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Going round in circles: how the earth’s rotation affects inertial navigation

Blogs September 9, 2023

Building an Inertial Navigation System (INS), or navigation engine, for an autonomous platform is tricky work. Not necessarily because of the hardware, but because of the myriad adjustments, calibrations and calculations that are required to properly integrate the IMU and the GNSS receivers so that the INS works anywhere on the planet (or, at the very least, in the places you want it to work).

In this article, we’re taking a closer look at what the earth’s rotation does to inertial navigation systems, and explaining a bit about how OxTS accounts for these things so that our devices give users accurate data anywhere on the planet.

 

What do I need to consider?

In summary, you need to be aware of the following things:

  • A high-quality IMU is sensitive enough to pick up the rotation of the earth.
  • The spin of the earth also causes the gyroscopes in the IMU to wander
  • IMUs are also affected by the Coriolis effect and the associated Eötvös effect

If these things aren’t accounted for, your IMU will provide bad data to the navigation engine, compromising the quality of its outputs. Which may compromise the structural integrity of your automated vehicle when it crashes because it can’t travel straight, or thinks it’s in the wrong place.

 

First things first: understanding the inertial frame

This one can be a bit tricky to wrap your head around, so let’s start here: are you moving right now?

The answer is yes. Even if you’re sat down. Even if you’re in bed. Even if you’re asleep. At this very moment, your body is clinging to a spinning rock that’s forever falling towards the sun – which itself is forever falling towards the centre of the Milky Way – which itself is forever falling towards the centre of the universe (probably).

Human beings can’t perceive these movements for two reasons:

  1. Because we are on the earth, we are moving at nearly the same (steady) speed as it is.
  2. The difference between the speed the earth is moving and the speed that the people on it are moving is so small that human beings can’t perceive it.

This means that our frame of reference when observing things is non-inertial. But our inertial measurement unit (as you may have guessed) is measuring in an inertial frame – meaning that it can detect that movement.

If you place your IMU on your desk, you will see that it measures some acceleration and some change in angular rate. It’s measuring two things:

  • The pull of gravity towards the centre of the earth (though we have a whole separate article on that here).
  • The spin of the earth.

The spin of the earth

As we’ve already covered, the earth is a chunk of rock screaming through space, spinning like a whirling dervish, pulling everything on its surface towards is molten core. Thankfully, the earth’s movement around the sun generates measurements that are so small they get lost in the background noise in your IMU. The spin of the earth, however, does not. The angular rate of the spin of the earth equates to about 15 degrees per hour – which, of course, is enough to throw off the heading calculations of your autonomous vehicle by an increasingly large amount over time.

The spin of the earth can also throw the gyroscopes in your IMU off in another way, known as apparent wander (one of the two types of wander your gyroscope will experience).

 

Gyroscope wander

Apparent wander

Let’s say you get up early and set your gyroscope to point North at 7am. If you checked that gyroscope again at 7pm, you’d find that it no longer pointed North. In truth, the gyroscope hasn’t moved. Instead, the earth has moved around the IMU – but because we are also moving with the earth, it looks to us as though the IMU has moved. Apparent wander is a constant phenomenon – and if you’re willing to get up early again, at 7am the next morning your gyroscope will be pointing North again – as long as you stay where you are. If your IMU moves, then you have another type of wander to contend with: transport wander.

Transport wander

If you were to align your gyroscope with local North, and then travel east or west across a line of longitude, it would look as though your gyroscope had moved so it was no longer aligned with North. As with apparent wander, it’s not the gyroscope that’s moved position – it’s the earth.

It’s important to note that both transport and apparent wander will affect your gyroscope at the same time (unless your INS was installed in a vehicle travelling westwards at the exact same speed as the earth’s rotation at your point on the planet, which isn’t likely to be a very commercially viable journey).

