In most instances in inertial navigation, we are going to be concerned with multiple reference frames. For example, the accelerometers may form their measurements with respect to a set of axes that are fixed in the vehicle; but navigation is usually performed with respect to a set of axes that is fixed in the Earth. This is certainly true for applications that are on or near the surface of the Earth.

First of all, we have to understand what these various reference frames are and how they are defined and used within inertial navigation. Once we have defined the various reference frames, then we will need to discuss how to convert between them. As we have discussed previously, Newton's laws are valid in an inertial reference frame and an inertial frame is one that is moving at fixed velocity with respect to the stars. The background stars in our galaxy serve as a reference frame and any frame that is moving at constant velocity (including zero) with respect to the background stars also is an inertial frame. The true inertial frame is the only frame in which Newton's laws are valid.

Previously we have covered the details of inertial navigation in the context of a fixed reference frame. Now we are going to begin addressing the fact that in most applications the vehicle is actually navigating with respect to the spheroidal, rotating earth. We will deal with these two effects (earth curvature and rotation) by taking into account the additional apparent forces that come into play. This will impact our navigation equation (i.e., the expression for the derivative of velocity). In the case of the fixed frame, the expression for the derivative of velocity was very simple. In the context of a rotating reference frame, however, it is a bit more complicated.

Previously we derived the body-frame update of the body-to-inertial DCM (Cⁱ _b ). This DCM was appropriate for use in a "fixed frame" environment (i.e., an inertial reference frame). For terrestrial, marine and aeronautical applications, however, we need to operate with respect the non-inertial earth frame. Our local-level reference frame, in many applications, is the navigation frame and thus we need to update the body-to-nav DCM (Cⁿ _b ). Since we have already derived the body-frame update, we need to derive the nav-frame update.

Up to this point, we have been treating accelerometers and gyroscopes as ideal instruments and have focused almost exclusively on the "strapdown processing" algorithms also known as the "free inertial mechanization" equations. Having accomplished that goal, we now turn our attention to the actual sensors that are used in real inertial systems. Prior to the proliferation of inertial measurement units (IMUs) based on micro-electro-mechanical systems (MEMS) fabrication techniques, the term "inertial navigation system" (INS) universally referred to inertial systems built with sensors capable of supporting stand-alone (i.e., unaided) navigation with low drift rates (e.g., a few nautical miles per hour).

However, the aforementioned explosion of low-cost, MEMS-based, IMUs (found, for example, in every consumer drone), has given rise to a far broader use of the phrase "inertial navigation system." A quick web search will reveal authors referring to low-cost units as an INS despite the fact that the units could not possibly be used for stand-alone positioning for more than a handful of seconds. Very low cost gyroscopes (i.e., priced well under one U.S. dollar in volume production) may have drift rates on the order of several degrees per second and, as we shall see in the next chapter, are completely unsuitable for long-duration stand-alone positioning. At the other end of the spectrum are so-called "navigation-grade" and "strategic-grade" sensors that have extremely low drift rates (i.e., less than 0.01 degree per hour).

This chapter will provide an overview of the principles of operation of the two technologies that have been the mainstay of navigation-grade gyro offerings since the 1980s as well as one of the most popular techniques used in the design of navigation-grade accelerometers. We will then discuss the major sensor errors.

In the previous chapter, we analyzed the Schuler error loop in the horizontal channel and derived closed form expressions for position and velocity errors given accelerometer and gyro biases. The closed form expressions are actually first-order approximations of the real error behavior. In this chapter, we will use simulations to illustrate the cross-coupling phenomena between the north-south and east-west channels and will go on to analyze the dominance of certain error sources as a function of mission duration.

A so-called "whole value" simulator generates simulated accel and gyro measurements and processes them using the same algorithms as would be used in an actual INS. In a computer simulation, however, the input errors (e.g., accel/gyros biases) are controlled and known. The analyst can simulate a single error source on a single sensor and evaluate its impact on the entire system.

In all of the simulation results to follow, the INS is modeled as being in a vehicle that is stationary, level, north-pointing and is at a latitude of 45 degrees North.

It is a reasonable question. Why would we want to go through the time and effort to integrate two different navigation systems? The simple answer is that we integrate multiple systems when no one system by itself will satisfy all the necessary requirements. More generally, we integrate multiple systems in order to construct a super-system that in some way outperforms any of the individual systems.

We are going to be spending a lot of time developing the framework behind the Kalman filter and ultimately driving toward the goal of developing an architecture that will enable us to integrate multiple navigation systems into a single integrated unit. We will start off by looking at navigation system integration and consider some key issues along with goals and requirements. We will investigate subsystem characteristics and will look at some challenges to integration.

Navigation engineers must be cognizant of the fact that their product is a component within a larger system. This larger system is typically a vehicle (specifically, a land, air, sea, or space vehicle). As I have told many students throughout my years of teaching: an inertial navigation system will not transport me from New York to Chicago. It will tell me where I am. It will tell me how fast I am going. It will tell me the current attitude of my vehicle and how fast it is turning. However, it will not answer the most basic navigation question that all drivers, pilots, and captains must answer in order to get where they want to go: Which way to steer?

The position, velocity, and attitude information determined by the navigation system, in virtually every case, is handed off to another system. Frequently it is either a flight management system or a flight control system or some kind of display for the pilot (or driver or captain...). It is also possible that it may be pointing a sensor or a weapon or providing the position and velocity needed to process synthetic aperture radar images. In any case, the navigation system needs to provide its information in a form and with a quality that satisfies the needs of the downstream user or system.

