A New Tree Graph Method for Synthesizing Planetary Gear Trains of Vehicle Powertrains

Abstract

The function of vehicle powertrains is to transmit power from the power source (engine or electric machine) to the output shaft. Different torques and speeds of the output shaft are achieved through the structure of planetary gear train (PGT). The performance of a powertrain depends on the choice of topological structure of the PGT to a large extent. The topological synthesis of PGTs is of great significance to the design of new powertrains. A tree graph method for the topological synthesis of PGTs is proposed in this paper. Firstly, the rules for generating (N + 1)-link tree graphs from N-link tree graphs are proposed. Then, the method for adding geared edges into the tree graph is presented, and topological graphs of 1-DOF PGTs are directly synthesized from their tree graphs. A unified algorithm applicable for both open-loop and closed-loop graphs is proposed to eliminate isomorphic tree graphs and topological graphs of PGTs. The complete atlases of tree graphs and topological graphs of 1-DOF PGTs with three to ten links are obtained for the first time.

1. Introduction

The function of vehicle powertrains is to transmit power from the power source (engine or electric machine) to the output shaft [1,2]. Different torques and speeds of the output shaft are achieved through the structure of planetary gear train (PGT). For example, Hoang and Yan [3,4] and Ho and Hwang [5,6] applied different PGTs to design hybrid powertrains and motor powertrains. The performance of a powertrain depends on the choice of topological structure of the PGT to a large extent. The topological synthesis of PGTs, which aims to obtain all feasible PGT structures, is of great significance to the design of new powertrains.

The topological synthesis of PGTs has been greatly developed in the last few decades. Hsu et al. [7] presented a method to synthesize PGTs based on the acyclic graph, and a structural code derived from degrees of vertices and neighbors was presented to detect isomorphic PGTs. Schmidt et al. [8] presented a grammar adaptation and a linear time algorithm for the synthesis and isomorphism detection of PGTs. Del Castillo [9] presented a method to synthesize PGTs with one multiple joint and without idle links, and the evaluation of determinant was applied for isomorphism detection. Rao and coworkers [10,11,12] synthesized 1-DOF PGTs with up to seven links based on the recursive method, and presented the Hamming string-based method and genetic algorithm-based method for the isomorphism detection of PGTs. Yang et al. [13] studied an ant algorithm-based algorithm to resolve the detection of isomorphic PGTs. Reddy et al. [14] presented a review and comparison of characteristic polynomial, eigenvector and eigenvalue, and Hamming number-based methods for the isomorphism detection of PGTs. Brumercik et al. [15] presented a simple method for the synthesis of two-linked PGTs. Kamesh et al. [16,17] synthesized 1-DOF PGTs with up to six links based on the recursive method, and presented the vertex incidence polynomial-based method and vertex distance-based method for detecting isomorphic PGTs. Yang et al. [18,19] presented a perimeter loop-based isomorphism detection method, based on which 1-DOF and 2-DOF PGTs with up to nine links were synthesized via the parent graph method. Shanmukhasundaram et al. [20] studied the synthesis of 1-DOF PGTs with up to seven links based on the acyclic graph method and the concept of kinematic unit, and the spectral method using eigenvalues and eigenvectors was applied to detect isomorphic PGTs. Xu et al. [21] proposed a matrix operation-based method for detecting the structural isomorphism of PGTs. Ding and Cai [22] investigated the characteristics of PGTs used in automatic transmissions and studied the synthesis of PGTs based on the corresponding parent graphs. Cui et al. [23] studied the synthesis of topological graphs of 1-DOF PGTs from their parent graphs, and a kind of characteristic constant was constructed to detect isomorphism.

The key issue in the topological synthesis of PGTs is how to obtain all possible topological structures of PGTs. The literature review shows that there exist three popular methods for the generation of PGTs, namely, the parent graph-based method [18,19,22,23], the recursive method [10,12,16], and the acyclic graph-based method [7,20]. The parent graph-based method includes steps of solving the matrix equation of the parent graph, and the synthesis of parent graphs, geared graphs, rotation graphs, and displacement graphs of PGTs. The disadvantage of this method is that it is complex and inefficient. Our tests indicate that due to the enormous computational load, when the parent graph-based method is applied to synthesize 10-link PGTs, the synthesis software on a personal computer crashes and cannot run. The recursive method aims to generate (N + 1)-link PGTs from N-link PGTs by adding one link with a geared pair and a revolute pair. A fatal disadvantage of this method is that it is impossible to generate a complete list of topological structures of PGTs, because a part of PGTs cannot be generated via this method. With regard to the acyclic graph-based method, a polygon and a vertex are used to represent a multiple joint and a link, respectively. Polygons and vertices are handled in different ways, and it is difficult to avoid overlaps of edges and polygons when sketching the topological graph of PGT.

In this paper, a new tree graph method is proposed for the topological synthesis of PGTs, and a unified algorithm applicable for both open-loop and closed-loop graphs is proposed to detect isomorphism. The complete atlases of tree graphs and topological graphs of 1-DOF PGTs with three to ten links are obtained for the first time. The merits of the present synthesis method are summarized and shown in

Table 1. Merits of the present synthesis method.

Existing Methods	Disadvantages of Existing Methods	Comparison
Parent graph-based method [18,19,22,23]	This method is complex and inefficient, and includes steps of solving matrix equation of parent graph, and the synthesis of parent graphs, geared graphs, rotation graphs, and displacement graphs of PGTs. When this method is applied to synthesize 10-link PGTs, the synthesis software on a personal computer crashes and cannot run.	The present method is efficient. Topological graphs of PGTs are generated directly from their tree graphs. The complete atlas of topological graphs of 1-DOF PGTs with 3 to 10 links are successfully generated.
Recursive method [10,12,16]	It is impossible to generate a complete list of topological structures of PGTs, because a part of PGTs cannot be generated via this method.	The present method is effective. Complete atlas of PGTs can be synthesized without missing any feasible one.
Acyclic graph-based method [7,20]	Polygons and vertices are handled in different ways, and it is difficult to avoid overlaps of edges and polygons when sketching the topological graph of PGT.	The present method is simple. Both solid vertices and hollow vertices in the topological graph are handled in the same way. There is no polygon in the topological graph of PGT.

