Structure Synthesis of Single-Driven Metamorphic Mechanisms Based on the Augmented Assur Groups

Abstract

This paper presents a new way of structure composition of single-driven metamorphic mechanisms to develop a systematic and modularized structure synthesis methodology of metamorphic mechanisms based on augmented Assur groups (AAGs). Planar metamorphic mechanisms can hence be constructed based on the developed AAGs by applying the structure composition rule of general planar mechanisms formed by Assur groups (AGs). First, the one-mobility AAGs are introduced based on class II and class III AGs; the structure formulation and composition methodology of planar metamorphic mechanisms are then proposed based on the AAGs, and the basic problems including mobility and synthesis of constrained metamorphic working mobility-configuration are investigated. This leads to the investigation of the degenerated equivalent AGs of AAGs in the metamorphic process and the corresponding kinematic characteristics, providing references for kinematic synthesis of metamorphic mechanisms. Further, a typical spatial metamorphic group is introduced based on the concept of AAGs, and the structure formation and composition of spatial metamorphic mechanisms are presented. Examples show that both planar and spatial metamorphic mechanisms can be constructed by utilizing the one-mobility blocks extended from the AGs. [DOI: 10.1115/1.4006741]

Full text

Shujun Li
College of Mechanical Engineering
and Automation,
Northeastern University,
Shenyang 110004, P. R. China
e-mail: shjunli@mail.neu.edu.cn
Jian S. Dai
Centre for Advanced Mechanisms
and Robotics,
Tianjin University,
Tianjin, P. R. China;
King’s College London,
University of London,
Strand, London WC2R 2LS, UK
e-mail: jian.dai@kcl.ac.uk
Structure Synthesis
of Single-Driven Metamorphic
Mechanisms Based on the
Augmented Assur Groups
This paper presents a new way of structure composition of single-driven metamorphic
mechanisms to develop a systematic and modularized structure synthesis methodology of
metamorphic mechanisms based on augmented Assur groups (AAGs). Planar metamor-
phic mechanisms can hence be constructed based on the developed AAGs by applying the
structure composition rule of general planar mechanisms formed by Assur groups (AGs).
First, the one-mobility AAGs are introduced based on class II and class III AGs; the
structure formulation and composition methodology of planar metamorphic mechanisms
are then proposed based on the AAGs, and the basic problems including mobility and
synthesis of constrained metamorphic working mobility-configuration are investigated.
This leads to the investigation of the degenerated equivalent AGs of AAGs in the meta-
morphic process and the corresponding kinematic characteristics, providing references
for kinematic synthesis of metamorphic mechanisms. Further, a typical spatial metamor-
phic group is introduced based on the concept of AAGs, and the structure formation and
composition of spatial metamorphic mechanisms are presented. Examples show that both
planar and spatial metamorphic mechanisms can be constructed by utilizing the one-
mobility blocks extended from the AGs. [DOI: 10.1115/1.4006741]
Keywords: augmented Assur group, structure synthesis, metamorphic degeneration, con-
strained configuration
1 Introduction
Since the metamorphic mechanisms were proposed in 1998 [1],
interests and attention are aroused in the field of mechanisms
study, and new metamorphic mechanisms and metamorphic ways
to reconfigure the mechanisms have been introduced. Dai and
Rees Jones [2] explored the matrix representation of topological
changes and operation of metamorphic mechanisms. Liu and
Yang [3] investigated the metamorphic ways of reconfiguring
mechanisms. Yan and Kuo [4] presented topological representa-
tion of variable joints. Wang and Dai [5] introduced a theoretical
foundation and structure synthesis method of metamorphic mech-
anisms. Li et al. [6] presented joint-gene based topological repre-
sentation and configuration transformation of metamorphic
mechanisms. The structure theory and synthesis of metamorphic
mechanisms are being developed. However, there is still a lack of
study on structure synthesis of metamorphic mechanisms with a
systematic and effective way as that of a general mechanism.
