Intelligent Control and Automation, 2011, 2, 430-449
doi:10.4236/ica.2011.24050 Published Online November 2011 (http://www.SciRP.org/journal/ica)
Copyright © 2011 SciRes. ICA
Particle Swarm Optimization Algorithm vs Genetic
Algorithm to Develop Integrated Scheme for
Obtaining Optimal Mechanical Structure
and Adaptive C o n troller of a Robot
Rega Rajendra, Dilip K. Pratihar
Soft Computing Laboratory, Department of Mechanical Engineering, Indian Institute of Technology,
Kharagpur, India
E-mail: regaraj@gmail.com, dkpra@mech.iitkgp.ernet.in
Received July 20, 2011; revised August 11, 2011; accepted September 15, 2011
Abstract
The performances of Particle Swarm Optimization and Genetic Algorithm have been compared to develop a
methodology for concurrent and integrated design of mechanical structure and controller of a 2-dof robotic
manipulator solving tracking problems. The proposed design scheme optimizes various parameters belong-
ing to different domains (that is, link geometry, mass distribution, moment of inertia, control gains) concur-
rently to design manipulator, which can track some given paths accurately with a minimum power consump-
tion. The main strength of this study lies with the design of an integrated scheme to solve the above problem.
Both real-coded Genetic Algorithm and Particle Swarm Optimization are used to solve this complex optimi-
zation problem. Four approaches have been developed and their performances are compared. Particle Swarm
Optimization is found to perform better than the Genetic Algorithm, as the former carries out both global and
local searches simultaneously, whereas the latter concentrates mainly on the global search. Controllers with
adaptive gain values have shown better performance compared to the conventional ones, as expected.
Keywords: Manipulator, Optimal Structure, Adaptive Controller, Genetic Algorithm, Neural Networks,
Particle Swarm Optimization
1. Introduction
In the age of high precision manufacturing of compo-
nents and parts, there is a need to develop precisely con-
trolled manipulator for handling micro-precision jobs
and machining applications. Also, the manipulator has to
traverse a given trajectory as accurately as possible to
avoid variations in trajectory causing damage to other
parts in assembly or distorted machining of geometric
components or inaccurate welding of jobs. Thus, trajec-
tory tracking is one of the most important tasks to be
performed by the manipulator.
A user specify motions as the sequences of points
through which a tool fixed to the end-effector of a ma-
nipulator has to pass. The effectiveness of such motion
specifying mechanism is greatly increased, if the tool
moves in a specified path between the user-specified
points. Intermediate points are interpolated along the
path at regular intervals of time during the motion, and
the manipulator’s kinematics equations are solved to
produce the corresponding joint parameter values. The
developed path interpolating function offers several ad-
vantages, including less computational cost and im-
proved motion characteristics. A second method uses a
motion planning phase to pre-compute enough interme-
diate points, so that the manipulator may be driven by
interpolation of joint parameter values, while keeping the
tool on an approximately pre-specified path. This tech-
nique allows a substantial reduction in real-time compu-
tation. The planning is done by an efficient recursive
algorithm, which generates enough intermediate points to
guarantee that the tool’s deviation from the path to be
tracked stays within a pre-specified error bounds.
Several attempts were made by various investigators
to design and develop suitable controllers for the robots.
The following model-based robot controllers had been
R. RAJENDRA ET AL.431
used: computed torque control, non-adaptive Propor-
tional Derivative (PD) control, PD control with feed-
back, and others. Some of those attempts are discussed
below.
Qu [1] developed PD control scheme to solve trajec-
tory tracking problem of a manipulator. He concluded
that PD control with time-varying gains could guarantee
global stability for the trajectory following problem of a
manipulator. Homsup and Anderson [2] proposed a
model-based PD control for a 2-dof planar manipulator.
Its performance was measured using system performance
ellipsoids drawn utilizing the information of control pa-
rameters, manipulator’s Jacobian and inertia matrices.
Thus, the performance was dependent not only on the
robot’s kinematics and dynamics, but also on the control
algorithm. A correlation between trajectory tracking er-
rors and workspace location was established. Kelly and
Salgado [3] developed a design procedure for selecting
gain values (that is,
p
K
and d
K
) of PD control with
computed feed-forward of robot dynamics and desired
trajectory. The performance of their approach was tested
through computer simulations on 2-dof manipulator. An
evolutionary PD control strategy was used by Ouyang
and Zhang [4] to improve trajectory tracking perform-
ance of a closed-loop robot manipulator. It could ensure
good trajectory tracking performance without using the
knowledge of robot dynamics. The performance of their
strategy was tested through computer simulations and
found to be better than conventional PD and non-linear
PD control strategies in terms of trajectory tracking per-
formance and fluctuation in the actuator’s torques.
Ravichandran et al. [5] developed a scheme for simul-
taneous plant-controller design optimization for a 2-dof
planar rigid manipulator and non-linear PD controller.
During optimization, a heuristic search technique named
evolution strategy and a weighted-sum problem formula-
tion were adopted to take care of multiple objectives and
generate a single Pareto-optimal solution. The perform-
ance of the developed scheme was demonstrated on
computer simulations, and the potential of the scheme to
yield desirable designs of the manipulator and controller
was realized.
Soft computing-based tools [6] had also been used to
design and develop suitable controller for the robots.
Some of those studies are discussed here. Ozaki et al. [7]
proposed a non-linear compensator using neural net-
works for trajectory control of a 2-dof manipulator. Its
performance was compared with that of adaptive con-
