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Abstract

The synthesis of the kinematic chain needs to obtain the information of the kinematic chain accurately and comprehensively. Isomorphism identification is an essential step in kinematic chain synthesis. In this article, a novel isomorphism determination method of planar kinematic chains with multiple joints based on joint–joint matrix description was proposed. First, a joint–joint matrix is presented to describe the kinematic chain, which can uniquely represent the kinematic chain structure. Then links and joints information were extracted from the matrix. And the link code and joint code were introduced to represent the link attributes and joint attributes, respectively. Furthermore, the standardization rules of joint–joint matrix are proposed. Isomorphism of kinematic chain is identified by comparing links, joints, and matrices. And the relationship between the links and the joints corresponding to the isomorphic kinematic chain is determined. Finally, the examples demonstrate that the method is novel and efficient.

Keywords

Joint–joint matrix planar kinematic chains multiple joints isomorphism identification joint code

Introduction

In 1964, the graph theory was introduced into the study of the topology of kinematic chains. Topological graph represents the link with a vertex and represents the joints with an edge. There is a corresponding relationship between the structure diagram and topological graph.¹ Due to the convenience of computer processing in matrix operation, the graph theory provides a powerful mathematical tool for the research and development of structural mechanics. In recent years, many scholars have made a lot of achievements in the description of the kinematic chain. H Ding et al.^2–5 presents a novel topological graph, namely the new bicolor topological graph, for the representation of the multiple joint kinematic chain. H. Ding attempts to establish unified topological models and corresponding mathematical representations for planar simple joint, multiple joint, and geared (cam) kinematic chains. J Liu⁶ suggested a new approach to convert the multiple joints to some parallel simple joints by using a pin-link to adjoin all the links on a multiple joint. The converted adjacent matrix is defined to represent the kinematic chains with multiple joints. W Zhang et al.⁷ introduced a comprehensive symbolic matrix representation for characterizing the topology of one of these mechanisms in a single configuration using general information concerning links and joints. S Li and JS Dai⁸ proposed an augmented adjacency matrix with the connectivity of links, the types of joint, and its axis orientation. The configuration transformation matrix for the link-annexing process is presented based on the augmented adjacency matrix operation. D Li et al.⁹ used the constraint graph of computational geometry rather than the traditional topological graph to characterize a metamorphic linkage in order to simplify the representation of its configuration changes. In the synthesis of planar kinematic chain, no matter what method is used, it is necessary to solve the problem in uniqueness and comprehensiveness of the kinematic chain with multiple joint. This problem is a bottleneck problem in the study of mechanism topology. And the topological description and isomorphism identification of kinematic chains with multiple joint need to be further studied.

Over the past several years, much work has been reported in the literature on the isomorphism identification of kinematic chains. Z Chang et al.¹⁰ proposed a new method based on eigenvectors and eigenvalues to identify isomorphism of mechanics kinematic chain. H Ding and Z Huang^11–13 proposed the canonical perimeter topological graph and the characteristic adjacency matrix approach to test isomorphism for both simple joint chains and multiple joint chains. K Zeng et al.¹⁴ used a fast deterministic algorithm called the dividing-and-matching algorithm for kinematic chain isomorphism identification. This method holds the high reliability and fast speed simultaneously. G Galán-Marín et al.¹⁵ designed a novel multivalued neural network that enables a simplified formulation of the graph isomorphism problem. V Dharanipragada and M Chintada¹⁶ proposed the split hamming method for isomorphism among KCs with prismatic pairs. There are many other methods.^14,17–20 Among them, there are sundry problems, such as complex methods, not intuitive, large calculation workload, and inconvenient to use computer processing. Isomorphism identification has been the focus study of planar kinematic chain synthesis.

In this article, a joint–joint matrix is presented to describe the kinematic chain, which can uniquely represent the kinematic chain structure. The kinematic chain diagram and the joint–joint matrix are one-to-one correspondence. The matrix includes links attribute, joints attribute, and multiple joints. Then the link code, array code, and joint code were introduced. Furthermore, the standardization rules of joint–joint matrix are proposed. Isomorphism of kinematic chain is identified by comparing links, joints, and matrices. And the relationship between the links and the joints corresponding to the isomorphic kinematic chain is determined. Finally, 22 8-link 1-DOF kinematic chains with one multiple joint are provided to demonstrate the effectiveness of this method.