One of the most important steps in designing kinematic chains
is to find all topological structures [7], detect isomorphism [8]in
these structures, and get all the kinematic chains required [9–13].
A classical approach for structure synthesis can be traced to Assur
[14] and the concept of Assur groups, which made contributions
in the field of structure synthesis [15] of planar mechanisms. The
structure theory and analysis method based on AGs (or dyads) is
one of the best available ways to compose and analyze planar ki-
nematic chains, and it has been used by scholars across the world
[14–24]. The process of structure design and kinematic analysis of
the chains based on AGs can be easily computerized [25–27].
From the kinematic structure [28] point of view, the metamor-
phic mechanism is a kind of multimobility mechanism; the differ-
ence is that the general multimobility mechanisms are controlled
by the driver links and have only one mobility-configuration as a
working configuration, but there are more mobility-configurations
as working configurations of a metamorphic mechanism which is
controlled by driver links and metamorphic elements of links or
joints.
Wang and Dai [4] introduced a procedure for structure synthesis
of multimobility metamorphic mechanisms via the topological ma-
trix operations. Li et al. [6] developed a joint-gene based topological
representation that the metamorphic mobility-configuration can be
formed by changing the orientation of the metamorphic joints. Zhang
et al. [29] presented an evolution methodology for synthesis and con-
figuration design of metamorphic mechanisms, which was developed
based on biological modeling and genetic evolution with biological
building blocks. They further introduced a kind of “metamorphic
reconfiguration unit” for the configuration design of metamorphic
mechanisms [30] and investigated spatial reconfiguration properties
based on Lie displacement subgroup concept, leading to reconfigura-
tion of spatial linkage chains with metamorphic modules [31].
Recently, for creating the metamorphic mechanisms, Zhang et al.
[32] invented a new metamorphic joint as variable axis (vA) joint
that has the ability to change the mobility between 1 and 3 to form
three distinct kinematic pairs for constructing metamorphic mecha-
nisms. Gan et al. [33] created a new metamorphic joint called recon-
figurable Hooke (rT) joint that changes the installation angle of the
joint to form various structures of parallel mechanisms [34]. The
limb synthesis and mobility-change aimed mechanism construction
were proposed by Gan et al. [35] for developing metamorphic mech-
anisms. Further to this, Zhang et al. [36] proposed a morphological
synthesis approach for metamorphic mechanism synthesis.
If we take the driver link of a general multimobility mechanism
and create a metamorphic process with a link that has the geometric
complement with another link to form mobility change, the general
multimobility mechanism [37,38] would be equivalent to a
mobility-configuration of the metamorphic mechanism. This
Contributed by the Mechanisms and Robotics Committee of ASME for publica-
tion in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received February 8,
2011; final manuscript received April 23, 2012; published online May 31, 2012.
Assoc. Editor: Qiaode Jeffrey Ge.
Journal of Mechanisms and Robotics AUGUST 2012, Vol. 4 / 031004-1Copyright V
C2012 by ASME

presents an idea of adding a link into an element of the AG to form
a one-mobility group to construct a metamorphic mechanism. If the
structures of the one-mobility group were to be formed, the struc-
ture of planar metamorphic mechanisms can be composed in the
same way as that of the general planar mechanism based on the
structure theory of AGs. This one-mobility group is called an AAG
or a basic metamorphic group in this paper.
An AAG will be degenerated into an equivalent AG after such
a link annexing metamorphic operation on the AAG. The kine-
matic characteristics of an AAG can be obtained by investigating
the corresponding degenerated equivalent AG as the essentials for
kinematic synthesis of metamorphic mechanisms. On the other
hand, the kinematic and dynamic analysis of metamorphic work-
ing phases can be implemented based on the degenerated AG to
provide a convenient and efficient way. In addition, the concept of
AG has not been applied to the structure synthesis of the spatial
mechanisms, and it is also an intention of the authors to check if
the concept of an AAG can be adopted for the structure synthesis
of spatial metamorphic mechanisms.
