Reducing spacecraft state uncertainty through indirect trajectory optimization
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.
Description
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