1.5.2. Rotation parametrization: axis and angle

Rotation tensor

(1.13)\[\mathbb{R} = \mathbb{I} + \sin \theta \, \hat{n}_\times + ( 1 - \cos \theta ) \, \hat{n}_\times \hat{n}_\times \ ,\]

Angular velocity

\[\vec{\omega}_\times = \dot{\mathbb{R}} \cdot \mathbb{R}^T = \]

Linearization

1.5.2.1. Rotation tensor

The expression of the rotation tensor \(\mathbb{R}\) that rotates vector \(\vec{v}^0\) into \(\vec{v}\),

\[\vec{v} = \mathbb{R} \cdot \vec{v}^0 \ ,\]

comes from little geometry and vector algebra. The rotation of an angle \(\theta\) around the axis determined by the unit vector \(\hat{n}\) reads

(1.14)\[\vec{v} = \hat{n} v^0_{\parallel} + \hat{x} v^0_{\perp} \cos \theta + \hat{y} v^0_{\perp} \sin \theta \ ,\]

with

• \(v^0_{\parallel} = \hat{n} \cdot \vec{v}^0\),

• \(\hat{n} \times \vec{v}^0 = \hat{y} v^0_{\perp}\), and thus \(\hat{y} = \frac{\hat{n} \times \vec{v}^0}{v^0_{\perp}}\),

• \(\hat{x} = \hat{y} \times \hat{n}\), and thus \(v^0_{\perp} \hat{x} = v^0_{\perp} \frac{(\hat{n} \times \vec{v}^0)}{v^0_{\perp}} \times \hat{n} = - \hat{n} \times \left( \hat{n} \times \vec{v}^0 \right) = - \hat{n}_{\times} \hat{n}_{\times} \cdot \vec{v}^0\), where the dot product is meant between the \(\hat{n}_\times\) tensors representing vector products.

Exploiting these expressions, the relation (1.14) between the vectors \(\vec{v}\), \(\vec{v}^0\) can be written in implicit tensor form. Different equivalent tensor expressions follows from the identity \(\hat{v}_\times \hat{v}_\times = \vec{v} \otimes \vec{v} - |\vec{v}|^2 \mathbb{I} \),

\[\begin{split}\begin{aligned} \vec{v} & = \hat{n} v^0_{\parallel} + \hat{x} v^0_{\perp} \cos \theta + \hat{y} v^0_{\perp} \sin \theta = \\ & = \hat{n} \hat{n} \cdot \vec{v}^0 - \cos \theta \hat{n}_\times \hat{n}_\times \cdot \vec{v}^0 + \sin \theta \hat{n}_\times \cdot \vec{v}^0 = \\ & = \left[ \hat{n} \hat{n} - \cos \theta \hat{n}_\times \hat{n}_\times + \sin \theta \hat{n}_\times \right] \cdot \vec{v}^0 = \\ \end{aligned}\end{split}\]

and thus

\[\begin{split}\begin{aligned} \mathbb{R} & = \hat{n} \hat{n} - \cos \theta \hat{n}_\times \hat{n}_\times + \sin \theta \hat{n}_\times = \\ & = \mathbb{I} + \sin \theta \, \hat{n}_\times + (1 - \cos \theta ) \, \hat{n}_\times \hat{n}_\times \ . \end{aligned}\end{split}\]

Proof of relation \(\ \hat{v}_\times \hat{v}_{\times} = \vec{v} \otimes \vec{v} - |\vec{v}|^2 \mathbb{I} \)

\[\begin{split}\begin{aligned} \hat{v}_\times \hat{v}_{\times} & = \hat{e}^i \hat{e}_m \varepsilon_{ijk} v_j \varepsilon_{klm} v_l = \\ & = \hat{e}^i \hat{e}_m \left( \delta_{il} \delta_{jm} - \delta_{im} \delta_{jl} \right) v_j v_l \ , & = \hat{e}^i \hat{e}_m \left( v_i v_m - v_j v_j \delta_{im} \right) = \\ & = \vec{v} \otimes \vec{v} - |\vec{v}|^2 \mathbb{I} \end{aligned}\end{split}\]

1.5.2.2. Angular velocity

\[ \vec{\omega} = \dot{\theta} \, \hat{n} + \sin \theta \, \dot{\hat{n}} - \left( 1 - \cos \theta \right) \, \hat{n}\times \dot{\hat{n}} \]

Proof

Using the expression of the angular velocity using unitary quaternion parametrization,

\[\mathbf{q} = q_0 + \vec{q} = \cos \frac{\theta}{2} + \sin \frac{\theta}{2} \, \hat{n} \ .\]

and

\[\vec{\omega} = - 2 \, \vec{q} \, \dot{q}_0 + 2 \, q_0 \, \dot{\vec{q}} - 2 \, \vec{q} \times \dot{\vec{q}}\]

with

\[\begin{split}\begin{aligned} \dot{q}_0 & = - \frac{\dot{\theta}}{2} \sin \frac{\theta}{2} \\ \dot{\vec{q}} & = \frac{\dot{\theta}}{2} \cos \frac{\theta}{2} \hat{n} + \sin \frac{\theta}{2} \dot{\hat{n}} \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} \vec{\omega} & = - 2 \, \vec{q} \, \dot{q}_0 + 2 \, q_0 \, \dot{\vec{q}} - 2 \, \vec{q} \times \dot{\vec{q}} = \\ & = - 2 \, \sin \frac{\theta}{2} \hat{n} \left( -\frac{\dot{\theta}}{2} \sin \frac{\theta}{2} \right) + 2 \cos \frac{\theta}{2} \left( \frac{\dot{\theta}}{2} \cos \frac{\theta}{2} \hat{n} + \sin \frac{\theta}{2} \dot{\hat{n}} \right) - 2 \, \sin \frac{\theta}{2} \hat{n} \times \left( \frac{\dot{\theta}}{2} \cos \frac{\theta}{2} \hat{n} + \sin \frac{\theta}{2} \dot{\hat{n}} \right) = \\ & = \dot{\theta} \, \hat{n} + \sin \theta \, \dot{\hat{n}} - \left( 1 - \cos \theta \right) \, \hat{n}\times \dot{\hat{n}} \end{aligned}\end{split}\]

having used \(\cos \theta = \cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2} = 1 - 2 \sin^2 \frac{\theta}{2}\), to transform \(\sin^2 \frac{\theta}{2}\), and \(\hat{n} \times \hat{n} = \vec{0}\).