A novel graphical joint-joint adjacent matrix method for the automatic sketching of kinematic chains with multiple joints

Mechanism and Machine Theory 150 (2020) 103847

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Research paper

A novel graphical joint-joint adjacent matrix method for the automatic sketching of kinematic chains with multiple joints Wei Sun a,b, Jianyi Kong a,b, Liangbo Sun c,∗ a

Key Laboratory of Metallurgical Equipment and Control Technology, Wuhan University of Science and Technology, 430080 Wuhan, China Engineering Research Center of Metallurgical Automation and Measurement Technology, Wuhan University of Science and Technology, Wuhan 430081, China c School of Machinery and Engineering, Wuhan Polytechnic University, 430080 Wuhan, China b

a r t i c l e

i n f o

Article history: Received 11 January 2020 Revised 7 February 2020 Accepted 11 February 2020

Keywords: Automatic sketching Graphical joint-joint adjacent matrix Kinematic chains Multiple joints Structure synthesis

a b s t r a c t The automatic sketching of kinematic chains (KCs) is an important part of the conceptual design of mechanisms, which is useful for the creative design of mechanisms. In this paper, a novel algorithm for automatic sketching of planar KCs with multiple joints was proposed based on joint-joint adjacent matrix (JJAM). Firstly, the standardization rules of JJAM were introduced, and the loops of planar KCs were extracted from the JJAM using breadth-first search algorithm (BFS). Further, the breadth-first spanning tree of the planar KCs was drawn by minimum crossing rules. Meanwhile, the rules of drawing structure diagram of a single link in JJAM were presented. Moreover, graphical joint-joint adjacent matrix (GJJAM) implying structure diagram between joints was expressed. This matrix was transformed into a graphical joint-joint adjacent matrix of planar KCs by matching structure diagram of the single link. This method has several advantageous features resulting in lower complexity relative to the methods presented in literatures. Finally, the examples demonstrate that the method is novel, efficient and convenient. © 2020 Published by Elsevier Ltd.

1. Introduction With the rapid development of economy, the simple 4- or 5-bar mechanisms cannot fully satisfy the need for industrial production. In recent years, the research of multi-degree-of-freedom mechanism has attracted the attention. Usually, the topological structure of kinematic chains is represented by the topological graph [1], symbol and text description for the conceptual design [2]. The topological graph is more intuitive and simple, which is often used to establish the mathematical relationship. And it is convenient for computer automatic processing when converted to a matrix. The special treatment of multiple joint is required in the analysis and synthesis of kinematic chains, which is unfavorable to the rapid development of modern mechanism research. The description and analysis of the topological structure including multiple joint have been a hot topic in the field of mechanisms. Yan and Hsu [3] used the contracted link adjacency matrix to describe and determine kinematic chain with multiple joints. Hsu [4] simplified a multiple joint into a single joint string by using a label tree. Ding et al. [5–7] proposed a new kind of bicolor topological graph to represent the topological structures of multiple joint kinematic chains. Represent links with solid vertices “●” and hollow vertices “◦” denote multiple joints.

∗

Corresponding author. E-mail addresses: [email protected] (W. Sun), [email protected] (J. Kong), [email protected] (L. Sun).

https://doi.org/10.1016/j.mechmachtheory.2020.103847 0094-114X/© 2020 Published by Elsevier Ltd.

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W. Sun, J. Kong and L. Sun / Mechanism and Machine Theory 150 (2020) 103847

