Doppler Orbit Determination: An Astrodynamics Project That Was Almost Fun...

Last semester I took a rather broad and general astrodynamics course and I thought I'd share a project I did in it, not because my writing was particularly good, but because I thought it was an interesting research problem... And I still don't have a good solution for it.The goal of my project was to use Doppler shift to develop a cheap, low-power positioning system system for satellites. Traditionally satellites are tracked with über-high precision optics, or a big radar array. But Doppler data is directionless, you only get scalar information from it. I wanted to find a way to take Doppler data from an array of sensors on the ground to pinpoint where exactly an overhead transmitter was (and how fast it was going).At first I tried to find a closed-form solution to the geometric problem of "Given X number of ground stations, each being approached-by or receding-from a satellite at a certain speed, could the location of the satellite be pinpointed?" In my case I went with 6 ground stations, assuming no redundancy. This was a simulation after all, I would have perfect data.But despite my best efforts at applying geometric intuition and substitution of obnoxious equations, I couldn't analytically solve for any of the satellite's states. So I went on to a brute force method of guess and check. And that worked, but it was extremely time-consuming. I was essentially iterating through x, y, z position and velocity challenges, and trying to find where the given and guessed data matched up best.There were probably a ton of ways I could have optimized my algorithm, like looking at the rates of convergence vs. each variable, dynamic meshing of my search grid, learning to code parallel execution threads... But the point is: even though I knew something could be done, I was at a loss for how to do it better. And that bugs me. I'm sick of the problem now (because it was for a grade, and involved late nights of coding), but it's this sort of nagging curiosity that keeps me interested in science and engineering. Life would be boring if there wasn't something left to achieve.And also, if you have any better ideas I'd love to hear them...-------Not that I want to encourage you to read this, but if you're at all curious about the technical challenges I faced, you can look at more crude figures I used... My quality of writing suffers greatly when I'm on a deadline. Re-reading this is making me cringe.>> Report in PDF form: Doppler Project Report_Final