1,212
edits
Changes
no edit summary
Where does this come into the satellite picture? Remember, the attitude of a satellite (an ‘extended’ object) gives its (more specially, its body frame’s) orientation with respect to a reference frame. And this requires three rotation parameters. Euler angles captures this idea. This is a set of three ‘ordered’ rotation parameters. So, each parameter corresponds to rotation about a specific axis the new coordinate system obtained post rotation. Each rotation is mathematically represented by a rotation matrix. And matrix multiplication of three rotation matrices (again, order is important) gives the net rotation.<br \>
[[File:Euler2.png|frame|center|Image reproduced from [http://keywordsuggest.org/gallery/427706.html here]]]
Shown above is a rotation following a particular convention of (Ф,Ө,ψ), the angles being rotation about (z,x,z) axes respectively. The rotation matrix A then is,<ref>http://mathworld.wolfram.com/EulerAngles.html</ref>
[[File:Equation67.png|frame|center]]
Euler angles, however, entails some limitations. Firstly, these values are not obvious. One cannot guess them by merely looking at the final orientation. Think of it, when you are restricted by convention to use a particular sequence of axis rotations, you have to in some sense have some ‘foresight’ to understand the consequence of each rotation on the following one(s) and correctly predict the right combination of values. In the case of our amateur astronomer who had to point to an airplane, if we were to ask him to make a rotation to account for its orientation first before pointing to it he would’ve had a hard time. He wouldn’t know how his action of pointing to it (i.e the subsequent rotations) affect this initial adjustment he has made for its orientation. <br \>
In the rotation we have made above, Ф and ψ have lost their unique identities. There are multiple values we could use to obtain the same result (as long as Ф +ψ is the same). Here, we have lost a degree of freedom, only two independent parameters are required to represent this rotation. In a mathematical sense, this is a problem, because we don’t have unique Ф and ψ values at hand. <br \>
Quaternions is an alternative to Euler angles. It uses the fact that one unique axis and a rotation about it can do this job. It is, in fact, more intuitive.
----
If you are done reading this page, you can go back to [[Attitude Determination and Control Subsystem]]
==References ==