troller proposed by Craig [8] in compensating unstruc-
tured uncertainties of the manipulator. The neural net-
work-based approach was found to be effective and effi-
cient in learning manipulator dynamics, and conse-
quently could track the trajectory accurately. Ghalia and
Alouani [9] designed a fuzzy logic-based controller of a
2-dof manipulator to determine appropriate gain values
of the compensator for tracking some trajectories accu-
rately. The performance of their approach was tested on
computer simulations. Rueda and Pedrycz [10] proposed
a hierarchical fuzzy-neural-PD controller for N-dof robot
manipulators to solve tracking problems accurately. In
their approach, a coordinator was implemented as a
fuzzy-neural network, whose purpose was to select acti-
vation levels for local regulators implemented as time-
varying PD controllers.
Park and Asada [11] developed a concurrent design
method of determining mechanical structure and suitable
controller for a 2-dof planar non-rigid manipulator. An
attempt was made to achieve high speed positioning by
optimizing arm link geometry, actuator locations and
feedback gains. In their study, optimal feed-back gains
minimizing the settling time were obtained as the fluc-
tuations of structural parameters, which were optimized
using a non-linear programming technique. Based on the
obtained optimal design, one prototype robot was built
and an outstanding performance was observed.
The concept of PSO algorithm was introduced by
Kennedy and Eberhart [12] in 1995. It is a popula-
tion-based search algorithm, which is initialized with the
population of random solutions, called particles, and the
population is known as swarm. Several modifications in
the PSO algorithm had been done by various researchers.
Shi and Eberhert [13] introduced a new parameter called
inertia weight into the original PSO algorithm, which
played an important role in balancing the global and lo-
cal searches. Clerc and Kennedy [14] analyzed how a
particle carries out its search in a complex problem space
and modified the original PSO on the basis of this analy-
sis. Chen et al. [15] improved the PSO algorithm with
adaptive inertia weight W and acceleration coefficients
in order to maintain population diversity and sustain
good convergence capacity to optimize back-propagation
neural networks.
PSO is simple in concept, as it has a few parameters
only to be adjusted. It has found applications in various
areas like constrained optimization problems, min-max
problems, multi-objective optimization problems and
many more. In addition to these application areas, it has
been applied to evolve weights and structure of some
neural networks (NNs). Han and Jiang [16] proposed an
endpoint prediction model of Basic Oxygen Furnace
(BOF) steelmaking based on PSO-tuned radial basis
function neural network. Braik et al. [17] developed a
mechanism to improve performance of NN in modeling a
chemical process through PSO. Abe and Komuro [18]
utilized an NN, tuned by PSO to save energy of the
flexible manipulator for point-to-point motion. Joint an-
Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.
Copyright © 2011 SciRes. ICA
432
gles generated by the NN suppressed the residual vibra-
tion and hence, minimized the motor torques, which was
kept as the objective function. The authors conducted
numerical studies and verified by the same with experi-
ments and concluded that PSO is an efficient optimizer.
Although a considerable amount of work has been
carried out in this field of research, there is still a need
for an integrated scheme for obtaining optimal mechani-
cal structure and controllers with adaptive gain values of
2-dof manipulator solving tracking problems. The aim of
this study is to obtain an optimal mechanical structure of
the manipulator along with adaptive controller that en-
ables high precision positioning. The links of the ma-
nipulator are treated as rigid bodies. To speed up opera-
tions, one needs powerful actuators and lightweight arm
links. In positioning, however, the major issue is to
minimize the settling time of the control system. The
settling time depends on a broad range of design pa-
rameters including mass and stiffness properties of the
links, gain values of the compensator, and others. These
parameters are coupled to each other and have intricate
interactions with respect to the robot’s settling time. For
instance, increasing the structural stiffness alone does not
always decrease the settling time. All the design pa-
rameters must be considered in an integrated manner in
order to optimize the performance. In the present paper,
an integrated scheme for obtaining optimal mechanical
structure and controller of a 2-dof manipulator solving
path tracking problems, has been proposed. Four ap-
proaches are developed, and their performances have
been compared on two path tracking problems.
The remaining part of this paper has been organized as
follows: Section 2 deals with mathematical formulation
of the problem. Tools and techniques used in the present
work are discussed and the proposed algorithms have
been explained in Section 3. Results are stated and dis-
cussed in the Section 4. Some concluding remarks are
made in Section 5 and the scope for future work is indi-
cated in Section 6.
2. Mathematical Formulation of the Problem
This section deals with mathematical formulation of the
problem. It has been posed as a constrained optimization
problem
2.1. Trajectory Analysis
A two degrees of freedom serial manipulator with hollow
circular cross-section links of lengths: 1 and 2
(where 1 > 2) has been considered, as shown in
Figure 1. A photograph of the set-up has been displayed
Figure 2. The links are connected using rotary joints.
The end-effector of the manipulator is directed to track
one straight and another circular paths separately, start-
ing from initial position (
L L
L L
,
ii
X
Y) up to final position
(,
f
f
X
Y) in time . The forward kinematics equations
of the manipulator can be written as follows:
t
(a) (b)
Figure 1. (a) A two degrees of freedom manipulator traversing a straight-line trajectory with torques applied at its joints; (b)
load distributions on two links of the manipulator .
R. RAJENDRA ET AL.433
Figure 2. A set-up of the 2-dof manipulator.
 
