the INS must rely on Newton's universal law of gravitation to take them into account in the navigation solution. Newton's law has the downward gravitational acceleration inversely proportional to the square of the radius from the Earth's center, which then falls off with increasing altitude. Therefore an INS resting stationary on the surface of the Earth with an upward navigational error in altitude would compute a downward gravitational acceleration smaller that the (measured) upward specific force countering gravity, which would result in an upward navigational acceleration error, which only makes matters worse. This would not be a problem for surface ships, it might have been a problem for aircraft if they did not already use barometric altimeters, and similarly for submarines if they did not already use depth sensors. It became an early example of sensor integration successfully applied to inertial navigation.
This is no longer a serious issue, now that we have chip‐scale barometric altimeters.
3.6.2.2 Strapdown Navigation Propagation
The basic signal processing functions for strapdown INS navigation are illustrated in Figure 3.21, where
G is the estimated gravitational acceleration, computed as a function of estimated position.
is the estimated position of the host vehicle in navigation coordinates.
is the estimated velocity of the host vehicle in navigation coordinates.
is the estimated acceleration of the host vehicle in navigation coordinates, which may be used for trajectory control (i.e. vehicle guidance).
is the estimated acceleration of the host vehicle in sensor‐fixed coordinates, which may be used for vehicle steering stabilization and control.
is the coordinate transformation matrix from sensor‐fixed coordinates to navigation coordinates, representing the attitude of the sensors in navigation coordinates.
is the estimated angular velocity of the host vehicle in sensor‐fixed (ISA) coordinates, which may be used for vehicle attitude stabilization and control.
is the estimated angular velocity of the host vehicle in navigation coordinates, which may be used in a vehicle pointing and attitude control loop.
Figure 3.21 Essential navigation signal processing for strapdown INS.
The essential processing functions include double integration (represented by boxes containing integration symbols) of acceleration to obtain position, and computation of (unsensed) gravitational acceleration as a function of position. The sensed angular rates also need to be integrated to maintain the knowledge of sensor attitudes. The initial values of all the integrals (i.e. position, velocity, and attitude) must also be known before integration can begin.
The position vector
Navigation functions that are not shown in Figure 3.21 include:
1 How initialization of the integrals for position, velocity, and attitude is implemented. Initial position and velocity can be input from other sources (GNSS, for example), and attitude can be inferred from some form of trajectory matching (using GPS, for example) or by gyrocompassing.
2 How attitude rates are integrated to obtain attitude, described in Section 3.6.1.1.
3 For the case that navigation coordinates are Earth‐fixed, the computation of navigational coordinate rotation due to earthrate as a function of position, and its summation with sensed rates before integration.
4 For the case that navigation coordinates are locally‐level, the computation includes the rotation rate of navigation coordinates due to vehicle horizontal velocity and its summation with sensed rates before integration.
5 Calibration of the sensors for error compensation. If the errors are sufficiently stable, it needs to be done only once. Otherwise, it can be implemented using the GNSS/INS integration techniques discussed in Chapter 12.
Figure 3.22 is a process flow diagram for the same implementation, arranged such that the variables available for other functions is around the periphery. These are the sorts of variables that might be needed for driving cockpit displays, antennas, weaponry, sensors, or other surveillance assets.
3.6.2.3 Gimbaled Navigation Propagation
The signal flowchart in Figure 3.23 shows the essential navigation signal processing functions for a gimbaled INS with inertial sensor axes aligned to locally level coordinates, where
Figure 3.22 Outputs (in angular brackets) of simple strapdown INS.
is the specific force (i.e. the sensible acceleration, which does not include gravitational acceleration) applied to the host vehicle.
is the instantaneous inertial rotation rate vector of the host vehicle.
denotes a specific force sensor (accelerometer).
denotes the ensemble of gimbal angle encoders, one for each gimbal angle. There are several possible formats for the gimbal angles, including digitized angles, three‐wire synchros signals, or pairs.
denotes an inertial rotation sensor (gyroscope).
Position is the estimated position of the host vehicle in navigation coordinates (e.g. longitude, latitude, and altitude relative to sea level).
Velocity is the estimated velocity of the host vehicle in navigation coordinates (e.g. east, north, and vertical).
Attitude is the estimated attitude of the host vehicle relative to locally level coordinates. For some three‐gimbal
