ENG1005 S2 2020 Assignment 2: Rotating Satellite
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ENG1005 S2 2020 Assignment 2: Rotating Satellite
A spherical satellite is flying through space while spinning on its axis. Unfortunately the gyroscope on the satellite is broken, and so the usual measurements for the axis and angular velocity of rotation are not available. In this assignment, you will find an alternative way to calculate the axis and angular velocity of the rotation of the satellite.
Suppose an inertial coordinate system1 is chosen so that the centre of the satellite is located at the origin. The locations of three antennas on the satellite are measured at time 0. They are located at
a(0) = (1, 0, 0); b(0) = (0, 1, 0); c(0) = (0, 0, 1)
The locations of the antennas are measured again 1 second later:
a(1) = ( 
, 
, 
) ; b(1) = ( − 
, 
, 0) ; c(1) = ( − 
, − 
, 
)
Part A: Axis of rotation
1. The rotation of the satellite is a linear transformation taking the triple of points {a(0), b(0), c(0)} to the triple {a(1), b(1), c(1)}. Determine the matrix A representing this transformation without the help of a calculator or computer. [2 marks]
2. Suppose v is a vector in the direction of the axis of rotation. Explain why v must be an eigenvector of A. What is the corresponding eigenvalue? [2 marks]
3. With the help of Matlab (or otherwise), find all eigenvalues of A. For each real eigenvalue of A, find the corresponding eigenvector(s). Hence determine the direction of the axis of rotation of the satellite.
[2 marks]
Part B: Angle of rotation
Having found the axis of rotation of the satellite, you will now find its angular velocity from your measure- ments.
1. With the help of a sketch, show geometrically that an anticlockwise rotation by an angle of θ about the z-axis is represented by the matrix
Rz = 
J
You may use the matrix for a 2-D rotation covered in the lectures.
2. Find all real and complex eigenvalues of Rz in terms of θ .
[2 marks]
[2 marks]
It is a fact that the matrix R for any rotation by an angle of θ about some axis of rotation can written as R = PRz P − 1
for some invertible matrix P, where P is independent of θ .2
3. Show that if P is an invertible matrix and X is any arbitrary matrix of the same size, then
det(X − λI) = det(PXP − 1 − λI)
You may find some of the results from the Week 3 applied class problem sheet useful here. [2 marks]
4. Hence explain why R and Rz must have the same eigenvalues, and express the sum of the eigenvalues of R in terms of θ . [2 marks]
5. Find the angular velocity of the satellite in radians per second, up to 4 decimal places of accuracy.
[2 marks]
You notice that there are three pieces of space debris near the satellite. You wonder if they are rotating uniformly around the satellite. You make measurements of the location of each piece of space debris at time
0 and again at time 1. The locations are related to your student number in a similar fashion to Assignment
1. If your student number is abcdefgh, then at time 0, the first debris is at position (cd,ef,gh)+(10, 10, 10),
the second is at position (cd,ef,gh)+(5, 10, 15), and the third at position (cd,ef,gh)+(20, 20, 20). At time
1, the first debris is at (100 , 100, 100) − (cd,ef,gh), the second at (105, 110, 100) − (cd,ef,gh) and the third
at (110, 105, 120) − (cd,ef,gh).
6. With the help of Matlab (or otherwise), determine B, the matrix representing the linear transformation taking the positions of each piece of debris at time 0 to their positions at time 1. [2 marks]
7. Does B represent a rotation? Justify your answer. Hint: you may want to look up and use some properties of rotation matrices, perhaps even ones you have just discovered! [2 marks]
There are also 2 additional marks given for the quality of the English, 2 additional marks for correct mathematical notation and 1 additional mark for appropriate use of sketches where relevant. These marks are easy to obtain but the markers will be instructed to be strict in awarding these marks.
2022-09-06