MSMechanical SciencesMSMech. Sci.2191-916XCopernicus PublicationsGöttingen, Germany10.5194/ms-7-39-2016A representation of the configurations and evolution of metamorphic mechanismsZhangW.zhangwuxiang@buaa.edu.cnDingX.LiuJ.School of Mechanical Engineering and Automation, Beihang University, Beijing, ChinaState Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, ChinaW. Zhang (zhangwuxiang@buaa.edu.cn)9February20167139477June20155January20166January2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://ms.copernicus.org/articles/7/39/2016/ms-7-39-2016.htmlThe full text article is available as a PDF file from https://ms.copernicus.org/articles/7/39/2016/ms-7-39-2016.pdf
Metamorphic mechanisms are members of the class of mechanisms that are able
to change their configurations sequentially to meet different requirements. The
paper introduces 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.
Furthermore, a matrix representation of an original metamorphic mechanism
that has the ability to evolve is proposed by uniting the matrices
representing all of the mechanism's possible configurations. The
representation of metamorphic kinematic joints is developed in accordance
with the variation laws of these mechanisms. By introducing the joint
variation matrices derived from generalized operations on the related
symbolic adjacency matrices, evolutionary relationships between mechanisms
in adjacent configurations and the original metmaorphic mechanism are made
distinctly. Examples are provided to demonstrate the validation of the method.
Introduction
In contrast to a traditional mechanism, a metamorphic mechanism is a
mechanism with variable topological structures and it is a good approach for
resolving the contradiction between economy, adaptation and efficiency. The
concept of metamorphic mechanisms was first introduced based on the idea of
reconfiguration in 1996 by Jian S. Dai and Rees Jones, which led to a new
era of modern mechanism development (Dai and Rees Jones, 1998).
Research on the metamorphic mechanism has been making significant
improvements in fundamentals and applications for nearly twenty years. The
essence and characteristics of metamorphic mechanisms as well as three
metamorphic approaches including variable components, adjacent relations and
kinematic joints were introduced by Dai et al. (2005a) and Liu and Yang (2004). In addition, some of
the basic constituent elements of these mechanisms, including links and
their connectivity relationships, remain unchanged to give the mechanism's
adjacent configuration complex coupling features. These two aspects are key
factors affecting the study of methods for configuring metamorphic
mechanisms (Zhang et al., 2011). Therefore, to create topological variations in the
characteristics of mechanisms in different configurations, the appropriate
structural representation for a metamorphic mechanism has been researched in
recent years.
Mechanism diagrams, topological graphs and conventional adjacency matrices
(Tsai, 2001) are simple and intuitive tools for describing the structure of a
mechanism in a single configuration. Dai et al. (2005b) and Dai and Rees Jones (2005) were the first to
propose an elementary transformation matrix that represented the variation
of a mechanism using the adjacency matrix method. Wang and Dai (2007) introduced
joint symbols into the adjacency matrix to express the variations of
kinematic joints. In the matrix, all of the links were numbered sequentially
and placed in principal diagonal positions; the off-diagonal elements were
expressed using joint symbols that represented the connectivity
relationship. Lan and Du (2008) used -1 as an element indicating a joint
frozen into a new adjacency matrix to represent the topological changes of
metamorphic mechanisms. Slaboch and Voglewede (2011) and Korves et al. (2012) proposed mechanism state
matrices as a novel way to represent the topological characteristics of
planar and spatial reconfigurable mechanisms. These matrices can be used as
an analysis tool to automatically determine the degrees of freedom of planar
mechanisms that only contain one degree of freedom (DOF) joint. Herve (2006)
showed how to create translational parallel manipulators using Lie-group
algebra, which can give reference to the related research. Yan and Kang (2009)
showed how to perform configuration synthesis of mechanisms with variable
topologies using graph theory.
