Strapdown Attitude Implementations

9.4.5.1 Strapdown Attitude Problem 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. 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 9.23 illustrates 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.

9.4.5.2 MATLAB Implementations The diagram in Fig. 9.24 shows four different representations used for relative attitudes, and the names of the MATLAB script m-files (i.e., with the added ending .m) on the accompanying CD-ROM for transforming from one representation to another.

9.4.5.3 Coning Motion 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 where the INS is mounted, and INS shock and vibration isolation is often designed to eliminate or substantially reduce this type of rotational vibration.

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

where θ_cone is the cone angle of the motion and Ω_coning is the coning frequency of the motion, as illustrated in Fig. 9.25.

The coordinate transformation matrix from body coordinates to inertial coordinates (Eq. C.112) will be

and the measured inertial rotation rates in body coordinates will be

The integral of ω_body

which is what a rate integrating gyroscope would measure.

The solutions for θ_cone = 0.1° and Ω_coning = 1 k Hz are plotted over one cycle (1 ms) in Fig. 9.26. The first two components are cyclical, but the third component accumulates linearly over time at about -1.9 10^-5 radians in 10^-3 second, which is a bit more than -1°/s. This is why coning error compensation is important.

9.4.5.4 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 [23]. It represents ISA attitude with respect to the reference inertial coordinate frame in terms of the rotation vector ρ required to rotate the reference

inertial coordinate frame into coincidence with the sensor-fixed coordinate frame, as illustrated in Fig. 9.27.

The Bortz dynamic model for attitude then has the form

where ω is the vector of measured rotation rates. The Bortz "noncommutative rate vector"

Equation 9.40 represents the rate of change of attitude as a nonlinear differential equation that is linear in the measured instantaneous body rates ω. Therefore, by integrating this equation over the nominal intersample period [0, Δt] with initial value ρ(0) = 0, an exact solution of the body attitude change over that period can be obtained in terms of the net rotation vector

that avoids all the noncommutativity errors, and satisfies the constraint of Eq. 9.42 as long as the body cannot turn 180° in one sample interval Δt. In practice, the integral is done numerically with the gyro outputs ω₁, ω₂, ω₃ sampled at intervals δt « Δt. The choice of δt is usually made by analyzing the gyro outputs under operating conditions (including vibration isolation), and selecting a sampling frequency 1/δt 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 fBrotz.m on the CD-ROM calculates f_Bortz (ω) defined by Eq. 9.41.

9.4.5.5 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 Δρ from the rotation vector representation, and the resulting values are used to generate the coordinate transformation matrix for accumulating velocity changes in inertial coordinates.

Converting Incremental Rotations to Incremental Quaternions An incremental rotation vector Δρ from the Bortz coning correction implementation of Eq. 9.43 can be converted to an equivalent incremental quaternion Δq by the operations

Quaternion Implementation of Attitude Integration If q_k-1 is the quaternion representing the prior value of attitude, Δq is the quaternion representing the change in attitude, and q_k is the quaternion representing the updated value of attitude, then the update equation for quaternion representation of attitude is

where the postsuperscript represents the conjugate of a quaternion.

9.4.5.6 Direction Cosines Implementation The coordinate transformation matrix inertial 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 .

Quaternions to Direction Cosines Matrices The direction cosines matrix from body-fixed coordinates to inertial coordinates can be computed from its equivalent unit quaternion representation

as