Modeling temperature effects in a MEMS ring gyroscope: toward physics-aware drift compensation

Abstract

Temperature plays an indispensable role in the long-term performance of MEMS gyroscopes, and despite extensive studies in the literature, analytical treatment of temperature effects is still an open problem. This paper, to the best of our knowledge, is the first attempt to address this gap for ring gyroscopes. We start with a superposition principle that disentangles thermal displacement fields from the gyroscope's nominal vibration. We set forth a geometrically nonlinear variational formulation to obtain the temperature-induced stiffness matrix. We conduct temperature tests on our 3.2 mm-diameter, 58 kHz ring gyroscopes equipped with 16 capacitive stress sensors. The experimental data validate our analytical modeling in the following key aspects: 1) The model accounts for not only changes in material properties but also a less explored factor, thermal stresses. Thanks to a strain interpolation module that leverages the measured stresses, the model predicts frequency variations consistently and captures hysteresis loops arising from residual stresses. Notably, we accurately estimate the deviation of the temperature coefficient of frequency (TCF) from the expected value -30 ppm/C-degrees (based on the widely known -60 ppm/C-degrees dependency of Young's modulus of silicon). 2) The model is able to capture stiffness couplings in the orders of less than 0.1 N/m (in a 7 kN/m device) and closely predicts the quadrature error and its leakage into the in-phase channel. Additionally, the model incorporates temperature variations of mechanical scale factor, drive mode's amplitude, damping coupling, and sense mode's phase in terms of their contribution to the in-phase error. Based on these merits, our model serves as a building block toward drift compensation algorithms encompassing the underlying physics of the temperature effects.