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Say I have a gyro that outputs degrees/second, and I take the euclidean magnitude of the x,y, and z rotations:

$$ \text{magnitude} = \sqrt{x^2 + y^2 + z^2} $$

Is this actually useful? Is magnitude now "total degrees/second"? Or does the weirdness surrounding non-commutative addition of rotations come into play?

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  • \$\begingroup\$ The axes are orthogonal; there is no weirdness there. \$\endgroup\$
    – Samuel
    Dec 2, 2013 at 18:27

2 Answers 2

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Think back to basic geometry and what the individual X, Y, and Z output values of the gyro actually mean.

One way to look at them is as the components of the rotation vector. This vector is parallel to the axis of rotatation with the magnitude being the speed of rotation. The square root of the sum of the squares of the individual components of this vector is its magnitude. So yes, your magnitude value is the speed of rotation. Draw a picture and this really should be obvious.

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The gyroscope sensor reports instantaneous angular velocity - the tiny rotation occuring from t to t+dt. This is a vector. Actually, pseudovector - it misbehaves around mirrors but that's not relevent here. You add angular velocities like normal. Pythagoras give the magnitude of this - how many radians per second the sensor is turning, without regard for in what direction.

It is the orientation of the object - the accumulation of all the instantaneous angular velocities over time - that shows noncommutative behaviour.

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