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I've been working with MEMS gyroscopes in hobbyist projects for years—but it's recently come to my attention that I don't have any principled understanding of what rotation rate measurements mean, physically speaking, and now it's bugging me.

My immediate motivation for asking is that I want to understand the design decisions in Paul Riseborough's Kalman filter for UAVs (see my earlier attempt at this question). Now that I'm aware of my confusion, I have the same issue with interpreting output from the Portland State Aerospace Society's inertial measurement unit, which has a similar package of three-axis accelerometers/gyroscopes/etc.

Paul states (at the above link) that it's standard in the literature to interpret the three-axis rate-gyro measurements as a kind of axis-angle representation of rotation, where the direction of the 3-vector is the axis of rotation, and the magnitude of the vector is the angular rotation rate around that axis. I just don't understand why that would be true in general, although it's clearly true in the special case where rotation is exactly around one gyro axis.

I think that if someone can show me the derivation of what each MEMS gyro axis should read, given an arbitrary 3-D rotation rate, that may lead to a deeper understanding of how I should interpret these sensor readings.

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You may be aware that both torque and angular momentum can be represented as a vector - and that such vectors follow the normal rules of vector addition.

Thus, if you have equal rotation about both the X and the Y axis, what you really have is rotation about the XY axis; and in general, rotation about an arbitrary axis can be projected onto the X, Y and Z axis to get the relative components of rotation.

Imagine a spherical top. If it is spinning perfectly vertically, there will be no angular momentum about the X or Y axis. But if you tilt it slightly so it is pointing towards the X axis, there will be a small component of the angular momentum about the X axis.

Now the earth's gravity will exert a torque on the top, but instead of it falling over, this torque causes the top to start precessing. This is the principle of the gyroscope, and understanding why and how that works will help you get a deeper understanding of the question you are asking (which is independent of the type of gyro sensor, by the way).

You can find a nice introduction to precession of gyroscopes and the vector representation of rotation and torque at http://www.freestudy.co.uk/dynamics/gyros.pdf

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