This paper focuses on forming the AAGs and developing a sys-
tematic way for synthesis of metamorphic mechanisms. It also pro-
vides an efficient way for kinematic and dynamic analysis. This
paper is organized as follows. The development of AAGs, the corre-
sponding degenerated AGs, and the kinematic characteristics of the
AAGs are presented in Sec. 2. Section 3studies the structure compo-
sition formula, the method of single-driven multimobility metamor-
phic mechanisms, and the metamorphic working mobility-
configurations based on the AAGs. Section 4investigates the compo-
sition of spatial metamorphic mechanisms by expanding the concept
of AAGs. The conclusions are given at the end of the paper.
2 Basic Metamorphic Groups
2.1 Assur Groups. Any planar mechanisms can be formed
by adding an element of the AGs to the driver link and the frame
and/or the former element of the AGs and the frame according to
the structure theory based on AGs. The class II AG is mostly used
in the planar mechanism design and an element of it is formed by
two binary links with three joints of revolute (R) type and/or pris-
matic (P) type. An element of three revolute R joints is called as
RRR-element in the group. The five elements of the class II AG
are shown in Fig. 1.
2.2 Class II Augmented Assur Group. If an additional
binary link and an R/P joint are inserted into the class II AG of
Fig. 1, mobility of the elements in the augmented group is one
instead of zero in the class II AG, and this augmented group is
called the class II AAG. The structure forms of the class II AAG
result in 12 elements following the permutation and combination
of three binary links and four R and/or P joints. However, two of
them are isomorphism, and one needs to be excluded since it has
four P joints resulting two mobility, therefore the total number of
the elements in the class II AAG of mobility one is reduced to
nine forms as in Fig. 2.
2.3 Class III and Class IV Augmented Assur Group
2.3.1 Class III and Class IV Assur Group. It is well known
that the element of class III AG is formed by one ternary link
connected to three binary links with R and/or P pairs. The element
of class IV AG is formed by two ternary links connected to two
binary links with R and/or P pairs. Some elements of typical class
III AG are presented in Fig. 3(a), and an element of class IV AG
is presented in Fig. 3(b).
2.3.2 Class III and Class IV Augmented Assur Group. If an
additional binary link and an R/P joint are inserted into an element
of the class III AG and class IV AG in Fig. 3, the mobility of ele-
ments in the group becomes one instead of zero. The group that
consists of these elements is called class III and class IV aug-
mented Assur group or AAG in short. Some typical elements of
class III and class IV AAG based on the class III and class IV AG
in Fig. 3are formed as in Figs. 4(a)and 4(b), respectively.
2.4 The Metamorphic Degeneration of AAGs and the
Kinematic Characteristics. The element of an AGG will degen-
erate into the equivalent AG during a metamorphosis operation.
The kinematic characteristics of elements in the AGG from the
corresponding degenerated AG are useful for kinematic synthesis
of metamorphic mechanisms. The AAG and the corresponding
degenerated AG allow a modularized approach of the analysis.
In general, one of the metamorphosis operations is implemented
by using geometric constraints and/or force constraints to make two
links to be overlapped, or to make the metamorphic joints to change
the mobility. In addition to the vA joint [32] and the rT joint [33],
some typical structures of metamorphic joints and links using geo-
metric constraints and force constraints are shown in Fig. 5(a)of a
Fig. 1 Elements of the class II Assur group
Fig. 2 The nine elements of the class II augmented Assur
group
Fig. 3 Some elements of typical class III and class IV Assur
groups
Fig. 4 Typical elements of class III and class IV augmented
Assur groups
031004-2 / Vol. 4, AUGUST 2012 Transactions of the ASME

turning joint with the geometric constraint to overlap two links, and
in Fig. 5(b)of a turning joint with the geometric constraint con-
trolled by the spring force. Figures 5(c)and 5(d)are prismatic joints
[39,40] with the geometric constraint and the spring force
constraint, respectively. Figure 5(e)is a metamorphic joint with the
geometric constraint controlled by the position of joint D with cam-
mechanism on the joint, to limit the clockwise turning of the link,
and it can be unlocked with the movement of prismatic joint con-
trolled by the cam-mechanism on the joint.
