The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.

1. The International Journal Of Engineering And Science (IJES)
|| Volume || 3 || Issue || 3 || Pages || 26-34 || 2014 ||
ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805
www.theijes.com The IJES Page 26
Identification of Isomorphism amongst Kinematic Chains and
Inversions Using Link - Link Connectivity and Adjacency Values
1,
Rajneesh Kumar Rai, 2,
Dr. Sunil Punjabi, 3,
Dr. Sunil Raj Madan
1,
Deptt. Of Mech. Engg. SDITS, Khandwa
2,
Deptt. of Mech. Engg. GEC, Ujjain
3,
Deptt. of Mech. Engg. MITS,Ujjain
----------------------------------------------------------ABSTRACT------------------------------------------------
The present work deals with problem of detection of isomorphism which is frequently encountered in
structural synthesis of kinematic chains. Using link-link connectivity and link- adjacency values, a new method
has been proposed to reveal simultaneously whether chain is isomorphic. Two new invariants, called first
adjacency link value (FALV) and second adjacency link value (SALV), are developed for identifying distinct
mechanisms of a planer kinematic chain. Another two invariants, called maximum first adjacency link value
(MFALV) and maximum second adjacency link value (MSALV), which are the by – products of the same
method, has been proposed to detect isomorphism among the kinematic chains. These invariants takes into
account the connectivity value of links and type of joints of kinematic chain and are used as the composite
identification number of a kinematic chain and mechanisms. The proposed method is easy to compute, reliable
and capable of detecting isomorphism in all types of kinematic chains. This study will help kinematician /
designer to select the best kinematic chain and mechanism to perform the specific task at conceptual stage of
design. The proposed method is applied to 9-links, 2DOF kinematic chains.
Key-Words: Inversion of kinematic chain, Isomorphism, Joint value, Kinematic chain, Link value.
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Date of Submission: 27 February 2014 Date of Acceptance: 25 March 2014
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I. INTRODUCTION
A major problem encountered during the structural synthesis of kinematic chains is the detection of
isomorphism among the kinematic chains. Various methods have been reported by the researchers in the past for
detection of isomorphism among the kinematic chains along with the methods to detect distinct inversions (1-
10). Each method has its own merits and demerits and involves lots of complex mathematics. There is always a
need of simple reliable and efficient method for detecting isomorphism among the kinematic chains. How ever,
almost much of the methods reported so far are based on 1st
adjacency of links. 1st
adjacency of a link deals with
links that are directly connected to it. These methods do not take care of higher order adjacency of links so these
methods do not specify necessary and sufficient conditions in the present work, this aspect is given more
importance and two structural invariants, 1st
adjacency link value (FALV) and 2nd
adjacency link value (SALV)
are developed on the basis of linked connectivity and link values to identify distinct inversions of a given chain.
Another two invariants, maximum1st adjacency link value (MFALV) and maximum 2nd
adjacency link value
(MSALV) which are the by-products of the same methods has been proposed to detect isomorphism among the
kinematic chains.
II. TERMINOLOGY
Following definitions are to be understood clearly before applying this method.
2.1 Link C o n n ec tiv ity
It is a numerical value given for each link which is based on its connectivity to other links. For example,
consider the adjacency matrix of kinematic chain 1 which is given below.

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All the diagonal elements represent degree of the corresponding links. If a link ‘j’ is directly connected to link
‘i’ then it is represented by the degree of link ‘j’ and if link ‘j’ is not directly connected to link ‘i’ then it is
represented by a number equal to zero (0).
However link connectivity of link 1 is represented by:
LC1 = 3 + 2 × 10 -3
+ 1 × 10 -2
= 3.012
Similarly, link connectivity for other links of kinematic chain 1 are:
LC2 = 3.012, LC3 = 3.012, LC4 = 3.012, LC5 = 2.011, LC6 = 2.011, LC7 = 2.011, LC8 = 2.011, LC9 = 2.02
Similarly, link connectivity for all the links of kinematic chain 2 are:
LC1 = 3.012, LC2 = 3.021, LC3 = 3.011, LC4 = 2.002, LC5 = 3.021, LC6 = 2.002, LC7 = 2.011, LC8 = 2.02, LC9 =
2.011
2.2 Joint Val u e
For a particular joint it is defined as the ratio of algebraic sum of all the link connectivity to the number of links
connected at that joint. It is denoted by Jv.
Jv = ( Σ all link connectivity of all links connected ) / number of links connected
For example, joint value of joint ‘a’ of kinematic chain 1 is given by:
Ja = (LC8 + LC9) / 2
= 2.0155
Similarly, joint value for other joints of kinematic chain 1 is:
Jb = 2.5115, Jc = 3.012, Jd = 3.012, Je = 2.5115, Jf = 2.0155, Jg = 3.012, Jh = 3.012, Ji = 2.5115, Jj = 2.011, Jk =
2.5115
Similarly, joint value for all the joints of kinematic chain 2 is:
Ja = 2.507, Jb = 2.5115, Jc = 3.016, Jd = 3.0115, Je = 3.0165, Jf = 2.5115, Jg = 2.5115, Jh = 2.516, Ji = 2.0155, Jj =
2.0155, Jk = 2.511
2.3 Link Va l u e
For a link it is defined as sum of joint values of all the joints of that link. For example, link value of link 1 of
kinematic chain as shown in figure 1, is given by:
LV1 = Jg + Jh + Ji
= 8.5355
3 0 3 3 2 0 0 0 0
0 3 3 3 0 2 0 0 0
3 3 3 0 0 0 2 0 0
3 3 0 3 0 0 0 2 0
3 0 0 0 2 2 0 0 0
0 3 0 0 2 2 0 0 0
0 0 3 0 0 0 2 0 2
0 0 0 3 0 0 0 2 2
0 0 0 0 0 0 2 2 2

3. Identification of Isomorphism amongst Kinematic Chains and…
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Similarly, link value of other links of kinematic chain 1 is:
LV2 = 8.5355 , LV3 = 8.5355 , LV4 = 8.5355 , LV5 = 4.5225 , LV6 = 4.5225 , LV7 = 4.527 , LV8 = 4.527 , LV9 =
4.031
Similarly, link value of all the links of kinematic chain 2 is:
LV1 = 8.535, LV2 = 8.039 , LV3 = 8.5385 , LV4 = 5.0185 , LV5 = 8.044 , LV6 = 5.023 , LV7 = 4.5265 , LV8 = 4.031
, LV9 = 4.5315
III. NEW STRUCTURAL INVARIANTS
Considering all essential features of the kinematic chains, two new structural invariants called first
adjacency link value (FALV) and second adjacency link value (SALV) are proposed. They are invariants for a
kinematic chain because they are independent of the labeling of links and joints of a chain.
