1.5.4. Rotation parametrization: quaternions#

Here quaternion parametrization of rotations is introducing the unitary quaternion

(1.15)#\[\mathbf{q} = q_0 + \vec{q} = \cos \frac{\theta}{2} + \sin \frac{\theta}{2} \vec{n} \ ,\]

into axis and angle parametrization.

Normalization condition.

\[1 = q_0^2 + \vec{q} \cdot \vec{q}\]

Rotation tensor

\[\mathbb{R} = \mathbb{I} + 2 q_0 \vec{q}_\times + 2 \vec{q}_{\times} \vec{q}_{\times} \ ,\]

Linearization

\[\Delta \mathbb{R} = 2 \Delta q_0 \vec{q}_\times + 2 q_0 \Delta \vec{q}_\times + 2 \Delta \vec{q}_\times \vec{q}_\times + 2 \vec{q}_\times \Delta \vec{q}_\times\]
\[\begin{split}\begin{aligned} \vec{\theta}_{\Delta, \times} := & \ \Delta \mathbb{R} \cdot \mathbb{R}^T \\ \vec{\theta}_{\Delta} = & - 2 \vec{q} \Delta q_0 + 2 q_0 \Delta \vec{q} - 2 \vec{q}_\times \Delta \vec{q} \end{aligned}\end{split}\]

Angular velocity

\[\begin{split}\begin{aligned} \vec{\omega}_\times := & \ \dot{\mathbb{R}} \cdot \mathbb{R}^T \\ \vec{\omega} = & - 2 \vec{q} \dot{q}_0 + 2 q_0 \dot{\vec{q}} - 2 \vec{q}_\times \dot{\vec{q}} \end{aligned}\end{split}\]

1.5.4.1. Rotation tensor#

Introducing the expression (1.15) of the unitary quaternion into expression (1.13) of the rotation tensor written using axis and angle parametrization, and using trigonometric identities

\[\begin{split}\begin{aligned} \sin \theta = \sin \left( 2 \frac{\theta}{2} \right) & = 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \\ \cos \theta = \cos \left( 2 \frac{\theta}{2} \right) & = \cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2} = \\ & = 2 \cos^2 \frac{\theta}{2} - 1 = \\ & = 1 - 2 \sin^2 \frac{\theta}{2} \end{aligned}\end{split}\]

the expression of the rotation tensor becomes

\[\begin{split}\begin{aligned} \mathbb{R} & = \mathbb{I} + \sin \theta \hat{n}_\times + ( 1 - \cos \theta ) \hat{n}_{\times} \hat{n}_{\times} = \\ & = \mathbb{I} + 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \hat{n}_\times + 2 \sin^2 \frac{\theta}{2} \hat{n}_{\times} \hat{n}_{\times} = \\ & = \mathbb{I} + 2 q_0 \vec{q}_\times + 2 \vec{q}_{\times} \vec{q}_{\times} \\ \end{aligned}\end{split}\]

