Presents the mathematical framework, technical language, and control systems know-how needed to design, develop, and instrument micro-scale whole-angle gyroscopes
This comprehensive reference covers the technical fundamentals, mathematical framework, and common control strategies for degenerate mode gyroscopes, which are used in high-precision navigation applications. It explores various energy loss mechanisms and the effect of structural imperfections, along with requirements for continuous rate integrating gyroscope operation. It also provides information on the fabrication of MEMS whole-angle gyroscopes and the best methods of sustaining oscillations.
Whole-Angle Gyroscopes: Challenges and Opportunities begins with a brief overview of the two main types of Coriolis Vibratory Gyroscopes (CVGs): non-degenerate mode gyroscopes and degenerate mode gyroscopes. It then introduces readers to the Foucault Pendulum analogy and a review of MEMS whole angle mode gyroscope development. Chapters cover: dynamics of whole-angle coriolis vibratory gyroscopes; fabrication of whole-angle coriolis vibratory gyroscopes; energy loss mechanisms of coriolis vibratory gyroscopes; and control strategies for whole-angle coriolis vibratory gyro- scopes. The book finishes with a chapter on conventionally machined micro-machined gyroscopes, followed by one on micro-wineglass gyroscopes. In addition, the book:
* Lowers barrier to entry for aspiring scientists and engineers by providing a solid understanding of the fundamentals and control strategies of degenerate mode gyroscopes
* Organizes mode-matched mechanical gyroscopes based on three classifications: wine-glass, ring/disk, and mass spring mechanical elements
* Includes case studies on conventionally micro-machined and 3-D micro-machined gyroscopes
Whole-Angle Gyroscopes is an ideal book for researchers, scientists, engineers, and college/graduate students involved in the technology. It will also be of great benefit to engineers in control systems, MEMS production, electronics, and semi-conductors who work with inertial sensors.
Doruk Senkal, PhD, has been working on the development of Inertial Navigation Technologies for Augmented and Virtual Reality applications at Facebook since 2018. Before joining Facebook, he was developing MEMS Inertial Sensors for mobile devices at TDK Invensense. He received his Ph.D. degree in 2015 from University of California, Irvine, with a focus on MEMS Coriolis Vibratory Gyroscopes. Dr. Senkal 's research interests, represented in over 20 international conference papers, 9 peer-reviewed journal papers, and 16 patent applications, encompass all aspects of MEMS inertial sensor development, including sensor design, device fabrication, algorithms, and control.
Andrei M. Shkel, PhD, has been on faculty at the University of California, Irvine since 2000, and served as a Program Manager in the Microsystems Technology Office of DARPA. His research interests are reflected in over 250 publications, 40 patents, and 2 books. Dr. Shkel has been on a number of editorial boards, including Editor of IEEE/ASME JMEMS and the founding chair of the IEEE Inertial Sensors. He was awarded the Office of the Secretary of Defense Medal for Exceptional Public Service in 2013, and the 2009 IEEE Sensors Council Technical Achievement Award. He is the IEEE Fellow.
Coriolis Vibratory Gyroscopes (CVGs) are mechanical transducers that detect angular rotation around a particular axis. In its most fundamental form, a CVG consists of two or more mechanically coupled vibratory modes, a forcing system to induce vibratory motion and a sensing system to detect vibratory motion. Angular rotation can be detected by sensing the energy transfer from one vibratory mode to another in the presense of Coriolis forces, Figure 1.1.
Historically, first examples of CVGs can be found in the Aerospace Industry, which were primarily used for navigation and platform stabilization applications. Later, advent of Micro-electromechanical System (MEMS) fabrication techniques brought along orders of magnitude reduction in cost, size, weight, and power (CSWaP), which made CVGs truly ubiquitous. Today CVGs are used in a wide variety of civilian applications, examples include:
- Industrial applications, such as robotics and automation;
- Automobile stabilization, traction control, and roll-over detection;
- Gesture recognition and localization in gaming and mobile devices;
- Optical image stabilization (OIS) of cameras;
- Head tracking in Augmented Reality (AR) and Virtual Reality (VR);
- Autonomous vehicles, such as self-driving cars and Unmanned Aerial Vehicles (UAVs).
1.1 Types of Coriolis Vibratory Gyroscopes
CVGs can be divided into two broad categories based on the gyroscope's mechanical element : degenerate mode (i.e. -axis) gyroscopes, which have - symmetry ( of 0 Hz), and nondegenerate mode gyroscopes, which are designed intentionally to be asymmetric in and modes (). Degenerate mode -axis gyroscopes offer a number of unique advantages compared to nondegenerate vibratory rate gyroscopes, including higher rate sensitivity, ability to implement whole-angle mechanization with mechanically unlimited dynamic range, exceptional scale factor stability, and a potential for self-calibration.
Figure 1.1 Coriolis Vibratory Gyroscopes, in their simplest form, consist of a vibrating element with two or more mechanically coupled vibratory modes. Illustration shows a -axis gyroscope and its vibratory modes along - and -axis.
