Structural synthesis of Assur groups with up to 12 links and creation of their classified databases

Mechanism and Machine Theory 145 (2020) 103668

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

Structural synthesis of Assur groups with up to 12 links and creation of their classified databases Peng Huang a, Huafeng Ding b,∗ a b

School of Control Engineering, Northeastern University at Qinhuangdao, Qinhuangdao, 066004, China School of Mechanical Engineering and Electronic Information, China University of Geosciences (Wuhan), Wuhan, 430074, China

a r t i c l e

i n f o

Article history: Received 22 July 2019 Revised 25 September 2019 Accepted 8 October 2019

Keyword: Assur groups Structural synthesis Baranov trusses Classified databases

a b s t r a c t The present study proposes a systematic method for synthesizing a complete set of Assur groups with different links. Moreover, the corresponding databases, including all classified topological structures of synthesized Assur groups, are created. In this regard, the structural correlation and the topological representation of Baranov trusses and Assur groups are initially reviewed. Then, the primary and the secondary vertex-classified sets are introduced to reduce the times of isomorphism identification in the Assur group synthesis. Based on the vertex-symmetry codes used for the isomorphism identification, the systematic synthesis procedure is proposed to achieve all non-isomorphic Assur groups derived from the corresponding Baranov trusses. Finally, a complete set of Assur groups with 2, 4, 6, 8, 10 and 12 links is obtained and three corresponding databases of all synthesized Assur groups are established in accordance with their classified structural characteristics. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The concept of Assur group [1] is an open kinematic chain with zero degree of freedom (DOF) so that it is not possible to obtain any simpler structure of the sort. It is known that a planar mechanism with certain motion is constituted by the frame link, driving links and Assur groups. Assur groups are widely applied in vast variety of mechanisms and play an important role in the analysis and synthesis of kinematic chains and mechanisms [2–5]. Manolescu [6] used Assur groups to generate planar kinematic chains with complex structures from simpler chains without rigid sub-chains. Tischler et al. [7] proposed a method to design multi-tipped fingers of a robot hand based on Assur groups. Moreover, Campos et al. [8] presented a method to synthesize kinematic chains and hybrid mechanisms based on Assur groups. Martins et al. [9] applied Assur groups to generate planar fractionated kinematic chains with up to 4 loops and 6 DOFs. Furthermore, Wohlhart [10] presented a method to perform the position analysis of planar mechanisms based on 8-link Assur groups. Li and Dai [11] developed a synthesis method for single-driven metamorphic mechanisms based on augmented Assur groups. Briot and Arakelian [12] investigated the corresponding shaking force and shaking moment for balancing 4-link mechanisms by adding Assur groups. Li et al. [13] presented a structural synthesis method to synthesize planar mechanisms by integrating Assur groups. Sun et al. [14] used structures of Assur groups to analyze the kinematics of planar mechanisms. Hahn and Shai [15] investigated the engineering characteristics of Assur groups and applied them in analyzing planar mechanisms and trusses. Quintero et al. [16] proposed a method to perform the position analysis of planar ∗

Corresponding author. E-mail address: [email protected] (H. Ding).

https://doi.org/10.1016/j.mechmachtheory.2019.103668 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668

mechanisms based on a 6-link Assur group. Moreover, Ramakrishna and Dibakar [17] applied Assur group addition method to design planar mechanisms. Based on Assur groups, Slavutin et al. [18] presented a method to detect singular positions and performed the singularity analysis in parallel mechanisms. Considering the extensive application of Assur groups in synthesizing and analyzing of kinematic chains and mechanisms, the structural synthesis of Assur groups has attracted researchers so that it has recently become a research hotspot. Dobrovolsky [19] listed all Assur groups with 2, 4 and 6 links. Baranov [20] presented a statically definable truss structure (later named as the Baranov truss) and investigated the structural correlation between N-link Baranov trusses and (N-1)-link Assur groups. Then he applied such structural correlation and synthesized 161 8-link Assur groups from 26 9-link Baranov trusses. However, two 9-link Baranov trusses were lost in Baranov’s synthesis, and the two missing 9-link Baranov trusses were later found by Manolescu and Erdelean [21]. Tartakovsky [22] replenished Baranov’s synthesis and enumerated 173 8-link Assur groups from 28 9-link Baranov trusses. Moreover, he achieved the structures of the whole 173 8-link Assur groups. Romaniak [23] proposed a link-replacement method to generated Assur groups. Furthermore, Peisach [24,25] summarized 13 structural conditions of Assur groups, and generated 5442 10-link and 251,638 12-link Assur groups, accordingly. Krokhmal and Krokhmal [26] developed an approach for structural synthesis of Assur groups based on the corresponding topological characteristics. Moreover, Han et al. [27] proposed a method to generate all 13 Assur groups with 2, 4 and 6 links from the kinematic chains with 4, 6 and 8 links. Reviewing the literature indicates that numerous methods have been presented continuously in the last several decades for synthesizing structures of Assur groups. However, only the synthesis results of 1, 2, 10, and 173, corresponding to 2link, 4-link, 6-link and 8-link Assur groups respectively, are verified so far. This is especially more pronounced for Assur groups with high number of links (e.g. 10-link or 12-link Assur groups), where only synthesis numbers rather than specific structures of the Assur groups are provided. This may be a challenge for further developments of synthesis and analysis methods based on Assur groups. Furthermore, although classified Assur groups regarding to the corresponding topological characteristics are helpful for synthesizing and analyzing kinematic chains and mechanisms, they are rarely referred in the existing literature. The main purpose of the present study is to propose a systematic method to synthesize all structures of Assur groups with up to 12 links. The remainder of this paper is arranged as follows. In Section 2, the structural correlation between Nlink Baranov trusses and (N-1)-link Assur groups is introduced and some associated basic concepts of graph representations are addressed. Then, a systematic method for synthesizing all non-isomorphic Assur groups is proposed in Section 3 based on the available set of Baranov trusses obtained from Ref. [28]. Synthesis results of Assur groups with up to 12 links are provided in Section 4, in accordance with link assortment arrays of the corresponding Baranov trusses. The discrepancies of synthesis results obtained from this study and the existing references are compared and discussed in Section 5. In Section 6, all synthesized Assur groups are classified by their structural characteristics in order to establish classified databases. 2. Basic theory and concepts 2.1. Correlation between Baranov trusses and Assur groups In order to illustrate the correlation between Baranov trusses and Assur groups, the two definitions of Baranov truss in Ref. [28] are reviewed as follows: Definition 1. A Baranov truss is a planar zero DOF kinematic chain from which an Assur group can be obtained by removing any one of its links. Definition 2. A Baranov truss is a planar zero DOF kinematic chain, which contains no rigid subchains. In Ref. [28], the Definition 2 of Baranov truss is applied to synthesize Baranov trusses, and the complete set of Baranov trusses with up to 13 links is obtained. In this present study, the main work is to synthesize the corresponding Assur groups from the Baranov trusses obtained in Ref. [28]. The Definition 1 of Baranov truss provides the correlation between N-link Baranov trusses and (N-1)-link Assur groups. In other words, removing each link of an N-link Baranov truss in order, results in N distinct (N-1)-link Assur groups. Thus, the Definition 1 of Baranov truss is considered in this study. However, identical links may exist in the given Baranov truss, which causes symmetry in links. Therefore, isomorphic Assur groups may be generated among the obtained N Assur groups. For example, consider the 9-link Baranov truss in Fig. 1a). It is observed that links 1 and 5 are identical because of the symmetry in the truss. Therefore, generated Assur groups corresponding to the removal of links 1 and 5 are isomorphic. Moreover, three couples of links in the given 11link Baranov truss (i.e. links 2 and 6, links 3 and 7, and links 4 and 8) are symmetric, while link 9 has no symmetry. Consequently, only five non-isomorphic 8-link Assur groups can be obtained from the given 9-link Baranov truss, which are illustrated in Fig. 1b)-f). 2.2. Graph representations of Baranov trusses and Assur groups In a recently published synthesis [28], the structure of a Baranov truss is represented by its topological graph. The method to obtain topological graph of a Baranov truss is addressed here: Links of the Baranov truss and joints are de-

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Fig. 1. (a) A 9-link Baranov truss; (b-f) Five corresponding 8-link Assur groups derived from Fig. 1a).

