 2 – Comment #34 – Response: We appreciate the reviewer's constructive\ncomment. We used the isotropic model because we previously tested both isotropic and\nanisotropic models and found their response results to be nearly identical. As Hopcroft et al.\n[Journal of microelectromechanical systems, 2010, 19(2): 229-238] stated, the results obtained by\nisotropic simulations of silicon material typically have an error within 30% compared to those\nproduced by anisotropic simulations. Tang et al. [Microsystems & Nanoengineering, 2019, 5(1): 45]\nalso used the isotropic model for simulation. Weprovide the FEA results of one F-AS parameter\nset, which is randomly selected from the optimization iterations, using two models. As illustrated\nin Fig.R48, it is evident that the FEA results of the two models coincide well with each other.\nThe sensitivity obtained from the two models is 72.5 and 74.0 μm/Gal, and the relative error is\naround 2% in this case. Choosing the isotropic model helped to simplify the parameter model\nandtomaintainconsistencywith theanalyticalcalculation in SupplementaryNote1.\nrelationshipsoftheF-ASusingtheisotropicandanisotropicmodels.\nHowever, we also recognize that using the anisotropic model is a more accurate choice.\nTherefore, in the simulations presented in the revised manuscript, we have replaced the model\nwith the anisotropic model for the simulations. Additionally, we have included the anisotropic\npropertiesof silicon usedin thesimulationsin therevised manuscript.\nChangesinthemanuscript.\n-Line 475 “The modulus tensor of silicon was set according to reference. In addition, the density\nwasset to2329kg/m3.”\n“Title - Multi-stage design of locally adjustable freeform anti-springs: a pathway\ntowards a μGal MOEMS gravimeter with 1 Hz resonant frequency and fabrication tolerance” Manuscript\nNCOMMS-23-62597A.\nWe would like to thank our Referees for their insightful, very detailed and useful comments.\nFollowing their suggestions and criticism, we clarified the novelty of the paper and provided more\ndetails regarding the analysis and method. We believe we made it clearer why the research\nreported here is an advance and achievement in the MEMS gravimeter, and we hope that the\nrevised manuscript meets the high standards of Nature Communications. Below is our point-by-\npoint response to our Reviewer’s comments.\nReviewer 2 – general assessment: The authors have indeed put a lot of effort into improving the manuscript\nand included more details. I really appreciate the authors for this. However, I am not entirely convinced by\nthe central theme of the paper (making a 1 Hz, more robust device). As I have indicated in my comments\nbelow, I liked that these devices are easier to fabricate owing to their better etching aspect ratio. I know that\nhigh aspect ratio simple anti-springs are challenging to fabricate. This point is not highlighted enough\nconsidering this is a critical issue.\nReviewer 2 – Comment #0 – Response 1: We appreciate our reviewer’s remark on the\nimprovement of our revised manuscript. We totally agree with the reviewer that fabrication\ntolerance should be highlighted. We have thoroughly revised the manuscript to highlight the\ncentral theme: achieving an ultra-low resonant frequency under the condition of low etching\naspect ratio in MEMS scale devices.\nFor example, we mentioned in the revised manuscript that fabrication tolerance is a critical\nconsideration in designing F-AS-based structures. Existing structures with resonant frequencies\nbelow 10 Hz often require stringent etching aspect ratios, as shown in Table R1. The aspect ratios\nare all larger than 18:1, and some even exceed 40:1. We constrained the beam width to be larger\nthan 30 µm in the multi-stage design, resulting in an aspect ratio of smaller than 400:30. This\ndesign achieved both an ultra-low resonant frequency and high fabrication tolerance, making it\nthe most easily fabricable structure among contemporary devices with resonant frequencies\nbelow 10 Hz.\nTable R1-Response (for Reviewer 2 – Comment #0) Typical MEMS structure and the required\naspect ratio\nResonant\nDevice Aspect ratio\nfrequency (Hz)\nMiddlemiss et al. 200:5\n2.3\n[Nature, 2016, 531(7596): 614] 40\nLiu et al.\n500:30\n[Microsystems & Nanoengineering, 2019, 5(1): 11] 2.9\n>16.7\n[IEEE Sensors Journal, 2021, 21(18): 19872]\nPike et al.\n525:28\n[2023 IEEE aerospace conference. IEEE, 2023: 1] 6\n>18.8\n[2018 IEEE MEMS. IEEE, 2018: 113]\nMansouri et al. 300:5.4\n8.7\n[Microsystems & Nanoengineering, 2019, 5(1): 14] >55.6\n400:30.86\nThis work 1.71\n<13.0\nIn the revised manuscript, we have compared the current state of fabrication tolerance of highly\nsensitive structures and clarified the width constraints we applied during the multi-stage design\nprocess. Additionally, the etching aspect ratio information from Table R1 has been updated in\nTable 2 of the manuscript.\nChanges in the manuscript.\n-Title.\n-Line 33. “… they still face challenges in achieving ultra-high sensitivity with fabrication tolerance\nand limited size of proof mass.”\n-Line 40. “… and a reasonable maximum stress with an etching aspect ratio of smaller than\n400:30.”\n-Line 66, 67. “… the biggest problem is designing a chip-scale structure with an ultra-low resonant\nfrequency and simultaneously maintaining structural robustness and fabrication tolerance …”\n-Line 69, 70. “usually requires slender beams with a high aspect ratio, which imposes stringent\ndemands on microfabrication.”\n-Line 82, 83, 84. “… balances ultra-high sensitivity, stress robustness, and fabrication tolerance.”\n“… and the required etching aspect ratio is always larger than 16, even exceeding 40.”\n-Line 109. “… the required etching aspect ratio is smaller than 13.”\n-Line 194, 195. “The fabrication tolerance, more specifically, a beam width larger than 30 μm, is\nset as the constraint.”\n-Line 259, 260. “… indicating that the F-AS obtained by multi-stage design had excellent\nperformance and fabrication tolerance.”\n-Line 442. “… with reasonable fabrication tolerance and robustness.”\n-Table 2.\ngeneral assessment #2: I am slightly intrigued by the spectral density plot of the device. There is big bump\nin the density below 100 mHz which is unexplained in the manuscript. It seems that the densities > 100 mHz\nare affected by a low pass filtering stage (corner at 100 mHz) which means that the true sensitivity is not 1.1\nugal/rtHz. Can the authors explain this? The authors plot the Allan-Deviation for the self-noise timeseries\nof the MOEMS device?\nsuggestions. This is a great question. First, the data presented is the spectral density plot of the\nraw data, and we did not apply a low pass filter.\nThe bump below 100 mHz is because we did not enable temperature control during fast sampling\n(at a sampling rate of 20 Hz). We initially thought the fast sampling testing duration was short,\nneglecting the temperature control. However, we later found that fluctuations in the ambient\ntemperature had a significant impact on the device, which is reflected in the noise level bump\nwithin this frequency range. As shown in Fig. R1a, we present the density plot for fast sampling\nwith temperature control. It is evident that temperature control effectively suppresses the noise\nlevel in the frequency range, compared with the density plot shown in the previous manuscript.