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Euler matrices numerical instability

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Hello everybody, it is known that when we try to extract euler angles from a rotation matrix, when the pitch is outside the interval of - pi/2 < 0 < pi/2 original values cannot be extracted. This leads to numerical instability of angles. On the book "Realtime Rendering" they say that there's a way to avoid this, paying the cost of more computation. They cite the following article: "Paul, Richard P.C., Robot Manipulators: Mathematics, Programming, and Control, MIT Press, Cambridge, Mass., 1981." I couldn't find it free online, does anybody know the tecnique or where I could find the solution for this problem? Thanks, E.R.

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I can recommend a different, related book which might also cover the same material: Sastry's Mathematical Introduction to Robotic Manipulation is very good.

As for your question... It's possible that I misunderstand you, but my gut reactions are (1) "it's not possible," and (2) "if you're converting back and forth to and from Euler angles you're doing something wrong."

To elaborate on point 2: Ordinarily people just bypass Euler angles altogether and use a global representation for SO(3) (like the unit quaternions modulo antipodes, or the rotation matrices) instead of a local coordinate chart. These are embeddings of SO(3) into higher-dimensional spaces. Euler angles, on the other hand, are just a local coordinate chart in the same way that a 2d map of the Earth necessarily involves some "cuts" and "warping" of the surface.

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