Fig. 2. (a) The topological graph of the Baranov truss in Fig. 1(a); (b-f) The bicolor topological graphs of the Assur groups in Fig. 1(b-f).

noted as the circle vertices of the graph and edges, respectively. For example, Fig. 2a) illustrates the topological graph of the 9-link Baranov truss in Fig. 1a). In the proposed computer-aided synthesis, the topological graph of a Baranov truss is expressed in the form of adjacency matrix M, where each element of the adjacency matrix xij is defined as follows:



 

M = xi j

n×n

=

1, i f vertices i and j are connected 0, otherwise

(1)

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P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668

Fig. 3. The parallel binary-vertex strings in the topological graph of a 9-link Baranov truss.

Where n is the number of vertices of the topological graph. For example, the adjacency matrix of the topological graph shown in Fig. 2a) can be written as:

⎡

0 1 ⎢ ⎢0 ⎢ ⎢1 ⎢ M =⎢1 ⎢0 ⎢ ⎢0 ⎣ 0 0

1 0 1 0 0 1 0 0 0

0 1 0 1 0 0 0 0 1

1 0 1 0 0 0 0 0 0

1 0 0 0 0 1 0 1 0

0 1 0 0 1 0 1 0 0

0 0 0 0 0 1 0 1 1

0 0 0 0 1 0 1 0 0

⎤

0 0⎥ 1⎥ ⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 1⎥ ⎦ 0 0

(2)

It should be indicated that an Assur group can be generated by removing a link from a given Baranov truss. Moreover, it can be generated by removing a vertex from the topological graph of the given Baranov truss. Thus, in order to distinguish Assur groups originating from Baranov trusses, the concept of the bicolor topological graph in the graph theory is applied to represent the topological structure of the Assur group. The bicolor topological graph of an Assur group is established as the following: (I) Obtaining the topological graph of the corresponding Baranov truss. (II) Using the square vertex to denote the removed vertex. For example, Fig. 1b) shows the Assur group, which originates from removal of link 1 in Fig. 1a). When the vertex 1 is replaced by square vertex 1, the bicolor topological graph of the Assur group shown in Fig. 2b) is obtained. Similarly, bicolor topological graphs of other four Assur groups in Fig. 1c)-f) are presented in Fig. 2c)-f), respectively. In the following work, Baranov trusses and Assur groups are substituted by their topological graphs and bicolor topological graphs (call topological graphs of Assur groups for short) to execute the synthesis study. 2.3. Basic concepts in topological graphs In the topological graphs of Baranov trusses, the degree of a vertex is defined as the number of edges that are directly connected to the vertex. Therefore, a binary vertex is defined as the vertex with two degrees, while a multiple-degree vertex (also called a non-binary vertex) is defined as the vertex that the corresponding degree is greater than two. A binary-vertex string is defined as a string formed by the binary vertices connected in series so that the length of a binary-vertex string is defined as the number of the binary vertices. It should be indicated that the first and last binary vertices are necessarily connected to multiple-degree vertices. Furthermore, when two binary-vertex strings with equivalent lengths are connected with the same multiple-degree vertices, the two binary-vertex strings are called parallel binary-vertex strings. According to the proved proposition in Ref. [28], it is concluded that in the topological graph of a given Baranov truss, each parallel binary-vertex string has only one binary vertex. For example, Fig. 3a) shows the topological graph of a 9-link Baranov truss. It indicates that four binary-vertex strings exist in this graph with length one, namely S1 =(-v2 -), S2 =(-v5 -), S3 =(-v8 -) and S4 =(-v9 -). It is observed that among these strings, S3 and S4 are parallel binary-vertex strings. Although positions of the two parallel binary-vertex strings in a topological graph are exchanged, the form of the topological graph remains unchanged. Thus, it is concluded that the binary vertices on parallel binary-vertex strings are symmetric. Based on the correlation between Baranov trusses and Assur groups, we have: Conclusion 1. If two binary vertices are located on two parallel binary-vertex strings, the two Assur groups generated by removing each of the two vertices are isomorphic.

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Fig. 4. An 11-link Baranov truss and its topological graph.

This conclusion will be utilized in Section 3.1 to reduce the times of the isomorphism identification in the Assur group synthesis. For example, Fig. 3a) and b) indicate that the two parallel binary-vertex strings (i.e. S3 and S4 ) are exchanged and the form of the topological graph remains unchanged. Thus, the two binary vertices v8 and v9 are symmetric. Therefore, it is concluded that the two Assur groups, generated by removing v8 and v9 respectively, are isomorphic. 2.4. Topological graph loops of a Baranov truss In the topological graph of a Baranov truss, vertices and edges are connected in series to form a path where any vertex or edge appears only once so that the starting and ending vertices are different. It should be indicated that the path transforms into a loop, when the starting and the ending vertices are the same. In our previous work [29], the depth-first search (DFS) algorithm was described in detail and it was applied to obtain topological graph loops of a kinematic chain. It is intended to apply the same technique in the present study to obtain all the loops of the synthesized topological graphs of Baranov trusses. For a specific vertex w in the topological graph of a Baranov truss, loops can be divided into two basic types according to whether the loop contains the vertex w or not. Moreover, these two basic types are used for the isomorphism identification of Assur groups in Section 3.2 and the classified databases of Assur groups in Section 6. 3. Structural synthesis method for Assur groups In our previous synthesis [28], a complete set of Baranov trusses with up to 13 links is achieved and the synthesized Baranov trusses are preserved by their simplified characteristic codes [28,30]. Based on these Baranov trusses, the present study is focused on the synthesis of the corresponding Assur groups. Theoretically, from each of the synthesized N-link Baranov trusses, N corresponding (N-1)-link Assur groups can be constructed by removing one link of the truss in order. However, the generation of numerous isomorphic Assur groups is inevitable in this way. Thus the whole synthesis process is concentrated on the isomorphism identification of obtained Assur groups. Since all available Baranov trusses synthesized in Ref. [28] are non-isomorphic, the isomorphism identification is only carried out among the Assur groups originated from a shared Baranov truss, causing by the symmetric links of the Baranov truss. It is intended to investigate the problem of the link symmetry identification for a given Baranov truss. In order to facilitate the computer-aided synthesis of Assur groups, what follows the structure of a Baranov truss is substituted by the corresponding topological graph. It should be indicated that the link symmetry identification of a Baranov truss is equivalent to the vertex symmetry identification in its topological graph. Aiming to resolve the problem of the vertex symmetry identification in the topological graph of a Baranov truss, the definitions of primary and secondary vertex-classified sets are introduced in Section 3.1 to classify the vertices for reducing the corresponding times of the isomorphism identification in the synthesis process of Assur groups. It should be indicated that vertex-symmetry codes are presented in Section 3.2. In order to eliminate the isomorphic Assur groups, the codes are applied to identify the symmetry of vertices in the topological graphs of Baranov trusses. The systematic synthesis procedure of structural synthesis of Assur groups with up to 12 links is addressed in Section 3.3. Fig. 4 shows an 11-link Baranov truss together with its topological graph, which is taken as an example to illustrate the algorithm in detail. 3.1. Vertex-classified sets In the topological graph of a given Baranov truss, the vertices with different numbers of degrees are non-symmetry with each other. In other words, all vertices with the same number of degrees are placed in a set, named as the primary vertex-classified set. A primary vertex-classified set with i degrees of vertices is defined by Vi . Considering the topological graph in Fig. 4b), it contains five vertices (v3 , v7 , v9 , v10 and v11 ) with two degrees, four vertices (v2 , v4 , v5 and v6 ) with three degrees and two vertices (v1 and v8 ) with four degrees. Thus three primary vertex-