. The flowchart and novelty of the present method are illustrated in Figure 1. The reminder of this paper is arranged as follows. In Section 2, the topological graph of PGT, the corresponding tree graph, and their adjacency matrices are introduced. In Section 3, rules for generating (N + 1)-link tree graphs from N-link tree graphs are proposed, and a unified isomorphism detection method applicable for both open-loop and closed-loop graphs is proposed and applied to eliminate isomorphic tree graphs. The atlas of non-isomorphic tree graphs with three to ten links is obtained. In Section 4, the generation of topological graphs of 1-DOF PGTs from the atlas of tree graphs is developed, and the unified isomorphism detection method is applied to eliminate isomorphic topological graphs. The complete atlas of non-isomorphic 1-DOF topological graphs with three to ten links is successfully synthesized. In Section 5, the merits of the present synthesis method are discussed in detail. In Section 6, the conclusions of the present research work are summarized.

2. Topological Graph of PGT and the Corresponding Tree Graph

For convenience of computer processing to realize the automation of topological synthesis, the structure of PGT can be represented by its topological graph, where a solid vertex, a hollow vertex, a solid edge, and a dashed edge represent a link, a multiple joint, a revolute pair, and a geared pair, respectively. For example, Figure 2a shows the structure of a six-link 1-DOF PGT, where different links are rendered with different colors, and Figure 2b shows the functional diagram of Figure 2a. Similarly, Figure 2c shows the functional diagram of a seven-link 1-DOF PGT. Topological graphs corresponding to Figure 2b,c are shown in Figure 2d and Figure 2e, respectively. In Figure 2d, hollow vertex 7 represents a multiple joint, and solid vertices represent links. A d-degree hollow vertex is equivalent to d − 1 revolute pairs. Letters a, b, c … are used to distinguish the levels (or locations) of the corresponding axes of rotation. For example, the revolute edges (solid edges) in Figure 2d have three different levels, which are denoted as a, b, and c, respectively. The levels of revolute edges in a PGT graph have the following characteristic.

Characteristic 1: All the revolute edges incident with the same hollow vertex have the same level, and all the other revolute edges have different levels. For example, in Figure 2d, revolute edges e_2,7, e_3,7, e_5,7, and e_6,7 are incident with hollow vertex 7; hence, these four revolute edges have the same level a. Revolute edges e_1,2 and e_3,4 have different levels b and c. There is no hollow vertex in Figure 2e; hence, all revolute edges have different levels, which are denoted as a, b, c, d, e, and f, respectively.

The adjacency matrix of a PGT graph is defined as Equation (1), where n is the total number of vertices (including solid and hollow vertices), x_i,j is the element in ith row and jth column of the matrix, and d is the degree of a hollow vertex. For example, the adjacency matrices of Figure 2d and Figure 2e are shown in Figure 3a and Figure 3b, respectively. The subgraph obtained by deleting all the geared edges (dashed edges) in a PGT graph is a tree. For example, the tree graphs corresponding to Figure 2d and Figure 2e are shown in Figure 4a and Figure 4b, respectively. A tree is a connected graph without any loop. A tree graph with n vertices has the following characteristics: Any two distinct vertices are connected by one and only one path. It contains (n − 1) edges. Connecting any two non-adjacent vertices with an edge yields a unique loop.

The adjacency matrix of a tree graph is defined as Equation (2). For example, the adjacency matrices of Figure 4a and Figure 4b are shown in Figure 5a and Figure 5b, respectively. If not necessary, the levels of revolute edges in the tree graph and topological graph of PGT are not labeled below.

$${\left[x_{i,j}\right]}_{n\times n}=\left\{\begin{array}{l}1, \mathrm{i}\mathrm{f} \mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x} i \mathrm{i}\mathrm{s} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{t}\mathrm{o} \mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x} j \mathrm{b}\mathrm{y} \mathrm{a} \mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d} \mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}\\ 2, \mathrm{i}\mathrm{f} \mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x} i \mathrm{i}\mathrm{s} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{t}\mathrm{o} \mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x} j \mathrm{b}\mathrm{y} \mathrm{a} \mathrm{d}\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{d} \mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}\\ d, \mathrm{i}\mathrm{f} i=j \mathrm{a}\mathrm{n}\mathrm{d} i \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{l}\mathrm{a}\mathrm{b}\mathrm{e}\mathrm{l} \mathrm{o}\mathrm{f} \mathrm{a} \mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w} \mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x}\\ 0, \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$$

(1)

$${\left[x_{i,j}\right]}_{n\times n}=\left\{\begin{array}{l}1, \mathrm{i}\mathrm{f} \mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x} i \mathrm{i}\mathrm{s} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{t}\mathrm{o} \mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x} j\\ d, \mathrm{i}\mathrm{f} i=j \mathrm{a}\mathrm{n}\mathrm{d} i \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{l}\mathrm{a}\mathrm{b}\mathrm{e}\mathrm{l} \mathrm{o}\mathrm{f} \mathrm{a} \mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w} \mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x}\\ 0, \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$$

(2)

3. Topological Synthesis of Tree Graphs

The topological synthesis of tree graphs is the primary and crucial task of topological synthesis of PGTs. A method for the synthesis of tree graphs in a recursive way is proposed. The main steps of this method are illustrated in Figure 1, and the detailed process is explained as follows.

3.1. Rules for Generation of Tree Graphs

An (N + 1)-link (solid vertex) tree graph can be generated by adding a solid vertex with a revolute edge into an N-link tree graph. Based on the Characteristic 1 introduced in Section 2, the rules for generating (N + 1)-link tree graphs from N-link tree graphs are proposed in this section. According to the level of the newly added revolute edge, the generation of (N + 1)-link tree graphs can be divided into the following two cases.

Case 1. The level of the added revolute edge is different from the levels of all revolute edges in the given N-link tree graph.

According to Characteristic 1, all the revolute edges incident with the same hollow vertex have the same level. Therefore, the connection method of the added revolute edge satisfies the following rule: The added revolute edge can be connected to any solid vertex, but not to any hollow vertex. Obviously, the number x₁ of (N + 1)-link tree graphs generated in Case 1 is equal to the number N of solid vertices in the given N-link tree graph, namely,

x₁ = N

(3)

For example, the six-link tree graph in Figure 4a contains six solid vertices, from which six seven-link tree graphs can be generated, as shown in Figure 6. The newly added vertex 8 and its incident revolute edge are connected to solid vertices 1, 2, 3, 4, 5, and 6, respectively.

