2.1. Introduction to the spatial structure of CH₄

In the chemical molecular structure, when an atom interacts with other four identical atoms, four chemical bonds can be formed. If each bond has the same bond length and bond angle, the spatial regular tetrahedral structure is formed. For example, methane molecule (CH₄), carbon tetrachloride molecule (CCl₄), etc., all belong to the spatial regular tetrahedral molecular configuration. Conversely, if there is a different bond parameter, it will belong to the spatial tetrahedral structure, such as monochloromethane molecule (CH₃Cl), trichloromethane molecule (CHCl₃), etc.

The regular tetrahedral structure of chemical molecules have a stable existence of three-dimensional structure compared to other spatial structure molecules. Its properties are more stable, and its structure is simpler. CH₄ is the most representative molecule among the regular tetrahedral structure chemical molecules, so this paper is based on CH₄ molecule to perform the topological evolution of the spatial structure.

The spatial regular tetrahedral structure of CH₄ [Reference Henri19] is shown in Fig. 5. It includes four identical C-H bonds with bond lengths of 109.1pm and bond angles of 109°28^′ [Reference Gillespie and Robinson3]. The molecular structure is a spatial symmetrical structure, and it exists in a stable state for a long time.

Figure 5. Structure of CH₄. (a) 3D model. (b) Spatial structure.

The properties of substances are the external performance of the molecular structure. In the formation of CH₄, due to the interaction between C atom and H atoms, each bond is formed with the same length and angle. Hence, a spatial regular tetrahedral molecule is obtained. In addition, the C atom of CH₄ is located in the center of regular tetrahedron structure, and the distance from C atom to each vertex is the C-H bond length. It can be described mathematically that each vertex of the regular tetrahedron structure of CH₄ is distributed on its circumscribed sphere, and the radius of the circumscribed sphere is the bond length.

To obtain mechanisms with stronger spatial structural stability, the spatial symmetrical molecular structure is used as the prototype in this paper, and the evolution from spatial structure to planar topology will be deduced.

2.2. Topological evolution for the spatial structure of CH₄

After describing the spatial structure characteristics of CH₄ in Section 2.1, the evolution of molecular configuration is deduced in this section. As shown in Fig. 6, it is the spatial regular tetrahedral structure of CH₄. The connection relationship between various element points for the spatial configuration of the chemical molecule is retained, and the spatial characteristics for the circumscribed circle of molecular structure are also reflected.

Figure 6. The spatial regular tetrahedral structure of CH₄.

Due to the intersection of multiple edges in a regular tetrahedron, the description of the spatial geometric relationship is complex. Thus, it is necessary to simplify the main geometric feature information and express the spatial connection relationship of each element clearly. The expression of edges and vertex positions for constituting the tetrahedral structure needed to be reduced. In the process of evolution, since the information of the four vertices of regular tetrahedron structure is known, the four structural edges of AB, AC, BD, and CD are used to express the spatial structure. The four structural edges are transformed from spatial structure into planar graph to express the basic configuration characteristics of the regular tetrahedral structure, and the reasons are as follows:

By projecting the regular tetrahedral structure onto plane, the top view is obtained, as shown in Fig. 7(a). According to the structural characteristics, it is necessary to determine any surface as the bottom surface of the regular tetrahedron firstly. Here, assuming that the bottom of the regular tetrahedron is △ABC, then three vertex information of the regular tetrahedron structure is expressed, and the bottom △ABC of the regular tetrahedron just can be determined by the edge AB and AC, as shown in Fig. 7(b). Secondly, on the basis of the determined bottom △ABC, it is also necessary to express the edge and vertex height information of regular tetrahedron structure. As shown in Fig. 7(b), the above information can be determined by the edge BD and CD. To sum up, the spatial feature information of regular tetrahedron structure is determined by these four edges, so they can be used to simplify the expression of the regular tetrahedron structure on the plane.

Figure 7. Top view and projection edges of the regular tetrahedral structure. (a) Top view. (b) Four projection edges.

It is known that two straight lines l ₁ and l ₂ in space intersect at point A, as shown in Fig. 8. Thus, it is necessary to prove that there is only one plane α passing through the straight lines l ₁ and l ₂, that is, the bottom surface of the above regular tetrahedron.

Figure 8. Two intersecting lines in space.

As shown in Fig. 8, taking another point C that does not coincide with A on the straight line l ₁, and then taking another point B that does not coincide with A on the straight line l ₂, it can be obtained that the three points A, B, and C are not collinear.

According to axiom three [Reference Shen and Wang20] in space solid geometry, it can be known that there is one and only one plane passing through three points that are not on a straight line. Hence, there is one and only one plane α through points A, B, and C.

On the contrary, according to the axiom one [Reference Shen and Wang20] in space solid geometry, it can be known that if two points on a straight line are in a plane, then all the points on the straight line are in this plane.

Since A∈ α , B∈ α , we can get: l ₂⊂ α , and l ₁⊂ α .

Thus, through two intersecting straight lines l ₁ and l ₂, there is one and only one plane α .

That is, two intersecting lines determine a plane.

Therefore, as shown in Fig. 9, the position of the bottom surface △ABC of regular tetrahedron structure can be determined by using the edges AB and AC. The proposed edges are proved.

Figure 9. The bottom surface of regular tetrahedron.

