digital-brain/content/zettels/angular_velocity.md

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2021-09-22 16:00:00 +02:00
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title = "Angular Velocity"
author = ["Thomas Dehaeze"]
draft = false
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## Non-integrability of the angular velocity vector {#non-integrability-of-the-angular-velocity-vector}
The non-integrability of the angular velocity vector is well described in ([Legnani et al. 2012](#orga01afc4)).
> 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](#org4ad23f3).
<a id="org4ad23f3"></a>
{{< 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 {#bibliography}
<a id="orga01afc4"></a>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):6481. <https://doi.org/10.1016/j.mechmachtheory.2012.07.008>.