Adjacency graph and matrix representation of scaling mechanisms

I. INTRODUCTION

Scaling mechanism is a kind of special deployable mechanism which maintains original shape scaling up and down in the motion [1, 2]. Topological structures of the scaling mechanism are different in three different states, completely deployable state, deployable progress and completely folded state. The evolution of these three states is consecutive. The description of each state reveals the evolution rules of the deployable process. Some researchers represent the topological evolution of metamorphic mechanisms by topological graph and its adjacency matrix. Dai [3, 4]

discussed the configuration transformation and its matrix evolution of metamorphic mechanisms. Adjacency matrix with symbol was put forward [5] . In this matrix, kinematic pairs were used as elements. Huston Lower Numbered Arrays were presented and configuration analysis of a typical metamorphic mechanism with this method was carried out [6]. A modified adjacency matrix was presented with component numbers putting at the diagonal of the matrix [7]. But this method was used to represent the metamorphic process step-by-step and the progress was complex. Incidence matrix with type of kinematic pairs was presented with the logical operation to represent the metamorphic process [8]. Extended adjacency matrix was presented to describe the compound hinges with hinge numbers as the matrix elements [9]. In this paper, adjacency graph is proposed to represent the topological transformation of the mechanism. The operation of matrix is used to represent the mergence or coincidence of the links and joints. It represents each state of the mechanisms continuously. Also this method can realize the evolution of the compound joints. So the application is much more widely.

II. ADJACENCY GRAPH AND ITS MATRIX REPRESENTATION AND EVOLUTION

Comparing the endpoint number and link number, it is found that the former is much more than the later, because each link has no less than two points. So the connectivity of the endpoints is used to represent the topological transformation of the mechanism. This method can represent the topological structure of the mechanism comprehensively. A. Adjacency graph of mechanisms In adjacency graph, end points of links are denoted by vertex (z), links are denoted by lines ( — ). The single joint is a joint with two links connecting to one point. The compound joint is a joint with three or more links connecting to one point, which is denoted by (o). lm denotes the link orders. vm denotes the vertex orders. This method used to represent the relationship of links and vertexes is coined adjacency graph. Figure 1 shows the spherical eight-bar linkage and its adjacency graph.

Figure 1. Spherical eight-bar linkage and its adjacency graph

B. Adjacency matrix representation of mechanisms The adjacency matrix representation of adjacency graph is different from the topology graph. The rows and columns of the matrix are denoted by vm .The coincidence of the vertexes is denoted by 1 2 ( , ,) ij j vv v …the elements of the matrix are denoted by lm , which represent the links of the mechanism. The mergence of the links after matrix operation is denoted by l l + +… . A kinematic chain is denoted by s×s square matrix, as is shown in (1).

( ) ij s s B l × = (1)

Where, if are in one link ;

0 if and are in defferent links; 0 a f j. nd i m ij m ij i j l vv l l vv i ­ ° = ® ° ¯ = The adjacency matrix of the spherical eight-bar linkage is shown in (2). 12 3 4 56 7 8 12 1 22 3 33 4 4 45 0 5 56 6 67 7 78 81 8 0 00000 0 00000 0 0 0000 00 0 000 000 0 00 0000 0 0 00000 0 00000 0 vv v v vv v v vl l vl l vl l v ll B v ll v ll v ll vl l ¯ ¡ ° ‑ ¢ ± (2) C. Matrix operation of mechanisms with topological