When the above concept and the metamorphic joints [32–34]
are to be used alternately in the metamorphosis operation instead
of the general joints of an AAG, the element of the AAG will be
degenerated into that of the equivalent AG. The possible elements
of the degenerated AG and the kinematic characteristics of classic
II AAG are presented in Table 1.
In the same way, the element of class III AGG degenerates into
that of the equivalent AG after the metamorphosis process. The
elements of the AG from the degenerated class III AAG are pre-
sented in Table 2. The kinematic characteristics of elements of the
class III AAG can be analyzed in the same way as that of class II
AAG in Table 1when the metamorphic joints in Fig. 5are to be
used instead of the general joints of the class III AGG alternately
in the metamorphosis operation.
3 Structure Synthesis of Metamorphic Mechanisms
Based on Augmented Assur Groups
3.1 Structure Synthesis of Planar Mechanisms Based on
Assur Groups. As stated above, any planar mechanisms can be
formed by adding an element of an AG to the driver link and frame,
Fig. 5 The proposed structures of metamorphic joints and
links using geometric and force constraints
Table1 Augmented Assur groups (AAGs) and possible degenerated corresponding Assur groups (AGs)
No./symbol Class II AAG
Possible degenerated equivalent AG after
metamorphic process using geometric constrains
Possible kinematic characteristics of
degenerated equivalent AG
1/ RRRR Rotation, oscillation, changing link
dimension and output direction
2/ RRPR Rotation, oscillation, reciprocating
3/ RPPR Rotation/oscillation and translation,
oscillation
4/ RRRP Rotation/oscillation, rotation and translation
5/ RRPP Rotation and translation,
rotation/oscillation and translation
6/ RPRP Oscillation, rotation/oscillation and translation,
rotation and translation
7/ RPPP Rotation/oscillation and translation,
translation and translation
8/ PRRP Translation and translation,
rotation and translation
9/ PRPP Translation and translation,
rotation/oscillation and translation
Journal of Mechanisms and Robotics AUGUST 2012, Vol. 4 / 031004-3

and/or former element of an AG and the frame according to the
structure theory based on AGs. A planar mechanism with mobility
mcan be assembled as follows. It starts by choosing the frame,
choosing mlink(s) as driver link(s) connected to the frame, and
adding elements of the AGs connected to the driver link(s) and
frame and/or connected to former element of the AG and frame in
sequence. A one-mobility six-bar linkage, for example, can be
assembled as follows. It starts by choosing a link connected to the
frame as a driver link with one RRR element of the AG connected
to the driver and the frame, and adding an RRP element of the AG
to connect to the RRR element and the frame. The composition pro-
cess and the assembled mechanism are presented in Fig. 6.
3.2 Structure Composition Theory and Structure Synthesis
of Metamorphic Mechanisms
3.2.1 Structure Formula and Composition Procedure Based
on Augmented Assur Groups. Planar metamorphic mechanisms
can be formed by structure blocks AAGs (and AGs) with the same
connective relationship and structure composition rules as that of
general planar mechanisms formed by AGs only. By adding an
element of an AAG (and AG) to the driver link and frame and/or
connected to the former element of the AAG/AG and frame in
sequence, a planar metamorphic mechanism can be constructed.
Thus, multimobility metamorphic mechanisms of great complex-
ity can be constructed by the sequential addition of elements of
the AAG (and AG) to the chains. This forms the structure compo-
sition theory of metamorphic mechanisms based on AAG. The
structure formulas and composition procedure for a multimobility,
l-loop, single-driven metamorphic mechanism are as follows:
(1) Calculate the number of elements of an AAG added to the
mechanism.
k¼m1(1)
where kis the number of elements of the AAG added to the
mechanism and mis the mobility of a source metamorphic
mechanism [31].
