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

This is why coning error compensation is important.

       Rotation Vector Implementation

      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.”

       Bortz Model for Attitude Dynamics

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).

      3.6.1.2 Quaternion Implementation

The quaternion representation of vehicle attitude is the most reliable, and it is used as the “holy point” of attitude representation. Its value is maintained using the incremental rotations from the rotation vector representation, and the resulting values are used to generate the coordinate transformation matrix for accumulating velocity changes in inertial coordinates.

      Quaternions represent three‐dimensional attitude on the three‐dimensional surface of the four‐dimensional sphere, much like two‐dimensional directions can be represented on the two‐dimensional surface of the three‐dimensional sphere.

       Converting Incremental Rotations to Incremental Quaternions

An incremental rotation vector from the Bortz coning correction implementation of Eq. (3.28) can be converted to an equivalent incremental quaternion by the operations

      (3.29)

      (3.30)

      (3.31)

       Quaternion implementation of attitude integration

       If is the quaternion representing the prior value of attitude,

        is the quaternion representing the change in attitude, and

        is the quaternion representing the updated value of attitude,

      then the update equation for quaternion representation of attitude is

      (3.32)

here “ represents quaternion multiplication (defined in Appendix B) and the superscript represents the conjugate of a quaternion,

      (3.33)

      3.6.1.3 Direction Cosines Implementation

      The