86. IJMPERDAPR 201986

International Journal of Mechanical and Production
Engineering Research and Development (UMPERD)

ISSN (P): 2249-6890; ISSN (E): 2249-8001
Vol. 9, Issue 2, Apr 2019, 879-894
© TJPRC Pvt. Ltd.

TOPOLOGICAL SYNTHESIS AND STRUCTURAL ANALYSIS
OF PLANAR PARALLEL MECHANISMS
P. VIJAY, A. SRINATH & PARVATINI SRI NAGA VENKAT

Department of Mechanical Engineering , Koneru Lakshmaiah Education Foundation,

Vaddeswaram, Andhra Pradesh, India

ABSTRACT

This paper presents the identijication of the best robot hand for the application of given task at the conceptual stage of
design, based on the characteristics of kinematic chains like stiffness and compactness of the structure of the
mechanism, The stiffness of the chain mainly depends upon the elasticity, supports and the dimension of the links of tlie
mechanism. The chain with stiffer links will have the greater stiffness and lighter in weight, which leads the designer to
think on load bearing capabilities of the chain. Compactness is the structural aspect of the chain, which tells about how
closely the links of the chain arranged. More closeness leads to more compactness of the chain structurally, and more
compactness leads to difficult in forward kinematics. The methodology adopted already by the Ashok dagar is been used
in the present work to identify the best among the nine robot hands (ten bar single degree of freedom) based on the
stiffness and compactness. Same characteristics are compared individually, and identified the best and high rated robot
hands.

KEYWORDS: Stiffness, Compactness, Rigidity, Mechanism, Compare & Characteristics

TRAN5

STELLAR

•Journal Publlcations • Research Consultancy

Received: Jan 31, 2019; Accepted: Feb 21, 2019; Published: Apr 05, 2019; Paper Id.: IJMPERDAPR201986

1. INTRODUCTION

The classification of robots based on the level of sophistication, whether low or medium and/or high, we
have grippers and end-effectors. The gripper consists of mechanism, which can be controlled with servo motors or
controlling methods. The gripper is same as that of human hand, which also consists of finger tips for grasping and
gripping of the objects. This is how grippers play a vital role in all aspects of robotic applications. Grippers find
their application where, hazardous work environments like handling radioactive materials, welding with high
temperature zones, special applications etc. The usage of the grippers may differ from geometry to geometry of the
objects. As per the requirement of the objects, the design of robot hand gripper may vary. The main limitations of
the robot hand grippers which we need to focus are difficulty in manipulation and stability of different irregular
objects. In the history, many of the robot hands are there with multi number degree of freedom and controlling of
this very much complex.

2. LITERATURE SURVEY

Based on the topology the kinematic chains and their characteristics can be read by the designer at the
conceptual design stage, and can able to select the best mechanism to do the further work as per the application [1].
Ashok dargar modelled the kinematic chains as springs which are connected in series and obtained the
characteristics as compactness and stiffness [2]. The performance of the kinematic chains was evaluated based on

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Original Article

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P. Vijay, A. Srinath & Parvatini Sri Naga Yenkat

the proposed concept of correlation by A. srinath and Rao[3]. A. C. Rao presented the different kinematic inversions using
fuzzy logics [4]. A. C. Rao presented the work of selecting the best kinematic chain at the design stage by the proposed
methodology of comparative study of chains and mechanisms using kinematic characteristics like stiffness and
compactness of the mechanism, and has given logical aspects of weakness and strength of the chain/mechanism
structurally to obtain the mechanical advantages of the chains/mechanisms. As a designer, one must know the ability of
static behaviour of the chain/mechanism in transferring the force or torque. A chain, in which the links are connected close
to each other resemble the compactness of the chain or mechanism. In the graph theory, the distance between the two links
in the mechanism is equal to the least number of joints that separate them. This is how the compactness of the each
mechanismis calculated. Now, the stiffness is the other characteristic considered and calculated. Stiffness depends on links
stiffness and elasticity of the links[5]. A. C. Rao proposed another methodology for topological characteristics based on
genetic algorithm [6]. Hong-Sen Yan and Chin-HsingKuo have focussed their attention in representing the kinematic
characteristics of the mechanisms and their analysis of variable kinematic joints. The proposed work results the logical
foundation of structural analysis of mechanisms with topological characteristics.[7] P. Vijay applied proposed
methodology and rated the best mechanism at the design stage itself for nine ten bar mechanisms of single degree of
freedom based on the joint matrix and chain value matrix like characteristics [8]. Shinji Nishiwaki proposed methodology
based on topology of optimal structure for homogenation and flexibility applied to mechanisms[9]. QiongJin and Ting-Li
Yang proposed a methodology for topology synthesis of parallel manipulators based actuation, built in it. The matrices of
output and input characteristics result in formulating the formula for mobility equation and output character equation.