Previously we explored the Kalman filter within the framework of a simple problem. In the bucketful of resistors problem, we were estimating a single static parameter. We pulled one resistor out of the bucket and attempted to estimate its resistance given some successive measurements. Obviously we need to go beyond that for most practical applications. We need to be able to handle cases where the parameter of interest is actually changing over time and we want to be able to form an estimate of its value over time as it is changing. Furthermore, we want to be able to handle cases where we are estimating multiple parameters simultaneously and those parameters are related to each other. It is not a simple matter of five parameters and thus the need for five single-parameter Kalman filters running at the same time. If we are trying to estimate multiple parameters that have some kind of relationship to each other, then we want to be able to exploit those relationships in the course of our estimation and thus we need to expand our Kalman filter accordingly.

We are thus transitioning from univariate parameter estimation to multivariate parameter estimation and from static systems to dynamic systems. Along the way we will explore state variables, state vectors, and state transition matrices. We will learn about the fundamental equations upon which the Kalman filter is based: the so-called "system equation" and the "measurement equation." We will need to learn about covariance matrices and then eventually we will develop the full Kalman recursion. We will then take our bucketful of resistors example and modify it slightly in order to ease into these new topics.

Up to this point, we have only considered the estimation of a single parameter (the univariate case) with a value that did not change over time (a static system). We need to generalize the Kalman filter to be able to handle dynamic multivariate cases. The single parameters are going to become vectors of parameters and the variances are going to become covariance matrices. We will model system dynamics within the filter (the state transition matrix) and will model complex relationships between the states and the measured quantities (the H matrix will no longer be equal to 1).

In the previous chapter, we introduced the linearized Kalman filter and the extended Kalman filter in the context of two-dimensional radionavigation-based positioning (specifically, multilateration DME). We can now apply these concepts to GNSS. It is useful to study GNSS-only Kalman filtering to understand its performance and limitations. Many GNSS receivers output a position solution produced by a Kalman filter and, within some inertial integration schemes, the system outputs three separate solutions: a GNSS-only solution, an inertial-only solution, and a GNSS-inertial integrated solution. Since our purpose here is not to delve into the details that differentiate the various GNSSs, we will focus our development primarily on GPS.

In an earlier chapter, we explored how the Kalman filter does an excellent job of taking the radionavigation system geometry into account when forming its estimates but does not do well in handling nonlinear system dynamics. In the previous chapter, we introduced the concept of the complementary filter and applied it to a simple one-dimensional aided-inertial navigation problem. Since the filter in the aided case is estimating inertial errors, instead of the total aircraft state, the non-linearities arising from a dynamic trajectory were not a problem. An aided-inertial architecture is founded on the principle that the inertial navigation system does an excellent job of determining the position, velocity and attitude of the vehicle in the short term and thus the Kalman filter is not needed for that task. The purpose of the aiding is to prevent the inertially determined vehicle state from drifting off over the long term.

In this chapter, we extend the inertial aiding concept to two dimensions. We will use a two-dimensional radionavigation system (DME-DME) to aid a simplified inertial navigation system. We will be estimating inertial errors and one way to view this is that the stand-alone inertial position solution provides the "nominal" trajectory used to linearize the system and measurement equations to make them suitable for the Kalman filter. This implies that the inertial errors may be modeled linearly. In order to introduce some key concepts in this chapter, we will use a very simple inertial error model. In the next chapter, we will develop a significantly more detailed inertial error model but the fact remains that the model must be linear in order to be suitable for the Kalman filter. We will discover in a later chapter that once the inertial error grows sufficiently large the linearity assumption breaks down. We will also discover that the typical solution to this problem is to feedback the estimated errors as corrections to the inertial system thus keeping the inertial errors small.

Having established a state-space model that describes inertial error propagation, we have at last arrived at the point where a complete aided-inertial Kalman filter can be described. Although we showed in a previous chapter that loose coupling is inferior to tight coupling (i.e., loosely coupled filters cannot de-weight aiding measurements under conditions of bad geometry), we will study it nevertheless since the measurement equation has a simpler form than it does in tight coupling and thus it allows us to ease into the complexity of the full aided-inertial filter. In addition, it must be remembered that in some instances, the designer is forced to use loose coupling. This occurs, for example, when the aiding source (e.g., GNSS receiver) that is to be utilized has already been installed on a given platform (e.g., aircraft or land vehicle) but provides only processed data (e.g., position, velocity) on the external interface.

We have dedicated a considerable number of pages to the description of the system and the measurement models in the Kalman Filter and, of course, we have applied that to GNSS/INS integration. We have developed the inertial error models and we have developed the GNSS measurement models. Recall the H matrix has a considerable number of zeroes in it and but yet it is the sole representation, within the filter, of the direct relationship between the measurements and the states. So the question is: How does the filter estimate all of the states and, in particular, those states that are not direct functions of the measurements? In a loosely coupled filter, for example, the input measurements consist of observed position differences. How does the filter use those observations to be able to estimate things like attitude error and gyro biases? What's up with that? How does that work?

In an earlier chapter, we introduced the topic of inertial initialization. We showed how with a stationary vehicle the accelerometer outputs could be processed to determine rough initial estimates of pitch and roll and, having done so, the gyro outputs could be processed to determine a rough estimate of yaw or heading (or wander angle depending on the chosen convention). Although it is common for the entire process to be referred to as "alignment," it is more precisely "leveling" and "alignment." Although enhanced versions of this kind of processing (with fixed gain filters) were used in early strapdown systems (i.e., 1970s), it is more common in modern practice to use a Kalman filter.

The random walk and first-order Gauss-Markov processes are two of the most important stochastic processes used to model inertial sensor errors. They are both used extensively in aided-inertial Kalman filters.