Description of the joint–joint matrix

The description of the kinematic chain plays a very important role in the analysis of the kinematic chain. For the traditional representation method, the kinematic chain diagram is transformed into a topological graph and then transformed into a matrix. When the structure diagram is transformed into a topological graph, if the kinematic chain contains any multiple joints, the topological graph representation is tedious. In this article, the joint–joint matrix is used to realize the expression of the kinematic chain with multiple joint directly to the matrix. The row label and column label of the joint–joint matrix are the serial number of joints $J_{n}$ . The size of the matrix is $n \times n$ . $n$ is the number of joints of the kinematic chain. The joint–joint matrix is expressed as

$A = (\begin{matrix} 0 & a_{1, 2} & \dots & a_{1, j} & \dots & a_{1, n - 1} & a_{1, n} \\ a_{2, 1} & 0 & \dots & a_{2, j} & \dots & a_{2, n - 1} & a_{2, n} \\ ⋮ & ⋮ & ⋱ & ⋮ & \dots & ⋮ & ⋮ \\ a_{i, 1} & \dots & \dots & 0 & \dots & \dots & a_{i, n} \\ ⋮ & ⋮ & \dots & ⋮ & ⋱ & ⋮ & ⋮ \\ a_{n - 1, 1} & a_{n - 1, 2} & \dots & a_{n - 1, j} & \dots & 0 & a_{n - 1, n} \\ a_{n, 1} & a_{n, 2} & \dots & a_{n, j} & \dots & a_{n, n - 1} & 0 \end{matrix})$ (1)

where, the value of the diagonal element $a_{i, j} (i = j)$ is “0.” The other element of the matrix $a_{i, j} (i \neq j; i = 1, \dots, n; j = 1, \dots, n)$ is the serial number of the links. If there is not any connection between two links, the value of the element is “0.”

The serial number of links and joints in the kinematic chain is not limited, and only needs to be labeled in sequence. The serial number of links and joints in 10 bar kinematic chain with multiple joints is composed, shown in Figure 1. The joint–joint matrix of a kinematic chain $C_{1}$ is expressed as

$A_{C_{1}} = {\begin{matrix} 0 & 2 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 2 \\ 2 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 3 & 0 & 4 & 0 & 0 & 0 & 4 & 8 & 0 & 0 \\ 1 & 0 & 4 & 0 & 1 & 1 & 0 & 4 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 6 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 7 & 0 & 0 \\ 0 & 0 & 0 & 0 & 6 & 0 & 0 & 5 & 0 & 0 & 0 \\ 0 & 0 & 4 & 4 & 0 & 0 & 5 & 0 & 0 & 0 & 0 \\ 0 & 0 & 8 & 0 & 0 & 7 & 0 & 0 & 0 & 9 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 10 \\ 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 0 \end{matrix}}$ (2)

Figure 1.

The structure diagram of kinematic chain $C_{1}$ .

The element $a_{i, j} (i \neq j)$ of joint–joint matrix $A_{C_{1}}$ expresses the connection relationship between joint $J_{i}$ and joint $J_{j}$ of the kinematic chain $C_{1}$ $(i, j = 1, 2, \dots, n)$ . For example, the value of the element $a_{3, 4}$ is “4,” which indicates that the joint 3 is connected to joint 4, and the serial number of the connected link is 4. Similarly, in the third row of the joint–joint matrix $A_{C_{1}}$ , there are the values “3,”“4,” and “8.” It conveys links 3, 4, and 8 connected in multiple joint 3.

The joint–joint matrix information expression

The joint–joint matrix $A_{C_{1}}$ is very simple, but the matrix contains all the information about the kinematic chain. Therefore, the joint–joint matrix has a great advantage in describing the kinematic chain.

The links attribute

In the joint–joint matrix, the serial number of $k (k \geq 3)$ element multiple links in the ith row element (or ith column element) corresponding to the joint $J_{i}$ appears $k - 1$ . If the link is binary link, the serial number of the link in the ith row element (or ith column element) corresponding to the joint $J_{i}$ appears one time. For example, the link 6 is a binary link. So the serial number 6 appears one time in the fifth row. If the link is ternary element link, the serial number of the link appears two times. For example, the link 2 is ternary element link. So the serial number 2 appears two times in the first row.

The attribute of the links with joints’ information is represented by $N_{i} - \times \times \times$ . $N_{i}$ expresses that the link is $i$ element multiple links. The subscript array indicates the connection of this link and binary link. Its rules:

The size of the subsequent array is equal to the elements of the link. For example, the link 2 is ternary element link. So the attribute of the ternary link $N_{3}$ with three digital numbers is represented as $N_{3} - \times \times \times$ .

The value of the subsequent array indicates the number of series binary link. When multi-link is connected to multi-link, the value is “0.” For example, the link 2 is connected to quaternary link 1 at joint 1. The value is “0.” The link 2 is connected to binary link 3 at joint 2, and the binary link 3 is connected to ternary link 4 at joint 3. The value is “1.” The link 2 is connected to binary link 10 in joint 11, the binary link 10 is connected to binary link 9 in joint 10, and the binary link 9 is connected to multiple joint 9. The value is “2.” The link code of link 2 is represented as $N_{3} - 012$ . The value of the array arranges from small to large.

