Metamorphic mechanisms belong to the class of mechanisms that are able to change their configurations sequentially to meet different requirements. In this paper, a holographic matrix representation for describing the topological structure of metamorphic mechanisms was proposed. The matrix includes the adjacency matrix, incidence matrix, links attribute and kinematic pairs attribute. Then, the expanded holographic matrix is introduced, which includes driving link, frame link and the identifier of the configurations. Furthermore, a matrix representation of an original metamorphic mechanism is proposed, which has the ability to evolve into various sub-configurations. And evolutionary relationships between mechanisms in sub-configurations and the original metamorphic mechanism are determined distinctly. Examples are provided to demonstrate the validation of the method.

Compared with the traditional mechanisms which usually have fixed mobility, metamorphic mechanisms are a multi-topological structure, in terms of economy, adaptability and efficiency, and prevails the traditional ones. Metamorphic mechanisms were proposed by Jiansheng Dai and John Rees Jones in 1996 based on the idea of reconfiguration (Dai and Rees John, 1999), which contributed to the new field of modern mechanical development and attracted the attention and interest of the theory of mechanism researchers around the world. In the past two decades, the research of principles and applications in metamorphic mechanisms has made a great breakthrough. The variation mode of metamorphic mechanism has been added to variable attribute of components, variable orientation of kinematic joints, friction self-locking joint units (Gan et al., 2009, 2010; Zhang et al., 2013; Ding and Li, 2015; Ye et al., 2016). Therefore, in order to create topological variations in the characteristics of mechanisms in different configurations, the appropriate structural representation for a metamorphic mechanism has been researched (Li et al., 2011a, 2016).

Mechanism diagram, topological graph and traditional adjacency matrix are
simple and intuitive tools for describing a mechanical structure in a single
sub-configuration. In order to analyze the configurations of the metamorphic
mechanism, a basic transformation matrix which uses adjacency matrices to
represent institutional changes was proposed by Dai et al. (2005). And a description method of displacement subgroup for structural
transformation of the metamorphic mechanism was proposed (Zhang and Dai,
2009). Li et al. (2011b) found that the constraint graph of computational
geometry rather than the traditional topological graph to characterize a
metamorphic linkage. The joint-gene based variable
topological representation is proposed, and the method and procedures for
the configuration transformations were presented based on the topological
representation (Li et al., 2009). The constraint function was defined
according to the constraint features of kinematic joints. With constraint
functions as elements, the adjacent matrix of the original kinematic chains
of metamorphic mechanism was proposed (Liu, 2012). Li and Dai (2011)
introduced an augmented adjacency matrix with the connectivity of links, the
types of joint and its axis-orientation. Due to the
lack of discussion on the variation of the metamorphic mechanism, the matrix
operations of these methods are still

In order to describe the changes between different configurations in the
process further and better, Zhang et al. (2016) introduced a comprehensive
symbolic matrix representation for describing the topological structure. An element “

Although the above-mentioned methods have their own characteristics, there are still some shortcomings in terms of simplicity, validity and accuracy. For the sake of describing the types and orientation of kinematic pair conveniently and distinguishing the types of mechanisms, it's necessary to give a more detailed description of the various types of information on the kinematic chain and the transformation process. It is difficult to describe the comprehensive information including the attributes of the kinematic pair and geometrical structure of the mechanism. Therefore the existing methods have limitations in the structural transformation description of the metamorphic mechanism.

The paper introduces a holographic matrix method to describe the topological
structure of metamorphic mechanism in all configurations, which relate to
the information such as the attributes of links, joints, frame links and
driving links. Meanwhile, multiple links and multiple joints can be
identified by serial numbers of links. Then an expanded matrix
representation of an original metamorphic mechanism is introduced. It is
able to evolve into any sub-configuration of the mechanism. Furthermore, a
planar 3-

A number of methods have been developed to describe the topological
structure of metamorphic mechanisms. For instance, topological graph and
matrix representation can be widely used in the field of mechanism analysis,
kinematic chain synthesis, isomorphic identification and topological
structure generation of metamorphic mechanisms. The expression of multiple
joints is always a difficult problem as shown in Fig. 1. Ding et al. (2009, 2010, 2013) proposed a new kind of
bicolor topological graph to represent the topological structures of
multiple joint kinematic chains. Represent links of kinematic chains with
solid vertices “

A kinematic chain with multiple joints and topology diagram.

At present, the matrix that represents the configuration changes of
metamorphic mechanism includes adjacency matrix and incidence
matrix (Akbari et al., 2009). The

When using a computer to simulate a mechanism, it is only necessary to know
the coordinates of the joints in the mechanisms. Therefore all the
structural information of the mechanisms is known and schematic diagram of
the mechanism can be drawn. Based on this idea, a holographic matrix was
proposed. The size of the matrix is

The structure diagram of a 10 bar kinematic chain

The serial number of links and kinematic pairs in 10 bar kinematic chain
with multiple joints is expressed as shown in Fig. 2. The holographic matrix
of 10 bar kinematic chain

The adjacency matrix is the matrix of the adjacent relations between
vertices, representing the relationship between kinematic pairs and links.
The element value of the non-zero is converted to ”1”. Diagonal elements are
omitted. The adjacency matrix

The number of non-zero and unequal values in the

The relationship between the two links can be divided into multi-link and multi-link, multi-link and binary link, binary link and binary link by the inductive method. If it's a multiple joints, it's a combination of conditions. The holographic contains the connection relation of the kinematic pairs, which is not available in other matrices.

