National Conference on Recent Trends in Engineering & Technology
Computerize Technique to Find the Best
Inversion of Given Kinematic Chain in Simple
Planar Mechanism
Yagnesh Gambhava#1, Dr.Anurag Verma*2, Alpesh daamor#3
#
BVM PG student Mechanical Department, Gujarat Technological University
V.V.Nagar, Gujarat, India
1
[email protected]
3
[email protected]
*
Principal, GCET
V.V.Nagar, Gujarat, India
2
[email protected]
Abstract
The scrupulous knowledge of the inversion for a
given kinematic chain is essential for the designer in
mechanical engineering and persistent effort has made to
know about best possible structure from inversion. The
existing research literature shows that methods have been
detected to find out the inversion of a given kinematic chain
in simple planar mechanism. The present paper shows the
new methodology which can be easily computerized, less time
consuming among many other related techniques. Here this
work shows the proposed method implemented on 8 links one
degree of freedom.
Keywords Degree of freedom, Inversion, Kinematic Chain,
I INTRODUCTION
A machine part which moves relative to some other part is
known as kinematic link or element. when two links or
elements of a machine are in contact with each other then
they are known as kinematic pair. As we know that, when
one of the links of a kinematic chain is fixed it is known as
mechanism. So we can obtain as many mechanisms as the
number of links in a kinematic chain by fixing it in turn
different links in a kinematic chain. This method of
obtaining different mechanisms by fixing different links in
a kinematic chain is known as inversion of a mechanism.
The relative motion between the various links is not
changed in any manner through the process of inversion,
but their absolute motions may be changed drastically.
The various methods to detect inversions one the simple
methods derive by A.C.Rao and D.Varada Raju[5] The
connectivity matrix of the various links, a matrix of zeros
13-14 May 2011
and ones, is first formed and the Hamming number matrix
is computed. Apart from such methods here a new link
Hamming string-(which is defined as the string obtained by
concatenating the link Hamming number and the frequency
of individual Hamming numbers in that row--is then
formed) is developed to know best possible arrangement.
The chain Hamming string, defined as the string obtained
by the concatenation of the chain Hamming number and
the link Hamming strings in descending order is formed.
Also, the link Hamming string of every link together with
those of its neighbors’ is an excellent test for among the
inversions of a given chain. These twin claims have been
verified on a computer for all four bar, six-bar and ten-bar
chains with one degree of freedom as well as ten-bar chains
with three-degrees of freedom. It is felt that the greatest
advantage of this method is that the chain Hamming string
reveals at a glance, without much additional computation,
how many inversions are possible out of a given chain.
Similarly, as per hamming number also detected the
isomorphism of given kinematic chain in simple planar
mechanism is possible with technique.
[9] The work presents a new method for the identification
of isomorphism and inversions or the distinct mechanisms
(DM) from a given kinematic chain (KC). The method was
based on weighted physical connectivity matrix (WPCM)
of the given KC. The two structural invariants, namely,
sum of absolute characteristic polynomial coefficients
[WPCMPS] and maximum absolute value of characteristic
polynomial coefficient [WPCMPmax] have been derived
and used as the identification numbers of the kinematic
chains.
B.V.M. Engineering College, V.V.Nagar,Gujarat,India
National Conference on Recent Trends in Engineering & Technology
[7]Synthesis of planar kinematic chains is in general
tedious and time consuming. This paper reports a very
simple and direct method for the generation of n-link fdegree-of-freedom (D.O.F.) chains from (n - 2)-link and fD.O.F chains. Hamming number technique reported earlier
[Raju, Mechanism and Machine Theory 26, 55-75 (4)] for
testing isomorphism among kinematic chains is extended to
reveal identity and symmetry among the links and joints of
a chain. This in turn enables generation of distinct n-link
chains directly from each distinct (n - 2)-link without
having to test for degeneration and isomorphism. The
formulae suggested will give the number of such chains.
Chains with (n - l)-links and (f+ l)-D.O.F, are obtained as a
by-product. The distinct n-link, f-D.O.F, chains obtained
from all the basic chains need, however, to be tested for
isomorphism among themselves.
II.
METHODOLOGY
0
1
0
0
0
0
0
1
1
0
1
0
0
0
0
0
0
1
0
1
0
0
0
1
0
0
1
0
1
0
1
0
0
0
0
1
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
1
0
1
0
1
1
0
1
0
0
0
1
0
Step 2:- To generating hamming number matrix Part
[5] The method is divided into two parts. The First part
shows the formation of Hamming matrix and second part
is the formation of the Hamming string and Hamming
value which can be further utilized to predict the
inversions. The 8 link one degree of freedom simple
jointed planner kinematic chain having 16 numbers of
possible configuration but this paper shows only one
configuration to present methodology.