For the joint connecting to multi-link is a multiple joint, and the multiple joint is treated as a multi-link, indicated by the value “–1.”

The links attribute of the kinematic chain $C_{1}$ is shown in Table 1.

Table 1.

The table on the attribute of the links.

The link no.	1	2	3	4	5	6	7	8	9	10
The link code	$N_{4}$ -0012	$N_{3}$ -012	$N_{2}$ -00	$N_{3}$ -102	$N_{2}$ -01	$N_{2}$ -01	$N_{2}$ -00	$N_{2}$ -00	$N_{2}$ -01	$N_{2}$ -01

$N_{i}$ expresses that the link is $i$ element multiple links.

The joints attribute

By induction, the relationship between the two links can be divided into binary link and binary link, multi-link and binary link, and multi-link and multi-link. If it is a multiple joint, it is a combination of conditions. The joint–joint matrix contains the connection relation of the joints

In the joint–joint matrix, the array codes are obtained by generalized operations on the ith row elements eliminating zero elements and keeping the non-zero elements. If same non-zero element appears more than twice only two of them are kept. For example, in the equation (6), one appears thrice and two twice in the first row. In the array code only two 1’s will be kept. The array code corresponding to the joints is shown in equation (3). The value of array code is arranged from small to large. Analyzing each array code, the table on the type of joints can be obtained as shown in Table 2.

Table 2.

The table on the type of joints.

The typeof joint	Simple binaryjoint	Simple ternaryjoint	Simple quaternaryjoint	Double jointB-B-B type	Double jointB-B-M type	Double jointB-M-M type
The array code	$(a, b)$	$(a, a, b)$	$(a, a, b, b)$	$(a, b, c)$	$(a, a, b, c)$	$(a, a, b, b, c)$
The diagram

B: binary; M: multiple.

In the array code, the serial number of the same link appears two times, which expresses the link is multi-link. And the serial number appears only one time, which expresses the link is a binary link. For example, the array code $(a, a, b, c)$ means multi-link a, binary link b, and binary link c are connected in the joint.

In order to better represent the connection relationship of joints, the joint code $J_{\times \times} - \times \times$ indicates the attribute of the joints. The meaning of its representative is as follows:

The size of the subscript array expresses the number of links connected in the joint. For example, the array code $(1, 1, 6)$ of the joint $J_{5}$ means link 1 and link 6 connecting in this point, so the size of the subscript array in the point $J_{5}$ is 2, represented as $J_{\times \times}$ .

The value of the subscript array indicates the number of series binary link in the joint. When the joint connects to multi-link, the value is “0.” It specifies that the values in the subscript array are arranged from small to large. For example, the multi-link 1 is quaternary link in the joint 5 $(1, 1, 6)$ . And the binary link 6 is connected to binary link 5, which in turn is connected to ternary link 4. The number of series binary link is 2. So the value of the subscript array is “02,” represented as $J_{02}$ . And the ternary link 2 is connected to binary link 3 in the joint 2(2,2,3). The ternary link 2 is multi-link, the value is “0.” The binary link 3 is connected ternary link 4. So the number of series binary link is 1. The the value of the subscript array is “01,” represented as $J_{01}$ .

The subsequent array corresponds to the subscript array. The size of subsequent array is equal to the size of the subscript array. But the value of subsequent array represents the link type at the end of series binary link. If the numbers of series binary link are the same, the value of the subsequent array arranges from large to small. For example, the link 1 in the joint 5 $(1, 1, 6)$ is quaternary link, so the value is “4.” The type of link at the end of series binary link 6 is ternary link 4, so the value is “3.” Therefore, the value of subsequent array is “43,” represented as $J_{02} - 43$ .

If the link type at the end of series binary link is multiple joints, the value corresponding to subsequent array is “0.” For example, the ternary link 2 in the joint 2(2,2,3) is ternary multi-link, the value is “3.” The type of link at the end of series binary link 3 is ternary link 4, but the joint 3 is multiple joint. The value corresponding to subsequent array is “0.” The value of subsequent array is “30,” represented as $J_{01} - 30$ .

The relationship between subscript array and subsequent array in joint code of $J_{5}$ is shown in Figure 2. The joint code can be obtained by retrieving the array codes. For example, in joint $3 (3, 4, 4, 8)$ , the link type at the end of the binary link 3 is ternary link 2, and the link type at the end of the binary link 8 is a multiple joint. The joint code in the joint $3 (3, 4, 4, 8)$ is $J_{011} - 330$ . Thus, the table of the joint code be obtained, as showed in Table 3.

Figure 2.

The joint code of $J_{5}$ .

Table 3.

The table of the joint code.