 

112 12
112 12
cos cos
sin sin
X
LtL t
YLt Ltt


 
 
t
(1)
The joint angles: and can be obtained
by carrying out inverse kinematics, as given below:

1t

2t

 

2 222
12
1
2
12
22
11
1
12 2
cos 2
sin
tan tancos
XYLL
tLL
Lt
Y
t
LL t









 
 
(2)
2.2. Determination of Power Consumption
Lagrange-Euler formulation [19] is used to determine
energy consumption of the manipulator at its two joints.
The manipulator’s joints torques:
1t
and
2t
consist of inertia, centrifugal and Coriolis and gravity
terms, as given below.







() ,tDtt htt ct
 

 
1
2
(3)
where , , and represent inertia, centrifugal and
Coriolis, and gravity terms, respectively. Now, torques
required at two joints of the manipulator (refer to Figure
1) can be determined as follows:
Dhc
11 12
11
12
hc
DD


 ,
21 22
22
12
hc
DD


 ,
where



22
111 2
122 2
11
222
212
2
1cos
3
11
34
Lt
mm m
DLL
L
mmm
Rr
 
 
L
,


2
22
12 2122
2
12
22
11
43
1
cos
2
L
mm
DD Rr
t
mLL
 
,

22
2
22 22
2
11
34
Lr
mm
DR

,


12
12 2
12
2
12
22
2
sin
0.5sin
t
hm
LL
t
mLL

,

2
12
22 2
1
0.5sin t
hm
LL
,

 



111
1
22 12
21 1
0.5cos
0.5cos
cos
mt
gL
c
mgLt t
mgLt


,
 

22212
0.5 coscmgL tt

.
Now, energy consumed by the manipulator (kW) in
tracing the trajectory can be determined as follows:

100 22
11
10.025
100 ij jiij
ij
Et


 , (4)
where the second part of this equation indicates the en-
Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.
434
ergy dissipated by the motors connected at the joints [20].
It is important to mention that the cycle time has been
assumed to be equal to 100 seconds and it is divided into
100 instants.
2.3. Structural Analysis: Bending of the Links
Structural analysis [21] is confined to Hooke’s Law of
analysis, for which the manipulator is used to verify its
behavior while tracking the trajectory with minimum
error. Due to the weight of the links and motors, there is
a chance of bending and deflection of the links and thus,
it may deviate from the desired trajectory. To prevent
mechanical failure of the links, the developed stress
should be less than the allowable stress
, that is, .
developed
allowable developed allowable
For minimum consumption of energy by the manipu-
lator, suitable hollow cross-section of the links possess-
ing sufficient strength and rigidity to withstand the
bending and deflection is to be determined. The links are
made up of aluminum. Figure 1(b) displays load distri-
bution, on two links of the manipulator, where 1 in-
dicates the weight of the second motor and second link,
and load due to 1

W
; 2 denotes the load due to 2
W
;
and w represents uniformly distributed load on the links
due to self-weight. Link1 can be considered as a beam of
hollow circular cross-section, which is subjected to ,
and a fixed moment of
1
W
w
 
2
12
2
0.5 cos
M
wLt t

 . Thus, it is to be de-
signed considering the effect of bending moment 1
M
,
which can be determined as follows:



2
1121
21 11
2
12
2
0.5 cos
0.5cos
mwt
WwL
MLLL
wLt t



 , (5)
where 2m represents the weight of second motor. As
the two links are assumed to have same cross-section and
link1 is more critical compared to the other one, an at-
tention has been focused on its safe design. The devel-
oped bending stress on link1 can be determined as fol-
lows:
W

max
22
4
π
developed
M
Rr



, (6)
where and represent the outer and inner radii of
the hollow circular link and the maximum bending mo-
ment Mmax can be determined as follows :
R r

22
112
21
max1 2
0.5
m
M
wwL
WLLL

L
.
It is important to mention that linear relationships for
elastic modulus (E) and yield strength (
y
) of aluminum
with its density (ρ) have been established according to
[22-28], as given below.
1.92554 92.56675Ee
(7)
7.52288 730638.91191
allowable E
(8)
It is to be noted that allowable stress allowable
has
been kept equal to
y
.
2.4. Stability and Response Analysis in
Trajectory Tracking
The control problem of a robotic manipulator is that of
determining the time history of joint inputs required to
cause its end-effectors to execute a commanded motion
typically specified either as a sequence of end-effector’s
positions and orientations or as a continuous path. De-
pending on the controller design, the joints inputs may be
in the form of joint forces, torques or inputs from actua-
tors. A particular controller design has a significant im-
pact on the performance of the manipulator and conse-
quently, on the range of its possible applications (either
point-to-point or continuous). In addition, mechanical
design of the manipulator itself will influence the type of
control scheme needed. It is to be noted that permanent
magnet DC motors with gear reduction are commonly
used in robots. The design objective is to choose the
compensator (either PD or PID) in such a way that the
plant output tracks or follows a desired path. For set
points, tracking is a problem of constant or step reference
command d
that arises in point-to-point motion [29],
[30]. For a PD compensator, the control input U(s) is
given in the Laplace domain as follows:
 

d
pd
UsKKsss
 
(9)
The resulting closed-loop system response is given by
the following expression:
   

pd
d
KKs Ds
ss
s
s
 

, (10)
where D(s) is the disturbance on the system (here, it has
been assumed to be equal to zero), and
s
is the
closed-loop characteristic polynomial given by

2
dp
s
JsB KsK  (11)
The closed-loop system will be stable for positive
values of
p
K
, d
K
and bounded disturbances, and the
joint error is given by

d
s
s
 . For the PD com-
pensator given in Equation (9), the step response is de-
termined by the closed-loop natural frequency
and
damping ratio
. The compensator’s gain values:
p
K
and d
K
can be obtained as follows:
2,
2,
p
d
KJ
K
JB

(12)
Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.435
where J is the inertia of the links, motor and gear, which
can be determined from the relationship: 2
1Js
;
represents effective damping of the system [31];
B
indicates the natural frequency. Now, the joint errors can
be calculated as follows:
 
, where 1,2,
d
ijterror i
itt i
 
(13)
The trajectory equations are found to be as follows af-
ter considering error in motors and deflection of links.



 

 

1
11 1
1
11 1
12
2211 2
12
22 11 2
cos ,
sin ,
cos ,
sin .
XL t
YL t
X
XLt t
YYLtt
 

 

(14)
Therefore, the trajectory error can be calculated as fol-
lows:
22
222
error
error
2
X
XX
YYY

 (15)
Total Trajectory error can be determined like
the following.
total
Err
100 100
11
totalerror error
ii
Err X Y



, (16)
Now, average error () can be obtained as fol-
lows:
avg
Err
2100
total
avg
Err
Err , (17)
This problem can be posed as a constrained optimiza-
tion as given below.
Minimize
avg
EErr
, (18)
subject to
developed allowable

