Cover Scheme Methods Applied To Solving Nonlinear Optimal Control Problems in Space Flight Applications

Author

• Bethany Hintz, University of Central FloridaFollow

ORCID

0009-0001-0019-789X

Keywords

Optimal Control, Cover Schemes, Coupled Radial Basis Functions

Abstract

In this work, a Coupled Radial Basis Function (CRBF) and cover scheme method is presented to solve nonlinear optimal control problems (OCPs) in space flight applications. Since OCPs are often very complex, numerical methods are used to solve them. Direct methods transcribe the OCP into a nonlinear programming problem (NLP) by discretizing the continuous-time problem.  Indirect methods use Pontryagin’s Maximum Principle to derive the necessary conditions for optimality, resulting in a two-point boundary value problem (TPBVP). These solving methods have inherent challenges: direct methods are reliant on the type of nodal distribution, whereas indirect methods require a priori information about the system and a good initial guess to obtain a solution. This work aims to overcome these challenges by adopting a hybrid formulation, where the necessary conditions for optimality are derived from Pontryagin’s Maximum Principle and the resulting TPBVP is transcribed using collocation techniques. CRBFs are used as the basis function for approximation, since they are insensitive to the shape parameter and agnostic to nodal distribution, allowing the addition of nodes to refine the approximation at areas of interest, such as constraints. The problem domain is locally collocated with segments called "elements'' with overlapping regions called "covers'' to enforce continuity between the states and control. Covers have been shown to improve the robustness of the solution with partial differential equations (PDEs). An error estimate model is derived to determine the ideal number of nodes for each problem, resulting in an adaptive CRBF-cover scheme algorithm. The adaptive CRBF-cover scheme method is studied to solve three fundamental space flight optimal control problems: (1) rendezvous and proximity operations problem, (2) the optimal attitude control problem, and (3) the Low-Thrust Orbit Transfer problem. Results have been validated against existing solutions in the literature, by propagating the dynamics using initial costates, or via commercial software tools, e.g. GPOPS-II. Overall, the accuracy is shown to be on the same order when compared to existing solutions. The effect of sharing nodes in overlapping cover regions is analyzed, and it is found that shared nodes reduce overall relative error in most cases and adding nodes at constraint entry and exit points reduces relative error compared to the locally collocated solution. Most importantly, the addition of shared nodes reduces relative error up to 90% in constrained regions compared to the locally collocated baseline solution. As a result, the present adaptive CRBF-cover scheme method contributes a new solving method to the literature that is agnostic to nodal distribution, allows for overlapping nodes between elements to increase robustness, and does not require a good initial guess nor a priori information about the system. Future work may include studies for real-time applications or model predictive control (MPC).

Completion Date

2026

Semester

Spring

Committee Chair

Elgohary, Tarek

Degree

Doctor of Philosophy (Ph.D.)

College

College of Engineering and Computer Science

Department

Department of Mechanical and Aerospace Engineering

Format

PDF

Document Type

Dissertation

Identifier

DP0053140

Release Date

5-15-2027

STARS Citation

Hintz, Bethany, "Cover Scheme Methods Applied To Solving Nonlinear Optimal Control Problems in Space Flight Applications" (2026). Graduate Studies Theses and Dissertations 2026. 83.
https://stars.library.ucf.edu/gradstudies_etd_2026/83