alt="images"/> in ISA‐fixed coordinates, which had historically been used to express its effect on a coordinate transformation matrix (3.14)
which is not particularly well‐conditioned for numerical integration.
The alternative representation in terms of quaternions is described in Section 3.6.1.2 and (in greater detail) in Appendix B.
3.5 Initializing The Navigation Solution
3.5.1 Initialization from an Earth‐fixed Stationary State
3.5.1.1 Accelerometer Recalibration
This is only possible with gimbaled systems and it adds to the start‐up time, but it can improve performance. Navigation accuracy is very sensitive to accelerometer biases, which can shift due to thermal transients in turn‐on/turn‐off cycles, and can also drift randomly over time for some accelerometer design. Fortunately, gimbals can be used to calibrate accelerometer biases in a stationary 1g environment. Bias and scale factors can both be determined by using the gimbals to point each accelerometer input axes straight up and then straight down (by nulling the horizontal accelerometer outputs). Then each accelerometer's bias is the average of the up and down outputs and scale factor is half the difference divided by the local gravitational acceleration.
3.5.1.2 Initializing Position and Velocity
Velocity. Zero, by definition.
Position. From GNSS, if available, otherwise from local sources – e.g. signs or local auxiliary sensors (see Section 3.5.1.3).
3.5.1.3 Initializing ISA Attitude
Gyrocompassing. Gyrocompass alignment of stationary vehicles uses the sensed direction of acceleration to determine the local vertical and the sensed direction of rotation to determine north, as illustrated in Figure 3.13.
Using auxiliary sensors. Non‐gyroscopic attitude sensors can also be used as aids in alignment. These include the following:
Figure 3.13 Gyrocompassing determines sensor orientations with respect to east, north, and up.
Magnetic sensors are used primarily for coarse heading alignment, to speed up INS alignment.
Star trackers are used primarily for space‐based or near‐space applications. The Snark cruise missile and the U‐2 spy plane used inertial‐platform‐mounted star trackers to maintain INS alignment on long flights, an idea attributed to Northrop Aviation.
Optical alignment systems have been used on some systems prior to launch. Some use Porro prisms mounted on the inertial platform to maintain optical line‐of‐sight reference through ground‐based theodolites to reference directions at the launch complex.
Quasi magnetostatic sensors have been used in virtual reality systems for determining the attitude and position of the headset relative to its environment. These use three orthogonal and independently coded low‐to‐medium frequency magnetic dipole sources to illuminate the local area where rotational and locational orientation data is required, and three‐axis AC magnetic sensors as receivers.
3.5.1.4 Gyrocompass Alignment Accuracy
Gyrocompass alignment is not necessary for integrated GNSS/INS navigation, although many INSs may already be configured for it.
Accuracy
A rough rule‐of‐thumb for gyrocompass alignment accuracy is
(3.15)
where
is the minimum achievable root‐mean‐square (RMS) alignment error in radians,
is the RMS accelerometer accuracy in 's,
is the RMS gyroscope accuracy in degrees per hour,
15 deg/h is the rotation rate of the Earth, and
is the latitude at which gyrocompassing is performed.
Alignment accuracy is also a function of the time allotted for it, and the time required to achieve a specified accuracy is generally a function of sensor error magnitudes (including noise) and the degree to which the vehicle remains stationary.
Gimbaled implementation. Gyrocompass alignment for gimbaled systems is a process for aligning the inertial platform axes with the navigation coordinates using only the sensor outputs while the host vehicle is essentially stationary. For systems using ENU navigation coordinates, for example, the platform can be tilted until two of its accelerometer inputs are zero, at which time both input axes will be horizontal. In this locally leveled orientation, the sensed rotation axis will be in the north–up plane, and the platform can be slewed about the vertical axis to null the input of one of its horizontal gyroscopes, at which time that gyroscope input axis will point east–west. That is the basic concept used for gyrocompass alignment, but practical implementation requires filtering5 to reduce the effects of sensor noise and unpredictable zero‐mean vehicle disturbances due to loading activities and/or wind gusts.
Strapdown implementation. Gyrocompass alignment for strapdown systems is a process for “virtual alignment” by determining the sensor cluster attitude with respect to navigation coordinates using only the sensor outputs while the system is essentially stationary.
Error‐free implementation. If the sensor cluster could be firmly affixed to the Earth and there were no sensor errors, then the sensed acceleration vector
(3.16)
(3.17)
