| Abstract: | 本論文探討了克普勒型軌道附近相對運動, 並將之應用於宇航器的編隊飛行上。我們首先討論在克普勒型軌道附近的相對運動的力學模型, 一般在討論此等問題時, 通常是採用Tschauner-Hempel Equation (T-H Equation) 來描述。有了飛行器的運動方程式之後, 我們接著探討了“局部時間逼近法(local timeapproximation)” 在這個問題的可應用性。本文證明此控制法則在近圓軌道上亦能對軌道做有效的控制。然後在這個概念下, 本文提出了兩種可能的控制器設計方法-包括在時域進行設計以及利用T-H Equation進行控制器設計; 我們不僅能穩定原本不穩定的相對運動軌道, 並且能夠重現“縮小版”的主軌道。
由於對軌道施加控制必需額外花費能量, 因此我們也探討了利用自然的週期性相對軌道, 去維持編隊的可能性。我們運用已知的二體問題完整解, 去求出兩條同週期的不同軌道之間的相對位置方程式, 再加以簡化。這個方法可以避免去解線性化之後的常微分方程, 並且得到了一個比直接去積分T-H Equation 更精確的解。我們可以把這個結果應用到宇航器的編隊, 並能降低維持編隊所需要的油耗。最後, 我們也利用數值模擬的方法來驗證我們的結果的正確性。
This thesis investigates relative trajectories about a Keplerian orbit with potential applications to the formation flight of spacecraft. We first consider a spacecraft formation about a nominal Keplerian orbit, whose dynamics is usually described by the Tschauner-Hempel Equation (T-H Equation). Briefly reviewing the results from the T-H Equation, we analytically prove the applicability of the “local time approximation” to the T-H Equation. With the guidance of local time approximation, we propose potential design methods of control law both in the time domain and in the true-anomaly domain. By designing the control law in the true-anomaly domain, we not only stabilize the unstable relative trajectory, but also“re-construct” the “scaled” nominal orbit for our formation of spacecraft.
We also present another methodology for determining relative motion initial conditions for periodic motion in the vicinity of a Keplerian orbit. In this method the spacecraft relative dynamics is derived by subtracting two neighborhood orbits, and simplifying the results. Our work avoids solving the linearized differential equations, and provides a more precise approximation to the relative motion than the T-H Equation did. This result can be used for lowering down the fuel usage for relative orbit maintenance. Numerical simulations are also presented to verify our results. |
