First of all, I'm not a physics student (even though I've always been quite fascinated by it), and this can possibly be an easy problem, but here's my doubt.
I was reading this Wiki's article about Inertial Navigation Systems (ING), it says, at the end of the paragraph and I'm linking you to:
However, by tracking both the current angular velocity of the system and the current linear acceleration of the system measured relative to the moving system, it is possible to determine the linear acceleration of the system in the inertial reference frame.
What I think I understood...
I understood what's an inertial reference frame, it's basically a reference frame (to simplify, let's just talk about classical mechanics) where the inertia law of Newton is valid, that is an object traveling at a constant velocity will remain so unless an external force acts upon it.
I understood roughly what's the angular velocity and I understood that linear acceleration here is the proper acceleration measured by an accelerometer (that is an acceleration not due to gravity).
What I really did not understand
What I'm having difficulties with is:
What does it mean that the linear acceleration was measured relative to the moving system? (Of course it may be more easy for you to read that paragraph than me to explain you)
What would be the linear acceleration of the system in the inertial reference frame? I'm not asking you how to calculate it, but the conceptual meaning of it.
Does this mean that the first linear acceleration measured by the accelerometer was not measured in an inertial reference frame? If yes, why? If yes, in which (type of) frame was it measured then?
Why do we care about the linear acceleration in the inertial reference frame?
Is the inertial frame of the system or of an another object?