Case 2. The level of the added revolute edge is the same as one of the levels of revolute edges in the given N-link tree graph.

The level of the added revolute edge has two possibilities: (a) The level of the newly added revolute edge is the same as the level of revolute edges which are incident with a hollow vertex (denoted as vertex k). (b) The level of the newly added revolute edge is the same as the level of a revolute edge (denoted as edge e_i,j) which is not incident with any hollow vertex. For possibility (a), the added revolute edge is directly connected to hollow vertex k. For possibility (b), edge e_i,j is deleted and then vertices i, j and the added vertex are connected to a same new hollow vertex. Obviously, the number x₂ of (N + 1)-link tree graphs generated in Case 2 is equal to the number L of levels of revolute edges in the given N-link tree graph, namely,

x₂ = L

(4)

For example, the six-link tree graph in Figure 4a contains three edge levels, from which three seven-link tree graphs can be generated, as shown in Figure 7. When the level of the newly added revolute edge is the same as the level (level a) of revolute edges which are incident with hollow vertex 7, the newly added revolute edge is directly connected to hollow vertex 7, as shown in Figure 7a. When the level of the newly added revolute edge is the same as the level (level b) of revolute edge e_1,2, edge e_1,2 is deleted and vertices 1, 2 and the added vertex 8 are connected to the new hollow vertex 9, as shown in Figure 7b. Similarly, when the level of the newly added revolute edge is the same as the level (level c) of revolute edge e_3,4, the generated seven-link tree graph is shown in Figure 7c.

The number of (N + 1)-link tree graphs generated from a given N-link tree graph is equal to the sum of tree graphs generated in the above two cases, namely,

x = x₁ + x₂ = N + L

(5)

For example, in the six-link tree graph shown in Figure 4a, the number N of solid vertices and the number L of edge levels are 6 and 3, respectively. Therefore, the number of seven-link tree graphs generated from Figure 4a is x = N + L = 6 + 3 = 9, as shown in Figure 6 and Figure 7.

The process of generating eight-link tree graphs from the seven-link tree graph without any hollow vertex shown in Figure 4b is briefly illustrated as follows. There are seven solid vertices in Figure 4b; hence, seven eight-link tree graphs can be generated in Case 1. The newly added vertex 8 and its incident revolute edge are connected to solid vertices 1, 2, 3, 4, 5, 6, and 7, respectively, as shown in Figure 8. There are six different edge levels in Figure 4b; hence, six eight-link tree graphs can be generated in Case 2, as shown in Figure 9. The level of the newly added revolute edge is the same as edge levels a, b, c, d, e, and f in Figure 4b, respectively. The total number of eight-link tree graphs generated from the seven-link tree graph in Figure 4b is x =N + L = 7 + 6 = 13.

All feasible tree graphs can be generated recursively. As is well known, there exists only one three-link 1-DOF PGT; hence, there is only one three-link tree graph. All possible four-link tree graphs can be generated from this three-link tree graph, as shown in Figure 10. These four-link tree graphs can be used to derive all possible five-link tree graphs, and these five-link tree graphs can be used to derive all possible six-link tree graphs, and so on. For example, Figure 11 and Figure 12 show all possible (N + 1)-link tree graphs which are generated from an N-link tree graph (N equals 4 and 8, respectively).

3.2. A Unified Isomorphism Detection Algorithm and the Atlas of Tree Graphs

When the number of links exceeds five, PGTs and their tree graphs have complex topological structures, and the number of isomorphic graphs is very large. Due to inaccuracy and inefficiency, manual isomorphism detection cannot be used for the synthesis of PGTs. Therefore, it is necessary to study an isomorphism detection algorithm that can be easily automated by computer. Most of the existing isomorphism detection algorithms are only applicable for closed-loop graphs, but not for open-loop graphs. Here, a unified isomorphism detection algorithm applicable for both open-loop and closed-loop graphs is proposed.

The topological graphs of PGTs derived from isomorphic tree graphs are also isomorphic. Therefore, removing isomorphic tree graphs can greatly reduce the number of isomorphic topological graphs of PGTs and improve the efficiency of topological synthesis of PGTs. In this section, the proposed isomorphism detection algorithm is used in the process of generating tree graphs to obtain the atlas of non-isomorphic tree graphs. The tree graph in Figure 6a is used as an example to explain the isomorphism detection algorithm step by step.

Step 1. Determine all the vertices with degree larger than one. For example, the degrees of vertices 1 to 8 in Figure 6a are 2, 2, 2, 1, 1, 1, 4, and 1, respectively. Vertices 1, 2, 3, and 7 have a degree larger than one.

Step 2. Determine the longest path. Among the paths consisting of vertices with degree larger than one, the path with the largest number of vertices is the longest path. If there exists more than one path with the largest number of vertices, all these paths are selected as the longest paths. For example, both paths 1-2-7-3 and 3-7-2-1 in Figure 6a are the longest paths, as shown in Figure 13.

Step 3. Determine the main path. Degrees of vertices on the longest path are concatenated to form a vertex degree string, and the path with the largest vertex degree string is defined as the main path. If there exists more than one path with the largest vertex degree string, all these paths are selected as the main paths. For example, the degrees of vertices 1, 2, 3, and 7 in Figure 6a are 2, 2, 2, and 4, respectively. Vertex degree strings of the longest paths 1-2-7-3 and 3-7-2-1 are 2242 and 2422, respectively. Obviously, the longest path 3-7-2-1 is the main path.

Step 4. Determine the labels of vertices on the main path. The previous labels of vertices in the tree graph are removed, and the m vertices on the main path are relabeled as 1, 2, 3 … m in order. For example, vertices 3, 7, 2, and 1 on the main path 3-7-2-1 in Figure 6a are relabeled as 1, 2, 3, and 4, respectively, as shown in Figure 14.

