Angle Random Walk Research Articles

Two-level and two-period modulation for closed-loop interferometric fiber optic gyroscopes.

We propose and experimentally demonstrate a modulation technique for closed-loop interferometric fiber optic gyroscopes that improves its scale factor control and allows for the construction of gyroscopes with optimized angle random walk. The proposed two-level and two-period modulation is composed of two intercalated periods of the square wave modulation, one with amplitude of ϕ rad and the other with amplitude of 2π-ϕ rad. As in the two-level modulation, the angular velocity is obtained from the difference of consecutive output levels. Modulation depth error is obtained from the difference of the mean levels of the output from two consecutive periods. The proposed technique eliminates the limitation of square wave modulation: inefficient scale factor control at low angular velocities. Additionally, it requires a slower analog-to-digital converter than four-level modulation.

A Dual-Mode Actuation and Sensing Scheme for In-Run Calibration of Bias and Scale Factor Errors in Axisymmetric Resonant Gyroscopes

This paper demonstrates a dual-mode actuation and sensing scheme for differential operation and self-calibration of axisymmetric Coriolis resonant gyroscopes. The proposed scheme actuates both modes of an axisymmetric gyroscope with two identical in-phase excitation signals, and senses both modes concurrently. This dual-mode actuation architecture utilizes the difference of the individual mode outputs to cancel out the common-mode bias terms, and provide rate-independent frequency split monitoring. The symmetry of the dual-mode architecture can be utilized to provide in-run scale factor calibration capability by mimicking the mechanical Coriolis force in the electrical domain, thereby providing a virtual electrical rotation to the gyroscope, to estimate the drift of the physical scale factor over time and temperature. The dual-mode architecture is demonstrated on a 2.625-MHz substrate-decoupled bulk acoustic wave gyroscope. The inherent bias cancellation provides sub-10°/hr bias instability, with $1.4\times $ improvement of angle random walk, as compared with the conventional single-mode actuation scheme. Benefiting from the in-run mode-matching capability, the dual-mode actuation scheme exhibits $45\times $ reduction of bias drift over a temperature range of [10–80] °C. In-run scale factor calibration achieves $150\times $ reduction of scale factor turn-on repeatability error down to 62 ppm, and over $100\times $ reduction of the temperature drift of scale factor over [10–50] °C.

Decaying Time Constant Enhanced MEMS Disk Resonator for High Precision Gyroscopic Application

This paper reports a new design strategy of adding lumped masses to the frame structure of a disk resonator gyroscope (DRG) to mitigate its figure of merit (FOM). A greatly enhanced decaying time constant τ can lead to very small FOM, which could then greatly mitigate the DRG's bias and bias drift. Additionally, this technique could also reduce the Brownian noise floor. A comprehensive investigation of the possible ways to add the lumped masses is presented, and a detailed design guideline is provided. A DRG prototype based on this design strategy is then presented and shows a τ of 38.5 s, a quality factor Q of 358 k, and a FOM of 1.9°/s. The prototype is operated in the force-to-rebalance mode, and an angle random walk of 0.012°/√h and a bias stability of 0.08°/h are demonstrated experimentally. This design strategy is suitable for use in batch fabrication of very high precision microelectromechanical systems devices such as gyroscopes.

Investigation, modeling, and experiment of an MEMS S-springs vibrating ring gyroscope

This study presents a microelectromechanical systems S-springs vibration ring gyroscope (MSVRG), which is driven by electrostatic force and detected by capacitance. First, a ring resonator structure with eight S-shaped symmetrical supporting springs is developed, and the capacitor electrodes are designed according to the vibration characteristics of the ring resonator. Then, a precise equivalent stiffness model of MSVRG is established based on the vibration mechanics and the Cassette theorem, which can be employed for any other ring gyroscope with differently shaped springs. Moreover, the mode resonant frequency error of between experiments, finite-element analysis (FEA) and theory calculation, is 10.54% and 3.76%, respectively. After that, the process of MSVRG structure is introduced and the structure is manufactured, and the theory calculation and FEA value with processed parameters have 1.19% and 4.59% difference with resonant frequency tested value, respectively. Finally, the static performance of the fabricated MSVRG is tested, and the bias instability is about 0.0119 deg/s and angle random walk is about 0.0359 deg/s1/2 at room temperature.