Due to the complexity and variety of multi-dof kinematic chains, the research on structural synthesis method is more necessary, which is important for the conceptual design of mechanisms. Manolescu [8] presented the structural synthesis method of kinematic chains using the concept of “Assur Structure”. Soni [9] conducted the structural analysis of space linkages with two general constraints by using Franke’s condensed notation. Mruthyunjaya [10] presented a general method of structural synthesis by the transformation of the corresponding “binary chains”. Sohn and Freudenstein [11] applied the dual graph concept to enumerate the kinematic structures of mechanisms. Hwang and Hwang [12] developed a method using the contracted link adjacency matrix to synthesize kinematic chains. Yan and Hwang [13,14] developed a systematic and precise approach for enumerating non-isomorphic kinematic chains based on the theory of permutation groups. Rao [15,16] presented the Hamming number technique to enumerate kinematic chains. Butcher and Hartman [17] proposed an algorithm to exhaustively enumerate and structurally classify planar simple-jointed kinematic chains using the hierarchical representation of Fang and Freudenstein. Sunkari and Schmidt [18] used a McKay-type algorithm in combination with an efficient degeneracy testing algorithm for the synthesis of planar mechanisms. Martins et al. [19] developed an Assur group based method of exclusive generation of fractionated kinematic chains. Gavoille [20,21] used unified planar linkage consisting of rigid blocks connected by stiffness-varying zero-length springs and formulated the synthesis problem as iterative design optimization problem. Nie et al. [22] presented an addition method with 2 links and 3 pairs based on graph theory for type synthesis of kinematic chains. Yan and Chiu [23,24] provided an enumeration algorithm based on graph theory for constructing various atlases of generalized kinematic chains with up to 16 links. H. Ding et al. [25,26] introduced an improved algorithm to obtain the characteristic number string of topological graphs with two multiple joints to enhance the efficiency of isomorphism identification and synthesized a complete set of planar non-fractionated simple joint mechanisms with up to 19 links. Eleashy [27] proposed a systematic methodology to generate all solutions of planar 8-bar kinematic chains with up to 3 prismatic pairs. The synthesis work is closely followed by the automatic sketching of kinematic chains for the convenience of designing mechanisms. Because there are tens of thousands of synthesis results, the sketching of kinematic chains is desired to be automated to improve the efficiency. Dobrjanskyj and Freudenstein [28] proposed a method for the automatic sketching of the graph of a mechanism defined by its incidence matrix. Olson et al. [29] presented the development of a computer algorithm for automatic sketching of kinematic chains as part of the computer-aided type synthesis process. Chieng and Hoeltzel [30] proposed the first automated method based on the exhaustive use of independent loops of the graphs, but also including the peripheral loop. Belfiore and Pennestri [31] described an original method for the fully automatic sketching of planar kinematic chains. This procedure may ease the interpretation of the results obtained during the enumeration of complex multi-loop kinematic chains. Mauskar and Krishnamurty [32] presented a simple and robust methodical procedure for automatically sketching every type of kinematic chains regardless of the number of links and degrees of freedom. Nie et al. [33] developed a method based on independent loops addition or subtraction to sketch kinematic chains. Pucheta et al. [34,35] developed a loop permutation method to sketch kinematic chains. Link crossings were tested by verifying all possible edge crossings in each two adjacent links, and eliminated by moving the involved joint. Bedi and Sanyal [36] presented a simple and reliable method to identify the joints and loops as the basic constituents of a kinematic chain. Hsieh et al. [37,38] applied the concept of a line graph to sketch the generalized kinematic chains, illustrated by those chains with up to 8 links. Recently, Huang et al. [39] proposed a new method for the connectivity calculation between two links, based on which an automatic method to acquire the connectivity matrix for planar closed kinematic chains was developed. W. Yang et al. [40,41] proposed a new method for the automatic sketching of planar kinematic chains by converting the corresponding topological graphs with the aid of a line graph. Most of the existing methods only focused the sketching of kinematic chains with simple joints. No matter what method is used for automatic sketching of planar kinematic chains, it is necessary in order to solve the problem of uniqueness and comprehensiveness of the topological structure of the kinematic chains, especially with multiple joints. The automatic sketching procedure will enable the designer to systematically visualize the link-joint inter-relationship of the enumerated mechanisms, thus enhancing the overall efficiency of the mechanism design process. The main purpose of the graph layout is to obtain a representation form to facilitatethe the visualization and understanding of the topological structure of the kinematic chains. In this paper, a joint-joint adjacent matrix is introduced to describe the multiple joints kinematic chains, which can uniquely represent the topological structure of kinematic chains. And this matrix can be also used as the data storage structure of kinematic chains. Firstly, the standardization rules of joint-joint adjacent matrix are presented. Then, relevant loops were derived from the joint-joint adjacent matrix using the breadth-first search algorithm. The breadth-first spanning tree of a kinematic chain was drawn by minimum crossing rule. Meanwhile, the rules of drawing structure diagram of a single link in a joint-joint adjacent matrix were presented. Further, a graphical joint-joint adjacent matrix implying structure diagram between joints was expressed. And this graphical matrix was transformed into a graphical joint-joint adjacent matrix of a kinematic chain by matching structure diagram of the single link. Finally, three groups cases are provided to demonstrate the effectiveness of this method.

