href="#fb3_img_img_4ecd0532-dbfd-5c81-9c65-69ed924e6905.png" alt="equation"/>
(3.18)
would define the initial value of the coordinate transformation matrix from sensor‐fixed coordinates to ENU coordinates:
(3.19)
Practical implementation. In practice, the sensor cluster is usually mounted in a vehicle that is not moving over the surface of the Earth, but may be buffeted by wind gusts or disturbed during fueling and loading operations. Gyrocompassing then requires some amount of filtering (Kalman filtering, as a rule) to reduce the effects of vehicle buffeting and sensor noise. The gyrocompass filtering period is typically on the order of several minutes for a medium‐accuracy INS but may continue for hours, days, or continuously for high‐accuracy systems.
3.5.2 Initialization on the Move
3.5.2.1 Transfer Alignment
This method is generally faster than gyrocompass alignment, but it requires another INS on the host vehicle and it may require special maneuvering of the host vehicle to attain observability of the alignment variables. It is commonly used for in‐air INS alignment for missiles launched from aircraft and for on‐deck INS alignment for aircraft launched from carriers. Alignment of carrier‐launched aircraft may also use the direction of the velocity impulse imparted by the steam catapult.
3.5.2.2 Initializing Using GNSS
This is an issue in GNSS/INS integration, which is covered in Chapter 12. In this case it must also estimate the INS orientation and velocity, the observability of which generally depends on the host vehicle trajectory.
3.6 Propagating The Navigation Solution
3.6.1 Attitude Propagation
Knowing the instantaneous rotational orientations of the inertial sensor input axes with respect to navigational coordinates is essential for inertial navigation to work. The integration of accelerations for maintaining the navigation solution for velocity and position depends on it.
3.6.1.1 Strapdown Attitude Propagation
Strapdown Attitude Problems
Early on, strapdown systems technology had an “attitude problem,” which was the problem of representing attitude rate in a format amenable to accurate computer integration over high dynamic ranges. The eventual solution was to represent attitude in different mathematical formats as it is processed from raw gyro outputs to the matrices used for transforming sensed acceleration to inertial coordinates for integration.
Figure 3.14 illustrates the resulting major gyro signal processing operations, and the formats of the data used for representing attitude information. The processing starts with gyro outputs and ends with a coordinate transformation matrix from sensor coordinates to the coordinates used for integrating the sensed accelerations.
Coning Motion
This type of motion is a problem for attitude integration when the frequency of motion is near or above the sampling frequency. It is usually a consequence of host vehicle frame vibration modes or resonances in the INS mounting, and INS shock and vibration isolation is often designed to eliminate or substantially reduce this type of rotational vibration.
Figure 3.14 Strapdown attitude representations.
Coning motion is an example of an attitude trajectory (i.e. attitude as a function of time) for which the integral of attitude rates does not equal the attitude change. An example trajectory would be
(3.20)
(3.21)
where
is the rotation vector,
is called the cone angle of the motion,
is the coning frequency of the motion,
as illustrated in Figure 3.15.
The coordinate transformation matrix from body coordinates to inertial coordinates will be
(3.22)
Figure 3.15 Coning motion.
and the measured inertial rotation rates in body coordinates will be
(3.23)
The integral of
(3.24)
which is what a rate integrating gyroscope would measure.
Figure 3.16 Coning error for 1° cone angle, 1 kHz coning rate.
The solutions for