(2) Calculate the number of elements of the AG added to the
mechanism.
Adding one element of class II AAG/AG forms a loop in
the chain, and adding one element of the class III and class
IV AAG/AG forms two loops in the chain. The relationship
between loops and AAG/AGs of class II–IV is
l¼gII þ2gIII þ2gIV (2)
where lis the number of loops of the mechanism, g
II
is ei-
ther an element of class II AG or an element of class II or
elements from both groups, g
III
is either an element of class
III AG or an element of class III AAG or elements from
both groups, and g
IV
is either an element of class IV AG or
an element of class IV AAG elements from both groups.
(3) Choose a link connected to the frame as the driver link.
(4) Add elements of the AAG/AG to the driver link and frame
and/or connected to the former AAG/AG and the frame in
sequence.
It is clear that at least one element of the AAG should be
added to the mechanism to provide metamorphosis mobility
to form the single-driven metamorphic mechanism.
3.2.2 Mobility of Metamorphic Mechanisms. According to
the structure theory of metamorphic mechanism based on AAG,
the mobility of the mechanism is the combination of the mobility
of driver and mobility of elements of the AGG. Add one element
of the AGG to the mechanism, one more mobility will produce
the mechanism because the AAG is a one-mobility structure
Fig. 6 Planar six-bar linkage and its composition process
Table 2 Class III augmented Assur group and possible degenerated corresponding Assur group
No. Class III AAG Possible degenerated equivalent AG after metamorphic process using geometric constrains
1
2
3
4
5
6
031004-4 / Vol. 4, AUGUST 2012 Transactions of the ASME

group; the mobility of single-driven metamorphic mechanism
based on AAG can be written as
m¼kþ1(3)
where kis the number of AAG elements. The mobility of meta-
morphic mechanism based on AAG can further be calculated by
Gru¨bler’s formula as that of general planar mechanism
m¼3n2pl(4)
where nis the number of moving links and p
l
is the number of
lower pairs in the mechanism.
3.2.3 Examples of Structure Synthesis of Metamorphic
Mechanisms. A two-mobility metamorphic mechanism which can
provide rotation/oscillation and the translation output movements
can be formed as follows. This starts by first choosing a link as
driver link (with frame) of the mechanism, and following Eq. (1),
choosing k¼m1¼1 elements of an AGG providing one mobil-
ity and one loop with rotation/oscillation and translation output
movement. For instance, a RRPR according to its degenerated
Assur group, and then connect the RRPR AAG to the driver and
frame to form the mechanism. The composition process and the
assembled source metamorphic mechanism are shown in Fig. 7.
The two working stages of the mechanism are discussed in Sec.
4.2 in the constraint analysis.
It should be pointed out that the difference of a two-mobility
source metamorphic mechanism from a two-mobility general
mechanism is that the metamorphic mechanism has single-driver
link and one of the mobility is constrained by kinematic geometry
arrangements, geometric constraints, designated profiles of links
and joints, etc. alternately according to the required working con-
figurations. The constrained metamorphic process of the mecha-
nism in Fig. 7is shown in Fig. 9, Sec. 4.2.
Further example is to design a one-loop, two-mobility meta-
morphic mechanism with translation input, which can be formed
as follows: first, choosing a prismatic link as driver link (with
frame) of the mechanism providing translation, and second,
according to Eq. (1), choosing k¼m1¼1, for instance, an
RRRP AGG providing one mobility with translation output move-
ments, and then connecting the RRRP AAG to the driver link and
frame to form the mechanism. The composition process and the
assembled source mechanism are shown in Fig. 8.
A typical metamorphic mechanism for metal forming and cut-
ting is analyzed in Sec. 4.2 of the constraint analysis of various
working stages.