3. OBJECTIVE OF THE WORK

The main objective of this work is to model the nine single degree of freedom, ten bar mechanism with the springs
as the link forming closed kinematic chain, for which the topological characteristics such as stiffness and compactness for
each mechanism are calculated by the application of proposed methodology by Ashok dargar [1] and are compared, and to
rare the best mechanism to be suited for optimally converting the given input to the required output. The stiffness and
compactness are formulated in terms of stiffness and distances matrices. The links of the mechanism are stiffer, then such
mechanism is stiffer leads to less weight and more elastic and more flexible. The compactness resembling asmore closely,
the links are connected then, the mechanisms is more compact and occupies less space. Based on these, the mechanisms
are compared and are rated as more stiffer and high compactness.

Figure 1: Robot Hand (a)

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4 . STRUCTURAL SYNTHESIS AND ANALYSIS

The chain is modelled like a system of springs connected in series; the stiiTness of the chain can be calculated as
the summation of the joint values j v [1]

Summation of degree of links connected

The joint values jv =-

Number of links connected at that joint

The stiffness of the chain can be calculated as,

1/k = l/kl+l/k2+.1/kn

For robot hand Figure 1 (a)

1/k = l/kl+l/k2+.1/klO

= ‘/ 4 + 1 /4+1 /4+1 /4+1 /4.5+1/5+1 /2.5+1 /2+1 /4+1 /4

= 2.82

The distance between two links is nothing but the least number of joints that separate them and the distance
between two joints is least number of links that separate them. [1]

Two distant matrices are calculated as D1 - link distance matrix and D2 -joint distance matrix and the link
distance value and joint distance vale are nothing but sum of all elements of D1 and D2 values respectively. The D1 values
are taken from the previous work of the author [7] for all the nine robot hands. Now the D2 for robot hand l(a) can be
computed as follows:

0 1 2 3 4 5 3 2 1

1 0 1 2 3 4 4 3 2

2 1 0 1 2 3 5 4 3

3 2 1 0 1 2 6 5 4

D2= 4 3 2 1 0 1 7 6 5

543210876
345678012
2 3 4 5 6 7 1 0 1

1 2 3 4 5 6 2 1 0

For robot hand (a),

Joint distance value, J. D. V = sum of all values of D2 matrix

=240

Link distance value, L. D. V = 329[7]

Compactness C, = J. D. V+L. D. V

=569

Similarly, the stiffness and compactness can be computed for other robot hands as follows:

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Figure 2: Robot Hand (b)

The chain is modelled like a system of springs connected in series; the stiffness of the chain can be calculated as
the summation of the joint values j v [ij

Summation of degree of links connected

The joint values j v =-

Number of links connected at that joint

The stiffness of the chain can be calculated as,

1/k = l/kl+l/k2+.1/kn

For robot hand Figure 1 (b)

1/k = l/kl+l/k2+.1/klO

= 14+1/4+1/4+1/4.5+1/5+1/4.5+1/2+1/2+1/4+1/4

= 2.89

The distance between two links is nothing but the least number of joints that separate them, and the distance
between two joints is least number of links that separate them. [1]

Two distant matrices are calculated as D1 - link distance matrix and D2 -joint distance matrix and the link
distance value and joint distance vale are nothing but sum of all elements of D1 and D2 values, respectively [10-13]. The
D1 values are taken from the previous work of the author [7] for all the nine robot hands. Now, the D2 for robot hand (b)
can be computed as follows:

0 1 2 3 4 5 3 2 1

1 0 1 2 3 4 4 3 2

2 1 0 1 2 3 5 4 3

3 2 1 0 1 2 6 5 4

D2= 4 3 2 1 0 1 7 6 5

543210876
345678082
2 3 4 5 6 7 1 0 1

1 2 3 4 5 6 2 1 0

For robot hand (b),

Joint distance value, J. D. V = sum of all values of D2 matrix

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883

=249

Link distance value, L. D. V= 332[7]
Compactness C, = J. D. V+L. D. V

=581

The chain is modelled like a system of springs connected in series; the stiffness of the chain can be calculated as
the summation of the joint values j v tl]

Summation of degree of links connected

The joint values j v =-

Number of links connected at that joint

The stiffness of the chain can be calculated as,

1/k = l/kl+l/k2+.1/kn

For robot hand Figure 3 (c)

