Improved Particle Swarm Optimization

978-1-5090-2084-3/16/$/31.00©2016 IEEE
An Improved Particle Swarm Optimization for
Multi-Robot Path Planning
Abstract — This paper proposes a new methodology to
optimize trajectory of the path for multi-robots using
Improved particle swarm optimization Algorithm (IPSO) in
clutter Environment. IPSO technique is incorporated into
the multi-robot system in a dynamic framework, which will
provide robust performance, self-deterministic cooperation,
and coping with an inhospitable environment. The robots on
the team make independent decisions, coordinate, and
cooperate with each other to accomplish a common goal
using the developed IPSO. A path planning scheme has been
developed using IPSO to optimally obtain the succeeding
positions of the robots from the existing position in the
proposed environment. Finally, the analytical and
experimental result of the multi-robot path planning were
compared with those obtained by IPSO, PSO and DE
(Differential Evolution) in a similar environment. The
simulation and the Khepera environment result show
outperforms of IPSO as compared to PSO and DE with
respect to the average total trajectory path deviation and
average uncovered trajectory target distance.
Keywords— Particle swarm optimization, multi-robot path
planning, average total trajectory path deviation, average
uncovered trajectory target distance, average path Length.
I. INTRODUCTION
Particle swarm optimization Algorithm (PSO) is effective
and efficient using an alternative approach to the multi robot
path planning. Although many algorithms [27] [28] have
been proposed and proven to be feasible and efficient for
robot motion planning and collision avoidance, classic
techniques for path planning problem [12] [25] are general
methods like Roadmap, Cell Decomposition, Potential
Fields, optical tweezers and Mathematical Programming.
Many authors have proposed multi-robot and the single
robot path planning problems using different classical
techniques [1] [5], Neural Network [3], artificial Immune
system [4] [11] and heuristic optimization algorithms [13]
[16] [17] [18] [19]. High time complexity in large problem
spaces and trapping in local optimum are drawbacks for
classic techniques and in many meta-heuristic algorithms.
These drawbacks cause the classical techniques and
inefficient in the various problem spaces. To improve the
efficiency of classical methods, probabilistic algorithms like
PRM and RRT are proposed for improving the local
optimization problem, many evolutionary algorithms like
Genetic Algorithms[27][2], PSO [24] [26], Bee colony
Optimization [14] and Differential evolution algorithm [15]
gravitational search algorithm[20] [27]are used in multi-
robot path planning problem. The fitness function of the
GSA [27] has two main components: first one is the
objective function describing the selection of next position
on an optimal trajectory based on velocity, and the second
one is the constraint on acceleration representing avoidance
of collision with other robots and with static obstacles.
PSO is a heuristic algorithm [7] [9] [10] is a recent
algorithm that has been inspired by the Newtonian’s law of
gravity and motion. GSA has undergone a lot of changes to
the algorithm itself and has been applied in various
applications. At present, there are various variants of GSA
[20] [21] [22] [23] which have been developed to enhance
and improve the original version. The algorithm has also
been explored in many areas [8] [9].
For realization multi robot path planning problem with
different goal of the respective robots with PSO [29-33] by
the considering a fitness function is constructed to determine
the next position of the robots that lie on optimal trajectories
leading towards the respective goals. Path-planning problem
considered here is formulated by a centralized approach,
where an iterative algorithm is invoked to determine the
next position of all the robots satisfying all the constraints
imposed on the multi-objective function. The algorithm is
iterated until all the robots reach their destination (goal
position).
In this paper, the implementation of the modified particle
swarm optimization algorithm has been enhanced to
determine the trajectory of path for multiple robots from
predefined initial positions to predefine target positions in
the environment with an objective to minimize the path
length of all the robots. The result shows that the algorithm
can improve the solution quality in a reasonable amount of
time. This paper improves the particle swarm optimization
algorithm (IPSO) for improving the global path planning
problem of the multi-robots by improving the convergence
rate. Finally, the efficiency of the IPSO has been proved
through the simulation as well as Khepera environment and
a result is compared with other evolutionary computing such
an PSO and DE.