4 Working Configuration Analysis of Planar
Metamorphic Mechanisms
4.1 Total Numbers of Metamorphic Working Configura-
tions. In fact, the metamorphic process of planar mechanisms
formed by elements of AAGs is the process to degenerate all AAG
elements into the corresponding AG elements. That means the
metamorphic process is taking place only on the AAGs and annex-
ing the links (or utilizing the metamorphic joints) of AAGs.
There are several metamorphic joints that connect annexed
links in one element of class II AAG, leading to four working
mobility-configurations. For two elements of class II AAG,
42¼16 mobility-configurations will appear, and so on. The num-
ber of metamorphic mobility-configurations of the mechanism
composed with class II AAG is
T¼4t(5)
where tis the number of elements in class II AAG in the
mechanism.
The general formula of total numbers of metamorphic working
configurations of single-driven metamorphic mechanism based on
AAG can be deduced as follows:
T¼Y
k
i¼1
Tt
i(6)
where Tt
iis the number of mobility-configurations of the loop
with the ith class of AAG and kis the number of AAG in the
mechanism.
4.2 Constrained Metamorphic Working Configuration
Analysis. To obtain all metamorphic mobility-configurations of a
mechanism, we can restrain joints of elements of AAG of the
mechanism alternately according to the required working stages
and working sequence using kinematic geometry arrangements,
geometric constraints, designated profiles of links and joints, etc.
The mechanism in Fig. 7, for example, has one element of class
II AAG; there are four working mobility-configurations according
to Eq. (5). All four metamorphic configurations can be obtained
by using geometric and force constraints of metamorphic joints as
in Fig. 5to constrain one mobility at a time. The corresponding
constrained metamorphic configurations are shown in Fig. 9. The
first two working phases are given in Figs. 9(a)and 9(b). Figure
9(a)is a configuration with mobility one and the movement of
slider D is constrained by spring force with spring forces being
larger than the joint frictions. In this case, the mechanism is a
four-bar linkage with only four revolute joints while the slider D
does not have a relative motion with the sleeve. When the mecha-
nism moves to another configuration in Fig. 9(b), the movement
of joint B is constrained by a geometric constraint to make two
links to become one. Therefore, the driving force presses the
spring to move the slider D. The mechanism becomes a crank-
slider mechanism. Figures 9(c)and 9(d)give another two working
stages with the force control. By actuating joint A counterclock-
wise as that in Fig. 9(c), joint C is constrained by a peg that the
mechanism is a crank-slider mechanism with joints A, B, and E
and slider D movements. When the slider D moves to a certain
position in the housing when the housing link is restricted by a
bolt at joint E in Fig. 9(d), the peg is released so that joint C is
free to move. The mechanism becomes another crank-slider
mechanism with the housing link becoming the ground link.
Figure 10(a)shows a mechanism of a practical steel ingot-
cutting machine. The main mechanism is a two-mobility source
Fig. 8 A RRRP source metamorphic mechanism and its com-
position process
Fig. 7 A two-mobility RRPR source metamorphic mechanism
and its composing process
Journal of Mechanisms and Robotics AUGUST 2012, Vol. 4 / 031004-5

metamorphic mechanism with initial working stage as shown in
Fig. 10(b).
Analyzing this mechanism, it can be seen that it has two work-
ing stages as shown in Fig. 11. Figure 11(a)gives the working
stage I—press holding, in which the movement of lower cutter 4
was restrained by spring 7, the upper cutter 1 moves downward
the ingot as in Fig. 10(a). At this stage, point D is static while
crank 2 moves the upper cutter 4 downward and also moves block
11 downward to the ingot 8 through links 9 and 10. When both
upper cutter 1 and block 11 reach the ingot as in Fig. 11(b), the
resisting force from the pressing is increased and overcomes the
spring force resulting from spring 7. At this constraint point, the
metamorphic mechanism changes its stage to working stage II
when the upper cutter becomes static but point D is moving
upward as in Fig. 11(c). This gives the working stage II for cut-
ting. At this stage, the lower cutter 4 moves upward toward the in-
got as in Fig. 11(d).
