Michela Mancini
[Advisor: Dr. John A. Christian]

will propose a doctoral thesis entitled

Algebraic Methods in Spacecraft Navigation

On

Friday, May 10 at 10:00 a.m.
Montgomery Knight Building 317

Abstract
Algebraic geometry is a branch of mathematics that studies curves and surfaces that are solutions to polynomial equations. In spacecraft navigation, Keplerian orbits, crater rims, and global planetary surfaces can all be modeled using a polynomial – more specifically, a polynomial of degree two. For this reason, tools from algebraic geometry find application in this area, providing useful means to develop solutions to many different navigation problems.

For example, this work uses algebraic geometry to develop an analytical description of the projection of an elliptical crater rim imaged with a pushbroom camera. Pushbroom cameras are common science instruments in planetary imaging and exploration, and craters are common features on the surface of rocky planetary bodies. Despite the pervasiveness of this problem, an analytical framework for the projection of a crater was missing in the literature. The projection geometry of a pushbroom camera depends on its dynamics and the image formation process is quite different from conventional (i.e., central projection) cameras. Importantly, conics do not project to other conics in the image plane unless special conditions are met.

Algebraic geometry can also be used to solve multiple initial orbit determination (IOD) problems, providing an algebraic description of the orbit that is both mathematically convenient and physically intuitive. Using this parameterization, for example, it is shown how the Gibbs problem can be solved by simply intersecting two planar lines, with the intersection point’s coordinates having a clear interpretation in terms of the orbit parameters.

Finally, it will be shown how algebraic geometry can be used to simplify the analysis of a conic in the image plane of a central projection camera. In particular, the common problem of intersecting two conics, which is encountered often in the computer vision world and is an important step of some camera-based terrain relative navigation (TRN) solutions, will be proved to be reducible to solving a pure quadratic equation, considerably simplifying the computations required by already established conic intersection techniques.

Committee

• Dr. John A. Christian – School of Aerospace Engineering (advisor)
• Dr. Brian C. Gunter – School of Aerospace Engineering
• Dr. Koki Ho – School of Aerospace Engineering