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Two-Body Problem and Torque-Free Attitude Mechanics Assignment

## Orbit Propagation via Analytical Solution of Two-Body Problem

The aim of this assignment is to facilitate and demonstrate understanding of the two-body problem and torque-free attitude mechanics. 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 Keplerâ€™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 given the longitude and latitude of a spacecraft.

The second part of the coursework deals with attitude kinematics and Eulerâ€™s equations. You will be asked to represent the orientation of a satellite with respect to different reference frames and to propagate both Eulerâ€™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. 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.

Orbit Propagation via Analytical Solution of Two-Body Problem
Given coe =
a e i ? ? ?T , where ? is the true anomaly calculated in W2.2,

1. Calculate the orbital period of the satellite, P, report its value, and create a time vector of 1000 equally spaced points t ? [0, P] (thus assuming t0 = 0 s). Use Eq. (1), (2) and (3) to propagate the state of the satellite using classic orbit elements, attaching a plot of the mean anomaly, eccentric anomaly, and true anomaly values as a function of time t (Tip: plot multiple lines on the same plot using the hold on
command to save space);

2. Create a MATLAB function [X] = COE2RV(coe, mu) to convert the state of a spacecraft from classic orbit elements to ECI position and velocity components; While looping over the time vector t, use [X] = COE2RV(coe, mu) to convert the analytical solution of the two-body problem into ECI position and velocity coordinates;Â

3. Report the initial and final values of X =x y z x? y? z?T and comment on whether the two vectorsare expected to be the same and why. Include a 3D plot of the spacecraft trajectory along with the Earth and the initial conditions of the satellite.

Using transport theorem,

1. Rewrite the left-hand side of Eq. (6) to derive the equations of motion of a satellite as seen from the Earth-Centered Earth-Fixed reference frame (ECEF). You may assume that the angular velocity of the ECEF frame as seen from the ECI frame is constant and parallel to the third axis of the ECEF frame. Include key passages in your report;

2. Rewrite your equations of motion as a system of first-order ordinary differential equation and write a MATLAB function [dXdt] = TBP_ECEF(t, X, mu) that outputs the right-hand side of the newly found system of first-order ordinary differential equations. Include your new system of first-order ordinary differential equations in the report and clearly indicate whether your new state vector includes the inertial or rotating velocity of the spacecraft;

3. Convert the initial conditions of the spacecraft from ECI to ECEF, knowing that at t = t0 = 0 s, the two reference frames are aligned. That is, F I = I3, where I3 is the 3 Ã— 3 identity matrix. Report your formulae and ECEF initial conditions in the manuscript; Then, integrate the equations of motion of the ECEF two-body problem in MATLAB using either ode45 or ode113 for 10 orbital periods;Â

4. Use the output of W5.4 to calculate the latitude and longitude of the satellite over ten orbital periods. Pass these values to the provided function PlotGroundTracks.m (type help PlotGroundTracks for info on inputs and outputs) and generate a 2D plot of the satelliteâ€™s ground-track. Enclose your figure in the report and discuss whether the satellite roughly passes over Guildford, UK, at some point along its orbit.