3.1 Fi r st Adj ac en cy Li n k Va l ue (FALV)
For a link it is defined as the sum of link values of all the links that are directly connected to it. For example,
FALV of link 1 of kinematic chain, as shown in figure, is given by:
Lf1 = LV3 + LV4 +LV5
= 21.5935
Similarly, FALV for other links of kinematic chain 1 are given by:
Lf2 = 21.5935, Lf3 = 21.598, Lf4 = 21.598, Lf5 = 13.058, Lf6 = 13.058, Lf7 = 12.5665, Lf8 = 12.5665, Lf9 = 9.054
Similarly, FALV for all the links of kinematic chain 2 are given by:
Lf1 = 21.601, Lf2 = 18.58, Lf3 = 21.1005, Lf4 = 16.574, Lf5 = 18.0895, Lf6 = 16.083, Lf7 = 12.5695, Lf8 = 9.058,
Lf9 = 12.075
3.2 Se c on d Adjacency Li n k Va l ue (SALV)
For a link it is defined as the sum of first adjacency link values of all the links that are directly connected to it.
For example, SALV of link 1 of kinematic chain, as shown in figure1, is given by:
LS1 = Lf3 + Lf4 + Lf5
= 56.254
Similarly, SALV for other links of kinematic chain 1 are given by:
LS2 = 56.254, LS3 = 55.7535, LS4 = 55.7535, LS5 = 34.6515, LS6 = 34.6515, LS7 = 30.652, LS8 = 30.652, LS9 =
25.133
Similarly, SALV for all the links of kinematic chain 2 are given by:
LS1 = 55.764, LS2 = 53.7575, LS3 = 52.7505, LS4 = 40.181, LS5 = 49.759, LS6 = 36.6695, LS7 = 30.1585, LS8 =
24.6445, LS9 = 27.1475
3.3 Ma xi mu m Fi r s t Ad jac en c y Link Va l u e (MFALV)
For a kinematic chain, it is defined as the maximum of first adjacency link value of all the links of a kinematic
chain. For example, MFALV of kinematic chain 1 and 2 are 21 .598 a nd 21.6 0 1 r e sp ec ti ve l y .
3 .4 Ma xi mu m S e c o nd Ad ja ce n cy Li n k Va lu e (MSALV)
For a kinematic chain, it is defined as the maximum of second adjacency link value of all the links of a
kinematic chain. For example, MFALV of kinematic chain 1 and 2 are 5 6.25 4 a nd 5 5.764
r e spe ct i vel y.
IV. TESTS FOR ISOMORPHISM AMONG INVERSIONS
The first and second adjacency values of the various links have the characteristics to declare the
number of inversions that can be obtained from a given chain and by fixing which links these inversions are
possible. If the first and second adjacency values of two links are identical then the inversions are equivalent and
constitute only one distinct mechanism.
V. TESTS FOR ISOMORPHISM AMONG KINEMATIC CHAINS
The invariants of MFALV and MSALV are definitive test for isomorphism among the chains. The test
is very simple based on comparison of these two structural invariants. If in comparison these invariants are same
then chains are declared to be isomorphic otherwise not. For example:

4. Identification of Isomorphism amongst Kinematic Chains and…
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For chain 1:-
MFALV = 21.598 MSALV = 55.7535
For chain 2:-
MFALV = 21.601 MSALV = 55.764
This method shows that chain 1 and chain 2 are non- isomorphic as the corresponding values of MFALV and
MSALV are different. Similar explanation can be given to other chains also.
VI. RESULTS
(1) The proposed structural invariants MFALV and MSALV of kinematic chain are able to detect isomorphism
among the kinematic chains. In the present paper, 9 links, 2 DOF kinematic chains have been tested successfully
for their non- isomorphism.
(2) Using the presented method the number of mechanisms derived from the family of 9 links, 2 DOF kinematic
chains are 254. This is also verified by the results given by A.Dargar. et. al. which is also 254.
VII. CONCLUSION
Authors strongly believe that method is unique as it takes care of nature and all inherent properties of
the mechanism. The present method is applicable to planar chains of any size and complexity even the
kinematic chain with co-spectral graph and also able to identify all distinct mechanisms derived from a given
kinematic chain. It is hoped that the presented method presents a new concept on which a new classification
system for distinct mechanisms can be based.