Figure 10. Evolution of edges. (a) Evolution of four edges. (b) Planar topological expression.

Figure 11. Preliminary topological graph for the spatial structure of CH₄.

It can be seen from the above analysis that the two edges AB and AC intersect, so it is shown that the points A, B, and C are coplanar, that is, the bottom surface of spatial regular tetrahedron structure is determined. When the three points A, B, and C are known, the information for the bottom surface of the regular tetrahedron structure can be expressed by the two edges AB and AC. In the same way, it can be proved that the two intersected edges BD and CD determine the plane, where △BCD is located. Then, the information for the vertex and high of spatial regular tetrahedron structure is determined. Hence, the four edges AB, AC, BD, and CD of the spatial regular tetrahedron structure can be used to express its characteristics. It is further shown that the edges selected in this section can be used to evolve and represent the spatial characteristics of regular tetrahedron structure.

In the process of evolution, to facilitate the distinction and analysis, the spatial tetrahedral structural features are decomposed into multiple triangles. The edges AB, AC, BD, and CD correspond to ∠AEB, ∠AEC, ∠BED, and ∠CED, respectively, as shown in Fig. 10(a). After the four edges of the structure are transformed into the circumcircle of the regular tetrahedron, the planar topological expression is generated as shown in Fig. 10(b).

Then, the redundant lines and color expressions are removed, and a preliminary topological graph for the spatial structure of CH₄ is obtained, as shown in Fig. 11. Among them, the core skeleton of CH₄ is retained, and the molecular spatial topological structure is simplified to the plane.

During the evolution of the above molecular spatial structure, the edge AB of △ABE, the edge AC of △ACE, the edge BD of △BDE, and the edge CD of △CDE have all been evolved to the circumcircle. And in the evolution of △ABE and △ACE, △BDE, and △CDE, there are two common edges AE and DE; that is, the other edge feature of the above triangle is also expressed. Based on the above information, each triangle in the molecular spatial structure is expressed with the existing topological relationship. Then, the redundant connection relationship can be removed, and the connection between B, C, and E is further simplified. Thus, the topological graph based on the molecular spatial structure is obtained, as shown in Fig. 12(a). On the other hand, since the expressions of two common edges AE and DE have already existed in ΔAED, the connection relationship is preserved. By sorting out, the final topological graph for the molecular spatial structure of regular tetrahedral is obtained, as shown in Fig. 12(b).

Figure 12. Topological graph for the spatial structure of CH₄. (a) The simplified graph. (b) The final topological graph.

2.3. Topological evolution results for the spatial structure of CH₄

After the evolution of the topological relationship in Section 2.2, the spatial regular tetrahedral structure of CH₄ has been expressed with the planar topological relationship, as shown in Fig. 13. It is a more simplified form to describe the spatial structure.

Figure 13. The spatial structure and topological graph of CH₄. (a) Spatial structure. (b) Topological graph.

Based on the molecular spatial structure of CH₄, as shown in Fig. 13(a), C is at the center of the molecular structure and H is at the two ends of each branch of the regular tetrahedral structure, that is: C is in the middle of each H-C-H branch, and C belongs to the common connecting part. Therefore, the topological connection form of H-C-H that exists in CH₄ is still preserved during the evolution process. Also, due to the symmetric form of the spatial regular tetrahedral structure, the structure of each branch is the same and can be completely overlapped by rotation, so it can be simplified to be one branch chain structure to analyze. And the shape of the external sphere of CH₄ spatial structure is preserved in the evolution process, so the topological graph is in a circular form. In this paper, by using such a connection relationship in the spatial structure of CH₄ for topological evolution expression, the elements in each branch are split and merged based on the expression form of shared connection, so the topological graph of the spatial structure as shown in Fig. 13(b) is obtained.

Through analyzing the topological graph of CH₄, it can be concluded that the left, middle, and right branches of the topological graph are exactly the same. The C atom of each branch is taken as the core, and it connects two H atoms up and down to generate two C-H bonds. The H atoms at the top and bottom of the topological graph are shared, so as to form three branches, and then, the topological relationship for the spatial structure of CH₄ is expressed.

Similarly, using the same topological expression form of CH₄, if the C-H bond is replaced by the C-Cl bond, a carbon tetrachloride molecule (CCl₄) is formed. Its topological graph is shown in Fig. 14(a), and the corresponding spatial structure is shown in Fig. 14(b); that is, it is also a spatial regular tetrahedron structure.

Figure 14. The carbon tetrachloride molecule. (a) Topological graph. (b) Spatial structure.

By comparing the topological graph and spatial structure of CH₄ and CCl₄, it can be found that: for the same type of CMSS, the element symbols in topological graphs are different, and the chemical bonds formed are different, so the types and properties of chemical molecules are different, but the expressions of topological graphs are the same; that is, the number of branches, the number of elements, and the connection relationship are the same. Hence, the same type of chemical molecular spatial structure can be expressed in the form of the same topological graph. It is distinguished by labeling element symbols, and the topological relationship expression for a class of chemical molecules with the same structure can be obtained.

