9.5.2 Navigation Error Propagation

The dynamics of INS error propagation are strongly influenced by the fact that gravitational accelerations point toward the center of the earth and decrease in magnitude with altitude and is somewhat less influenced by the fact that the earth rotates.

9.5.2.1 Schuler Oscillations of INS Errors The dominant effect of alignment errors on free-inertial navigation is from tilts, also called leveling errors. These are errors in the estimated direction of the local vertical in sensor-fixed coordinates. The way in which tilts translate into navigation errors is through a process called Schuler oscillation. These are oscillations at the same period that Schuler had identified for gyrocompassing errors (see Section 9.1.1.) The analogy between these Schuler oscillations of INS errors and those of a simple gravity pendulum is illustrated in Figure 9.28. The physical force acting on the mass of a gravity pendulum is the vector sum of gravitational acceleration (mg) and the tension T in the support are, as shown in Figure 9.28a. The analogous acceleration driving inertial navigation errors is the difference between the modeled gravitational acceleration (which changes direction with estimated location) and the actual gravitational acceleration, as shown in Figure 9.28b. In the case of the gravity pendulum, the physical mass of the pendulum is oscillating. In the case of INS errors, only the estimated position, velocity, and acceleration errors oscillate. The gravity pendulum is a physical device, but the Schuler pendulum is a theoretical model to illustrate how INS errors behave.

In either case, the restoring acceleration is approximately related to displacement δ from the equilibrium position by the harmonic equation

the solution for which is

where g is the acceleration due to gravity at the surface of the earth, L is the length of the support arm of the gravity pendulum, R is the radius to the center

of the earth, δ_max is the peak displacement, ω is the oscillation frequency (in radians per second), and φ is an arbitrary oscillation phase angle.

In the case of the gravity pendulum, the period of oscillation

and for the Schuler pendulum

at the surface of the earth.

This ≈ 84.4-min period is called the Schuler period. It is also the orbital period of a satellite at that radius (neglecting atmospheric drag), and the exponential time constant of altitude error instability in pure inertial navigation (see Section 9.5.2.2).

The corresponding angular frequency (Schuler frequency)

Dependence on Position and Direction. A spherical earth model was used to illustrate the Schuler pendulum. The period of Schuler oscillation actually depends on the radius of curvature of the equipotential surface of the gravity model, which can be different in different directions and vary with longitude, latitude, and altitude. However, the variations in Schuler period due to these effects are generally in the order of parts per thousand, and are usually ignored.

Impact on INS Performance Schuler oscillations include variations in the INS errors in position, velocity, and acceleration (or tilt), which are all related harmonically. Thus, if the peak position displacement from the median location is

Note, however, that when the initial INS error is a pure tilt error (i.e., no position error), the peak position error will be 2 δ_max after ≈ 42.2 min from the starting location, and the RMS position error will be (1 + 1/√2) δ_max. If the initial INS error is a pure tilt error, then the true INS position would at one end of the Schuler pendulum swing not the middle and the peak and RMS position errors

as plotted in Fig. 9.29. This shows why alignment tilt errors are so important in free inertial navigation. Tilts as small as one milliradian can cause peak position excursions as big as 10 km after 42 min.

Figure 9.30 is a plot generated using the MATLAB INS Toolbox from GPSoft, showing how an initial northward velocity error of 0.1 ms excites Schuler oscillations in the INS navigation errors, and how coriolis accelerations rotate the direction of oscillation just like a Foucault pendulum with Schuler period. The total simulated time is about/14 h, enough time for 10 Schuler oscillation periods.

For a maximum velocity error δ_max = 0.1 m/s, the maximum expected position error would be

which is just about the maximum excursion seen in Fig. 9.30.

9.5.2.2 Vertical Channel Instability The modeled vertical acceleration due to gravity in the downward direction, as a function of height h above the reference geoid, is dominated by the first term

where R is the reference radius and GM is the model gravitational constant, ≈ 398,600 km³/s² for the WGS84 gravity model. Consequently, the vertical gradient of downward gravity in the downward direction will be

which is positive. Therefore, if δh is the altitude estimation error, it satisfies a differential equation of the form

which is exponentially unstable with solution

Not surprisingly, the exponential time constant of this vertical channel instability

the Schuler period (see Section 9.5.2.1).

9.5.2.3 Coriolis Coupling The coriolis effect couples position error rates (velocities) into into their second derivative (accelerations) as

where φ is geodetic latitude. Adding Schuler oscillations yields the model

where δE is east position error and δN is north position error. This effect can be seen in Fig. 9.30, in which the coriolis acceleration causes the error trajectory to swerve to the right. The sign of sin φ in Eq. 9.86 is negative in the southern hemisphere, so it would swerve to the left there.

Note that, through coriolis coupling, vertical channel errors couple into east channel errors.