Rotation Matrix Construction

This a quick guide to creating a rotation matrix, mainly for the purpose of coordinate transformations.

Basic Introduction

Rotation matrix in \(SO(3)\) is a linear transformation from one coordinate frame to another. Let \(e\) and \(b\) be some arbitrary Euclidean coordinate frames, and \(x^e\) and \(x^b\) be vectors represented in those coordinate frame, respectively.

The rotation \(R^e_b\) represents the rotation of frame \(b\), relative to frame \(e\). Though counter-intuitive, if we pre-multiply a vector in \(b\) frame (\(x^b\)) by this matrix, the resulting vector is how the \(x^b\) is represented in frame \(e\). That is,

\[x^e = R^e_b · x^b.\]

Given the properties of \(SO(3)\), the inverse of \(R^e_b\) is simply the transpose of it. That is,

\[R^b_e = (R^e_b)^T.\]

This \(R^b_e\) can be used transform a vector in frame \(e\) to frame \(b\).

Constructing the Rotation Matrix

Let’s construct the matrix \(R^e_b\).

Consider the frames defined above. In this figure, each of \(e_i\) and \(b_i\) are unit vectors. As mentioned above, \(R^e_b\) represents the rotation of the \(b\) frame relative to the \(e\) frame. In other words, \(i\)-th column of \(R^e_b\) represents the direction of \(b_i\) with respect to the frame \(e\).

Let’s start with \(b_1\). This vector is facing \(-e_3\) direction, and no projection on \(e_1\) or \(e_2\). So, the vector \(b_1\) in \(e\) frame can be written as

\[b_1 = \begin{bmatrix} 0 \\ 0 \\ -1 \end{bmatrix}.\]

Similarly, \(b_2\) can be written as

\[b_2 = \begin{bmatrix} 0 \\ -1 \\ 0 \end{bmatrix},\]

and \(b_3\) can be written as

\[b_3 = \begin{bmatrix} -1 \\ 0 \\ 0 \end{bmatrix}.\]

Combining all above vectors, the rotation of \(b\) frame relative to \(e\) frame, or \(R^e_b\), can be written as

\[R^e_b = \begin{bmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix}.\]