The joint no.	The array code	The joint code
1	$(1, 1, 2, 2)$	$J_{00} - 43$
2	$(2, 2, 3)$	$J_{01} - 30$
3	$(3, 4, 4, 8)$	$J_{011} - 330$
4	$(1, 1, 4, 4)$	$J_{00} - 43$
5	$(1, 1, 6)$	$J_{02} - 43$
6	$(1, 1, 7)$	$J_{01} - 40$
7	$(5, 6)$	$J_{11} - 43$
8	$(4, 4, 5)$	$J_{02} - 34$
9	$(7, 8, 9)$	$J_{112} - 403$
10	$(9, 10)$	$J_{11} - 30$
11	$(2, 2, 10)$	$J_{02} - 30$

Isomorphism identification

The process of isomorphism identification is shown in Figure 3. First, we can write the joint–joint matrix of the kinematic chain. Second, the links attribute is obtained from the matrix. If the information is different, the kinematic chain is not isomorphic. Third, the joints attribute is obtained. If this information is not same, the kinematic chain is not isomorphic. Finally, the joint–joint matrices are standardized. The relationship between the links is determined. When the corresponding link numbers are replaced, if the matrces are the same, the kinematic chain is isomorphic. Otherwise the kinematic chain is not isomorphic.

Figure 3.

The flow chart of isomorphism identification.

The 10 bar kinematic chain with multiple joints is shown in Figure 4. According to the description of the joint–joint matrix in section “Description of the joint-joint matrix,” the expression of the kinematic chain $C_{2}$ and $C_{3}$ is carried out.

Figure 4.

The kinematic chain $C_{2}$ , $C_{3}$ . (a) Structure diagram of kinematic chain C₂ and (b) structure diagram of kinematic chain C₃.

The joint–joint matrix of the kinematic chain $C_{2}$

$A_{C_{2}} = {\begin{matrix} 0 & 2 & 2 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 2 & 0 & 2 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 6 \\ 2 & 2 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 4 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 5 \\ 1 & 0 & 0 & 0 & 1 & 1 & 0 & 8 & 8 & 0 & 0 \\ 0 & 7 & 0 & 0 & 0 & 0 & 8 & 0 & 8 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 8 & 8 & 0 & 9 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 10 \\ 0 & 6 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 10 & 0 \end{matrix}}$ (4)

The joint–joint matrix of the kinematic chain $C_{3}$

$A_{C_{3}} = {\begin{matrix} 0 & 2 & 2 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \\ 2 & 2 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 9 & 0 & 8 & 0 & 0 & 0 & 10 & 0 & 0 \\ 0 & 0 & 0 & 8 & 0 & 7 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 7 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 6 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 4 & 4 & 0 \\ 0 & 3 & 0 & 10 & 0 & 0 & 0 & 4 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 5 & 0 \end{matrix}}$ (5)

The links judgment

The links type of the kinematic chain $C_{2}$ and $C_{3}$ is the same. However, the attribute of the links with joint information is different as shown in Table 4. The kinematic chain $C_{2}$ and the kinematic chain $C_{3}$ are not isomorphism.

Table 4.

The table on the attribute of the links.

The link no.	The link codeof KC $C_{2}$	The link codeof KC $C_{3}$
1	$N_{4}$ -0012	$N_{4}$ -0022
2	$N_{3}$ -102	$N_{3}$ -011
3	$N_{2}$ -01	$N_{2}$ -00
4	$N_{2}$ -01	$N_{3}$ -102
5	$N_{2}$ -00	$N_{2}$ -01
6	$N_{2}$ -00	$N_{2}$ -01
7	$N_{2}$ -00	$N_{2}$ -01
8	$N_{3}$ -012	$N_{2}$ -01
9	$N_{2}$ -01	$N_{2}$ -00
10	$N_{2}$ -01	$N_{2}$ -00

Table 5.

The joint codes of the KC C2 and KC C3.

The joint no.	The KC $C_{2}$		The KC $C_{3}$
The joint no.	The arraycode	The jointcode	The arraycode	The jointcode
1	$(1, 1, 2, 2)$	$J_{00} - 43$	$(1, 1, 2, 2)$	$J_{00} - 43$
2	$(2, 2, 6, 7)$	$J_{011} - 330$	$(2, 2, 3)$	$J_{01} - 30$
3	$(2, 2, 3)$	$J_{02} - 34$	$(2, 2, 9)$	$J_{01} - 30$
4	$(3, 4)$	$J_{11} - 43$	$(8, 9, 10)$	$J_{112} - 304$
5	$(1, 1, 4)$	$J_{02} - 43$	$(7, 8)$	$J_{11} - 40$
6	$(1, 1, 5)$	$J_{01} - 40$	$(1, 1, 7)$	$J_{02} - 40$
7	$(1, 1, 8, 8)$	$J_{00} - 43$	$(1, 1, 6)$	$J_{02} - 43$
8	$(7, 8, 8)$	$J_{01} - 33$	$(1, 1, 4, 4)$	$J_{00} - 43$
9	$(8, 8, 9)$	$J_{02} - 30$	$(3, 4, 4, 10)$	$J_{011} - 330$
10	$(9, 10)$	$J_{11} - 30$	$(4, 4, 5)$	$J_{02} - 34$
11	$(5, 6, 10)$	$J_{112} - 403$	$(5, 6)$	$J_{11} - 43$

KC: kinematic chain.