2. Description of the joint-joint adjacent matrix The joint-joint adjacent matrix has been introduced in our previous work [42]. A 10-bar kinematic chain with two multiple joints as an example is shown in Fig. 1.

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Fig. 1. A 10-bar kinematic chain C with two multiple joints.

Fig. 2. The joint code of joint 5.

The joint-joint adjacent matrix AC of kinematic chain C is expressed as:

AC =

⎧ 0 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎨1 1

⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪0 ⎪ ⎪ ⎪ ⎪0 ⎪ ⎪ ⎪ ⎩0

2

2 0 3 0 0 0 0 0 0 0 2

0 3 0 4 0 0 0 4 8 0 0

1 0 4 0 1 1 0 4 0 0 0

1 0 0 1 0 1 6 0 0 0 0

1 0 0 1 1 0 0 0 7 0 0

0 0 0 0 6 0 0 5 0 0 0

0 0 4 4 0 0 5 0 0 0 0

0 0 8 0 0 7 0 0 0 9 0

0 0 0 0 0 0 0 0 9 0 10

⎫

2⎪ ⎪ 2⎪ ⎪ ⎪ ⎪ 0⎪ ⎪ ⎪ 0⎪ ⎪ ⎪ ⎬ 0⎪ 0 ⎪ 0⎪ ⎪ ⎪ 0⎪ ⎪ ⎪ ⎪ 0⎪ ⎪ ⎪ 10⎪ ⎪ ⎭ 0

(1)

In order to aggregate the serial number of multiple links in joint-joint adjacent matrix, the standardized rules of jointjoint adjacent matrix are proposed as follows: (1) The link having larger degree is arranged in priority. If the multi-links have the same degree, the multi-link having smaller subsequent array of link code is arranged in priority. (2) The joints of multi-link having larger subscript array and subsequent array of joint code are arranged in priority. (3) The joints of binary link having larger subscript array and subsequent array of joint code are arranged in priority. The joint code of joint 5 in kinematic chain C is shown in Fig. 2. The joint-joint adjacent matrix is standardized by the above rules and the serial number of the joints of kinematic chain C is re-numbered as shown in Table 1. The normalized joint-joint adjacent matrix ACS of the kinematic chain C is obtained by transforming the row number and column number of the original joint-joint adjacent matrix.

ACS =

⎧ 0 ⎪ ⎪ ⎪ ⎪1 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎨0 0

⎪ ⎪ 2 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩0 0

1 0 1 1 4 4 0 0 0 0 0

1 1 0 1 0 0 0 0 7 0 0

1 1 1 0 0 0 0 0 0 6 0

0 4 0 0 0 4 3 0 8 0 0

0 4 0 0 4 0 0 0 0 5 0

2 0 0 0 3 0 0 2 0 0 0

2 0 0 0 0 0 2 0 0 0 10

0 0 7 0 8 0 0 0 0 0 9

0 0 0 6 0 5 0 0 0 0 0

⎫

0⎪ ⎪ 0⎪ ⎪ ⎪ ⎪ 0⎪ ⎪ ⎪ 0⎪ ⎪ ⎪ ⎬ 0⎪ 0 ⎪ 0⎪ ⎪ ⎪ 10⎪ ⎪ ⎪ ⎪ 9⎪ ⎪ ⎪ 0⎪ ⎪ ⎭ 0

The structure diagram of kinematic chain C with renumbered joints is shown in Fig. 3.

(2)

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W. Sun, J. Kong and L. Sun / Mechanism and Machine Theory 150 (2020) 103847 Table 1 The new number of joints. The new serial number of joints

The original serial number

The link code

The joint code

1 2 3 4 5 6 7 8 9 10 11

1 4 6 5 3 8 2 11 9 7 10

N4 -0012

/ / /

J00 − 43 J00 − 43 J01 − 40 J02 − 43 J011 − 330 J02 − 34 J01 − 30 J02 − 30 J112 − 403 J11 − 43 J11 − 30

No. of joints

The array code

The first layer

1 2 3 4

(1,1,2,2) (1,1,4,4) (1,1,7) (1,1,6)

1 1 1 1

5

(3,4,4,8)

6 7 8

(4,4,5) (2,2,3) (2,2,10)

9

(7,8,9)

10 11

(5,6) (9,10)

N3 -002 N3 -012

Table 2 The loop information.

