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:

  1. 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)

  2. 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.

  3. 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?

  4. Why do we care about the linear acceleration in the inertial reference frame?

  5. Is the inertial frame of the system or of an another object?

By definition, a rotating frame of reference is not "inertial" - because objects in different locations that are stationary with respect to each other experience different "forces" (in particular, the Coriolis and centrifugal "force": these are fictitious forces that are needed to explain how objects appear to move as seen from a rotating frame of reference).

So if you have an accelerometer in a rotating object, that device will detect these fictitious forces. If you are trying to figure out how much your rotating frame of reference is moving "in the real world" you need to keep track of the rotation, and figure out what you would see if you could subtract these fictitious accelerations from the readings of the accelerometer.

Further - imagine you rotate your sensor by 90 degrees. Now your X is pointing in the Y direction, and the Y is pointing in the -X direction. So if you are recording an "X acceleration" you were actually moving in Y. Again, if you keep track of the motion / orientation of the rotating frame, you can correct for this.

I hope that clears up some of your confusion. I realize I didn't go point by point through your questions - but I think this should get you going.

  • So, in summary, you're saying that in the real world we do not have exactly an inertial reference frame since we can also rotate and thus the initial angular velocity and the linear acceleration measured respectively by the gyroscope and accelerometer are not in an inertial frame, but we can somehow combine the results to obtain a linear acceleration in an imaginary inertial frame. Why are we interested in the measurements in the inertial frame? Is it because it simplify things somehow? – nbro Mar 6 '17 at 23:10
  • Whether you can ignore the rotation of the earth depends on the scale of the problem you are solving. Shooting a rocket to the moon, the rotation matters. Driving to the grocery store it does not. But if your sensor is inside an object that rotates much faster than the earth, these things matter a lot more. – Floris Mar 6 '17 at 23:12

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