The joints judgment

From the array code, the kinematic chain $C_{2}$ and the kinematic chain $C_{3}$ are exactly the same. But the joint code $J_{01} - 40$ in the joint 6 of the kinematic chain does not exist in the joint codes of the kinematic chain $C_{3}$ as shown in Table 5. The joint code at joints 8, 9,10, and 11 are also different. As long as one attribute is different, then the two kinematic chains are different. It is indisputable that the kinematic chain $C_{2}$ and the kinematic chain $C_{3}$ are not isomorphism.

The matrices information judgment

In order to match the joints better and faster, a standardized joint–joint matrix is proposed. The serial number of the joints is renumbered. The rules are as follows:

The links are sorted from large to small according to the number of element. When there are many multi-links with the same number of element, the multi-link is sorted with multiple joints priority.

When sorting the joints of the multi-links, the joints are arranged from small to large according to subscript array and subsequent array of joint code.

Then the multiple joints are sorted from large to small according to the number of element.

Finally, the joints of binary link are arranged from large to small according to subscript array and subsequent array of joint code.

The joint–joint matrix is standardized by the above rules. And the joints are re-ordered as showed in Table 6.

Table 6.

The table on the new number of joints.

The new serialnumber of joints	The originalserial number	The link code	The joint code
1	1	$N_{4}$ -0012	$J_{00} - 43$
2	4		$J_{00} - 43$
3	6		$J_{01} - 40$
4	5		$J_{02} - 43$
5	3	$N_{3}$ -002	$J_{011} - 330$
6	8	$N_{3}$ -002	$J_{02} - 34$
7	2	$N_{3}$ -012	$J_{01} - 30$
8	11	$N_{3}$ -012	$J_{02} - 30$
9	9	/	$J_{112} - 403$
10	7	/	$J_{11} - 43$
11	10	/	$J_{11} - 30$

Through the elementary transformation of the matrix, the normalized joint–joint matrix of the kinematic chain $C_{1}$ is obtained

$A_{C_{1} S} = {\begin{matrix} 0 & 1 & 1 & 1 & 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 4 & 4 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 7 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 \\ 0 & 4 & 0 & 0 & 0 & 4 & 3 & 0 & 8 & 0 & 0 \\ 0 & 4 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 5 & 0 \\ 2 & 0 & 0 & 0 & 3 & 0 & 0 & 2 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 10 \\ 0 & 0 & 7 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 9 \\ 0 & 0 & 0 & 6 & 0 & 5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 9 & 0 & 0 \end{matrix}}$ (6)

The standardized joint–joint matrix of the kinematic chain $C_{2}$

$A_{C_{2} S} = {\begin{matrix} 0 & 1 & 1 & 1 & 2 & 2 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 8 & 8 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 5 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 \\ 2 & 0 & 0 & 0 & 0 & 2 & 7 & 0 & 6 & 0 & 0 \\ 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 8 & 0 & 0 & 7 & 0 & 0 & 8 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 9 \\ 0 & 0 & 5 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 10 \\ 0 & 0 & 0 & 4 & 0 & 3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 10 & 0 & 0 \end{matrix}}$ (7)

The relationship between the links and the joints corresponding to the kinematic chain is determined as showed in Table 7.

Table 7.

The table on corresponding links.

The joint no.	The joint code	The array code of KC $C_{1}$	The array code of KC $C_{2}$	Corresponding relations between links
1	$J_{00} - 43$	$(1, 1, 2, 2)$	$(1, 1, 2, 2)$	1–1
2	$J_{00} - 43$	$(1, 1, 4, 4)$	$(1, 1, 8, 8)$	/
3	$J_{01} - 40$	$(1, 1, 7)$	$(1, 1, 5)$	7–5
4	$J_{02} - 43$	$(1, 1, 6)$	$(1, 1, 4)$	6–4
5	$J_{011} - 330$	$(3, 4, 4, 8)$	$(2, 2, 6, 7)$	4–2
6	$J_{02} - 34$	$(4, 4, 5)$	$(2, 2, 3)$	5–3
7	$J_{01} - 33$	$(2, 2, 3)$	$(7, 8, 8)$	2–8, 3–7
8	$J_{02} - 30$	$(2, 2, 10)$	$(8, 8, 9)$	10–9
9	$J_{112} - 403$	$(7, 8, 9)$	$(5, 6, 10)$	8–6
10	$J_{11} - 43$	$(5, 6)$	$(3, 4)$	/
11	$J_{11} - 30$	$(9, 10)$	$(9, 10)$	9–10

KC: kinematic chain.