1/k = l/kl+l/k2+.1/klO

= 14+1/4+1/4.5+1/5+1/4.5+1/4+1/2+1/2+1/4+1/4

= 2.92

The distance between two links is nothing but the least number of joints that separate them, and the distance
between two joints is least number of links that separate them. [1]

Two distant matrices are calculated as D1 - link distance matrix and D2 -joint distance matrix, and the link
distance value and joint distance value are nothing but sum of all elements of D1 and D2 values, respectively [14]. The D1
values are taken from the previous work of the author [7] for all the nine robot hands. Now, the D2 for robot hand (c) can
be computed as follows:

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0 1 2
1 0 1

3

2

D2=

2

3

4

5
3
2
1

1 0 1

4

3

2

2

3

4
4
3
2

1 0 1

5

4

3

2

2

3
5

4
3

1 0 1

3

4

5

6
7

2

6

5

4

1 0 8

2

3

4

5

6
7

8 0 1

1

2

3

4

5

6
2

1 0 1
2 1 0

For robot hand (c),

Joint distance value, J. D. V = sum of all values of D2 matrix

=240

Link distance value, L. D. V= 332[7]

Compactness C, = J. D. V+L. D. V

=572

Figure 4: Robot Hand (d)

The chain is modelled like a system of springs connected in series; the stiffness of the chain can be calculated as
the summation of the joint values j v [ij

Summation of degree of links connected

The joint values j v

Number of links connected at that joint
The stiffness of the chain can be calculated as,

1/k = l/kl+l/k2+.1/kn

For robot hand Figure 4 (d)

1/k = l/kl+l/k2+.1/klO

= 14+1/4+1/4+1/4.5+1/7.5+1/2.5+1/2.5+1/2+1/4+1/4
= 2.9

Impact Factor (JCC): 7.6197

SCOPUS Indexed Journal

NAAS Rating: 3.11

Topological Synthesis and Structural Analysis ofPlanar Parallel Mechanisms

885

The distance between two links is nothing but the least number of joints that separate them and the distance
between two joints is least number of links that separate them. [1]

Two distant matrices are calculated as D1 - link distance matrix and D2 -joint distance matrix and the link
distance value and joint distance vale are nothing but sum of all elements of D1 and D2 values, respectively. The D1
values are taken from the previous work of the author [7] for all the nine robot hands. Now, the D2 for robot hand (d) can
be computed as follows:

0 1 2 3 4 4 3 2 1

1 0 1 2 3 3 4 3 2

2 1 0 1 2 2 5 4 3

3 2 1 0 1 1 6 5 4

D2= 432102765

432120765
345677012
2 3 4 5 6 6 1 0 1

1 2 3 4 5 5 2 1 0

For robot hand (d),

Joint distance value, J. D. V = sum of all values of D2 matrix

=228

Link distance value, L. D. V= 330[7]

Compactness C, = J. D. V+L. D. V

=558

Figure 5: Robot Hand(e)

The chain is modelled like a system of springs connected in series; the stiffness of the chain can be calculated as
the summation of the joint values j v tl]

Summation of degree of links connected

The joint values j v =-

Number of links connected at that joint

The stiffness of the chain can be calculated as,

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P. Vijay, A. Srinath & Parvatini Sri Naga Yenkat

1/k = l/kl+l/k2+.1/kn

For robot hand Figure 5 (e)

1/k = l/kl+l/k2+.1/klO

= 14+1/4+1/4.5+1/7.5+1/2.5+1/4.5+1/2+1/2+1/4+1/4
= 2.97

The distance between two links is nothing but the least number of joints that separate them, and the distance
between two joints is least number of links that separate them. [1]

Two distant matrices are calculated as D1 - link distance matrix and D2 -joint distance matrix and the link
distance value and joint distance vale are nothing but sum of all elements of D1 and D2 values, respectively. The D1
values are taken from the previous work of the author [7] for all the nine robot hands. Now, the D2 for robot hand (e) can
be computed as follows:

0 1 2 3 3 4 3 2 1

1 0 1 2 2 3 4 3 2

2 1 0 1 1 2 5 4 3

321023654

D2= 3 2 1 2 0 1 6 5 4

432310765
345667012
2 3 4 5 5 6 1 0 1

1 2 3 4 4 5 2 1 0

For robot hand (e),

Joint distance value, J. D. V = sum of all values of D2 matrix

=217

Link distance value, L. D. V= 330[7]

