# Reducing spacecraft state uncertainty through indirect trajectory optimization

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## Abstract

The exact position and velocity of a spacecraft is never known; instead, an estimate of the spacecraft location is determined based on observations of the spacecraft state. The accuracy of this state estimate depends on numerous factors including the number, quality, frequency, and types of measurements; the accuracy with which the equations of motion are modeled; and the trajectory of the spacecraft relative to the observer. Many choices of trajectories are available to transfer a spacecraft from an initial set of constraints to a final set of constraints. Most efforts to optimize these transfers involve determining the minimum propellant or minimum time transfer. This dissertation provides a technique to determine trajectories that lead to a more accurate estimate of the spacecraft state. The calculus of variations is used to develop the necessary theory and derive the optimality conditions for a spacecraft to transfer between a set of initial and final conditions while minimizing a combination of fuel consumption and a function of the estimation error covariance matrix associated with the spacecraft Cartesian position and velocity components. The theory is developed in a general manner that allows for multiple observers, moving observers, a wide variety of observation types, multiple gravity bodies, and uncertainties in the spacecraft equations of motion based on the thrust related parameters of the spacecraft. A series of example trajectories from low Earth orbit (LEO) to a near geosynchronous Earth orbit (GEO) shows that either the trace or the integral of the trace of the covariance matrix associated with the Cartesian position and velocity can be reduced significantly with a small increase in the integral of the spacecraft thrust acceleration squared. A method to minimize the uncertainty of the spacecraft state in a set of coordinates other than the one in which the spacecraft equations of motion and covariance are expressed is also introduced. The technique allows one to minimize the uncertainty in non-Cartesian components such as the spacecraft semimajor axis, flight path angle, or range without developing the equations of motion for the spacecraft or covariance in a non-Cartesian frame. Example problems with transfers from LEO to near GEO and LEO to lunar orbit demonstrate that the covariance associated with the semimajor axis can be reduced significantly with only a slight increase in fuel consumption.