REFERENCES
[1] J . J . U i c k e r a n d A . R a i e u , A m e t h o d f o r i d e n t i f i c a t i o n a n d r e c o g n i t i o n o f e q u i v a l e n c e o f
k i n e m a t i c c h a i n s , M e c h . M a c h . T h e o r y , V o l 1 0 , 1 9 7 5 , p p 3 7 5 - 3 8 3
[2] T . S . M r u t h y u n j a y a a n d M . R . R a g h a v a n , S t r u c t u r a l A n a l y s i s o f k i n e m a t i c c h a i n s a n d
M e c h a n i s m b a s e d o n M a t r i x R e p r e s e n t a t i o n , T r a n s a c t i o n o f A S M E , J o u r n a l o f
M e c h a n i c a l D e s i g n , V o l . 1 0 1 , 1 9 7 9 , p p 4 8 8 - 5 1 8
[ 3 ] H . S . Y a n a n d A . S . H a l l , L i n k a g e c h a r a c t e r i s t i c p o l y n o m i a l s ; d e f i n i t i o n , c o e f f i c i e n t s b y
i n s p e c t i o n , T r a n s a c t i o n o f A S M E , J o u r n a l o f M e c h a n i c a l D e s i g n , V o l . 1 0 3 , 1 9 8 1 , p p
5 7 8 - 5 8 4
[ 4 ] H . S . Y a n a n d A . S . H a l l , L i n k a g e c h a r a c t e r i s t i c p o l y n o m i a l s a s s e m b l y , T r a n s a c t i o n o f
A S M E , J o u r n a l o f M e c h a n i c a l D e s i g n , V o l . 1 0 4 , 1 9 8 2 , p p . 1 1 - 2 0
[ 5 ] H . S . Y a n a n d W . M . H w a n g , L i n k a g e P a t h C o d e , M e c h . M a c h . T h e o r y , V o l . 1 9 , 1 9 8 4 , p p
4 2 5 - 4 2 9
[ 6 ] R . K . D u b e y a n d A . C . R a o , N e w c h a r a c t e r i s t i c p o l y n o m i a l - A r e l i a b l e i n d e x t o d e t e c t
i s o m o r p h i s m b e t w e e n k i n e m a t i c c h a i n s , P r o c e e d i n g o f t h e N a t i o n a l C o n f e r e n c e o n
M a c h i n e a n d M e c h a n i s m , I I S c , B a n g l o r e , 1 9 8 5 , p p . 3 6 - 4 0
[ 7 ] T . S . M r u t h y u n j a y a a n d H . R . B a l a s u b r a m a n i a n , I n q u e s t o f a r e l i a b l e a n d e f f i c i e n t
c o m p u t a t i o n a l t e s t f o r d e t e c t i o n o f i s o m o r p h i s m i n k i n e m a t i c c h a i n s , M e c h a n i s m a n d
M a c h i n e T h e o r y , V o l . 2 2 , 1 9 8 7 , p p 1 3 1 - 1 3 9
[ 8 ] J . K . K i m a n d B . M . K w a k , A n a l g o r i t h m o f t o p o l o g i c a l o r d e r i n g f o r u n i q u e
r e p r e s e n t a t i o n o f g r a p h , T r a n s a c t i o n o f A S M E , J o u r n a l o f M e c h a n i c a l D e s i g n , V o l . 1 1 4 ,
1 9 9 2 , p p . 1 0 3 - 1 0 8
[ 9 ] J . K . S h i n a n d S . K r i s h n a m u r t h y , O n i d e n t i f i c a t i o n a n d c a n o n i c a l n u m b e r i n g o f p i n -
j o i n t e d k i n e m a t i c c h a i n s , T r a n s a c t i o n o f A S M E , J o u r n a l o f M e c h a n i c a l D e s i g n , V o l . 1 1 6 ,
1 9 9 4 , p p . 1 8 2 - 1 8 8
[ 1 0 ] E . R . T u t t l e , G e n e r a t i o n o f p l a n e r k i n e m a t i c c h a i n s , M e c h . M a c h . T h e o r y , V o l . 3 1 , 1 9 9 6 ,
p p . 7 2 9 - 7 4 8
Table 1: Structural Invariants of 9 links, 2 dof Kinematic Chains
K i n e m a t i c
C h a i n N o .
F A L V M F A L V S A L V M S A L V
1 2 1 . 5 9 8 / 2 1 . 5 9 8 / 2 1 . 5 9 3
5 / 2 1 . 5 9 3 5 / 1 3 . 0 5 8 / 1 3 .
0 5 8 / 1 2 . 5 6 6 5 / 1 2 . 5 6 6 5 /
9 . 0 5 4
2 1 . 5 9 8 5 6 . 2 5 4 / 5 6 . 2 5 4 / 5 5 . 7 5 3 5 / 5 5 . 7
5 3 5 / 3 4 . 6 5 1 5 / 3 4 . 6 5 1 5 / 3 0 . 6 5 2
/ 3 0 . 6 5 2 / 2 5 . 1 3 3
5 6 . 2 5 4
2 2 1 . 6 0 1 / 2 1 . 1 0 0 5 / 1 8 . 0 8
9 5 / 1 8 . 5 8 / 1 6 . 0 8 3 / 1 2 . 5
6 9 5 / 1 2 . 0 7 5 / 9 . 0 5 8
2 1 . 6 0 1 5 5 . 7 6 4 / 5 3 . 7 5 7 5 / 5 2 . 7 5 0 5 / 4 9 .
7 5 9 / 4 0 . 1 8 1 / 3 6 . 6 6 9 5 / 3 0 . 1 5 8 5
/ 2 7 . 1 4 7 5 / 2 4 . 6 4 4 5
5 5 . 7 6 4
3 2 1 . 1 0 6 5 / 2 1 . 1 0 6 5 / 1 8 . 5
8 6 / 1 8 . 5 8 6 / 1 6 . 0 7 9 / 1 6 .
0 7 9 / 1 2 . 5 7 1 / 1 2 . 5 7 1 / 9 .
0 5 4
2 1 . 1 0 6 5 5 3 . 2 6 4 5 / 5 3 . 2 6 4 5 / 5 2 . 2 6 3 5 / 5 2
. 2 6 3 5 / 3 7 . 1 7 2 / 3 7 . 1 7 2 / 3 0 . 1 6 0
5 / 3 0 . 1 6 0 5 / 2 5 . 1 4 2
5 3 . 2 6 4 5

5. Identification of Isomorphism amongst Kinematic Chains and…
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4 2 1 . 5 9 8 / 2 1 . 5 9 8 1 9 . 0 9 8 /
2 1 . 5 9 3 5 / 2 1 . 5 9 3 5 / 1 3 . 0
5 8 / 1 3 . 0 5 8 / 1 2 . 5 5 6 5 / 1 2
. 5 5 6 5 / 9 . 0 5 4
2 1 . 5 9 8 5 6 . 2 4 9 5 / 5 6 . 2 4 9 5 / 5 5 . 7 5 8 / 5 5 .