Method part 1
Step:-1:- To generating connectivity matrix
0
4
1
5
4
4
3
4
0
5
3
4
4
4
1
5
0
6
3
5
3
5
3
6
0
5
1
6
4
4
3
5
0
4
1
4
4
5
1
4
0
5
3
5
2
6
1
5
0
5
1
6
2
5
3
6
26
26
28
28
26
26
28
5
1
6
2
5
3
6
0
28
216
Part 2:-To Generating hamming string and Hamming
Value
Figure 1 8 link one degree of freedom
The connectivity of Figure is given below
1-2, 8
2-1, 3
3-2, 4, 8
4-3, 5, 7
5-4, 6
6-5, 7
7-4, 6, 8
8-1, 3, 7
13-14 May 2011
For the generation of hamming string one link has to be
fixed , so , here the explanation for the first row is, link is
total of 26 comprising of zero 6, two 5s, three 4s, one
3s,zero 2s, one 1 and one 0.Likewise, the link is total of 28
containing two 6, one 5s, one 4s, two 3s, one 2s, one 1 and
one 0 Whenever one another links are going to be fixed All
other rows have been deduced from the Hamming matrix in
a similar manner.
Hamming string for this 8-bar link fixing a link 1 in given
above fig.1 is below
216, 28, 002201111, 28, 002201111, 28, 002201111, 28,
002201111
26, 000231011, 26, 000231011, 26, 000231011, 26,
000231011
B.V.M. Engineering College, V.V.Nagar,Gujarat,India
National Conference on Recent Trends in Engineering & Technology
Similarly, by fixing remaining seven link the following
string can be developed for a given structure.
Sr.No
Link
Hamming String
1
link 1
28 002201111
III.
RESULTS
As we know that 8 link one degree of freedom having 16
different possible structures. To predict best possible
inversion out of same must be based on the hamming
values which are given below.
26 000231011
2
link 2
TABLE:I
26 000231011
STRUCTURE
1
2
3
4
5
6
7
8
9
10
11
28 002201111
12
26 000231011
13
28 002201111
14
15
16
28 002201111
26 000231011
3
link 3
28 002201111
28 002201111
26 000231011
4
link 4
28 002201111
28 002201111
5
link 5
6
link 6
26 000231011
7
link 7
HAMMING VALUE(FIX LINKS)
180(1-5-6-7), 160(2-3-4-8)
180(1-3-4-6-7), 164(5), 160(2-8)
180(1-4-5-6), 160(2-3-7-8)
180(1-5), 176(3-4),164(6-7), 160(2-8)
180(1-5), 176(6-7), 164(2-4), 160(3-8)
180(1-4), 176(5-6), 164(3-7), 160(2-8)
180(4-7), 176(1-2), 168(5),160(3-6-8)
176(2-3-6-7), 164(1-4-5-8)
176(1-2-5-6), 164(3-4-7-8)
180(1-4-5-7), 176(1), 152(2-6), 140(8)
180(1-4-5), 176(7), 172(6), 156(2-3),140(8)
180(5-7), 176(1-4), 172(2), 160(3), 152(6),
144(8)
180(4), 176(1-7), 172(2-6), 164(3),
156(5),144(8)
176(1-4-7), 172(2-6), 160(3-5), 148(8)
180(4-5), 172(1-2-6-7), 136(3-8)
164(1-2-4-5-6-7), 144(3-8)
28 002201111
28 002201111
IV.
FLOW CHART
The flow chart is the base on which the coding can be
developed. The flow chart based on the proposed method is
given below.
26 000231011
8
link 8
28 002201111
28 002201111
26 000231011
The hamming valve by fixing different link is given as
Link No
Hamming Value
1
176
2
176
3
164
4
164
5
176
6
176
7
164
8
164
The hamming valve by fixing one of the link 1-2-5-6 is
176 the connectivity matrix and hamming matrix will
change but hamming value remain same .the hamming
value by fixing one of the link 3-4-7-8 is 164 the
connectivity matrix and hamming matrix will be change
but the hamming values remain the same .the best inversion
obtained by fixing one of the link 1-2-5-6 because
hamming value is maximum.
13-14 May 2011
B.V.M. Engineering College, V.V.Nagar,Gujarat,India
National Conference on Recent Trends in Engineering & Technology
V.
APPLICATION
it has been reported that different configuration of simple
jointed kinematic chain use in robot hand application and
different inversion can be find out for particular objectives.
VI.
CONCLUSION
This paper shows a universal methodology to find out best
possible inversion of a structure of simple jointed planar
kinematic chain. This method can be easily implemented
whenever the number of links and degree of freedom
increases, as well as with some modification it can be
suitable for n-number of link and f- number of degree of
freedom. Such method is also computerized and coding can
be developed to reduce time and acquire results faster.
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13-14 May 2011
B.V.M. Engineering College, V.V.Nagar,Gujarat,India