From the above analysis results, it is shown that the topological graph can be used to express the structure and element connection relationship of molecular spatial structure. The chemical bonds between atoms are used to form a stable spatial structure. For the mechanism, a spatial mechanism is formed through the kinematic pairs between components. The kinematic pairs play the same role as the chemical bonds in spatial structure. Thus, the chemical molecular structure and mechanism configuration can be analogized in the evolution. For example, the CMSS is expressed by the topological graph, and different chemical molecules under the same spatial structure are obtained by replacing chemical bonds. These methods can be further applied to the expression of spatial mechanisms. A similar form of the topological graph for molecular structure can be used to express spatial PMs. By changing the types of kinematic pairs between components in the topological graph, different PM configurations are generated, and the configuration design and expression of the same type of mechanism can be realized.

To sum up, combined with the expression of topological graph for CMSS and the graph theory method of planar topological structure, the two are introduced into the expression of topological relationship of the spatial PM. Hence, a new topological method is used to represent PM, simplify the expression, and establish the connection between PMs and topological models. The new topological graph method is to relieve the limitation of the application of graph theory and realize the research goal of mechanism configuration design.

3.1. Topological evolution results for the spatial structure of CH₄

Through the topological transformation of molecular spatial structure, the above evolution results show that the spatial structure and planar topological graph can be effectively transformed. Hence, researching the relationship between spatial structure and topological graph helps to generate new spatial configurations quickly.

The spatial topological structure of CH₄ shown in Fig. 15 is analyzed, and it has the following characteristics:

1. 1. Since the molecular structure of CH₄ belongs to a spatial symmetrical structure, the evolution retains this feature. Thus, the topological graph is still symmetrical, that is, the three branches are exactly the same, and can be overlapped by rotation.

2. 2. The expression of spatial structure is simplified by the topological graph, but its characteristics are still retained, that is, the element arrangement of branches in the topological graph is the same as molecular spatial structural characteristics, and the corresponding relationship is consistent.

3. 3. In each closed-loop triangle of the spatial structure, the C atom is taken as the center, and two H atoms are connected to form each branch. The regular tetrahedral structure is formed by three branches. The same characteristics are still in the corresponding topological graph, and the chemical bond is represented by a solid line.

4. 4. The common elements and chemical bonds in the spatial structure are merged and split in the topological graph. The branch distribution is simplified in these ways.

5. 5. The shape of the topological graph is obtained from the evolution of the circumscribed sphere of spatial regular tetrahedron structure, so the shape of the topological graph is round. The spatial structure can be better represented by this feature of a round.

PM has many configurations, and each configuration has multiple chains. Considering that the spatial structure of CH₄ is symmetrical. Thus, the topological graph of CH₄ is compared with the structure of the symmetrical parallel mechanism (SPM). There are many similarities between them, as shown in Table I.

Table I. Comparison for topological graph of CH₄ and SPM.

Figure 15. The spatial topological structure of CH₄. (a) Spatial structure. (b) Topological graph. (c) Topological graph prototype.

It can be seen from Table I that the spatial structural topological graph of CH₄ is very similar to the structural characteristics of SPM. They are both composed of upper parts, lower parts, and multiple branches, and the branches in their respective structures are the same. They both have multiple spatial closed-loop chains. SPM is composed of multiple branch chains with the same structural form, and branch chains are connected to the moving platform and the fixed platform through kinematic pair. These branch chains are evenly distributed, and each branch chain can coincide after rotation, so it belongs to the spatial symmetrical mechanism. In fact, the spatial structural topological graph of CH₄ is exactly the same expression form. Therefore, in the expression of the topological relationship of spatial mechanism, SPM is taken as the research object. It is combined with the spatial structural topological graph of CH₄, the topological relationship of SPM is described, and the configuration information of SPM is expressed, so as to realize the configuration analysis and design of SPM.

3.2. Definition for the topological graph of spatial SPM

A graph is a collection of vertices or nodes connected together by lines or edges. In the topological graph of planar mechanism, the components are represented by points and the kinematic pairs are represented by edges [Reference Ding and Huang21]. The topological graph has a certain corresponding relationship with the structural diagram of the planar mechanism. According to the basic definition of the graph, combined with the spatial structural topological graph of CH₄ and the representing method of planar mechanism, the topological graph of spatial SPM is defined as follows:

Assuming that the spatial SPM is Ms. For the mechanism Ms, the number of branch chains of Ms is m, and the number of kinematic pairs of each branch is k. Then, the number of vertices of the topological graph is $n=m\cdot (k-1)$ , and the number of edges of the topological graph is $e=m\cdot k$ . To facilitate the analysis, each component and kinematic pair are uniformly numbered here. It is stipulated that the fixed platform serial number is set as 1, and the moving platform serial number is set as n. The components and kinematic pairs of each branch chain are numbered sequentially from left to right and from bottom to top (from the fixed platform to the moving platform). Similarly, the components are also represented by circles (in which the hollow circle represents the movable component, the solid circle represents the fixed component), and the kinematic pairs are represented by edges. At the same time, in order to reflect the feature of the spatial structure, the circular contour is still adopted in the topological graph, and thus the topological graph of the mechanism Ms is obtained.

Table II. Types of topological graph for spatial SPMs.

According to the above definition of the topological graph of SPM, the expression form of the topological graph for spatial SPM can be obtained, as shown in Fig. 16. The unified form of spatial SPM is represented by this topological graph, that is, the topological graph of spatial SPM with m branch chains, and each branch chain contains k kinematic pairs. Meanwhile, from the characteristics of the spatially symmetric parallel mechanism, it is known that the DOF of the spatially symmetric parallel mechanism is equal to the number of branch chains of the mechanism topological graph, that is, the DOF is m.

