Inertial Navigation Systems — Page 244, Lesson 281

Let’s start with the heart of this system — the Schuler Period — because everything about INS errors hangs off it. Imagine a simple pendulum. If you displace it and let go, it swings back and forth with a natural period that depends on its length. Now imagine a pendulum whose length is the radius of the Earth — about 6,371 kilometres. Such a pendulum would have a natural period of 84.4 minutes. That’s the Schuler Period. In an inertial navigation system, we don’t have a physical pendulum that long, of course. Instead, we mechanize the platform — we build the control system — so that the platform stays horizontal as the vehicle moves. The control signals that keep it horizontal are the V/R and U/R terms. Let me unpack those. V is the vehicle’s velocity, U is another velocity component, and R is the radius of the Earth. So V/R and U/R are angular rates — the rate at which the platform must tilt to stay level as the vehicle moves over the curved Earth. By mechanizing the platform to remain horizontal in this way, we produce an analogue — an electronic and mechanical equivalent — of that Earth pendulum with its 84.4-minute period. Here’s the key consequence: if the platform is displaced from the horizontal for any reason, it will not just stay tilted. It will oscillate — swing back and forth — with a period of 84.4 minutes. That oscillation is the Schuler Period. Now, why does this matter? Because it gives us a natural way to classify the errors of the INS. Errors fall into three headings: bounded errors, unbounded errors, and inherent errors. Let’s take bounded errors first. A bounded error is one that builds up to a maximum, then returns to zero within the 84.4-minute Schuler cycle. So it’s self-correcting — it oscillates and comes back. The main causes are three. First, platform tilt due to initial misalignment — if the platform starts slightly tilted, the Schuler oscillation will carry it back to level. Second, inaccurate measurement of acceleration by the accelerometers — a small error in sensed acceleration feeds in, but again it’s bounded by the Schuler cycle. Third, integrator errors in the first stage of integration. In an INS, acceleration is integrated once to get velocity, and a second time to get distance. Errors in that first integration stage are bounded. Now unbounded errors. These are different — they do not return to zero. They are either cumulative track errors or cumulative distance errors. Two causes give cumulative track errors: initial azimuth misalignment of the platform — that’s a heading error at start-up — and wander of the azimuth gyro, meaning the gyro that defines the vertical axis drifts over time. Then there are errors that give cumulative errors in the recording of distance run. First, wander in the levelling gyros. This causes a Schuler oscillation of the platform — so the platform does oscillate — but here’s the subtle part: the mean recorded value of distance run is increasingly divergent from the true distance run. So even though the platform oscillates, the average of what it records drifts further and further from reality. Second, integrator errors in the second stage of integration — that’s the stage that integrates velocity into distance, so an error there accumulates directly into distance. So the contrast to hold onto: bounded errors oscillate and return to zero within 84.4 minutes; unbounded errors accumulate without returning to zero. And the third heading, inherent errors, is listed but not expanded here — it’s a category we’ll meet when the material develops it. Let me show you the geometry of that Schuler oscillation. That figure shows the platform oscillating about the horizontal, with R as the radius of the Earth, and the labelled values 63.3, 21.1, 42.2, and 84.4 — those are minutes, marking points along the 84.4-minute Schuler cycle. The platform swings through the horizontal, overshoots, and returns — that’s the bounded behaviour we just discussed. So the whole picture is: the platform is kept level by V/R and U/R control signals, which makes it behave like an 84.4-minute Earth pendulum. Any disturbance sets it oscillating at that Schuler Period. Errors that ride on that oscillation and return to zero are bounded; errors that accumulate — from azimuth misalignment, gyro wander, or second-stage integrator drift — are unbounded.