1,212
edits
Changes
no edit summary
[[File:EulerAngles.png|thumb| Azimuth is measured from the north point, or sometimes from the south point, of the horizon around to the east; altitude is the angle above the horizon. Image reproduced from [https://upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Azimuth-Altitude_schematic.svg/300px-Azimuth-Altitude_schematic.svg.png here]]]
An amateur astronomer was staring into the dark, shiny night sky. Far away from usual hustle-bustle of the city and its pollution clouded sky, he was trying to locate some stars on his sophisticated telescope. He had a list of stars and some numbers scribbled onto a sheet of paper. Against each star, he had written two values- an azimuth angle and an angle of elevation or altitude. To direct to a point in a 3D space, after all, two ‘rotation’ values should suffice. You first indicate the azimuth angle (angle swept on a horizontal reference plane starting from a reference line on it). Then the elevation angle tells how high above this horizontal plane you ‘lift’. <br \>
Now if he were to locate an airplane instead (it is quite unlikely that he would find one in that desolate paradise of his!) he would require another ‘rotation’ value to specify its orientation. Think of it, just pointing to the airplane as if it were a star will not tell the direction it is travelling in. That validates the necessity for a third parameter. The lesson learnt from the story is this- for completely specifying the orientation of an ‘extended object’ like the airplane with respect to a reference frame, three rotation parameters are a minimum. <br \>
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:Euler2Euler6.png|framethumb|center|800px|Rotations along z, x and z axes respectively. Image reproduced inspired from [http://keywordsuggestmathworld.wolfram.org/gallerycom/427706EulerAngles.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]]