Figure 16. Topological graph of spatial SPM.

From the definition of graph theory and the topological graph of SPM, it can be known that: for all spatial SPMs with m branch chains and the number of kinematic pairs of each branch chain is k, the corresponding topological graphs are isomorphic graphs. Namely, the topological graph has the same structural form, and the same kind of SPM can be represented by any one of the topological graphs. Hence, by sorting out and giving examples of the structural types of the topological graph for spatial SPM, as shown in Table II, the topological graph of SPM with three to six branch chains can be obtained.

3.3. Mathematical expression and properties of the topological graph

After transforming the spatial SPM to a topological graph, to facilitate subsequent automatic processing and analysis by computer [Reference Ding, Huang, Liu and Cao22], it is necessary to express the topological graph of SPM in a mathematical form and describe the topological relationship. Since the matrix is an effective mathematical tool to study graph theory, and the general adjacency matrix and the correlation matrix [Reference Yan and Kuo23, Reference Lu24] can describe all topological characteristics of the topological graph. Hence, the adjacency matrix is taken as the algebraic representation of topological graph for configuration design and analysis in this paper. However, the general adjacency matrix only expresses the connection relationship between components. The mechanism topology information is described less, and the correspondence between the matrix and configuration is poor. Therefore, a kinematic pair adjacency matrix M _KPA is constructed to describe the PM in this section, as shown in formula (1). The information contained in the matrix M _KPA includes the number of components, the number of kinematic pairs, the type of kinematic pairs, the spatial position relationship of kinematic pairs, and the connection relationship between components and kinematic pairs, etc. The configuration characteristics of PM are expressed more comprehensively in the matrix M _KPA.

(1) \begin{equation} M_{\textrm{KPA}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} L_{1} & _{pq}^{\mathit{C}2}{J}{_{}^{12}} & _{pq}^{\mathit{C}3}{J}{_{}^{13}} & \cdots & _{pq}^{\mathit{C}\mathit{i}}{J}{_{}^{1i}} & \cdots & _{pq}^{\mathit{C}(\mathit{n}{-}1)}{J}{_{}^{1,n-1}} & _{pq}^{\mathit{C}\textit{n}}{J}{_{}^{1,n}}\\[4pt] a_{21} & L_{2} & _{pq}^{\mathit{C}3}{J}{_{}^{23}} & \cdots & _{pq}^{\mathit{C}\mathit{i}}{J}{_{}^{2i}} & \cdots & _{pq}^{\mathit{C}(\mathit{n}{-}1)}{J}{_{}^{2,n-1}} & _{pq}^{\mathit{C}\textit{n}}{J}{_{}^{2,n}}\\[4pt] a_{31} & a_{32} & L_{3} & \ddots & \cdots & _{pq}^{\mathit{C}\textit{j}}{J}{_{}^{3j}} & \cdots & \cdots \\[4pt] \vdots & \vdots & \ddots & \ddots & _{pq}^{\mathit{C}\textit{i}}{J}{_{}^{i-1,i}} & \vdots & \vdots & \vdots \\[4pt] a_{i1} & a_{i2} & \cdots & a_{i,i-1} & L_{i} & _{pq}^{\mathit{C}({i}{+}1)}{J}{_{}^{i,i+1}} & \cdots & _{pq}^{\mathit{C}\textit{n}}{J}{_{}^{i,n}}\\[4pt] \cdots & \cdots & a_{i+1,j} & \cdots & a_{i+1,i} & \ddots & \ddots & \cdots \\[4pt] a_{n-1,1} & a_{n-1,2} & \cdots & \cdots & \cdots & \ddots & L_{n-1} & _{pq}^{\mathit{C}\textit{n}}{J}{_{}^{n-1,n}}\\[4pt] a_{n,1} & a_{n,2} & \cdots & \cdots & a_{n,i} & \cdots & a_{n,n-1} & L_{\mathit{n}} \end{array}\right] \end{equation}

where i is the row of the matrix and j is the column of the matrix, i, j, p, q represent the No. of the mechanism component, respectively, (i, j, p, q = 1, 2, …, n); $\mathit{L}_{\mathit{i}}$ represents the ith component of the mechanism according to the defined serial number; $a_{ij}$ represents the connection relationship between various components of the mechanism; the left upper corner mark Cj represents the constraint type.

Through the constructed kinematic pair adjacency matrix M _KPA for the topological graph of PM, not only the degrees of freedom (DOF) of the mechanism can be obtained but also the diagram of the mechanism can be inversely described. Under the condition that only the matrix M _KPA of the mechanism is known, the information can be used to draw the mechanism diagram. These will be verified in the subsequent sections.

