| Abstract: | 多次進出大氣層所產生最大軌道平面轉變 之最佳軌跡及次佳軌跡具備了軌道之特性。因 此,引發了吾人探討此解之軌道參數變化特性 。 首先,利用適當之關係將最佳軌跡之軌跡 參數(.phi.,.psi.,.theta.,h,u,.gamma.)轉換為軌道參數(.alpha.,.OMEGA.,I,a,e,.omega.)。接著,引用Lagrange's軌 道方程式直接推導軌道參數變化之方程式。並 且利用上述之方程式及次佳控制法則來計算軌 跡。當然,在引用次佳控制法則及橫斷條件時 已經將它們以軌道參數來表示了。 依據上述之結果,吾人發覺.omega.及.OMEGA.之 變化很小,因而使得.alpha..apprxeq.f。所以,忽略 .omega.,.OMEGA.及.alpha.三個式子,僅考慮I,a,及e,三 式即可,而使得計算大氣輔助軌道平面轉變更 為簡單。
The properties of the optimal and sub-optimal solutions to multiple-pass aeroassisted plane change are studied in terms of the trajectory variables in Refs. (6) and (8). The solutions show the strong orbital nature. Therefore, it is interested to have the solutions in terms of the elements of the orbit. We shall use the relations between the trajectory variables .phi., .psi. and .theta. and the orbital elements .alpha., .OMEGA. and I as given by the spherical trigonometry and the relations between the trajectory variables h, u and .gamma. and the orbital elements a, e and .omega. as given by the orbital theory to derive the differential equations for the variations of the orbital elements (.alpha., .OMEGA., I, a, e, .omega.) along the optimal trajectories. Then, it is proposed to obtain the variations of the orbital elements by direct integration of their equations and by using the classical Lagrange's planetary equations. Finally, we shall use these equations and the approximate control derived in Ref. (8) to calculate the trajectories. In this respect, the approximate control law and the transversality condition are transformed in terms of the orbital elements. Following the above results, we can reduce the computational task by further simplification. With .omega. and .OMEGA. being small and returning to the value of zero after each revolution, we neglect the equations for .omega., and .OMEGA., Also, since .omega..apprxeq.0, that is .alpha..apprxeq.f, we can neglect the equation for the .alpha. and have only three state equations for the integration. |