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P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668 Table 1 Degree-sequences of vertices in Fig. 4b). Primary vertex-classified sets

Vertices

Adjacent vertices

Degree-sequences of vertices

V2

v3 v7 v9 v10 v11 v2 v4 v5 v6 v1 v8

v2 , v6 , v1 , v1 , v5 , v1 , v1 , v4 , v2 , v2 , v7 ,

33 43 44 44 43 432 432 332 332 3322 2222

V3

V4

v4 v8 v8 v8 v8 v3 , v3 , v6 , v5 , v4 , v9 ,

v6 v5 v11 v7 v9 , v10 v10 , v11

classified sets are obtained as follows:



V2 = (v3 , v7 , v9 , v10 , v11 ) V3 = (v2 , v4 , v5 , v6 ) V4 = (v1 , v8 )

(3)

For a vertex v in the topological graph of a given Baranov truss, the degree-sequence of the vertex v is generated by degrees of the adjacent vertices, where the degrees of the adjacent vertices are ranked from the largest to the smallest. For example, the vertex v2 in Fig. 4b) has three adjacent vertices namely v1 , v3 and v6 . Thus the degree-sequence of the vertex v2 is 432. Similarly, the degree-sequences of each vertex in Fig. 4b) can be obtained, as listed in Table 1. In a primary vertex-classified set, all vertices with the same degree-sequence are placed in a set named as the secondary vertex-classified set. For example, in the primary vertex-classified set V2 in Table 1, the degree-sequences of vertices v3 , v7 , v9 , v10 and v11 are 33, 43, 44, 44 and 43, respectively. Thus three secondary vertex-classified sets, named W1 = (v3 ), W2 = (v7 , v11 ) and W3 = (v9 , v10 ), can be obtained. Similarly, all primary vertex-classified sets in Fig. 4b) are obtained as follows.

⎧ ⎪ ⎪W1 = (v3 ) ⎪ W2 = (v7 , v11 ) ⎪ ⎪ ⎪W = (v , v ) ⎨ 3 9 10 W4 = (v2 , v4 ) ⎪ ⎪ W5 = (v5 , v6 ) ⎪ ⎪ ⎪ ⎪ ⎩W6 = (v1 ) W7 = (v8 )

(4)

It should be indicated that for the topological graph of a given Baranov truss, the vertices in different secondary vertexclassified sets are non-symmetric, so the Assur groups generated by removing each of these vertices are not isomorphic. In other words, the isomorphism identification for synthesized Assur groups is only concentrated on generated Assur groups by removing each of vertices in a same secondary vertex-classified set. Therefore, based on the definitions of the primary and the secondary vertex-classified sets, two conclusions are utilized to reduce the times of the isomorphism identification. It should be indicated that generated Assur groups derived from the two conclusions are unnecessary for the isomorphism identification discussed in Section 3.2. The conclusion 1 discussed in Section 2.3 indicates that the Assur groups generated by removing each of the binary vertices, which are located on parallel binary-vertex strings respectively, are isomorphic. Thus, in the secondary vertexclassified set W3 , binary vertices v9 and v10 are located on two parallel binary-vertex strings namely S1 =(-v9 -) and S2 =(v10 -), respectively. Based on the discussed conclusion, the two Assur groups generated by removing vertices v9 and v10 , as shown in Fig. 5, are isomorphic. It is concluded that only the Assur group generated by removing v9 can be stored. Conclusion 2. If only one vertex exists in a secondary vertex-classified set, the specific Assur group generated by removing this vertex will not be isomorphic with other Assur groups. For example, Eq. (4) shows that three secondary vertex-classified sets can be found, which contain only one vertex. These set are called W1 =(v3 ), W6 =(v1 ), and W7 =(v8 ). Fig. 6 shows that three non-isomorphic Assur groups are obtained by removing the vertices v3 , v1 and v8 , respectively. Moreover, these three Assur groups do not need to identify isomorphism with other generated Assur groups. The conclusions 1 and 2 indicate that among Assur groups generated from the 11-link Baranov truss in Fig. 4, only 3 times of isomorphism identification are carried out for Assur groups generated by removing each of vertices in a same secondary vertex-classified set (namely W2 , W4 , and W5 ). 3.2. Vertex-symmetry codes In the topological graph of a Baranov truss, the isomorphism identification of synthesized Assur groups is only focused on ones generated by removing each of the vertices in a same secondary vertex-classified set, which discussed in Section 3.1.

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Fig. 5. Two isomorphic Assur groups synthesized from the 11-link Baranov truss in Fig. 4.

Fig. 6. Three non-isomorphic Assur groups synthesized from the 9-link Baranov truss in Fig. 4.

In this part, it is assumed that the vertex w is one of the vertices in a secondary vertex-classified set, and the steps to obtain the vertex-symmetry code of the vertex w are addressed as follows in detail. Moreover, the vertex-symmetry code of the vertex w can uniquely represent the Assur group generated by removing the vertex w from the topological graph of the corresponding Baranov truss. Step 1. Select the maximum loops of the vertex w Our previous work [29] applied the depth-first search (DFS) algorithm to obtain all loops of the topological graph of a kinematic chain. It should be indicated that the same technique is applied in the present study for obtaining all loops of the topological graph of a given Baranov truss. In the topological graph of a given Baranov truss, according to whether loops contain the vertex w or not, they are divided into two basic types. Moreover, among all loops containing the vertex w, the loop with the largest number of vertices is defined as the maximum loop of the vertex w. In this step, all the maximum loops of the vertex w are preserved. For example, Table 2 lists all loops of the topological graph in Fig. 4b). Considering the vertex v2 in the secondary vertex-classified set W3 , among the 23 loops, 17 loops contain the vertex v2 which are signed by ‘∗ ’ in Table 2. By definition, there are four maximum loops of the vertex v2 , namely L2 , L3 , L12 , and L13 , which contain 9 vertices, respectively. Step 2. Determine perimeter loops of the vertex w For each of the obtained maximum loops of the vertex w, starting with the vertex w, degree-permutations of the maximum loop are generated twice through sequencing degrees of vertices one by one along the maximum loop, one time in the clockwise direction and the other time in the anticlockwise direction. With obtained the two degreepermutations of the maximum loop regarded as numbers, the bigger one is defined as the degree-permutation number of the maximum loop.

