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+++ title = "Angular Velocity" author = ["Thomas Dehaeze"] draft = false +++
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Non-integrability of the angular velocity vector
The non-integrability of the angular velocity vector is well described in (Legnani et al. 2012).
It is well known that the angular velocity vector is not the time derivative of any set of angular coordinates. In other words, it is impossible to define a set of three coordinates representing the 3D angular position of a body whose time derivative is equal to the angular velocity vector.
This is illustrated in Figure 1.
{{< figure src="/ox-hugo/angular_nonintegrability.png" caption="Figure 1: Effect of different sequences of rotations of a rigid body. In both cases we get Rot(x)=0, Rot(y)=90deg and Rot(z)=90deg" >}}
Bibliography
Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” Mechanism and Machine Theory 58 (nil):64–81. https://doi.org/10.1016/j.mechmachtheory.2012.07.008.