However, the axial orientation of a joint and information on link variations
were not epitomized in the aforementioned research. Therefore, Yang (2004)
introduced the concept of a geometric constraint for expressing the relative
positions and orientations of the joint axes and generalized it into six
types: parallelism, coincidence, intersection, perpendicularity, coplanarity
and randomness. Li et al. (2010) suggested using a constraint graph from
computational geometry rather than the traditional topological graph to
characterize a metamorphic linkage to simplify the representation of its
configuration changes. The adjacency submatrix of the constraint graph
provides a convenient description of changes in the topology of links and
joints in the operation of the metamorphic linkage. Li et al. (2009) and Li and Dai (2010a, b)
developed a topological representation matrix with information on loops,
types of links and joints that included orientation information, which has
been used in subsequent research. They also introduced a joint-orientation
interchanging metamorphic method based on the matrix.
This paper presents a novel method of characterizing the topology of
metamorphic mechanisms in all configurations that involves information about
links and joints, including their types and axial orientations. Furthermore,
a method of constructing an original matrix that represents the original
metamorphic mechanism is proposed. Next, the paper proposes two matrix
operations that are useful for representing topological changes and evolving features.
Configuration characteristics of metamorphic mechanisms
A metamorphic mechanism is a mechanism with variable topological structures
that can be transformed from one structure to another continuously. There
are variable parts and coupling parts, giving the metamorphic mechanism a
variable topological structure and coupling relationship. In particular,
variability is the distinguishing feature that separates metamorphic
mechanisms from common mechanisms; this is an important area of research.
The incorporation of links, the changing relationships of adjacent links and
the changing properties of kinematic pairs have been explored to summarize
the variable features of metamorphic mechanisms. In essence, a metamorphic
kinematic joint is the essential prerequisite for changing the number of and
connective relationships among its active links, leading to a transformation
of the configuration of the entire mechanism (Zhang et al., 2011).
An example of a planar five-bar metamorphic linkage which has five
configurations is shown in Fig. 1. Transformations between them are
performed by locking different kinematic joints sequentially. When the
mechanism is in configuration 1, as shown in Fig. 1a, slider c is locked at
the top end of the slot in link d. In this configuration, the mechanism can
be treated as a four-bar mechanism. When the mechanism is in configuration 2,
as shown in Fig. 1b, links a and b are fixed together by locking the
revolute joint B as well as slider c and link d are unlocked. Therefore the
mechanism is transformed into a guide bar mechanism. When revolute joint C is
locked, links b and c are fixed to transform mechanism into another guide
bar mechanism, as shown in Fig. 1c. When the mechanism is in configuration 4,
links a and e are fixed together by locking revolute joint A, as shown in
Fig. 1d. Link b becomes the driving link of the mechanism. When link d arrives
at the location shown in Fig. 1e, joint D is locked to fix links d and e. This
transforms mechanism into a crank slider mechanism. Therefore, the mechanism
realizes transformations between different configurations by locking its
kinematic joints in particular sequences.
From Fig. 1, we conclude that the structure of the mechanism can be
transformed from one to another by locking 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. In
addition, metamorphic kinematic joints are able to change their types and
motion orientations to realize configuration transformations (Yan and Kuo, 2006). In a
metamorphic mechanism, there is at least one metamorphic kinematic joint,
which can change the number and connectivity relationships of active links.
There are some basic constituent elements, including links and
their connectivity relationships, which remain unchanged to create complex
coupling features among the links in adjacent configurations, as shown in Fig. 1.
Therefore, to understand the configuration characteristics of metamorphic
mechanisms, it is necessary to present a configuration representation that
can express not only the characteristics of the mechanism in all of its
configurations but also the variations during the transformation process
intuitively with the help of specific operations.
A five-bar planar metamorphic linkage.
Representing the configurations of metamorphic mechanisms
It is known that an adjacency matrix can be used to represent the
topological structures of metamorphic mechanisms. This matrix and an
EU-elementary matrix operation were introduced for expressing a configuration
transformation (Dai et al., 2005b; Dai and Rees Jones, 2005). Furthermore, a symbolic adjacency matrix was
constructed by introducing information on link variations and joint
orientations (Li and Dai, 2010a; Zhang and Ding, 2012). The variations and coupling features of the
metamorphic mechanism in adjacent configurations can be determined by
applying the generalized difference and intersection operations to the
corresponding symbolic matrices.