5 Structure Synthesis of Spatial Metamorphic
Mechanisms
The method of forming planar metamorphic mechanisms by
AAGs can be adopted and developed to studying the structure
synthesis of spatial metamorphic mechanisms. The idea is to de-
velop the one-mobility spatial metamorphic group (SMG) to form
the spatial metamorphic mechanism as that of planar metamorphic
mechanism formed by AAGs.
5.1 The Spatial Metamorphic Group (SMG). The one-
mobility SMG is proposed referring to the AAGs of planar meta-
morphic mechanisms. The SMG can be converted from one-
mobility spatial mechanisms. Some typical one-mobility SMGs
(except the local mobility) are formed as in Fig. 12 but further
work needs to be done to study this group.
5.2 Structure Synthesis of Single-Loop Spatial Metamorphic
Mechanisms. A spatial metamorphic mechanism can be com-
posed by connecting an element of the SMG to the driver and
Fig. 11 The working stages of the metamorphic mechanism
extracted from a steel-ingot cutting machine
Fig. 9 Four metamorphic working-stage configurations with
one class II AAG linkage
Fig. 10 The mechanism of a steel ingot-cutting machine
031004-6 / Vol. 4, AUGUST 2012 Transactions of the ASME

frame in sequence, which is in the similar way as composition of
a planar metamorphic mechanism by AAGs. A two-mobility spa-
tial metamorphic mechanism can be formed as follows: First,
choosing a link as driver link with frame jointed together and add-
ing an element of the SMG to the driver link and the frame to
form a spatial metamorphic mechanism. A two-mobility 5R spa-
tial spherical metamorphic mechanism, for example, can be
formed by a driver link with the frame, and a RRRR (spherical)
SMG connected to the driver link and frame as shown in Fig. 13.
One of the metamorphic working mobility-configurations of the
5R spherical mechanism is shown in Fig. 14 when two links are
being annexed. The 5R spherical mechanism was applied to the
metamorphic hand [41,42] as shown in Fig. 15.
A two-mobility spatial RRSSR metamorphic mechanism can be
formed by a driver link with frame, and a RSSR SMG joint to the
driver and frame in the same way as shown in Fig. 16. One of the
metamorphic mobility-configurations of the RRSSR spherical
mechanism is shown in Fig. 17.
6 Conclusions
This paper presents a systematic and modularized structure syn-
thesis method of single-driven multimobility planar and spatial
source metamorphic mechanisms based on the developed AAGs and
SMG. The structure composition method and rules are presented in
the same way as that of the general planar mechanisms composed by
AG. The structure composition of metamorphic mechanisms is
developed and the characteristics of the AAGs are investigated to
provide the references for kinematic synthesis of metamorphic
mechanism. The study of the degenerated AGs further proposes a
way for the kinematic and dynamic analysis of mobility-
configurations as that of traditional planar mechanisms. On the other
hand, extending the AAGs of planar metamorphic mechanisms to
the spatial metamorphic group (SMG) of spatial ones presents a new
approach to the study of spatial metamorphic mechanisms and kine-
matic analysis of the metamorphic working configurations.
Acknowledgment
The authors would like to acknowledge the financial support of
the National Natural Science Foundation of China (NSFC)
Fig. 14 The one-mobility metamorphic working configuration
Fig. 15 The metamorphic hand [43,44]
Fig. 16 Spatial RRSSR source metamorphic mechanism and
its composition process
Fig. 17 The one-mobility working configuration with the links
being annexed
Fig. 12 Some typical elements of the one-mobility spatial
metamorphic group
Fig. 13 5R spherical source metamorphic mechanism and its
composing process
Journal of Mechanisms and Robotics AUGUST 2012, Vol. 4 / 031004-7

(Grant No. 50875038, 51175069) and the support of the Engineer-
ing and Physical Science Research Council (EPSRC) of the
United Kingdom.
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031004-8 / Vol. 4, AUGUST 2012 Transactions of the ASME