Quaternions to Direction Cosines Matrices
The direction cosines matrix
(3.34)
as
(3.35)
Strapdown with Whole‐angle Gyroscopes
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 3.18 Coordinates for strapdown navigation with whole‐angle gyroscopes.
Figure 3.19 Attitude representation formats and MATLAB® transformations.
3.6.1.4 MATLAB® Implementations
Figure 3.19 shows 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/gnss for transforming from one representation to another.
3.6.1.5 Gimbal Attitude Implementations
The primary function of gimbals is to isolate the ISA from vehicle rotations, but they are also used for other INS functions.
Vehicle Attitude Determination
The gimbal angles determine the vehicle attitude with respect to the ISA, which has a controlled orientation with respect to navigation coordinates. Each gimbal angle encoder output determines the relative rotation of the structure outside gimbal axis relative to the structure inside the gimbal axis, the effect of each rotation can be represented by a
Figure 3.20 Simplified control flow diagram for three gimbals.
ISA Attitude Control
Gimbals control ISA orientation. This is a three‐degree‐of‐freedom problem, and the solution is unique for three gimbals. That is, there are three attitude‐control loops with (at least) three sensors (the gyroscopes) and three torquers. Each control loop can use a PID controller, with the commanded torque distributed to the three torquers according to the direction of the torquer/gimbal axis with respect to the gyro input axis, somewhat as illustrated in Figure 3.20, where
Disturbances includes the sum of all torque disturbances on the individual gimbals and the ISA, including those due to ISA mass unbalance and acceleration, rotations of the host vehicle, air currents, torque motor errors, etc.
Gimbal dynamics is actually quite a bit more complicated than the rigid‐body torque equation
which is the torque analog of
Desired rates refers to the rates required to keep the ISA aligned to a moving coordinate frame (e.g. locally level).
Resolve to gimbals is where the required torques are apportioned among the individual torquer motors on the gimbal axes.
The actual control loop is more complicated than that shown in the figure, but it does illustrate in general terms how the sensors and actuators are used.
For systems using four gimbals to avoid gimbal lock, the added gimbal adds another degree of freedom to be controlled. In this case, the control law usually adds a fourth constraint (e.g. maximize the minimum angle between gimbal axes) to avoid gimbal lock.
3.6.2 Position and Velocity Propagation
3.6.2.1 Vertical Channel Instability
The INS navigation solution for altitude and altitude rate is called its vertical channel. It might have become the Achilles heel of inertial navigation if it had not been recognized (by physicist George Gamow [10]) and resolved (by Charles Stark Draper and others) early on.
The reason for this is that the vertical gradient of the gravitational acceleration is negative. Because accelerometers cannot sense gravitational accelerations, the INS must rely on Newton's universal law of gravitation to take them into account in the navigation solution. Newton's law has the downward gravitational acceleration inversely proportional to the square of the radius from the Earth's center, which then falls off with increasing altitude. Therefore an INS resting stationary on the surface of the Earth with an upward navigational error in altitude would compute a downward gravitational acceleration smaller that the (measured) upward specific force countering gravity, which would result in an upward navigational acceleration error, which only makes matters worse. This would not be a problem for surface ships, it might have been a problem for aircraft if they did not already use barometric altimeters, and similarly for submarines if they did not already use depth sensors. It became an early example