7 5 8 / 3 4 . 6 5 1 5 / 3 4 . 6 5 1 5 / 3 0 . 6 5 2
/ 3 0 . 6 5 2 / 2 5 . 1 3 3
5 6 . 2 4 9 5
5 2 1 . 5 9 8 / 2 1 . 5 9 8 / 1 8 . 0 8 5
5 / 1 8 . 0 8 5 5 / 1 6 . 5 7 9 5 / 1 6
. 5 7 9 5 / 1 2 . 0 7 5 / 1 2 . 0 7 5 /
9 . 0 6 3
2 1 . 5 9 8 5 6 . 2 6 3 / 5 6 . 2 6 3 / 5 0 . 2 5 2 5 / 5 0 . 2
5 2 5 / 3 9 . 6 8 3 5 / 3 9 . 6 8 3 5 / 2 7 . 1 4 8
5 / 2 7 . 1 4 8 5 / 2 4 . 1 5
5 6 . 2 6 3
6 2 4 . 4 6 9 5 / 1 9 . 0 8 2 / 1 8 . 5 9
0 5 / 1 8 . 5 9 0 5 / 1 6 . 0 7 4 5 / 1
6 . 0 7 4 5 / 1 2 . 0 7 0 5 / 1 2 . 0 7
0 5 / 9 . 0 6 3
2 4 . 4 6 9 5 5 6 . 6 1 8 5 / 5 6 . 2 6 3 / 5 2 . 6 1 4 5 / 5 2 .
6 1 4 5 / 3 7 . 6 7 2 5 / 3 7 . 6 7 2 5 / 2 7 . 6 5
3 5 / 2 7 . 6 5 3 5 / 2 4 . 1 4 1
5 6 . 6 1 8 5
7 2 1 . 1 1 1 / 1 8 . 5 9 0 5 / 1 8 . 0 9
/ 1 6 . 0 8 8 / 1 6 . 0 9 2 5 / 1 5 . 5
9 2 / 1 4 . 5 8 2 / 1 2 . 0 7 9 5 / 1 2
. 5 8
2 1 . 1 1 1 5 2 . 7 9 1 / 5 2 . 7 7 3 / 4 9 . 7 7 9 / 4 3 . 7 6
4 / 3 6 . 6 8 0 5 / 3 5 . 6 9 3 / 3 3 . 1 7 2 5 / 3
0 . 1 6 9 5 / 2 7 . 1 6 2
5 2 . 7 9 1
8 2 1 . 1 0 6 5 / 2 1 . 1 0 6 5 / 1 8 . 0
9 / 1 8 . 0 9 / 1 6 . 0 8 8 / 1 3 . 0 6
7 / 1 3 . 0 6 7 / 1 2 . 5 6 6 5 / 1 2 .
5 6 6 5
2 1 . 1 0 6 5 5 2 . 2 6 3 5 / 5 2 . 2 6 3 5 / 4 9 . 7 6 1 / 4 9 .
7 6 1 / 3 3 . 6 7 3 / 3 3 . 6 7 3 / 3 1 . 1 5 7 / 3
1 . 1 5 7 / 3 6 . 1 8
5 2 . 2 6 3 5
9 2 1 . 6 0 7 / 2 1 . 1 0 2 / 1 8 . 0 8 5
5 / 1 7 . 5 8 9 5 / 1 6 . 5 7 9 5 / 1 3
. 0 6 7 / 1 2 . 5 7 5 5 / 1 2 . 5 7 1 /
1 2 . 5 7 1
2 1 . 6 0 7 5 5 . 2 7 1 / 5 2 . 7 5 9 5 / 5 0 . 2 5 2 5 / 3 9 .
6 9 2 5 / 4 6 . 7 5 3 5 / 3 0 . 6 6 1 / 3 3 . 6 7 3
/ 3 0 . 1 6 0 5 / 3 0 . 6 6 1
5 5 . 2 7 1
1 0 2 4 . 1 2 3 / 1 8 . 5 8 6 / 1 8 . 5 8 6
/ 1 8 . 0 9 / 1 8 . 0 9 / 1 6 . 0 7 9 /
1 2 . 5 7 1 / 1 2 . 5 7 1 / 1 2 . 5 6 6
5 / 1 2 . 5 6 6 5
2 4 . 1 2 3 5 5 . 2 6 2 / 5 2 . 7 6 8 5 / 5 2 . 7 6 8 5 / 4 9 .
2 6 5 / 3 7 . 1 7 2 / 3 1 . 1 5 7 / 3 1 . 1 5 / 3 0
. 6 5 6 5 / 3 0 . 6 5 6 5
5 5 . 2 6 2
1 1 2 1 . 1 0 6 / 2 1 . 1 0 6 / 1 8 . 0 9 /
1 8 . 0 9 / 1 6 . 0 8 8 / 1 3 . 0 6 2 5
/ 1 3 . 0 6 2 5 / 1 2 . 5 7 1 / 1 2 . 5
7 1
2 1 . 1 0 6 5 2 . 2 5 9 / 5 2 . 2 5 9 / 4 9 . 7 6 5 5 / 4 9 . 7
6 5 5 / 3 6 . 1 8 / 3 4 . 1 6 9 / 3 4 . 1 6 9 / 3 0
. 6 6 1 / 3 0 . 6 6 1
5 2 . 2 5 9
1 2 2 0 . 6 0 1 5 / 1 8 . 5 9 5 / 1 8 . 5 9
5 / 1 6 . 0 7 9 / 1 6 . 0 7 9 / 1 5 . 5
9 2 / 1 5 . 5 9 8 / 1 4 . 5 8 6 5 / 1 3
. 0 7 6 / 1 2 . 0 7 5
2 0 . 6 0 1 5 5 2 . 2 7 2 5 / 5 2 . 2 7 2 5 / 5 0 . 2 6 6 / 4 3 .