In the matrix of formula (1), the lower triangular element represents whether there is a connection relationship between various components of the mechanism, and its specific value is defined as follows:

(2) \begin{equation} a_{ij}=\left\{\begin{array}{l} 0,\quad \text{ No connection between the components }i\text{ and }j\\[5pt] 1,\quad \text{There is }\textrm{a}\text{ connection between the components }i\text{ and }j \end{array}\right. \end{equation}

The diagonal element $\mathit{L}_{\mathit{i}}$ represents the ith component of the mechanism according to the defined serial number. To express the information of the driving pair, the fixed platform, and the moving platform of mechanism, the diagonal element is defined and assigned as follows:

(3) \begin{equation} L_{i}=a^{ii}=\left\{\begin{array}{l} 7,\quad \text{Component }i\text{ is the driving part}\\[3pt] 8,\quad \text{Component }i\text{ is the passive part}\\[3pt] 9,\quad \text{Component }i\text{ is the fixed platform}\\[3pt] 10,\quad \text{Component }i\text{ is the moving platform} \end{array}\right. \end{equation}

The upper triangular element $_{\mathit{pq}}^{\mathit{Cj}}{\mathit{J}}{_{}^{\mathit{ij}}}$ indicates that the kinematic pair J is connected to the current component i and j, and the positional relationship between this axis of kinematic pair and the axis of connecting component p and q is the constraint type Cj. The left upper corner mark Cj represents whether there is a position connection between the axes of the kinematic pair in mechanisms. The lower left corner mark pq represents the axis of kinematic pair for connecting component p and q. There is a positional relationship between the axis of connecting component p and q and the axis of connecting component i and j. Among them, there are six kinds of positional relationships included in Cj, namely the axes in space are parallel, the axes are coaxial, the axes intersect at one point, the axes are perpendicular, the axes are parallel to the same plane, and the axes are in any direction [Reference Yang, Shen, Liu and Hang6, Reference Yang9]. The specific definition and assignment are shown in Table III.

Table III. Definition of position constraint type Cj for kinematic pair axis.

For the kinematic pair $\mathit{J}^{\mathit{ij}}$ of the element $_{\mathit{pq}}^{\mathit{Cj}}{\mathit{J}}{_{}^{\mathit{ij}}}$ , it represents the connection relationship between the component i and j, and the specific definition and assignment are shown in Table IV.

Table IV. Definition of connection relationship.

In the kinematic pair adjacency matrix M _KPA, the expression elements $_{\mathit{pq}}^{\mathit{Cj}}{\mathit{J}}{_{}^{\mathit{ij}}}$ for the axis position connection relationship of the kinematic pair are specifically described as follows:

$R^{12}$ means that the component 1 and 2 are connected by the revolute pair. $P^{34}$ means that the component 3 and 4 are connected by the prismatic pair. $_{23}^{a}{R}{_{}^{12}}$ means that the component 1 and 2 are connected by the revolute pair, and the axis of the revolute pair is parallel to the axis of component 2 and 3 in space. $_{45}^{b}{P}{_{}^{34}}$ means that the component 3 and 4 are connected by the prismatic pair, and the axis of the prismatic pair is coaxial (coincident) to the axis of component 4 and 5 in space. $_{67}^{f}{S}{_{}^{78}}$ means that the component 7 and 8 are connected by the spherical pair, and the axis of the spherical pair is arbitrary with the axis of component 6 and 7 in space.

Moreover, the kinematic pair adjacency matrix M _KPA is analyzed. It has the following properties:

1. 1. The lower triangular element a _ij in the matrix M _KPA described the connection relationship between components i and j. It is judged by the values of 0 and 1. If there is a connection relationship, the element value is 1; otherwise, it is 0.

2. 2. The diagonal element L _i in the matrix M _KPA described the serial number and quantity of components in the mechanism. According to the size of the matrix M _KPA n×n, it can be known that there are n components in the mechanism.

3. 3. The upper triangular element $_{\mathit{pq}}^{\mathit{Cj}}{\mathit{J}}{_{}^{\mathit{ij}}}$ in the matrix M _KPA described the types of kinematic pair between component i and j, and the spatial position relationships for the axis of kinematic pairs.

4. 4. The DOF of the mechanism can be calculated through the information in the matrix M _KPA, such as the number of mechanism components, the number of kinematic pairs, and the type of kinematic pairs, etc.

5. 5. The matrix M _KPA can be used to draw the spatial mechanism. With the help of the information of the number of mechanism components, the number of kinematic pairs, the type of kinematic pairs, and the spatial position relationship for the axis of kinematic pairs in the matrix, the topological graph for spatial mechanism can be drawn, and the diagram of mechanism can be further obtained.

6. 6. The adjacency matrix of mechanism can be derived from the lower triangular element a _ij in the matrix M _KPA. The elements of the adjacency matrix are symmetrical about the diagonal. The diagonal elements are all 0, and the adjacency matrix belongs to the real symmetrical matrix.

4.1. Verification and analysis based on the topological graph

To verify the corresponding consistency and uniqueness of the proposed new topological expression method in this paper with spatial SPM, the spatial SPM is obtained through the topological graph and kinematic pair adjacency matrix in this section. Here, the 3-RPS PM [Reference Hunt25] is taken as an example for verification. In Section 3.2, different structural types of topological graph for spatial SPMs are obtained. When m = 3, as shown in Fig. 17, the topological graph of SPM with three identical branched structures in space is obtained based on Fig. 16.

Figure 17. Topological graph of spatial SPM (m = 3).

After determining the number of branches of the spatial SPM, if the connection relationship between the components is uncertain, many similar topological relationships of configurations can be generated. For example, the 3-RPS, 3-PRS, 3-SPR, 3-RRC, 3-RRPR, and other three-branch SPMs can be represented by Fig. 18. Hence, the kinematic pairs and the spatial position relationships of axes of each branch chain structure needed to be further clarified to determine the mechanism configuration. When the connection relationships between different components are given in the topological graph, a specific topological graph of SPM is generated. Here, the number of kinematic pairs in the topological graph Fig. 17 is set as k = 3, and the kinematic pairs of each branch chain adopt R, P, and S. Then, the topological relationship of the spatial SPM is formed, as shown in Fig. 18.