Compactness C, = J. D. V+L. D. V

=547

Figure 6: Robot Hand (f)

Impact Factor (JCC): 7.6197

SCOPUS Indexed Journal

NAAS Rating: 3.11

Topological Synthesis and Structural Analysis ofPlanar Parallel Mechanisms

887

The chain is modelled like a system of springs connected in series; the stiiTness of the chain can be calculated as
the summation of the joint values j v ^

Summation of degree of links connected

The joint values j v =-

Number of links connected at that joint

The stiffness of the chain can be calculated as,

1/k = l/kl+l/k2+.1/kn

For robot hand Figure 6 (f)

1/k = l/kl+l/k2+.1/klO

= 14+1/4.5+1/7.5+1/2.5+1/4.5+1/4+1/2+1/2+1/4+1/4

= 2.97

The distance between two links is nothing but the least number of joints that separate them, and the distance
between two joints is least number of links that separate them. [1]

Two distant matrices are calculated as D1 - link distance matrix and D2 -joint distance matrix and the link
distance value and joint distance vale are nothing but sum of all elements of D1 and D2 values, respectively. The D1
values are taken from the previous work of the author [7] for all the nine robot hands. Now, the D2 for robot hand (f) can
be computed as follows:

0 1 2 2 3 4 3 2 1

1 0 1 1 2 3 4 3 2

2 1 0 1 2 3 5 4 3

2 1 1 0 1 2 5 4 3

D2=

322101654
433210765
345567012
2 3 4 4 5 6 1 0 1

1 2 3 3 4 5 2 1 0

For robot hand (f),

Joint distance value, J. D. V = sum of all values of D2 matrix

=220

Link distance value, L. D. V= 332[7]

Compactness C, = J. D. V+L. D. V
=552

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6

E

Figure 7: Robot Hand (g)

The chain is modelled like a system of springs connected in series; the stiffness of the chain can be calculated as
the summation of the joint values j v tl j

Summation of degree of links connected

The joint values j v =-

Number of links connected at that joint

The stiffness of the chain can be calculated as,

1/k = l/kl+l/k2+.1/kn

For robot hand Figure 7 (g)

1/k = l/kl+l/k2+.1/klO

= 14+1/4.5+1/7.5+1/4.5+1/2+1/4.5+1/2+1/2+1/4+1/4

= 3.04

The distance between two links is nothing but the least number of joints that separate them and the distance
between two joints is least number of links that separate them. [1]

Two distant matrices are calculated as D1 - link distance matrix and D2 -joint distance matrix and the link
distance value and joint distance vale are nothing but sum of all elements of D1 and D2 values, respectively. The D1
values are taken from the previous work of the author [7] for all the nine robot hands. Now, the D2 for robot hand (g) can
be computed as follows:

0

1

2

3

2

3

3

2

1

1

0

1

2

1

2

4

3

2

2

1

0

1

2

2

5

4

3

3

2

1

0

3

3

6

5

4

D2= 4

1

1

2

0

1

5

4

3

3

2

2

3

1

0

6

5

4

3

4

5

6

5

6

0

1

2

2

3

4

5

4

5

1

0

1

1

2

3

4

3

4

2

1

0

For robot hand (g).

Impact Pactor (JCC): 7.6197

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NAAS Rating: 3.11

Topological Synthesis and Structural Analysis of Planar Parallel Mechanisms

889

Joint distance value, J. D. V = sum of all values of D2 matrix

=201

Link distance value, L. D. V= 316[7]

Compactness C, = J. D. V+L. D. V
=517

The chain is modelled like a system of springs connected in series; the stiffness of the chain can be calculated as
the summation of the joint values j v [ij

Summation of degree of links connected

The joint values j v =-

Number of links connected at that joint

The stiffness of the chain can be calculated as,

1/k = l/kl+l/k2+.1/kn

For robot hand Figure 4 (d)

1/k = l/kl+l/k2+.1/klO

= 14.5+1/7.5+1/4.5+1/2+1/4.5+1/4+1/2+1/2+1/4+1/4

= 3.04

The distance between two links is nothing but the least number of joints that separate them, and the
between two joints is least number of links that separate them. [1]

Two distant matrices are calculated as D1 - link distance matrix and D2 -joint distance matrix and
distance value and joint distance vale are nothing but sum of all elements of D1 and D2 values, respectively.
values are taken from the previous work of the author [7] for all the nine robot hands. Now, the D2 for robot hand (i) can
be computed as follows:

distance

the link
The D1

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Vijay, A. Srinath & Parratini Sri Naga Venkat

0

1

2

1

2

3

4

3

2

1

0

1

2

3

4

5

4

3

2

1

0

3

4

5

6

5

4

1

2

3

0

1

2

3

2

1

D2= 2

3

4

1

0

1

4

3

2

3

4

5

2

1

0

5

4

3

4

5

6

3

4

5

0

1

2

3

4

5

2

3

4

1

0

1

2

3

4

1

2

3

2

1

0

For robot hand (i).

