How does inertial guidance work?
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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.
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