→ → → →

2 4 7 6

The second layer

The third layer



4→3 4→8

4→5 2→3 2 → 10 7→8 7→9 6→5 9 → 10

Fig. 3. The structure diagram of kinematic chain Cs.

It is necessary to determine the basic loop set for the mathematical description of the loop relationship, including the rank of the loop set and the determinable conditions. The hierarchical analysis of links is performed on the information extracted from the standardized joint-joint adjacent matrix based on Breadth-First-Search algorithm [43–46]. The multi-link with the largest degree becomes as the starting point. The links connected to this multi-link are the first layer, and the links connected to the links of the first layer are the second layer and so on. When the same serial number of a link appears, a complete loop is found. And the leaves are not rooting at the end of the branch any more. Loops of kinematic chain Cs is shown in Table 2. Where, the arrow indicates the connection relationship between links. For example, link 1, a quaternary link, is selected as a root node by analyzing the link code. The leaves of the first layer contain link 2, link 4, link 6, link 7, which are connected to the root node 1. The leaves of the second layer contain link 3, link 5, link 8, link 9, link 10. In this layer, the link 3, link 5, link 8 appear twice, which means the leaves knot and a loop has been found. Those links do not continue

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Fig. 4. The flow-chart of automatic sketching.

to grow branches. In the third layer, only the link 10 corresponds to the link 10 of the second layer, which also confirms a loop. So loops of kinematic chain Cs in Fig. 4 as follows:

L1 = {1, 2, 3, 4} L2 = {1, 6, 5, 4} L3 = {1, 4, 8, 7}

(3)

L4 = {1, 7, 9, 10, 2} Where, the numbers in Eq. (3) represent the serial number of links. All loops can be obtained by the generalized operation “” of loops [47]. Other loops of kinematic chain Cs as follows:

L5 =L3  L4 = {1, 4, 8, 9, 10, 2} L6 = L1  L3  L4 = {2, 3, 8, 9, 10}

(4)

One of larger loops is the largest loop, which the sum number of degree of all links is largest. The largest loop of kinematic chain Cs is L5 {1, 4, 8, 9, 10, 2}.

3. Automatic sketching process Sketching work aims to generate visual structure diagram of kinematic chains. It arises frequently in the conceptual design of mechanisms, especially in the enumeration process where a large number of type synthesis generated. One of the problems in the automatic sketching of kinematic chains is to avoid crossing the edges and find the best representation of the graph. The topological graphs of planar kinematic chains can be classified into planar and non-planar graphs. A planar graph can be drawn in the plane without edge crossing, whereas, a non-planar graph contains at least one edge crossing. Therefore, the optimal layout of the topological graph should satisfy the minimum crossing rules. The specific process is shown in Fig. 4.

3.1. Drawing breadth-first spanning tree In order to better observe the layout of a kinematic chain, it is transformed into a breadth-first spanning tree. A graph data structure of table headers and nodes is established as shown in Fig. 5. The left side of Fig. 5 is the list, whose fourth column represents the number of nodes that this node is connected to the next level. For instance, the next layer nodes of link 4 are link 3, link 5 and link 8, so the number of nodes in the table is 3. The right side of Fig. 5 is a data structure of each node. The first number represents the serial number of links, and the second number represents the serial number of the joint which is connected to this link by the previous link. For example,

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Fig. 5. The graph data structure of kinematic chains.

Fig. 6. Tree diagram of kinematic chain Cs.

Fig. 7. Tree diagram with serial number of joints.

the link 2 is connected to link 1 in layer 0 at joint 1, so the node of link 2 is represented as “2, 1, pointer”. The tree diagram of kinematic chain Cs is illustrated in Fig. 6. The minimum crossing rules of drawing tree as follows: Rule 1: the links of the kinematic chains are classified by layer. The layered links of kinematic chain Cs are shown in Fig. 6(a). The degree of the nodes in the first layer and second layer is full. Where, link 7 is a ternary link in the tree diagram as shown in Fig. 6(b). In fact, it is a binary link in the kinematic chain Cs because link 7 has a multiple joint 9 as shown in Fig. 7. In the tree diagram, the link 3, link 5, link 8 and link 9 appear twice, which means the leaves knot and a loop has been found. And the number of leaves is equal to the degree of this link.

