Simple Analytic Solutions to Optimal Trajectory in Formation Flying with Linearizing Transformation of Original Nonlinear Dynamics without Losing Nonlinearity
- Paper number
IAC-09.C1.1.5
- Author
Mr. Sangjin Lee, Yonsei Univ, Korea, Republic of
- Coauthor
Prof. Sang-Young Park , Yonsei Univ, Korea, Republic of
- Coauthor
Prof. Kyu-Hong Choi, Yonsei University, Korea, Republic of
- Year
2009
- Abstract
Analytical solutions to the optimal trajectory problems in spacecraft formation flying are studied in this paper. So far, many linearized relative dynamics have been used to describe relative motions in spacecraft formation flying. These linearized dynamics, however, have some drawbacks in the sense that they neglect some nonlinearities which must be considered. Hill-Clohessy-Wiltshire(HCW) equations, for example, yield significant errors when being used for the high-eccentric orbit, because it assumes that the reference orbit is a circular orbit. In these sense, original nonlinear dynamics have been also used in order to compensate for the flaws of those linearized dynamics. However, these nonlinear dynamics as well have some shortcomings since they make it almost impossible to find analytical solutions in many cases. So as to deal with the problems, we apply a newly devised linearinzing transformation that transforms the original nonlinear dynamic equations into a linear-state-space dynamics equations without losing any nonlinearity. This transformation is achieved by describing the dynamics in terms of newly devised variables instead of original ones. The calculus of variation is then applied to solve the optimization problems which are set in the new variables. Especially, the method which does not need the inverse of fundamental matrix associated with dynamic equations is used. Unlike many previous researches, simple analytic solutions for optimization problems are obtained by using the method. Finally, the obtained solutions in new variables are transformed back into those in original variables. Thus, we eventually avoid the inaccuracy of the previous linearized dynamics and the complexity of the original nonlinear dynamics when finding analytical solutions. To confirm the validity of the newly devised relative dynamics and the method used for solving optimization problems, numerical simulations are performed with some possible cost functions.
- Abstract document
- Manuscript document
(absent)