本研究之目的在於應用組合理論中的數學模式，發展出一套系統化的演算方法，針對機構具有不同的設計限制，計算出其可行機構的數目。再者，本研究亦利用所提出的演算方法，來計算與驗證經由特殊化過程得到的可行特殊化機構的數目。
首先探討設計限制對機構概念設計的重要性，藉由所收集各種現有機構之設計限制，瞭解機構的拓樸構造特性，並加以整理與歸納，再依照物體與其屬性的關係，將設計限制分為：機構本身的屬性、機構內具某種屬性的物體、機構內兩種物體的關係、及機構內的特定物體等四種類型。然後提出一個系統化演算方法，計算可行機構的數目。此方法的步驟一為確認運動鏈的排列群。步驟二為決定循環指式。步驟三為將設計限制數學模式化，並代入循環指式項中；先定義與說明尋求修正排列群的步驟，應用連桿組路徑碼表示設計限制中某些特定的接頭必須相對應於某些特定的連桿，再藉由變數來表示修正排列群中剩餘的桿件或接頭，並利用生成函數的概念來表示設計限制中的鄰接關係、附隨關係、以及特定的接頭必須相對應於特定的連桿。步驟四為依據Polya’s 理論之樣式多項式的係數，求得可行機構的數目。此外，本研究並提出一套運動鏈特殊化的程序，將各種類型的連桿和接頭分配至運動鏈上，以產生合乎所有設計限制的的特殊化機構，並且利用已提出演算方法驗證可行特殊化機構的數目。最後，本研究利用許多的例子來說明所提出的演算方法以及如何得到特殊化機構。
藉由本研究所提出的演算方法，不只可以精確地計算可行機構的數目，並且可以簡化機構概念設計的程序。再者，本研究的結果，可取代傳統用觀察的方式來檢測機構具有的同構性，進而得到所有非同構的機構數目。

This work is presents a systematic approach based on combinatorial theory for counting the number of non-isomorphic planer mechanisms from kinematic chains subject to various design constraints in the conceptual design of mechanisms. And, this study provides a procedure for generating non-isomorphic specialized mechanisms. By applying the proposed approach, the number of all non-isomorphic feasible specialized mechanisms can be counted and verified.
This work starts by discussing the significance of design constraints for the conceptual design of mechanisms and studying the design constraints of various examples to conclude the characteristics of topological structures. Based on the attributes of the objects, design constraints are classified into four types. Next, a 4-step procedure is proposed for counting the number of mechanisms from kinematic chain with required design constraints. Step 1 is to identify the permutation groups. Step 2 is to determine the cycle index of permutation groups. Step 3 is to assign the design constraints to the cycle index. A procedure for obtaining the defined modified permutation groups is presented. The concept of the linkage path code is employed to include the design constraints that a number of specified joints must correspond to a number of specified links. The remaining links or joints are applied to the variables to indicate the modified permutation groups, and the generating function is used to present the design constraints with the adjacency relation, the incident relation, or some joints corresponding to some links into the cycle index. Step 4 is to obtain the number of feasible mechanisms from the corresponding coefficients that are identified from the pattern inventory polynomial. Furthermore, a procedure for specialization is presented such that the required specialized mechanisms can be obtained, and the proposed approach is applied to count and verify the number of all non-isomorphic feasible specialized mechanisms. Finally, several examples are provided to illustrate the proposed approach and how to obtain the specialized mechanisms.
The proposed approach not only can accurately count the number of non-isomorphic mechanisms, but also can simplify the procedure for the conceptual design of mechanisms. The result of this work can also be used, in a systematic manner, to determine isomorphic, mechanisms. And, all non-isomorphic mechanisms can be obtained exactly and systematically.

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