A plane kinematic chain inversion refers to a plane kinematic
chain with one link fixed (assigned as the ground link). In the creative
design of mechanisms, it is important to select proper ground links.
The structural synthesis of plane kinematic chain inversions is helpful for
improving the efficiency of mechanism design. However, the existing
structural synthesis methods involve isomorphism detection, which is
cumbersome. This paper proposes a simple and efficient structural
synthesis method for plane kinematic chain inversions without detecting
isomorphism. The fifth power of the adjacency matrix is applied to recognize
similar vertices, and non-isomorphic kinematic chain inversions are directly
derived according to non-similar vertices. This method is used to
automatically synthesize 6-link 1-degree-of-freedom (DOF), 8-link 1-DOF, 8-link 3-DOF, 9-link
2-DOF, 9-link 4-DOF, 10-link 1-DOF, 10-link 3-DOF and 10-link 5-DOF plane
kinematic chain inversions. All the synthesis results are consistent with
those reported in literature. Our method is also suitable for other kinds of
kinematic chains.
Introduction
In the early stage of the creative design of mechanisms, the design work was
accomplished mainly based on researchers' experience and inspiration,
resulting in a low design efficiency. Moreover, the derived mechanism
configurations were very limited. In order to resolve this problem, many
systematic methods have been proposed since the 1960s to create novel mechanisms
(Olson et al., 1985; Yan, 1992; Al-Dweiri, 2010; Yan and Chiu, 2014). An effective
mechanism design method is based on the atlas of topological structures of
kinematic chains (Rao, 1997; Butcher and Hartman, 2005; Sunkari and Schmidt,
2006; Ding et al., 2012, 2016).
A plane kinematic chain inversion refers to a plane kinematic chain with
one link fixed (assigned as the ground link). When designing mechanisms
based on kinematic chains, it is essential to select proper ground
links. The structural synthesis of plane kinematic chain inversions plays an
important role in the improvement of mechanism design efficiency. A cumbersome part of the structural synthesis process is to detect
and eliminate isomorphic kinematic chain inversions. Meanwhile, kinematic
chain components are represented by vertices in graph theory, and similarity
analysis of kinematic chain components can be transformed into a similarity
recognition problem of vertices in topological graphs. The similarity of
kinematic chain components should be analyzed first to reduce redundant
design schemes and improve the efficiency of innovative design.
In the early stage of research on the structural synthesis, the
characteristic polynomial-based method was applied to synthesize 10-link
3-degree-of-freedom (DOF) and 11-link 2-DOF kinematic chain inversions (Mruthyunjaya, 1984;
Mruthyunjaya and Balasubramanian, 1987). Subsequently, the hamming number
technique was developed to detect isomorphic inversions of given kinematic
chains
(Rao and Prasad Raju Pathapati, 2000; Rao and Varada Raju, 1991). Chu and Cao (1994)
utilized the link's adjacent-chain table to distinguish kinematic chain
inversions and confirmed that there are 1834 10-link 1-DOF inversions.
Vijayananda (1994) applied the representation set of links to detect
isomorphism and enumerated a part of kinematic chain inversions with up to
13 links. Tuttle (1996) presented an isomorphism detection method based on
the theory of finite symmetry groups and synthesized inversions from the
corresponding non-fractionated kinematic chains.
An approach based on permutation groups was proposed to derive
non-isomorphic inversions and mechanisms considering specific design
constraints (Yan and Hwang, 1991; Yan and Hung, 2006; Hung et al., 2008).
Based on genetic algorithms, Liu and McPhee (2007) studied the automatic
kinematic synthesis of planar mechanisms with revolute joints by considering
mechanism topology and geometric parameters. Simoni et al. (2009) applied
the group theory technique to avoid the generation of isomorphic inversions
and enumerated planar inversions with up to four independent loops. Dargar et al. (2013) and Rizvi et al. (2016) developed methods to determine structural invariants
and identification numbers of kinematic chains to detect isomorphic
kinematic chains inversions. Their methods were applied to synthesize
kinematic chain inversions with up to 10 links. Mruthyunjaya (2003) and
Simoni et al. (2011) reviewed the synthesis methods of kinematic chain
inversions and analyzed the contradiction in the synthesis results. Yang et
al. (2017) presented an automatic method to synthesize kinematic chain
inversions, including those having complex topology. Ding and Huang (2007)
improved the perimeter-loop-graph-based method to detect isomorphism. Sun et
al. (2018) applied the joint–joint matrix to represent planar kinematic
chains with multiple joints and studied the isomorphism detection of
kinematic chains by comparing links, joints and matrices.
The literature review shows that the existing methods for structural
synthesis of inversions involve isomorphism detection, which is
cumbersome. This paper presents a simple and efficient structural
synthesis method without detecting isomorphism. The fifth power of the adjacency
matrix is applied to recognize similar vertices, and non-isomorphic
kinematic chain inversions are directly derived according to non-similar
vertices. This method is used to automatically synthesize 6-link 1-DOF,
8-link 1-DOF, 8-link 3-DOF, 9-link 2-DOF, 9-link 4-DOF, 10-link 1-DOF,
10-link 3-DOF and 10-link 5-DOF plane kinematic chain inversions. All the
synthesis results are consistent with those reported in literature. The
present method can also be applied to kinematic chain inversions with more
links.