Step 5. Determine the labels of vertices not on the main path. First, for each vertex V_x not on the main path, its corresponding path P_x is determined. The path begins at vertex V_x and ends at vertex V_s on the main path closest to vertex V_x. The number of vertices on path P_x is denoted as t, and the label of vertex V_x is determined as V_s + t − 1.

In order to clearly explain the above concept, the path P_x corresponding to vertex V_x in an example 10-link tree graph is shown in Figure 15a. This path begins at the vertex V_x and ends at vertex 3 on the main path. The number of vertices on this path is t = 3. Therefore, the label of vertex V_x is V_s + t − 1 = 3 + 3 − 1 = 5, as shown in Figure 15b. Continuing the example of Figure 14, paths P₁, P₂, P₃, and P₄ corresponding to vertices V₁, V₂, V₃, and V₄ not on the main path are shown in Figure 16a. Path P₁ begins at vertex V₁ and ends at vertex 1 on the main path. The number of vertices on path P₁ is t = 2. Therefore, the label of vertex V₁ is V_s + t − 1 = 1 + 2 − 1 = 2. Similarly, the labels of vertices V₂, V₃, and V₄ are 3, 3, and 5, respectively, as shown in Figure 16b.

Step 6. Determine the codes of vertices in the given tree graph. For each vertex V in a tree graph, its code is formed by adding a suffix to its label. The suffix consists of three numbers enclosed in a parenthesis. The first number is the label of vertex on the main path closest to vertex V. If vertex V is on the main path, the first number is 0. The second number is the number of vertices in the sub-tree containing vertex V. If vertex V is on the main path, the second number is 0. The third number is the degree of vertex V.

Definition of a sub-tree. A sub-tree is a tree consisting of the vertices not on the main path and the edges connecting these vertices, and this tree is connected to a vertex on the main path. For Figure 15a, the three sub-trees ST₁, ST₂, and ST₃ connected to vertices 1, 3, and 6 on the main path are marked with dashed boxes and shown in Figure 15c. Sub-trees ST₁ and ST₃ are the special trees containing only one vertex, and sub-tree ST₂ contains three vertices. For Figure 16b, four sub-trees ST₁, ST₂, ST₃, and ST₄ connected to vertices 1, 2, and 4 on the main path are marked with dashed boxes and shown in Figure 16c, and each sub-tree contains only one vertex.

Codes of vertices in Figure 16b are obtained as follows. For vertex 1 on the main path in Figure 16b, the first and second numbers of the suffix of its code are 0. The degree of vertex 1 is 2; hence, the third number of the suffix is 2. The code of vertex 1 is obtained as 1(002). Similarly, the codes of vertices 2, 3, and 4 on the main path in Figure 16b are 2(004), 3(002), and 4(002), respectively. There are two vertices 3 not on the main path in Figure 16b, and both are connected to vertex 2 on the main path; hence, the first number of the suffix is 2. These two vertices 3 are in sub-trees ST₂ and ST₃ containing only one vertex; hence, the second number of the suffix is 1. The degree of these two vertices 3 is 1; hence, the third number of the suffix is 1. The codes of these two vertices 3 are obtained as 3(211). Similarly, the codes of vertices 2 and 5 not on the main path in Figure 16b are 2(111) and 5(411), respectively. The codes of all the vertices in Figure 16b are shown in Figure 17.

Step 7. Relabel the vertices in the given tree graph. According to the ascending order of vertex codes, n vertices in the tree graph are relabeled as 1, 2, 3 … n in sequence. If there exists more than one vertex with the same code, these vertices are permuted to obtain all possible ways of relabeling. For example, the codes of vertices in Figure 17 are arranged in ascending order as 1(002), 2(004), 2(111), 3(002), 3(211), 3(211), 4(002) and 5(411). Two of these vertex codes are 3(211); hence, the number of ways to relabel vertices in Figure 17 is $A_2^2=2$, as shown in Figure 18.

Step 8. Determine the characteristic code (c-code) of the given tree graph and carry out the task of isomorphism detection. The numerical code formed by concatenating the upper triangular elements (including diagonal elements) in the adjacency matrix of a relabeled tree graph is defined as its tree-code, and the largest tree-code is defined as the c-code of the given tree graph. The detection of isomorphic tree graphs is carried out by comparing their c-codes. Isomorphic tree graphs have the same c-code, and non-isomorphic tree graphs have different c-codes.

For example, the process of obtaining the tree-code of Figure 18a is shown in Figure 19, and the derived tree-code of Figure 18a is C_a = 01100000-4011100-000000-00010-0000-000-01-0. Similarly, the tree-code of Figure 18b can be acquired as C_b = 01100000-4011100-000000-00010-0000-000-01-0. By definition, the larger tree-code is the c-code of the tree graph in Figure 6a. Because C_a is the same as C_b, tree-code C_a = 01100000-4011100-000000-00010-0000-000-01-0 is taken as the c-code of Figure 6a. Similarly, c-codes of all the tree graphs in Figure 6 and Figure 7 are determined and shown in

Table 2. C-codes of the tree graphs in Figure 6 and Figure 7.

Figures	C-Codes
Figure 6a,d	01100000-4011100-000000-00010-0000-000-01-0
Figure 6b,c	01110000-4001110-000000-00000-0001-000-00-0
Figure 6e,f	01100000-4011100-000000-00001-0000-010-00-0
Figure 7a	01100000-5011110-000000-00001-0000-000-00-0
Figure 7b,c	311100000-00010000-0000000-000000-41110-0001-000-00-0

. The c-codes of the tree graphs in Figure 6a,d are the same; hence, the tree graphs in Figure 6a,d are isomorphic. Similarly, the tree graphs in Figure 6b and c are isomorphic, the tree graphs in Figure 6e,f are isomorphic, and the tree graphs in Figure 7b,c are isomorphic. Therefore, only five of the nine tree graphs in Figure 6 and Figure 7 are non-isomorphic, which means that five non-isomorphic seven-link tree graphs can be generated from the six-link tree graph in Figure 4a.