Figure 18. Topological graph of 3-RPS PM.

The 3-RPS PM is represented by the topological graph in Fig. 18. It is a typical representative of the 3-DOF PM and belongs to the spatial SPM. It can be seen from the topological graph that the mechanism is composed of three identical branch chains, a moving platform, and a fixed platform. Each branch chain contains three kinematic pairs: R, P, and S. The upper end is connected to the moving platform through the spherical pair, the lower end is connected to the fixed platform through the revolute pair, and the prismatic pair is in the middle of the branch chain.

According to the definition of the kinematic pair adjacency matrix M _KPA in Section 3.3, with the spatial position relationships of axes by $\{\left.-\textrm{R}\bot \textrm{P}-\textrm{S}-\}\right.$ in each branch chain, the matrix M _KPA for the topological graph of 3-RPS PM can be obtained as follows:

(4) \begin{equation} M_{\textrm{KPA}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} L_{1} & \textrm{R}^{12} & 0 & \textrm{R}^{14} & 0 & \textrm{R}^{16} & 0 & 0\\[4pt] 1 & L_{2} & _{12}^{d}{\textrm{P}}{_{}^{23}} & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & 1 & L_{3} & 0 & 0 & 0 & 0 & _{23}^{f}{\textrm{S}}{_{}^{38}}\\[4pt] 1 & 0 & 0 & L_{4} & _{14}^{d}{\textrm{P}}{_{}^{45}} & 0 & 0 & 0\\[4pt] 0 & 0 & 0 & 1 & L_{5} & 0 & 0 & _{45}^{f}{\textrm{S}}{_{}^{58}}\\[4pt] 1 & 0 & 0 & 0 & 0 & L_{6} & _{16}^{d}{\textrm{P}}{_{}^{67}} & 0\\[4pt] 0 & 0 & 0 & 0 & 0 & 1 & L_{7} & _{67}^{f}{\textrm{S}}{_{}^{78}}\\[4pt] 0 & 0 & 1 & 0 & 1 & 0 & 1 & L_{8} \end{array}\right] \end{equation}

From the 8 × 8 square matrix of formula (4), it can be known that the PM is composed of eight components (L ₁, L ₂, …, L ₈), three revolute pairs (R), three prismatic pairs (P), and three spherical pairs (S). At the same time, information such as the connection relationship between each component, the type of kinematic pair, the spatial position relationship for the axis of kinematic pair, etc., are also expressed in the matrix. Among them, $\textrm{R}^{12}$ means that the component 1 and 2 are connected by the revolute pair. $_{12}^{d}{\textrm{P}}{_{}^{23}}$ means that the component 2 and 3 are connected by the prismatic pair, and its axis is in a d-type constraint relationship with the axis of revolute pair of component 1 and 2; that is, the two axes are perpendicular to each other. $_{23}^{f}{\textrm{S}}{_{}^{38}}$ means that the component 3 and 8 are connected by spherical pair, and its axis is in the f-type constraint relationship with the axis of rotation pair of component 2 and 3, that is, the two axes intersect and can be in any orientation.

Combined with the information of the matrix M _KPA, according to the calculation formula of spatial degrees of freedom [Reference Huang, Zhao and Zhao26, Reference Huang, Liu and Li27], the DOF of this mechanism is obtained as follows:

\begin{equation*} M=6\left(n-g-1\right)+\sum _{i=1}^{g}f_{i}+\mu -\xi =6\times \left(8-9-1\right)+\left(1\times 3+1\times 3+3\times 3\right)=3 \end{equation*}

The above kinematic pair adjacency matrix M _KPA of the 3-RPS PM is further analyzed. By splitting the matrix M _KPA, the upper triangular matrix M _KPAU and the lower triangular matrix M _KPAL are obtained. The upper triangular matrix M _KPAU of the kinematic pair adjacency matrix is shown in formula (5).

(5) \begin{equation} M_{\textrm{KPAU}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & {}^{}{\textrm{R}}{_{}^{12}} & 0 & {}^{}{\textrm{R}}{_{}^{14}} & 0 & {}^{}{\textrm{R}}{_{}^{16}} & 0 & 0\\[4pt] 0 & 0 & _{12}^{d}{\textrm{P}}{_{}^{23}} & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & 0 & 0 & 0 & 0 & 0 & 0 & _{23}^{f}{\textrm{S}}{_{}^{38}}\\[4pt] 0 & 0 & 0 & 0 & _{14}^{d}{\textrm{P}}{_{}^{45}} & 0 & 0 & 0\\[4pt] 0 & 0 & 0 & 0 & 0 & 0 & 0 & _{45}^{f}{\textrm{S}}{_{}^{58}}\\[4pt] 0 & 0 & 0 & 0 & 0 & 0 & _{16}^{d}{\textrm{P}}{_{}^{67}} & 0\\[4pt] 0 & 0 & 0 & 0 & 0 & 0 & 0 & _{67}^{f}{\textrm{S}}{_{}^{78}}\\[4pt] 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \end{equation}

From the element information of the matrix M _KPAU, the connection order and branch chain loop between components are obtained, as shown in Table V.