2 5 9 / 3 7 . 1 9 / 3 3 . 1 8 1 5 / 3 3 . 1 8 1 5 /
3 2 . 6 7 6 5 / 2 7 . 6 6 2 5
5 2 . 2 7 2 5
1 3 2 0 . 6 0 6 / 1 8 . 5 9 9 5 / 1 8 . 0 9
9 / 1 5 . 5 8 7 5 / 1 5 . 5 8 7 5 / 1 5
. 5 9 2 / 1 5 . 0 8 2 5 / 1 3 . 0 7 1 5
/ 1 2 . 5 6 6 5
2 0 . 6 0 6 5 1 . 7 8 1 / 4 9 . 7 7 / 4 8 . 7 6 4 5 / 4 6 . 7 6
7 / 3 3 . 6 8 2 / 3 3 . 6 8 2 / 3 3 . 1 8 1 5 / 3 3
. 1 7 2 5 / 3 1 . 1 7 0 5
5 1 . 7 8 1
1 4 1 8 . 0 9 4 5 / 1 8 . 0 9 4 5 / 1 7 . 5
9 4 / 1 7 . 5 9 4 / 1 6 . 0 9 7 / 1 6 .
0 8 8 / 1 6 . 0 8 8 / 1 2 . 5 7 5 5 / 1
2 . 5 7 5 5
1 8 . 0 9 4 5 4 9 . 7 7 / 4 9 . 7 7 / 4 6 . 7 6 7 / 4 6 . 7 6 7 /
3 6 . 1 8 9 / 3 6 . 1 8 9 / 3 0 . 1 6 9 5 / 3 0 . 1
6 9 5 / 1 6 . 0 9 7
4 9 . 7 7
1 5 1 8 . 0 9 4 5 / 1 8 . 0 9 4 5 / 1 7 . 5
9 4 / 1 7 . 5 9 4 / 1 6 . 0 9 2 5 / 1 6
. 0 9 2 5 / 1 6 . 0 8 8 / 1 2 . 5 7 5 5
/ 1 2 . 5 7 5 5
1 8 . 0 9 4 5 4 9 . 7 7 4 5 / 4 9 . 7 7 4 5 / 4 6 . 7 6 2 5 / 4 6
. 7 6 2 5 / 3 6 . 1 8 9 / 3 5 . 6 8 8 5 / 3 5 . 6 8
8 5 / 3 0 . 1 6 9 5 / 3 0 . 1 6 9 5
4 9 . 7 7 4 5
1 6 4 2 . 2 2 2 / 3 6 . 1 8 9 / 3 6 . 1 8 9
/ 3 2 . 1 7 6 / 3 1 . 1 8 4 / 3 1 . 1 8
4 / 3 0 . 1 5 6 / 2 5 . 1 4 2 / 2 5 . 1
4 2
4 2 . 2 2 2 5 2 . 2 7 7 / 4 9 . 2 7 4 / 4 9 . 2 7 4 / 4 7 . 2 7
2 / 3 6 . 1 8 9 / 3 3 . 1 7 2 4 / 3 3 . 1 7 2 4 / 3
0 . 6 6 5 5
5 2 . 2 7 7
1 7 2 1 . 1 0 6 5 / 2 1 . 1 0 6 5 / 1 8 . 0
9 / 1 8 . 0 9 / 1 6 . 0 8 8 / 1 3 . 0 6
7 / 1 3 . 0 6 7 / 1 2 . 5 6 6 5 / 1 2 .
5 6 6 5
2 1 . 1 1 1 5 2 . 2 6 3 5 / 5 2 . 2 6 3 5 / 4 9 . 7 6 1 / 4 9 .
7 6 1 / 3 6 . 1 8 / 3 3 . 6 7 3 / 3 3 . 6 7 3 / 3 1
. 1 5 7 / 3 1 . 1 5 7
5 2 . 2 6 3 5
1 8 1 8 . 0 9 9 / 1 8 . 0 9 9 / 1 5 . 5 9 2
/ 1 5 . 5 9 2 / 1 5 . 5 9 2 / 1 5 . 5 9
2 / 1 5 . 0 9 6 / 1 5 . 0 8 7 / 1 5 . 0
8 7
1 8 . 0 9 9 4 9 . 2 8 3 / 4 9 . 2 8 3 / 4 6 . 2 8 / 4 6 . 2 8 /
3 0 . 1 7 8 5 / 3 0 . 6 7 9 / 3 0 . 1 7 4 / 3 0 . 6
7 9 / 3 0 . 6 7 9
4 9 . 2 8 3
1 9 1 8 . 0 9 9 / 1 8 . 0 9 9 / 1 5 . 5 9 2
/ 1 5 . 5 9 2 / 1 5 . 5 9 2 / 1 5 . 5 9
2 / 1 5 . 0 9 6 / 1 5 . 0 8 7 / 1 5 . 0
8 7
1 8 . 0 9 9 4 8 . 2 5 6 / 4 8 . 2 5 6 / 4 6 . 2 8 / 4 6 . 2 8 /
3 3 . 1 8 6 / 3 3 . 1 8 6 / 3 3 . 1 8 6 / 3 3 . 1 8
6 / 3 0 . 1 7 4
4 8 . 2 5 6
2 0 2 5 . 1 4 7 8 5 / 2 4 . 6 4 1 9 5 / 2
0 . 6 2 1 3 / / 1 9 . 1 0 1 8 / 1 7 . 1
0 4 7 5 / 1 6 . 6 0 8 7 5 / 1 3 . 0 7
8 7 / 1 3 . 0 7 4 2 / 9 . 5 6 2 1 5
2 5 . 1 4 7 8 5 4 6 . 7 5 8 / 4 6 . 2 5 8 / 3 3 . 6 8 / 3 2 . 1 8 8
/ 3 0 . 1 6 3 / 2 9 . 6 8 1 / 2 7 . 1 4 8 / 2 4 . 6
4 6 / 2 3 . 6 4 9
4 6 . 7 5 8

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2 1 2 4 . 6 5 0 9 5 / 2 2 . 1 3 6 3 / 2 0 .