However, the matrices representing mechanisms in different configurations do
not have the same dimension and need to be normalized, increasing the
complexity of the representation and operations. Simultaneously, the upper
off-diagonal elements in the matrix are the same as the lower off-diagonal
elements, which means that the matrix contains information in duplicate.
Therefore, to decrease the complexity of expressing the matrix and
subsequent operations on it, we improve the symbolic matrix for the
mechanism in configuration m and express it as follows:
A(m)=L1J1,2(m)⋯J1,i(m)⋯J1,k-1(m)J1,k(m)a2,1(m)L2⋯J2,i(m)⋯J2,k-1(m)J2,k(m)⋮⋮⋱⋮⋮⋮⋮ai,1(m)⋯⋯Li⋯⋯Ji,k(m)⋮⋮⋮⋮⋱⋮⋮ak-1,1(m)ak-1,2(m)⋯ak-1,i(m)⋯Lk-1Jk-1,k(m)ak,1(m)ak,2(m)⋯ak,i(m)⋯ak,k-1(m)Lk,
where the principal diagonal element Li represents the link whose
sequence number in the mechanism is i. The numbers of rows and columns are
both k, which indicates the number of links in all configurations. Normally,
k is greater than or equal to the maximum number of effective links in every
configuration. The upper off-diagonal element Ji,j(m) denotes the
connectivity relationship between links Li and Lj. It can be
represented by a symbol with subscript where the symbol denotes the joint
type and the subscript expresses the geometric constraint relationship of
the joint axes located at the ends of the link. It is noted that the rule is
also applicable for analyzing tertiary links for the essence of the proposed
matrix is to record the connectivity relationship between links. A special
element -1 is employed here to represent a frozen joint between two links
(Lan and Du, 2008) and the element 0 represents the two links that are not connected.
Specific expressions for the geometric constraints, including parallelism,
intersection, coincidence, perpendicularity and randomicity, are given in
Li et al. (2009). The lower off-diagonal element aj,i(m) is the sequence
number of the configuration if the state of the corresponding upper
off-diagonal element Ji,j(m) is changed in configuration m. It
should be noted that, for the first configuration matrix A(1), if there
is a joint constraint between links i and j, the value of aj,i(1) is 1.
If there is no such constraint, its value is 0. For a matrix
A(m), when only the joint constraint between links i and j is
changed in configuration m, the value of aj,i(m) is m. This value
is also assigned to the other lower off-diagonal elements to be consistent
with the corresponding elements in the previous matrix,
A(m-1). Therefore, dimensional consistency of the matrices for
the mechanisms in different configurations is one of the advantages of the
proposed symbolic matrix representation. In addition, the symbolic matrix
can describe the connectivity relationship of all links synthetically as
well as their corresponding variations and provides sufficient information
for the subsequent matrix operations.
By applying Eq. (1), we express the five-bar metamorphic linkage, which has
the five configurations shown in Fig. 1, as
A(1)=eR00R∥R1aR∥R0001bR∥R0001c-11001d,A(2)=eR00R∥R1a-10002bR∥R0001cP⊥R1002d,A(3)=eR00R∥R1aR∥R0003b-10003cP⊥R1002d,A(4)=e-100R∥R4aR0003bR∥R0004cP⊥R1002d,A(5)=eR00-15aR∥R0003bR∥R0004cP⊥R5002d,
following the configuration transformation sequence. The numbers of rows and
columns in all of these matrices are 5, a result that depends on the number
of links a, b, c, d, and e occurring in these five configurations. The upper and
lower off-diagonal elements record information on the joint constraints and
their variations. For example, comparing A(4) and A(5), the elements
J1,2(4), J1,2(5) and J1,5(4), J1,5(5)
differ because they show that joint A and joint D have changed from -1 to
R and R∥R to -1, respectively. Meanwhile, the
corresponding elements a2,1(5) and a5,1(5) have changed
from 4 to 5 and 1 to 5 to record the sequence number of the
configuration in which joints A and D are in these positions in a working cycle.
Matrix operations for metamorphic mechanisms
The proposed symbolic matrix describes the topology of the mechanism in a
single configuration. However, exploring the variation laws of these
mechanisms in different configurations is very important for developing
novel metamorphic mechanisms. Therefore, it is feasible to take advantage of
matrix operations for constructing the original metamorphic mechanism and
determining the features of its topological variations.
