Concept of the sensing paradigm of A2F based on PRC

In conventional sensing paradigm, the signal feature extraction for classification tasks usually relies on physically separated sensing part and processing part. The sensing part usually consists of specific sensors that are only responsible for signal acquisition, while the signal feature extraction also requires a series of analog signal pre-processing (filtering, noise reduction, normalization, etc.), time-frequency domain transformation, analog-to-digital conversion (ADC), and computing in the processing part, as schematically illustrated in Fig. 1a. This physically separated sensing and processing paradigm inevitably imposes a heavy burden on transmission bandwidth and power consumption. In this work, we use a stiffness modulated differential resonant accelerometer with two identical resonators as an edge device to execute the signal acquisition and processing, which significantly reduces data transfer and simplifies the system structure, as schematically illustrated in Fig. 1b and Fig. 1c. The accelerometer can respond to the acceleration perturbation along its sensitive direction by the proof mass, which changes the axial stiffness of the resonant beams connected to the proof mass by micro-leverage structures. The change in axial stiffness causes a shift in the resonance frequency of the resonant beam, which is usually detected by a closed-loop interface circuit. Then the acceleration can be calculated by the relationship between the acceleration and the frequency shift. Meanwhile, we utilize its inherent nonlinearity and short-term memory characteristics to build a physical reservoir to realize feature extraction and classification for analog input36, which is referred to as Analog-to-Feature (A2F) conversion. Owing to the superiority of the physical reservoir without adaptive updating in training and the feasibility of designing a reservoir using a micromechanical resonator with nonlinear dynamics, the reservoir-based sensing paradigm of A2F can serve as an ideal candidate for a near-sensor or edge computing paradigm with high speed, low cost, and energy efficient. For another resonator in this differential resonant accelerometer, we use it to monitor the actual acceleration input corresponding to the calibrated electrostatic force input, as shown in Fig. 1d.

The nonlinear dynamics of the stiffness modulated MEMS accelerometer

As the key device that serves the dual purpose of sensing and processing for the reservoir-based sensing paradigm, the differential resonant accelerometer with specially designed sensitivity enhancing structure and tuning structure has been reported in our previous work41–43, as schematically illustrated in Fig. 1C. The details of the fabrication and the size information for the device is shown in Materials and Methods part. Here, we focus on its nonlinear dynamics that are crucial to the physical reservoir. The sensitive beam connected to the proof mass can be simplified as a second-order mass-damper-spring system, whose motion equation can be written as,

$${m}_{r}\ddot{z}+c\dot{z}+\left({k}_{m}+{k}_{axial}\right)z+{k}_{m3}{z}^{3}=\frac{1}{2}\frac{{C}_{0}d}{{\left(d-z\right)}^{2}}{\left({V}_{dc}+{V}_{ac}\text{cos}{\omega }_{d}t\right)}^{2}-\frac{1}{2}\frac{{C}_{0}d}{{\left(d+z\right)}^{2}}{\left({V}_{dc}\right)}^{2}$$

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where \(z,\dot{z},\ddot{z}\) are the displacement, velocity and acceleration of the resonant beam in transverse motion, respectively, \({m}_{r}\) is the effective mass of the resonant beam, \(c=\frac{{m}_{r}{w}_{n}}{Q}\) is the equivalent damping coefficient, \({k}_{m}\) is the linear stiffness parameter, \({k}_{axial}\) is the axial stiffness perturbation caused by the external acceleration, \({k}_{m3}\) is the nonlinear (cubic) stiffness parameter, \(d,{C}_{0}\) are the initial distance and the corresponding capacitance between the resonant beam and electrode, respectively, \({V}_{dc}\) is the bias voltage applied on the resonant beam, \({V}_{ac},{\omega }_{d}\) are the amplitude and angular frequency of the driven signal for the resonant beam.