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P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668 Table 2 All loops of the topological graph in Fig. 4b). Loop

Number of vertices ∗

L 1 =( v 1 , v 2 , v 3 , v 4 , v 1 ) L 2 =( v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 , v 8 , v 9 , v 1 )∗ L3 =(v1 , v2 , v3 , v4 , v5 , v6 , v7 , v8 , v10 , v1 )∗ L4 =(v1 , v2 , v3 , v4 , v5 , v11 , v8 , v9 , v1 )∗ L5 =(v1 , v2 , v3 , v4 , v5 , v11 , v8 , v10 , v1 )∗ L 6 =( v 1 , v 2 , v 6 , v 5 , v 4 , v 1 )∗ L7 =(v1 , v2 , v6 , v5 , v11 , v8 , v9 , v1 )∗ L8 =(v1 , v2 , v6 , v5 , v11 , v8 , v10 , v1 )∗ L 9 =( v 1 , v 2 , v 6 , v 7 , v 8 , v 9 , v 1 )∗ L10 =(v1 , v2 , v6 , v7 , v8 , v10 , v1 )∗ L11 =(v1 , v2 , v6 , v7 , v8 , v11 , v5 , v4 , v1 )∗ L12 =(v1 , v4 , v3 , v2 , v6 , v5 , v11 , v8 , v9 , v1 )∗

Loop L13 =(v1 , L14 =(v1 , L15 =(v1 , L16 =(v1 , L17 =(v1 , L18 =(v1 , L19 =(v1 , L20 =(v1 , L21 =(v2 , L22 =(v2 , L23 =(v5 , –

4 9 9 8 8 5 7 7 6 6 8 9

Number of vertices v4 , v4 , v4 , v4 , v4 , v4 , v4 , v9 , v3 , v3 , v6 ,

v3 , v3 , v3 , v5 , v5 , v5 , v5 , v8 , v4 , v4 , v7 ,

v2 , v6 , v5 , v11 , v8 , v10 , v1 ) v 2 , v 6 , v 7 , v 8 , v 9 , v 1 )∗ v2 , v6 , v7 , v8 , v10 , v1 )∗ v6 , v7 , v8 , v9 , v1 ) v6 , v7 , v8 , v10 , v1 ) v11 , v8 , v9 , v1 ) v11 , v8 , v10 , v1 ) v10 , v1 ) v 5 , v 6 , v 2 )∗ v5 , v11 , v8 , v7 , v6 , v2 )∗ v8 , v11 , v5 )

∗

9 8 8 7 7 6 6 4 5 8 5 –

Fig. 7. The topological graphs obtained by re-sketching the four maximum loops as the outmost loops. Table 3 Degree-permutations of the maximum loops L2 , L3 , L12 , and L13 . The maximum loop

Direction

Vertices along the maximum loop

Degree-permutations of the maximum loop

L2

Clockwise Anticlockwise Clockwise Anticlockwise Clockwise Anticlockwise Clockwise Anticlockwise

v2 , v2 , v2 , v2 , v2 , v2 , v2 , v2 ,

323,332,424 342,423,332 323,332,424 342,423,332 333,242,432 323,424,233 333,242,432 323,424,233

L3 L12 L13

v3 , v1 , v3 , v1 , v6 , v3 , v6 , v3 ,

v4 , v5 , v6 , v7 , v8 , v9 , v1 , v2 v9 , v8 , v7 , v6 , v5 , v4 , v3 , v2 v4 , v5 , v6 , v7 , v8 , v10 , v1 , v2 v10 , v8 , v7 , v6 , v5 , v4 , v3 , v2 v5 , v11 , v8 , v9 , v1 , v4 , v3 , v2 v4 , v1 , v9 , v8 , v11 , v5 , v6 , v2 v5 , v11 , v8 , v10 , v1 , v4 , v3 , v2 v4 , v1 , v10 , v8 , v11 , v5 , v6 , v2

Moreover, among all maximum loops of the vertex w, the one with the largest degree-permutation number is defined as the perimeter loop of the vertex w. With vertices on the perimeter loop sketched on the outmost loop and remaining vertices placed inside of the outmost loop, the obtained topological graph is called a perimeter graph of the vertex w. In this step, all obtained perimeter graphs of the vertex w are preserved. Table 2 indicates that four maximum loops of the vertex v2 , namely L2 , L3 , L12 and L13 , are obtained. With the vertex v2 as the starting vertex, Fig. 7 shows the topological graphs obtained by re-sketching the four maximum loops, including L2 , L3 , L12 and L13 as the outermost loops derived from Fig. 4b). Furthermore, Table 3 shows that the degreepermutation number of the maximum loop L2 in clockwise direction is 323,332,424, while the one in anticlockwise direction is 342,423,332. Therefore, it is concluded that the degree-permutation number of L2 is 342,423,332, which starts from the vertex v2 in anticlockwise direction. Similarly, the degree-permutation numbers of L3 , L12 , and L13 are 342,423,332, 333,242,432 and 333,242,432, respectively. It should be indicated that the maximum loops L2 and L3 (both in the anticlockwise direction) are determined as perimeter loops of the vertex v2 . Fig. 7a) and b) show the corresponding perimeter graphs. Step 3. Obtain the canonical perimeter graphs of the vertex w According to obtained perimeter loops of the vertex w, the perimeter graphs, whose vertices are marked with new labels (step 3.1 and step 3.2), are defined as the canonical perimeter graph of the vertex w. Moreover, the adjacency matrix of a canonical perimeter graph is defined as the canonical adjacency matrix of the vertex w. In this step, all obtained canonical perimeter graphs of the vertex w are preserved. Step 3.1. Starting from the vertex w, new labels of vertices on each perimeter loop are assigned one by one with natural numbers ranging from ‘[1]’ to ‘[k]’, in accordance with the specific direction of the perimeter loop.

P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668

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Fig. 8. The new labels of the vertices on the perimeter loops L2 and L3 .

Fig. 9. The new labels of the vertices inside of the perimeter loops L2 and L3 .

Fig. 10. The shared canonical perimeter graph.

For example, Fig. 7a) and b) show two perimeter loops of the vertex v2 , namely L2 and L3 in the anticlockwise direction. Beginning with the vertex v2 relabeled as ‘[1]’, other eight vertices on two perimeter loops are relabeled from ‘[2]’ to ‘[9]’, as shown in Fig. 8. Step 3.2. Starting from the number ‘[k + 1]’, new labels of vertices inside each perimeter loop are assigned in accordance with nine rules proposed by Ref. [31]. Considering two topological graphs illustrated in Fig. 8a) and b), according to the rule 4 of Ref. [31], the new labels of two vertices inside each perimeter loop are relabeled as ‘[10]’ and ‘[11]’, shown in Fig. 9a) and b), respectively. Moreover, it is found that the two perimeter graphs in Fig. 7a) and b) have a shared canonical perimeter graph, which is illustrated in Fig. 10. Step 4. Acquire the characteristic adjacency matrix of the vertex w For each canonical perimeter graph of the vertex w, the number string (NS) is generated by concatenating elements of the upper-right triangle of its canonical adjacency matrix row by row. Then, the canonical adjacency matrix, which contains the largest value of number strings, is defined as the characteristic adjacency matrix (CAM) and the corresponding canonical perimeter graph is defined as the characteristic perimeter graph of the vertex w. For example, only one canonical perimeter graph is available for the topological graph illustrated in Fig. 4b), where the corresponding number string is NS=10 0 010 010 010 0 0 0101010 0,0 0 0 0 010 0 0 01110 0 0 0 010 0,0 010 0110 0 0 0 0. Therefore, the canonical perimeter graph shown in Fig. 10 is the characteristic perimeter graph. Moreover, the characteristic

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P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668

adjacency matrix is expressed as:

⎡

0 ⎢1 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ CAM=⎢1 ⎢0 ⎢ ⎢0 ⎢ ⎢1 ⎣0 0

{1} 0 1 0 0 0 0 1 0 1 0

0 {1} 0 1 0 0 0 0 0 0 0

0 0 {1} 0 1 0 0 0 0 1 1

0 0 0 {1} 0 1 0 0 0 0 0

[1 ] 0 0 0 {1} 0 1 0 0 0 0

0 0 0 0 0 {1} 0 1 0 0 1

0 (1 ) 0 0 0 0 {1} 0 1 0 0

{1} 0 0 0 0 0 0 {1} 0 0 0

0 (1 ) 0 (1 ) 0 0 0 0 0 0 0

⎤

0 0 ⎥ 0 ⎥ ⎥ (1 )⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ (1 )⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎦ 0