P.K Das1
, B.M. Sahoo2
, H.S.Behera3
, S Vashisht4
1,3
Dept. of Comp.Sc. &Engg. and Information Technology,VSSUT,Burla,Odisha,India
2,4
Dept. of Comp. Sc. & Engg., Amity University, Greater Noida
Email: 1
daspradipta78@gmail.com, 2
biswamohans@gmail.com, 3
hsbehera_india@yahoo.com,
4
svashisht@gn.amity.edu
2016 1st International Conference on Innovation and Challenges in Cyber Security (ICICCS 2016)
97

The rest of the paper is outlined as follows. Section II
briefly describes the improved particle swarm optimization
Algorithm. Formulation of the problem for multi-robot path
planning has been elaborated in the section III. Multi-
objective optimization problem solving using Improved
IPSO is described in detail in section IV. Section V
demonstrated the result of path planning for multi-robot
through simulation. In section VI, the experiment has been
conducted in Khepera II environment and finally,
conclusions of the work is presented in the section VII.
II. Particle swarm optimization (PSO)
A. Classical PSO (CPSO)
The CPSO is a stochastic population based, bio-inspired
evolutionary optimization algorithm, which was originally
introduced by Kennedy and Eberhart (1995), which utilizes
swarm intelligence to achieve the goal of optimization. It is
based on intelligent collective behavior of schools fish or
bird flocks. In the classical PSO algorithm, each member of
the population is known as a particle in a D-dimensional
search and set of particles is called a swarm. The velocity
parameter of the CPSO is dynamically updated by the
particles own experience and flying experience of its
accompaniment. The members of the entire population share
the information among individual to change each particle
position to find the best position in the search space. The
advantage of the CPSO over other optimization algorithm is
easy to implement and few parameters are to be adjusted.
Let N be the population size. In each generation k, the
velocity and position of the particles are updated using Eq
(1).
)1()()1(
))(.(
2
.
2
))(.(
1
.
1
)()1(
td
iVtd
ixtd
ix
td
ixd
gbest
xCtd
ix
i
d
pbest
xCtd
iVtd
iV
(1)
Where, ),......
2
,
1
(
iD
x
i
x
i
xix represent the current
position vector of the particle i )1( Ni in a D –
dimensional search space, ),......
2
,
1
(
iD
V
i
V
i
ViV represent
the velocity of the th
i particle,
1
C ( )0
1
C and
2
C ( )0
2
C
are the acceleration constants,
1
and
2
are two random
numbers in the range [0, 1].
pbest
x is the previous best
position of the thi particle in generation k,
gbest
x is the
previous global best position among all the particles in
generation k . If 0
1
C , then PSO algorithm is converted to
social-only model. Similarly, if 0
2
C , then it becomes a
cognition-only model.
B. Improved particle swarm optimization
To bring a balance between the exploration and exploitation
characteristics of PSO, Shi and Eberhated proposed a PSO
on inertia weight in which velocity of each particle is
updated and claimed that a larger the value of the inertia
weight will be provided a global search, while smaller the
value will be provided local search. So, it needs to change
the inertia weight dynamically to adjust the search capability
dynamically. Therefore, there are several proposals to
modify the PSO algorithm by changing the inertia weight
value in an adaptive manner in each iteration. In this paper,
we have improved PSO (IPSO) in the terms of adaptive
weight adjustment and acceleration coefficients to increase
the convergence rate to optimum value in PSO, the classical
PSO equation modified according to the following form.
(3))1()()1(
(2)))()((
3
.
3
))(.(
2
.
2
))(.(
1
.
1
)()1(
td
iVtd
ixtd
ix
td
ixt
i
PC
td
ixd
gbest
xCtd
ixd
i
pbest
xCtd
iV
i
wtd
iV
Here, )(t
i
P is the position the robot, where no obstacle is
present.