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Fig. 8. Structure diagrams of a ternary link.

Rule 2: branches with the same joint must be put together. For example, the joint 5 in the second layer appears twice in Fig. 7. Joint 5 is easily judged to be a multiple joint. So, link 3 and link 8 in layer 2 must be put together. The cross-check function of lines between leaves is set during leaf or leaf node movement. If the topological graphs of kinematic chains are planar, the final cross value of lines between leaves must be 0. If ones are non-planar, the cross value is greater than or equal to 1. The smaller the value, the better. The cross value of lines between leaves in Fig. 6(b) is 2. If link 3 and link 5 in layer 2 are exchanged in the branch of link 4, the cross value of lines between leaves increases to 3. If link 5 and link 8 swap positions, the cross value is reduced to 1. Where, the number without a circle in Fig. 7 is the serial number of links and the number with a circle indicates the serial number of joints. Rule 3: the same links of the same layer should be put together. And links which have a same leaf are put together as much as possible. According to the loop characteristic, the serial numbers of joints in each layer are adjusted as shown in Fig. 7. The minimal independent loops of kinematic chains are used to layout the spanning tree considering edge crossing avoidance. For instance, one of the minimal independent loops of kinematic chain Cs isL1 = {1, 2, 3, 4}, so the link 3 in layer 2 are placed together, and the link 2 is adjacent to link 4 in layer 1.If the topological graphs of kinematic chains are non-planar, lines between leaves must intersect in tree diagram. If the cross value is greater than 1, the leaves on the loop containing the cross are adjusted. If there are intersections in the independent loop set, the links in layer 1 are rearranged. As the leaves are adjusted in each layer, the goal is to reduce the cross value of lines between leaves. 3.2. Drawing structure diagram of single link The position of links in a kinematic chain can be obtained from the breadth-first spanning tree diagram. However, the specific shape of each link cannot be obtained. Therefore, structure diagram of each individual link needs to be drawn. A binary link has only one shape. The link 4 is a ternary link in kinematic chain Cs. The joint-joint adjacent sub-matrix of the link 4 is represented as a1 , a1 ⊆ACS .



a1 =

0 4 4

4 0 4

4 4 0



(5)

The diagonal elements ai, i of joint-joint adjacent matrix ACS express joints of kinematic chain Cs. The diagonal elements ai, i are moved to the boundary of this matrix which can restore shape of the link. Four shapes of link 4 (a ternary link) can be obtained as shown in Fig. 8. Each circle represents a joint of link 4 in Fig. 8. The links have different structural shapes by transforming different row number and column number of the matrix. Likewise, link 1 is a quaternary link in kinematic chain Cs. The joint-joint adjacent sub-matrix of the link 1 is represented as a2 , a2 ⊆ACS .

a2 =

⎧ ⎪ ⎨0

1

⎪ ⎩1 1

1 0 1 1

1 1 0 1

⎫

1⎪ ⎬ 1 1⎪ ⎭ 0

(6)

Fourteen shapes of link 1(a quaternary link) can be obtained as shown in Fig. 9. The type of three points in a straight line is not listed, which is equivalent to the shapes of ternary links. This method avoids the complex calculation of interior angles of the components in the literature [41]. 3.3. Drawing structure diagram of kinematic chain The layout of the joints must need to be known to draw the structure diagram of a kinematic chain. And the spanning tree diagram with the serial number of joints is shown in Fig. 7. The joint-joint adjacency matrix implies the structure diagram of a kinematic chain. Then, the rules of the row number and column number arrangement of graphical matrix are established. Rule 4 (row numbering rules): the row numbers of the graphical matrix are arranged according to the postorder traversal (LRD) method of the tree. In a breadth-first tree, the left sub-tree (leaf) is traversed firstly, then the right sub-tree (leaf) is traversed, and finally the root node is traversed. For example, the row number of the graphical matrix for the 10-bar kinematic chain Cs is “8, 7, 1, 5, 6, 2, 10, 4, 9, 11, 3”.

