Control of satellite imaging arrays in multi-body regimes

Lindsay DeMoore Millard, Purdue University

Abstract

In the current study, control strategies are investigated for spacecraft imaging formations in multi-body regimes. The specific focus of the analysis is spacecraft motion as modeled in the circular restricted three-body problem, where two large gravitational bodies affect the motion of spacecraft in their vicinity. Five equilibrium points, or libration points, exist as solutions to the differential equations of motion in the circular restricted three-body problem. A specific periodic solution to these equations is an orbit in the vicinity of a libration point, i.e., a halo orbit. Halo orbits are ideal locations for spacecraft imaging arrays as they remain at a nearly fixed distance from the larger, or primary, bodies in the system. For example, if the Sun and Earth are considered the primary bodies, a spacecraft array can be placed near a libration point on the far side of the Earth, protected from the harsh radiation of the Sun at all times. A model of image reconstruction is developed for two common satellite imaging platform designs: an interferometric sparse aperture array and an occulter-telescope formation. The resolution of an image produced by an array is largely determined by the corresponding coverage of the (u, v) plane. The (u, v) plane is not a physical plane, but rather a relationship between frequencies and amplitudes in the Fourier expansion of the electromagnetic signal from the object of interest. Coverage of the (u, v) plane is derived based on several characteristics of the spacecraft configuration and the motion in physical space. Therefore, to determine formation motion history that may be advantageous to imaging, a mathematical model relating spacecraft motion in physical space to coverage of the (u, v) plane, and thus image reconstruction, is necessary. From these models, two control algorithms are developed that increase the resolution of the images produced by the formation while exploiting multi-body dynamics to reduce satellite fuel usage. The first method incorporates nonlinear optimal control techniques to determine constellation motion that maximizes resolution of an image while minimizing fuel. Specifically, the problem is formulated using an augmented Lagrange multiplier method and numerically solved using a sequential quadratic programming algorithm. The second approach is a geometric control algorithm that is developed based on the characteristics of the dynamical phase space near periodic orbits in the circular restricted three-body problem. This algorithm incorporates natural quasi-periodic motion in the problem to reduce control costs and produce relative spacecraft motion advantageous for imaging arrays. These two new methods are compared and contrasted with more traditional methods, including time-varying linear quadratic regulators, impulsive targeting, and input feedback linearization. Methods for state estimation are also explored. The control algorithms are implemented (numerically) on satellite constellations of differing size and function, including examples similar to the following National Aeronautics and Space Administration missions: Terrestrial Planet Finder, the Micro-Arcsecond X-ray Interferometry Mission, and the Terrestrial Planet Finder-Occulter. Notional image reconstruction is demonstrated for varying formation size, maximum baseline, distance to the object of interest, and wavelength of electromagnetic radiation.

Degree

Ph.D.

Advisors

Howell, Purdue University.

Subject Area

Aerospace engineering|Optics

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