1.5.4.2. Angular velocity#

\[\begin{split}\begin{aligned} \vec{\omega}_{\times} & = \dot{\mathbb{R}} \cdot \mathbb{R}^T = \\ & = \left( 2 \dot{q}_0 \vec{q}_\times + 2 q_0 \dot{\vec{q}}_{\times} + 2 \dot{\vec{q}}_\times \vec{q}_\times + 2 \vec{q}_\times \dot{\vec{q}}_\times \right) \cdot \left( \mathbb{I} - 2 q_0 \vec{q}_\times + 2 \vec{q}_{\times} \vec{q}_{\times} \right) = \\ & = 2 \dot{q}_0 \vec{q}_\times + 2 q_0 \dot{\vec{q}}_{\times} + 2 \dot{\vec{q}}_\times \vec{q}_\times + 2 \vec{q}_\times \dot{\vec{q}}_\times + \\ & \ \ - 4 \dot{q}_0 q_0 \vec{q}_\times \vec{q}_\times - 4 q_0^2 \dot{\vec{q}}_{\times} \vec{q}_\times - \underbrace{4 q_0 \dot{\vec{q}}_\times \vec{q}_\times \vec{q}_\times}_{(a)} - \underbrace{4 q_0 \vec{q}_\times \dot{\vec{q}}_\times \vec{q}_\times}_{-4 q_0 \left( \dot{\vec{q}} \cdot \vec{q} \right) \vec{q}_\times} + \\ & \ \ + 4 \dot{q}_0 \underbrace{\vec{q}_\times \vec{q}_\times \vec{q}_\times}_{-|\vec{q}|^2 \vec{q}_\times} + \underbrace{4 q_0 \dot{\vec{q}}_{\times} \vec{q}_\times \vec{q}_\times}_{(a)} + 4 \dot{\vec{q}}_\times \underbrace{\vec{q}_\times \vec{q}_\times \vec{q}_\times}_{-|\vec{q}|^2 \vec{q}_\times} + \underbrace{4 \vec{q}_\times \dot{\vec{q}}_\times \vec{q}_\times \vec{q}_\times}_{-\left( \dot{\vec{q}} \cdot \vec{q} \right) \vec{q}_\times \vec{q}_\times} = \\ & = \underbrace{ 2 \dot{q}_0 \vec{q}_\times}_{(e)} + 2 q_0 \dot{\vec{q}}_{\times} + \underbrace{2 \dot{\vec{q}}_\times \vec{q}_\times}_{(c.1)} + 2 \vec{q}_\times \dot{\vec{q}}_\times + \\ & \ \ - 4 \underbrace{\dot{q}_0 q_0 \vec{q}_\times \vec{q}_\times}_{-(b)} - \underbrace{4 q_0^2 \dot{\vec{q}}_{\times} \vec{q}_\times}_{(c.2)} + \underbrace{4 q_0 \left( \dot{\vec{q}} \cdot \vec{q} \right) \vec{q}_\times}_{(d)} + \\ & \ \ - \underbrace{4 \dot{q}_0 |\vec{q}|^2 \vec{q}_\times}_{ 4\dot{q}_0 \vec{q}_\times - 4 \dot{q}_0 q_0^2 \vec{q}_\times = 2(e)-(d)} - \underbrace{4 |\vec{q}|^2 \dot{\vec{q}}_\times \vec{q}_\times}_{(c.3)} - \underbrace{4\left( \dot{\vec{q}} \cdot \vec{q} \right) \vec{q}_\times \vec{q}_\times}_{(b)} = \\ & = -2 \dot{q}_0 \vec{q}_\times + 2 q_0 \dot{\vec{q}}_\times - 2 \dot{\vec{q}}_\times \vec{q}_\times + 2 \vec{q}_\times \dot{\vec{q}}_\times = \\ & = \left[ - 2 \vec{q} \dot{q}_0 + 2 q_0 \dot{\vec{q}} - 2 \vec{q}_\times \dot{\vec{q}} \right]_\times \end{aligned}\end{split}\]

having used

  • \(\vec{q}_\times^T = - \vec{q}_\times\), \(\left( \vec{q}_\times \vec{q}_\times \right)^T = \vec{q}_\times \vec{q}_\times\)

  • \(1 = q_0^2 + |\vec{q}|^2\), and thus \(\vec{q} \cdot \dot{\vec{q}} = - q_0 \dot{q}_0\)

  • \(\vec{q}_\times \vec{q}_\times = \vec{q} \otimes \vec{q} - |\vec{q}|^2 \mathbb{I}\)

  • \(\vec{q}_\times \vec{q}_\times \vec{q} = \underbrace{\vec{q}_\times \vec{q} \otimes \vec{q}}_{=\mathbb{0}} - |\vec{q}|^2 \vec{q}_{\times} = -|\vec{q}|^2 \vec{q}_{\times} \)

  • \(\dot{\vec{q}}_\times \vec{q}_\times = \vec{q} \, \dot{\vec{q}} - \left( \dot{\vec{q}} \cdot \vec{q} \right) \, \mathbb{I}\)

  • \(\vec{q}_\times \dot{\vec{q}}_\times = \dot{\vec{q}} \, \vec{q} - \left( \dot{\vec{q}} \cdot \vec{q} \right) \, \mathbb{I}\)

Proof - todo

  • \((\vec{a}_\times \vec{b})_{\times} = \vec{a}_{\times} \vec{b}_\times - \vec{b}_\times \vec{a}_\times \)

Proof \(\ (\vec{a}_\times \vec{b})_{\times} = \vec{a}_{\times} \vec{b}_\times - \vec{b}_\times \vec{a}_\times \)

Using

\[(\vec{a} \times \vec{b}) \times \vec{c} + (\vec{b} \times \vec{c}) \times \vec{a} + (\vec{c} \times \vec{a}) \times \vec{b} = \vec{0} \]

(todo, prove it with index notation) it immediately follows the desired relation

\[\begin{split}\begin{aligned} (\vec{a}_\times \vec{b})_{\times} \vec{v} & = ( \vec{a} \times \vec{b} ) \times \vec{v} = \\ & = - ( \vec{b} \times \vec{v} ) \times \vec{a} - ( \vec{v} \times \vec{a} ) \times \vec{b} = \\ & = \vec{a} \times ( \vec{b} \times \vec{v} ) - \vec{b} \times ( \vec{a} \times \vec{v} ) = \\ & = \left( \vec{a}_{\times} \vec{b}_\times - \vec{b}_\times \vec{a}_\times \right) \vec{v} \end{aligned}\end{split}\]