Rotation Matrix

A rotation matrix is a way to describe a relationship between cartesian coordinate system.

Reference coordinate system is referred as Global | World | Inertial | Fixed

The other one is referred as Local | Body | Relative | Moving

In this section, we refer Inertial as a reference coordinate system, and Body as the other coordinate system.

Here, we denote Inertial and Body frame as

\[ \vec{I}= \begin{bmatrix} \vec{I}_x \\ \vec{I}_y \\ \vec{I}_z \\ \end{bmatrix},\ \vec{B}= \begin{bmatrix} \vec{B}_x \\ \vec{B}_y \\ \vec{B}_z \\ \end{bmatrix} \]

A (arbitrary) vector expressed in each frame, \(v_I\) and \(v_B\) is the component of vector expressed in each frame.

\[\vec{v}=\vec{I}^Tv_I=\vec{B}^Tv_B\]

Relationship between \(v_I\) and \(v_B\)

\[ v_B=B \cdot \vec{I}^T v_I = R_{BI} v_I = \begin{bmatrix} \vec{B}_x \cdot \vec{I}_x \ \ \ \ \vec{B}_x \cdot \vec{I}_y \ \ \ \ \vec{B}_x \cdot \vec{I}_z \\ \vec{B}_y \cdot \vec{I}_x \ \ \ \ \vec{B}_y \cdot \vec{I}_y \ \ \ \ \vec{B}_y \cdot \vec{I}_z \\ \vec{B}_z \cdot \vec{I}_x \ \ \ \ \vec{B}_z \cdot \vec{I}_y \ \ \ \ \vec{B}_z \cdot \vec{I}_z \\ \end{bmatrix} v_I \]
\[ v_I=I \cdot \vec{B}^T v_B = R_{IB} v_B = \begin{bmatrix} \vec{I}_x \cdot \vec{B}_x \ \ \ \ \vec{I}_x \cdot \vec{B}_y \ \ \ \ \vec{I}_x \cdot \vec{B}_z \\ \vec{I}_y \cdot \vec{B}_x \ \ \ \ \vec{I}_y \cdot \vec{B}_y \ \ \ \ \vec{I}_y \cdot \vec{B}_z \\ \vec{I}_z \cdot \vec{B}_x \ \ \ \ \vec{I}_z \cdot \vec{B}_y \ \ \ \ \vec{I}_z \cdot \vec{B}_z \\ \end{bmatrix} v_B \]

Hence,

\[R_{BI}=R_{IB}^T= R_{IB}^{-1}\]
\[R_{IB}=R_{BI}^T= R_{BI}^{-1}\]

ACTIVE(ALIBI) and PASSIVE(ALIAS) interpretation

ACTIVE Interpretation

We want to rotate a vector from \(\vec{v}\) to \(\vec{v}'\), to have same component in body frame. $$ \vec{v} = \vec{I}^T\vec{v}_I $$

\[ \vec{v}' = \vec{B}^T\vec{v}_I = \vec{I}^T\vec{v}'_I \]

Hence,

\[ \vec{v}'_I = \vec{I} \cdot \vec{B}^T\vec{v}_I = R_{IB} \vec{v}_I = R \vec{v}_I \]

PASSIVE Interpretation

We want to express a (same) vector in the body frame.

\[ v_B=B \cdot \vec{I}^T v_I = R_{BI} v_I = R v_I \]

Understandably, Rotation Matrix \(R\) is different depend on the interpretations.

You need to properly select one of interpretation from context, when you read documents without explicit explanations.

Principal Rotations

ACTIVE PASSIVE
\(R_x(\theta)\) \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \\ \end{bmatrix}\) \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \\ \end{bmatrix}\)
\(R_y(\theta)\) \(\begin{bmatrix}\cos \theta & 0 & \sin \theta \\0 & 1 & 0 \\-\sin \theta & 0 & \cos \theta \\\end{bmatrix}\) \(\begin{bmatrix}\cos \theta & 0 & -\sin \theta \\0 & 1 & 0 \\\sin \theta & 0 & \cos \theta \\\end{bmatrix}\)
\(R_z(\theta)\) \(\begin{bmatrix}\cos \theta & -\sin \theta & 0 \\\sin \theta & \cos \theta & 0 \\0 & 0 & 1 \\\end{bmatrix}\) \(\begin{bmatrix}\cos \theta & \sin \theta & 0 \\-\sin \theta & \cos \theta & 0 \\0 & 0 & 1 \\\end{bmatrix}\)

Exponential & Logarithm

The matrix exponential is \(\exp(A) = 1 + A + \frac{1}{2!}A^2 + \frac{1}{3!}A^3+\cdots=\sum_{n=0}^{\infty}\frac{1}{n!}A^n\)

\(R=\exp({\mathbf{\hat{\phi}}})=\exp(\phi \hat{a})=\cos\phi+(1-\cos\phi)aa^T+\sin\phi\hat{a}\)

\(\phi=\cos^{-1}(\frac{tr(R)-1}{2}) + 2\pi m\)

\(\hat{a}=\frac{R-R^T}{2\sin\theta}\)

\(R^\alpha=\exp(\hat{\phi})^{\alpha}=\exp(\alpha\hat{\phi})\)

Interpolation

\(R = R_1(R_1^TR_2)^{\alpha}\) (ACTIVE)

\(R = (R_2R_1^T)^{\alpha}R_1\) (PASSIVE)

Normalization

\(R'=(RR^T)^{-\frac{1}{2}}R\)

(\(A^{-\frac{1}{2}}=\frac{1}{\sqrt{\lambda_1}}e_1e_1^T+\frac{1}{\sqrt{\lambda_2}}e_2e_2^T+\frac{1}{\sqrt{\lambda_3}}e_3e_3^T\))

EigenPair of 3x3 Symmetric Matrix

Normalization of a (perturbed) rotation matrix is, find a new rotation matrix, close to the perturbed rotation matrix.

This can be formulated by,

maximize

\[J(R')=tr(R'R^T)-\frac{1}{2}\sum^{3}_{i=1}\sum^{3}_{j=1}\lambda_{ij}(r^{'T}_{i} r_j^{'} - \delta_{ij})\]

\(\delta_{ij}\) is Kronecker delta, \(R^T=[r_1 \ \ r_2 \ \ r_3]\), \((R')^T=[r_1^{'} \ \ r_2^{'} \ \ r_3^{'}]\)

Take the derivative of \(J\) with respect to the three rows of \(R'\) and set to Zero.

\[ [r_1^{'} \ \ r_2^{'} \ \ r_3^{'}] \begin{bmatrix} \lambda_{11} & \lambda_{12} & \lambda_{13} \\ \lambda_{21} & \lambda_{22} & \lambda_{23} \\ \lambda_{31} & \lambda_{32} & \lambda_{33} \\ \end{bmatrix} = [r_1 \ \ r_2 \ \ r_3] \]

due to symmetry of lagrange multiplier terms, \(\lambda_{ij}=\lambda_{ji}\)

\[\lambda R' = R, \ \ \lambda = \lambda ^ T\]
\[ \lambda ^ 2 = \lambda \lambda^T = RR^T \rightarrow \lambda = (RR^T)^\frac{1}{2} \]
\[\ \ \therefore R'=(RR^T)^{-\frac{1}{2}}R\]