# How does inertial guidance work?

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How does inertial guidance work?

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When you ride in a car, you feel it speed up and slow down, as well as turn. From this information, you can figure out where the car is even if your eyes are closed. You can even try to give the driver navigation instructions based off of this alone.

That is inertial guidance.

Computers can get pretty precise with it. Certainly better than you from the back seat of a car. Between gyroscopes and accelerometers, a microchip can figure out approximately how much it has moved from its current location.

There’s a few different ways to do this.
The systems I use at work are called IMU’s (Inertial Motion Units). They typically use fibre optic gyroscopes to monitor movement around a 3D environment.
Over the last few decades, we’ve gone from buying gyroscopes out of decommissioned cruise missiles (the paperwork was something else), to tiny little units that are built directly into the end unit.

The units detect acceleration along XYZ axis and rotation around those axis.
The accuracy is roughly determined by the cost, though there’s a huge number of variables that can affect that.
An example of the accuracy: we used a back pack lidar scanner, mounted on a car, to scan a few kilometres of tunnel. We had precise (mm accurate datums) at each end and the scanners IMU had only lost a few hundred mm. That’s not a particularly accurate IMU, it’s kinda more meant for small UAV guidance in enclosed spaces.

If you’re having trouble understanding how the gyroscope can measure motion, think about how a bicycle wheel resists changing its plane of rotation. Or a spinning top remains standing. The gyroscope measures the force needed to change its position. The gyroscopes spin at known speeds and are calibrated at the factory to output correct values.
Fibre optic gyroscopes are basically the same, but rather than physical force, they measure the change in travel time of a light signal around a fibre loop as the gyroscope moves.

I’m sure there are other types of IMUs out there, but these are the ones I’ve got end user experience with

wall of text coming up, it’s not ELI5 because 5 year olds don’t know what inertial guidance is and basic terms like pitch roll and yaw; you should look up doppler shift if you’re unfamiliar.

Inertial Guidance works on the most basic method of navigation called dead reckoning which is a process of calculating your position based on knowing your previous position or some kind of fix which is a landmark which you know the position of. It’s the way you navigate through your house when the lights are off. If you know where you are, you could probably navigate to the light switch without using senses other than your inner ear telling you which way is up and which way you just turned. To do this, you need to have a good estimate of your speed, heading and how much time has elapsed.

If you know where you’re starting from and know which way you’re facing and I told you you needed to walk 10 steps forward, then turn left 45° then walk 12 steps and the light switch will be there, you could easily navigate to it. In order for a plane or ship to accurately navigate using dead reckoning, it needs know it’s starting location, it’s heading (which way the nose is pointing) and it’s speed. To do this you need some special hardware.

Before we had GPS satellites that told you where you were anywhere on the face of the earth, we had to rely on gyros and accelerometers. Gyros tell you you’re rate of turn and accelerometers tell you how fast you’re changing velocity which is speed and direction. You need 3 or each axis of rotation. Old gyros relied on spinning mass to determine rate of turn, more advanced ones use either lasers or fiber optics to determine it based on doppler shift. With a fiber optic gyro, you have a spool of fiber optic wire wrapped around counterclockwise and another spool wrapped clockwise. As the spool rotates clockwise light entering the clockwise wound wire goes towards the end of the spool while light going into the counterclockwise spool goes away from the end of the spool; this difference in speed causes a doppler shift which is detected by a sensor at the end of the spool which tells you the rate at which the spool is rotating. The spool, light emitter, detector and circuitry all make up a single gyro. You have 3, one for each axis of rotation. You also have 3 accelorometers that measure change in velocity that work not too differently from the accelerometers inside your phone which it uses to determine which way is up by sensing which way gravity is pulling on the accelerometers. The problem with accelerometers is that they can’t tell the difference between g forces caused by turning or changes in speed, but you have a device that detects turning, the gyros. A computer analyzes the inputs from each gyro and accelerometer, the inputs from the gyros are subtracted from the inputs from the accelerometers so it can determine what your changes in speed are. Before you can make any movements you need to do an alignment so the computer can determine it’s heading based on inputs from the gyros. It does this by sensing the rotation of the earth. If you were to park a plane with an inertial guidance system on the north pole and do an alignment the only gyro that would sense rotation would be the one that detects changes in yaw because over a 24 hour period the yaw rate gyro would sense that it has rotated 360° or 0.25°/minute. If you were to park that same plane on the equator on the equator and faced it east/west the only gyro that would sense rotation would be the ones that sensed changes in pitch as over a 24 hour period it would sense it’s rotated 360°. If you parked that same plane pointing north/south on the equator the roll rate gyros would sense rotation. So anywhere else on earth, all three gyros would sense a slightly different rate of turn.

“[The missile knows where it is](https://www.youtube.com/watch?v=bZe5J8SVCYQ)” is actually describing inertial guidance.

Inertial guidance is all about trying to figure out where you are based on where you last knew you were with your knowledge of how you’ve moved.

An example of inertial guidance is you closing your eyes and trying to walk around on an open field and your brain guesstimating to work out where you are after having walked around a little. You know where you started and you guesstimate “well, I think from my starting point I traveled forward about 5 feet and then a little left from there 2 feet, so I must be in so and so position.”

In reality, this is terribly inaccurate, which is why inertial guidance on its own is very poor as errors accumulate the further you go.

Start with a one dimensional example. You start by figuring out where you are and what your speed is–this is your initial position. Every so often I measure my acceleration and then calculate my new position and velocity using the measurement (assuming constant acceleration).

Let’s assume I start at position 0 at rest. With a sample period of 1 s (this means once every second I measure my acceleration and calculate my new velocity and position).

Measurement 1 is 1 m/s/s. This means my new velocity is 1 m/s and my new position is 0.5m.

Measurement 2 is -5 m/s/s. My new velocity is -4 m/s and I traveled -1.5m, so my new position is -1m.

Measurement 3 is -10 m/s/s. My new velocity is -14 m/s and I traveled -9m, so my new position is -10.5 m.

and so on an so on.

You can go to three dimensions and things get a bit more complicated because you need to deal with vectors. The constant acceleration equations become a series of linear equations so you traditionally use matrices to solve the velocity and position vectors. Furthermore you need to worry about rotations yaw, roll, and pitch which are additional series of equations that you need to solve.

This is a very simple model. Actual systems will apply some fancy filters like Kalman filters., but the basic idea is you know where you are and calculate where you’re going. You can also directly measure velocity to check with your calculations and increase accuracy.

This allows you to navigate without access to more sophisticated systems like GPS. It is very common on submarines since most radio signals travel a very limited distance through water and GPS signals are so weak they barely penetrate the surface.

The downside to this method of navigation is that your error is cumulative. Over time your measurement errors can compound increasing your uncertainty as to position. With GPS your uncertainty is constant with time.