\nWe also conducted three repeated fast sampling measurements and compared them with PET\ndata, as shown in Fig. R1b to d. It can be observed that the noise levels after temperature control\nare lower than those in the previous manuscript. Furthermore, the results of the three\nmeasurements exhibit a high correlation with PET in the frequency range of the earth tremor peak\n(0.1–0.5 Hz), demonstrating the sensitivity of µGal level. Therefore, we believe it is safe to\nconclude that the true sensitivity is around 1 μGal Hz-1/2.\nMOEMS gravimeter and the reference gravimeter with temperature control. The red line\nrepresents the MOEMS gravimeter, the blue line represents the reference gravimeter, PET, and\nthe orange line represents the noise data of the MOEMS device. The red and the orange series\nwere recorded at a sampling rate of 20 Hz. The green series represent data that were recorded\n5\nat a sampling rate of 0.1 Hz. The Earth tremor peak can be observed at around 0.2 Hz. b to d\nAcceleration spectral density plot of three repeated fast sampling measurements. The lavender\nline represents the result without temperature control.\nIn the revised manuscript, we have revised Fig. 5c to include the new fast sampling results.\nAdditionally, the discussion about the bump was added in the main text.\nRegarding the Allan-Deviations, we did not plot the self-noise time series but plotted the raw\noutput data at a sampling rate of 0.1 Hz. The data presented in Fig. 5 in the manuscript are all\nincluded in the updated source data file.\nChanges in the manuscript.\n-Fig. 5c.\n-Line 390-392. “The noise bump around 0.1 Hz could be due to insufficiently precise temperature\ncontrol. Finer temperature control and temperature compensation are expected to suppress the\nnoise level in this frequency range.”\ngeneral assessment #3: There are better sensitivity and stability devices out there, which dampens the main\nmetrics of the MOEMS device. Their optimization algorithms is indeed an interesting addition to designing\nnovel springs, but it is difficult to make a computation time argument when there is no real-time need for\nrunning the routine (you could essentially run algorithms for days on a PC). I am in no way suggesting that\nthis work is not important, it is highly publishable, but seems to be more directed towards a MEMS\njournal (JMEMS, Nature Micro Nano).\nReviewer 2 – Comment #0 – Response 3: We appreciate the reviewer’s remark on the\nimportance of our work. Although a significant portion is related to MEMS, it is a highly\ninterdisciplinary and comprehensive work.\n1) the proposed freeform multi-stage design method balances a large design parameter space\nwith relatively low computational cost in both modeling and optimization. This approach is not only\napplicable to MEMS structure design but also holds value for addressing other complex design\nproblems.\n2) Admittedly, for non-real-time design tasks, someone can extend runtime to complete any\noptimizations. However, the scale of some tasks may not be solvable within days. For example,\nwe found that a single-stage optimization of an F-AS-based structure with 500 iterations would\n6\ntake over a month, even running on a high-performance server with two Intel Xeon Platinum\n8350C CPUs. Additionally, there is a potential risk of non-convergence. In such cases, the multi-\nstage design and designated AELPSO algorithm becomes effective. This algorithm is applicable\nbeyond MEMS.\n3) This work involves a highly integrated optical displacement measurement unit. It is also a\ncrucial component of the study beyond MEMS.\nIn summary, this work not only designs and fabricates a high-performance MOEMS gravimeter\nwith fabrication tolerance but also involves the design method, algorithm, and optical\nmeasurement techniques that have the potential to impact non-MEMS fields. Therefore, we\nbelieve this work is suitable for publication in a broad-scope journal like Nature Communications.\nWe hope our revised manuscript can meet your high standards.\nReviewer 2 - Comment #1: “I can see what the authors are trying to say and the importance of reducing\nthe resonant frequency. I like that the aspect ratio is better than the contemporary devices. In my opinion,\nthis is a big challenge that the authors have addressed. This is more significant than a resonant frequency\nof 1 Hz (achievable using simple geometry) and the grating readout sensitivity (only slightly better than the\ncapacitive schemes). The author should put this at the centre of their work. I am not sure why this point has\nnot been highlighted considering fabrication of high aspect ratio structures is a difficult challenge while\nmaking devices with simple geometries.”\nReviewer 2 – Comment #1 – Response: We appreciate the reviewer's suggestion. Referring\nto Comment #0 – Response 1, we have thoroughly revised the manuscript to highlight the central\ntheme. The title of the manuscript has been revised to emphasize the significance of fabrication\ntolerance. In the revised manuscript, we have compared the current state of fabrication tolerance\nof highly sensitive structures and clarified the width constraints we applied during the multi-stage\ndesign process. Additionally, the etching aspect ratio information from Table R1 has been updated\nin Table 2 of the manuscript.\nReviewer 2 - Comment #2: “As far as I know, the dynamic range in the contemporary capacitive devices\nis set by the capacitive readout, and less so by the displacement mechanics.”\nReviewer 2 – Comment #2 – Response: We appreciate the reviewer's suggestion. Our original\nintent was to convey that for mechanics with high sensitivity (slope), such as F-AS, the linear\nrange of displacement also affects the device's dynamic range. However, we agree with the\nreviewer’s point that, for most capacitive devices, the dynamic range is primarily limited by the\nnonlinear output of the capacitive readout. Therefore, we have removed some statements\nregarding dynamic range in the revised manuscript.\nChanges in the manuscript.\n-Line 69, 70, 71 in previous manuscript. “In addition, the trade-off between the acceleration-\ndisplacement sensitivity and the linear range poses a challenge to designing a MEMS device with\ngood linearity in a reasonable dynamic range.” was deleted.\n-Line 82 in previous manuscript. “… and a reasonable linear range” has been replaced by “… and\nfabrication tolerance”.\n…\nReviewer 2 - Comment #3: “Thanks for adding the modal results. Can the authors comment on “It is not\nclear what the disadvantages of the design are”?”\nReviewer 2 – Comment #3 – Response: We appreciate the reviewer's suggestion. Of course,\nthe design has some disadvantages:\n1) Even with the multi-stage design method, the optimization of the F-AS-based structure still\nrequires several days. Most of this time is spent on FEA simulations, which are essential for\naccurately predicting the performance of the sensing unit. In the future, semi-analytical or fully\nanalytical methods could be considered to improve design efficiency.\n2) The design uses a manual compression mechanism. It is simple and relatively robust but\ncannot be adjusted or reversed once compressed. Additionally, the existence of the compression\nloading hole places higher demands on packaging. Future designs could explore alternative\ncompression mechanisms or methods to reach the near-zero stiffness region of the flexure.