In this tutorial, we will be learning about how to use the concept of Euler angles to perform coordinate system changes. We will then relate these changes back directional cosine matrices. Let's start!
Let's say there is a frame S and frame E, shown in Figure 1 below, and we want to figure out how to rotate frame E to align it with frame S. Note that frame S has axes $ x_s $, $ y_s $, and $ z_s $, while frame E has axes $ x_e $, $ y_e $, and $ z_e $.
Fig. 1 - Image of frames S and E.
We can start off by viewing the axes in the negative $ z_s $ and negative $ y_e $ axes to get an idea of how much we need to rotate about each axis.
Fig. 2 - Frames S and E in different perspectives.
In Figure 2, on the left, we can see the view of both frames looking down $ -z_s $, and on the right, we can see the frames looking down $ - y_e $. Now since we are using the rotation sequence, 3-2-1, we will want to first rotate about the $ z_e $ axis first.
Notice that in the left image in Figure 2, the x axes are separated by $ 180° - \mu $. After making this rotation, then the x axes would be aligned. The next step in the sequence is to rotate about the y axis, if needed. We can now move our attention over to the right image in Figure 2. Notice that the z axes are separated by a degree of $ 90° - \lambda $. After making this rotation, the y axes are now aligned.
What is interesting here is when you align two axes together, the third will be aligned as a result. So all we needed to do here was align the z and y axes, and the x aligned itself with the other frame's x axis.
I should note though that we did not have to rotate about the $ x_e $ axis because the $ x_e $-$ z_e $ plane was perpendicular to the origin. If this was not true, then a rotation about the $ x_e $ axis would have been needed.
We can connect this back to the directional cosine matrix. Since we made a rotation about $ z_e $ of $ 180° - \mu $, we would say $ \psi = 180° - \mu $, and since we rotated about $ y_e $ of $ 90° - \lambda $, $ \theta = 90° - \lambda $. However, since we made no rotation about $ x_e $, we would let $ \phi = 0° $.