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

(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.mon www.wiley.com/go/grewal/gnsscalculates defined by Eq. (3.26).

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.

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)

The coordinate transformation matrix from body‐fixed coordinates to inertial coordinates is needed for transforming discretized velocity changes measured by accelerometers into inertial coordinates for integration. The quaternion representation of attitude is used for computing .

The direction cosines matrix from body‐fixed coordinates to inertial coordinates can be computed from its equivalent unit quaternion representation

(3.34)

as

(3.35)

These are momentum wheel gyroscopes whose spin axes are unconstrained, such as they are for Foucault's gimbaled gyroscope or electrostatic gyroscopes. In either case, the direction of the spin axis serves as an inertial reference – like a star. Two of these with quasi‐orthogonal spin axes provide a complete inertial reference system. The only attitude propagation required in this case is to compensate for the spin axis drifts due to bearing torques and rotor axial mass unbalance (an acceleration‐sensitive effect). The different coordinate systems involved are illustrated in Figure 3.18.

Figure 3.18Coordinates for strapdown navigation with whole‐angle gyroscopes.

Figure 3.19Attitude representation formats and MATLAB® transformations.

Figure 3.19shows four different representations used for relative rotational orientations, and the names of the MATLAB® script m‐files (i.e. with the added ending .m) on www.wiley.com/go/grewal/gnssfor transforming from one representation to another.