1.7. Rotations

1.7.1. Rotation tensor

Given 2 Cartesian bases \(\{ \hat{e}^0_i \}_{i=1:3}\), \(\{ \hat{e}^1_j \}_{j=1:3}\), the rotation tensor providing the transformation

\[\hat{e}^1_i = \mathbb{R}^{0 \rightarrow 1} \cdot \hat{e}^0_i \ ,\]

is

\[\mathbb{R}^{0 \rightarrow 1} = R_{ij}^{0 \rightarrow 1} \hat{e}^0_i \otimes \hat{e}^0_j = R_{ij}^{0 \rightarrow 1} \hat{e}^1_i \otimes \hat{e}^1_j \]

with \(R^{0 \rightarrow 1}_{ij} = \hat{e}^0_i \cdot \hat{e}^1_j\).

Angular velocity.

\[\vec{\omega}^{01}_{\times} = \mathbb{\Omega}^{01} = \dot{\mathbb{R}}^{01} \cdot \mathbb{R}^{01,T}\]

Using index notation

\[\varepsilon_{ijk} \omega_j = \dot{R}_{ij} R_{kj}\]

and the identities

\[\varepsilon_{ijk} \varepsilon_{lmk} = \delta_{il} \delta_{jm} - \delta_{jl} \delta_{im}\]

\[\varepsilon_{ijk} \varepsilon_{ljk} = \delta_{il} \delta_{jj} - \delta_{ij} \delta_{jl} = 3 \delta_{il} - \delta_{il} = 2 \delta_{il}\]

it follows

\[\varepsilon_{ilk} \varepsilon_{ijk} \omega_j = \varepsilon_{ilk} \dot{R}_{ij} R_{kj}\]

\[2 \delta_{lj} \omega_j = \varepsilon_{ilk} \dot{R}_{ij} R_{kj}\]

\[\omega_l = \frac{1}{2} \varepsilon_{ilk} \dot{R}_{ij} R_{kj} = - \frac{1}{2} \varepsilon_{lik} \dot{R}_{ij} R_{kj} = -\frac{1}{2} \varepsilon_{lij} \Omega_{ij}\]

Angular acceleration. Angular acceleration, \(\vec{\alpha}\), is the time derivative of angular velocity, \(\vec{\omega}\),

\[\vec{\alpha} = \dot{\vec{\omega}} \ .\]

1.7.2. Successive rotations

Orientation. Given 3 Cartesian bases \(\{ \hat{e}^0_i \}_{i=1:3}\), \(\{ \hat{e}^1_j \}_{j=1:3}\), \(\{ \hat{e}^2_k \}_{k=1:3}\),

\[\begin{split}\begin{aligned} \hat{e}^2_i & = \mathbb{R}^{1 \rightarrow 2} \cdot \hat{e}^1_i = \\ & = \mathbb{R}^{1 \rightarrow 2} \cdot \mathbb{R}^{0 \rightarrow 1} \cdot\hat{e}^0_i \end{aligned} \ , \end{split}\]

i.e composition of rotations holds

\[\mathbb{R}^{0 \rightarrow 2} = \mathbb{R}^{1 \rightarrow 2} \cdot \mathbb{R}^{0 \rightarrow 1} \ .\]

Angular velocity. Time derivative w.r.t. reference frame 0 is indicated as the standard time derivative

\[\dot{a} = \dfrac{d a}{d t} = \dfrac{{}^0 d a}{d t} = \dfrac{{}^1 d}{dt} + \vec{\omega}_{1/0} \times \ ,\]

\[\begin{split}\begin{aligned} \dfrac{d}{dt} \mathbb{R}^{21} & = \frac{d}{dt} \left[ R^{21}_{ik} \hat{e}^1_i \otimes \hat{e}^1_k \right] = \\ & = \dot{R}^{21}_{ik} \hat{e}^1_i \otimes \hat{e}^1_k + \mathbb{\Omega}^{10} \cdot \mathbb{R}^{21} - \mathbb{R}^{21} \cdot \mathbb{\Omega}^{10} = \\ & = \dfrac{{}^1 d}{dt} \mathbb{R}^{21} + \mathbb{\Omega}^{10} \cdot \mathbb{R}^{21} - \mathbb{R}^{21} \cdot \mathbb{\Omega}^{10} = \\ \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} \mathbb{\Omega}^{20} & = \dot{\mathbb{R}}^{20} \cdot \mathbb{R}^{20, T} = \\ & = \frac{d}{dt} \left( \mathbb{R}^{21} \cdot \mathbb{R}^{10} \right) \cdot \mathbb{R}^{20, T} = \\ & = \left\{ \left[ \dfrac{{}^1 d}{dt} \mathbb{R}^{21} + \mathbb{\Omega}^{10} \cdot \mathbb{R}^{21} - \mathbb{R}^{21} \cdot \mathbb{\Omega}^{10} \right] \cdot \mathbb{R}^{10} + \mathbb{R}^{21} \cdot \dot{\mathbb{R}}^{10} \right\} \cdot \mathbb{R}^{01} \cdot \mathbb{R}^{12} = \\ & = \dfrac{{}^1 d}{dt} \mathbb{R}^{21} \cdot \mathbb{R}^{12} + \mathbb{\Omega}^{10} = \\ & = \mathbb{\Omega}^{21} + \mathbb{\Omega}^{10} \ . \end{aligned}\end{split}\]

so that addition of relative angular velocity holds

\[\mathbb{\Omega}^{20} = \mathbb{\Omega}^{21} + \mathbb{\Omega}^{10} \qquad , \qquad \vec{\omega}_{2/0} = \vec{\omega}_{2/1} + \vec{\omega}_{1/0} \ .\]

Angular acceleration. Time derivative of angular velocity composition provides the addition of relative angular accelerations

\[\dfrac{{}^0 d}{dt} \vec{\omega}_{2/0} = \dfrac{{}^0 d}{dt} \vec{\omega}_{2/1} + \dfrac{{}^0 d}{dt} \vec{\omega}_{1/0} \ ,\]

or

\[\vec{\alpha}_{2/0} = \vec{\alpha}_{2/1} + \vec{\alpha}_{1/0} \ .\]

1.7.3. Linearization of rotations

\[\mathbb{I} = \mathbb{R} \cdot \mathbb{R}^T\]

Increment

\[\mathbb{0} = \delta \mathbb{R} \cdot \mathbb{R}^T + \mathbb{R} \cdot \delta \mathbb{R}^T\]

and thus the antisymmetric tensor can be written as

\[\delta \theta_{\times} := \delta \mathbb{R} \cdot \mathbb{R}^T = \delta \mathbb{\Theta} \ ,\]

so that

\[\delta \theta_l = -\frac{1}{2} \varepsilon_{lij} \delta R_{ik} \, R_{jk} = - \frac{1}{2} \varepsilon_{lij} \delta \Theta_{ij}\]

1.7.4. Parametrizations

Minimal sets of parameters to represent rotations have 3 parameters. However these sets of parameters are not regular over all the possible rotations, and the transformation becomes singular somewhere. Quaternions provide a set of 4 parameters for a regular parametrization of rotations

1.7.4.1. Euler angles

1.7.4.2. Axis and rotation angle

1.7.4.3. Quaternions