Graphic representation of kinematic chain inversion
The concept of graph theory is frequently adopted to represent the
topological structures of kinematic chains and mechanisms. In the graphic
representation of kinematic chains, a vertex denotes a link, and an edge
denotes a kinematic pair. For example, Fig. 1a shows an 8-link 1-DOF
kinematic chain, and Fig. 1b shows its graphic representation. For
convenience of developing computer-aided structural synthesis program, the
graphic representation is further denoted by its adjacency matrix, which is
defined as in Eq. (1). For example, the adjacency matrix of Fig. 1b is
shown in Eq. (2).
1AM=[xi,j]n×n=1,if vertexiis adjacent to vertexj0,otherwise2AM=0100010110101000010100000010100001010100100010100000010110000010
A kinematic chain inversion refers to a kinematic chain with one link
fixed (assigned as the ground link). For example, two inversions derived
from Fig. 1a are shown in Fig. 2a and b, where links 1 and 3 are
assigned as ground links, respectively. Their graphic representations are
shown in Fig. 2c and d, where the ground link is denoted by a vertex
marked with a circle.
(a) An 8-link 1-DOF kinematic chain and (b) its graphic representation.
Inversions derived from Fig. 1a and their graphic
representations.
Detection of similar vertices
Similar vertices in the graphic representation have the same topological
attribution and characteristics. If similar vertices are assigned as ground
links, the derived kinematic chain inversions are isomorphic. In other
words, they have duplicate topological structures.
We developed a simple and efficient method to detect similar vertices based
on the fifth power of the adjacency matrix. Given two matrices AMa= [ai,j]n×n and AMb= [bi,j]n×n, their multiplication is
AMc= AMa⋅ AMb= [ci,j]n×n, and the element of matrix AMc is ci,j=∑k=1nai,k⋅bk,j.
In particular, the multiplication of matrix AM and itself is called the power of
matrix and is denoted as AM2=AM⋅AM=[ci,j]n×n, and its element is
ci,j=∑k=1nai,k⋅ak,j. For example, the
power of the matrix in Eq. (2) is shown in Eq. (3). Similarly, the
third, fourth and fifth powers of the matrix in Eq. (2) are shown in
Eqs. (4), (5) and (6), respectively.
3AM2=AM⋅AM=30102020030202011020200002020100202030100201030220001020010002024AM3=AM2⋅AM=06030705605070300504030130405010070506037030605003010504501030405AM4=AM3⋅AM=180901601200180120160990901204001209090416012018090016090180121204090900904012096AM5=AM4⋅AM=043025046030430300460250030021025013250210300130046030043025460250430300025013030021300130250210
For the fifth power of matrix AM5, its rearranged matrix RAM5 is
acquired by arranging the elements of each row in descending order. For
example, the rearranged fifth power of matrix RAM5 corresponding to
Eq. (6) is shown in Eq. (7). If the elements in the ith row and the jth
row of RAM5 are the same, vertices i and j in the corresponding graphic
representation are detected to be similar vertices. For example, the
elements in the first, second, fifth and sixth rows in Eq. (7) are (46,
43, 30, 25, 0, 0, 0, 0); hence vertices 1, 2, 5 and 6 in Fig. 1b are
similar vertices. The elements in the third, fourth, seventh and eighth rows
in Eq. (7) are (30, 25, 21, 13, 0, 0, 0, 0); hence vertices 3, 4, 7 and
8 in Fig. 1b are similar vertices. In fact, similar vertices in Fig. 1b can be directly detected by visual inspection. However, using visual
inspection, it is very hard or impossible to precisely detect similar
vertices in other complex cases. Our detection method based on the fifth
power of the adjacency matrix AM5 is suitable for both simple and complex
kinematic chains. Moreover, it is very easy to achieve automatic detection
by developing a computer program.
RAM5=464330250000464330250000302521130000302521130000464330250000464330250000302521130000302521130000
Structural synthesis of kinematic chain inversions
The existing methods for structural synthesis of kinematic chain inversions
involve isomorphism detection, which is cumbersome. In this
paper, the kinematic chain inversions are synthesized based on the
information of similar vertices, without detecting isomorphism. If similar
vertices are assigned as ground links, the derived kinematic chain
inversions are isomorphic. Non-isomorphic kinematic chain inversions can be
directly derived according to non-similar vertices. For example, vertices 1
and 3 in Fig. 1b are non-similar vertices. Only two non-isomorphic
inversions can be derived from Fig. 1b, as illustrated in Fig. 2c
and d. Another example 10-link 1-DOF kinematic chain is shown in Fig. 3.