\nReviewer 2 - Comment #4: “What is the value of g that the authors are using? From back calculations,\nit seems it is 10 m/s2. The authors should mention this if they want to use this value, or, use the standard\ngravity value of 9.80665 m/s2 (and mention it)”\nReviewer 2 – Comment #4 – Response: We appreciate the reviewer's suggestion. The value\nof g used in this paper is 9.7944 m/s2, which is the reference gravitational acceleration obtained\nfrom the cave laboratory. We have appended the g-value in the revised manuscript.\n-Line 378. “The reference gravitational acceleration obtained from the cave laboratory is 9.7944\nm/s2.”\nReviewer 2 - Comment #5: “32/33: the opening sentence needs to be rephrased. Success in what\napplications? What do you by maintaining high sensitivity? What does limited volume mean?”\nReviewer 2 – Comment #5 – Response: We appreciate the reviewer's suggestion. This\nsentence has been split into two sentences: MEMS gravimeters have found success in\ngeophysical and space applications. However, they still face challenges in maintaining ultra-high\nsensitivity with fabrication tolerance and limited size of proof mass.\nLimited volume means the limited size of the proof mass and die, always resulting in large k, small\nm, and large resonant frequency.\nChanges in the manuscript.\n-Line 32, 33. “MEMS gravimeters have found success in geophysical and space applications.\nHowever, they still face challenges in achieving ultra-high sensitivity with fabrication tolerance\nand limited size of proof mass.”\nReviewer 2 - Comment #6: “36: What is multi-stage directional design scheme? It would be useful to\nkeep the sentences simple and avoid using special terms that readers might not be familiar with”\nReviewer 2 – Comment #6 – Response: We appreciate the reviewer's suggestion. The term\n\"multi-stage directional design scheme\" refers to a scheme that combines both multi-stage and\ndirectional design approaches. \"Multi-stage\" implies that the process is divided into independent\nbut interconnected stages, while \"directional\" refers to having a clear optimization objective in\neach stage.\nWe agree with the reviewer’s opinion that the statement was overly lengthy, and it has been\nsimplified to “multi-stage design” in the revised manuscript.\nChanges in the manuscript.\n-“multi-stage directional design” has been revised to “multi-stage design”.\nReviewer 2 - Comment #7: “37: what is low-stress manufacturing?”\nstress manufacturing\" we originally meant introducing minimal stress during the fabrication\nprocess. However, we acknowledge that this phrasing is unconventional. We have adjusted the\nterminology of the entire fabrication and packaging as \"designated manufacturing and compact\npackaging\" in the revised manuscript.\nChanges in the manuscript.\n-Line 37. “Designated manufacturing and compact packaging are implemented …”.\nReviewer 2 - Comment #8: “38/39: what does “ever-reported” mean? Do you mean the lowest reported\nresonant frequency? Is it the lowest reported resonant frequency in MEMS scale devices?”\nReviewer 2 – Comment #8 – Response: We appreciate the reviewer's suggestion. Yes, what\nwe intended to convey is the lowest reported resonant frequency. In fact, it is safe to say that our\ndevice is one of the lowest reported resonant frequencies in MEMS scale devices. We found two\narticles claiming that the resonant frequency of their devices could be less than or equal to 1 Hz.\nFirst is the positive and negative stiffness combination mechanism designed by Wu et al. [Journal\nof microelectromechanical systems, 2020, 20(5): 1322-1331]. However, the structure had an\nextremely large etching aspect ratio, close to 50:1, making fabrication highly challenging.\nAdditionally, their spectral density plot (their Fig. 13(b)) did not present the claimed sub-1 Hz\nresonance peak. Regarding the second structure proposed by Yi et al. [2024 IEEE International\nSymposium on Inertial Sensors and Systems (INERTIAL). IEEE, 2024: 1-4], their testing methods and\nthe frequency range selected for the spectral density plot (their Fig. 8) both have some issues.\nHowever, to maintain rigor, we have revised the statement in the manuscript to “one of the lowest\nreported resonant frequencies”, which we believe is a safer expression.\nChanges in the manuscript.\n-Line 38, 39. “… achieves one of the lowest reported resonant frequencies of 1.71 Hz, …”.\n-Line 83. “… the lowest reported resonant frequencies of MEMS mechanisms are around 2~3\nHz …”.\n-Line 239. “… has one of the lowest reported resonant frequencies …”.\nReviewer 2 - Comment #9: “41/42: what is long-period measurement?”\nperiod measurement\" refers to measurements taken over an extended duration, used to generate\nthe density plot in Fig. 5. However, we acknowledge the ambiguity and have revised the term to\n\"the noise-level and tidal measurement\" for clarity.\nChanges in the manuscript.\n-Line 41, 42. “… the noise-level and tidal measurement.”\nReviewer 2 - Comment #10: “51/52/53: Not all relative gravimeters use mechanical springs. For example,\nsuperconducting gravimeters (iGrav) use magnetically levitated mass (however, the magnetic counter force\ncan be modelled as an equivalent spring)”\nReviewer 2 – Comment #10 – Response: The reviewer is correct. Strictly speaking, it should\nbe \"mechanical relative gravimeters\". The corresponding modification has been made in the\nrevised manuscript.\nChanges in the manuscript.\n-Line 51, 52. “… mechanical relative gravimeters measure …”.\nReviewer 2 - Comment #11: “54: what do you mean by “successful devices”? successful in what?”\nReviewer 2 – Comment #11 – Response: We appreciate the reviewer's suggestion. We meant\nthat MEMS gravimeters have proved themselves as successful devices in geophysical and space\napplications. We also added some relevant references to support this statement.\nChanges in the manuscript.\n-Line 54, 55. “… have proved themselves as successful devices in geophysical and space\napplications6,16,17,18,19”\nReviewer 2 - Comment #12: “55: MEMS gravimeters are not salient topics (?). I think this sentence needs\nrephrasing”\nReviewer 2 – Comment #12 – Response: We appreciate the reviewer's suggestion. This\nstatement has been removed, and the sentence has been revised to “MEMS gravimeters have\n11\nproved themselves as successful devices in geophysical and space applications in terms of their\nfurther miniaturization, mass production, and cost reduction.”\nChanges in the manuscript.\n-Line 55, 56. “MEMS gravimeters have proved themselves as successful devices in geophysical\nand space applications in terms of their further miniaturization, mass production, and cost\nreduction.”\nReviewer 2 - Comment #13: “56: What do you mean by extremely high acceleration sensitivity? When\ndoes an acceleration sensitivity become extremely high?”\nReviewer 2 – Comment #13 – Response: We appreciate the reviewer's suggestion. \"extremely\nhigh acceleration sensitivity\" refers to μGal-level acceleration sensitivity. We intended to express\nthat achieving μGal-level sensitivity requires improvements in two units. This sentence has been\nrevised in the revised manuscript to “μGal-level acceleration sensitivity should be achieved by\nimproving both the acceleration-displacement sensitivity and displacement measurement\nsensitivity.”\nChanges in the manuscript.\n-Line 56, 57. “μGal-level acceleration sensitivity should be achieved by improving both the\nacceleration-displacement sensitivity and displacement measurement sensitivity.”