According to our previous work36, a specifically designed resonant beam exhibits hybrid nonlinear dynamics including the duffing nonlinearity and the intrinsic transient exponential nonlinear response characteristics when it is seen as a typical underdamped second-order oscillation system, which guarantees the dynamic nonlinear features and fading memory of the reservoir. Thanks to the complex nonlinear phenomena caused by the scale effect and fabrication tolerance in the MEMS resonant accelerometers, the nonlinear dynamics of the resonant frequency and amplitude of the resonant beams with the external acceleration can act as a physical reservoir able to map temporal inputs into a feature space that can be analyzed by a trained readout layer. Figure 2a shows the simulation result for the frequency shift (delta-f = f-f0) dependent dynamical nonlinearity of the resonant beam with the external acceleration input. Figure 2b shows the simulation result for the amplitude dependent dynamical nonlinear hysteresis of the resonant beam with the external acceleration input, which is similar to the electrostatic spring softening effect in the duffing nonlinearity. As we all know, the hysteresis region between the two bifurcation points ([a=-54g, a=-1g]) reserves complex nonlinear dynamics. But as an edge sensor, the differential resonant accelerometer should be driven to an appropriate nonlinear region to ensure not only the nonlinear transformation of the input signal, but also the stable sensing of the external acceleration. Considering the resonant frequency of the resonant beam needs to be tracked and locked by a Phase Lock Loop (PLL), which is incompatible with the frequency detection through the closed-loop interface circuit. Therefore, we use the nonlinear response between the amplitude of the resonant beam and the external acceleration as the reservoir dynamics in this work. The grey region represents the optimal dynamic range of the input acceleration for our system, which corresponds to the classification accuracy over 90% for different motion postures, as shown in Fig. 2d. Experimentally, we use electrostatic force instead of the external acceleration to perturbate the stiffness of the resonant beam to verify the dynamical nonlinearity of the device. The simulation and experimental results are shown in the Fig. 2c.

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The experimental setup for the PRC with a stiffness modulated accelerometer

The differential resonant accelerometer is packaged and integrated on a printed circuit board (PCB), which is driven by the electrostatic force instead of the external acceleration, and detected by an interface circuit, as shown in Fig. 3b and **c**. To calibrate the external acceleration dependent electrostatic force, we fix the PCB with the packaged accelerometer on a triaxial rotating stage, and simulate the acceleration of ± 1g by changing the orthogonal position between gravity and the sensitive axis, as shown in Fig. 3a. A lock-in amplifier (Zurich Instruments MFLI500kHz/5MHz) is used to sweep frequency and detect the frequency shift. A power source (Keysight U8032A ) is used to apply electrostatic force to equilibrate the gravity. Results show that the electrostatic force corresponding to the 27.6V DC voltage is equal to the inertial force generated by 1g acceleration, as shown in Fig. 2c. Figure 3d shows the optical microscopy image of the differential resonant accelerometer with a scale bar of 1 mm. The comb driving structure, clamped-clamped beam structure, and micro lever structure are highlighted in red dotted frames 1, 2, 3, respectively, which correspond to the SEM images Fig. 3e, Fig. 3f, and Fig. 3g.

The classification for the acceleration input via the MEMS accelerometer-integrated RC

As a novel physical reservoir without digital operation such as delay and mask, the designed resonant beam with hybrid nonlinear dynamics has shown excellent performance on feature extraction and classification based on some standard benchmarks36. In this work, we migrate this novel physical reservoir on the differential resonant accelerometer to make it an edge sensor. The priority is to verify its performance for the acceleration input corresponding to the different motion postures by the customized dataset acquired from a homemade six-axis IMU sensor, which is elaborated in our previous work36. Figure 4a shows the different acceleration waveforms in the time domain corresponding to the 8 motion postures, including “Jump”, “Walk”, “Jog”, “Squat”, “Stretch”, “ChestE”, “ArmC”, and “BodyC. Thanks to the nonlinear dynamical mechanism that stiffness perturbation caused by the external acceleration and the non-delay architecture without mask operation in the digital domain, the input acceleration can be directly fed into the dynamics of the reservoir and be classified by feature extraction from the analog domain with low power consumption. Figure 4b shows the color block diagram for the results of the different features corresponding to the 8 motion postures extracted by the nonlinear transformation of our MEMS accelerometer-integrated RC. We perform ten times tenfold cross-validation tests and use the confusion matrix to characterize the classification performance on the customized dataset for motion posture recognition. Each sample of the test dataset is divided into 15 groups and tested by ten times. Results show a classification accuracy of (97.33 ± 0.136)%, as shown in Fig. 4c. A box plot inset displays the discrete distribution of the data. Figure 4d shows the results of the false-color confusion matrix. The color bar represents the number of samples whose predicted result are consistent with the target result. The power consumption of the core device including the accelerometer and its driving interface circuit in our prototype system is estimated to be less than 300 ± 1mW, as shown in Fig. 3. In our experimental setup, the power consumption of the interface circuit is approximated by Pc = 5V×0.06A = 300mW, while the power consumption of the differential resonant accelerometer is approximated by Pa = 100mV×0.01A + 10V×0.01uA = 1mW.