Temperature Energy Influence Compensation for MEMS Vibration Gyroscope Based on RBF NN‐GA‐KF Method

This paper proposed three methods to compensate the temperature energy influence drift of the MEMS vibration gyroscope, including radial basis function neural network (RBF NN), RBF NN based on genetic algorithm (GA), and RBF NN based on GA with Kalman filter (KF). Three‐axis MEMS vibration gyroscope (Gyro X, Gyro Y, and Gyro Z) output data are compensated and analyzed in this paper. The experimental results proved the correctness of these three methods, and MEMS vibration gyroscope temperature energy influence drift is compensated effectively. The results indicate that, after RBF NN‐GA‐KF method compensation, the bias instability of Gyros X, Y, and Z improves from 139°/h, 154°/h, and 178°/h to 2.9°/h, 3.9°/h, and 1.6°/h, respectively. And the angle random walk of Gyros X, Y, and Z was improved from 3.03°/h1/2, 4.55°/h1/2, and 5.89°/h1/2 to 1.58°/h1/2, 2.58°/h1/2, and 0.71°/h1/2, respectively, and the drift trend and noise characteristic are optimized obviously.

A novel MEMS S-springs vibrating ring gyroscope with atmosphere package

This study presents a new MEMS vibrating ring gyroscope (VRG), which is driven by electrostatic force and detected by capacitance. A novel ring resonator with eight S-shaped symmetrical supporting springs is developed based on the advantageous characteristics of a thin-shell vibrating gyroscope. The capacitance electrodes, including the drive electrodes, the sense electrodes and the mode control electrodes, are designed according to the vibration characteristics of the ring resonator and the shape of the supporting springs. In addition, the operational principle of these electrode capacitors and the electrostatic force of the drive electrodes are discussed in detail. The gyroscope with high aspect-ratio structures is manufactured through an efficient fabrication process. Finally, the performance characteristics of the fabricated VRG are tested, and the experimentally obtained the zero-bias instability is about 0.0167°/s and angle random walk (ARW) is about 0.1363°/s1/2 at room temperature. Experimental results show that the VRG has a simple structure and relatively better performance characteristics for low and medium angular velocity measurements.

Temperature‐dependence improvement for a MEMS gyroscope using triangular‐electrode based capacitive detection method

This work presents analysis and test results on temperature-dependence improvement for a microelectromechanical systems (MEMS) gyroscope based on triangular-electrode based (TEB) capacitive detection method. To reduce the effects of temperature variations on MEMS gyroscopes, TEB capacitive detection method is applied because the extracted amplitude and phase are robust to system parameter fluctuations. In the first place, analytical results for the scale factors and zero-rate outputs, respectively, utilising linear electromechanical amplitude modulation (linear-EAM) method and TEB method are derived. Then experimental results of the temperature characteristics for gyroscopes are conducted. The temperature coefficient of scale factor over the temperature range of −10 to 60°C is reduced from −8845 ppm/°C for linear-EAM method to 1660 ppm/°C for TEB method with an improvement factor of 5.3. Moreover, the temperature-induced zero-rate output variation is decreased from −68.05°/s for linear-EAM method to −29.09°/s for TEB method with an improvement factor of 2.3. Experimental results of temperature-dependence improvement agree well with the theoretical analysis. In the end, Allan variance analysis demonstrates a bias instability of 2.85°/h, and an ARW (angle random walk) of 0.16°/√h with a vacuum-packaged gyroscope using TEB method, which is significantly better than the bias instability 5.47°/h and ARW 0.22°/√h tested using linear-EAM method.

Interferometric Optical Gyroscope Based on an Integrated Si3N4 Low-Loss Waveguide Coil

We describe the measurement and characterization of an interferometric optical gyroscope that uses an integrated 3-m large-area silicon nitride waveguide coil with waveguide loss <0.78 dB/m that is compatible with wafer-scale integration. The angle random walk and bias instability were measured to be 8.52°/h1/2 and 58.7°/h, respectively. The measured performance is comparable to that of a commercial rate grade gyroscope, demonstrating that chip-scale integration is a feasible path to low cost, low power, compact implementation of optical gyroscope, and realization of longer coils will expand use of this technology to more demanding navigation applications.

