# Optimal Trajectories For Constrained Station Change in Geo Using A Legendre Pseudospectral Method

Open Access
Author:
Kim, Seung Pil
Aerospace Engineering
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
June 22, 2012
Committee Members:
• Robert Graham Melton, Dissertation Advisor
• Robert Graham Melton, Committee Chair
• David Bradley Spencer, Committee Member
• Joseph Francis Horn, Committee Member
• Julio Urbina, Committee Member
Keywords:
• station change
• Legendre pseudospectral methods
• low-thrust
Abstract:
In this dissertation a method for determining optimal trajectories for constrained geostationary station change is presented. If a satellite failure occurs, an emergency station relocation maneuver could be required to replace a malfunctioning satellite and continue to maintain the operational capabilities for a particular mission. The algorithm for numerical approximation of the optimal control is formulated using a Legendre pseudospectral method. The state equations are enforced at each node using a differentiation matrix and forcing the state derivatives from the Legendre polynomial representation to be equal to the state equation derivatives evaluated at the Legendre-Gauss-Lobatto points. An advantage of a pseudospectral method is that it can determine the optimal control using fewer unknown parameters (the states and controls and the node points) than other direct methods. The spacecraft dynamics equations are formulated in 2-D polar form; and perturbation forces are neglected except for the low-thrust control. The collision avoidance term, which is included in the objective function as an integral form, is considered during transfer by specifying a maximum or minimum radius depending on whether the transfer is in the east or west-direction. Several east-and-west-direction station change transfer maneuvers are simulated; the histories of the states and control behavior are obtained and all the variables are found to be within feasible ranges. Multiple revolutions and large longitude station change cases are also demonstrated using this method. The transfer time is found to be only weakly affected by the initial thrust acceleration for the multi-revolution change cases. This method would be possible to implement using very low-thrust engines, in particular, by employing attitude control thrusters. iv