Enhancements to Chebyshev-Picard Iteration Efficiency for Generally Perturbed Orbits and Constrained Dynamical Systems
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Modified Chebyshev Picard Iteration (MCPI) is an iterative numerical method for solution of Ordinary Differential Equations (ODEs). This dissertation presents a body of work that serves to enhance the overall performance and the algorithmic automation of MCPI, applied to the problem of perturbed orbit propagation. Additionally, an MCPI framework is derived that greatly improves MCPI performance for ODE systems that intrinsically have associated conserved quantities. Leveraging these developments, software libraries are presented that are designed to make MCPI more accessible and automated, both for the problem of orbit propagation, and for general ODE systems. The work outlined in this document is the result of an effort to promote MCPI from an algorithm of academic interest to a broadly applicable toolset for general use by researchers worldwide in all disciplines. MCPI is able to numerically propagate perturbed orbits to arbitrarily high solution accuracy, bounded by the limits of numerical precision. The improvements to MCPI for orbit propagation are focused on decreasing the computational cost of high-accuracy propagation in a two-fold manner; by reducing the number of required iterations necessary to achieve convergence, and decreasing the computational cost per iteration. Typically, the spherical harmonic gravity function evaluations are the most computationally expensive part of perturbed orbit propagation, so the strategies for reducing the cost per iteration focus on techniques for reducing the cost of gravity series evaluations. Additionally, automated tuning parameter selection logic is introduced to enable MCPI to propagate large batches of perturbed orbits, without the necessity of a user in the loop. By making use of an associated conserved quantity within applicable ODE systems, MCPI is shown to be able to achieve much higher performance. A first order and a second order constrained MCPI formulation are developed that are able to vastly reduce the required number of iterations for convergence, increase the achievable segment length, and increase the overall solution accuracy for a given convergence threshold. Software libraries are presented with the goal of encouraging widespread use of the MCPI method. Serial libraries are available for general ODE systems, akin to the Matlab ODE** methods. More specialized libraries, making use of the computational improvements and automated tuning, are available for perturbed orbit propagation. A parallel framework based upon the orbit propagation libraries is presented that is designed for space catalog maintenance, uncertainty propagation, or conjunction analysis.
Ordinary Differential Equations
Space Situational Awareness
Macomber, Brent David (2015). Enhancements to Chebyshev-Picard Iteration Efficiency for Generally Perturbed Orbits and Constrained Dynamical Systems. Doctoral dissertation, Texas A & M University. Available electronically from