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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 3.17 Rotation 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

(3.28)

that avoids all the noncommutativity errors, and satisfies the constraint of Eq. (3.27) so long as the body cannot turn 180° in one sample interval . In practice, the integral is done numerically with the gyro outputs sampled at intervals . The choice of is usually made by analyzing the gyro outputs under operating conditions (including vibration isolation), and selecting a sampling frequency well above the Nyquist frequency for the observed attitude rate spectrum. The frequency response of the gyros also enters into this design analysis.

The MATLAB® function fBortz.m on www.wiley.com/go/grewal/gnss calculates defined by Eq. (3.26).