These discontinuities are caused by the existence of many-to-one mappings between the Euler angle parameterization of the set of 3D rotations. This allows the data set to flip between different Euler angle combinations which correspond to a single 3D rotation, which, although remaining continuous in the space of rotation, are discontinuous in the Euler angle parameter space.
The Euler filter chooses on a sample-by-sample basis between the possible Euler angle representations of each 3D rotation in the data set in such a way as to preserve the continuity of the Euler angle time series, without changing the actual 3D rotations.
Euler filtering is available in a number of 3D animation packages.
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