Some help with this exercise please.
If you could give an explanation and not only the answers it would be great!!!
Besides the 3-2-1 rotation sequence, another very common sequence in spacecraft applications is the 3-1-3 rotation sequence in which the z axis is used twice. This particular sequence is namely used to represent the orientation of the orbital reference frame denoted FO and defined by its orthonormal unit vectors {Ox,Oy,Oz} with respect to the Earth-Centered Inertial (ECI) reference frame denoted FI and defined by its orthonormal unit vectors n {Ix,Iy,Iz} o, that is F~O = CoiF~I
Both reference frames have their origin at the center of the Earth. The unit vectors of FI are all fixed with respect to stars, with ~Ix pointing in the direction of the first point of Aries, ~Iz in the direction of the Earth’s polar rotation axis, and ~Iy completing the triad. For this reason, the inertial reference frame is also called the
celestial reference frame. On the other hand, the unit vectors of FO are rotating in the orbital plane as the spacecraft travels around the Earth. Specifically, O~x is always pointing in the direction of the spacecraft (i.e., along the Earth-spacecraft position vector ~r), O~z points in the normal direction of the orbital plane, and O~y completes the triad. In this context, the 3-1-3 rotation sequence is very useful to inertially locate a spacecraft, and the first rotation is through an angle called the right ascension of the ascending node (RAAN), the second rotation is through the inclination angle, and finally the third rotation is through the argument of latitude, respectively denoted by Ω, i, and u.
(a) Calculate the rotation matrix COI .
(b) As mentioned above, O~x is always pointing in the direction of the spacecraft.
Thus, in FO, the spacecraft position vector is expressed by: r = rO~x
where r is the distance from the center of the Earth to the spacecraft, i.e.,
n
the module of the position vector ~r. Express the position vector ~r in terms of {~Ix,~Iy,~Iz}.
(c) An unfortunate consequence of using three angles to describe a rotation matrix is that a singularity occurs. Demonstrate the rotation sequence 3-1-3 has a singularity when i = 0, i.e., Ω and u cannot be uniquely determined.
(d) Denoting COI as the following matrix:
COI =
Cxx Cxy Cxz
Cyx Cyy Cyz
Czx Czy Czz
demonstrate that, outside the singularity, the three angle can be uniquely determined as
Ω = − tan−1(Czx/Czy)
i = cos−1 Czz
u = tan−1(Cxz/Cyz)
Thanks a lot!
1 answer