The corresponding relations between links in the kinematic chain $C_{1}$ and $C_{2}$ can be determined by retrieving the array code. For example, comparing with the array code $(1, 1, 7)$ at the joint 3 in the kinematic chain $C_{1}$ and the array code $(1, 1, 5)$ in the kinematic chain $C_{2}$ , the link 7 of the kinematic chain $C_{1}$ corresponds to the link 5 of the kinematic chain $C_{2}$ .

The link numbers of kinematic chain $C_{2}$ are replaced by corresponding link numbers of kinematic chain $C_{1}$ . The new joint–joint matrix of kinematic chain $C_{2}$ is

$A_{C_{2} S^{'}} = {\begin{matrix} 0 & 1 & 1 & 1 & 4 & 4 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 7 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 \\ 4 & 0 & 0 & 0 & 0 & 4 & 3 & 0 & 8 & 0 & 0 \\ 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 5 & 0 \\ 0 & 2 & 0 & 0 & 3 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 10 \\ 0 & 0 & 7 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 9 \\ 0 & 0 & 0 & 6 & 0 & 5 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 9 & 0 & 0 \end{matrix}}$ (8)

The joint–joint matrices are symmetric matrices, including $A_{C_{1} S}$ and $A_{C_{2} S^{'}}$ . The matrix $A_{C_{2} S^{'}}$ can be transformed into the matrix $A_{C_{1} S}$ by the corresponding row and column transformation. There is a reversible matrix $P$ , which satisfies the equation $P^{T} A_{C_{2} S^{'}} P = A_{C_{1} S}$ . The meaning of the row and column transformation is the change of the joint number. The equivalent of the eigenvalues of the two matrices can embody a matrix that can be transformed into another matrix through the row and column transformation.²¹ The eigenvalue of matrix $A_{C_{1} S}$ and $A_{C_{2} S^{'}}$ are same, $eig (A_{C_{1} S}) = eig (A_{C_{2} S^{'}})$ . So they are isomorphic.

Case analysis

22 8-link, 1-DOF kinematic chains with one multiple joint shown in Figure 5.

Figure 5.

The 8-link kinematic chain with one multiple joint.

Due to the large number of kinematic chains, the joint–joint matrix of each kinematic chain is normalized in order to better illustrate the effectiveness of the method. The link attributes and the joint attributes of 8-link kinematic chains are shown in Tables 8 and 9.

Table 8.

The table on link code.

The joint no.	1	2	3	4	5	6	7	8
D ₁	N₃-001	N₃-011	N₃-012	N₂-00	N₂-01	N₂-01	N₂-00	N₂-00
D ₂	N₃-101	N₃-002	N₃-012	N₂-01	N₂-01	N₂-01	N₂-01	N₂-00
D ₃	N₃-100	N₃-012	N₃-012	N₂-01	N₂-01	N₂-01	N₂-01	N₂-00
D ₄	N₃-102	N₃-002	N₃-012	N₂-01	N₂-01	N₂-01	N₂-01	N₂-00
D ₅	N₃-002	N₃-011	N₃-011	N₂-00	N₂-00	N₂-01	N₂-01	N₂-00
D ₆	N₃-111	N₃-011	N₃-012	N₂-00	N₂-01	N₂-01	N₂-00	N₂-00
D ₇	N₃-111	N₃-011	N₃-012	N₂-00	N₂-00	N₂-00	N₂-01	N₂-01
D ₈	N₃-112	N₃-011	N₃-012	N₂-00	N₂-00	N₂-00	N₂-01	N₂-01
D ₉	N₃-101	N₃-012	N₃-111	N₂-00	N₂-00	N₂-00	N₂-01	N₂-01
D ₁₀	N₃-001	N₃-012	N₃-112	N₂-00	N₂-00	N₂-00	N₂-01	N₂-01
D ₁₁	N₃-012	N₃-012	N₃-111	N₂-00	N₂-00	N₂-00	N₂-01	N₂-01
D ₁₂	N₃-102	N₃-002	N₃-012	N₂-01	N₂-01	N₂-01	N₂-01	N₂-00
D ₁₃	N₃-102	N₃-112	N₃-012	N₂-01	N₂-01	N₂-01	N₂-01	N₂-00
D ₁₄	N₃-111	N₃-111	N₃-111	N₂-00	N₂-00	N₂-00	N₂-00	N₂-00
D ₁₅	N₃-011	N₃-011	N₃-111	N₂-00	N₂-00	N₂-00	N₂-00	N₂-00
D ₁₆	N₄-0122	N₃-012	N₂-00	N₂-01	N₂-01	N₂-00	N₂-01	N₂-01
D ₁₇	N₄-0222	N₃-102	N₂-01	N₂-01	N₂-01	N₂-01	N₂-01	N₂-01
D ₁₈	N₄-1112	N₃-111	N₂-00	N₂-01	N₂-01	N₂-00	N₂-00	N₂-00
D ₁₉	N₄-1122	N₃-111	N₂-01	N₂-01	N₂-01	N₂-01	N₂-00	N₂-00
D+ ₂₀	N₄-1122	N₃-112	N₂-01	N₂-01	N₂-00	N₂-00	N₂-01	N₂-01
D ₂₁	N₃-012	N₃-012	N₂-01	N₂-01	N₂-00	N₂-00	N₂-01	N₂-01
D ₂₂	N₄-1222	N₂-00	N₂-01	N₂-01	N₂-01	N₂-01	N₂-01	N₂-01