The present isomorphism detection algorithm is also applicable for tree graphs without any hollow vertex. Taking Figure 8a for instance, vertices 1, 2, 3, and 7 have the degree larger than one; hence, six longest paths 1-2-3, 1-2-7, 3-2-1, 3-2-7, 7-2-1, and 7-2-3 can be acquired, as shown in Figure 20. The vertex degree strings of paths 3-2-1 and 3-2-7 are both 332, which is larger than those of the other paths. Hence, paths 3-2-1 and 3-2-7 are determined as the main paths. For the main path 3-2-1, vertices 3, 2, 1 are labeled as 1, 2, 3, respectively, as shown in Figure 21a. Similarly, the labels of vertices on the main path 3-2-7 are shown in Figure 21b. Paths P₁, P₂, P₃, P₄, and P₅ corresponding to vertices V₁, V₂, V₃, V₄, and V₅ not on the main path in Figure 21a are shown in Figure 22a. Taking path P₃ for instance, path P₃ begins at vertex V₃ and ends at vertex 2 on the main path, and the number of vertices on path P₃ is t = 3. Therefore, the label of vertex V₃ is V_s + t − 1 = 2 + 3 − 1 = 4. Similarly, the labels of vertices V₁, V₂, V₄ and V₅ in Figure 22a are 2, 2, 3, and 4, respectively, as shown in Figure 22b, and the labels of vertices not on the main path in Figure 21b are shown in Figure 22c.

For vertex 1 on the main path in Figure 22b, the first and second numbers of the suffix of its code are 0. The degree of vertex 1 is 3; hence, the third number of the suffix is 3. The code of vertex 1 is obtained as 1(003). Similarly, the codes of vertices 2 and 3 on the main path are 2(003) and 3(002), respectively. The four sub-trees ST₁, ST₂, ST₃, and ST₄ in Figure 22b are shown in Figure 23a. Taking vertex 3 not on the main path in Figure 22b for instance, it is connected to vertex 2 on the main path; hence, the first number of the suffix of its code is 2. Sub-tree ST₃ contains two vertices; hence, the second number of the suffix is 2. The degree of vertex 3 is 2; hence, the third number of the suffix is 2. Therefore, the code of vertex 3 not on the main path in Figure 22b is 3(222). Similarly, the codes of all the vertices in Figure 22b can be determined, as shown in Figure 23b. The codes of all the vertices in Figure 22c are shown in Figure 23c. The codes of vertices in Figure 23b are arranged in ascending order as 1(003), 2(003), 2(111), 2(111), 3(002), 3(222), 4(221), and 4(311). Two of these vertex codes are 2(111); hence, the number of ways to relabel vertices in Figure 23b is $A_2^2=2$, as shown in Figure 24a,b. Similarly, the number of ways to relabel vertices in Figure 23c is also $A_2^2=2$, as shown in Figure 24c,d. All the tree-codes of the tree graphs in Figure 24 are 01110000-0001100-000000-00000- 0001-010-00-0; hence, the c-code of the tree graph in Figure 8a is acquired as 01110000-0001100-000000-00000-0001-010-00-0.

Similarly, the c-codes of all the tree graphs in Figure 8 and Figure 9 can be acquired, as shown in

Table 3. C-codes of the tree graphs in Figure 8 and Figure 9.

Figures	C-Codes
Figure 8a	01110000-0001100-000000-00000-0001-010-00-0
Figure 8b	01110000-0001110-000000-00000-0001-000-00-0
Figure 8c	01111000-0000110-000000-00000-0000-001-00-0
Figure 8d,e	01100000-0011000-000000-00110-0000-001-00-0
Figure 8f	01110000-0001100-000000-00000-0010-000-01-0
Figure 8g	01110000-0001100-000000-00000-0011-000-00-0
Figure 9a	311100000-00011000-0000000-000000-00011-0100-000-00-0
Figure 9b	011100000-30011000-0000000-000000-00110-0000-001-00-0
Figure 9c	011100000-00011000-0000000-000000-30110-0000-001-00-0
Figure 9d,e	311100000-00011000-0000000-000000-00110-0000-001-00-0
Figure 9f	011100000-00011000-0000000-000000-00100-0000-311-00-0

. According to the data in this table, the tree graphs in Figure 8d,e are isomorphic and the tree graphs in Figure 9d,e are isomorphic. Therefore, eleven of the thirteen tree graphs in Figure 8 and Figure 9 are non-isomorphic, which means that eleven non-isomorphic eight-link tree graphs can be generated from the seven-link tree graph in Figure 4b.

(N + 1)-link tree graphs generated from isomorphic N-link tree graphs are also isomorphic. All isomorphic tree graphs can be eliminated by applying the present isomorphism detection algorithm to the process of generating tree graphs in Section 3.1, and the atlas of non-isomorphic tree graphs can be obtained. As is well known, there exists only one three-link tree graph, from which five four-link tree graphs can be generated, as shown in Figure 10. After eliminating isomorphism, three non-isomorphic four-link tree graphs can be obtained, as shown in Figure 25. All feasible five-link tree graphs can be generated from these non-isomorphic four-link tree graphs, and eight non-isomorphic five-link tree graphs can be obtained after eliminating isomorphism, as shown in Figure 26. All feasible six-link tree graphs can be generated from these non-isomorphic five-link tree graphs, and 21 non-isomorphic six-link tree graphs can be obtained after eliminating isomorphism, as shown in Figure 27. Similarly, all non-isomorphic seven-, eight-, nine-, and ten-link tree graphs can be obtained in turn. For example, part of the non-isomorphic 10-link tree graphs are shown in Figure 28. We have developed a computer program to automatically synthesize the compete atlas of tree graphs. The synthesis results of tree graphs are shown in

Table 4. Synthesis result of tree graphs.

No. of Links	3	4	5	6	7	8	9	10
No. of Tree Graphs	1	3	8	21	58	164	495	1539

. Because the adjacency matrix of a tree graph is symmetric, only the upper triangular elements of the adjacency matrices of tree graphs are stored in the database. The complete database of tree graphs with three to ten links is available at Mendeley Data. Readers can download the database by visiting the URL “https://doi.org/10.17632/kt77nz942x.1”.

4. Topological Synthesis of 1-DOF PGTs

A method for synthesizing topological graphs of 1-DOF PGTs directly from the atlas of non-isomorphic tree graphs is proposed. The main steps of this method are illustrated in Figure 1, and the detailed process is explained as follows.

