機械臂在今日工業及社會上的使用已愈來愈廣泛，為了提高機械臂的靈活度、使用壽命或達到某些特定要求等，所以我們增加了機械臂的自由度來解決這些問題；除此之外，我們亦需要一套良好的控制理論來提高機械臂的精確性、增加系統強健性以及降低外界雜訊干擾，剛度控制即為其中一種。
Salisbury提出了著名的剛度矩陣轉換式，然近年來被發現該轉換式無法能量守恆的錯誤，並提出新的保守剛度矩陣轉換式(CCT)。提出該式以來，平面三自由度及空間六自由度的串、並聯式機械臂都已證實其CCT的正確性。本文進一步將CCT的概念應用到過自由度機械臂上，並使用Pseudoinverse來解決非方陣的反矩陣問題，且藉由機構的對偶原理來討論過自由度機械臂的對偶並聯式機械臂；觀察數值模擬的結果，比較傳統剛度矩陣轉換和CCT矩陣轉換的差異。
由平面4R及空間7R的串聯式機械臂和對偶的平面4RPR、空間7SPS並聯式機械臂的數值模擬結果可知，傳統剛度矩陣轉換仍無法達到能量守恆，而CCT則可得到能量守恆的正確結果。

Manipulators have been widely used for decades. In order to improve the flexibility of manipulators and to achieve certain tasks, we have to increase the degree of freedom of manipulators and use the so-called redundant manipulators. In addition, good control algorithms are needed to improve the accuracy of manipulators and to reduce the interference of miscellaneous signals from the environment. Stiffness control is one of the well-known control strategies, mainly used in assembly tasks. Stiffness mapping is an essential operation for stiffness control.
The widely used formula of stiffness mapping was first proposed by Salisbury. It was then discovered that Salisbury’s formulation did not obey the low of energy conservation. A conservative congruence transformation (CCT) has been proposed to correct Salisbury’s formulation. Research in the past few years has confirmed the correctness of the CCT in non-redundant serial and parallel manipulators. In this thesis, CCT is applied to redundant manipulators. Pseudoinverse is used to deal with the inverse of non-square matrices. The dual parallel manipulators of redundant serial manipulators are also discussed. The conventional stiffness mapping and the CCT are compared in the numerical simulations of two redundant serial manipulators and their dual parallel manipulators.
According to the numerical simulations of planar 4R and spatial 7R serial manipulators and their dual parallel manipulators, planar 4RPR and spatial 7SPS manipulators, CCT give the accurate results that obey the law of energy conservation. On the other hand, the results from the conventional stiffness mapping do not obey the law of energy conservation.

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