Algebraic Methods in Spacecraft Navigation
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Mancini, Michela
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Abstract
The technological advancements of recent decades and the enduring drive to explore the universe are enabling new types of space missions, each presenting unique challenges. Deep space missions increasingly require autonomous navigation, for which Optical Navigation (OPNAV) offers a promising solution. As a result, developing and refining OPNAV algorithms is a vibrant area of research.
Algebraic geometry provides a convenient mathematical framework for OPNAV, as many objects and phenomena in spacecraft navigation can be described in polynomial form. Keplerian orbits, elliptical crater rims, ellipsoidal celestial bodies, atmospheric bands, planetary rings, are all curves and surfaces that can be modeled as degree-two polynomials. Additionally, phenomena and constraints such as the Doppler effect, stellar aberration, image distortion, and line-of-sight observations are naturally expressed in polynomial form.
This dissertation explores the application of techniques from algebraic and projective geometry to address navigation-related challenges. In the first part, analytical tools are leveraged to obtain the closed-form expression of a crater rim imaged with a pushbroom camera. Crater rims are often visible in the images captured by such sensors, and knowledge of their analytical shape enables interesting capabilities, such as crater reconstruction and spacecraft state estimation directly processing the image.
Subsequently, projective geometry is used to develop an analytical framework for parameterizing Keplerian orbits. This led to a novel velocity propagation technique, and to a solution to the problem of fitting a conic with constrained focus location to three points. Additionally, parameter homotopy continuation is used to initialize the state of an orbiting transmitter.
Finally, algebraic geometry is used to develop a technique for partially calibrating a camera from the image of a single celestial body. Then, the conic intersection problem is revisited and an algorithmic framework providing a simpler solution compared to current techniques is developed.
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2025-04-25
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Dissertation