# MATLAB

Project description:

This project is assisting the Naval Research Laboratory with analysis of data it collected on a satellite mission. The posted EXCEL file when loaded provides the vectors, tcm, xcm, ycm, zcm, t, x, y, and z. The t vector contains the times of the observations in seconds. The x, y, and z vectors contain the 3D position data in kilometers. The center of mass of the system are the vectors with cm in the name. The first observation consists of the first element of each vector, the second observation consists of the second element of each vector, and so on. The data for the center of mass are in uniform steps of 1 second. The data for the observations of the endmass are not in uniform time steps because there were instances where the data was not collected. Make sure that when you are comparing the position of the endmass to the position of the center of mass that the two positions are at the same time.

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Define the difference between the endmass and the center of mass as:

where for each endmass observation time the differences are calculated.

The relative displacements next should be converted to a spherical coordinate system:

where the relative distance, ρ = k[∆x,y,z]k2 and the angles α and θ define the coordinates the problem will be working in.

Your analysis needs to determine appropriate equations for ρ¨, α¨, and θ¨. In addition, your analysis needs to determine the initial values of ρ˙, α˙, and θ˙. These three equations and three values are to be included in the text of your report. Time plots of the calculated (using the second derivative equations and initial conditions) range and angular measurements are to be included as well as time plots of the rates of the three values. In addition, time plots of the residuals of the range and angles should be included. Based on the calculated values, the cartesian coordinates of the endmass should be determined and those residuals as well should be plotted as a time plots. The sum of the square of the position error needs to be calculated and included in the report. The project with the lowest sum of the square of the position error will receive extra credit.

Report outline:

1. A brief abstract that explains the purpose of the project, the methods used, and a summary of results.(The abstract should be roughly 100-300 words.)
2. A discussion of your methodology. This section should include a description of the methods you are using to solve the problem. Figures and equations should be used in this section as appropriate.
3. A discussion of your results. Present calculations as appropriate, but at the very least the final results should be included as mentioned above as well as references to the equations used to obtain them. Your discussion of the results should include whether or not the results were as expected or not. Also, comment on any particular insights gained from your results. You can also demonstrate differences to other methods to help to explain why you chose the method you chose.
4. A conclusions section. This section should consists of a few paragraphs that summarize the problem you were solving, what techniques you applied, what your findings were, why your findings are valuable, and any follow up work that you would recommend.
5. An appendix of all the software you used.

Pages (550 words)
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