1 2 5 3 / 2 0 . 1 2 5 3 / 1 6 . 6 0 8
7 5 / 1 6 . 0 9 9 2 5 / 1 6 . 0 9 0 2
5 / 1 2 . 5 8 2 7 / 9 . 5 6 6 6 5
2 4 . 6 5 0 9 5 7 8 . 9 9 5 6 5 / 5 6 . 3 4 9 8 5 / 5 3 . 3 3 2 9 /
4 0 . 7 4 1 2 / 4 0 . 7 4 1 2 / 3 8 . 2 2 6 5 5 / 3
4 . 2 1 7 6 / 3 1 . 7 0 2 9 5 / 2 9 . 1 9 1 4 5
7 8 . 9 9 5 6 5
2 2 2 1 . 6 6 9 4 / 2 0 . 1 3 4 7 5 / 2 0 .
1 2 9 8 / 2 0 . 1 2 9 8 / 1 9 . 1 1 2
5 / 1 8 . 1 0 6 2 / 1 6 . 1 1 7 2 5 / 1
2 . 0 7 8 6 / 9 . 5 7 1 6
2 1 . 6 6 9 4 7 6 . 5 0 6 6 5 / 5 8 . 3 6 5 8 / 5 1 . 3 2 4 6 / 4
0 . 7 8 0 6 5 / 4 0 . 7 8 0 6 5 / 3 9 . 7 7 5 6 / 3
1 . 2 4 1 / 2 8 . 1 9 5 8 5 / 1 7 . 2 8 5
7 6 . 5 0 6 6 5
2 3 2 7 . 1 8 7 5 / 2 2 . 6 2 2 8 5 / 2 2 .
6 2 2 8 5 / 1 7 . 1 0 4 3 / 1 7 . 1 0
4 3 / 1 7 . 0 9 5 3 / 1 3 . 0 7 4 2 / 1
3 . 0 7 4 2 / 1 0 . 0 6 2 2
2 7 . 1 8 7 5 7 9 . 4 5 4 3 / 5 7 . 3 2 7 / 5 7 . 3 2 7 / 4 5 . 2
4 5 7 / 3 9 . 7 2 7 1 5 / 3 7 . 2 1 9 7 / 3 7 . 2 1
9 7 / 3 5 . 6 9 7 0 5 / 3 5 . 6 9 7 0 5
7 9 . 4 5 4 3
2 4 1 6 . 0 9 6 1 / 1 6 . 0 9 6 1 / 1 6 . 6
1 1 4 5 / 1 6 . 6 1 1 4 5 / 2 2 . 6 3
6 3 5 /
2 2 . 6 3 6 3 5 3 1 . 6 8 2 2 5 / 3 1 . 6 8 2 2 5 / 3 4 . 2 2 5 3 /
3 4 . 2 2 5 3 / 5 6 . 2 4 2 2 / 7 5 . 9 8 8 1 5 / 3
9 . 7 4 1 1 5 / 4 2 . 3 2 1 1 / 3 3 . 2 2 2 9
7 5 . 9 8 8 1 5
2 5 2 5 . 1 4 3 3 5 / 2 0 . 6 2 1 3 / 1 9 .
1 0 2 2 5 / 1 6 . 6 0 8 3 / 1 6 . 6 0
8 3 / 1 4 . 0 8 3 7 5 / 1 3 . 5 7 5 1
5 / 1 2 . 5 6 5 1 5 / 1 0 . 0 7 1 2
2 5 . 1 4 3 3 5 8 0 . 9 9 2 2 5 / 5 8 . 3 2 9 8 / 5 7 . 3 2 3 8 5 /
4 3 . 7 4 8 7 / 3 7 . 7 0 8 6 5 / 3 5 . 3 1 7 6 / 3
5 . 3 1 7 6 / 3 5 . 2 2 7 6 / 3 2 . 6 7 7 4
8 0 . 9 9 2 2 5
2 6 2 7 . 1 5 3 / 2 2 . 6 2 7 3 5 / 2 2 . 6
2 2 4 / 1 7 . 6 0 0 3 / 1 7 . 1 0 4 7
5 / 1 7 . 0 9 5 3 / 1 3 . 5 7 4 2 5 / 1
2 . 5 7 8 2 / 9 . 5 6 2 1 5
2 7 . 1 5 3 7 9 . 9 5 4 8 / 5 7 . 8 2 2 5 5 / 5 6 . 8 2 6 5 / 4
5 . 2 4 9 7 5 / 4 0 . 7 2 7 2 5 / 4 0 . 2 2 2 7 / 3
6 . 7 1 5 1 5 / 3 2 . 1 8 9 5 / 2 9 . 6 8 2 9 5
7 9 . 9 5 4 8
2 7 2 5 . 1 4 2 9 / 2 4 . 6 4 1 9 5 / 2 0 .
6 2 1 3 / 1 9 . 1 0 6 7 5 / 1 7 . 0 9
9 8 / 1 6 . 6 0 8 7 5 / 1 4 . 0 7 9 2
5 / 1 2 . 0 7 3 6 5 / 9 . 5 6 7 1
2 5 . 1 4 2 9 7 9 . 4 6 3 8 / 7 7 . 8 1 8 9 5 / 5 7 . 8 2 8 9 / 4
3 . 7 3 9 7 / 4 2 . 2 4 2 7 3 8 . 7 1 6 7 / 3 4 . 1
9 5 5 5 / 2 8 . 6 6 4 5 / 2 8 . 6 6 0 3 5
7 9 . 4 6 3 8
2 8 2 2 . 1 4 4 4 / 2 0 . 1 3 3 8 / 2 0 . 1
3 3 8 / 2 0 . 1 3 3 8 / 2 0 . 1 3 3 8 /
1 5 . 6 0 3 7 / 1 5 . 6 0 3 7 8 / 1 2 .
