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!
wooooooooooow to much stuff
To help you with this exercise, let's go step-by-step through each part.
(a) Calculate the rotation matrix COI.
In this case, we are given the 3-1-3 rotation sequence with angles represented as Ω, i, and u. To calculate the rotation matrix COI, we need to use Euler angles and the following sequence of rotations:
1. Rotate around the z-axis by Ω (right ascension of the ascending node)
2. Rotate around the x-axis by i (inclination angle)
3. Rotate around the z-axis again by u (argument of latitude)
The rotation matrix COI can be calculated by multiplying the individual rotation matrices for each rotation. The matrix representation for each rotation is as follows:
1. Rotation around z-axis by angle Ω:
⎡cos(Ω) -sin(Ω) 0⎤
⎢sin(Ω) cos(Ω) 0⎥
⎣ 0 0 1⎦
2. Rotation around x-axis by angle i:
⎡ 1 0 0⎤
⎢ 0 cos(i) -sin(i)⎥
⎣ 0 sin(i) cos(i)⎦
3. Rotation around z-axis again by angle u:
⎡cos(u) -sin(u) 0⎤
⎢sin(u) cos(u) 0⎥
⎣ 0 0 1⎦
To obtain the final rotation matrix COI, multiply these matrices in the order given above:
COI = (Rotation around z by Ω) * (Rotation around x by i) * (Rotation around z by u)
(b) Express the position vector ~r in terms of {~Ix,~Iy,~Iz}.
In this part, we are asked to express the spacecraft position vector ~r in terms of the orthonormal unit vectors {~Ix, ~Iy, ~Iz} of the Earth-Centered Inertial (ECI) reference frame.
From the information provided, we are told that O~x always points in the direction of the spacecraft. So, the position vector ~r is simply the distance from the center of the Earth to the spacecraft, multiplied by the unit vector O~x.
Therefore, the position vector ~r can be expressed as:
~r = r * O~x
(c) Demonstrate the rotation sequence 3-1-3 has a singularity when i = 0, i.e., Ω and u cannot be uniquely determined.
In this part, we need to show that when the inclination angle (i) is equal to 0, the rotation sequence 3-1-3 has a singularity, which means that the angles Ω and u cannot be uniquely determined.
To demonstrate this, we can consider the rotation matrix COI and try to solve for Ω and u when i = 0.
When i = 0, the rotation matrix becomes:
COI = (Rotation around z by Ω) * (Rotation around x by 0) * (Rotation around z by u)
Since the rotation around x by 0 doesn't change anything, we can simplify the equation to:
COI = (Rotation around z by Ω) * (Rotation around z by u)
To find Ω and u, we need to invert this equation. However, because the order of rotations is important, we cannot determine unique values for both Ω and u. This is the singularity in the 3-1-3 rotation sequence when i = 0.
(d) Demonstrate that, outside the singularity, the three angles can be uniquely determined as Ω = − tan−1(Czx/Czy), i = cos−1 Czz, u = tan−1(Cxz/Cyz).
We are asked to show that, outside the singularity (when i is not equal to 0), the three angles Ω, i, and u can be uniquely determined using the rotation matrix COI.
To demonstrate this, we need to express the rotation matrix COI in terms of the three angles Ω, i, and u and then solve for these angles.
The rotation matrix COI can be expressed using the matrix notation given in part (d):
COI =
Cxx Cxy Cxz
Cyx Cyy Cyz
Czx Czy Czz
From this matrix, we can see that:
Czx = sin(i)*cos(Ω)
Czy = sin(i)*sin(Ω)
Cxz = -sin(u)*cos(i)
Cyz = cos(u)*cos(i)
Using these equations, we can express the three angles as follows:
Ω = -tan^(-1)(Czx/Czy)
i = cos^(-1)(Czz)
u = tan^(-1)(Cxz/Cyz)
I hope this explanation helps you understand and solve the exercise! Let me know if you need any further clarification.