The rate at which transport wander affects your IMU is called a LOT of different things. A few of them include:

  • Transport rate (that’s the term we use)
  • Transport theorem
  • Transport equation
  • Rate of change transport theorem
  • Basic kinematic equation

And, while we’re on the subject of phenomena that affect your IMU while it’s in motion, there are two more that we need to talk about while discussing the spin of the earth: the Coriolis Effect, and the Transport Rate.

 

The Coriolis effect

If you dredge up those memories of your physics lessons, you’ll remember that the Coriolis effect is an inertial force. It acts on objects that move on a surface that’s rotating in a non-inertial frame (indeed, the fact that we experience Coriolis forces at all proves that the surface of the earth is not an inertial frame).

The Coriolis effect is strong enough to alter the course of an object’s travel across the globe (such as an artillery shell or a bullet from a sniper rifle) – and also strong enough to alter the readings from your IMU.

We’ve created a handy table below to summarise how the Coriolis effect will influence your IMU. Note that if you’re travelling upwards or downwards, the curve could be left, right, away from you or towards you depending on your orientation, so we’ve just stuck with the cardinal direction:

 

Direction of travelEffect in Northern hemisphereEffect in Southern hemisphere
NorthCurve to the right (East)Curve to the left (East)
SouthCurve to the left (West)Curve to the right (West)
EastCurve to the right (South)Curve to the left (North)
WestCurve to the left (North)Curve to the right (South)
UpCurve to the WestCurve to the East
DownCurve to the EastCurve to the West

 

The rate at which Coriolis affects your readings does of course depend on your latitude (the further from the equator you are, the stronger the effect). And, unfortunately, Coriolis forces aren’t the only forces that might throw off your IMU as you travel across the earth…

The Eötvös effect

The Eötvös effect is a vertical component of Coriolis forces. The simplest summary of it is that objects travelling westward across the earth experience an increase in gravity, while objects travelling eastwards experience a decrease in gravity.

On top of the effects of earth’s gravity (which we’ll be discussing in a different article), you’ll need to make sure you’ve worked out whether the Eötvös effect is strong enough to affect your readings (unlike horizontal Coriolis forces, the Eötvös effect gets weaker the further away from the equator you go). If it does affect your readings, you’ll need to account for it.

Secondary Coriolis effects

As if all that wasn’t enough, you actually need to account for two sets of Coriolis effects. This is because your IMU is actually moving across two different frames:

  • The earth frame, which is spinning (these are the primary Coriolis forces).
  • The local frame of your IMU, which is also moving relative to the earth frame, causing a secondary Coriolis effect.

This effect will also cause small but consequential drift in your IMU readings unless you account for it. Unfortunately, the table above won’t be much help for secondary Coriolis effects, as they bleed out to affect all three axes!

 

How do I stop these phenomena from wrecking my INS?

To make sure your INS gives you reliable information, there’s only one solution – and that’s to spend some time modelling. No, not that kind.

At OxTS, we’ve spent hundreds of hours building and testing models of corrections for the accelerometers and gyroscopes in our devices which compensate for each of the effects we’ve detailed here. It gets especially complex when you consider that each force, to an extent, combines and competes with the others. There are no shortcuts, really – you just need to sit and do the maths (though, unlike your secondary school days, you can use a calculator).

 

Questions?

Hopefully, this article has given you an idea of how to stop the earth’s rotation from interfering with the performance of the IMU in your INS. If it’s raised more questions for you, then we’d be very happy to chat through things with you in more detail – just click here to get in touch:

Referenced works:

Transport Wander of a Gyroscope (sphaera.co.uk)

Apparent Wander of a Gyroscope (sphaera.co.uk)

https://stratus.ssec.wisc.edu/courses/gg101/coriolis/coriolis.html

How does the coriolis effect work when travelling west – Search (bing.com)

meteorology – Does the Coriolis Effect affect a wind from East to West? – Earth Science Stack Exchange

Morton et al, Position, Navigation, and Timing Technologies in the 21st Century: Integrated Satellite Navigation, Sensor Systems, and Civil Applications, Volume 2, Wiley-IEEE Press; 1st edition, 2021

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