\nReviewer 2 - Comment #14: “61/62/63: the use of “have kept appearing” is not suitable here. You should\nuse “have been published…””\nReviewer 2 – Comment #14 – Response: We appreciate the reviewer's suggestion. It has\nbeen revised to “have been published …”.\nChanges in the manuscript.\n-Line 62. “… have been published …”.\nReviewer 2 - Comment #15: “65: what is the stability numbers for CG5 and PET?”\nstability of Scintrex CG5 is better than 20 µGal/day [https://scintrexltd.com/wp-\ncontent/uploads/2018/11/CG-5-Brochure-R3.pdf], and the stability of LaCoste PET is better than\n16.67 µGal/day [https://microglacoste.com/wp-content/uploads/2017/03/MgL_gPhoneX-Brochure.pdf].\nWe have added corresponding information in the revised manuscript.\nChanges in the manuscript.\n-Line 65. “… and good long-term stability of around 20 µGal/day.”\nReviewer 2 - Comment #16: “70: is there a trade-off between the sensitivity and linearity?”\nReviewer 2 – Comment #16 – Response: We appreciate the reviewer's suggestion. To be\nprecise, there is a trade-off between the sensitivity and linear range. As shown in Fig. R2, when\nthe sensitivity (slope) of a spring-mass structure increases, the linear range decreases.\nHowever, we have removed the related statements in the revised manuscript to avoid ambiguity.\nFig. R2-Response (for Reviewer 2 – Comment #0). Schematic of the displacement-acceleration\nrelationship of two structures with different sensitivity, derived from Fig. 1e in the manuscript.\nChanges in the manuscript.\n13\n-Line 68 to 71. The statement has been removed and replaced with “For example, achieving a\nlow resonant frequency usually requires slender beams with a high aspect ratio, which imposes\nstringent demands on microfabrication. In addition, low resonant frequencies always bring the\nmaximum stress within the springs dangerously close to the tolerable limits.”\nReviewer 2 - Comment #17: “74: what do you mean by recently emerged? (would be useful to cite the\noriginal anti-spring works)”\nReviewer 2 – Comment #17 – Response: We appreciate the reviewer's suggestion. Anti-spring\nis not recently emerged. We intended to convey that anti-spring has been recently introduced into\nMEMS accelerometers and gravimeters. This sentence has been revised to “As an effective way\nto reduce stiffness, the anti-spring mechanism has been recently introduced into MEMS\naccelerometers and gravimeters”. The references involving anti-springs were also added.\nChanges in the manuscript.\n-Line 73, 74. “As an effective way to reduce stiffness, the anti-spring mechanism1,15,31 has been\nrecently introduced into MEMS accelerometers and gravimeters31,32,33.”\nReviewer 2 - Comment #18: “83: do you mean best-ever?”\nReviewer 2 – Comment #18 – Response: We appreciate the reviewer's suggestion. Yes, what\nwe originally intended to express was the best-ever or lowest reported. Referring to Comment #8,\nwe have revised it to “lowest reported resonant frequencies”.\nChanges in the manuscript.\n-Line 82. “… the lowest reported resonant frequencies …”.\nReviewer 2 - Comment #19: “84: cite works related to freeform geometries”\nReviewer 2 – Comment #19 – Response: We appreciate the reviewer's suggestion. We have\nadded references related to freeform geometries, including freeform optical components [Optica,\n2021, 8(11): 1501-1506], freeform shells [Thin-Walled Structures, 2023, 185: 110559], and freeform\nbeams [Thin-Walled Structures, 2024, 195: 111418].\n-Line 83, 84. “The introduction of freeform geometries35,36,37 …”.\nReviewer 2 - Comment #20: “85: please use the correct accent in Bezier”\nReviewer 2 – Comment #20 – Response: We appreciate the reviewer's suggestion. It has\nbeen revised to “Bézier”.\nChanges in the manuscript.\n-Line 85. “… polynomial and Bézier curves …”.\nReviewer 2 - Comment #21: “86/87: please rephrase this sentence to explain the work of Wang et al in\nsimple terms and how that work is different than the work reported by the authors”\nReviewer 2 – Comment #21 – Response: We appreciate the reviewer's suggestion. This\nsentence has been revised to “For example, Wang et al. used polynomial and Bezier curves to\nachieve global optimization, while the lack of local adjustability made it challenging to perform\nfine-tuning of the design.”\nChanges in the manuscript.\n-Line 85, 86, 87. “For example, Wang et al. used polynomial and Bezier curves to achieve global\noptimization, while the lack of local adjustability made it challenging to perform fine-tuning of the\ndesign.”\nReviewer 2 - Comment #22: “88: what do you mean by “anti-spring usually experience nonlinear stiffness\ntuning during work””\nReviewer 2 – Comment #22 – Response: We appreciate the reviewer's suggestion. What we\nintended to convey was that there would be a change in stiffness during the loading process of\nthe anti-spring, and this stiffness change is often nonlinear. In the revised manuscript, this\nsentence has been revised to “Anti-springs undergo dramatic changes in stiffness during the\nloading process.”\nChanges in the manuscript.\n15\n-Line 87, 88. “Anti-springs undergo dramatic changes in stiffness during the loading process.”\nReviewer 2 - Comment #23: “88-95: this are very important sentences. The authors make several claims\nhere which need to be simplified. What is main problem that the authors are trying to solve? Reduce\ncomputation? Local adjustability?”\nReviewer 2 – Comment #23 – Response: We appreciate the reviewer's suggestion. We have\nsignificantly revised these sentences. The main problem we aim to address is the high\ncomputation cost and difficulty in achieving convergence in a single optimization run due to the\nvast parameter space of the F-AS. Local adjustability is necessary to decompose the expensive\noptimization problem.\nIn the revised manuscript, these sentences have been revised to “Therefore, although F-AS can\nprovide a larger design parameter space, the vast parameter space results in a high computation\ncost, making it tough to find an optimal design in a single-stage optimization, even with\nconventional metaheuristic algorithms. The local adjustability of freeform geometry and feasible\nalgorithms are both necessary to decompose the expensive optimization problem of F-ASs.”\nChanges in the manuscript.\n-Line 89-93. “Therefore, although F-AS can provide a larger design parameter space, the vast\nparameter space results in a high computation cost, making it tough to find an optimal design in\na single-stage optimization, even with conventional metaheuristic algorithms. The local\nadjustability of freeform geometry and feasible algorithms are both necessary to decompose the\nexpensive optimization problem of F-ASs.”\nReviewer 2 - Comment #24: “96: the sentence needs rephrasing. What do you mean by “crucial factor\nof the MEMS gravimeter”?”\nReviewer 2 – Comment #24 – Response: We appreciate the reviewer's suggestion. This\nsentence has been revised to “High displacement sensitivity is also crucial to achieving μGal-level\nacceleration sensitivity of the MEMS gravimeter.”\nChanges in the manuscript.\n-Line 94, 95. “High displacement sensitivity is also crucial to achieving μGal-level acceleration\nsensitivity of the MEMS gravimeter.”\nReviewer 2 – Comment #25 – Response: We appreciate the reviewer's suggestion. It has\nbeen revised to “… could be further pushed …”.\nChanges in the manuscript.\n-Line 97. “… could be further pushed …”.