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# Collocation and Nonlinear Programming

Optimal Finite-Thrust Spacecraft Trajectories Using Collocation and Nonlinear Programming Background Recently there has been great interest by NASA and the ESA in developing satellite constellations that will serve to act as interferometers for use in gravity wave detection These satellite constellations would be composed of six satellites arranged in varying orbits such that they form an equilateral triangle with each triangle vertex containing two satellites at close proximity Each vertex would send two laser signals to the satellites at the other vertices The satellites then calculate the distance between themselves and the other vertices based on the time it took the laser beam to reach them This distance is compared with the known distance between the satellites and any increase would suggest that a gravity wave has passed through the detector The job of the astrodynamicist is to keep the satellites in their orbits so that the distance between each vertex remains fixed This starts off as a problem of forced-Keplerian motion but then gets a little more complicated when the constraint of minimizing thruster firing time is thrown in When the satellite thrusters are firing to maintain their orbit the interferometer cant take accurate data because the disturbance due to the movement is far greater than the disturbances caused by gravity waves This dilemma brings about the problem of needing to minimize the thruster firing time so that you can maximize the data collection time One method of optimizing finite-thrust spacecraft trajectories is the Direct Collocation with Nonlinear Programming approach DCNLP Direct Collocation with Nonlinear Programming Direct Collocation with Nonlinear Programming is a method of approximating the continuous optimization problem with a parameter optimization problem This method has been developed over the course of the last thirty-five years and a summary of its history can be found in reference 1 The DCNLP approach approximates the spacecraft trajectory with piecewise polynomials that are represented by state and control variables at a given

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