Mechanical and Electrical Noise in Sense Channel of MEMS Vibratory Gyroscopes.

This paper presents a theoretical analysis of mechanical and electrical noise in the sense channel of micro-electromechanical systems (MEMS) vibratory gyroscopes. Closed-form expressions for the power spectral density (PSD) of the noise equivalent rate (NER) of gyroscopes in the open-loop and the force-rebalance operations are derived by using an averaged PSD model and an equivalent transfer function. The obtained expressions are verified through numerical simulations, demonstrating close agreements between the analytic and the numerical models. Based on the derived expressions for the PSD of the NER, the impacts of the modal frequency split, quality factor, and the gain of the feedback forcer, as well as the gain of the signal conditioning circuit, on the gyroscope noise characteristics are theoretically analyzed. In addition, the angle random walk (ARW) and the standard deviation of the NER are also discussed through the PSD models. Finally, the effects of the loop closing, the mode matching, and the gain of the feedback forcer on the PSD of the NER were verified via a MEMS vibratory gyroscope with a tunable modal frequency split.

Fundamental limitations of sensitivity of whispering gallery mode gyroscopes

We analyze theoretically both the fundamental and the technical quantum limitations of the sensitivity of a passive resonant optical gyroscope based on a high finesse monolithic optical microcavity. We show that the quantum back action associated with the resonantly enhanced optical cross- and self-phase modulation results in the standard quantum limit of the angle random walk of the gyroscope, which reaches approximately 0.2 deg/hr1/2 for a millimeter scale CaF2 whispering gallery mode resonator based device.

Signal integrity in capacitive and piezoresistive single- and multi-axis MEMS gyroscopes under vibrations

The work investigates effects of external mechanical vibrations, with a frequency up to 40kHz, on the operation of micro-electromechanical gyroscopes of different architecture and sensing technology. The analyzed devices differ (i) in the number of sensing axes (one, two or three) per single drive frame, so in the number of resonant modes within the tested vibration range, and (ii) in the used sense transduction technology (parallel-plate capacitances or piezoresistive nano-gauges). After theoretically modeling effects of vibrations occurring around the frequency of in-plane and out-of-plane resonant modes, in presence of process imperfections and nonlinearities, the work benchmarks the predictions through an extensive experimental campaign: using a shaker, vibrations of controlled amplitude and frequency are applied, in operation, to the devices along both in-plane and out-of-plane axes. Results show that, within the tested sensors, the best tolerance to vibrations, quantitatively measured in terms of angle random walk (ARW) worsening, is achieved for single-axis structures based on piezoresistive sensing. Their ARW degrades in the worst case by 1.2 times, compared to a 230-fold degradation observed in 3-axis capacitive sensors.

Operation of a high quality-factor gyroscope in electromechanical nonlinearities regime

This paper describes the operation of a high quality factor gyroscope in various regimes where electromechanical nonlinearities introduce different forms of amplitude–frequency (A–f) dependence. Experiments are conducted using an epitaxially-encapsulated 2 × 2 mm2 quad-mass gyroscope (QMG) with a quality factor of 85 000. The device exhibits third-order Duffing nonlinearity at low bias voltages (15 V) due to the mechanical nonlinearity in the flexures and at high bias voltages (35 V) due to third-order electrostatic nonlinearity. At intermediate voltages (~26 V), these third-order nonlinearities cancel and the amplitude–frequency dependence is greatly reduced. A model is developed to demonstrate the connection between the electromechanical nonlinearities and the amplitude–frequency dependence, also known as the backbone curve. Gyroscope operation is demonstrated in each nonlinear operating regime and the key performance measures of the gyroscope’s performance, angle random walk (ARW) and bias instability, are measured as a function of drive-mode vibration amplitude. We find that low ARW can be achieved even though the gyroscope’s drive mode exhibits large amplitude–frequency dependence, and that bias instability is largely independent of the operating regime.