Constructing the original metamorphic mechanism
The original metamorphic mechanism is able to evolve into any configuration
of the mechanism and contains all of the topological elements found in all
of configurations in a working cycle. A method for constructing original
metamorphic mechanisms from biological modeling and genetic evolution was
introduced in Wang and Dai (2007) and Zhang et al. (2008). In this paper, based on Eq. (3), an original matrix
A(0) for representing the original metamorphic mechanism is given by
A(0)=A(1)∪A(2)∪⋯∪A(m)∪⋯∪A(n)=L1⋯⋯⋯⋯⋯⋯⋮⋱⋮⋮⋮⋮⋮⋯⋯Li⋯∏m=1n-1Ji,j(m)∪Ji,j(m+1)⋯⋯⋮⋮⋮⋱⋮⋮⋮⋯⋯aj,i(1),⋯,aj,i(m),⋯,aj,i(n)⋯Lj⋯⋯⋮⋮⋮⋮⋮⋱⋮⋯⋯⋯⋯⋯⋯Lk,
where the operator ∪ represents the union of its arguments. The
result, A(0), has the same form as Eq. (1). All of 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. (3). Details of the operative principles are as follows:
The principal diagonal elements of A(0) are the same as
those of A(i) (i= 1, …, n), indicating the links
remain unchanged.
The operation that unites the lower off-diagonal elements and records
the sequence numbers of the configuration is performed by uniting the
elements in these matrices as a set of results in A(0), which
can be expressed asA(0)(j,i)=aj,i(1),…,aj,i(m),…,aj,i(n)(i<j≤k),where the number 0 is ignored. If the values of the adjacent elements are
same in this set, only one of them should be kept. The information given by
this set is very helpful for constructing the matrices for a single
configuration of the mechanism.
The physical meaning of uniting the upper off-diagonal elements of these
matrices is to achieve the most variability in the kinematic joints. The
operation starts from the upper off-diagonal elements in the first matrix,
A(1); then, the joint type and orientation are expanded based
on the elements of the next adjacency configuration matrix in the sequence.
We express the operation asA(0)(i,j)=∏m=1k-1A(m)(i,j)∪A(m+1)(i,j)=∏m=1k-1Ji,j(m)∪Ji,j(m+1)(i<j≤k).
Basically, the uniting operator is equivalent to an extension of the type
and axial orientation of a kinematic joint. If the adjacent elements are
same, it represents the corresponding connectivity relationship between the
related links keeps unchanged. So these same numbers in the operation result
need to be omitted just keeping one.
For example, according to Eqs. (2)–(5), elements A(0)(4, 3) and
A(0)(3, 4) of matrix A(0) can be calculated as follows:
A(0)(4,3)={1,1,3,4,4}={1,3,4}A(0)(3,4)=∏m=14J3,4(m)∪J3,4(m+1)=R∥R∪R∥R∪-1∪R∥R∪R∥R=R∥R∪-1∪R∥R.
Therefore, the joint between links b and c changes twice during the
configuration transformations from 1 to 3 and from 3 to 4 while its axial
orientation remains unchanged during the working cycle.
The construction procedure of the metamorphic kinematic joints can be
illustrated in Fig. 2. Firstly, the joints should be listed according to the
sequence indicated in the corresponding operation result. Further, geometric
limit is used to realize the transformations between these adjacent joints
in sequence. Geometric limit is a most common way of making the type of
kinematic joints to be changed by releasing or adding appropriate
constraints at suitable geometric locations. Such as in Fig. 2a, the
kinematic joint between links a and b is a revolute joint whose axis is
parallel to the adjacent revolute joint, R1. In the next configuration,
the revolute joint is locked. Therefore, two limiting stoppers are laid on
the two links a and b, respectively. When the two stoppers are contacted, the
two links are fixed together and the number of DOF of the revolute joint is
changed to zero in Fig. 2a. Figure 2b shows that the joint is performing
translating motions with arrows denoting the direction of pin's motion and
indicating the number of DOFs the joint possesses. When the pin reaches the
position shown in the second figure, it stops translating but remains
rotating as shown. This is identified as a typical metamorphic kinematic
joint that varies from a prismatic joint to a rotating pair. Similarly, Fig. 2c
demonstrates a series of varying orientations of a revolute pair
undergoing the orientations about different axes, successively.