Table V. Correspondence between the matrix M _KPAU and the branches of topological graph.

Further, the information in the matrix M _KPAU can be used to obtain the connection relationship of components, as shown in Fig. 19(a). Starting from component 1 to get the topological relationship diagram, as shown in Fig. 19(b). Then, combining the kinematic pair information from the matrix M _KPAU and expressing it to form the topological graph of SPM, as shown in Fig. 19(c).

Figure 19. The topological graph of 3-RPS PM obtained by M _KPAU. (a) The connection relationship. (b) Topological graph. (c) Topological graph of 3-RPS PM.

To sum up, by combining the topological graph of the 3-RPS PM with the information expressed by the kinematic pair adjacency matrix M _KPA of formula (4), the spatial symmetrical 3-RPS PM is uniquely determined. Then, the topological graph of mechanism is evolved from plane to space, and the corresponding spatial 3-RPS PM is obtained, as shown in Fig. 20.

Figure 20. Topological graph and corresponding 3-RPS PM. (a) Topological graph. (b) 3-RPS parallel mechanism.

In addition, based on the kinematic pair adjacency matrix M _KPA, the lower triangular matrix M _KPAL is obtained, as shown in formula (6).

(6) \begin{equation} M_{\textrm{KPAL}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\[4pt] 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\[4pt] 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\[4pt] 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\[4pt] 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \end{array}\right] \end{equation}

Then, the adjacency matrix M _A is further derived from the lower triangular matrix M _KPAL, as shown in formula (7). The matrix M _A is an 8 × 8 square matrix, so the corresponding mechanism has eight components. The connection relationship between each component is also expressed. Among them, the diagonal elements are all 0, the elements are symmetrical about the diagonal, and the matrix M _A belongs to a real symmetrical matrix.

(7) \begin{equation} M_{\textrm{A}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0\\[4pt] 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1\\[4pt] 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\[4pt] 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\[4pt] 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\[4pt] 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1\\[4pt] 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \end{array}\right] \end{equation}

After uniformly assigning values to the matrix elements, the corresponding matrix M _KPA for the topological graph of 3-RPS PM is obtained, as shown in formula (8).

(8) \begin{equation} M_{\textrm{KPA}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 9 & 1^{12} & 0 & 1^{14} & 0 & 1^{16} & 0 & 0\\[2pt] 1 & 7 & _{12}^{4}{2}{_{}^{23}} & 0 & 0 & 0 & 0 & 0\\[2pt] 0 & 1 & 8 & 0 & 0 & 0 & 0 & _{23}^{6}{3}{_{}^{38}}\\[2pt] 1 & 0 & 0 & 7 & _{14}^{4}{2}{_{}^{45}} & 0 & 0 & 0\\[2pt] 0 & 0 & 0 & 1 & 8 & 0 & 0 & _{45}^{6}{3}{_{}^{58}}\\[2pt] 1 & 0 & 0 & 0 & 0 & 7 & _{16}^{4}{2}{_{}^{67}} & 0\\[2pt] 0 & 0 & 0 & 0 & 0 & 1 & 8 & _{67}^{6}{3}{_{}^{78}}\\[2pt] 0 & 0 & 1 & 0 & 1 & 0 & 1 & 10 \end{array}\right] \end{equation}

To facilitate computer processing, according to the upper triangular matrix M _KPAU of formula (5), the symmetry of the matrix about the diagonal elements is used to obtain the adjacency matrix M _A. After uniformly assigning values to the matrix M _A and simplifying the expression information, the numerical matrix M _D is derived, as shown in formula (9).

To facilitate computer processing, according to the upper triangular matrix M _KPAU of formula (5), by taking advantage of the symmetry of the matrix M _KPAU about the diagonal elements, that is, making the matrix elements $a_{ij}=a_{ji},(i\neq j)$ , and then symmetrizing the upper triangular matrix elements and simplifying the information of the upper and lower corner markers of the matrix elements, the matrix M _D based entirely on the numerical expression is derived, as shown in formula (9).

(9) \begin{equation} M_{\textrm{D}}=\left[\begin{array}{l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l@{\quad}l} 9 & 1 & 0 & 1 & 0 & 1 & 0 & 0\\[2pt] 1 & 7 & 2 & 0 & 0 & 0 & 0 & 0\\[2pt] 0 & 2 & 8 & 0 & 0 & 0 & 0 & 3\\[2pt] 1 & 0 & 0 & 7 & 2 & 0 & 0 & 0\\[2pt] 0 & 0 & 0 & 2 & 8 & 0 & 0 & 3\\[2pt] 1 & 0 & 0 & 0 & 0 & 7 & 2 & 0\\[2pt] 0 & 0 & 0 & 0 & 0 & 2 & 8 & 3\\[2pt] 0 & 0 & 3 & 0 & 3 & 0 & 3 & 10 \end{array}\right] \end{equation}

where i is the row of the matrix and j is the column of the matrix, i and j represent the No. of the mechanism component, respectively, (i = 1, 2, … , 8; j = 1, 2, … , 8). The diagonal values 7, 8, 9, 10 represent that the ith component of the mechanism is the driving component, the passive component, the fixed platform, and the moving platform, respectively; the values 1, 2, 3 in the matrix represent that the components i and j are connected with each other through the revolute pair R, the prismatic pair P, and the spherical pair S, respectively; the value 0 in the matrix represents that there is no topological connection between the components i and j.