(5)

The characteristic perimeter graph of the vertex w is the unique representation for the Assur group generated by removing the vertex w from the topological graph of a given Baranov truss. In the computer-aid procedure, the characteristic perimeter graph of the vertex w is represented by the characteristic adjacency matrix of the vertex w. If two generated Assur groups have a shared characteristic adjacency matrix, it can be concluded that the two Assur groups are isomorphic. Step 5. Obtain vertex-symmetry code of the vertex w To increase the efficiency of the isomorphism identification in our synthesis procedure, the vertex-symmetry code of the vertex w is developed to replace characteristic adjacency matrix for identifying isomorphism. The steps to acquire the vertex-symmetry code of the vertex w are addressed as follows. If more than one Assur groups have the equal vertex-symmetry codes, it is concluded that these vertices are symmetric with each other. Therefore, Assur groups generated by removing each of the vertices are isomorphic so that only one of them should be preserved. In these steps, alphabetic letters are used to represent multi-digital numbers (A represents 10, B represents 11, C represents 12, and so on). Moreover, it is assumed that there are k vertices on the outmost loop of the obtained characteristic perimeter graph. Step 5.1. The upper-right triangle elements ‘1 in the first row of the CAM, excluding the elements x12 and x1 k , are represented by their column subscripts, denoted by C1 . Especially, if the degree of the vertex w (named vertex [1] in the characteristic perimeter graph) is two, the string C1 is a default string. Step 5.2. Since k vertices on the outmost loop of the obtained characteristic perimeter graph are connected in order, the k upper-right triangle elements x12 , x23 , x34 , …, x( k -1) k , and x1 k of the CAM are 1, denoted by k. Step 5.3. The rest upper-right triangle elements ‘1 of the CAM are represented by their row and column subscripts, denoted by C2 . Step 5.4. All upper-right triangle elements ‘0 of the CAM are omitted. In this step, the character string C1 _k_C2 is defined as the vertex-symmetry code of the vertex w. For example, nine vertices are adjacent in turn on the outmost loop of Fig. 10, and in its CAM (as specified in Eq. (5)): Excluding the elements x12 and x19 , only one upper-right triangle element ‘1 exists, namely x16 enclosed by ‘[]’. So C1 =6 is obtained. The nine upper-right triangle elements x12 , x23 , x34 , …, x89 , and x19 , enclosed by ‘{}’ in Eq. (5), are denoted by number 9. The remaining upper-right triangle elements ‘1 , namely x28 , x2A , x4A , x4B , x7B enclosed by ‘()’ are denoted by their row and column subscripts, including 28, 2A, 4A, 4B and 7B All ‘0 elements in upper-right triangle of CAM are omitted. So the vertex-symmetry code of the vertex v2 is 6_9_282A4A4B7B, which can represent the Assur group generated by removing the vertex v2 from the topological graph illustrated in Fig. 4b). Similarly, dealing with vertices in secondary vertexclassified sets W2 , W3 and W5 in Eq. (4), all vertex-symmetry codes are listed in Table 4. For vertices v7 and v11 in W2 , two vertex-symmetry codes are the same as _9_2A2B474A598B and Assur groups generated by removing v7 and v11 are isomorphic. Thus, only the Assur group generated by removing v7 can be stored, shown in Fig. 11a). For vertices v2 and v4 in W3 , two vertex-symmetry codes are the same as 6_9_282A4A4B7B and the Assur groups generated by removing v2 and v4 are isomorphic. Thus, only the Assur group generated by removing v2 can be stored, shown in Fig. 11b). Moreover, for vertices v5 and v6 in W5 , two vertex-symmetry codes are the same as 5_9_2A366B8A8B and the Assur groups generated by removing v5 and v6 are isomorphic. Thus, only the Assur group generated by removing v5 can be stored, shown in Fig. 11c). Consequently, derived from the topological graph of the 11-link Baranov truss illustrated in Fig. 4, only seven Assur groups are synthesized, which are generated by removing v1 , v2 , v3 , v5 , v7 , v8 and v9 , respectively.

P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668

11

Table 4 Vertex-symmetry codes of vertices derived from the secondary vertex-classified set (SVCS) W2 , W3 and W5 in Eq. (4). SVCS

Vertex

Maximum loops of the vertex

Perimeter loops of the vertex

Vertex-symmetry code of the vertex

W2

v7

L2 , L3

_9_2A2B474A598B

v11

L12 , L13

v2

L2 , L3 , L12 , L13

v4

L2 , L3 , L12 , L13

v5

L2 , L3 , L12 , L13

v6

L2 , L3 , L12 , L13

L 2 =( v 7 , v 8 , v 9 , v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 ) L3 =(v7 , v8 , v10 , v1 , v2 , v3 , v4 , v5 , v6 , v7 ) L12 =(v11 , v8 , v9 , v1 , v4 , v3 , v2 , v6 , v5 , v11 ) L13 =(v11 , v8 , v10 , v1 , v4 , v3 , v2 , v6 , v5 , v11 ) L 2 =( v 2 , v 1 , v 9 , v 8 , v 7 , v 6 , v 5 , v 4 , v 3 , v 2 ) L3 =(v2 , v1 , v10 , v8 , v7 , v6 , v5 , v4 , v3 , v2 ) L12 =(v4 , v1 , v9 , v8 , v11 , v5 , v6 , v2 , v3 , v4 ) L13 =(v4 , v1 , v10 , v8 , v11 , v5 , v6 , v2 , v3 , v4 ) L12 =(v5 , v6 , v2 , v3 , v4 , v1 , v9 , v8 , v11 , v5 ) L13 =(v5 , v6 , v2 , v3 , v4 , v1 , v10 , v8 , v11 , v5 ) L 2 =( v 6 , v 5 , v 4 , v 3 , v 2 , v 1 , v 9 , v 8 , v 7 , v 6 ) L3 =(v6 , v5 , v4 , v3 , v2 , v1 , v10 , v8 , v7 , v6 )

W4

W5

_9_2A2B474A598B 6_9_282A4A4B7B 6_9_282A4A4B7B 5_9_2A366B8A8B 5_9_2A366B8A8B

Fig. 11. Three Assur groups generated by removing v7 , v2 and v5 .

3.3. Systematic synthesis procedure In our previous work [28], structural synthesis method for topological graphs of Baranov trusses are proposed, and synthesis results of Baranov trusses with up to 13 links are presented according to the link assortment arrays. In the present study, on the basis of the structural correlation between Baranov trusses and Assur groups, a computer-aid method for synthesizing all Assur groups with up to 12 links is developed and the detailed steps of the whole synthesis process are listed as follows. Step 1. Begin with the synthesis task for N-link Assur groups. Step 2. Obtain all link assortment arrays [28] of (N + 1)-link Baranov trusses, namely L={L1 , L2 , …, Li ,…, Lr }, and initialize i = 1. Step 3. Acquire all topological graphs of Baranov trusses of Li , namely G= {G1 , G2 , …, Gj , …, Gs }, and initialize j = 1. Step 4. By following the method presented in Section 3.1, obtain the secondary vertex-classified sets of Gj , namely W= {W1 , W2 , …, Wk , …, Wt }, and initialize k = 1. Step 5. Determine whether there exist two or more than two vertices in Wk satisfying the Conclusion 1 in Section 3.1. If so, the Assur groups generated by removing these vertices are isomorphic and go to Step 6; otherwise, go to Step 7. Step 6. Among all the vertices obtained from Step 5, keep the first vertex in Wk , and other vertices are ruled out. Step 7. Rearrange the vertices in Wk as Wk ={v1 , v2 , …, vm ,…, vp }, and initialize m = 1.