Its adjacency matrix, the fifth power of the adjacency matrix AM5 and the
rearranged fifth power of the adjacency matrix RAM5 are shown in Eqs. (8)–(10), respectively. According to Eq. (10), vertices 1, 4, 7 and 8
are similar vertices, vertices 2 and 3 are similar vertices and vertices 5,
6, 9 and 10 are also similar vertices. Vertices 1, 2 and 5 are non-similar
vertices, and three non-isomorphic inversions can be derived from Fig. 3,
as shown in Fig. 4.
8AM=01010000011010001000010100010010101000000001010000000010100001000101000010001010000000010110000000109AM5=04404002003602444051025044025005104402504402540044024036020002502401502001120025015024011004403602404002036044020040024002502001102401524025011020015010RAM5=444036242000000514444252500000514444252500000444036242000000252420151100000252420151100000444036242000000444036242000000252420151100000252420151100000
An example 10-link 1-DOF kinematic chain.
Non-isomorphic inversions derived from Fig. 3.
The atlas of plane non-fractionated kinematic chains can be derived using
the existing method (Ding et al., 2016). Based on this atlas, plane
kinematic chain inversions can be synthesized using the present method. For
example, there are two 6-link 1-DOF kinematic chains, from which five
non-isomorphic inversions can be synthesized, as shown in Fig. 5; there
are five 8-link 3-DOF kinematic chains, from which 18 non-isomorphic
inversions can be synthesized, as shown in Fig. 6. A part of the 8-link 1-DOF,
10-link 1-DOF and 10-link 3-DOF kinematic chain inversions are shown in
Figs. 7–9, respectively. We have developed a computer program based on
C++, and the synthesis process is fully automatic. All the numerical
representations of kinematic chain inversions are generated automatically
and stored in the database. For example, the complete database of 517
10-link 3-DOF kinematic chain inversions is shown in Appendix A. Taking the
number string “1-100000011101000001000001100000100001000100100”, for
instance, the first number “1” represents the ground link, and the
remaining numbers are the upper triangular elements of the adjacency matrix
of a 10-link 3-DOF kinematic chain. The detailed synthesis results of
kinematic chain inversions are listed in Table 1. All the results are
consistent with those reported in literature (Tuttle, 1996; Simoni et al.,
2009; Yang et al., 2017). The running time of our computer program is also
listed in Table 1, which is measured on a personal laptop Intel (R) Core
(TM) i5-8265U CPU @ 1.60 GHz 8 GB RAM. It is clear that our method can be
used to synthesize kinematic chain inversions efficiently.
The atlas of 6-link 1-DOF kinematic chain inversions.
The atlas of 8-link 3-DOF kinematic chain inversions.
A part of the 8-link 1-DOF kinematic chain inversions.
A part of the 10-link 1-DOF kinematic chain inversions.
A part of the 10-link 3-DOF kinematic chain inversions.
Synthesis results of kinematic chain inversions.
Number ofNumber ofNumber ofNumber of kinematicRunning time of thelinksDOFskinematic chainschain inversionscomputer program(seconds)61250.9898116711.178835181.94292352202.176946281.90410123018342.209103745171.5371058390.874Conclusions
The existing structural synthesis methods for plane kinematic chain
inversions involve isomorphism detection, which is cumbersome.
This paper proposes a simple and efficient synthesis method without
detecting isomorphism. The fifth power of the adjacency matrix is applied to
recognize similar vertices, and non-isomorphic kinematic chain inversions
are directly derived according to non-similar vertices. We have developed a
computer program based on C++, and the synthesis process is fully
automatic. This method is used to synthesize 6-link 1-DOF, 8-link 1-DOF,
8-link 3-DOF, 9-link 2-DOF, 9-link 4-DOF, 10-link 1-DOF, 10-link 3-DOF and
10-link 5-DOF plane kinematic chain inversions. All the synthesis results
are consistent with those reported in literature, demonstrating the
reliability of our method. The present research is helpful for improving the
efficiency of mechanism design.
Database of kinematic chain inversions
The complete database of 517 10-link 3-DOF kinematic chain
inversions.
The data are available upon request from the corresponding author.
The supplement related to this article is available online at: https://doi.org/10.5194/ms-12-1061-2021-supplement.
Author contributions
JC is the lead author of this article. He was responsible for collecting the
research literature, organizing the paper structure and writing the paper.
JD and WH are coauthors of this paper. They provided suggestions for the
revision and correction of this paper. RC is the corresponding author of
this paper. He presented the idea of this research and was responsible for
the whole process of writing and revising this paper.
Competing interests
The contact author has declared that neither they nor their co-authors have any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
The authors thank the reviewers for their critical and constructive review
of the article.
Financial support
This research has been supported by the Natural Science
Foundation of China (grant no. 51805306) and the Launching Project of High-level
Talents of Hangzhou Vocational & Technical College (grant no. HZYGCC202103).
Review statement
This paper was edited by Daniel Condurache and reviewed by three anonymous referees.
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