# Patent Analysis and Structural Synthesis of Epicyclic Gear Trains Used in Automatic Transmissions
## 2 Representation of EGTs  
### Graph-Based Representation  
The topological graph in the graph theory can conveniently represent the topological structures of all kinds of kinematic chains. Based on the unified topological representation models proposed by [PERSON] et al. [16], the topological graph of EGTs can be obtained as follows:  
(1) Use solid vertices (\"\(\bullet\)\") and hollow vertices (\"\(\circ\)\") to denote links and multiple joints, respectively.  
(2) Use solid lines (\"--\") and dash lines (\"--\") to present revolute pairs and gear pairs, respectively.  
It should be indicated that topological graphs of EGTs are double bicolor graphs (DBGs) in the view of the graph theory.  
For an EGT, only coaxial links, namely central gears (i.e., sun and ring gears) and carriers, can be used as the input, output, or fixed members to obtain the desired speed ratios [29]. The coaxial links form a multiple joint based on the definition of multiple joint [30], which means that the degree of the hollow vertex is equal to the number of coaxial links. The number of actual revolute pairs at the multiple joint is one fewer than that of coaxial links. There are the following quantitative relationships between EGTs and corresponding DBGs.  
\[N_{l}+N_{m}=v \tag{1}\]  
\[N_{r}=N_{s}-N_{m} \tag{2}\]  
Figure 1: Some typical automatic transmission (AT) products. Note: (**a**) A four-gear AT, (**b**) A six-gear AT, (**c**) A six-gear AT, (**d**) An eight-gear AT, (**e**) A nine-gear AT, (**f**) A nine-gear AT.  
\[N_{g}=N_{d}, \tag{3}\]  
where \(N_{l}\), \(N_{m}\), \(N_{g}\), \(N_{r}\) denote the number of links, multiple joints, gear pairs and revolute pairs in an EGT respectively; \(v\), \(N_{s}\), \(N_{d}\) denote the number of vertices, solid lines and dash lines of corresponding DBG respectively.  
According to the Chebychev-Grubler-Kutzbach criterion for planar mechanisms [31], the DOF of EGT and corresponding DBG can be obtained.  
\[F_{E}=3(N_{l}-1)-2N_{r}-N_{g} \tag{4}\]  
\[F_{D}=3(v-1)-2N_{s}-N_{d}, \tag{5}\]  
where \(F_{E}\) denotes the DOF of EGT, \(F_{D}\) denotes the DOF of corresponding DBG.  
According to Equations (1)-(5), there is the following relationship between the DOF of EGT and the DOF of DBG.  
\[F_{E}=F_{D}-N_{m}. \tag{6}\]  
For the EGTs with one main shaft, there is only one multiple joint so that the DOF of corresponding DBGs is one bigger than that of EGTs. For example, the EGT shown in Figure 2a has six links (i.e., labels 2-7) and four of them (i.e., labels 2, 4, 6, 7) are coaxial links forming a multiple joint (i.e., label 1). There are five revolute pairs and four gear pairs in total. According to the Equation (4), the DOF of the EGT can be obtained.  
\[F_{E}=3\times(6-1)-2\times 5-4=1 \tag{7}\]  
According to the Equations (1)-(3), the corresponding DBG has six solid lines, four dash lines and seven vertices including one hollow vertex. Then, the DOF of the DBG can be obtained based on Equation (5).  
\[F_{D}=3\times(7-1)-2\times 6-4=2 \tag{8}\]  
It meets the relationship obtained in Equation (6). The corresponding seven-vertex two-DOF DBG is shown in Figure 2b. By transforming the dash lines into solid lines, DBGs can be transformed into multiple joint topological graphs, which have been synthesized by [PERSON] et al. [32]. In the present study, the multiple joint topological graphs are defined as the basic graphs of DBGs. Figure 2c illustrates the basic graph of the DBG shown in Figure 2b.