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Can we say that the Accelerometer's values are somewhat the integral of the Gyroscope's values ?

I think that because I know that the velocity is the derivative of the position with respect to time and the acceleration is the derivative of the velocity with respect to time.

Let's say I turn in circles. If my assessment is true, then my accelerometer should tell me my velocity is getting bigger and bigger. And it will, even if my velocity doesn't change.

So, I know my assessment is false, but I don't understand why ...

Would you happen to know ?

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Velocity is the derivative of position. Acceleration is the derivative of velocity. If you turn in a perfect constant velocity circle the magnitude of velocity (speed) is constant however the direction is changing. Because the direction is changing there will be a constant magnitude acceleration pointing towards the center of the circle. –  OSE May 15 '13 at 13:47

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No. Mathematically, the two are completely independent measures of motion (though practical measurements may be imperfect).

The accelerometer measures the motion of a specific point -- let's say the center of mass. The gyroscope measures the orientation of everything else around that point.

Imagine a big, old-fashioned spinning gyroscope. You can accelerate it in a straight line by pushing in the same direction on both the top and bottom of the spin axis. There is no net torque on the gyro, so it won't want to turn at all. Plenty of acceleration; no change in the gyro.

On the other hand, you can also imagine pulling and pushing oppositely on the spin axis. The center of mass of the gyro won't move, so an ideal accelerometer would measure zero. But there would be a torque, so the gyro would rotate. No acceleration; plenty of change in the gyro.

So, you see, the two are completely independent.

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