\nReviewer 2 - Comment #26: “99: it is okay to write picometer-level and femtometer as not everyone will\nimmediately know what fm is.”\nReviewer 2 – Comment #26 – Response: We appreciate the reviewer's suggestion. We have\nincluded the full terms, picometer and femtometer, when \"pm\" and \"fm\" first appear in the revised\nmanuscript.\nChanges in the manuscript.\n-Line 97. “… via picometer- or femtometer-level displacement …”.\nReviewer 2 - Comment #27: “107: use “to meet the design requirements of the MOEMS gravimeter”\nmaybe?”\nReviewer 2 – Comment #27 – Response: We appreciate the reviewer's suggestion. It has\nbeen revised to “… to meet the design requirements of the MOEMS gravimeter”.\nChanges in the manuscript.\n-Line 104, 105. “… to meet the design requirements of the MOEMS gravimeter.”\nReviewer 2 - Comment #28: “135/136: “The uncompressed beams do not deform…”: this statement is\nredundant”\nReviewer 2 – Comment #28 – Response: We appreciate the reviewer's suggestion. This\nstatement has been deleted.\n-Line 135, 136 in the previous manuscript. “The uncompressed beams do not deform when the\ninput force F is zero, corresponding to a displacement of zero.” has been removed.\nReviewer 2 - Comment #29: “128/157: the exposition on the mechanics of anti-spring could be moved to\nsupplementary/appendix. (you could similarly modify Fig 1 )”\nReviewer 2 – Comment #29 – Response: We appreciate the reviewer's suggestion. We have\nmoved the section on the mechanics of the anti-spring to the appendix and modified Fig. 1\naccordingly.\nChanges in the manuscript.\n-Appendix.\nReviewer 2 - Comment #30: “162/175: the design of the MOEMS gravimeter is introduced abruptly. I\nwould suggest that you could start the results section with this and then go on to talk about anti-spring and\nfree-form anti-springs”\nReviewer 2 – Comment #30 – Response: We appreciate the reviewer's suggestion. The result\nsection was thoroughly revised. This section now begins with the overall packaged device of the\nMOEMS gravimeter, followed by the configuration of the F-AS, and then a discussion of the\nmechanics of the anti-spring.\nChanges in the manuscript.\n-Line 124-146.\nReviewer 2 - Comment #31: “Eq 2: I am not sure this is an equation. Also, “s.t. {S,G,R2}” is not a\ncondition. Clearly, you are trying to maximize S and G but minimizing R2. Frame the optimization problem\nusing correct mathematical language.”\nReviewer 2 – Comment #31 – Response: We appreciate the reviewer's suggestion. Eq. 2 has\nbeen revised using the correct mathematical language.\nChanges in the manuscript.\n18\n-Line 190.\nReviewer 2 - Comment #32: “242: can you plot the acceleration-displacement transfer function for a\nrepresentative 1.03 Hz device?”\nReviewer 2 – Comment #32 – Response: We appreciate the reviewer's suggestion. The\nacceleration-displacement transfer function for a 1.03 Hz device is presented in Fig. R3a and b\nvia simulation. Additionally, to demonstrate the transfer function plot obtained using this method\nis accurate, we compared the simulated transfer function of the fabricated structure with the\nexperimental result, as shown in Fig. R3c. The experimental result for the amplitude-frequency\nresponse was obtained by combining the acceleration output with the optical displacement\nresponse gain. It is found that the simulation result is consistent with the experimental result. The\ndiscrepancies are primarily due to temperature/vibration noise and the true signal from the earth\ntremor. Therefore, it can be considered that the transfer function for a 1.03 Hz device we obtained\nfrom the simulation is reliable, as depicted in Fig. R3a and b.\nfrequency curve and b Phase-frequency curve for a 1.03 Hz device. c Acceleration-displacement\namplitude-frequency curve, as a comparison with the experimental result, and d Phase-frequency\ncurve for a fabricated device with a resonant frequency of around 1.6 Hz.\nReviewer 2 - Comment #33: “268: best ever”\nReviewer 2 – Comment #33 – Response: We appreciate the reviewer's suggestion. It has\nbeen revised to “one of the lowest reported …”.\nChanges in the manuscript.\n-238. “… one of the lowest reported …”.\nReviewer 2 - Comment #34: “289: fabrication tolerance instead of fabrication feasibility?”\nReviewer 2 – Comment #34 – Response: We appreciate the reviewer's suggestion. It has\nbeen revised to “fabrication tolerance”.\nChanges in the manuscript.\n-Line 258, 259. “… had excellent performance and fabrication tolerance.”\nReviewer 2 - Comment #35: “296: use the scientific notation for 0.0015673”\nReviewer 2 – Comment #35 – Response: We appreciate the reviewer's suggestion. We have\nused the scientific notation to express the FOM.\nChanges in the manuscript.\n-Line 266. “… with the final FOM being 1.5673 × 10-3.”\nReviewer 2 - Comment #36: “315: “For a certain cavity length…” rather than “With…””\nReviewer 2 – Comment #36 – Response: We appreciate the reviewer's suggestion. It has\nbeen revised to “For a certain cavity length, …”.\n-Line 285. “For a certain cavity length, …”.\nReviewer 2 - Comment #37: “329: the use of the word “performs” doesn’t seem right here”\nReviewer 2 – Comment #37 – Response: We appreciate the reviewer's suggestion. We have\nreplaced “performs” with “has” in the revised manuscript.\nChanges in the manuscript.\n-Line 299. “… the simulation results and has a sensitivity of 1.28 V/µm, …”.\nReviewer 2 - Comment #38: “330: where is the st. dev. Error coming from? (why does the response get\nmore noisy for certain positions?)”\nReviewer 2 – Comment #38 – Response: We appreciate the reviewer's suggestion. The\nstandard deviation error primarily arises from environmental vibrations and disturbances, which\nare transmitted through structural components to the cavity, leading to fluctuations in the cavity\nlength. At certain positions, specifically those with a higher slope, the response becomes noisier\nbecause a greater slope indicates relatively higher sensitivity. This increased sensitivity makes\nthe system more susceptible to the aforementioned disturbances, resulting in a noisier response.\nWe have added a brief discussion to the revised manuscript.\nChanges in the manuscript.\n-Line 300-303. “The standard deviation error primarily arises from environmental vibrations and\ndisturbances, which are transmitted through structural components to the cavity. At certain\npositions, specifically those with a larger slope, the response becomes noisier because a larger\nslope indicates higher sensitivity to the disturbances.”\nReviewer 2 - Comment #39: “452: Is it 1.71 Hz or 1.61 Hz (line# 452)?”\nReviewer 2 – Comment #39 – Response: We appreciate the reviewer's suggestion. In fact,\n1.61 refers to the optimized simulation value, and 1.71 is the experimental value obtained from\nthe turntable experiment. To avoid ambiguity, we have added a statement clarifying that the\nexperimental value is 1.71 Hz.\n-Line 427. “… the turntable experimental result of 1.71 Hz, …”.\nReviewer 2 - Comment #40: “453: explain what 90 degree is”\nReviewer 2 – Comment #40 – Response: We appreciate the reviewer's suggestion. 90° means\nthat the device is under the nominal operation condition. We have deleted the redundant\nexpression of 90° but added the operation condition in the revised manuscript.\nChanges in the manuscript.\n-Line 426, 427. “… of 1.61 Hz under the nominal operation condition, …”.\nReviewer 2 - Comment #41: “483: what is the density of?”