and
0.001 0.0029t
 ,
2600.0 2800.00
 ,
5.0 < ω1 < 20.0,
10.0 < ω2 < 15.0,
1.0 < s1 < 1.0,
1.0 < s2 < 1.0.
3. Proposed Algorithm
The said constrained optimization problem has been
solved using a real-coded Genetic Algorithm (GA) [32],
as the variables are real in nature. Moreover, a real-coded
GA has some advantages over the binary-coded GA,
such as its ability to provide more precise solutions and
there is no hamming cliff problem [6]. A GA is a popula-
tion-based search and optimization technique, which
works based on the mechanics of natural genetics and
principle of natural selection [33,34]. It starts with a
population of solutions created at random, which are
further modified using the operators like reproduction,
crossover and mutation. In the present study, tournament
selection, polynomial mutation and simulated binary
crossover have been used. Interested readers may refer to
[6] for a detailed description of the algorithm.
Particle Swarm Optimization (PSO), introduced by
Kennedy and Eberhart in 1995 [12] is a stochastic popu-
lation-based evolutionary computation technique, which
has also been used to solve the said optimization problem.
It can be linked to bird flocking, fish schooling or socio-
logical behavior of a group of people. It has been used to
solve a variety of optimization problems. In PSO, the
population of solution is known as swarm. It uses a
number of agents known as particles that constitute a
swarm moving around in the search space looking for the
best solution. Each particle is treated as a point in an
N-dimensional space, which adjusts its flying according
to its own flying experience and that of other particles.
Each particle keeps track of its coordinates in the solu-
tion space, which are associated with the best solution in
terms of fitness that has been achieved so far by it. This
value is called personal best, Pbest. Another best value
that is tracked by the PSO is the best value obtained so
far by any particle lying in its neighborhood. This value
is called global best, Gbest. The basic concept of PSO
lies in accelerating each particle toward its Pbest and the
Gbest locations, with a random weighted acceleration at
each time step. Four approaches have been developed as
explained below.
3.1. Approach 1
Optimizati o n u si ng a re al -coded GA only
Figure 3 shows the working cycle of a real-coded GA.
A GA-solution carries real values of six variables, such
as t
, ρ, 1
, 2
, 1
s
, 2
s
. The constrained optimization
problem has been solved using a penalty function ap-
proach [6]. As it is a minimization problem, a fixed posi-
tive penalty of P
100 has been added for each vio-
lated constraint (if any). Thus, the fitness (f) of the
GA-solution can be represented as follows:
=
avg
fEErr P
 (19)
The GA tries to find optimal solution iteratively.
3.2. Approach 2
Optimization using a combined neural network and
Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.
Copyright © 2011 SciRes. ICA
436
Figure 3. A schematic view representing working cycle of real-coded GA (that is, approach 1).
real-coded GA
In approach 1, one GA-solution supplies one set of
values of 1
, 2
, 1
s
, 2
s
for the entire cycle time of
100 seconds. However, it may be required to adopt dif-
ferent sets of the said parameters for different instants of
the cycle time to track the trajectory accurately. It has
been tried in this approach. For each duration, error
and error are determined by comparing the calculated
values of
X
Y
X
and with their corresponding target
values, and these are fed as inputs to a feed-forward
neural network (refer to Appendix A), which consists of
three layers, namely input, hidden and output layers.
There are two and four neurons in the input and output
layers of the network, as decided by the number of inputs
and outputs, respectively. A thorough parametric study
has been carried out to determine the numbers of hidden
neurons, which has come to be equal to four. The first
and third output neurons have log-sigmoid transfer func-
tion, and the second and fourth neurons are assumed to
have tan-sigmoid transfer function. The neurons of input
and hidden layers are assumed to have linear and
tan-sigmoid transfer functions, respectively. In this ap-
proach, the GA carries information of thickness of the
links (t), density of the link material (
Y
) and thirty real
variables related to the neural network, such as eight and
sixteen connecting weights between the input and hidden
layers (that is, [V]) and hidden and output layers (that is,
[W]), respectively; bias and coefficients of transfer func-
tion of hidden neurons; four different coefficients of
transfer functions of the output neurons. The values of
1
, 2
, 1
s
and 2
s
are determined as the outputs of
the neural network. Figure 4 displays the flowchart of
approach 2. The fitness of a solution has been calculated
using Equation (19).
3.3. Approach 3
Optimization using Particle Swarm Optimization
(PSO) only
Similar to other population-based optimization meth-
ods, such as GAs, the PSO starts with the random ini-
tialization of population particles in the search space.
The PSO algorithm works based on the social behaviour
of the particles in the swarm. Therefore, it finds the
global best solution by simply adjusting the trajectory of
each individual toward its own best location and toward
the best particle of the entire swarm at each time step
(generation) [12,14]. In PSO algorithm, the trajectory of
each individual in the search space is adjusted by dy-
namically changing the velocih particle, accord- ty of eac
R. RAJENDRA ET AL.437
Figure 4. Flowchart of approach 2.
ing to its own flying experience and that of other parti-
cles in the search space. The position and velocity vec-
tors of particle in the d-dimensional search space
can be represented as
th
i

1,,
ii id
X
xx and
ii id
, respectively. The value of i
V vector
can be varied in the range of
1,,
Vv v
max max
,vv to reduce the
tendency of particles to leave the search space. The value
of ma is usually chosen to be equal to maxx
v,kx
where [35]. According to a user defined
fitness function, let us say that the best position of each
particle (which corresponds to the best fitness value ob-
tained by that particle at time t) is 1ii
and the fittest particle found so far at time t is
0.1 1k.0

,,
id
pPp
1,,
g
gg
Pp pd
. The new velocities and positions of
the particles for the next fitness evaluation are calculated
using the following two equations:


 

1
,
id id
gd id
t bRandpxt
andpx t
 


2
1
id id
vt Wv
bR

(20)

