在機構設計中，雙接頭連桿導引剛體至數個指定位置之方法，一直是機構合成問題之研究重點。在所有的空間雙接頭連桿中，方程式與未知數數目相同的接頭僅有以下四種︰平面及空間雙旋轉對(R-R)、空間雙圓球對(S-S)、空間雙圓柱對(C-C)及空間圓柱圓球對(C-S)。前三種類型的接頭已有學者解出解析解或數值解，對於圓柱圓球對(CS)連桿機構導引剛體通過八個位置之合成問題，至今尚未有學者能找出其所有解。本文分析CS連桿機構之合成方程式，並指定八個剛體位置，解出空間中所有CS連桿機構導引剛體通過指定八個位置之有限解。
為了解出CS連桿機構之合成問題，本文前段先介紹螺旋剛體位移，及新多項式連續法之原理，接著探討CS連桿機構之螺旋合成及幾何合成方程式，並利用新多項式連續法，將設定之九個變數做適當地分群，計算Bezout Number可由原本未分群之65536減少至17920。觀察CS連桿機構之幾何合成方程式，可利用數學運算之方法，將整個合成方程式齊性化，此法可將Bzout Number由17920減少至8960。最後，建構與目標方程式相關之初始系統(Start System)及同倫系統(Homotopy System)。使用套裝軟體Bertini及新多項式連續法做同倫路徑追蹤，使用數種Bertini之參數設定。在延伸應用的部份，本文將求得之實數解，用於建構CS平行平台機構，在定義S接頭之極限位置空間以及CS連桿桿長後，將其當作設計參數，設定權重為1比1，並從所有實數解中選出最合適的解，設計並聯平台機構，探討其可行性。
本研究中代入至少十組不同的位移螺旋參數，進行CS連桿機構之最大指定八個位置合成問題。均由8960條同倫路徑中，得到804組有限解，並且可由804組有限解中得到30～50組實數解，其運算時間視設計問題複雜度而定，約為3～30小時不等。
本論文解出CS連桿機構之最大指定八個位置合成問題之數值解，無論位移螺旋參數設定為實數或複數，其解集合數目固定為804組有限解，而實數解數目不為一定值，隨位移螺旋參數之設定而變動。運算時間亦隨設計問題之複雜度而變。在所得之30～50組實數解中，選出最合適的解3～5組設計並聯平台機構，會產生干涉及機構鎖死問題尚待後人努力克服。

The dimensional synthesis of spatial dyads for rigid body guidance is the central problem in mechanism design. In computational kinematics, we are interested in the solutions to dyad synthesis problems which result in the same number of equations and unknowns when maximum number of positions is specified. The only unsolved problem to date is the eight-position synthesis of the C-S dyad. This thesis seeks to solve the design equations of the C-S dyad and find all of their solutions numerically.
This thesis utilizes screw geometry and employs the numerical continuation method. In addition to screw geometry, we also investigate geometric constraints and derive the eight-position synthesis equations for the C-S dyad. We then employ the new polynomial continuation method by using the software package Bertini. The number of tracked paths is reduced from 65536 to 8960 by introducing multiple homogeneous variables.
The result shows that there are 804 finite solutions, including real and complex ones, to the eight-position synthesis problem. After trying more than 10 sets of randomly specified screw parameters and obtaining the same number of 804 finite solutions, we are confident that the number is correct. The number of real finite solutions is unpredictable because it depends on the specified screws. The computing time for each case varies and depends on the complexity of the specified screws. Furthermore, this thesis uses the acquired real finite solutions to construct and analyze 3CS and 5CS parallel linkages.

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