Achieving Long-term Bias Stability in High-q Inertial Mems by Temperature Self-sensing with a 0.5 Millicelcius Precision

We present long-term bias drift compensation in high-quality (Q) factor MEMS gyroscopes using real-time temperature selfsensing. The approach takes advantage of linear temperature dependence of the drive-mode resonant frequency for selfcompensation of temperature-induced sense-mode drifts. The approach was validated by a vacuum packaged silicon quadruple mass gyroscope, with signal-to-noise ratio (SNR) enhanced by isotopic Q-factors of 1.2 million. Owing to high Q-factors, a measured frequency stability of 0.01 ppm provided a temperature self-sensing precision of 0.0004 C, on par with the state-of-the-art MEMS resonant thermometers. Real-time self-compensation yielded a total bias error of 0.5 /hr and total scale-factor error of 700 ppm over temperature variations. This enabled repeatable long-term rate measurements required for MEMS gyrocompassing with a milliradian azimuth precision. INTRODUCTION In recent years several groups have reported silicon MEMS gyroscopes with sub-degree per hour Allan deviation of bias [1-3]. However, long-term bias and scale-factor drifts limit their potential in real-world missions. The drift source for most MEMS is their inherent sensitivity to temperature variations. An uncompensated bias sensitivity of 500 (/hr)/C is typical for MEMS gyroscopes [4]. Whereas, quartz and silicon MEMS resonators used in timing applications exhibit orders of magnitude higher long-term stability, due to more advanced compensation techniques. For instance, high-stability dual-mode oscillators use the secondary mode as a thermometer for the compensation of primary mode drifts [5]. In contrast, conventional approaches for gyroscope's calibration rely on third-order thermal models and external temperature sensors, which suffer from such effects as thermal lag and temperatureinduced hysteresis. These limitations motivate the development of new real-time self-calibration methods for inertial MEMS. Fused quartz hemispherical resonator gyroscope utilizes a temperature compensation technique which uses the resonant frequency as a measure of gyroscope temperature [6]. Successful implementation of this self-sensing technique relies on two main factors. The first is linearity of the temperature-frequency dependence of the resonator material, and the second is high frequency stability, brought forth by a high Q-factor. Recently, we introduced a MEMS quadruple mass gyroscope (QMG) [2,3,7], which satisfies these requirements with Q-factor above 1 million, and linear temperature coefficient of frequency (TCF), thanks to the single crystalline silicon resonator body, Fig. 1 inset. Frequency-based measurements of temperature also provides inherently better stability then amplitude-based (voltage) readings commonly employed in temperature sensors (1 ppm stability and repeatability is easy in frequency domain, but almost impossible in analog signal domain). These make high-resolution self-sensing potentially possible in silicon MEMS technology. In this paper, we demonstrate that the resonant temperature self-sensing can be used in real-time for high-Q MEMS gyroscopes (Fig. 2) to yield a subdegree per hour total bias error over temperature variations. We also demonstrate that the long-term stability provided by the selfcompensation approach allows for repeatable measurements of small angular rates, required for gyrocompassing applications. TEMPERATURE SENSITIVITY ANALYSIS Here we analyze temperature-induced drift sources in high-Q gyroscopes and show the importance of temperature compensation. Temperature-Induced Drifts Operation of vibratory z-axis angular rate gyroscopes is based on energy transfer between two vibratory modes, Fig. 2. The drivemode is continuously excited at resonance, and the sense-mode is used for the rate detection. The amplitude (y) of the sense-mode is proportional to the rate (z), with scale-factor (SF) and bias (B): ). ( B SF y z     (1) Assuming a worst-case scenario, the sense-mode is openloop, and thus more susceptible to temperature variations. Scalefactor and bias are functions of the angular gain (k  1), the sensemode natural frequency y, and the drive amplitude (x) [8]: , 2 y eff x kQ SF   . 2 ) 2 sin( ) / 1 (      B (2) Here, (1/) is the damping mismatch between vibratory modes,  is the principal axis of damping, and Qeff is the effective Q-factor: , ) ( 4 1 2 2 y y y eff Q Q Q      (3) which reaches maximum Qy at zero frequency mismatch ( = 0). 9780964002494/HH2012/$25©2012TRF 287 Solid-State Sensors, Actuators, and Microsystems Workshop Hilton Head Island, South Carolina, June 3-7, 2012 30 40 50 60 70 80 −1200 −1000 −800 −600 −400 −200 0 200 Temperature, °C D riv e− m od e fr eq ue nc y Δf /f, p pm Linear fit Measured TCF: −24 ±0.04 ppm/°C Figure 3: Linear frequency-temperature dependence, revealing a −24 0.04 ppm/°C TCF over temperature range of 30 C to 75 C. 10 10 10 10 10 10 10 10 10 Integration time τ, s Fr eq ue nc y st ab ili ty σ y ( τ) , p pb Allan deviation fit τ−1/2 slope fit τ0 slope