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Fig. 9. Structure diagrams of link 1. Table 3 The layered joints. No. of layer

The leaf

The joints

1 2.1 2.2 2.3 3

link link link link link

1,2,4,3 7,5 6,10 5,9 8,11

1 3 5 8 10

The joints of different layers need further be analyzed for the column number arrangement of the graphical matrix. So layers of joints are sorted according to the spanning tree diagram in Fig. 7. The joints of link 1 (root node) is used as the first layer. The joints of link 3, link 5 and link 8 belong to the second layer. The joints of link 10 are the third layer. The joints of each layer having shorter length of the loop are arranged in priority. For example, the shortest loops of kinematic chain Cs are L1 = {1, 2, 3, 4} andL2 = {1, 6, 5, 4}. So the joints of link 3 and link 5 are arranged firstly. Layered joints of kinematic chain Cs are shown in Table 3. Where, the joint numbers in each row in Table 3 are arranged from left to right according to the breadth-first tree in Fig. 7. Rule 5 (column numbering rules): the column number of the graphical matrix is arranged according to the layered joints. If there is a multiple joint, record it once. For example, the row number of graphic matrix for kinematic chain Cs is “1, 2, 4, 3, 7, 6, 10, 5, 9, 8, 11”. The joint-joint adjacency matrix ACS is transformed by the corresponding row number and column number in rule 4 and rule 5. So the graphical joint-joint adjacency matrix AG of kinematic chain Cs is:

8

⎧ 1⎪2 ⎪ 2⎪ 0 ⎪ ⎪ ⎪0 4⎪ ⎪ ⎪ 3⎪ ⎪0 ⎪ ⎨2 7⎪

AG = 6 0 ⎪ 10⎪ 0 ⎪ ⎪0 5⎪ ⎪ ⎪ ⎪ 9⎪ 0 ⎪ ⎪0 8⎪ ⎪ ⎩ 11 10

7 2 0 0 0 0 0 0 3 0 2 0

1 0 1 1 1 2 0 0 0 0 2 0

5 0 4 0 0 3 4 0 0 8 0 0

6 0 4 0 0 0 0 5 4 0 0 0

2 1 0 1 1 0 4 0 4 0 0 0

10

4

9

11

3

0 0 6 0 0 5 0 0 0 0 0

1 1 0 1 0 0 6 0 0 0 0

0 0 0 7 0 0 0 8 0 0 9

0 0 0 0 0 0 0 0 9 10 0

1⎪ ⎪ 1⎪ ⎪ ⎪ ⎪ 1⎪ ⎪ ⎪ 0⎪ ⎪ ⎪ ⎬ 0⎪ 0 ⎪ 0⎪ ⎪ ⎪ 0⎪ ⎪ ⎪ ⎪ 7⎪ ⎪ ⎪ 0⎪ ⎪ ⎭ 0

⎫

(7)

W. Sun, J. Kong and L. Sun / Mechanism and Machine Theory 150 (2020) 103847

Fig. 10. The position of joints of kinematic chain Cs.

Fig. 11. The structure diagram of kinematic chain Cs.

Fig. 12. The program interface.

9

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Fig. 13. An atlas of 8-link kinematic chains with 1 multiple joint.

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Fig. 14. An atlas of 8-link kinematic chains with 2 multiple joint.

In Eq. (7), the row number and column number of graphical matrix is the serial number of joints. This graphical matrix contains the position of each joint of kinematic chain Cs as shown in Fig. 10. Where, the number 0 in a circle which is the diagonal element of the normalized joint-joint adjacent matrix ACS represents the joints of kinematic chain Cs. Although the graphical matrix contains the relationships between joints of a kinematic chain, it still needs to match the types of individual links.

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Fig. 15. 14-link kinematic chains.

Rule 6: the joints of each link are adjusted according to the structure diagram of each link. The row number and column number of the graphical matrix are adjusted according to the types of individual links. For instance, link 2 is a ternary link that needs to be connected to the links on the right side in Fig. 10. The type A-3 and type A-4 in Fig. 7 can be selected. If link 2 matches the type A-3, the row number "7 1 of the graphical matrix AG needs to be converted to "1 7 . Similarly, link 1 matches type B-2 because it is connected to the links below itself. The column number "1, 2, 4, 3 of graphical matrix AG

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Fig. 16. 19-link kinematic chains.

needs to be transformed into "1, 3, 2, 4 . Finally, the graphical joint-joint adjacency matrix BG of kinematic chain Cs is:

8

⎧ 1⎪2 ⎪ 3⎪ 0 ⎪ ⎪ ⎪0 2⎪ ⎪ ⎪ 4⎪ ⎪0 ⎪ ⎨2 7⎪

BG = 6 0 ⎪ 10⎪ 0 ⎪ ⎪0 5⎪ ⎪ ⎪ ⎪ 9⎪ 0 ⎪ ⎪0 8⎪ ⎪ ⎩ 11 10

1 0 1 1 1 2 0 0 0 0 2 0

7 2 0 0 0 0 0 0 3 0 2 0

5 0 0 4 0 3 4 0 0 8 0 0

2 1 1 0 1 0 4 0 4 0 0 0

6 0 0 4 0 0 0 5 4 0 0 0

10

4

9

11

3

0 0 0 6 0 5 0 0 0 0 0

1 1 1 0 0 0 6 0 0 0 0

0 7 0 0 0 0 0 8 0 0 9

0 0 0 0 0 0 0 0 9 10 0

1⎪ ⎪ 0⎪ ⎪ ⎪ ⎪ 1⎪ ⎪ ⎪ 1⎪ ⎪ ⎪ ⎬ 0⎪ 0 ⎪ 0⎪ ⎪ ⎪ 0⎪ ⎪ ⎪ ⎪ 7⎪ ⎪ ⎪ 0⎪ ⎪ ⎭ 0

⎫

(8)

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Where, the row number and column number of the graphical matrix BG represent the serial numbers of joints. In Eq. (8), the structure diagram of kinematic chain Cs has been graphed as shown in Fig. 11. 4. Case analysis According to the automatic sketching theory, a dialog-based MFC visualization program is developed in C++ language. The joint-joint matrix of a kinematic chain is stored in the TXT type file. Then, an algorithm based on Breadth-First-Search (BFS) is used to identify all loops of a kinematic chain. Further, the graphical joint-joint adjacent matrix is created. And the structure diagram of the kinematic chain is sketched in the program interface as shown in Fig. 12. Where, the last row and last column elements corresponding to the symbol "99 in Fig. 12(a) represent the serial numbers of joints. Other elements of the matrix represent graphical joint-joint adjacent matrix. Take the cases of the 8-link kinematic chains with 1 multiple joint and 2 multiple joints in reference [7], 14-link kinematic chains with nonplanar topological graph and several complex multi-link kinematic chains. Sketching results satisfy the aesthetic criterion of minimizing link crossings. 4.1. Case 1: 22 8-link, 1-DOF kinematic chains with 1 multiple joint The results for an atlas of 8-link kinematic chains with 1 multiple joint are shown in Fig. 13. 4.2. Case 2: 18 8-link, 1-DOF kinematic chains with 2 multiple joints The results for an atlas of 8-link kinematic chains with 2 multiple joint are shown in Fig. 14. 4.3. Case 3: 5 14-link 1-DOF kinematic chains with non-planar topological graph The results for five 14-link kinematic chains with non-planar topological graph are shown in Fig. 15. 4.4. Case 4: 5 19-link, 8-DOF kinematic chains The results for five 19-link kinematic chains are shown in Fig. 16. These kinematic chains contain five independent loops. And the 19-link kinematic chain G3 is a non-planar topological graph. 5. Conclusions In this paper, a novel method converted the joint-joint adjacent matrix to graphic representation of planar kinematic chains was presented. There is a one-to-one correspondence between structure diagram of kinematic chains and this matrix. The structure diagram of kinematic chains is displayed in the graphical joint-joint adjacent matrix and stored conveniently. Firstly, the standardized rules of joint-joint adjacent matrix were introduced. The loops were extracted from the matrix base on Breadth-First-Search algorithm. Meanwhile, the breadth-first spanning tree was created. Then, rules of drawing structure diagram of a single link were presented. According to the tree diagram, the graphical matrix is realized by matching the type of individual link. This method possesses the ability of automatic sketching of kinematic chains with multiple joints. The time complexity of this method is O(n3 ). The automatic sketching of the results of the structural synthesis as visual kinematic chains is helpful for the creative design of mechanisms. Declaration of Competing Interest The authors declare that they have no conflict of interest. Acknowledgment This work was supported in part by the National Natural Science Foundation of China (No. 51875418). Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2020.103847.

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