Mohinder S. Grewal - Global Navigation Satellite Systems, Inertial Navigation, and Integration

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 3.14illustrates 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.

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 3.14Strapdown 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 3.15Coning 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 3.16Coning error for 1° cone angle, 1 kHz coning rate.

The solutions for and are plotted over one cycle (1 ms) in Figure 3.16. The first two components are cyclical, but the third component accumulates linearly over time at about radians in one millisecond, which is a bit more than deg/s. This is why coning error compensation is important .

This implementation is primarily used at a faster sampling rate than the nominal sampling rate (i.e. that required for resolving measured accelerations into navigation coordinates). It is used to remove the nonlinear effects of coning and skulling motion that would otherwise corrupt the accumulated angle rates over the nominal intersample period. This implementation is also called a “coning correction.”

This exact model for attitude integration based on measured rotation rates and rotation vectors was developed by John Bortz (1935–2013) [9]. It represents ISA attitude with respect to the reference inertial coordinate frame in terms of the rotation vector required to rotate the reference inertial coordinate frame into coincidence with the sensor‐fixed coordinate frame, as illustrated in Figure 3.17.

Figure 3.17Rotation vector representing coordinate transformation.

The Bortz dynamic model for attitude then has the form

(3.25)

where is the vector of measured rotation rates. The Bortz “noncommutative rate vector”

(3.26)

(3.27)

Equation ( 3.25) represents the rate of change of attitude as a nonlinear differential equation that is linear in the measured instantaneous body rates . Therefore, by integrating this equation over the nominal intersample period with initial value , an exact solution of the body attitude change over that period can be obtained in terms of the net rotation vector