Table 9.

The table of the joint code.

The joint no.	1	2	3	4	5	6	7	8	9
D ₁	J₀₀-33	J₀₀-33	J₀₁-30	J₀₁-33	J₀₁-30	J₀₁-33	J₀₂-30	J₁₁₂-333	J₁₁-30
D ₂	J₀₂₂-333	J₀₀-33	J₀₁-30	J₀₀-33	J₀₂-30	J₀₁-33	J₀₂-30	J₁₁-30	J₁₁-30
D ₃	J₀₂₂-333	J₀₀-33	J₀₀-33	J₀₁-33	J₀₂-30	J₀₁-33	J₀₂-30	J₁₁-30	J₁₁-30
D ₄	J₀₁₂-333	J₀₀-33	J₀₂-33	J₀₀-33	J₀₂-33	J₀₁-30	J₀₂-30	J₁₁-30	J₁₁-33
D ₅	J₀₀-33	J₀₀-33	J₀₂-30	J₀₁-30	J₀₁-33	J₀₁-30	J₀₁-33	J₁₁₂-333	J₁₁-30
D ₆	J₀₁₂-333	J₀₁-33	J₀₁-33	J₀₀-33	J₀₁-33	J₀₁-33	J₀₁-30	J₀₂-30	J₁₁-30
D ₇	J₀₁₂-333	J₀₁-33	J₀₁-33	J₀₀-33	J₀₁-30	J₀₁-33	J₀₁-33	J₀₂-30	J₁₁-30
D ₈	J₀₁₁-333	J₀₁-33	J₀₂-33	J₀₀-33	J₀₁-33	J₀₁-30	J₀₁-30	J₀₂-33	J₁₁-33
D ₉	J₀₁₂-333	J₀₀-33	J₀₁-33	J₀₁-33	J₀₂-30	J₀₁-30	J₀₁-33	J₀₁-33	J₁₁-30
D ₁₀	J₀₀-33	J₀₀-33	J₀₁-30	J₀₁-30	J₀₂-33	J₀₁-30	J₀₂-33	J₁₁₁-333	J₁₁-33
D ₁₁	J₀₀₁-333	J₀₁-33	J₀₂-33	J₀₁-33	J₀₂-33	J₀₁-33	J₀₁-33	J₀₁-30	J₁₁-33
D ₁₂	J₀₁₂-333	J₀₀-33	J₀₂-33	J₀₀-33	J₀₂-30	J₀₁-30	J₀₂-33	J₁₁-33	J₁₁-30
D ₁₃	J₀₀₂-333	J₀₀-33	J₀₂-33	J₀₁-33	J₀₂-33	J₀₁-33	J₀₂-30	J₁₁-30	J₁₁-33
D ₁₄	J₀₁₁-333	J₀₁-33	J₀₁-33	J₀₁-30	J₀₁-33	J₀₁-33	J₀₁-30	J₀₁-33	J₀₁-33
D ₁₅	J₀₀-33	J₀₁-30	J₀₁-33	J₀₁-30	J₀₁-33	J₀₁-30	J₀₁-33	J₀₁-33	J₁₁₁-333
D ₁₆	J₀₀-43	J₀₁-40	J₀₂-43	J₀₂-43	J₀₁-30	J₀₂-34	J₁₁₂-434	J₁₁-43	J₁₁-43
D ₁₇	J₀₀-43	J₀₂-40	J₀₂-40	J₀₂-43	J₀₂₂-344	J₀₂-34	J₁₁-40	J₁₁-40	J₁₁-43
D ₁₈	J₀₁-40	J₀₁-43	J₀₁-43	J₀₂-40	J₀₁-30	J₀₁-34	J₀₁-34	J₁₁₂-434	J₁₁-40
D ₁₉	J₀₁-43	J₀₁-43	J₀₂-40	J₀₂-40	J₀₂₂-344	J₀₁-34	J₀₁-34	J₁₁-43	J₁₁-40
D ₂₀	J₀₁-40	J₀₁-43	J₀₂-43	J₀₂-40	J₀₁₂-344	J₀₁-34	J₀₂-34	J₁₁-43	J₁₁-40
D ₂₁	J₀₀-33	J₀₁-30	J₀₂-30	J₀₁-30	J₀₂-30	J₁₁₂₂-3333	J₁₁-30	J₁₁-30	–
D ₂₂	J₀₁-40	J₀₂-40	J₀₂-40	J₀₂-40	J₁₂₂₂-4444	J₁₁-40	J₁₁-40	J₁₁-40	–