Stress Effects and Compensation of Bias Drift in a MEMS Vibratory-Rate Gyroscope

Long-term gyroscope drift can be effectively removed by employing simultaneous on-chip stress and temperature compensation. Stress effects are significant and their inclusion augments the commonly applied temperature compensation. A silicon-on-insulator matched-mode z-axis vibratory-rate gyroscope, as a prototype testbed to study these effects, includes released silicon resistors connected in a Wheatstone bridge as on-chip stress sensors. The gyroscope is ovenized within 300 K ± 20 mK using an external heater and an on-chip temperature sensor to suppress the temperature effects. The gyroscope is in-house vacuum packaged and operated at matched closed-loop drive and sense modes. Stress compensation significantly suppresses long-term drift resulting in 9°/h/√Hz angle random walk and 1°/h bias instability at 10000 s (around 3 h) averaging time, which is seven times improvement over the uncompensated gyroscope output. The sensitivity of zero-rate offset to stress is -0.22°/day/Pa and -0.045°/day/Pa for the tests with and without externally applied stress, respectively.

Resonant pitch and roll silicon gyroscopes with sub-micron-gap slanted electrodes: Breaking the barrier toward high-performance monolithic inertial measurement units

This paper presents the design, fabrication, and characterization of a novel high quality factor (Q) resonant pitch/roll gyroscope implemented in a 40 μm (100) silicon-on-insulator (SOI) substrate without using the deep reactive-ion etching (DRIE) process. The featured silicon gyroscope has a mode-matched operating frequency of 200 kHz and is the first out-of-plane pitch/roll gyroscope with electrostatic quadrature tuning capability to fully compensate for fabrication non-idealities and variation in SOI thickness. The quadrature tuning is enabled by slanted electrodes with sub-micron capacitive gaps along the (111) plane created by an anisotropic wet etching. The quadrature cancellation enables a 20-fold improvement in the scale factor for a typical fabricated device. Noise measurement of quadrature-cancelled mode-matched devices shows an angle random walk (ARW) of 0.63° √h−1 and a bias instability of 37.7° h−1, partially limited by the noise of the interface electronics. The elimination of silicon DRIE in the anisotropically wet-etched gyroscope improves the gyroscope robustness against the process variation and reduces the fabrication costs. The use of a slanted electrode for quadrature tuning demonstrates an effective path to reach high-performance in future pitch and roll gyroscope designs for the implementation of single-chip high-precision inertial measurement units (IMUs).

Research on Reducing the Influence of Laser Frequency Noise on Resonator Optical Gyro

For resonator optical gyro (ROG), widely used frequency locking method is to adjust the frequency of the light source, which may cause line width enhancement, resulting in low accuracy of the gyro. To avoid this situation, a frequency locking scheme for ROG based on a piezoelectric (PZT) cylinder in the resonator was researched. A transmission-type R-FOG with a PZT cylinder for frequency locking was constructed. After that, contrast experiment was conducted. By using the PZT frequency locking scheme, angle random walk was increased from 0.0069°/ $\sqrt {\textrm {h}} $ to 0.0050°/ $\sqrt {\textrm {h}} $ , which has significance in the future work of ROG.

Resonant microphotonic gyroscope

It has been known for four decades that high finesse millimeter-scale optical cavities can produce high-performance, compact optical gyroscopes. Yet, a practical implementation of such a device has been hindered by multiple technical challenges, including Rayleigh scattering and optical nonlinearity of the cavity material. In this Letter we report on the implementation of an integrated passive gyroscope using a monolithic cavity characterized by 7 mm diameter, finesse of 105, and Rayleigh backscattering less than 10 ppm. The device is characterized with quantum noise limited angle random walk of 0.02 deg/h1/2, and bias drift of 3 deg/h, corresponding to detection of rotation-originated optical path change of 1.3×10−16 cm.

