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_eb represents the rotation of frame e, relative to frame b. 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_eb · x_b .

Given the properties of SO(3), the inverse of R_eb is simply the transpose of it. That is,

R_be = R_eb^T .

This R_be can be used transform a vector in frame e to frame b.

Constructing the Rotation Matrix

Let’s construct the matrix R_eb.

Consider the frames defined above. In this figure, each of ei and bi are unit vectors. As mention above, this represents the rotation of the e frame relative to the b frame. All we have to do is define each of the axes of e frame in b frame.

Let’s start with e1. This vector is facing -b3 direction, and no projection on b1 or b2. So, the vector e1 in b frame can be written as

[ 0]
[ 0]
[-1].

Similarly, e2 can be written as

[ 0]
[-1]
[ 0],

and e3 can be written as

[-1]
[ 0]
[ 0].

Combining all above vectors, e frame relative to b frame, or R_eb, can be written as

       [ 0,  0, -1]
R_eb = [ 0, -1,  0]
       [-1,  0,  0].

Verifying the Rotation Matrix

If you are still not sure, the easiest way to verify your rotation matrix is to do a transformation. We know that, if we pre-multiply a vector in b frame by R_eb, it is transformed to the e frame.

Let’s take b1. This in b frame is

[ 1]
[ 0]
[ 0].

Pre-multiply this by R_eb, we get

[ 0]
[ 0]
[-1],

which makes sense. If you repeat the same calculation for b2 and b3, you will see similar results.