0 8 3 1 / 1 2 . 0 8 3 1
2 2 . 1 4 4 4 8 0 . 5 3 5 2 / 5 2 . 3 5 0 7 / 5 2 . 3 5 0 7 / 3 7
. 7 4 8 1 / 3 7 . 7 4 8 1 / 3 7 . 7 4 8 1 / 3 7 . 7
4 8 1 / / 2 7 . 6 8 6 8 / 2 7 . 6 8 6 8
8 0 . 5 3 5 2
2 9 2 4 . 1 4 9 5 / 2 2 . 1 3 5 8 5 / 2 0 .
1 1 9 2 5 / 1 7 . 1 0 8 8 / 1 7 . 0 9
8 2 5 / 1 6 . 1 0 3 7 5 / 1 5 . 0 7 4
2 / 1 3 . 5 8 3 2 5 / 1 2 . 5 6 7 6 5
2 4 . 1 4 9 5 7 6 . 4 9 1 8 / 5 3 . 8 3 2 5 / 4 8 . 8 2 0 3 / 3 9
. 2 4 4 6 5 / 3 9 . 2 3 4 3 / 3 7 . 7 2 8 7 5 / 3 7
. 2 2 4 6 5 / 3 6 . 7 2 7 7 5 / 3 2 . 2 0 1 6 5
7 6 . 4 9 1 8
3 0 2 7 . 1 5 7 5 / 2 2 . 1 2 6 8 5 / 2 2 .
1 2 6 8 5 / 1 7 . 6 0 0 3 / 1 7 . 6 0
0 3 / 1 3 . 0 7 8 7 / 1 3 . 0 7 8 7 / 1
3 . 5 7 8 7 5 / 1 3 . 5 7 8 7 5
2 7 . 1 5 7 5 7 9 . 4 5 4 3 / 5 3 . 8 1 4 9 5 / 5 3 . 8 1 4 9 5 /
4 0 . 7 3 6 2 5 / 4 0 . 7 3 6 2 5 / 3 9 . 7 2 7 1
5 / 3 9 . 7 2 7 1 5 / / 3 5 . 2 0 5 5 5 / 3 5 . 2 0
5 5 5
7 9 . 4 5 4 3
3 1 2 5 . 1 4 3 4 5 / 2 4 . 1 3 2 5 5 / 1
8 . 0 9 6 8 / 1 7 . 1 0 9 2 5 / 1 7 . 1
0 9 2 5 / 1 7 . 0 9 5 4 / / 1 4 . 0 7 0
3 5 / 1 3 . 0 7 9 1 5 / 1 3 . 0 7 9 1
5
2 5 . 1 4 3 4 5 7 5 . 4 5 7 3 5 / 5 6 . 2 9 9 7 / 5 1 . 3 0 1 7 5 /
4 2 . 2 3 8 8 5 / 3 8 . 2 0 2 9 / 3 7 . 2 1 1 7 / 3
7 . 2 1 1 7 / 3 5 . 2 0 6 0 5 / 3 5 . 2 0 6 0 5
7 5 . 4 5 7 3 5
3 2 2 1 . 6 4 8 4 / 1 9 . 6 4 2 8 / 1 9 . 6
4 2 8 / 1 9 . 6 3 8 7 5 / 1 6 . 6 1 7
7 5 / 1 6 . 1 0 8 7 / 1 5 . 1 0 3 6 5 /
1 5 . 0 9 9 1 5 / 1 2 . 5 8 3 1 5
2 1 . 6 4 8 4 7 5 . 4 6 2 1 / 5 4 . 3 4 4 7 5 / 4 7 . 2 9 6 0 5 /
3 7 . 7 2 7 1 / 3 7 . 7 2 7 1 / 3 6 . 7 2 7 0 5 / 3
4 . 2 0 6 5 / 3 1 . 6 9 6 4 / 3 1 . 1 9 7 3 5
7 5 . 4 6 2 1
3 3 2 4 . 6 5 0 8 5 / 2 0 . 1 2 9 8 / 2 0 .
1 2 9 8 / 1 7 . 1 0 4 3 / 1 5 . 5 9 4
6 / 1 4 . 5 8 4 2 / 1 3 . 5 8 7 7 5 / 1
3 . 0 8 8 1 5 / 1 2 . 0 7 8 1 5
2 4 . 6 5 0 8 5 7 1 . 9 4 8 1 / 5 2 . 3 3 7 7 5 / 4 6 . 8 0 5 7 / 4
0 . 2 4 5 4 5 / 4 0 . 2 4 5 4 5 / 3 8 . 2 3 8 6 / 3
1 . 6 8 8 5 / 2 8 . 6 8 2 7 5 / 2 6 . 6 6 2 3 5
7 1 . 9 4 8 1
3 4 2 1 . 1 3 4 4 / 2 0 . 1 3 4 3 / 2 0 . 1
3 4 3 / 1 9 . 1 1 6 2 / 1 6 . 6 1 7 7
5 / 1 6 . 6 1 7 7 5 / 1 5 . 5 9 6 2 / 1
3 . 0 8 8 6 / 1 3 . 0 8 8 6
2 1 . 1 3 4 4 6 7 . 5 0 4 1 / 5 2 . 8 6 4 8 / 3 9 . 7 9 3 4 / 3 6
. 7 5 0 6 / 3 6 . 7 5 0 6 / 3 1 . 7 2 3 / 3 1 . 7 2
3 / 3 0 . 7 1 3 9 5 / 3 0 . 7 1 3 9 5
6 7 . 5 0 4 1
3 5 2 1 . 1 3 4 4 / 2 0 . 1 3 9 2 5 / 2 0 .