11
id id
xt vt
id
xt
, (21)
where id is the velocity of dth dimension of the ith
particle, W is a constant known as inertia weight [36], 1
and denote the acceleration coefficients, and
v
2
b
b
rand
and
Rand
are two separately generated
uniformly distributed random numbers lying in the range
Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.
438
of
0,1 . The first part of Equation (20) represents the
previous velocity, which provides the necessary mo-
mentum for the particles to roam across the search space.
The second part of Equation (20) is known as the cogni-
tive component that represents the personal thinking of
each particle. The cognitive component encourages the
particles to move toward their own best positions found
so far. The third part of Equation (20) is known as the
social component, which indicates the collaborative ef-
fect of the particles in finding the global optimal solution.
The social component always pulls the particles toward
the global best particle found so far. The PSO is becom-
ing very popular due to its simple architecture, ease of
implementation and ability to quickly converge to a rea-
sonably good solution. The flowchart of PSO algorithm
is shown in Figure 5. The values of six variables,
namely t
,
, 1
, 2
, 1
s
and 2
s
are supplied by
the PSO during optimization and the fitness of a solution
Figure 5. The flowchart of PSO algorithm.
Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.
Copyright © 2011 SciRes. ICA
439
has been determined using Equation (19).
3.4. Approach 4
Optimization using combined neural network and
PSO algorithm
In approach 3, one PSO-solution gives one set of val-
ues of ω1, ω2, s1, s2 for the entire cycle time of 100 sec-
onds. However, it may be required to choose separate
sets of the said parameters for different instants of the
cycle time to track the trajectory accurately. It has been
attempted in this approach. For each duration, error
and error values are obtained by comparing the calcu-
lated values of
X
Y
X
and with their corresponding
target values, and these are fed as inputs to a feed-for-
ward neural network (refer to Appendix A), which con-
sists of three layers, namely input, hidden and output
layers. The input and output layers of the network con-
tain two and four neurons, respectively. A detailed pa-
rametric study has been carried out to decide the num-
bers of hidden neurons, which has turned out to be equal
to four. The first and third output neurons have log-sig-
moid transfer function, and the second and fourth neu-
rons are assumed to have tan-sigmoid transfer function.
The neurons of input and hidden layers are assumed to
have linear and tan-sigmoid transfer functions, respec-
tively. In this approach, the PSO solution carries infor-
mation of the thickness of the links (t), density of the link
material (ρ) and thirty real variables related to the neural
network, such as eight and sixteen connecting weights
between the input and hidden layers (that is, [V]) and
hidden and output layers (that is, [W]), respectively; bias
and coefficients of transfer function of hidden neurons;
four different coefficients of transfer functions of the
output neurons. The values of ω1, ω2, s1, and s2, are ob-
tained as the outputs of the neural network. The fitness of
a solution has been determined using Equation (19).
Y
4. Results and Discussion
The performances of the developed four approaches have
been tested through computer simulations. The lengths of
two links: 1 and 2 are assumed to be equal to 0.3m
and 0.2 m, respectively. The terms: B (effective damping)
and
L L
(damping ratio) [31] of Equation (12) have been
set equal to 0.02 and 1.0, respectively. The outer radius
of two circular links is considered as R = 0.03 m. The
motor connected to the second link weighs 0.568 kg.
Simulations are conducted on a P-IV PC. Results of four
developed approaches are stated and discussed below for
tracking of one straight path and another circular path
separately.
4.1. Straight Path Tracking
The 2-dof manipulator will have to track a straight path
as accurately as possible after consuming minimum
power and ensuring enough mechanical strength. Inverse
kinematics calculations have been carried out using
Equation (2) to determine joint angles corresponding to
the movement of the end-effector along a straight path in
Cartesian coordinate system. A fourth-order polynomial
is found to be suitable to represent the trajectory of
1t
as follows:
23
1101112131
taatatatat
 4
4
,
where the values of the coefficients have turned out to be
as a10 = 0.183987352, a11 = 0.008541301
a12 = 0.000193741, a13 = 0.000001798 and
a14 = 0.000000007. Similarly, the trajectory function
2t
is seen to be a cubic polynomial as follows:
23
212223
atat 
22
0
ta at , where
a20 = 2.279365512, a21 = 0.011757362,
a22 = 0.000026833, a23 = 0.000000209. Angular velo-
cities and accelerations have been determined for both
the joints by differentiating angular displacement with
respect to time (t) for once and twice, respectively. Re-
sults of the developed four approaches are stated, dis-
cussed and compared below.
4.1.1. Results of Approach 1
As the performance of a GA depends on its parameters, a
thorough parametric study is carried out to determine the
values of optimal GA-parameters.
Figure 6 shows the results of the said parametric study,
in which one parameter has been varied at a time keeping
the others fixed.
Thus, the following GA-parameters are found to yield
the best results: c = 0.76, m = 0.0055, Population
size = 150, maximum number of generations max
= 150. The optimized values of the parameters:
P P
P G
t
, ρ,
1
, 1
s
, 2
and 2
s
are found to be equal to 0.001m,
2600.00 kg/m3, 5.002, 0.02612, 10.00 and 0.0141,
respectively. The obtained results will be discussed in
Figures 7 to 11, at the end of this section.
4.1.2. Results of Approach 2
The set of optimal GA-parameters has been obtained
using an approach explained earlier. The following
GA-parameters are found to give the best results: c =
0.86, m = 0.0055, Population size = 220, Maxi-
mum number of generations max = 240. During opti-
mization, the ranges of
P
P P
G
t
and ρ have been kept the