1.1.1 Nondegenerate Mode Gyroscopes
Nondegenerate mode CVGs are currently being used in a variety of commercial applications due to ease of fabrication and lower cost. Most common implementations utilize two to four vibratory modes for sensing angular velocity along one to three axes. This is commonly achieved by forcing a proof mass structure into oscillation in a so-called "drive" mode and sensing the oscillation on one or more "sense" modes. For example, the -axis of the gyroscope in Figure 1.1, can be instrumented as a drive mode and the -axis can be instrumented as a sense mode. When a nonzero angular velocity is exerted (i.e. along the -axis in Figure 1.1), the resultant Coriolis force causes the sense mode (i.e. the mode along the -axis in Figure 1.1) to oscillate at the drive frequency at an amplitude proportional to input angular velocity.
Resonance frequency of sense modes are typically designed to be several hundreds to a few thousand hertzs away from the drive frequency. The existence of this so-called drive-sense separation () makes nondegenerate mode gyroscopes robust to fabrication imperfections. However, a trade-off between bandwidth and transducer sensitivity exists since smaller drive-sense separation frequency leads to higher transducer sensitivity, while the mechanical bandwidth of the sensor is typically limited by drive-sense separation ().
Nondegenerate mode gyroscopes are typically operated using open-loop mechanization. In open-loop mechanization, "drive" mode oscillation is sustained via a positive feedback loop. The amplitidue of "drive" mode oscillations are controlled via the so-called Amplitude Gain Control (AGC) loop. No feedback loop is employed on the "sense" mode, which leaves "sense" mode proof mass free to oscillate in response to the angular rate input.
1.1.2 Degenerate Mode Gyroscopes
Degenerate mode gyroscopes utilize two symmetric modes for detecting angular rotation. For an ideal degenerate mode gyroscope, these two modes have identical stiffness and damping; for this reason typically an axisymmetric or - symmetric structure is used, such as a ring, disk, wineglass, etc. Degenerate mode gyroscopes are commonly employed in two primary modes of instrumentation: (i) force-to-rebalance (FTR) (rate) mechanization and (ii) whole-angle mechanization.
In FTR mechanization, an external force is applied to the vibratory element that is equal and opposite to the Coriolis force being generated. This is a rate measuring gyroscope implementation, where the magnitude of externally applied force can be used to detect angular velocity. The main benefit of this mode of operation is to boost the mechanical bandwidth of the resonator, which would otherwise be limited by the close to zero drive-sense separation () of the degenerate mode gyroscope.
In the whole-angle mechanization, the two modes of the gyroscope are allowed to freely oscillate and external forcing is only applied to null the effects of imperfections such as damping and asymmetry. In this mode of operation the mechanical element acts as a "mechanical integrator" of angular velocity, resulting in an angle measuring gyroscope, also known as a Rate Integrating Gyroscope (RIG).
Whole-angle gyroscope architectures can be divided into three main categories based on the geometry of the resonator element: (i) lumped mass systems, (ii) ring/disk systems, and (iii) micro-wineglasses. Ring/disk systems are further divided into three categories: (i) rings, (ii) concentric ring systems, and (iii) disks. Whereas, micro-wineglasses are divided into two categories according to fabrication technology: surface micro-machined and bulk micro-machined wineglass gyroscope architectures, Figure 1.2 .
String and bar resonators can also be instrumented to be used as whole-angle gyroscopes, even though these types of mechanical elements are typically not used at micro-scale due to limited transduction capacity. In principle, any axisymmetric elastic member can be instrumented to function as a whole-angle gyroscope.
1.2 Generalized CVG Errors
Gyroscopes are susceptible to a variety of error sources caused by a combination of inherent physical processes as well as external disturbances induced by the environment.
Error sources in a single axis rate gyroscope can be generalized according to the following formula: (1.1)
where is the measured gyroscope output, is scale factor error, is bias error, and is noise. Without loss of generality, for a whole-angle gyroscope the error sources can be written as: (1.2)
Figure 1.2 Micro-rate integrating gyroscope (MRIG) architectures.
where is the measured gyroscope output, corresponding to total angular read-out, including the actual angle of rotation, errors in scale factor, bias, and noise.
1.2.1 Scale Factor Errors
Scale factor (or sensitivity) errors represent a deviation in gyroscope sensitivity from expected values, which results in a nonunity gain between "true" angular rate and "perceived" angular rate. Scale factor errors can be caused by either an error in initial scale factor calibration or a drift in scale factor postcalibration due to a change in environmental conditions, such as a change in temperature or supply voltages, application of external mechanical stresses to the sensing element, or aging effects internal to the sensor, such as a change in cavity pressure of the vacuum packaged sensing element.
1.2.2 Bias Errors
Bias (or offset) errors can be summarized as the deviation of time averaged gyroscope output from zero when there is no angular rate input to the sensor. Aside from initial calibration errors, bias errors can be caused by a change in environment conditions. Examples include a change in temperature, supply voltages or cavity pressure, aging of materials, and application of external mechanical stresses to the sensing element. An additional source of bias errors is external body loads, such as quasi-static acceleration, as well as vibration.
1.2.3 Noise Processes
Noise in gyroscopes can be grouped under white noise, flicker () noise, and quantization noise. The most common numerical tool for representing gyroscope noise processes is Allan Variance.
184.108.40.206 Allan Variance
Originally created to analyze frequency stability of clocks and oscillators, Allan Variance analysis is also widely used to represent various noise processes present in inertial sensors, such as gyroscopes . Allan Variance analysis consists of data acquisition of gyroscope output over a period of time at zero rate input and constant temperature. This is followed by binning the data into groups of different integration times: (1.3)
where is the sampling time, is the sample number,...