\nReviewer 2 – Comment #41 – Response: We appreciate the reviewer's suggestion. The\ndensity is of single-crystal silicon. We have added it to the revised manuscript.\nChanges in the manuscript.\n-Line 458, 459. “… the density of single-crystal silicon …”.\nReviewer 2 - Comment #42: “Table 2: I think writing “@90 deg” is redundant as the devices are expected\nto be operated for the vertical orientation.”\nReviewer 2 – Comment #42 – Response: We appreciate the reviewer's suggestion. We have\ndeleted the redundant statements in the revised Table 2.\nChanges in the manuscript.\n-Table 2.\nWe really appreciate our referee’s comments.\nFirst, we have to apologize for the delayed response. The Editor contacted us almost\none month ago. However, the corresponding author's inbox was filled with messages,\ncausing our secretary to miss this email and fail to inform us in time. We received this\nmessage on October 30, requesting us to supplement the time-series data and code.\nIn the previous manuscript, we used the time-series of Fig. 5f to plot the PSD of the\nslow sampling data, as presented in Fig. 5c. The only difference between Fig. 5f and\nFig. 5d was that the linear drift was removed in Fig. 5f. It can be easily verified by using\nthe time-series data from Fig. 5f to reproduce the slow sampling curve shown in\nprevious Fig. 5c.\nHowever, we realized and totally agree with our referee that using the raw data,\nspecifically the data from Fig. 5d, would be more suitable for plotting the slow sampling\nPSD in Fig. 5c. Now, we have replotted the slow sampling PSD in Fig. 5c by using the\nraw data shown in Fig. 5d. The revised Fig. 5c is identical to the plot that our referee\nobtained (a minor difference may be due variations in the PSD processing parameters),\nand has been attached in the zip file. Additionally, we have provided all the raw time-\nseries data and the plotting codes of Fig. 5c and Fig. R1-Response (for Reviewer 2-\ncomment#0).\nRegarding the year, it was an input error, and we apologize for any confusion caused.\nThe date data can be confirmed with the Shaanxi Earthquake Agency, China.\nThe source data and clean version of the manuscript have been updated. We hope\nthis response can help to address the issue.\nResponse to Reviewers.\n“Title - Multi-stage design of locally adjustable freeform anti-springs: a pathway\ntowards a μGal MOEMS gravimeter with 1 Hz resonant frequency and fabrication tolerance” Manuscript\nNCOMMS-23-62597C.\nWe are deeply grateful to our Referees for such detailed and constructive comments.\nFollowing their suggestions and criticism, we polished the manuscript by identifying the contributions and\nrestructuring the flow. We believe we made it clearer why the research reported here is an\nadvance and achievement in the MEMS gravimeter, and we hope that the revised manuscript\nmeets the high standards of Nature Communications. Below is our point-by-point response to our\nReviewer’s comments.\nReviewer 2 – general assessment: Thanks a lot for making the changes and I appreciate it!\nThere are a certain areas where the manuscript needs quite a bit of improvement. The manuscript still lacks\na coherent flow and tend to confuse the reader (at least, in my case).\nThe authors need to exactly identify their key contributions and then have subheadings and a flow around\nthat.\nReviewer 2 – Comment #0 – Response 1: We appreciate our reviewer’s remark on the\nimprovement of our manuscript. We agree with the reviewer that the previous manuscript needs\nfurther improvement, especially in the presentation. In the revised version, we have restructured\nthe manuscript's flow based on the suggestions provided by the reviewer. We rewrote the abstract\nand introduction, clearly identified the key contributions, added subheadings related to these key\ncontributions, and made significant revisions to the figures and the main text. We hope that this\nrevised version meets the high standards.\nReviewer 2 - Comment #1: “The current abstract is a good example which can be used to highlight the\nissue with the structure/flow.\nAbstract\nThe authors straight away go from from saying that MEMS gravimeters have issues (which readers, who are not\nactively working on these technologies, will not understand) to what they have done.\nWhile doing this, the authors have made general assumptions that may be incorrect. As per my view, MEMS\ngravimetry is an emerging field with \"potential\" applications. There is not a single paper where MEMS\ngravimeters have been used to carry out real studies. It is a bit of stretch to say that they have found success\nin geophysical and space applications.\nIn my opinion the abstract should follow this structure:\n1: What is gravimetry and what are its uses? (1 sentence)\n2: Issues with current gravimeters (1 sentence)\n3: How MEMS gravimeters solve these challenges? (1 sentence)\n4: What are the challenges with current MEMS demonstrations? (1 sentence)\n5: Then introduce your work and key contributions and very simple words\n2\n6: Follow this by key results”\nReviewer 2 – Comment #1 – Response: We are deeply grateful to the reviewer for providing\nsuch a detailed and constructive comment. The abstract was thoroughly rewritten as follows.\nChanges in the manuscript.\n-Abstract\n“Gravimeter measures gravitational acceleration, which is valuable for geophysical applications\nsuch as hazard forecasting and prospecting. Gravimeters have historically been large and\nexpensive instruments. Micro-Electro-Mechanical-System (MEMS) gravimeters feature small size\nand low cost through scaling and integration, which may allow large-scale deployment. However,\ncurrent MEMS gravimeters face challenges in achieving ultra-high sensitivity under fabrication\ntolerance and limited size. Here, we demonstrate a μGal-level Micro-Opto-Electro-Mechanical-\nSystem (MOEMS) gravimeter by combining a freeform anti-spring (F-AS) design and an optical\nreadout. A multi-stage design approach driven by an algorithm is proposed to achieve high\nacceleration sensitivity without making high-aspect ratio springs. An optical grating-based readout\nis integrated, offering pm-level displacement sensitivity. Measurements reveal that the chip-scale\nsensing unit achieves an ultra-low resonant frequency of 1.71 Hz and acceleration-displacement\nsensitivity of over 95 μm/Gal with an etching aspect ratio of smaller than 400:30. The gravimeter\nprototype is benchmarked with a commercial gravimeter PET by the noise-level and tidal\nmeasurement. It demonstrates a self-noise of 1.1 μGal Hz-1/2 at 0.5 Hz, sub-1 μGal Hz-1/2 at 0.45\nHz, and a drift rate down to 153 μGal/day. Such a design approach with locally adjustable F-ASs\npaves the way to extreme design situations of complex structures. The unprecedented high\nperformance and small size of the MOEMS gravimeter suggest potential applications in industrial,\ndefense, and geophysics.”