same with those of approach 1. The connecting weights:
V,
W are varied in the range of 1.0 to 1.0. The
coefficients of transfer functions of the hidden neurons
(1_ hid) and those of first through fourth neurons of out-
ut layers (that is, , , , ) have
a
p
2_out
a3_out
a
4_out
a5_out
a
R. RAJENDRA ET AL.
Copyright © 2011 SciRes. ICA
440
(a) (b)
(c) (d)
Figure 6. Results of GA-parametric study: (a) fitness vs Pc; (b) fitness vs Pm; (c) fitness vs population size P; (d) fitness vs
maximum number of generations Gmax.
been optimized in the ranges of (0.1, 1.0), (2.0, 3.0), (1.0,
2.0), (1.0, 2.0) and (1.0, 2.0), respectively. Moreover, the
bias value: b has been varied from 0.000001 to 0.0001.
The optimized values of the variables, which are ob-
tained using this approach, are shown in Table 1. Results
have been discussed and compared with those of other
approaches with the help of Figures 7 to 11.
4.1.3. Results of Approach 3
In this approach, the following PSO-parameters are
found to yield the best results: number of particles inter-
acting with each particle = 3; dimension of the
search space d = 6; number of runs =100; number of
executions =1000. The optimized values of the parame-
ters: , ρ, 1
k
t
, 1
s
, 2
, and 2
s
are found to be equal
to 0.001m, 2600.00 kg/m3, 13.715, 0.078, 14.271, and
0.00556, respectively. Results of this approach have been
explained in detail with the help of Figures 7 to 11, at
the end of this section.
4.1.4. Results of Approach 4
The following PSO-parameters are seen to give the best
results: = 3; d = 36; number of runs =100; number of
executions = 1000. The optimal values of the parameters:
k
t
, ρ, 1
, 1
s
, 2
and 2
s
are found to be equal to
0.001 m, 2600.00 kg/m3, 11.715, 0.078, 10.271 and
0.00551, respectively. Table 2 shows the optimized val-
ues of other variables obtained by the PSO algorithm in
approach 4. Results of this approach have been compared
with that of other approaches in Figures 7 to 11.
4.1.5. Comp arisons
The paths tracked and energy consumed by the manipu-
lator using approaches 1, 2, 3 and 4 are compared here.
The values of energy (kW) consumption, and average
absolute percent deviation in trajectory tracking by four
approaches are found to be equal to 0.241371, 0.241371,
0.238906, 0.23859, and 0.000000763, 0.000000009,
.000000585, 0.00000000128, respectively, while trac- 0
R. RAJENDRA ET AL.
Copyright © 2011 SciRes. ICA
441
(a) (b)
Figure 7. Variations of PD controller’s gain values: (a) Kd; (b) Kp for Joint 1.
Table 1. Optimized values of the variables obtained by the GA in approach 2 for straight path tracking.
Variable Range Optimized value Variable Range Optimized value
t' 0.001, 0.03 0.001 W23 1.0, 1.0 1.00
ρ 2600.0, 2800.0 2600.0 W24 1.0, 1.0 0.99
V11 1.0, 1.0 1.00 W31 1.0, 1.0 0.99
V12 1.0, 1.0 1.00 W32 1.0, 1.0 1.00
V13 1.0, 1.0 1.00 W33 1.0, 1.0 0.99
V14 1.0, 1.0 0.99 W34 1.0, 1.0 0.89
V21 1.0, 1.0 1.00 W41 1.0, 1.0 0.99
V22 1.0, 1.0 1.00 W42 1.0, 1.0 0.99
V23 1.0, 1.0 1.00 W43 1.0, 1.0 0.99
V24 1.0, 1.0 1.00 W44 1.0, 1.0 0.99
W11 1.0, 1.0 0.99 a1_hid 0.1, 1.0 0.099
W12 1.0, 1.0 0.99 a2_out 2.0, 3.0 1.999999
W13 1.0, 1.0 0.99 a3_out 1.0, 2.0 0.99999
W14 1.0, 1.0 1.00 a4_out 1.0, 2.0 1.000
W21 1.0, 1.0 0.99 a5_out 1.0, 2.0 0.999999
W22 1.0, 1.0 0.99 b 0.000001 to 0.0001 0.00000023
ing the straight path. Figures 7 and 8 show the variations
of d
K
,
p
K
values for joint 1 and joint 2, respectively,
while tracing the given straight trajectory. Comparisons
of these approaches are shown on the entire trajectory in
Figure 9(a), whereas Figure 9(b) displays the same
comparison on a finer scale within the smaller ranges of
X
and . Figures 10 and 11 display the variations of
percent deviation in predictions of
Y
X
and Yvalues in
a cycle, respectively, for the four approaches. The devia-
tions in tracking the trajectory are found to be less in
approaches 2 and 4 compared to that in approaches 1 and
3. Approach 4 is found to perform better than other ap-
proaches in terms of accuracy in path tracking and power
consumption. The better performance of approach 4
R. RAJENDRA ET AL.
442
Table 2. Optimized values of the variables obtained by the PSO in approach 4 for straight path tracking.
Variable Range Optimized value Variable Range Optimized value
t' 0.001, 0.03 0.001 W23 1.0, 1.0 1.00
ρ 2600.0, 2800.0 2600.0 W24 1.0, 1.0 0.7125
V11 1.0, 1.0 0.38296 W31 1.0, 1.0 0.9599
V12 1.0, 1.0 0.1 W32 1.0, 1.0 0.911465
V13 1.0, 1.0 0.1 W33 1.0, 1.0 0.62298
V14 1.0, 1.0 1.00 W34 1.0, 1.0 0.67631
V21 1.0, 1.0 1.00 W41 1.0, 1.0 0.049
V22 1.0, 1.0 0.622 W42 1.0, 1.0 0.955
V23 1.0, 1.0 0.761 W43 1.0, 1.0 0.84
V24 1.0, 1.0 0.761 W44 1.0, 1.0 0.8364
W11 1.0, 1.0 0.84 a1_hid 0.1, 1.0 0.2437
W12 1.0, 1.0 0.44 a2_out 2.0, 3.0 2.1013
W13 1.0, 1.0 0.85 a3_out 1.0, 2.0 2.00
W14 1.0, 1.0 0.35 a4_out 1.0, 2.0 2.00
W21 1.0,1.0 0.71 a5_out 1.0, 2.0 1.744
W22 1.0, 1.0 0.6924 b 0.000001 to 0.0001 0.000023
(a) (b)
Figure 8. Variations of PD controller’s gain values: (a) Kd; (b) Kp for Joint 2.
could be due to the use of a PSO algorithm in place of a
GA. The latter is a potential tool for global search only
but its local search capability is poor, whereas the former
carries out both the global and local searches simultane-
ously. Moreover, PSO is a greedier algorithm compared
to the GA.
4.2. Circular Path Tracking
The end-effector of the manipulator will have to trace a
circular path in Cartesian coordinate system. The corre-
sponding joint angle values are determined using Equa-
tion (2). A second-order polynomial is found to be suit-
Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.443
(a) (b)
Figure 9. Comparisons of the paths tracked by the robot using approaches 1, 2, 3 and 4: (a) entire trajectory; (b) a part of the
trajectory shown on the finer scales.
Figure 10. Variations of percent deviation in prediction of X
values in a cycle for four approaches.
able to represent the trajectory of as
, where 10 = 0.39713758259,
11 = 0.01657138221, 12 = 0.0000095. Similarly,
the trajectory function of 2 has been expressed
using another second-order polynomial
as 20 21
2, where
a20 = 0.92207514343, a21 = 0.00000008916,
a22 = 0.00000000085.