Joint distance value, J. D. V = sum of all values of D2 matrix

=194

Link distance value, L. D. V= 313[7]
Compactness C, = J. D. V+L. D. V
=507

Figure 9: Robot Hand (i)

The chain is modelled like a system of springs connected in series; the stiffness of the chain can be calculated as
the summation of the joint values j v [ij

Summation of degree of links connected

The joint values j v =-

Number of links connected at that joint

The stiffness of the chain can be calculated as,

1/k = l/kl+l/k2+.1/kn

For robot hand Figure 4 (d)

1/k = l/kl+l/k2+.1/klO

= 1/6+1 /4+1 /4+1/2+1 /4+1 /4+1/2+1/2+1 /4+1 /4

= 3.16

The distance between two links is nothing but the least number of joints that separate them, and the distance
between two joints is least number of links that separate them. [1]

Impact Factor (JCC): 7.6197

SCOPUS Indexed Journal

NAAS Rating: 3.11

Topological Synthesis and Structural Analysis ofPlanar Parallel Mechanisms

891

Two distant matrices are calculated as D1 - link distance matrix and D2 -joint distance matrix and the link
distance value and joint distance vale are nothing but sum of all elements of D1 and D2 values, respectively. The D1
values are taken from the previous work of the author [7] for all the nine robot hands. Now, the D2 for robot hand 8(h) can
be computed as follows:

0 1 1 2 3 3 3 2 1

1 0 2 1 2 3 4 3 2

1 1 0 2 1 2 4 3 2

212034543

D2= 2 2 1 3 0 1 5 4 3

332410654
344556012

2 3 3 4 4 5 1 0 1

1 2 2 3 3 4 2 1 0

For robot hand (h),

Joint distance value, J. D. V = sum of all values of D2 matrix

=200

Link distance value, L. D. V= 300[7]

Compactness C, = J. D. V+L. D. V
=500

5. RESULTS AND DISCUSSIONS

The table gives the mechanisms with the stiffness values and compactness values, D1 and D2 distant matrices and
robot hands (a) to (i). The rating of robot hands were done as per the stiffness value and based on the compactness value.
The links are stiffer than the robot hand that is more rigid, and the links are closer, and more is the compactness i. e.
structurally rigid.

6. CONCLUSIONS

A simple method to compute the stiffness of the mechanisms and compactness of the mechanism is applied
successfully, for ten bar single degree of freedom of such nine robot hands, and are rated by comparing the stiffness and
compactness. The stiffness and compactness will be more for the robot hands, whose value is low [1]. Out of the nine robot
hands, robot hand (a) is stiffer having stiffness value 2.82, and robot hand (i) is more compact or rigid, having compactness
value 500.

Table 1: Robot Hands with Compactness Yalues

Robot Hand

D1

D2

Compactness

i

200

300

500

h

194

313

507

g

201

316

517

e

217

330

547

f

220

332

552

d

228

330

558

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Table 1: Contd..

a

240

329

569

c

240

332

572

b

249

332

581

Table 2: Robot Hands with Stiffness Yalues

Robot hand

Stiffness

a

2.82

b

2.89

d

2.9

c

2.92

e

2.97

f

2.97

g

3.04

h

3.04

i

3.16

Table 3: Robot Hands Rating based on Compactness Yalues

Robot hand

D1

D2

Compact

Rated high/low

i

200

300

500

1

h

194

313

507

2

g

201

316

517

3

e

217

330

547

4

f

220

332

552

5

d

228

330

558

6

a

240

329

569

7

c

240

332

572

8

b

249

332

581

9

Table 4: Robot Hands Rating based on Stiffness Yalues

Robot hand

Stiffness

Rated high/low

a

2.82

1

b

2.89

2

d

2.9

3

c

2.92

4

e

2.97

5

f

2.97

5

g

3.04

6

h

3.04

6

i

3.16

7

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Impact Factor (JCC): 7.6197

SCOPUS Indexed Journal

NAAS Rating: 3.11

Topological Synthesis and Structural Analysis ofPlanar Parallel Mechanisms

893

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