Development of General-Purpose Root-Finding Module for General Mission Analysis Tool
- 2 Downloads
A general-purpose root-finding module for designing spacecraft trajectories is developed to have similar accuracy to that of other well-known root-finding modules, and greater speed. Three quasi-Newton root-finding algorithms are implemented: the Newton–Raphson method, the Broyden’s method and the Generalized Broyden’s method. Based on the proposed root-finding module, General Mission Analysis Tool (GMAT) of National Aeronautics and Space Administration’s Goddard Space Flight Center (NASA/GSFC) is functionally extended by integrating the Broyden’s method and the Generalized Broyden’s method into its differential corrector module. Non-trivial spacecraft trajectory design problems, such as Lambert’s problem for trans-lunar trajectory and minimum-time transfer problem to the Mars are solved to analyze the performance of the proposed module. The numerical performances of each root-finding algorithm are quantitatively analyzed by the total number of function evaluations, the total number of iterations, convergence error, mean convergence rate, and running time. The overall comparative analysis shows that the Broyden’s method and the Generalized Broyden’s method are about 20–40% faster than the Newton–Raphson method and solutions from each algorithm have similar numerical accuracy. We also show that for selected test problems, the Generalized Broyden’s method converges in less running time than others with similar numerical accuracy in GMAT. The updated differential corrector module was released in R2014a version of GMAT.
KeywordsBroyden’s method Root-finding methods General mission analysis tool Space trajectory design
This study was supported by the National Research Foundation of Korea through the Space Core Technology Development Program funded by the Ministry of Science, ICT & Future Planning (project number: 2013M1A3A3A02042448). This research was also supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (2013R1A1A2013091).
- 1.The GMAT Development Team (2013) General mission analysis tool: GMAT user guide R2013a. National Aeronautical and Space Administration, Goddard Space Flight Center, GreenbeltGoogle Scholar
- 4.Crittin F, Bierlaire M (2003) A generalization of secant methods for solving nonlinear systems of equation. In: 3rd Swiss transport research conference, Monte Verita/Ascona, 19–21 March 2003Google Scholar
- 6.Paprzycki M, Dent D, Kucaba-Piętal A (2002) Solvers for nonlinear algebraic equations; where are we today? In: Wyrzykowski R, Dongarra J, Paprzycki M, Waśniewski J (eds) Parallel Processing and Applied Mathematics. PPAM 2001. Lecture Notes in Computer Science, vol 2328. Springer, Berlin, Heidelberg, pp 719–728Google Scholar
- 9.Berry MM (2012) Comparisons between Newton–Raphson and Broyden’s method for trajectory design problems. Adv Astronaut Sci 142:1177–1194Google Scholar
- 12.van de Rotten BA (2003) A limited memory Broyden method to solve high-dimensional systems of nonlinear equations. Ph. D. Dissertation, Leiden UniversityGoogle Scholar
- 14.Song YJ, Park SY, Choi KH, Shim ES (2008) Development of Korean preliminary lunar mission design software. J Korean Soc Aeronaut Space Sci 36:357–367 (in Korean) Google Scholar
- 18.Bryson AE Jr (1999) Dynamic optimization. Addison-Wesley, BostonGoogle Scholar