| Summary: | Increasing satellite attitude requirements demand high accuracy estimation methods capable of operating under significant constraints. To meet these demands, dynamical modeling has been used as an effective alternative to ADCS rate hardware for small satellite missions. However, current dynamical models and estimators generally do not respect the Hamiltonian nature of attitude dynamics. This nature confers unique geometric properties to the system which must be preserved for accurate modeling. This thesis addresses the inconsistency by considering estimation from a geometric point of view. It proves that the nonlinear and first-order attitude equations form a single Hamiltonian system with joint structure. This motivates the first known unified geometric treatment of nonlinear and first-order Hamiltonian equations. The resulting nonlinear maps and state transition matrices are shown to offer considerable advantages, including better state accuracies and integral invariant preservation for comparable computational expense compared to standard solutions of the same order. Additionally, a new unified attitude integrator is presented that exactly preserves the Hamiltonian structure of the nonlinear/first-order attitude system. These geometric methods are employed in the Kalman Filter, and it is demonstrated that qualitative properties preserved by the geometric maps are preserved with high accuracy by the estimators. Simulations reveal distinct geometric properties of the geometric filters' state estimates and error bounds that nongeometric estimators do not possess. Substantially improved state estimates are also demonstrated. A geometric investigation of the general nonlinear estimation problem follows. Probability density functions are shown to be conserved properties of deterministic Hamiltonian systems, and appropriate geometric integrators exactly preserve the functions as they evolve in time. Based on these insights, a new iterative filter is derived which preserves qualitative properties of nonlinear dynamics, first-order dynamics, and the general estimation problem. Comparisons with a benchmark iterative filter demonstrate substantially reduced computational burden and superior convergence properties given high nonlinearity.
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