12

P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668 Table 5 The results of 2-link Assur groups synthesized from 3-link Baranov trusses. No.

Link assortment arrays

1 [3] Total number

No. of Baranov trusses

No. of Assur groups

1 1

1 1

Table 6 The results of 4-link Assur groups synthesized from 5-link Baranov trusses. No.

Link assortment arrays

1 [3,2] Total number

No. of Baranov trusses

No. of Assur groups

1 1

2 2

Table 7 The results of 6-link Assur groups synthesized from 7-link Baranov trusses. No.

Link assortment arrays

1 [3,4,0] 2 [4,2,1] Total number

No. of Baranov trusses

No. of Assur groups

2 1 3

7 3 10

Table 8 The results of 8-link Assur groups synthesized from 9-link Baranov trusses. No.

Link assortment arrays

1 [3,6,0,0] 2 [4,4,1,0] 3 [5,2,2,0] 4 [6,0,3,0] 5 [5,3,0,1] Total number

No. of Baranov trusses

No. of Assur groups

9 13 4 1 1 28

52 91 23 2 5 173

Table 9 The results of 10-link Assur groups synthesized from 11-link Baranov trusses. No.

Link assortment arrays

1 [3,8,0,0,0] 2 [4,6,1,0,0] 3 [5,4,2,0,0] 4 [6,2,3,0,0] 5 [7,0,4,0,0] 6 [5,5,0,1,0] 7 [6,3,1,1,0] 8 [7,1,2,1,0] 9 [6,4,0,0,1] Total number

No. of Baranov trusses

No. of Assur groups

62 222 183 30 1 35 24 3 2 562

545 2269 1780 263 5 327 217 23 9 5438

Step 8. If p > 1, go to Step 9; otherwise, there is only one vertex in Wk and according to the Conclusion 2 in Section 3.1, the Assur group generated by removing this vertex is non-isomorphic with other Assur groups. So save this Assur group and go to Step 10. Step 9. Based on the method proposed in Section 3.2, obtain the vertex-symmetry code of the vertex vm . If it is identical with one of existing vertex-symmetry codes, delete the code; otherwise save vertex-symmetry code of the vertex vm . Step 10. if m
P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668

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Table 10 The results of 12-link Assur groups synthesized from 13-link Baranov trusses. No.

Link assortment arrays

1 [3,A,0,0,0,0] 2 [4,8,1,0,0,0] 3 [5,6,2,0,0,0] 4 [6,4,3,0,0,0] 5 [7,2,4,0,0,0] 6 [8,0,5,0,0,0] 7 [5,7,0,1,0,0] 8 [6,5,1,1,0,0] 9 [7,3,2,1,0,0] 10 [8,1,3,1,0,0] 11 [7,4,0,2,0,0] 12 [8,2,1,2,0,0] 13 [9,0,2,2,0,0] 14 [6,6,0,0,1,0] 15 [7,4,1,0,1,0] 16 [8,2,2,0,1,0] 17 [9,0,3,0,1,0] 18 [7,5,0,0,0,1] Total number

No. of Baranov trusses

No. of Assur groups

591 4299 7510 3368 367 9 1159 2101 686 42 83 26 2 104 86 16 1 2 20,452

6907 54,380 93,774 40,938 4065 63 14,393 25,569 7892 446 874 253 10 1148 937 143 4 15 251,811

Fig. 12. The synthesis software of 10-link Assur groups from 11-link Baranov trusses corresponding to link assortment [5,4,2,0,0].

of Baranov trusses synthesized in Ref. [28] and corresponding Assur groups are classified and listed by the link assortment arrays [28] of Baranov trusses. Table 9 indicates the synthesis of 10-link Assur groups. Altogether 5438 10-link Assur groups are synthesized from 562 11-link Baranov trusses. The obtained Assur groups and the corresponding Baranov trusses are distributed by 9 link assortment arrays. For example, derived from 183 topological graphs of Baranov trusses corresponding to link assortment array [5,4,2,0,0], altogether 1780 topological graphs of Assur groups can be synthesized. Fig. 12 shows an excerpt (first 15 topological graphs) of 10-link Assur groups corresponding to the discussed link assortment array, which is synthesized and drawn by the automatic software used in this study. Table 10 indicates synthesis results of 12-link Assur groups. Altogether 251,811 12-link Assur groups are synthesized from 20,452 13-link Baranov trusses. Obtained Assur groups and corresponding Baranov trusses are distributed by 18 link assortment arrays. For example, derived from the 7510 topological graphs of Baranov trusses corresponding to link assortment array [5,6,2,0,0,0], altogether 93,774 topological graphs of Assur groups can be synthesized. Fig. 13 shows an excerpt (the

14

P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668

Fig. 13. The synthesis software of 12-link Assur groups from 13-link Baranov trusses corresponding to link assortment [5,6,2,0,0,0].

first 15 topological graphs) of 12-link Assur groups corresponding to the discussed link assortment array, which is synthesized and drawn by the automatic software used in this study. 5. Comparative analysis and discussion Based on the automatic synthesis method developed in this study, all feasible Assur groups can be synthesized from the corresponding Baranov trusses. The synthesis results of Assur groups and the corresponding Baranov trusses derived from our synthesis and the existing references are summarized and compared in Table 11. With the aim to demonstrate the validity of our synthesis, it is necessary to conduct a comparative analysis and discuss the discrepancies found in the existing literature. The synthesis results of 2-link, 4-link, 6-link and 8-link Assur groups are 1, 2, 10 and 173, respectively. Moreover, these synthesized Assur groups are synthesized from 1, 1, 3 and 28 Baranov trusses with 3, 5, 7 and 9 links. The obtained synthesis results, including the numbers of Assur groups and Baranov trusses, are generally verified and accepted in the existing literature. Thus, it is reasonable to make the conclusion that the obtained synthesis results in the present study are accurate. As displayed in Table 11 for the synthesis results of 10-link and 12-link Assur groups, although numerous methods for synthesizing Assur groups are addressed in literature, so far to the best of our knowledge, there is only one synthesis (namely Peisach’s synthesis in Refs. [24,25]), in which the synthesis results of 10-link and 12-link Assur groups are reported. In Ref. [24], thirteen structural conditions of Assur groups are summarized initially. Then the classification and the structural characteristics of Assur groups are analyzed based on their structural conditions. Moreover, the (N + 1) × (N + 1) dimensions structural matrix is applied to generate N-link Assur groups and isomorphism identification for generated Assur groups is executed by their individual symbol codes. Based on the synthesis process, as a result, 5442 10-link and 251638 12-link Assur groups are obtained and reported in Ref. [25]. Compared with the synthesis method in Refs. [24,25], however, this study develops an indirect method to synthesize all Assur groups from the corresponding Baranov trusses. In our previous work [28], 562 11-link Baranov trusses are synthesized from 43 valid contracted graphs (see Section 2 in Ref. [28] for the concept of valid contracted graph). The numbers of Baranov trusses and contracted graphs are confirmed by Ref. [32]. Derived from these 562 confirmed 11-link Baranov trusses, this study synthesizes 5438 10-link Assur groups. Similarly in Ref. [28], 20452 13-link Baranov trusses are synthesized from 406 valid contracted graphs. But as provided in Ref. [32], there are 20429 13-link Baranov trusses and 405 corresponding valid contracted graphs, with one valid contracted graph and 23 13-link Baranov trusses less than ours. It is found that there exists one and only one valid contracted graph in Ref. [28], from which 23 13-link Baranov trusses can be synthesized. The numbers are exactly equal to the differences between our results and Ref. [32]. Thus, the discrepancy between these two results may be attributed to the fact that the found contracted graph is lost in Ref. [32], leading to that the 23 corresponding

