2020/21 EEE3039 - MATLAB Assignment
N. Baresi, H. Holt & N. Bernardini
• Introduction: The aim of this assignment is to facilitate and demonstrate understanding of the two-body
problem and torque-free attitude dynamics. The coursework is divided into two parts.
The first part of the assignment focuses on orbital mechanics and the equations of the two-body problem.
You will be challenged with the task of integrating the equations of motion in the Earth-Centered Inertial
reference frame (ECI) and comparing your solution with the analytical propagations enabled by Kepler’s
equation. You will also be asked to derive the equations of motion in the Earth-Centered Earth-Fixed
reference frame (ECEF) and to generate ground tracks using a provided MATLAB function that outputs
the Earth sub-satellite point of a spacecraft. Finally, you will be tasked with designing a Sun-synchronous
orbit and the orbital maneuver required to reach the desired orbit configuration.
The second part of the coursework deals with attitude kinematics and Euler’s equations. You will be asked
to represent the orientation of a satellite with respect to different reference frames and to propagate both
Euler’s equations and the kinematics equations for a desired time interval. This numerical experiment will
provide you with the opportunity to investigate spacecraft orientations and witness the advantages and
disadvantages of the different attitude representations that will be introduced throughout the module.
• Assignment Task: Write a report describing your results, derivations, and observations. The report
should be no longer than 20 pages including Figures and without counting the Appendix.
When writing your report, please make an effort to explain your reasoning and the equations being used.
Your MATLAB code should be included in the Appendix and as part of your submission on SurreyLearn
(either in a “.zip” file or as “.m” files). Do not include screenshots, as these do not comply with the
professional standards of industry and research reports.
• Due Date: Please submit a PDF of your final report online on SurreyLearn by Tuesday, November
24, 16:00 (BST). A “Coursework Assignment” folder is available for your submissions on SurreyLearn
under “Assessment”.
Coursework Data
• µ = 398600.4418 [km3 /s2 ];
• R⊕ = 6378.137 [km];
• J2 = 1082.63 × 10−6 ;
• Ixx = 2500 [kg m2 ];
• Iyy = 5000 [kg m2 ];
• Izz = 6500 [kg m2 ];
Ixx 0 0
• Spacecraft inertia matrix in principal axes frame: I = 0 Iyy 0 .
0 0 Izz
1
, 1 The Two-Body Problem
A satellite is being launched into a highly-inclined Earth orbit with initial orbit elements:
a = 7178.137 km
e = 0.07
i = 96 deg
Ω = 30 deg
ω = 65 deg
M0 = 182 deg
1.1 Solve Kepler’s Equation using Newton’s Method and Find Initial True Anomaly
Your lectures have introduced Kepler’s equation as a way to link the mean anomaly of a satellite (M ) with its
eccentric anomaly (E) and eccentricity (e):
M = E − e sin E. (1)
Later on, we learned how to relate the mean anomaly of a satellite to the time along its orbit using the time
equation
M = M0 + n (t − t0 ), (2)
p
where n = GM/a3 is the mean motion of the satellite.
Using Newton’s method, Eq. (1) can be solved numerically to find the value of E that satisfies Eq. (1)
for any suitable pair of e ∈ [0, 1) and M values. Once the value of E is found, the true anomaly of the spacecraft
can be calculated via
s
θ 1+e E
tan = tan . (3)
2 1−e 2
Newton’s method is an iterative method for solving nonlinear equations of the form f (x) = 0 via successive
approximations. Starting from a reliable initial guess, x(0) , one can expand f (x(0) + δx(0) ) using a Taylor’s
series approximation and find
f (x(0) + δx(0) ) = f (x(0) ) + f 0 (x(0) ) δx(0) + H.O.T. (4)
Neglecting high-order terms and assuming f (x(0) + δx) = 0 yields
!
f (x (0) )
δx(0) = − . (5)
f 0 (x(0) )
Using Eq. (5), it is possible to update the value of the initial guess as x(1) = x(0) + δx(0) and repeat the
process until f (x(k) ) < T ol, i.e, f (x(k) ) is smaller than a given threshold (we recommend using T ol = 10−10 )
1. Create a MATLAB function [E]=Kepler(e, M, tol) to solve Kepler’s equation using Newton’s method.
Use E (0) = M to initialize the iterative algorithm and output the value of the Eccentric anomaly in
radians; [6 marks]
2. Report the value of the initial Eccentric Aanomaly of the satellite found via Newton’s method as well as
the number of iterations needed to achieve convergence. Then, apply Eq. (3) to find the initial value of
θ and comment on whether the satellite is closer to its apoapsis or periapsis point. [4 marks]
2
N. Baresi, H. Holt & N. Bernardini
• Introduction: The aim of this assignment is to facilitate and demonstrate understanding of the two-body
problem and torque-free attitude dynamics. The coursework is divided into two parts.
