Uncertainty propagation using differential algebra: Applications in astrodynamics

This thesis explores the application of Differential Algebra (DA) for nonlinear uncertainty propagation in dynamical systems, with a focus on astrodynamics. After establishing the theoretical principles and implementing the DA technique using the DACE library in C++, the study reformulates three classical orbital propagation techniques - numerical integration, Kepler's equation, and Lagrange coefficients - within the DA framework. Rigorous comparisons are performed between the DA methods and traditional Monte Carlo simulations. The results demonstrate that DA-based polynomial maps closely encapsulate the statistical envelopes produced by extensive Monte Carlo runs, but with far greater computational efficiency. DA’s analytic enclosure property provides robust upper and lower bounds for the uncertainty evolution in orbital problems, even for strongly nonlinear regimes. The work extends these methods to advanced satellite modeling cases, such as J2 perturbed orbit propagation and collision-risk estimation, underscoring the technique’s relevance for timely uncertainty assessment in space-situational-awareness applications.