以離散角動量守恆演算法則從事衛星系統動態分析

Abstract

在一般的控制器設計及系統分析中，基於成本及人力物力的考量，電腦模擬已經漸漸成為一種不可或缺的工具。因此，如何獲得正確而有效地系統動態特性模擬乃成為一非常重要的研究課題。在此論文中，我們提出一線性角動量守恆的格式來修正Park and Chiou在1992年所提的歐拉方程式(Euler Equations)積分演算法則。此修正演算法則與Simo和Wang 在1991年所提的演算法則比較，可獲得較好的結果。此外，我們也針對拘束性剛體動態系統建立一積分演算法則，並將其應用於保齡球系統動態模擬。另外，我們也將此演算法應用到控制系統設計的模擬上以展示其對模擬結果的影響。歸納而言，此演算法則包括下列優點：(1)歐拉方程式得以線性角動量守恆定律表示；(2)積分演算法則之積分間隔大小可以放大而模擬結果依然收斂，也就是說，計算效率得以提升；(3)由於截止誤差及積分估計誤差經由角動量守恆定律之強制條件平衡，可以獲得更精確之模擬結果。

Computer simulation becomes popular in conrol system design and system anarysis because of its cost reduction and effort-saving. In thsi thesis, a linear momentum conservation form is addressed to modify the Euler equations integration algorithms proposed by Park and Chiou in 1992. Compared with the algorithm proposed by Simo and Wang in 1991, the modified algorithm provides better performance in Euler rotational equations of motion. A constrained rigid body motion integration algorithm is applied to analyze the dynamics of the motion equation of a bowling ball. Furthermore, the modified algorithm is also applied to a controller design problem to demonstrate the effectiveness of the algorithm. To sumarize, the modified algorithm provides the following attractive features: (i) The Euler equation can be exactly fitted by a linear momentum conservation law; (ii) The step size of the integration algorithm can be magnified to a larger number and the simulation results still converge. That is, the computational efficiency can be promoted; and (iii) The truncation error and integration approximation error can be balanced through the momentum conservation law enforcement, preciser simulation results can be obtained.