4.1. Generation of Topological Graphs of PGTs Based on the Atlas of Tree Graphs

An N-link F-DOF PGT contains N-1 revolute pairs and N-1-F geared pairs. In particular, an N-link 1-DOF PGT contains N-2 geared pairs. Therefore, topological graphs of N-link 1-DOF PGTs can be obtained by adding N-2 geared edges into an N-link tree graph. The process of generating topological graphs of 1-DOF PGTs is as follows.

Step 1. For a given tree graph, determine all possible solid vertex pairs between which a geared edge can be added. The fundamental loop of a PGT graph has the following characteristic.

Characteristic 2. A unique fundamental loop can be obtained by adding a geared edge into a tree graph, and a fundamental loop contains two different edge levels.

According to Characteristic 2, a reasonable solid vertex pair is defined as follows. If the path connecting two solid vertices in a tree graph contains two different edge levels, these two solid vertices form a reasonable solid vertex pair. Take the eight-link tree graph in Figure 8a for instance; the labels of edge levels are shown in Figure 29. The path connecting solid vertices 1 and 3 in Figure 29 contains two different edge levels, namely, levels a and b; hence, vertices 1 and 3 form a reasonable solid vertex pair, denoted as (1, 3). The loop acquired by adding a geared edge between vertices 1 and 3 is shown in Figure 30a, and this loop satisfies Characteristic 2. Similarly, vertices 3 and 7 also form a reasonable solid vertex pair, denoted as (3, 7). The loop acquired by adding a geared edge between vertices 3 and 7 is shown in Figure 30b. The path connecting solid vertices 1 and 4 in Figure 29 contains three edge levels, namely, levels a, b, and e. The loop acquired by adding a geared edge between vertices 1 and 4 is shown in Figure 30c. This loop does not satisfy Characteristic 2; hence, vertices 1 and 4 do not form a reasonable solid vertex pair. Finally, eight reasonable solid vertex pairs in Figure 29 can be acquired, namely, (1, 3), (1, 7), (2, 4), (2, 5), (2, 6), (2, 8), (3, 7), and (4, 5).

Step 2. Select N-2 reasonable solid vertex pairs, and add a geared edge between each reasonable solid vertex pair to obtain topological graphs of PGTs. The number of reasonable solid vertex pairs in the tree graph is denoted as K, and there are ${\mathrm{C}}_K^{N-2}$ choices of N-2 reasonable solid vertex pairs, which means that the number of derived topological graphs is ${\mathrm{C}}_K^{N-2}$. For example, the number of reasonable solid vertex pairs in Figure 29 is K = 8, and the number of solid vertices is N = 8; hence, the number of choices of N-2 solid vertex pairs is ${\mathrm{C}}_K^{N-2}={ \mathrm{C}}_8^6=28$, as shown in

Table 5. Choices of solid vertex pairs in Figure 29.

Serial No.	Choice of Solid Vertex Pairs	Serial No.	Choice of Solid Vertex Pairs
1	(1, 3), (1, 7), (2, 4), (2, 5), (2, 6), (2, 8)	15	(1, 3), (1, 7), (2, 4), (2, 5), (2, 8), (4, 5)
2	(1, 3), (1, 7), (2, 4), (2, 6), (2, 8), (4, 5)	16	(1, 3), (1, 7), (2, 4), (2, 5), (3, 7), (4, 5)
3	(1, 3), (1, 7), (2, 5), (2, 6), (2, 8), (4, 5)	17	(1, 3), (1, 7), (2, 4), (2, 6), (3, 7), (4, 5)
4	(1, 3), (2, 4), (2, 5), (2, 6), (2, 8), (3, 7)	18	(1, 3), (1, 7), (2, 5), (2, 6), (3, 7), (4, 5)
5	(1, 3), (2, 4), (2, 6), (2, 8), (3, 7), (4, 5)	19	(1, 3), (2, 4), (2, 5), (2, 6), (3, 7), (4, 5)
6	(1, 3), (2, 5), (2, 6), (2, 8), (3, 7), (4, 5)	20	(1, 7), (2, 4), (2, 5), (2, 6), (3, 7), (4, 5)
7	(1, 7), (2, 4), (2, 5), (2, 6), (2, 8), (3, 7)	21	(1, 3), (1, 7), (2, 4), (2, 8), (3, 7), (4, 5)
8	(1, 7), (2, 4), (2, 6), (2, 8), (3, 7), (4, 5)	22	(1, 3), (1, 7), (2, 5), (2, 8), (3, 7), (4, 5)
9	(1, 7), (2, 5), (2, 6), (2, 8), (3, 7), (4, 5)	23	(1, 3), (2, 4), (2, 5), (2, 8), (3, 7), (4, 5)
10	(1, 3), (1, 7), (2, 4), (2, 5), (2, 6), (3, 7)	24	(1, 7), (2, 4), (2, 5), (2, 8), (3, 7), (4, 5)
11	(1, 3), (1, 7), (2, 4), (2, 5), (2, 8), (3, 7)	25	(1, 3), (2, 4), (2, 5), (2, 6), (2, 8), (4, 5)
12	(1, 3), (1, 7), (2, 4), (2, 6), (2, 8), (3, 7)	26	(1, 7), (2, 4), (2, 5), (2, 6), (2, 8), (4, 5)
13	(1, 3), (1, 7), (2, 5), (2, 6), (2, 8), (3, 7)	27	(1, 3), (1, 7), (2, 6), (2, 8), (3, 7), (4, 5)
14	(1, 3), (1, 7), (2, 4), (2, 5), (2, 8), (4, 5)	28	(2, 4), (2, 5), (2, 6), (2, 8), (3, 7), (4, 5)

. The corresponding 28 topological graphs are shown in Figure 31. Taking the first choice (1, 3), (1, 7), (2, 4), (2, 5), (2, 6), (2, 8) in Table 5 for instance, a geared edge is added between solid vertices 1 and 3, 1 and 7, 2 and 4, 2 and 5, 2 and 6, and 2 and 8, respectively. The corresponding topological graph is the first graph in Figure 31.

Step 3. Delete unreasonable topological graphs of PGTs. There are two cases of unreasonable topological graphs.