The result of uniting different kinematic joints.
According to the construction process described above, the matrix of the
original metamorphic mechanism for the five-bar metamorphic linkage shown in
Fig. 1 is
A0=eR∪-1∪R00R∥R∪-1{1,5}aR∥R∪-1∪R∥R000{1,2,3}bR∥R∪-1∪R∥R000{1,3,4}c-1∪P⊥R{1,5}00{1,2}d.
Therefore, the original metamorphic mechanism can be generated by applying
the uniting operator to all of the mechanism's configurations and using link
and joint information. In particular, the matrix which includes the
information of all links and their connectivity relationships can make us
identify all possible combinations between links for creating different
mechanisms. So the mechanism is helpful to develop novel metamorphic
mechanisms using the representation method.
The joint variation matrix
The essential method for realizing configuration transformation of
metamorphic mechanisms is to change the characteristics of kinematic joints,
which lead to variations in the topology of the entire mechanism. Therefore,
to determine the joint variation rule for two adjacent configurations of a
mechanism, a generalized difference operation for two adjacency matrices is proposed.
Let Avar(m+1,m) be the joint variation matrix, which can
be described as the result of applying the generalized difference operator
to the topological representation matrices A(m+1) and
A(m), that is
Avar(m+1,m)=A(m+1)-A(m),
where – represents the generalized difference operator (Lan and Du, 2008; Li et al., 2010). The
resulting matrix contains information about the joint variation when the
mechanism is transformed from configuration m to configuration m+ 1. If the
mechanism is transformed from configuration m+ 1 to configuration m, the
joint variation matrix Avar(m,m+1) can be expressed as
Avar(m,m+1)=A(m)-A(m+1).
The purpose of this operation is to record variations in the upper and lower
off-diagonal elements. And the joint variation matrix achieved is the key
procedure for constructing the matrix represents the original metamorphic
mechanism which will be discussed in Sect. 4.3. If the two elements
located at the same position in matrices A(m) and
A(m+1) are equal, the corresponding element in matrix
Avar(m+1,m) is assigned the number 0. Conversely, the
elements in the minuend matrix are reserved directly, and the principal
diagonal elements remain unchanged for recognition purposes. The same rule
is used for the lower off-diagonal elements. If an upper off-diagonal
element is unchanged, the corresponding lower off-diagonal element needs to
be assigned the value 0 regardless of its actual value. The joint
variation matrix can be constructed directly from the physical meaning of
the joint variation rule. In addition, analysing the existing joints with
the characteristic of metamorphosis is one of the most important approaches
for achieving the principle of constructing the corresponding joint variation matrix.
Therefore, joint variation matrices for configurations 1 to 5 are given as follows:
Avar(2,1)=e00000a-10002b00000cP⊥R0002d,Avar(3,2)=e00000aR∥R0003b-10003c00000d,Avar(4,3)=e-10004a00000bR∥R0004c00000d,Avar(5,4)=eR00-15a00000b00000c05000d.
The relationship between the original metamorphic mechanism and the mechanism in any configuration
Because 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 is as
shown in Fig. 3.
The relationship between adjacent configurations:
the two adjacent matrices shown in Fig. 3 can be transformed into each other
using a joint variation matrix. From Eq. (9), the matrix
A(m+1) can be expressed as
A(m+1)=A(m)+Avar(m+1,m),
where + represents the generalized addition operator, which changes the
elements in matrix A(m) according to the corresponding
elements in the joint variation matrix of Avar(m+1,m).
Comparing the corresponding elements in the two matrices, the lower
off-diagonal elements in Avar(m+1,m) containing the value
m are selected, with the corresponding symmetrical upper triangular elements,
to replace the corresponding elements in matrix A(m) while
leaving the other elements unchanged. Similarly, matrix A(m)
can be expressed as
A(m)=A(m+1)+Avar(m,m+1).