The array codes of the matrix are obtained by generalized operations on the

The type of kinematic pairs.

Note: the type of multiple joint cancontain more links, the number of array code can continue to increase.

In array code, the serial number of the same link appears twice, which
expresses the link is multi-link. And the serial number appears only once,
which expresses the link is a binary link. The different serial number in
array code indicates the different links are connected to this joint. For
example, the array code

In order to represent the connection between kinematic pairs better, the
kinematic pair code

The size of the subscript array expresses the number of links in the
kinematic pair. For example, the array code

The value of the subscript array indicates the number of series binary
link in a kinematic pair. The values in the subscript array are arranged
from small to large. When multi-link is connected to multi-link, the value
is “0”. For example, the link 1 is quaternary link and link 6 is binary link
connected in the joint 5

The subsequent array corresponds to the subscript array. The size of
subsequent array is equal to the subscript array. But the value of
subsequent array represents the type of the link at the end of the binary
link. If the numbers of series binary link are the same, the value of the
subsequent array arranges from large to small. For intance, the link 1 in
the joint 5

If the type of the link at the end of the binary link is multiple joints, the value corresponding to subsequent array is “0”.

The kinematic pair code.

The serial number of links in the

The attribute of the links with kinematic pair information is represented
by

The size of the subsequent array is equal to the elements of the link.
For example, the attribute of the ternary link

The value of the array indicates the number of series binary link. When multi-link is connected to multi-link, the value is “0”. 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 attribute of the links.

Note:

The element

Some parameters must be given when the topological structure changes in the design of metamorphic mechanism. At the same time, the change of topological structure may lead to the change of the degree of freedom of the mechanism. It must be given new parameters of the driving link, or remove the parameters of the failed driving link. So the expanded holographic matrix is proposed based on the holographic matrix. Expressed as in Eq. (6).

The

The holographic matrix is used to describe the topological structure of the metamorphic planar mechanism. When the link sequence and the joint sequence are determined, a definite unique holographic matrix is obtained. The structure of the metamorphic mechanism can be obtained from the holographic matrix as well. For instances, a packaging mechanism is designed using the metamorphic principle which is a recirculating metamorphic mechanism as showed in Fig. 2. The main features of the mechanism are topological structure continues to change during the cycle of the driving link works and continues to occur and repeat the same function of the cycle changes.

A packaging mechanism.

From Fig. 3, it can be concluded that the topological structure of the
mechanism can be transformed from one to another by locking at different
kinematic joints accordingly. By applying modes such as the geometric limit,
force limit, and variation of the driving kinematic joint, the working
conditions of these kinematic joints can be switched between active and
locked states. Therefore, it is very important to accurately describe the
process of reconfiguration in metamorphic mechanisms. The holographic matrix
shown in Eq. (6) is used to represent the various configurations of the
packaging mechanism in Fig. 3.

The relationship between the original metamorphic mechanism and sub-configurations.

A planar 3-

A holographic matrix representation for describing the topological structure of metamorphic mechanisms in a single configuration is introduced. However, exploring variation laws of these mechanisms in different configurations is very important for developing novel metamorphic mechanisms (Zhang et al., 2016). The advantage of matrix operations is taken for constructing the original metamorphic mechanism.

The original metamorphic mechanism is able to evolve into any configuration
of the mechanism and contains all the topological elements found in all
sub-configurations. An original matrix

The matrix can identify all possible combinations between links for creating
different mechanisms. The original metamorphic mechanism can evolve into any
topological structure of the metamorphic mechanism. The transformation of a
single structure of the metamorphic mechanism can also be operated. The
relationship between the various matrices can be expressed. Therefore, the
information on the mechanism in configuration

An original metamorphic mechanism provides a foundation for a mechanism to transform itself into any configuration and expresses the joint variation characteristics from the symbolic adjacency matrices and the corresponding operations. The relationships between these matrices are as shown in Fig. 4.

A planar 3-

The metamorphic mechanism has two configurations. When the mechanism is in
configuration 1 as shown in Fig. 4a, the revolute pair

The serial number 4 appears twice in joint 3 or joint 7. Then it can be judged that the link 4 is a ternary link. The original matrix of the original metamorphic mechanism can be expressed as Eq. (11).

The value of the element

An expanded holographic matrix was proposed to describe the topological structure of the metamorphic mechanism in this paper. The expanded two columns of the matrix are the property of the frame link and driving link. That is an indispensable part of information in the metamorphic mechanism. In addition, the upper triangular matrix denotes the distance between any two joints. It is very important for the kinematics analysis and dynamics analysis of the metamorphic mechanism in the next step. The down triangular matrix is the serial number of the links in the kinematic chain. This implies a lot of information including the adjacency matrix, incidence matrix, links attribute and kinematic pairs attribute. And the accurate judgment for multiple links and multiple joints is given by the serial number of the links. Finally, an example shows that this method can reflect the effectiveness of the metamorphic mechanism.

The data required to reproduce these findings are available to download from

WS and LS designed the matrix; JK analyzed experimental results. WS wrote the manuscript.

The authors declare that they have no conflict of interest.

This work is partially supported by the National Natural Science Foundation of China under Grant (No. 51875418).

This research has been supported by the National Natural Science Foundation of China under Grant (grant no. 51875418).

This paper was edited by Doina Pisla and reviewed by Daniel Condurache and one anonymous referee.