Case 1. The topological graph contains any pendent vertex, namely, one-degree vertex. The pendent vertex is not in any of the fundamental loops; hence, this kind of graph does not satisfy the essential requirements of PGT. For example, the 10th graph in Figure 31 contains one pendent vertex, namely, vertex 8; hence, this graph is unreasonable. Similarly, the 11th to 24th topological graphs in Figure 31 are also unreasonable.

Case 2. The topological graph contains any rigid sub-chain. A rigid sub-chain, whose DOF is less than or equal to zero, is a part of the PGT. The links in a rigid sub-chain do not have relative motion. Rigid sub-chains can be detected effectively via the method in Ref. [24]. For example, the rigid sub-chain in the 25th graph in Figure 31 is shown in Figure 32a. This sub-chain contains four solid vertices, three revolute edges, and three geared edges. The DOF formula of a PGT is

F = 3(N − 1) − 2R − G

(6)

where N, R, and G are the numbers of links, revolute pairs, and geared pairs, respectively. Hence, the DOF of the sub-chain in Figure 32a is F = 3(N − 1) − 2R − G = 3 × (4 − 1) − 2 × 3 − 3 = 0. Similarly, the rigid sub-chains in the 26th to 28th topological graphs in Figure 31 are shown in Figure 32b, Figure 32c, and Figure 32d, respectively.

Therefore, only nine of the twenty-eight topological graphs in Figure 31 are reasonable, namely, the first to ninth topological graphs in Figure 31. This means that nine reasonable topological graphs of eight-link 1-DOF PGTs can be generated from the eight-link tree graph in Figure 8a.

4.2. Isomorphism Detection and Atlas of Topological Graphs of PGTs

Isomorphic PGTs have a repetitive topological structure and obviously have the same characteristic and function. If two PGTs are isomorphic, one can be completely replaced by the other. Therefore, isomorphism must be detected to avoid duplication and guarantee the uniqueness of each PGT. Topological graphs of PGTs generated from different tree graphs are obviously non-isomorphic; hence, it is only necessary to detect whether the topological graphs generated from the same tree graph are isomorphic. The isomorphism detection algorithm proposed in Section 3.2 is also applicable for topological graphs of PGTs, and the process of isomorphism detection is briefly described as follows.

Step 1. Determine the new labels of vertices in the tree graph corresponding to the given topological graph by applying steps 1 to 5 in Section 3.2, and obtain the relabeled topological graph. Taking the first topological graph in Figure 31, for instance, its corresponding tree graph is shown in Figure 8a. The two ways of relabeling of vertices in this tree graph are shown in Figure 22b,c. Then, geared edges in the first topological graph in Figure 31 are added into these two tree graphs, and the relabeled topological graphs are obtained and shown in Figure 33a,b.

Step 2. Similar to step 6 in Section 3.2, determine the codes of vertices in the relabeled topological graph. For each vertex V in a topological graph, its code is formed by adding a suffix to its label. The suffix consists of five numbers enclosed in a parenthesis. The first number is the label of vertex on the main path closest to vertex V. If vertex V is on the main path, the first number is 0. The second number is the number of vertices in the sub-tree containing vertex V. If vertex V is on the main path, the second number is 0. The third number is the total number of geared edges incident with those vertices in the sub-tree containing vertex V. If vertex V is on the main path, the third number is 0. The fourth number is the degree of vertex V. The fifth number is the total number of geared edges incident with those vertices which are connected to vertex V by geared edges.

For example, the four sub-trees in Figure 33a are shown in Figure 34a. For vertex 2 on the main path in Figure 33a, the first, second, and third numbers of the suffix of its code are 0. The degree of vertex 2 on the main path is 7; hence, the fourth number of the suffix is 7. Vertex 2 on the main path is connected to two vertices 2 and two vertices 4 not on the main path by geared edges, and the total number of geared edges incident with these four vertices is 4; hence, the fifth number of the suffix is 4. The code of vertex 2 on the main path in Figure 33a is obtained as 2(00074). For vertex 3 not on the main path in Figure 33a, it is connected to vertex 2 on the main path; hence, the first number of the suffix is 2. There are two vertices in sub-tree ST₃ containing vertex 3 not on the main path; hence, the second number of the suffix is 2. The number of geared edges incident with vertices 3 and 4 in sub-tree ST₃ is 2; hence, the third number of the suffix is 2. The degree of vertex 3 not on the main path is 3; hence, the fourth number of the suffix is 3. Vertex 3 not on the main path is connected to vertex 3 on the main path by a geared edge, and the number of geared edges incident with vertex 3 on the main path is 2; hence, the fifth number of the suffix is 2. The code of vertex 3 not on the main path in Figure 33a is obtained as 3(22232). Similarly, the codes of vertices in Figure 33a and Figure 33b are shown in Figure 34b and Figure 34c, respectively.

Step 3. Similar to steps 7 and 8 in Section 3.2, relabel all vertices in the topological graph and determine the c-code of the given topological graph. According to the ascending order of vertex codes, n vertices in the topological graph are relabeled as 1, 2, 3 … n in sequence. If there exists more than one vertex with the same code, these vertices are permuted to obtain all possible ways of relabeling. The numerical code formed by concatenating the upper triangular elements (including diagonal elements) in the adjacency matrix of a relabeled topological graph is defined as its code, and the largest code of relabeled topological graph is defined as the c-code of the given topological graph. The detection of isomorphic topological graphs is carried out by comparing their c-codes. Isomorphic topological graphs have the same c-code, and non-isomorphic topological graphs have different c-codes.

For example, the codes of vertices in Figure 34b are arranged in ascending order as 1(00042), 2(00074), 2(11124), 2(11124), 3(00042), 3(22232), 4(22224), 4(31124). Two of these vertex codes are 2(11124); hence, the number of ways to relabel vertices in Figure 34b is $A_2^2=2$, as shown in Figure 35a and b. Similarly, the number of ways to relabel vertices in Figure 34c is also 2, as shown in Figure 35c and d. The codes of Figure 35a,b are the same, namely, C_a = C_b = 01112000-0221122-000000-00000-0201-010-00-0. The codes of Figure 35c,d are the same, namely, C_c = C_d = 01110200-0221122-000000-00000-0201- 010-00-0. The value of C_a (or C_b) is larger than C_c and C_d; hence, C_a is the c-code of the first topological graph in Figure 31, namely, 01112000-0221122-000000-00000-0201-010-00-0.