For example, the relationship between matrices A(1) and A(2) is
A(2)=A(1)+Avar(2,1)A(1)=A(2)+Avar(1,2).
The relationships of the original metamorphic mechanism and the
mechanism in a single configuration: the original metamorphic mechanism is able to evolve into any configuration.
Therefore, the information on the mechanism in configuration m can be
extracted from the matrix A(0) to construct the corresponding
matrix A(m). The process of evolution from A(0)
to A(m) follows from Eq. (3).
The relationship between the original metamorphic mechanism and the
mechanism in any configuration.
First, the principal diagonal elements denoting the links in
A(0) are placed in their corresponding positions in
A(m) directly. Then, the lower off-diagonal elements
containing the value 1 and their corresponding upper off-diagonal
elements, which represent constraints on the joints of links in matrix
A(0), are similarly mapped to positions in A(m)
as long as the value of the corresponding element is not m. The next
important step is to select a number m from the elements comprising sets of
numbers and then, to identify its sequence number in the set
{aj,i(1), …, aj,i(m), …, aj,i(n)}. The sequence number can be used to
determine the corresponding joint constraint conveniently using the element
∏m=1n-1Ji,j(m)∪Ji,j(m+1). These
elements are then placed into A(m), the other elements of
which are assigned a value of 0.
For example, the elements marked by black triangles ▾ in Eq. (16) are
extracted to construct the matrix A(2) , which represents the
topology of the mechanism in configuration 2 according to the above procedure.
A0=e▾R▾∪-1∪R00R∥R▾∪-1{1▾,5}a▾R∥R∪-1▾∪R∥R000{1,2▾,3}b▾R∥R▾∪-1∪R∥R000{1▾,3,4}c▾-1∪P⊥R▾{1▾,5}00{1,2▾}d▾
The diagram in Fig. 3 shows that the evolutionary relationships between the
original metamorphic mechanism and all of its configurations can be
determined by applying matrix operations to the appropriate matrices.
Case study
A spatial four-bar metamorphic mechanism that has two configurations is
shown in Fig. 4. When the mechanism is in configuration 1, as shown in Fig. 4a,
it can be treated as an RSSR mechanism. The axis of joint D between links c
and d is perpendicular to the axis of joint A between links a and d. When
revolute joint D is transformed into a prismatic joint, the mechanism becomes
an RSSP mechanism, as shown in Fig. 4b.
A four-bar spatial metamorphic mechanism.
The topological structures of the metamorphic mechanism can be expressed in
matrix form as follows:
A(1)=dR0R⊥R1aS001bS101cA(2)=dR0P∥R1aS001bS201c.
The origin matrix of the original metamorphic mechanism and the joint
variation matrix can be expressed as
Avar(2,1)=A(2)-A(1)=d00R⊥R0a0000b0200cA(0)=A(1)∪A(2)=dR0R⊥R∪P∥R1aS001bS{1,2}01c.
The element R⊥R∪P∥R in matrix A(0)
represents the way in which both the axial orientation and the type of joint D
have changed. There, the joint can be considered a metamorphic kinematic
joint and be developed according to the variation sequence for the kinematic
behaviours of the entire mechanism.
Conclusions
The paper proposed a comprehensive symbolic matrix for characterizing the
topology of a metamorphic mechanism that involved information on the
variations of links and the axial orientations of the kinematic joints. In
addition, operations on the matrices of the adjacent configuration
mechanisms are defined to construct an origin matrix and joint variation
matrices. In particular, the construction and evolution of the matrix
representation for an original metamorphic mechanism show how it can be
transformed into any configuration matrix. The relationship between the
original metamorphic mechanism and all of its possible configurations and
methods of moving between them were presented. Examples illustrate the
effectiveness of this approach in characterizing metamorphic mechanisms. The
configuration representation of metamorphic mechanisms provides a foundation
for the analysis and synthesis of novel metamorphic mechanisms.
Acknowledgements
The authors gratefully acknowledge the support of the National Natural
Science Foundation of China (Project No. 51575018, No. 51275015 and No. 51175494)
and the Foundation of State Key Laboratory of Robotics (No. 2014).
Edited by: J. Schmiedeler
Reviewed by: two anonymous referees
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