P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668

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Table 12 The results of Assur groups classified by the number of external joints. Links of AGs

Corresponding BTs

Number of external joints

Links

LAAs

2

3

4

5

6

7

2

3 [3] Total number

1 1

– –

– –

– –

– –

– –

4

5 [3,2] Total number

1 1

1 1

– –

– –

– –

– –

6

7

[3,4,0] [4,2,1] Total number

3 1 4

4 1 5

– 1 1

– – –

– – –

– – –

8

9

[3,6,0,0] [4,4,1,0] [5,2,2,0] [6,0,3,0] [5,3,0,1] Total number

19 37 12 1 2 71

33 41 5 – 2 81

– 13 6 1 – 20

– – – – 1 1

– – – – – –

– – – – – –

10

11

[3,8,0,0,0] [4,6,1,0,0] [5,4,2,0,0] [6,2,3,0,0] [7,0,4,0,0] [5,5,0,1,0] [6,3,1,1,0] [7,1,2,1,0] [6,4,0,0,1] Total number

156 799 767 126 3 136 103 12 3 2105

389 1248 672 54 – 156 66 3 4 2592

– 222 341 83 2 – 24 5 – 677

– – – – – 35 24 3 – 62

– – – – – – – – 2 2

– – – –

13

1634 16,498 35,028 17,865 2012 33 5317 11,058 3848 240 427 135 6 480 446 75 2 6 95,110

5273 33,583 43,956 13,103 681 – 7917 10,309 2001 42 292 45 – 564 319 25 – 7 118,117

– 4299 14,790 9970 1372 30 – 2101 1357 122 – 26 2 – 86 27 1 – 34,183

– – – – – – 1159 2101 686 42 155 47 2 – – – – – 4192

– – – – – – – – – – – – – 104 86 16 1 – 207

– – – – – – – – – – – – – – – – – 2 2

12

[3,A,0,0,0,0] [4,8,1,0,0,0] [5,6,2,0,0,0] [6,4,3,0,0,0] [7,2,4,0,0,0] [8,0,5,0,0,0] [5,7,0,1,0,0] [6,5,1,1,0,0] [7,3,2,1,0,0] [8,1,3,1,0,0] [7,4,0,2,0,0] [8,2,1,2,0,0] [9,0,2,2,0,0] [6,6,0,0,1,0] [7,4,1,0,1,0] [8,2,2,0,1,0] [9,0,3,0,1,0] [7,5,0,0,0,1] Total number

– – – – –

13-link Baranov trusses cannot be obtained. In general, it can be concluded that our synthesis results of 11-link and 13-link Baranov trusses are correct. Furthermore, based on the correlation between N-link Assur groups and (N + 1)-link Baranov trusses, it is reasonable to believe that our results for 10-link and 12-link Assur groups are reliable. Since only synthesis numbers rather than specific structures of 10-link and 12-link Assur groups are provided in Ref. [25], it is unable to verify discrepant Assur groups from the obtained synthesis results in this study and Ref. [25]. In order to aid comparisons with our synthesis results of Assur groups, the specific structures of all the synthesized Assur groups with up to 12 links are collected and listed in a PDF file as a data set, which is uploaded to Mendeley Data and linked with this study. This PDF file is also published with this study and it will be used for the following methods to compare the results of Assur groups with our synthesis in future. To our best knowledge, the structures of 10-link and 12-link Assur groups are reported for the first time. And it is believed that the results of 10-link and 12-link Assur groups in our synthesis are correct, and will be confirmed and accepted by the following synthesis methods. 6. Classified databases of Assur groups Based on the computer-aided synthesis procedure, the complete set of non-isomorphic Assur groups with up to 12 links is synthesized from corresponding Baranov trusses. For convenient application in the mechanism-based creative design process, all synthesized Assur groups are classified by their structural characteristics, including number of external joints,

16

P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668 Table 13 The results of Assur groups classified by the maximum length of the open paths. Links of AGs

The corresponding BTs

Maximum length of the open paths

Links

LAAs

4

5

6

7

8

9

10

11

12

13

2

3 [3] Total number

1 1

– –

– –

– –

– –

– –

– –

– –

– –

– –

4

5 [3,2] Total number

2 2

– –

– –

– –

– –

– –

– –

– –

– –

– –

6

7

[3,4,0] [4,2,1] Total number

– – –

– 3 3

3 – 3

4 – 4

– – –

– – –

– – –

– – –

– – –

– – –

8

9

[3,6,0,0] [4,4,1,0] [5,2,2,0] [6,0,3,0] [5,3,0,1] Total number

– – – – – –

– – 1 – 1 2

– – 3 2 4 9

– 12 14 – – 26

11 61 5 – – 77

41 18 – – – 59

– – – – – –

– – – – – –

– – – – – –

– – – – – –

10

11

[3,8,0,0,0] [4,6,1,0,0] [5,4,2,0,0] [6,2,3,0,0] [7,0,4,0,0] [5,5,0,1,0] [6,3,1,1,0] [7,1,2,1,0] [6,4,0,0,1] Total number

– – – – – – – – – –

– – – – – – – – – –

– – – 1 – – 8 1 5 15

– – 2 6 1 2 15 9 4 39

– – 59 75 4 44 96 13 – 291

– 101 641 161 – 153 88 – – 1144

90 1245 916 20 – 106 10 – – 2387

455 923 162 – – 22 – – – 1562

– – – – – – – – – –

– – – – – – – – – –

12

13

– – – – – – – – – – – – – – – – – – –

– – – – – – – – – – – – – – – – – – –

– – – – 1 – – – 4 1 2 – 1 – 3 1 – 2 15

– – 1 1 1 – 1 6 26 1 16 11 – 1 25 10 – 8 108

– – 9 24 27 – 9 78 138 33 54 47 9 27 95 30 4 5 589

– – 10 278 217 31 11 737 1114 251 161 143 – 175 315 76 – – 3519

– – 850 4375 1898 32 538 5209 4043 160 444 52 – 400 337 26 – – 18,364

– 907 18,558 20,196 1741 – 4540 12,482 2375 – 179 – – 355 150 – – – 61,483

827 24,387 54,255 14,542 180 – 7041 6303 192 – 18 – – 164 12 – – – 107,921

6080 29,086 20,091 1522 – – 2253 754 – – – – – 26 – – – – 59,812

[3,A,0,0,0,0] [4,8,1,0,0,0] [5,6,2,0,0,0] [6,4,3,0,0,0] [7,2,4,0,0,0] [8,0,5,0,0,0] [5,7,0,1,0,0] [6,5,1,1,0,0] [7,3,2,1,0,0] [8,1,3,1,0,0] [7,4,0,2,0,0] [8,2,1,2,0,0] [9,0,2,2,0,0] [6,6,0,0,1,0] [7,4,1,0,1,0] [8,2,2,0,1,0] [9,0,3,0,1,0] [7,5,0,0,0,1] Total number

maximum length of open paths and number of vertices on the maximum loop. Moreover, three corresponding classified databases are established and proposed in Tables 12–14, respectively. (In Tables 12–14, AG denoting Assur group, BT denoting Baranov truss, LAA denoting link assortment array of Baranov trusses, and alphabetic letter “A” in link assortment array [3,A,0,0,0,0] denoting multi-digital number “10 )

6.1. The database of Assur groups classified by the number of external joints Consider that an Assur group is generated by removing a link from the corresponding Baranov truss. Moreover, the joints of the generated Assur group, which are connected with the removed link, are defined as external joints of the Assur group. Table 12 displays the results of Assur groups with up to 12 links, which are classified by numbers of external joints. It is observed that classified results are according to link assortment arrays of corresponding Baranov trusses. Here is an example. Example 1. . Derived from the 2101 13-link Baranov trusses in the link assortment array [6,5,1,1,0,0], altogether 25569 12link Assur groups can be synthesized (see Table 10). Table 12 indicates that among these 25569 Assur groups, classified results of Assur groups corresponding to 2, 3, 4 and 5 external joints are 11058, 10309, 2101 and 2101, respectively.

