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 EU operations. The EU-elementary matrix operation consists of the steps of applying an Ui,j-elementary matrix operation and an elimination Ej-elementary matrix operation used in pairs (Dai and Rees John, 2005).

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 “-1” was introduced to indicate the frozen kinematic pairs, which given an easy way to represent the transformation and the dimension of the adjacency matrix was kept unchanged after the transformation (Lan and Du, 2008). The transformation of metamorphic mechanism can be described clearly without complex matrix operations.

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-RRR parallel mechanism is provided to demonstrate the validation of the method.

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 “•” and hollow vertices “∘” denote joints as shown in Fig. 1b, and then remove hollow vertices “∘” denote simple joints as shown in Fig. 1c.

At present, the matrix that represents the configuration changes of metamorphic mechanism includes adjacency matrix and incidence matrix (Akbari et al., 2009). The EU elementary matrix is used to describe the transformation of metamorphic mechanisms. The adjacency matrix method and the incidence matrix method describe the matrix operation of the configuration transformation in metamorphic mechanisms clearly and they are beneficial to the realization of computer automation. However, it cannot fully express the size structure, kinematic pairs and other information. The description of the metamorphic mechanisms in sub-configurations is not comprehensive. This paper introduces a holographic matrix representation for describing the topological structure of metamorphic mechanisms. And the structural information can be described accurately and distinctly. The structural description of the metamorphic mechanism is an important part of kinematics and dynamics analysis.

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 A(0) for representing the original metamorphic mechanism is given as in Eq. (8). Where, the operator ∪ represents the union of its arguments (Zhang et al., 2016). The original matrix A(0) has the same elements form as Eq. (6). All the elements located in the same position in the set of related matrices from A(1) to A(n) gradually become united as shown in Eq. (8). Therefore, the matrix of the original metamorphic mechanism for the packaging mechanism shown in Fig. 2 is as in Eq. (9).

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 m can be extracted from the matrix A(0) to construct the corresponding matrix A(m). For example, the elements marked by triangles Δ in Eq. (9) are extracted to construct the matrix A(2). The topological structure of the mechanism in configuration 2 according to the above procedure was represented in Eq. (7).

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-RRR parallel mechanism is shown in Fig. 5. All R joints of the mechanism are in parallel.

The metamorphic mechanism has two configurations. When the mechanism is in configuration 1 as shown in Fig. 4a, the revolute pair F and the revolute pair I are locked. The link 1 is frame link. And the link 2 is the driving link. When the mechanism is in configuration 2 as shown in Fig. 5b, the revolute pair A and the revolute pair F are locked. The link 1 is always frame link. But the link 8 is the driving link. The topological structures of the metamorphic mechanism can be expressed in matrix form as in Eq. (10).

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 A(0)(9,9) in matrix -1∪R represents the revolute pair I has been changed. The value of the element A(0)(2,10) in the original matrix 2,8 explains the driving link has been changed.

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.