1t
a

2
10 1112
1taatat
 
a

22
taata
 
a
2
t
t
4.2.1. Results of Approach 1
The following GA-parameters (obtained through a care-
ful parametric study as explained earlier) are found to
Figure 11. Variations of percent deviation in prediction of Y
values in a cycle for four approaches.
yield the best results: c = 0.96, m = 0.0055, Popu-
lation size = 180, Maximum number of generations
ma = 240. The optimized values of the parameters:
P P
P
x
Gt
,
ρ, 1
, 1
s
, 2
and 2
s
are seen to be equal to 0.001m,
2600.00 kg/m3, 7.017, 0.02, 10.00, and 0.0116, re-
spectively.
4.2.2. Results of Approach 2
The best results have been obtained with the following
GA-parameters: c = 0.96, m = 0.0055, Population
size = 150, Maximum number of generations max
= 240. The optimized values of the parameters:
P P
PG
t
, ρ,
Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.
444
1
, 1
s
, 2
and 2
s
are seen to be equal to 0.001 m,
2600.00 kg/m3, 7.017, 0.02, 10.00, and 0.0116, re-
spectively. The optimized values of the variables ob-
tained by the GA are shown in Table 3.
4.2.3. Results of Approach 3
The following PSO-parameters (obtained through a
careful parametric study as explained earlier) are found
to yield the best results: = 3; d = 6; number of runs =
100; number of executions = 1000. The optimized values
of the parameters: , ρ, 1
k
t
, 1
s
, 2
and 2
s
are seen
to be equal to 0.001 m, 2600.00 kg/m3, 20.00, 0032556,
10.00 and 0.004085, respectively.
4.2.4. Results of Approach 4
The following PSO-parameters (obtained through a sys-
tematic study as explained earlier) are found to yield the
best results: k = 3; d = 36; number of runs = 100; number
of executions = 7000. The optimized values of the vari-
ables obtained by the PSO algorithm are shown in Table
4.
4.2.5. Comp arisons
Results of the above four approaches have been com-
pared here. The variations of d
K
and
p
K
values for
the joints: 1 and 2 are displayed in Figures 12 and 13,
respectively. The manipulator is found to consume
0.864591, 0.771185, 0.755222, 0.752206 kW of energy
using approaches 1, 2, 3 and 4, respectively. Figure 14(a)
shows the circular path tracked by the manipulator using
the above four approaches. In order to clearly distinguish
the path traced by the robot, a segment of this path has
been displayed in Figure 14(b) The values of percent
deviation in prediction of X- and Y-values have been de-
termined as obtained by the said four approaches and
these are shown in Figures 15 and 16, respectively. Ap-
proaches 1, 2, 3 and 4 have yielded the values of average
absolute percent deviation in prediction of the path as
0.000001103, 0.00000032, 0.00000038, and 0.00000023,
respectively. Once again, approach 4 has proved its su-
premacy over other approaches in terms of accuracy in
path tracking and energy consumption. It has happened
so, due to the reasons explained above.
4.3. Comparisons with Others’ Studies
Path tracking problems of a 2-dof manipulator had been
solved by various researchers. In this connection, the
studies of Qu [1], Homsup and Anderson [2], Kelly and
Salgado [3] and Ouyang and Zhang [4] are worth men-
tioning. Soft computing-based approaches had also been
developed for the said purpose [7,9,10]. This study also
deals with trajectory tracking problems of a manipulator.
In the present paper, an integrated scheme has been de-
Table 3. Optimized values of the variables obtained by the GA in approach 2 for circular path tracking.
Variable Range Optimized value Variable Range Optimized value
t' 0.001, 0.03 0.001 W23 1.0, 1.0 1.00
ρ 2600.0, 2800.0 2600.00 W24 1.0, 1.0 1.00
V11 1.0, 1.0 0.906785 W31 1.0, 1.0 1.00
V12 1.0, 1.0 0.861631 W32 1.0, 1.0 1.00
V13 1.0, 1.0 1.00 W33 1.0, 1.0 1.00
V14 1.0, 1.0 1.00 W34 1.0, 1.0 1.00
V21 1.0, 1.0 1.00 W41 1.0, 1.0 1.00
V22 1.0, 1.0 1.00 W42 1.0, 1.0 1.00
V23 1.0, 1.0 1.00 W43 1.0,1.0 1.00
V24 1.0, 1.0 1.00 W44 1.0, 1.0 1.0
W11 1.0,1.0 1.00 a1_hid 0.1, 1.0 0.1
W12 1.0, 1.0 1.00 a2_out 2.0, 3.0 2.00
W13 1.0,1.0 1.00 a3_out 1.0, 2.0 0.999999
W14 1.0, 1.0 1.00 a4_out 1.0, 2.0 1.00
W21 1.0, 1.0 1.00 a5_out 1.0, 2.0 0.999999
W22 1.0, 1.0 1.00 b 0.000001 to 0.0001 0.000003
Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.445
Table 4. Optimized values of the variables obtained by the PSO in approach 4 for circular path tracking.
Variable Range Optimized value Variable Range Optimized value
t' 0.001, 0.03 0.001 W23 1.0, 1.0 0.99
ρ 2600.0, 2800.0 2600.00 W24 1.0, 1.0 0.99
V11 1.0, 1.0 0.999 W31 1.0, 1.0 1.00
V12 1.0, 1.0 0.98 W32 1.0, 1.0 1.00
V13 1.0, 1.0 0.99 W33 1.0, 1.0 1.00
V14 1.0, 1.0 0.99 W34 1.0, 1.0 0.99
V21 1.0, 1.0 0.99 W41 1.0, 1.0 0.99
V22 1.0, 1.0 0.99 W42 1.0, 1.0 0.99
V23 1.0, 1.0 1.00 W43 1.0,1.0 1.00
V24 1.0, 1.0 0.99 W44 1.0, 1.0 1.00
W11 1.0,1.0 0.99 a1_hid 0.1, 1.0 0.10
W12 1.0, 1.0 0.99 a2_out 2.0, 3.0 2.00
W13 1.0, 1.0 1.0 a3_out 1.0, 2.0 1.00
W14 1.0, 1.0 .0.99 a4_out 1.0, 2.0 1.00
W21 1.0, 1.0 0.99 a5_out 1.0, 2.0 1.00
W22 1.0, 1.0 0.99 b 0.000001 to 0.0001 0.000 0001691
(a) (b)
Figure 12. Variations of PD contr oll er’s gain values: (a) Kd; (b) Kp for Joint 1.
veloped to obtain optimal mechanical structure and PD
controllers simultaneously for a 2-dof manipulator, so
that it can track the trajectories accurately after consum-
ing the minimum power. To the best of the authors’
knowledge, this study is unique for a rigid link manipu-
lator, although an attempt (not exactly the same) was
made by Park and Asada [11] for a non-rigid link ma-
nipulator. It had been reported in Abe et al. [18], Braik et
al. [17], Chen et al. [15] that computational cost of the
PSO is less compared to that of a GA. The performance
of PSO algorithm has been compared with that of a GA,
in the present study.
Copyright © 2011 SciRes. ICA
R. RAJENDRA ET AL.
Copyright © 2011 SciRes. ICA
446
(a) (b)
Figure 13. Variations of PD contr oll er’s gain values: (a) Kd; (b) Kp for Joint 2.
(a) (b)
Figure 14. Comparisons of the circular paths tracked by the robot in approaches 1, 2, 3 and 4: (a) entire trajectory; (b) a
segment of the trajectory shown in finer scales.
5. Conclusions
An integrated scheme for obtaining optimal mechanical
structures and adaptive PD controller for a 2-dof ma-
nipulator has been developed and its performance has
been tested through computer simulations on two trajec-
tory tracking problems. The robot studied in this paper is
a simple one. However, the main strength of this study
lies with the design and development of the above inte-
grated scheme. The robot should be able to track the tra-
jectory accurately, after consuming the minimum power
and ensuring no mechanical failure of the same. Four
approaches have been developed. In approaches 1 and 3,
natural frequency
and stability locus point
s
have
been kept constant throughout the cycle, whereas these
values are selected adaptively in the cycle in approaches
2 and 4. Approach 4 has outperformed other three ap-
proaches in terms of both power consumption as well as
accuracy in trajectory tracking due to the reasons ex-
plained earlier. Moreover, approach 4 has yielded a more
stable system compared to other approaches. The better
performance of the PSO algorithm than that of the GA
R. RAJENDRA ET AL.447
Figure 15. Variations of percent deviation in prediction of X
values in a cycle for four approaches.
Figure 16. Variations of percent deviation in prediction of Y
values in a cycle for four approaches.
could be due to its inherent ability to carry out the global
and local searches simultaneously. On the other hand, the
GA is a potential tool for global search, although it may
not be so much powerful in local search.
6. Scope for Future Work
In the present study, the performances of developed ap-
proaches have been tested through computer simulations.
However, it will be more interesting to test their per-
formances in real-experiments. An improved version of
PSO algorithm [37,38] may also be used in future to
solve the said problem. The authors are working on these
issues.
7. Acknowledgements
Rega Rajendra thanks financial support of the All India
Council of Technical Education (AICTE), New Delhi,
India, under the Quality Improvement Programme (QIP)
Scheme, to carry out this study.
8. References
[1] Z. Qu, “Global Stability of Trajectory Tracking of Robot
under PD Control,” Dynamics and Control, Vol. 4, No. 1,
1994, pp. 59-71. doi:10.1007/BF02115739
[2] W. Homsup and J. N. Anderson, “PD Control Perform-
ance of Robotic Mechanisms,” Proceedings of American
Control Conference, Minneapolis, 10-12 June 1987, pp.
472-475.
[3] R. Kelly and R. Salgado, “PD Control with Computed
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R. RAJENDRA ET AL.449
List of Symbols and Abbreviated Terms
10 14
,,aa Coefficients of fourth-order polynomial
20 32
,,aa Coefficients of cubic polynomial
1_
ahid Coefficient of transfer function of hidden
neurons
2_
aout Coefficient of transfer function for first
output neuron
3_
aout Coefficient of transfer function for sec-
ond output neuron
4_
aout Coefficient of transfer function for third
output neuron
5_
aout Coefficient of transfer function for fourth
output neuron
b Bias value
12
,bb
B
Acceleration coefficients
Effective damping value
12
,cc
d
Gravity terms of torque
Dimension of search space
D Inertia term of torque