Design of High- $Q$ Compact Passive Ring Resonators via Incorporating a Loss-Compensated Structure for High Performance Angular Velocity Sensing in Monolithic Integrated-Optical-Gyroscopes

A compact InP/InGaAsP passive ring resonator (PRR), in which a loss-compensation section is vertically integrated, for angular velocity sensing is theoretically studied in this paper. The loss compensation is realized by current injection into the optical gain section, which is optically coupled with the PRR. Based on the proposed loss-compensation scheme, the propagation loss in the PRR can be reduced down to 0.01 dB/cm. Consequently, the quality factor of up to $1.5 \times 10^{7}$ and resonance depth of more than 7 dB are theoretically obtained in such resonators with a radius of larger than 5 mm. Furthermore, when these compact PRRs are used as angular velocity sensors, the resolution of better than 10°/h and the angle random walk of less than $0.075 {^{\circ }}/\sqrt {h} $ are theoretically predicted. These characteristics indicate that our proposed compact PRRs are capable of acting as an angular velocity sensor in high efficient monolithically integration of resonant optical gyroscope applications in proof of principle.

Stiffness-mass decoupled silicon disk resonator for high resolution gyroscopic application with long decay time constant (8.695 s)

We propose a stiffness-mass decoupling concept for designing large effective mass, low resonant frequency, small size, and high quality factor micro/nanomechanical resonators. This technique is realized by hanging lumped masses on the frame structure. An example of a stiffness-mass decoupled silicon disk resonator for gyroscopic application is demonstrated. It shows a decay time constant of 8.695 s, which is at least 5 times longer than that of the pure frame silicon disk resonator and is even comparable with that of the micromachined three-dimensional wine-glass resonators made from diamond or fused silica. The proposed design also shows a Brownian noise induced angle random walk of 0.0009°/√h, which is suitable for making an inertial grade MEMS gyroscope.

A novel oscillation control for MEMS vibratory gyroscopes using a modified electromechanical amplitude modulation technique

This paper reports a novel oscillation control algorithm for MEMS vibratory gyroscopes using a modified electromechanical amplitude modulation (MEAM) technique, which enhances the robustness against the frequency variation of the driving mode, compared to the conventional EAM (CEAM) scheme. In this approach, the carrier voltage exerted on the proof mass is frequency-modulated by the drive resonant frequency. Accordingly, the pick-up signal from the interface circuit involves a constant-frequency component that contains the amplitude and phase information of the vibration displacement. In other words, this informational detection signal is independent of the mechanical resonant frequency, which varies due to different batches, imprecise micro-fabrication and changing environmental temperature. In this paper, the automatic gain control loop together with the phase-locked loop are simultaneously analyzed using the averaging method and Routh–Hurwitz criterion, deriving the stability condition and the parameter optimization rules of the transient response. Then, a simulation model based on the real system is set up to evaluate the control algorithm. Further, the proposed MEAM method is tested using a field-programmable-gate-array based digital platform on a capacitive vibratory gyroscope. By optimizing the control parameters, the transient response of the drive amplitude reveals a settling time of 45.2 ms without overshoot, according well with the theoretical prediction and simulation results. The first measurement results show that the amplitude variance of the drive displacement is 12 ppm in an hour while the phase standard deviation is as low as 0.0004°. The mode-split gyroscope operating under atmospheric pressure demonstrates an outstanding performance. By virtue of the proposed MEAM method, the bias instability and angle random walk are measured to be 0.9° h−1 (improved by 2.4 times compared to the CEAM method) and 0.068° (√h)−1 (improved by 1.4 times), respectively.

A New Continuous Rotation IMU Alignment Algorithm Based on Stochastic Modeling for Cost Effective North-Finding Applications

Based on stochastic modeling of Coriolis vibration gyros by the Allan variance technique, this paper discusses Angle Random Walk (ARW), Rate Random Walk (RRW) and Markov process gyroscope noises which have significant impacts on the North-finding accuracy. A new continuous rotation alignment algorithm for a Coriolis vibration gyroscope Inertial Measurement Unit (IMU) is proposed in this paper, in which the extended observation equations are used for the Kalman filter to enhance the estimation of gyro drift errors, thus improving the north-finding accuracy. Theoretical and numerical comparisons between the proposed algorithm and the traditional ones are presented. The experimental results show that the new continuous rotation alignment algorithm using the extended observation equations in the Kalman filter is more efficient than the traditional two-position alignment method. Using Coriolis vibration gyros with bias instability of 0.1°/h, a north-finding accuracy of 0.1° (1σ) is achieved by the new continuous rotation alignment algorithm, compared with 0.6° (1σ) north-finding accuracy for the two-position alignment and 1° (1σ) for the fixed-position alignment.