TABLE OF CONTENTS From the above we have four forms for expressing coordinate transformations: the DCM, the Euler angles, the rotation vector and the quaternion. Next we discuss and summarize the conversion between the different forms. Due to the close relation between the quaternion and the rotation vector, conversion to/from the rotation vector is not addressed. If needed, conversion to/or from the quaternion will be carried and then followed by using Eq. (3.17) or its inverse.
Given the Euler angles, their sines and cosines can be computed and substituted in Eq. (3.5) to give the DCM. Conversely, when the DCM is known then Eq. (3.5) gives
where C ij is the ( i th row, j th column) element of the matrix C.
Given the quaternion, its elements can be substituted in Eq. (3.15) to generate the DCM. Expanding yields
Now we compute Q when C is given. The trace of C (the sum of its diagonal terms) is given by
In view of Eq. (3.12) the above becomes
from which
Selecting (-180,180) to be the range of a rotation angle implies that the cosine of one half of any value in this range will always be positive and hence q 0, from Eq. (3.16), will be positive.
Continuing with Eq. (3.15) it can be seen that
from which
As mentioned before, to convert a DCM to a rotation...