Ds
E
Disturbance on the system
Energy, kW
E Elastic Modulus, N/m2
total
Err
Err
Total error
avg
Fitness
Average error
f
g
Acceleration due to gravity, m/s2
Gbest Swarm global best solution
max
Dynamic Coriolis term
G
h
Maximum number of generations
I
Moment of inertia, kg.m2
k Number of particles interacting with each
particle
jterr Joint error
J
Inertia of links, motor and gear, kg·m2
d
K
Derivative gain
p
K
Proportional gain
12
L
m
,L Length of the links, m
Mass of the link, kg
1
M
Bending moment of first link
M
Moment, N-m
c
P
P
Probability of crossover
i ith Particle best fitness
g
P Swarm’s global best fitness
m
P
P
Probability of mutation
Population size
P
Penalty term
Pbest Particle best solution
R Outer radius of hollow circular link
r Inner radius of hollow circular link
s
Stability locus point
t Time, s
t
Thickness, m
vid Particle’s updated velocity in dth dimen-
sion
xid Particle’s updated position in dth dimen-
sion
w Uniformly distributed load, N/m
U(s) Control input
Vi Velocity Vector
V Connecting weights between input and
hidden layers
W Connecting weights between hidden and
output layers
W Inertia weight
1
W
W
Concentrated load acting on first link, N
2 Concentrated load acting on second link,
N
2m
W Weight of second motor
i
X
Position Vector
,
ii
X
Y
,
Coordinates of initial position
f
f
X
Y, Coordinates of final position
Damping ratio
s
System response in Laplace Transform
Joint angle, rad
s
Angle in Laplace domain, rad
Density, kg/m3
y
Yeild Stress, N/m2
Torque, N-m
Natural frequency
s
Closed-loop characteristic polynomial
GA Genetic Algorithm
PD Proportional Derivative
PID Proportional Integral Derivative
PSO Particle Swarm Optimization
SBX Simulated binary crossover
Rand( ) Random number generator in the range
of (0,1)
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