Inertial Navigation Systems — Page 244, Lesson 281

Let's pick up with the heart of the inertial navigation system — the Schuler Period, and then the error picture that defines how good this system really is. First, the Schuler Period. I want you to picture a simple pendulum — a weight on a string. If you displace it, it swings back and forth with a natural period. Now, imagine a pendulum whose length is equal to the radius of the Earth. That's the concept here. The radius of the Earth is labelled R in Figure 18.15. Such an imaginary pendulum would have a natural period of 84.4 minutes. That number — 84.4 minutes — is the Schuler Period. Now, how does this apply to an inertial platform? The platform is mechanized to remain horizontal. The control signals that keep it horizontal are the V/R and U/R terms for vehicle movement. Let me unpack those. V is the vehicle's velocity, U is another velocity component — and dividing each by R, the Earth's radius, gives you an angular rate. So V/R and U/R are the angular rates the platform needs to command to stay level as the vehicle moves over the curved Earth. By mechanizing the platform this way, we create an analogue of that Earth pendulum. The result is beautiful: if the platform is displaced from the horizontal, it doesn't just drift off — it oscillates back and forth with a period of 84.4 minutes. That oscillation is the Schuler Period. This is the key design feature that makes an INS self-correcting for horizontal errors, because the platform behaves like a pendulum that always wants to return to level. Now let's move to the errors of INS. They're conveniently grouped under three headings: bounded errors, unbounded errors, and inherent errors. Bounded errors are the ones that build up to a maximum, then return to zero within the 84.4-minute Schuler cycle. So they're self-correcting over that period. The main causes are three. First, platform tilt due to initial misalignment — the platform starts slightly tilted. Second, inaccurate measurement of acceleration by the accelerometers — the sensors themselves aren't perfect. Third, integrator errors in the first stage of integration. Remember, an INS integrates acceleration to get velocity — that's the first stage. Errors there are bounded. Unbounded errors are different. They're either cumulative track errors or distance errors. There are two causes listed. First, initial azimuth misalignment of the platform — the platform is rotated the wrong way around the vertical at start-up. Second, wander of the azimuth gyro — the gyro that defines the heading drifts over time. Both of these give cumulative track errors. Then there are errors that give rise to cumulative errors in the recording of distance run. Two causes here. First, wander in the levelling gyros. This causes a Schuler oscillation of the platform — so the platform oscillates — but the mean recorded value of distance run is increasingly divergent from the true distance run. So even though it oscillates, the average drifts away from reality. Second, integrator errors in the second stage of integration. The second stage integrates velocity to get distance — errors there are unbounded, they accumulate. So the contrast to hold onto: bounded errors come and go within the 84.4-minute Schuler cycle, while unbounded errors accumulate over time and never return to zero. The first stage of integration — acceleration to velocity — gives bounded errors. The second stage — velocity to distance — gives unbounded errors. And the gyros: azimuth gyro wander gives track errors, levelling gyro wander gives distance errors. That's the complete error picture for the INS.