The NAVSTAR Global Positioning System. New York: Van Nostrand Reinhold.3 3 Grewal, M.S. and Andrews, A.P. (2019). Application of Kalman Filtering to GPS, INS, & Navigation, Short Course Notes. Anaheim, CA: Kalman Filtering Consultant Associates.
4 4 Grewal, M.S. and Andrews, A.P. (2015). Kalman Filtering: Theory and Practice Using MATLAB®, 4e. New York: Wiley.
5 5 Allan, D.W., Ashby, N., and Hodge, C.C. (1997). The Science of Timekeeping, Hewlett Packard, Application Note 1289. Palo Alto, CA: Hewlett‐Packard.
3 Fundamentals of Inertial Navigation
An inertial system does for geometry…what a watch does for time.1
Charles Stark Draper (1901–1987)
Charles Stark Draper was the American pioneer in inertial navigation who founded the Instrumentation Laboratory at MIT in 1932 to develop aircraft instrumentation technologies. In his analogy previously quoted, watches keep track of time by being set to the correct time, then incrementing that time according to the inputs from a “time sensor” (a frequency source) to update that initial value.
An inertial navigation system (INS) does something similar, only with different variables – and it increments doubly. An INS needs to be set to the correct position and velocity. Thereafter, they use measured accelerations to increment that initial velocity, and use the resulting velocities to increment position.
3.1 Chapter Focus
The overview of inertial navigation in Section 1.3 alluded to the history and terminology of the technology. The focus here is on how inertial sensors function and how they are integrated into navigation systems, including the following:
1 Terminology for the phenomenology and apparatus of inertial navigation
2 Technologies used for sensing rotation and acceleration
3 Error characteristics of inertial sensors
4 Sensor error compensation methods
5 How to compensate for unsensed gravitational accelerations
6 Initializing and propagating navigation solutions for attitude (rotational orientation), velocity, and position
7 Carouseling and indexing as methods for mitigating the effects of sensor errors
8 System‐level testing and evaluation
9 INS performance metrics and standards
How this all affects navigation performance is discussed in Chapter 11.
Scope.
The technology of inertial navigation has been evolving for nearly a century, its diversity and sophistication have grown enormously, and the scale of inertial systems has shrunk by orders of magnitude – from the unbearable to the wearable. As a result, we cannot cover every aspect of every technical approach to every application in full detail in this one chapter. The focus here will be limited to just those INS applications involving global navigation satellite system (GNSS) navigation, which limits it to the terrestrial environment with GNSS availability at least part of the time, and to those aspects of the application necessary for GNSS/INS integration. We will endeavor to cover here the essential design and implementation issues involved in these applications.
3.2 Terminology
Much of the terminology for inertial navigation evolved when the technology was highly classified and being developed by independent design teams, the result of which has been considerable diversity. The terminology used throughout the book, listed in the following text, generally follows a standardized terminology for inertial sensors [1] and systems [2].
Inertia is the propensity of bodies to maintain constant translational and rotational velocity, unless disturbed by forces or torques, respectively (Newton's first law or motion).
Inertial reference frames are coordinate frames in which Newton's laws of motion are valid. They cannot be rotating or accelerating. They are not necessarily the same as the navigation coordinates, which are typically dictated by the navigation problem at hand. We live in a rotating and accelerating environment here on Earth, and that defines an Earth‐fixed locally level coordinate system we already feel comfortable with – even though it is accelerating (to counter gravity) and rotating. These rotations and accelerations must be taken into account in the practical implementation of inertial navigation.
Navigation coordinates are those used for representing the position of the inertial sensors with respect to its environment. In GNSS/INS integration, this will generally be the same as that used by the GNSS, representing the near‐Earth environment. See Appendix B (www.wiley.com/go/grewal/gnss) for descriptions of navigation coordinates and the transformations involved.
The navigation solution for inertial navigation includes the instantaneous values of position, velocity, and rotational orientation of the inertial sensors with respect to navigation coordinates. It must be sufficient for propagating the solution forward in time, given the inertial sensor outputs.
Inertial sensors measure inertial accelerations and rotations, both of which are vector‐valued variables.
Accelerometers measure specific force, the point being that accelerometers do not measure gravitational acceleration. Specific force is modeled by Newton's second law as
Gyroscopes (often shortened to “gyros”) are sensors for measuring rotation.
Displacement gyros (also called whole‐angle gyros) measure accumulated rotation angle, in angular units (e.g. radians or degrees).
Rate gyros measure rotation rates in angular rate units (e.g. radians per second, degrees per hour, etc.).
Inertial navigation depends on gyros for maintaining knowledge of how the accelerometers are oriented in inertial and navigational coordinates.
Input axes of an inertial sensor define which vector component(s) of acceleration, rotation, or rotation rate it measures. These are illustrated by arrows in Figure 3.1, with rotation arrows wrapped around the input axes of gyroscopes to indicate the direction of rotation. Multi‐axis sensors measure more than one component.
Calibration