P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668

17

Table 14 The results of Assur groups classified by the number of vertices on the maximum loop. Links of AGs

The corresponding BTs

Number of vertices on the maximum loop

Links

LAAs

0

4

5

6

7

8

9

10

11

12

2

3 [3] Total number

1 1

– –

– –

– –

– –

– –

– –

– –

– –

– –

4

5 [3,2] Total number

1 1

1 1

– –

– –

– –

– –

– –

– –

– –

– –

6

7

[3,4,0] [4,2,1] Total number

– 1 1

2 1 3

2 1 3

3 – 3

– – –

– – –

– – –

– – –

– – –

– – –

8

9

[3,6,0,0] [4,4,1,0] [5,2,2,0] [6,0,3,0] [5,3,0,1] Total number

– – – – 1 1

– 7 3 1 1 12

4 15 6 – 1 26

16 30 9 1 2 58

18 24 4 – – 46

14 15 1 – – 30

– – – – – –

– – – – – –

– – – – – –

– – – – – –

10

11

[3,8,0,0,0] [4,6,1,0,0] [5,4,2,0,0] [6,2,3,0,0] [7,0,4,0,0] [5,5,0,1,0] [6,3,1,1,0] [7,1,2,1,0] [6,4,0,0,1] Total number

– – – – – – – – 2 2

– 6 15 7 – 6 10 2 1 47

– 25 60 21 1 20 17 4 1 149

2 101 202 56 1 45 57 5 3 472

27 334 374 60 1 80 56 8 2 942

151 685 526 76 2 90 58 4 – 1592

220 738 464 41 – 67 18 – – 1548

145 380 139 2 – 19 1 – – 686

– – – – – – – – – –

– – – – – – – – – –

12

13

– – – – – – – – – – – – – – – – – 2 2

– – 9 26 5 – 18 50 36 4 9 5 1 12 17 4 – – 196

– 36 197 228 49 2 65 255 169 16 25 15 – 25 32 7 – 2 1123

– 159 698 758 177 7 177 781 491 47 75 28 3 65 89 19 2 3 3579

– 235 1771 2085 449 11 498 1891 1004 96 120 58 – 142 179 33 – 5 8577

10 1274 6918 5891 871 14 1475 4183 1798 96 217 66 6 243 237 43 2 3 23,347

226 6574 18,710 9927 976 18 3216 6721 2102 141 237 68 – 280 228 31 – – 49,455

1723 16,476 28,445 11,808 1143 11 4287 6874 1837 46 160 13 – 247 130 6 – – 73,206

3013 19,564 27,156 8610 377 – 3482 4110 439 – 28 – – 111 24 – – – 66,914

1935 10,062 9870 1605 18 – 1175 704 16 – 3 – – 23 1 – – – 25,412

[3,A,0,0,0,0] [4,8,1,0,0,0] [5,6,2,0,0,0] [6,4,3,0,0,0] [7,2,4,0,0,0] [8,0,5,0,0,0] [5,7,0,1,0,0] [6,5,1,1,0,0] [7,3,2,1,0,0] [8,1,3,1,0,0] [7,4,0,2,0,0] [8,2,1,2,0,0] [9,0,2,2,0,0] [6,6,0,0,1,0] [7,4,1,0,1,0] [8,2,2,0,1,0] [9,0,3,0,1,0] [7,5,0,0,0,1] Total number

Table 11 The comparison of the synthesis results of Assur groups and Baranov trusses. Assur group synthesis

Baranov truss synthesis in our previous work (namely Ref. [28])

Links

Our results

Confirms

Other results

Links

Our results

Confirms

Other results

2 4 6 8 10 12

1 2 10 173 5438 251,811

Well Well Well Well – –

– – – – 5442[25] 251,638[25]

3 5 7 9 11 13

1 1 3 28 562 20,452

Well Well Well Well [32] –

– – – – 239 [33], 568 [34] 20,429 [32]

known known known known

known known known known

6.2. The database of Assur groups classified by the maximum length of the open paths For a specific vertex w in the topological graph of a given Baranov truss, all loops can be divided into two basic types according to whether the loops contain the vertex w or not. For the generated Assur group by removing the vertex w from the topological graph, loops with the vertex w in the corresponding topological graph are changed into open paths. It should be indicated that the number of edges in an open path is defined as the length of the open path.

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P. Huang and H. Ding / Mechanism and Machine Theory 145 (2020) 103668

Table 13 displays the results of Assur groups with up to 12 links, which classified by the maximum length of the open paths. It is observed that all classified results are according to link assortment arrays of corresponding Baranov trusses. Here is an example. Example 2. Derived from 4299 13-link Baranov trusses in the link assortment array [4,8,1,0,0,0], altogether 54380 12-link Assur groups can be synthesized (see Table 10). Table 13 indicates that among these 54380 Assur groups, classified results of Assur groups with the maximum length of open paths being as 11, 12 and 13, are 907, 24387 and 29,086, respectively. 6.3. The database of Assur groups classified by the number of vertices on the maximum loop For a specific vertex w in the topological graph of a given Baranov truss, all loops can be divided into two basic types according to whether the loops contain the vertex w or not. For the Assur group generated by removing the vertex w from the topological graph, loops without the vertex w remain unchanged. It should be indicated that the loop with largest number of vertices is defined as a maximum loop. Table 14 displays the classified synthesis results of Assur groups with up to 12 links, which are classified by number of vertices on the maximum loop. It is observed that classified results are according to link assortment arrays of corresponding Baranov trusses. Moreover, if generated Assur groups contain no loops, it is stipulated that the number of vertices on the maximum loop of these Assur groups is equal to zero. Here is an example. Example 3. Derived from 591 13-link Baranov trusses in the link assortment array [3,A,0,0,0,0], altogether 6907 12-link Assur groups can be synthesized (see Table 10). Table 14 indicates that among these 6907 Assur groups, classified results of Assur groups with the number of vertices on the maximum loop being as 8, 9, 10, 11 and 12, are 10, 226, 1723, 3013 and 1935, respectively. 7. Conclusions The present study proposes a systematic method for synthesizing all Assur groups with 2, 4, 6, 8, 10 and 12 links. Moreover, three corresponding classified databases of synthesized Assur groups according to their structural characteristics are established. The structural synthesis begins with the review of the structural correlation between Assur groups and their corresponding Baranov trusses. Then, the primary and secondary vertex-classified sets are addressed to classify vertices in the topological graph of a given Baranov truss. Assur groups generated by removing vertices in different secondary vertex-classified sets do not require identifying isomorphism with each other. Moreover, based on vertex-symmetry codes used for isomorphism identification, the systematic synthesis procedure is proposed to obtain all Assur groups derived from corresponding Baranov trusses. Furthermore, the complete set of Assur groups with 2, 4, 6, 8, 10 and 12 links is obtained, and the results between ours and the ones in the existing literature are compared and discussed in detail. Besides, three corresponding databases of all synthesized Assur groups are also established, which are classified by their structural characteristics. It is believed that all synthesized Assur groups with up to 12 links and the corresponding classified databases will make a great contribution for the synthesis and analysis of kinematic chains and mechanisms. Declaration of Competing Interest We have no conflicts of interest to declare. 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