\nReviewer 2 - Comment #2: “Again, the authors should first identify their key contributions, which could\nbe (in no specific order):\n1: Using an algorithmic approach to spring design (providing users with more parameters to play with\ncompared to current designs)\n2: Achieving a high acceleration sensitivity without making high-aspect ratio springs (making fabrication\neasier and more tolerant to fabrication related imperfections)\n3\n3: Integrating an optical readout that has a better displacement sensitivity (to overcome capacitive sensing\nlimits)”\nReviewer 2 – Comment #2 – Response: We fully agree with and greatly appreciate the key\ncontributions summarized by our reviewer. In the revised manuscript, we made minor adjustments\nto the aforementioned contributions and included them in the abstract and introduction. We have\nsummarized two main challenges of the MEMS gravimeter in the first paragraph of the\nintroduction: (1) achieving high acceleration sensitivity with an easily fabricated chip-scale\nstructure, and (2) integrating a readout with improved displacement sensitivity without\ncompromising miniaturization and process compatibility. Then, we outlined the key contributions\nin the third paragraph of the introduction, including: (1) A multi-stage algorithmic approach is\ndeveloped to spring design, providing a larger parameter space and feasible optimization. (2) A\nchip-scale spring-mass system is optimized to achieve unprecedented high acceleration-\ndisplacement sensitivity over 95 μm/Gal without making high-aspect ratio springs. (3) The optical\ngrating-based readout allows displacement measurement with 1.2 pm Hz-1/2 sensitivity and shows\nhigh integration and compatibility with the process.\nChanges in the manuscript.\n-Line 38-40. “A multi-stage design approach driven by an algorithm is proposed to achieve high\nacceleration sensitivity without making high-aspect ratio springs. An optical grating-based readout\nis integrated, offering pm-level displacement sensitivity.”\n-Line 91-98. “A multi-stage algorithmic approach is developed to spring design, providing a larger\nparameter space and feasible optimization. … . A chip-scale spring-mass system is optimized to\nachieve unprecedented high acceleration-displacement sensitivity over 95 μm/Gal (resonant\nfrequency lower than 2 Hz) without making high-aspect ratio springs. The optical grating-based\nreadout allows displacement measurement with 1.2 pm Hz-1/2 sensitivity and shows high\nintegration and compatibility with the process.”\n…\nReviewer 2 - Comment #3: “Then, have headings/subheadings that clearly are related to the key\ncontributions. The content inside these headings should then be clear and succinct. Sometimes having more\nfigurers and sub-figures within the manuscript muddle the key takeaways.”\nReviewer 2 – Comment #3 – Response: We appreciate the reviewer's suggestion. In the\nrevised manuscript, we revised/added subheadings as follows. Level 1 subheadings: Sensing unit\nof the MOEMS gravimeter with F-ASs and optical readout; Multi-stage design of highly sensitive\nanti-springs with fabrication tolerance. Experimental Results Level 2 subheadings: Displacement\nsensitivity measurement of the optical grating-based readout; Acceleration sensitivity and\nresonant frequency measurement of the mechanical sensing unit; Noise level measurement and\ncomparison of gravimeters. In addition, we have significantly streamlined both the text and the\nimages, removing more than six pages. The text has been reduced by more than 1500 words,\nand some sub-figures have been removed based on the reviewer's suggestions.\nChanges in the manuscript.\n-Headings and subheadings.\n-Fig. 1, Fig. 2, Fig. 3, Fig. 5.\n-Main text.\nReviewer 2 - Comment #4: “For example,\nFig 1: do the authors need d and e? can these be moved to the supplementary document? Also, the individual\ncomponents of the device are also annotated alphabetically ((a), (b), ..) which might confuse the readers as\nthe sub-figures are also annotated as a, b, ...\nAlso, there are two deisgns of the proof-mass. One rectangular, while the other one with rectangular\nextensions. I got a bit lost why two designs were used. It would be immensely helpful to maintain consistency.\nThe compression mechanism has not mention anywhere in the script (apart from a few close up images in\nthe supplementary). Were these ever used?”\nReviewer 2 – Comment #4 – Response: We appreciate the reviewer's suggestion. In the\nrevised Fig. 1, the previous sub-figures d and e have been moved to the supplementary document.\nThe individual components of the device, shown in Fig. 1b, were annotated using Roman\nnumerals.\nIn addition, we have revised the geometry of the proof mass to maintain consistency. Actually,\nthe proof mass was adjusted with rectangular extensions to avoid collisions. Refer to our previous\nresponse to Reviewer 1 – Comment #5, the adjustment did not change the center and area of the\nproof mass, nor did it affect the device's modes and mechanical properties. The change in area\nis approximately 0.4%. The modes (the first three modes) before and after the mass shape\nadjustment are shown in Fig. R1b (for structure presented in the original manuscript). It can be\nobserved that the frequencies of the first three modes are nearly identical. The tiny differences\nare likely due to mesh errors. It had almost no effect on its tilting effects and cross-axis sensitivity.\nFig. R1-Response (for Reviewer 2 – Comment #4). a Shape of proof mass before and after adjustment.\nb First three modes of the mechanical sensing unit before and after the proof mass adjustment. c Cross-\naxis sensitivity in the x-direction of two mechanical sensing units.\nRegarding the compression mechanism, they were used to actively compress the anti-springs\nwith a certain distance to lower the stiffness. During fabricating and transportation, we did not\napply the compression, which ensured higher stiffness and enhanced robustness. During use,\ncompression was applied by using the mechanism (Supplementary Fig. 1), placing the beams in\n6\na compressed state, thereby enabling it to achieve quasi-zero stiffness. In the revised manuscript,\nwe have appended the description of the usage of the compression mechanism in the main text.\nChanges in the manuscript.\n-Fig. 1.\nReviewer 2 - Comment #5: “Fig 2: the subfigures related to modal analysis could be moved to the\nsupplementary. I might have suggested to include these but currently they muddle the key messages from\nthe figure.”\nReviewer 2 – Comment #5 – Response: We appreciate the reviewer's suggestion. The sub-\nfigures and discussion related to modal analysis have been moved to Supplementary Note 2.\nChanges in the manuscript.\n-Fig. 2.\n-Supplementary Note 2. Modal analysis results of the optimized structure.\nReviewer 2 - Comment #6: “Fig 3: This is generally good. However, maybe you could combine the Std\nDev and the sensitivity plot (error bars representing the worst std dev figures)”\nReviewer 2 – Comment #6 – Response: We appreciate the reviewer's suggestion. We have\ncombined the Std Dev and the sensitivity plot in Fig. 3c by adding the error bars and Std Dev inset\ninto the sensitivity plot.\nChanges in the manuscript.\n-Fig. 3.\nf, and h (a single timeseries figure) and combine e, g, and i (a single allan deviation figure)?\nAlso, there are blue color series overlaid on each other and two orange color shades on top of each other.\nWhat are these?”\nReviewer 2 – Comment #7 – Response: We appreciate the reviewer's suggestion. We have\ncombined previous Fig. 5d and f, Fig. 5e and g, respectively, obtaining new Fig. 5d and Fig. 5e.\nWe did not merge previous Fig. 5h and Fig. 5i as well because the lines are too numerous and\nthe colors are difficult to distinguish (in our settings, the data of our device use red tones).