Global Navigation Satellite Systems, Inertial Navigation, and Integration. Mohinder S. Grewal

inherently unstable, it can achieve very low bearing torques. Perhaps the most accurate momentum wheel gyroscopes to date were the superconducting electrostatic gyroscopes used in a theoretical physics experiment named “Gravity Probe B” [4], a NASA‐funded program to resolve two fine points of Einstein's theory. It was able to achieve drift rate accuracies in the order of deg/h, but only in a zero‐g environment, and at enormous cost. Unfortunately, scaling down the size of momentum wheel gyroscopes tends to scale up the ratio of surface area to angular momentum, which scales up angular drift rates due to bearing torques.

       Whole‐angle Gyroscopes

Foucault's 1852 gyroscope was mounted inside two sets of gimbals that freed its spin axis to remain pointing in a fixed inertial direction, held there by the conservation of angular momentum in inertial coordinates. Electrostatic gyroscopes have spherical rotors supported by electrostatic forces inside spherical suspension cavities, which gives their spin axes freedom to remain in fixed inertial directions. These are called whole‐angle gyroscopes, and they can use the directions of their spin axes as inertial reference directions.

       Rate Gyroscopes

      These use torques applied to the spinning rotor to keep its spin axis aligned with its enclosure. The spin axis rotational slewing rate is then proportional to the applied torque. There are also rate gyroscopes that do not use momentum wheels.

       Axial Mass Unbalance Torques

      If the center of mass of the rotor of a momentum wheel gyroscope is not concentric with its center of support, then the offset between the downward gravitational force on its mass and the upward force supporting it will create a torque. The component of that torque perpendicular to the spin axis of the rotor will then cause the rotor angular momentum to precess about the applied vertical force. It is an acceleration‐sensitive error torque due to axial mass unbalance that is difficult to avoid within manufacturing tolerances. It is commonly mitigated by calibrating its magnitude and compensating for it during operation.

      3.3.1.2 Coriolis Vibratory Gyroscopes (CVGs)

       Tuning Fork Gyroscopes

The tuning fork is the paradigmatic Coriolis vibratory gyroscope, long used as voice‐band frequency source before being pressed into service as a gyroscope, and it has since inspired miniaturized MESG designs using the same general principles. These mechanical principles are illustrated in Figure 3.3, showing how the normally counterbalanced synchronized motions of the two tines are coupled through the Coriolis effect due to rotations about the handle into an unbalanced vibratory torsion mode that is output through the handle. Counterbalanced momentum of the tines is an essential design feature for Coriolis vibratory gyroscopes (CVGs), here using two masses traveling in opposite directions to mitigate the effects of Newton's third law. Vibrations tend to launch elastic waves into the surrounding materials, so control of the distribution of vibration is a critical design issue for all CVGs – keeping it where it belongs and eliminating it where it does not belong. Operation requires methods for controlling the tuning fork motion and sensors for detecting the output mode. Tuning fork gyroscopes fabricated from quartz can perform both functions using the piezoelectric properties of quartz. Tuning forks fabricated in silicon are discussed in the following text.

Figure 3.3 Tuning fork gyroscope.

       MEMS Tuning Fork Gyroscope

A design originally developed at the Charles Stark Draper Laboratories in the 1980s and 1990s [5] does not physically resemble a tuning fork, but it uses the same principle of two masses resonating in the plane of their silicon substrate, and in synchrony to maintain zero net momentum. The input rotation axis is in the plane of the substrate, as illustrated in Figure 3.4. Vibratory motion is controlled by electrostatic “comb drives” (interdigitated electrodes) developed at the University of California at Berkeley. The input axis is in the plane of the substrate and the output vibration mode is normal to the substrate surface. Many improvements have been made in this original design. There are also devices using rotational vibrations coupled with Coriolis effects.

       Hemispherical Resonator Gyroscopes

These are also called wine glass gyroscopes, referring to resonant modes of wine glass rims, the nodes of which move when the wine glass is rotated about its stem – caused by Coriolis coupling. These devices can continue to operate through short‐term radiation events when electronic devices are shut down.

Figure 3.4 MEMS tuning fork gyroscope.

      3.3.1.3 Optical Gyroscopes (RLGs and FOGs)

      There are two essential designs for optical gyroscopes, both of which depend on the Sagnac effect, a phenomenon studied by Franz Harress in 1911 [6] and Georges Sagnac in 1913. The effect has to do with the relative delay of two light beams from the same source traveling in opposite directions around the same closed loop, and how their relative delay depends on the rotation rate of the apparatus in the plane of the loop. The effect has been named for Sagnac, who showed that the delay difference was proportional to the rotation rate, and the effect scaled as the area of the loop. The effect was not used for sensing rotation until after a working laser was demonstrated in 1960, first with the lasing cavity in the closed optical path – the ring laser gyroscope (RLG) – and later using a kilometers‐long coil of optical fiber – the fiber optic gyroscope (FOG).

      Ring laser gyroscopes are rate integrating gyroscopes. Their output interferometric phase rate is proportional to rotation rate, so each output phase shift represents an incremental inertial angular rotation angle. To minimize temperature and pressure sensitivities, their closed‐loop optical paths are typically machined into very stable materials. Early designs exhibited a “lock‐in” problem near zero input rates, due to backscatter off the mirrors. Later designs avoided this by using out‐of‐plane optical paths and multi‐frequency lasing cavities.

      Fiber optic gyroscopes were first developed after single‐mode optical fibers became available, about a decade after the first laser. Unlike RLGs, FOGs are rate gyros. Their output is proportional to the input rotation rate, and must be integrated to get rotation angles. The optical loop in this case is a very long coil of optical fiber with an external laser source.

      3.3.2 Accelerometers

      Accelerometers used in inertial navigation measure the force required to keep a proof mass stationary with respect to its enclosure, which is called specific force to distinguish it from unsensed gravitational accelerations. Accelerometer designs differ in how that force is measured, and how that force is distributed. Examples of these different design approaches are given in the following text.

      3.3.2.1 Mass‐spring Designs

      These