Higher Order Polynomial Approximation For Applications in Space Situational Awareness
Open Access
- Author:
- Hall, Zachary Joseph
- Graduate Program:
- Aerospace Engineering
- Degree:
- Master of Science
- Document Type:
- Master Thesis
- Date of Defense:
- November 16, 2018
- Committee Members:
- Puneet Singla, Thesis Advisor/Co-Advisor
David Bradley Spencer, Committee Member - Keywords:
- space situational awareness
conjugate unscented transformation
polynomial approximation
higher order sensitivity
uncertain lambert problem
reachability sets - Abstract:
- The uncertain Lambert problem and computation of reachability sets are in many ways complimentary problems with important applications in Space Situational Awareness (SSA). Formulating the solution to these problems in a stochastic framework is an important intermediate step to enabling the next generation of collision assessment, satellite tracking and characterization capabilities. Traditionally, accurately computing solutions to these problems in a probabilistic manner was either computationally infeasible due to the large number of simulations required to capture the statistical properties of the solution, or was only accurate near the nominal solution due to the inability of linear variational methods to capture higher order statistical moments of the solution. Using a Taylor series expansion of the deterministic solution, a polynomial approximation method is proposed to calculate the higher order sensitivity matrices. An orthogonal polynomial basis function expansion is used to represent the solution of the uncertain Lambert and reachability problems. The coefficients of these orthogonal polynomials represent the higher order sensitivity matrices. The Conjugate Unscented Transformation (CUT) method is utilized for the purposes of computing the multi-dimensional expectation integrals required to determine the least squares coefficients, and offers extensive computational savings compared to other quadrature methods. Additionally, the CUT method allows the computation of these sensitivity matrices without the cumbersome need to explicitly take higher order partial derivatives, as well as having the advantage of achieving higher accuracy over a larger input domain than was previously available using other numerical integration methods. The mathematical framework for the polynomial approximation method is laid out and test case simulations for both the uncertain Lambert problem and the reachability set problem are presented. Discussion of results, as well as the relative advantages and limitations of the method for each simulation is included.