The process of acquiring the c-code of the second topological graph in Figure 31 is briefly described as follows. The tree graph corresponding to this topological graph is also Figure 8a. The relabeling of vertices in the tree graph is shown in Figure 22b,c. Geared edges in the second topological graph in Figure 31 are added into these two tree graphs, and the relabeled topological graphs are obtained and shown in Figure 36a,b. The codes of vertices in these two topological graphs are shown in Figure 37. All vertex codes in Figure 37a are different; hence, there is only one way of relabeling Figure 37a, as shown in Figure 38a. There is also only one way of relabeling Figure 37b, as shown in Figure 38b. The code of Figure 38a is C_a = 01112000-0021122-020000-00000-0201-010-00-0. The code of Figure 38b is C_b = 01110200-0021122-020000-00000-0201-010-00-0. Since C_a is larger than C_b, C_a is the c-code of the second topological graph in Figure 31, namely, 01112000-0021122-020000-00000-0201-010-00-0.

The c-code of each topological graph can be determined via the above method. C-codes of the first to ninth topological graphs in Figure 31 are shown in

Table 6. C-codes of the nine reasonable topological graphs in Figure 31.

Figures	C-Codes
the 1st and 7th topological graphs	01112000-0221122-000000-00000-0201-010-00-0
the 2nd, 3rd, 8th, and 9th topological graphs	01112000-0021122-020000-00000-0201-010-00-0
the 4th topological graphs	01112200-0221122-000000-00000-0001-010-00-0
the 5th and 6th topological graphs	01112200-0021122-020000-00000-0001-010-00-0

. According to Table 6, the first and seventh topological graphs in Figure 31 are isomorphic, the second, third, eighth, and ninth topological graphs are isomorphic, and the fifth and sixth topological graphs are isomorphic. Therefore, only four of the nine reasonable topological graphs in Figure 31 are non-isomorphic. This means that only four non-isomorphic topological graphs of eight-link 1-DOF PGTs can be generated from the eight-link tree graph in Figure 8a.

Based on the atlas of N-link non-isomorphic tree graphs acquired in Section 3.2, all reasonable topological graphs of N-link 1-DOF PGTs can be derived. Then, isomorphic topological graphs can be eliminated via the present isomorphism detection algorithm, and the complete atlas of non-isomorphic topological graphs can be synthesized. The atlases of non-isomorphic topological graphs of four- and five-link 1-DOF PGTs are shown in Figure 39 and Figure 40, respectively. A part of non-isomorphic topological graphs of six- and ten-link 1-DOF PGTs are shown in Figure 41 and Figure 42, respectively. We have developed a computer program to automatically synthesize the compete atlas of topological graphs of PGTs. The upper triangular elements of all the adjacency matrices of three- to ten-link 1-DOF PGT graphs are stored in the database, which is available at Mendeley Data. Readers can download the database by visiting the URL “https://doi.org/10.17632/kt77nz942x.1”. Each topological graph corresponds to a unique topological structure of PGT. The structure representations of example six-link 1-DOF PGTs derived from their topological graphs are illustrated in Figure 43. In this study, the complete atlases of tree graphs and topological graphs of 1-DOF PGTs with three to ten links are obtained for the first time. Our synthesis results of topological graphs of 1-DOF PGTs are listed in

Table 7. Synthesis results of 1-DOF PGTs.

No. of Links	3	4	5	6	7	8	9	10
No. of Topological Graphs	1	3	13	81	647	6360	71,833	904,208

.

5. Merits of the Present Synthesis Method

In the literature, there exist three popular methods for the generation of PGTs, namely, the parent graph-based method [18,19,22,23], the recursive method [10,12,16], and the acyclic graph-based method [7,20]. The present method for generating topological graphs of PGTs is efficient and effective. N-link topological graphs are directly generated from their corresponding N-link tree graphs. Compared to the parent graph-based method and acyclic graph-based method, the present method has the advantages of simplicity and efficiency. In addition, compared to the recursive method, the present method has the advantage of effectiveness. A complete atlas of PGTs can be successfully generated without missing any feasible one. The PGTs suitable for designing vehicle powertrains can also be synthesized via the present method. For example, this type of PGT includes the famous Ravigneaux PGT and Wolfrom PGT [6,25], as shown in Figure 44a,b. The corresponding topological graphs shown in Figure 44c,d are included in our database of PGTs.

Furthermore, the present isomorphism detection algorithm is efficient, reliable, and versatile. Firstly, topological graphs of PGTs generated from different tree graphs are obviously non-isomorphic; hence, it is only necessary to detect whether the topological graphs generated from the same tree graph are isomorphic. Moreover, the labels of vertices in the topological graph can be directly determined according to the corresponding tree graph. For example, the labels of vertices in the topological graphs in Figure 33 are directly determined according to the tree graphs shown in Figure 22b,c. These characteristics are greatly helpful for improving the efficiency of isomorphism detection. Secondly, many existing isomorphism detection methods are not reliable when applied to PGTs with more than six links. The present isomorphism detection algorithm is proven to be still reliable when applied to complex PGTs with more than six links. Thirdly, most existing methods are only applicable for closed-loop graphs but not for open-loop graphs. The present algorithm is versatile and can be applied to both open-loop and closed-loop graphs.

6. Conclusions

A tree graph method for the topological synthesis of PGTs is proposed in this paper. Firstly, the rules for generating (N + 1)-link tree graphs from N-link tree graphs are proposed. Then, the method for adding geared edges into the tree graph is presented, and the topological graphs of 1-DOF PGTs are directly generated from the atlas of tree graphs. The code of vertex is constructed, and a unified algorithm applicable for both open-loop and closed-loop graphs is proposed to eliminate isomorphic tree graphs and topological graphs of PGTs. The present synthesis method is automatically implemented by developing a computer program. As a result, the complete atlases of tree graphs and topological graphs of 1-DOF PGTs with three to ten links are obtained for the first time. The present synthesis method has been verified as being reliable and efficient, and it can also be applied to the synthesis of multi-DOF PGTs. The present study is of great significance to the design of new powertrains.