1 3 9 2 5 / 1 8 . 1 0 6 2 / 1 8 . 1 0
6 2 / 1 6 . 6 1 7 7 5 / 1 6 . 6 1 7 7
5 / 1 3 . 0 8 3 6 5 / 1 3 . 0 8 3 6 5
2 1 . 1 3 4 4 7 3 . 5 1 4 / 5 1 . 3 2 8 9 5 / 5 1 . 3 2 8 9 5 / 3
9 . 2 4 0 6 / 3 9 . 2 4 0 6 / 3 4 . 7 2 3 9 5 / 3 4
. 7 2 3 9 5 / 3 4 . 2 1 8 0 5 / 3 4 . 2 1 8 0 5
7 3 . 5 1 4
3 6 2 4 . 1 8 0 6 / 2 4 . 1 8 0 6 / 2 2 . 1
5 5 6 5 / 2 2 . 1 5 5 6 5 / 1 7 . 1 2
5 4 5 / 1 7 . 1 2 5 4 5 / 1 6 . 1 2 0
4 / 1 6 . 1 2 0 4 / 1 0 . 0 8 0 2
2 4 . 1 8 0 6 8 1 . 6 0 7 0 5 / 8 1 . 6 0 7 0 5 / 4 4 . 3 1 1 3 /
4 4 . 3 1 1 3 / 3 9 . 2 8 1 1 / 3 9 . 2 8 1 1 / 3 2
. 2 3 5 8 5 / 3 2 . 2 3 5 8 5 / 3 2 . 2 4 0 8
8 1 . 6 0 7 0 5

7. Identification of Isomorphism amongst Kinematic Chains and…
www.theijes.com The IJES Page 32
3 7 2 4 . 1 9 0 5 / 1 7 . 1 3 0 4 / 1 7 . 1
3 0 4 / 1 7 . 1 3 0 4 / 1 7 . 1 3 0 4 /
1 7 . 1 3 0 4 / 1 7 . 1 3 0 4 / 2 1 . 1
4 5 6 5 / 2 1 . 1 4 5 6 5
2 4 . 1 9 0 5 7 5 . 5 8 1 7 / 7 5 . 5 8 1 7 / 4 2 . 2 9 1 3 / 3 8
. 2 7 6 0 5 / 3 8 . 2 7 6 0 5 / 3 8 . 2 7 6 0 5 / 3
8 . 2 7 6 0 5 / 3 8 . 2 7 6 0 5 / 3 8 . 2 7 6 0 5
7 5 . 5 8 1 7
3 8 2 8 . 1 8 5 9 5 / 2 8 . 1 8 5 9 2 / 1
8 . 1 2 0 6 / 1 8 . 1 2 0 6 / 1 8 . 1 2
0 6 / 1 8 . 1 2 0 6 / 1 7 . 1 2 0 5 / 1
7 . 1 2 0 5 / 1 0 . 0 7 0 3
2 8 . 1 8 5 9 5 8 1 . 5 4 7 6 5 / 8 1 . 5 4 7 6 5 / 4 6 . 3 0 6 5
5 / 4 6 . 3 0 6 5 5 / 4 6 . 3 0 6 5 5 / 4 6 . 3 0 6
5 5 / 3 8 . 2 5 6 2 5 / 3 8 . 2 5 6 2 5 / 3 4 . 2 4
1
8 1 . 5 4 7 6 5
3 9 2 8 . 7 1 2 5 5 / 2 5 . 1 9 2 5 3 5 /
2 5 . 1 9 2 5 3 5 / 2 1 . 6 7 1 0 3 5
/ 2 1 . 6 7 1 0 3 5 / 2 2 . 1 7 2 0 3 /
1 6 . 6 1 3 0 2 5 / 1 3 . 0 9 2 0 2 /
1 1 . 0 9 0 0 2
2 8 . 7 1 2 5 5 1 1 5 . 8 9 9 1 7 / 6 3 . 4 7 7 0 9 / 4 3 . 3 4 2
0 7 / 4 3 . 3 2 5 5 7 5 / 4 3 . 3 2 5 5 7 5 / 4 1 .
8 0 4 5 7 / 3 9 . 8 0 2 5 7 / 3 9 . 8 0 2 5 7 / 3
8 . 7 8 5 0 5 5
1 1 5 . 8 9 9 1 7
4 0 3 1 . 1 1 4 5 5 5 / 2 7 . 1 8 4 5 4 5
/ 2 2 . 6 5 8 5 3 5 / 2 2 . 6 5 8 5 3
5 / 2 2 . 1 6 2 5 3 5 / 2 2 . 1 6 2 5
3 5 / 1 4 . 5 9 7 5 2 5 / 1 4 . 5 9 7
5 2 5 / 1 0 . 9 8 1 0 2
3 1 . 1 1 4 5 5 5 1 1 6 . 8 2 6 6 8 5 / 6 0 . 3 0 9 6 0 5 / 4 4 . 3
2 5 0 7 / 4 9 . 8 4 3 0 8 / 4 9 . 8 4 3 0 8 / 4 5 .
7 1 2 0 8 / 4 5 . 7 1 2 0 8 / 4 2 . 0 9 5 5 7 5 /
4 2 . 0 9 5 5 7 5
1 1 6 . 8 2 6 6 8
5
Table 2: Inversions of Kinematic Chains Shown in Figures
Ki n e m a t i c
Ch a i n
No .
No . o f In v e r s i o n s Ki n e m a t i c Ch a i n
No .
No . o f In v e r s i o n s
01 05 21 08
02 09 22 08
03 05 23 06
04 05 24 07
05 05 25 08
06 06 26 09
07 09 27 09
08 05 28 04
09 08 29 09
10 06 30 05
11 05 31 07
12 07 32 08
13 08 33 08
14 05 34 06
15 05 35 05
16 06 36 05
17 05 37 03
18 04 38 04
19 04 39 07
20 09 40 06
FIGURES OF 9 LINKS 2 DOF KINEMATIC CHAINS

8. Identification of Isomorphism amongst Kinematic Chains and…
www.theijes.com The IJES Page 33

9. Identification of Isomorphism amongst Kinematic Chains and…
www.theijes.com The IJES Page 34