Rotation Matrices

Overview

Rotation matrices are matrices which are used to describe the rotation of a rigid body from one orientation to another.

In \(\mathbb{R3}\) they form a 3x3 matrix.

$$ \mathbf{R} = \begin{bmatrix} \hat{u_x} & \hat{v_x} & \hat{w_x} \\ \hat{u_y} & \hat{v_y} & \hat{w_y} \\ \hat{u_z} & \hat{v_z} & \hat{w_z} \end{bmatrix} $$

such that if \(\vec{\b{a}}\) was a point in 3D space, then we can rotate \(\vec{\b{a}}\) to \(\vec{\b{b}}\) around the origin by applying \(\b{R}\) in the following manner:

$$ \b{R}\vec{\b{a}} = \vec{\b{b}} $$

where:
\(\b{a}\) and \(\b{b}\) are 1x3 vectors

Combining Rotations

Two successive rotations represented by \(\b{R_1}\) and \(\b{R_2}\) can be represented by a single rotation matrix \(\b{R_3}\) where:

$$ \b{R_3} = \b{R_2} \b{R_1} $$

Pay careful attention to the order of the matrix multiplication, successive rotation matrices are multiplied on the left.

How To Find The Rotation Matrix Between Two Coordinate Systems

Suppose I have the frame with the following unit vectors defining the first coordinate system \( X1Y1Z1 \):

$$ X1=\begin{bmatrix}x_x\\x_y\\x_z\end{bmatrix} \quad Y1=\begin{bmatrix}y_x\\y_y\\y_z\end{bmatrix} \quad Z1=\begin{bmatrix}z_x\\z_y\\z_z\end{bmatrix} $$

And a second coordinate system \( X2Y2Z2 \) defined by the unit vectors:

$$ X2=\begin{bmatrix}x_x\\x_y\\x_z\end{bmatrix} \quad Y2=\begin{bmatrix}y_x\\y_y\\y_z\end{bmatrix} \quad Z2=\begin{bmatrix}z_x\\z_y\\z_z\end{bmatrix} $$

The rotation matrix \( R \) which rotates objects from the first coordinate system \(X1Y1Z1\) into the second coordinate system \(X2Y2Z2\) is:

$$ R = \begin{bmatrix} X1 \cdot X2 & X1 \cdot Y2 & X1 \cdot Z2\\ Y1 \cdot X2 & Y1 \cdot Y2 & Y1 \cdot Z2\\ Z1 \cdot X2 & Z1 \cdot Y2 & Z1 \cdot Z2\\ \end{bmatrix} $$

where:
\( \cdot \) is the matrix dot product
and everything else as above

Creating A Rotation Matrix From RPY Values

Roll-pitch-yaw (RPY) values can be easily converted into a rotation matrix. To represent a extrinsic rotation with Euler angles \( \alpha \), \( \beta \), \( \gamma \) are about axes \( x \), \( y\), \( z \) can be formed with the equation:

$$ \b{R} = \b{R}_z(\gamma) \b{R}_y(\beta) \b{R}_x(\alpha) $$

where:

$$ \b{R}_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \\ \end{bmatrix} \\ \b{R}_y(\theta) = \begin{bmatrix} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta) \\ \end{bmatrix} \\ \b{R}_z(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \end{bmatrix} $$