Based on the mathematic representation of loops of kinematic chains, this paper proposes the " ⊕ " operation of loops and its basic laws and establishes the basic theorem system of the loop algebra of kinematic chains. Then the basis loop set and its determination conditions, and the ways to obtain the crucial perimeter topological graph are presented. Furthermore, the characteristic perimeter topo-logical graph and the characteristic adjacency matrix are also developed. The most important characteristic of this theory is that for a topological graph which is drawn or labeled in any way, both the resulting characteristic perimeter topological graph and the characteristic adjacency matrix obtained through this theory are unique, and each has one-to-one correspondence with its kinematic chain. This character-istic dramatically simplifies the isomorphism identification and establishes a theoretical basis for the numeralization of topological graphs, and paves the way for numeralization and computerization of the structural synthesis and mechanism design further. Finally, this paper also proposes a concise isomorphism identifica-tion method of kinematic chains based on the concept of characteristic adjacency matrix.
It is well known that the traditional Grübler-Kutzbach formula fails to calculate the mobility of some classical mechanisms or many modern parallel robots,and this situation seriously hampers mechani-cal innovation.To seek an efficient and universal method for mobility calculation has been a heated topic in the sphere of mechanism.The modified Grübler-Kutzbach criterion proposed by us achieved success in calculating the mobility of a lot of highly complicated mechanisms,especially the mobility of all recent parallel mechanisms listed by Gogu,and the Bennett mechanism known for its particular difficulty.With wide applications of the criterion,a systematic methodology has recently formed.This paper systematically presents the methodology based on the screw theory for the first time and ana-lyzes six representative puzzling mechanisms.In addition,the methodology is convenient for judgment of the instantaneous or full-cycle mobility,and has become an effective and general method of great scientific value and practical significance.In the first half,this paper introduces the basic screw theory,then it presents the effective methodology formed within this decade.The second half of this paper presents how to apply the methodology by analyzing the mobility of several puzzling mechanisms.Finally,this paper contrasts and analyzes some different methods and interprets the essential reason for validity of our methodology.
