+++
title = "Angular Velocity"
author = ["Dehaeze Thomas"]
draft = false
category = "Mechanics and Physics"
+++

Tags
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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 (<a href="#citeproc_bib_item_1">Legnani et al. 2012</a>).

> 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--fig:angular-nonintegrability).

<a id="figure--fig:angular-nonintegrability"></a>

{{< figure src="/ox-hugo/angular_nonintegrability.png" caption="<span class='figure-number'>Figure 1: </span>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}

<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” <i>Mechanism and Machine Theory</i> 58: 64–81. doi:<a href="https://doi.org/10.1016/j.mechmachtheory.2012.07.008">10.1016/j.mechmachtheory.2012.07.008</a>.</div>
</div>