Hence, based on the numerical matrix M _D, the configuration of PM can be designed and analyzed with the help of the computer programming.

4.2. Verification and analysis based on the spatial SPM

To verify the corresponding consistency of the spatial SPM with the new topological expression method proposed in this paper, the 3-RRC PM is taken as an example to obtain its corresponding topological graph and kinematic pair adjacency matrix in this section.

The 3-RRC PM is a spatial SPM, and it can realize three-dimensional translational motion, as shown in Fig. 21. From the diagram of the 3-RRC PM, it consists of three identical branches, a moving platform and a fixed platform, with a total of eight components. Each branch chain includes three kinematic pairs: R, R, and C, and the axes of these kinematic pairs are parallel to each other. The lower end of each branch chain is connected to the fixed platform through the first revolute pair, and the upper end is connected to the moving platform through the cylindrical pair. Based on the above information, the topological graph of 3-RRC PM can be further obtained, as shown in Fig. 22.

Figure 21. The diagram of 3-RRC PM.

Figure 22. The topological graph of 3-RRC PM.

Combining the information in the diagram with the topological graph of 3-RRC PM, the kinematic pair adjacency matrix M _KPA for the topological graph of 3-RRC PM can be obtained, as shown in formula (10).

(10) \begin{equation} M_{\textrm{KPA}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} L_{1} & \textrm{R}^{12} & 0 & \textrm{R}^{14} & 0 & \textrm{R}^{16} & 0 & 0\\[2pt] 1 & L_{2} & _{12}^{a}{\textrm{R}}{_{}^{23}} & 0 & 0 & 0 & 0 & 0\\[2pt] 0 & 1 & L_{3} & 0 & 0 & 0 & 0 & _{23}^{a}{\textrm{C}}{_{}^{38}}\\[2pt] 1 & 0 & 0 & L_{4} & _{14}^{a}{\textrm{R}}{_{}^{45}} & 0 & 0 & 0\\[2pt] 0 & 0 & 0 & 1 & L_{5} & 0 & 0 & _{45}^{a}{\textrm{C}}{_{}^{58}}\\[2pt] 1 & 0 & 0 & 0 & 0 & L_{6} & _{16}^{a}{\textrm{R}}{_{}^{67}} & 0\\[2pt] 0 & 0 & 0 & 0 & 0 & 1 & L_{7} & _{67}^{a}{\textrm{C}}{_{}^{78}}\\[2pt] 0 & 0 & 1 & 0 & 1 & 0 & 1 & L_{8} \end{array}\right] \end{equation}

After the same assignment, the corresponding kinematic pair adjacency matrix M _KPA and the numerical adjacency matrix M _D are obtained, as shown in formulas (11) and (12).

(11) \begin{equation} M_{\textrm{KPA}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 9 & 1^{12} & 0 & 1^{14} & 0 & 1^{16} & 0 & 0\\[2pt] 1 & 7 & _{12}^{1}{1}{_{}^{23}} & 0 & 0 & 0 & 0 & 0\\[2pt] 0 & 1 & 8 & 0 & 0 & 0 & 0 & _{23}^{1}{5}{_{}^{38}}\\[2pt] 1 & 0 & 0 & 7 & _{14}^{1}{1}{_{}^{45}} & 0 & 0 & 0\\[2pt] 0 & 0 & 0 & 1 & 8 & 0 & 0 & _{45}^{1}{5}{_{}^{58}}\\[2pt] 1 & 0 & 0 & 0 & 0 & 7 & _{16}^{1}{1}{_{}^{67}} & 0\\[2pt] 0 & 0 & 0 & 0 & 0 & 1 & 8 & _{67}^{1}{5}{_{}^{78}}\\[2pt] 0 & 0 & 1 & 0 & 1 & 0 & 1 & 10 \end{array}\right] \end{equation}

(12) \begin{equation} M_{\textrm{D}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 9 & 1 & 0 & 1 & 0 & 1 & 0 & 0\\[4pt] 1 & 7 & 1 & 0 & 0 & 0 & 0 & 0\\[4pt] 0 & 1 & 8 & 0 & 0 & 0 & 0 & 5\\[4pt] 1 & 0 & 0 & 7 & 1 & 0 & 0 & 0\\[4pt] 0 & 0 & 0 & 1 & 8 & 0 & 0 & 5\\[4pt] 1 & 0 & 0 & 0 & 0 & 7 & 1 & 0\\[4pt] 0 & 0 & 0 & 0 & 0 & 1 & 8 & 5\\[4pt] 0 & 0 & 5 & 0 & 5 & 0 & 5 & 10 \end{array}\right] \end{equation}

So far, through the diagram of 3-RRC PM, the 3-RRC PM is transformed into the expression of the planar topological relationship, and the corresponding topological graph and kinematic pair adjacency matrix are established. By using the kinematic pair adjacency matrix, various connection relationships between components in the topological graph of the 3-RRC PM are described in detail. The corresponding consistency of the 3-RRC PM with the proposed new topological expression method is verified. At the same time, combined with the numerical matrix, it will be convenient to use for digital and automated design of the SPM configuration.