The terminology used there will be elaborated upon in Chapter 3.
The relative input axis directions of all the inertial sensors in an INS is controlled by hard‐mounting them all to a common base, forming what is called an inertial sensor assembly (ISA) or IMU.4 This way, all the sensor input axes will have a fixed direction relative to one another. The ISA is essentially the “holy point” of inertial navigation, just as the receiver antenna is the holy point for GNSS navigation. It is the reference point for the navigation solution.
1.3.2.2 Sensor Attitude Control
Methods by which the information from gyroscopes is used for resolving acceleration inputs into suitable coordinates for integration can be divided into two general approaches:
1 Inertial stabilization of the accelerometer input axes, using the gyroscopes. The first precise and reliable integrating accelerometer was invented in Germany by Fritz Mueller (1907–2001) in the 1930s [18]. Its proof mass is a MWG mounted such that its center of mass is offset axially from its center of support within its enclosure, causing the gyroscope to precess about the input acceleration axis at an angular rate proportional to specific force applied orthogonal to its spin axis, and accumulating a net precession angle proportional to velocity change in that direction. However, because it contains a MWG, it is also sensitive to rotation. That error source could be eliminated by maintaining Mueller's gyroscope in a fixed inertial direction, and the German visionary Johannes Boykow (1879–1935) played a significant role in developing an approach by which the outputs of rotation sensors could be fed back in servo control loops to gimbals nulling the inertial rotations of the ISA [18]. The result is now called an inertially stabilized platform or inertial platform, and the resulting INS is called “gimbaled.”
2 Strapdown systems use software to replace hardware (gimbals), processing the gyro outputs to maintain the coordinate transformation between accelerometer‐fixed coordinates and (essentially) inertial platform coordinates. The accelerometer outputs can then be transformed to what would have been inertial platform coordinates and processed just as they had been with a gimbaled system – without requiring an inertial platform. Getting rid of gimbals reduces INS weight and cost, but it also increases computer requirements. As a consequence, strapdown system development had to await the essential computer development, which did not happen until the mid‐twentieth century. However, as silicon‐based technologies advanced to produce chip‐based sensors and computers, only high‐end inertial systems would require gimbals.
1.3.2.3 Initialization
The initial solution to the inertial navigation problem is the starting position and orientation of the ISA. The starting position must usually be entered from other sources (including GNSS). If the ISA is sufficiently stationary with respect to the Earth, its starting orientation with respect to local vertical can be determined from the accelerometer outputs. If it is not at the poles and its rotation sensors are sufficiently accurate, its orientation with respect to North, East, South, and West can be determined by sensing the direction of Earth rotation. Otherwise, if the local Earth magnetic field is not too close to being vertical, it can be oriented using magnetometers. Other alternatives are mentioned in Chapter 3.
1.3.2.4 Integrating Acceleration and Velocity
Integration of gyro outputs (to maintain the attitude solution) and accelerometer outputs (to maintain the velocity and position solution) is done digitally. This is relatively straightforward for integrating accelerations and velocities in inertially stabilized Cartesian coordinates for gimbaled systems, and gimbal servo control takes care of the attitude solution. Otherwise, more sophisticated integration methods are required for maintaining the required coordinate transformation matrices in strapdown systems.
Because sensor output errors are being integrated, the resulting navigation solution errors tend to accumulate over time. Performance of INSs is generally specified in terms of navigation error growth rates.
1.3.2.5 Accounting for Gravity
Because accelerometers cannot measure gravitational accelerations, these must be calculated using the position and attitude solutions and added to the sensed acceleration. Otherwise, an INS resting on the surface of the Earth would measure only the upward support force countering gravity and think it was being accelerated upward.
The need to calculate gravity causes vertical dynamic instability of the navigation solution, because gravitational acceleration decreases with altitude (Newton's law of universal gravitation). Therefore, a navigation error in altitude leads to an error in calculating the unsensed downward gravitational acceleration. Upward navigational errors cause decreased downward gravity estimates, leading to a net upward acceleration error and to exponential growth of vertical velocity error and even faster growth in altitude error. This may not be a problem for ships at sea level, but it is serious enough otherwise that inertial navigators require auxiliary sensors (e.g. altimeters or GNSS) to stabilize altitude errors.
Gravity modeling also alters the dynamics of horizontal errors in position and velocity, but this effect is not unstable. It causes horizontal errors to behave like a pendulum with a period of about 84.4 minutes, something called the Schuler period.5 When Coriolis force is taken into account, this error behaves like a Foucault pendulum with that period.
1.4 GNSS/INS Integration Overview
1.4.1 The Role of Kalman Filtering
The Kalman filter has been called “navigation's integration workhorse,” [30] for the essential role it has played in navigation, and especially for integrating different navigation modes. Ever since its introduction in 1960 [31], the Kalman filter has played a major role in the design and implementation of most new navigation systems, as a statistically optimal method for estimating position using noisy measurements. Because the filter also produces an estimate of its own accuracy, it has also become an essential part of a methodology for the optimal design of navigation systems. The Kalman filter has been essential for the design and implementation of every GNSS.
Using the Kalman filter, navigation systems designers have been able to exploit a powerful synergism between GNSSs and INSs, which is possible because they have very complementary error characteristics:
Short‐term position errors from the INS are relatively small, but they degrade significantly over time.
GNSS position accuracies, on the other hand, are not as good over the short term, but they do not degrade with time.
The Kalman filter takes advantage of these characteristics to provide a common, integrated navigation implementation with performance superior to that of either subsystem (GNSS or INS). By using statistical information about the errors in both systems, it is able to combine a system with tens of meters position uncertainty (GNSS) with another system whose position uncertainty degrades at kilometers per hour (INS) and achieve bounded position uncertainties in the order of centimeters (with differential GNSS) to meters.
1.4.2 Implementation
The Kalman filter solves for the solution with the least mean‐squared error by using data‐weighting proportional to statistical information content (the inverse of uncertainty) in the measured data. It combines GNSS and INS information to the following:
1 Track