The first part of the assignment focuses on orbital mechanics and the equations of the two-body problem.
You will be challenged with the task of integrating the equations of motion in the Earth-Centered Inertial
reference frame (ECI) and comparing your solution with the analytical propagations enabled by Kepler’s
equation. You will also be asked to derive the equations of motion in the Earth-Centered Earth-Fixed
reference frame (ECEF) and to generate ground tracks using a provided MATLAB function that outputs
the Earth sub-satellite point of a spacecraft. Finally, you will be tasked with designing a Sun-synchronous
orbit and the orbital maneuver required to reach the desired orbit configuration.
The second part of the coursework deals with attitude kinematics and Euler’s equations. You will be asked
to represent the orientation of a satellite with respect to different reference frames and to propagate both
Euler’s equations and the kinematics equations for a desired time interval. This numerical experiment will
provide you with the opportunity to investigate spacecraft orientations and witness the advantages and
disadvantages of the different attitude representations that will be introduced throughout the module.
• Assignment Task: Write a report describing your results, derivations, and observations. The report
should be no longer than 20 pages including Figures and without counting the Appendix.
When writing your report, please make an effort to explain your reasoning and the equations being used.
Your MATLAB code should be included in the Appendix and as part of your submission on SurreyLearn
(either in a “.zip” file or as “.m” files). Do not include screenshots, as these do not comply with the
professional standards of industry and research reports.
• Due Date: Please submit a PDF of your final report online on SurreyLearn by Tuesday, November
24, 16:00 (BST). A “Coursework Assignment” folder is available for your submissions on SurreyLearn
under “Assessment”.
Coursework Data
• µ = 398600.4418 [km3 /s2 ];
• R⊕ = 6378.137 [km];
• J2 = 1082.63 × 10−6 ;
• Ixx = 2500 [kg m2 ];
• Iyy = 5000 [kg m2 ];
• Izz = 6500 [kg m2 ];
Ixx 0 0
• Spacecraft inertia matrix in principal axes frame: I = 0 Iyy 0 .
0 0 Izz
1
, 1 The Two-Body Problem
A satellite is being launched into a highly-inclined Earth orbit with initial orbit elements:
a = 7178.137 km
e = 0.07
i = 96 deg
Ω = 30 deg
ω = 65 deg
M0 = 182 deg
1.1 Solve Kepler’s Equation using Newton’s Method and Find Initial True Anomaly
Your lectures have introduced Kepler’s equation as a way to link the mean anomaly of a satellite (M ) with its
eccentric anomaly (E) and eccentricity (e):
M = E − e sin E. (1)
Later on, we learned how to relate the mean anomaly of a satellite to the time along its orbit using the time
equation
M = M0 + n (t − t0 ), (2)
p
where n = GM/a3 is the mean motion of the satellite.
Using Newton’s method, Eq. (1) can be solved numerically to find the value of E that satisfies Eq. (1)
for any suitable pair of e ∈ [0, 1) and M values. Once the value of E is found, the true anomaly of the spacecraft
can be calculated via
s
θ 1+e E
tan = tan . (3)
2 1−e 2
Newton’s method is an iterative method for solving nonlinear equations of the form f (x) = 0 via successive
approximations. Starting from a reliable initial guess, x(0) , one can expand f (x(0) + δx(0) ) using a Taylor’s
series approximation and find
f (x(0) + δx(0) ) = f (x(0) ) + f 0 (x(0) ) δx(0) + H.O.T. (4)
Neglecting high-order terms and assuming f (x(0) + δx) = 0 yields
!
f (x (0) )
δx(0) = − . (5)
f 0 (x(0) )
Using Eq. (5), it is possible to update the value of the initial guess as x(1) = x(0) + δx(0) and repeat the
process until f (x(k) ) < T ol, i.e, f (x(k) ) is smaller than a given threshold (we recommend using T ol = 10−10 )
1. Create a MATLAB function [E]=Kepler(e, M, tol) to solve Kepler’s equation using Newton’s method.
Use E (0) = M to initialize the iterative algorithm and output the value of the Eccentric anomaly in
radians; [6 marks]
2. Report the value of the initial Eccentric Aanomaly of the satellite found via Newton’s method as well as
the number of iterations needed to achieve convergence. Then, apply Eq. (3) to find the initial value of
θ and comment on whether the satellite is closer to its apoapsis or periapsis point. [4 marks]
2