\nRegarding the issue of overlapping lines with different colors, we have changed all semi-\ntransparent colors to opaque colors and rearranged the order of the legends to align with the\nsequence of colored lines appearing in Fig. 5b and 5c. For example, in the revised Fig. 5c, the\norder of the legends is NLNM-NHNM-MOEMS fast sampling-MOEMS slow sampling-\nPET-Self-noise floor of MOEMS device. Excluding the background lines (NLNM and NHNM)\nand the MOEMS slow sampling series, the order of the different series appearing in the right side\nof the Fig. 5c is as follows: the red color series (MOEMS fast sampling) is at the bottom, the blue\ncolor series (PET) is above it, and the orange color series (Self-noise floor of the MOEMS device)\nis above the blue color series. This order actually corresponds to the sequence of our data\n9\nprocessing. We first obtained the MOEMS fast sampling data. After comparing and processing it\nwith the PET data, we then derived the self-noise floor of the MOEMS device.\nChanges in the manuscript.\n-Fig. 5.\nLine 44/45: MEMS gravimeters have not found success yet in geophysical and space applications. What\nare the space applications? I did not understand.”\nReviewer 2 – Comment #8 – Response: We thank our reviewer for the comment. It is indeed\ninappropriate to say that MEMS gravimeters have found success in geophysical and space\napplications. We used “space applications” in the previous manuscript to refer uses such as space\nstations and extraterrestrial planet exploration. Because Pike et al. sometimes referred to their\nseismometers as gravimeters [Pike, W. T., et al. A compact MEMS gravimeter with sub ng performance.\nNo. EGU24-19904. Copernicus Meetings, 2024.], and their seismometers were previously used as\nthe Lunar/Mars seismometers.\nTo avoid ambiguity, we have removed the description of “space applications” and instead\ndescribed MEMS gravimeters as having potential applications in geophysical applications in the\nrevised manuscript.\nChanges in the manuscript.\n-Line 32. “Gravimeter measures gravitational acceleration, which is valuable for geophysical\napplications such as hazard forecasting and prospecting.”\n-Line 55, 56. “Micro-Electro-Mechanical-System (MEMS) gravimeter is an emerging field with\npotential applications in terms of …”.\n-Line 392, 393. “… proves to be a potential candidate in industrial, defense, civil, and scientific\napplications.”.\nReviewer 2 - Comment #9: “Line 65: Is the drift rate of 20 ugal/day for CG-5/PET corrected or\nuncorrected?”\nReviewer 2 – Comment #9 – Response: We appreciate the reviewer's suggestion. The drift\nrate of around 20 μGal/day is the corrected stability. We have revised the description and added\nthe reference.\nChanges in the manuscript.\n-Line 60. “… and corrected long-term stability of around 20 μGal/day10,11.”\nReviewer 2 - Comment #10: “Line 143: You can make the discussion on geometric anti-springs even\nshorter. Move most things to supplementary (incl Eq 1)”\ndiscussion on geometric anti-springs have been moved to Supplementary Note 1.\nChanges in the manuscript.\n-Results-Sensing unit of the MOEMS gravimeter with F-ASs and optical readout.\nChanges in the Supplementary Information.\n-Supplementary Note 1. Mechanics and analytical calculation of the response of the F-AS.\nReviewer 2 - Comment #11: “Line 227: Why are you trying to get a W shaped response? Explain clearly.”\nReviewer 2 – Comment #11 – Response: In the previous revision, we mentioned achieving a\nW-shaped response, primarily considering that for turntable experiments, a W-shaped response\noffers better linearity compared to a V-shaped response, albeit at the cost of slightly reduced\nmaximum sensitivity at 90°. However, this design consideration was mainly for the turntable\nexperiments and would introduce misleading in understanding. Therefore, in the revised\nmanuscript, we have removed this description and simplified the explanation to that the second-\nstage design aims to achieve a linear acceleration-displacement response. The new revision\nsignificantly streamlines and clarifies the purpose and description of the multi-stage design, and\nwe hope it make the main text clear and succinct.\nChanges in the manuscript.\n-Line 200, 201. “The second-stage design was performed to adjust the anti-springs locally to\naddress the issue of relatively low linearity and high local stress.”\nReviewer 2 - Comment #12: “Line 395: why do you have day^1?”\nReviewer 2 – Comment #12 – Response: This superscript was intended to indicate a reference.\nHowever, it did indeed cause misunderstanding. We have removed the superscript in the revised\nmanuscript to eliminate this confusion.\nChanges in the manuscript.\n-Line 344, 345. “The drift is comparable to other MEMS gravimeters, which is around 100-300\nμGal/day.”\nWhat is limiting the short-period performance of the MOEMS gravimeter (I am not talking about the bump\ndue to thermal control)? Is it the optical readout? Would you be able to add the acceleration noise floor of\nthe optical readout so it is more clear to readers the fundamental limits of the MOEMS gravimeter.”\nReviewer 2 – Comment #13 – Response: We thank our reviewer for the constructive question.\nRegarding the short-period performance limit, we suppose that the optical readout can be\nconsidered the main limiting factor. However, the optical readout noise here, as shown in Fig. R2,\nencompasses the noise from both the optical displacement transducer and the signal processing\nsystem. Fig. R3 depicts the acceleration noise floor of the optical readout and the mechanical\nthermal noise in the PSD plot. It demonstrates that, apart from the bump caused by thermal control,\nthe optical readout noise is close to the self-noise floor, while the mechanical thermal noise is\nsignificantly lower than the self-noise floor. Note that the optical readout noise is a bit larger than\nthat of the standalone setup of the optical displacement sensing unit. This is because it coupled\ndisturbances such as environmental vibrations through mechanisms that were different from the\nstructural components of the optical displacement measurement setup. Many of the noise sources\nwere coupled, including environmental vibrations and noise from temperature control, which were\nusually difficult to separate. It can be stated that optical readout noise is indeed one of the main\ncontributors.\nWe have included the above image in Supplementary Fig. 6. We have also added a brief\ndiscussion on the fundamental limits of the MOEMS gravimeter in the main text.\nFig. R2-Response (for Reviewer 2 – Comment #13). The noise model sources for the MOEMS gravimeter\nshowing the topology and domain of their injection into the signal path.\ndata of MOEMS gravimeter and the reference gravimeter, including the optical readout noise and\nmechanical thermal noise.\nChanges in the manuscript.\n-Line 336-341. “The noise bump around 0.1 Hz could be due to insufficiently precise temperature\ncontrol. Finer temperature control and temperature compensation are expected to suppress the\nnoise level in this frequency range. Supplementary Fig. 6 depicts the acceleration noise floor of\nthe optical readout and the mechanism in the power spectral density plot. It demonstrates that,\napart from the bump caused by thermal control, the optical readout noise is close to the self-noise\nfloor, which can be considered the main limiting factor.”\nChanges in the Supplementary Information.\n-Supplementary Fig.6.\n14"}]

Decision letter