臺灣博碩士論文加值系統

　　剛體位移的基本問題為討論如何在已知剛體的起始以及最終位置時，如何描述此位移，許多的研究是利用非線性代數的方法去計算剛體上點資料的座標，以得到完全指定位移的螺旋，但是當資料出現誤差或是考慮的情形是過指定位移時，在計算上就會變得比較複雜且費時。而且除此之外，大多數的方法只能以點的幾何特徵為資料處理這方面的問題，僅有少數的研究是利用線和面的幾何特徵為資料，造成在應用上的不方便，因此本文提出ㄧ種新的概念，利用線性代數的方法推導出一種更具效率且應用更廣的一種方法來計算空間中剛體的位移。

　　在近十年中，有限扭轉螺旋已被證明出可以形成螺旋系統，許多不完全指定位移的線性表示式也陸陸續續被推導出來，並且形成投影空間，本文將一個完全指定位移或過指定位移分解成多個不完全指定位移，再交集這些不完全指定位移的投影空間，可以得到一交集線性方程式，若考慮的是資料正確的完全指定位移時，此方程式的解就是該位移的螺旋。在實際應用上所得到的往往是資料出現誤差的過指定位移，此時投影空間的交集會形成空集合。本文的目的在求解這些問題，本文中利用螺旋的定義修改原本的交集線性方程式以代入最小平方法得到描述此位移的最佳螺旋。

　　本文所提出的方法應用的範圍不只侷限於以點資料為基礎的位移上，還可以應用於線資料，面資料以及點與面的配合，並且以線性代數取代一般代數計算空間的螺旋，使螺旋理論的應用更加完整以及簡便。

　One of the fundamental problems related to the displacements of rigid bodies is to determine the displacement when the initial and finial positions of a rigid body are known. Most researchers utilize point data on a rigid body to compute the screw for a completely specified displacement. The least-square approximation is used when the displacement is redundantly specified. Some other researchers utilize different geometric entities, such as lines or planes to determine a displacement. To date, all the algorithms for computing the screw of a displacement require solving nonlinear algebraic equations. This thesis takes another approach by using linear algebra to compute completely and redundantly specified displacements with specified points, lines, or planes.

　Finite screws have been proved to form finite screw systems for over a decade. The screw systems of many kinds of incompletely specified displacement were derived, and their operations correspond to those of projective spaces. This thesis decomposes a displacement into several incompletely specified displacements, and the displacement screw is obtained by intersecting the screw system associated with the incompletely specified displacements. As a result, the determination of the screw requires solving only linear equations. If the motion of the rigid body is completely specified, the solution of the linear equations is the screw of the displacement. However, in practice, if the specified data of the positions of the body are inaccurate. The intersection of the projective spaces becomes an empty set. The goal of the thesis is to deal with this kind of problems.

　This thesis utilizes the properties of screws to modify the linear equations obtained from the intersection of screw systems, and then the displacement screw is obtained by using the least-square approximation.
In conclusion, this thesis deals with not only point specifications but also line and plane specifications of displacements. The proposed method computes the displacement screw by using linear algebra, and it substantially simplifies the calculation of rigid-body displacements.

摘要 Ⅰ
ABSTRACT Ⅱ
致謝 Ⅲ
目錄 Ⅳ
表目錄 Ⅶ
圖目錄 Ⅷ
符號說明 Ⅸ
第一章 緒言 1
1.1 前言 1
1.2 文獻回顧 2
1.3 研究動機與目的 4
1.4 本文架構 5
第二章 基本概念 6
2.1 螺旋理論 6
2.1.1 蒲朗克座標與螺旋座標 6
2.1.2 扭轉螺旋與螺距的定義 8
2.1.3 螺旋系統 11
2.2 不完全指定位移 11
2.3 螺旋三角形及其線性表示式 13
2.4 有限螺旋系統之投影空間 15
2.5 誤差下指定位移的螺旋 16
第三章 以線性代數方法求空間中 點位移之螺旋 19
3.1 點的不完全指定位移的投影空間 19
3.1.1 點的不完全指定位移的線性表示式 19
3.1.2 螺旋系統形成的向量子空間 21
3.1.3 點投影空間的交集 24
3.2 誤差下的交集線性方程式 26
3.2.1 不精確的完全指定位移 26
3.2.2 最小平方法 27
3.2.3 修改交集線性方程式 28
3.2.4 過指定位移 30
3.3 螺旋的參數 31
3.3.1 螺旋的方向、位置以及螺距 31
3.3.2 旋轉量以及平移量 32
3.4 物理意義 35
3.5 數值例 38
3.5.1 無誤差下空間中六點位移之螺旋 39
3.5.2 誤差下空間中六點位移之螺旋 41
第四章 線以及面之過指定位移螺旋 45
4.1 線的交集 45
4.1.1 線的不完全指定位移的線性表示式 45
4.1.2 線位移的向量子空間的交集 47
4.1.3 數值例 51
4.2 面的交集 54
4.2.1 面的不完全指定位移的線性表示式 54
4.2.2 面位移的向量子空間的交集 56
4.2.3 面位移與點位移的向量子空間的交集 60
4.2.4 數值例 62
第五章 結論與展望 69
參考文獻 71
自述 75
附錄A 線性方程式的解 76
附錄B 空間位移螺旋的程式碼 79
B.1 誤差下空間中六點位移之螺旋 79
B.2 誤差下六點位移螺旋之參數 82
B.3 誤差下空間中四線位移的螺旋 84
B.4誤差下空間中面與面位移之螺旋 88
B.5 誤差下空間中面與兩點位移之螺旋 95

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