Pseudospectral collocation is a widely used direct transcription method that discretizes continuous systems with spectral accuracy and high fidelity while requiring relatively few collocation points. The special orthogonal group, SO(3), is both a Lie group and a manifold, and thus demands on-manifold optimization techniques to preserve its geometric structure. Riemannian optimization is enabled by mapping problems to the tangent space through equivariant transformations, with the Cayley map providing a computationally efficient coordinate representation. The resulting discretized system is then formulated within a Riemannian optimization framework, ensuring geometric consistency. This work introduces a novel formulation of the Legendre–Gauss–Lobatto (LGL) pseudospectral collocation method using a Runge-Kutta-Munthe-Kaas geometric integrator tailored for on-manifold constrained optimization. Constrained Riemannian optimization is carried out via an augmented Lagrangian method adapted to manifold settings. The proposed formulation improves numerical stability and computational efficiency by combining the spectral properties of LGL collocation with the computational advantages of the Cayley transform. The LGL-based constrained optimization preserves the geometry and is more computationally efficient than trapezoidal collocation. The combination of LGL collocation and the Cayley map demonstrates that the proposed approach provides a computationally efficient and geometrically consistent solution.
","abstract_html":"<p>Pseudospectral collocation is a widely used direct transcription method that discretizes continuous systems with spectral accuracy and high fidelity while requiring relatively few collocation points. The special orthogonal group, SO(3), is both a Lie group and a manifold, and thus demands on-manifold optimization techniques to preserve its geometric structure. Riemannian optimization is enabled by mapping problems to the tangent space through equivariant transformations, with the Cayley map providing a computationally efficient coordinate representation. The resulting discretized system is then formulated within a Riemannian optimization framework, ensuring geometric consistency. This work introduces a novel formulation of the Legendre–Gauss–Lobatto (LGL) pseudospectral collocation method using a Runge-Kutta-Munthe-Kaas geometric integrator tailored for on-manifold constrained optimization. Constrained Riemannian optimization is carried out via an augmented Lagrangian method adapted to manifold settings. The proposed formulation improves numerical stability and computational efficiency by combining the spectral properties of LGL collocation with the computational advantages of the Cayley transform. The LGL-based constrained optimization preserves the geometry and is more computationally efficient than trapezoidal collocation. The combination of LGL collocation and the Cayley map demonstrates that the proposed approach provides a computationally efficient and geometrically consistent solution. </p>","abstract_has_math":false,"creators":["Whelton, Thomas"],"institution":null,"degree_name":"Master of Science in Aerospace Engineering","degree_level":"Thesis - ERAU Login Required","degree_discipline":"Aerospace Engineering","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2026,"date_issued":"2026-05-04T07:00:00Z","date_published":"2026-05-04T07:00:00Z","updated_at":"2026-07-27T19:26:22Z","subjects":["Optimization; Riemannian; Manifold; Topology; Gradient; Descent; Pseudospectral Collocation; LGL; Geometric Integration","Geometry and Topology","Navigation, Guidance, Control and Dynamics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://commons.erau.edu/edt/988","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Whelton, Thomas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Aerospace Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis - ERAU Login Required"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science in Aerospace Engineering"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Optimization; Riemannian; Manifold; Topology; Gradient; Descent; Pseudospectral Collocation; LGL; Geometric Integration","Geometry and Topology","Navigation, Guidance, Control and Dynamics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://commons.erau.edu/edt/988"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Pseudospectral collocation is a widely used direct transcription method that discretizes continuous systems with spectral accuracy and high fidelity while requiring relatively few collocation points. The special orthogonal group, SO(3), is both a Lie group and a manifold, and thus demands on-manifold optimization techniques to preserve its geometric structure. Riemannian optimization is enabled by mapping problems to the tangent space through equivariant transformations, with the Cayley map providing a computationally efficient coordinate representation. The resulting discretized system is then formulated within a Riemannian optimization framework, ensuring geometric consistency. This work introduces a novel formulation of the Legendre–Gauss–Lobatto (LGL) pseudospectral collocation method using a Runge-Kutta-Munthe-Kaas geometric integrator tailored for on-manifold constrained optimization. Constrained Riemannian optimization is carried out via an augmented Lagrangian method adapted to manifold settings. The proposed formulation improves numerical stability and computational efficiency by combining the spectral properties of LGL collocation with the computational advantages of the Cayley transform. The LGL-based constrained optimization preserves the geometry and is more computationally efficient than trapezoidal collocation. The combination of LGL collocation and the Cayley map demonstrates that the proposed approach provides a computationally efficient and geometrically consistent solution.
"]},{"key":"dc:title","label":"Title","values":["Constrained Optimization on SO(3), via Pseudospectral Collocation"]}]}],"canonical_facts":{"dc:creator":["Whelton, Thomas"],"dc:description.abstract":["Pseudospectral collocation is a widely used direct transcription method that discretizes continuous systems with spectral accuracy and high fidelity while requiring relatively few collocation points. The special orthogonal group, SO(3), is both a Lie group and a manifold, and thus demands on-manifold optimization techniques to preserve its geometric structure. Riemannian optimization is enabled by mapping problems to the tangent space through equivariant transformations, with the Cayley map providing a computationally efficient coordinate representation. The resulting discretized system is then formulated within a Riemannian optimization framework, ensuring geometric consistency. This work introduces a novel formulation of the Legendre–Gauss–Lobatto (LGL) pseudospectral collocation method using a Runge-Kutta-Munthe-Kaas geometric integrator tailored for on-manifold constrained optimization. Constrained Riemannian optimization is carried out via an augmented Lagrangian method adapted to manifold settings. The proposed formulation improves numerical stability and computational efficiency by combining the spectral properties of LGL collocation with the computational advantages of the Cayley transform. The LGL-based constrained optimization preserves the geometry and is more computationally efficient than trapezoidal collocation. The combination of LGL collocation and the Cayley map demonstrates that the proposed approach provides a computationally efficient and geometrically consistent solution.
"],"dc:identifier":["https://commons.erau.edu/edt/988"],"dc:subject":["Optimization; Riemannian; Manifold; Topology; Gradient; Descent; Pseudospectral Collocation; LGL; Geometric Integration","Geometry and Topology","Navigation, Guidance, Control and Dynamics"],"dc:title":["Constrained Optimization on SO(3), via Pseudospectral Collocation"],"thesis:degree_discipline":["Aerospace Engineering"],"thesis:degree_level":["Thesis - ERAU Login Required"],"thesis:degree_name":["Master of Science in Aerospace Engineering"]},"updated_at":"2026-07-27T19:26:22Z"}