In Table 8, the two groups of kinematic chains D₄ and D₁₂, and D₆ and D₇ cannot be distinguished. Other kinematic chains can be detected non-isomorphic. This also cannot be distinguished by joint attributes in Table 9. Further judgment is required

$A_{D_{4} S} = {\begin{matrix} 0 & 1 & 1 & 0 & 0 & 8 & 0 & 5 & 0 \\ 1 & 0 & 1 & 2 & 2 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 7 \\ 0 & 2 & 0 & 0 & 2 & 3 & 3 & 0 & 0 \\ 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 6 \\ 8 & 0 & 0 & 3 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 3 & 0 & 4 & 0 \\ 5 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 \\ 0 & 0 & 7 & 0 & 6 & 0 & 0 & 0 & 0 \end{matrix}}$ (9)

$A_{D_{12} S^{'}} = {\begin{matrix} 0 & 1 & 1 & 0 & 0 & 8 & 0 & 0 & 5 \\ 1 & 0 & 1 & 2 & 2 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 7 & 0 \\ 0 & 2 & 0 & 0 & 2 & 3 & 3 & 0 & 0 \\ 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 4 \\ 8 & 0 & 0 & 3 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 3 & 0 & 6 & 0 \\ 0 & 0 & 7 & 0 & 0 & 0 & 6 & 0 & 0 \\ 5 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 \end{matrix}}$ (10)

$A_{D_{6} S} = {\begin{matrix} 0 & 1 & 1 & 0 & 0 & 0 & 7 & 0 & 6 \\ 1 & 0 & 1 & 0 & 4 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 8 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 2 & 3 & 3 & 0 \\ 0 & 4 & 0 & 2 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 8 & 2 & 2 & 0 & 0 & 0 & 0 \\ 7 & 0 & 0 & 3 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 & 3 & 0 & 5 \\ 6 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 \end{matrix}}$ (11)

$A_{D_{7} S^{'}} = {\begin{matrix} 0 & 1 & 1 & 0 & 7 & 0 & 0 & 0 & 6 \\ 1 & 0 & 1 & 0 & 0 & 4 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 8 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 2 & 3 & 3 & 0 \\ 7 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 \\ 0 & 4 & 0 & 2 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 8 & 3 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 & 3 & 0 & 5 \\ 6 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 \end{matrix}}$ (12)

The eigenvalue of matrix $A_{D_{4} S}$ and $A_{D_{12} S^{'}}$ are different, $eig (A_{D_{4} S}) \neq eig (A_{D_{12} S^{'}})$ . So they are not isomorphic. Neither do the eigenvalue of matrix $A_{D_{6} S}$ and $A_{D_{7} S^{'}}$ , $eig (A_{D_{6} S}) \neq eig (A_{D_{7} S^{'}})$ . They are not isomorphic, too. In the same way, 18 8-link 1-DOF kinematic chains with two multiple joint and 83 9-link 2-DOF kinematic chains with one multiple joint are validated by this method.

Conclusion

In this article, a joint–joint matrix is proposed to describe the kinematic chain, which can represent the structure of the kinematic chain. The matrix can realize the expression of the kinematic chain diagram directly to the matrix calculation, omitting the need for expression of the topology diagram. Meanwhile, the serial numbers of the links and joints are retained. This is not available in other descriptive methods. The joint–joint matrix implies a lot of information about the kinematic chain, including links attribute and joints attribute. According to the extracted information, the multiple joint, multi-link can be identified. The standardization rules of joint–joint matrix are proposed. Isomorphism of kinematic chain is identified by comparing links, joints, and matrices. This method is both reliable and simple. The time complexity of this method is $O (n^{2})$ . The greatest advantage is more efficient than other methods without much additional computation. And the relationship between the links and the joints corresponding to the isomorphic kinematic chain is determined.

Footnotes

Handling Editor: Jan Torgersen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) received no financial support for the research,authorship,and/or publication of this article.

ORCID iDs

Wei Sun

Liangbo Sun

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