3
C is the object sensitive co-efficient and
2
is
random number in the range [0, 1]. Each robot equipped
with sensors, if the robot not sense the obstacle with in the
radius r, then 0
3
C
The local best value in IPSO can be computed as:
))(())1((),1(
))(())1((),(
)1(
tipbest
bj
Otix
bj
Oiftix
tipbest
bj
Otix
bj
Oiftipbest
tipbest ( 4)
Where
bj
O stands for the fitness function of the moving
particles and the global best position is obtained as:
))(()),......(
2
()),(
1
(min)( t
N
pbest
bj
Otpbest
bj
Otpbest
bj
Otgbest
(5)
The convergence rate of the PSO has been improved by fine
tuning of its parameter with the help of several techniques.
These techniques usually change the PSO update equations
without altering the inherent structure of the algorithm. The
velocity during the previous time step is scale it by a scale
factor inertia weight (w) to update a new velocity every
98

time as the particle moves the search space. The large value
of the inertia weight provides the global search where as the
small value of the inertia weight makes a local search. The
search ability is adjusted dynamically by dynamic change of
inertia weight. Here, the adaptive change of inertial weight
proposed in [12] is used and presented in the Eq.(6).
Empirical experiments have been performed in the past with
an inertia weight varies in [0.9, 0.4].
Finally the value of inertia weight which is used in Eq. (2)
can be determined by Eq. (5):
)( s
i
kiniw
i
ziw (6)
Where )(t
i
k is the evolutionary speed factor for each robot
is determined in Eq.(6) and store history of each robot.
iniw is the initial value for ineria weight and value of the
and is with in range of [0,1]. In the experiment, we
have choosen for better result.
i
z is the coefficient for
inertia weight. this term helps a better adaptation to robotic
domain which define in Eq(8).
))(()),1((max(
))(()),1((min(
1)(
t
i
pbestfitt
i
pbestfit
t
i
pbestfitt
i
pbestfit
t
i
k (7)
Where ))(( t
i
pbestfit is the best fitness value of )(t
i
pbest at
t iteration for ith
robot and value of )(t
i
k lies in between
[0,1]. The larger the value )(t
i
k , gets faster the evaluation
speed. If ,0)(t
i
k indicated the robot has no evaluation
after this iteration, which cause no changes in
))(( t
i
pbestF .
ttfi
tbest
F
ttfi
tbest
fit
s
),max(
),min(
(8)
Here, ttfi is the mean fitness of all robots and
tbest
F is the
best fitness at tth
iteration. The value of s lies in [0.1]. the
larger the value of s indicate robots are closer to each other.
otherwise,
robotbesttheisiif
iter
t
LUU
t
i
z
1
,
max
)(
)( (9)
Here, LU, are the upper and lower bounds for )(t
i
z ,
which is 2 and 0.5 respectively and maxiter is the maximum
iteration.
Similarly, the fixed value set for the acceleration
coefficients (conventionally fixed to 2.0). With the large
value of the social component
2
C in comparison with a
cognitive component
1
C leads particles to a local optimum
prematurely and relatively high value of cognitive
components results to wander the particles around the search
space. The quality of the solution quality is improved by
modifying cognitive and social coefficient term in such a
way that the cognitive component is reduced and social
component is increased as generation proceeds. The
modification of the coefficients are made (for tth
generation)
using the following Eq. (9) and Eq(10).
t
IterMax
f
C
i
C
i
CC
_
11
11
(9)
t
IterMax
i
C
f
C
i
CC
_
22
22
(10)
Where,
f
C
i
C
f
C
i
C
2
and
2
,
1
,
1
are initial and final values
of cognitive and social component acceleration factors
respectively and Max_Iter is the maximum number of
allowable iterations.
The accelerating and inertial weights are the major factor in
the numerical results [12, 23] to improve the accuracy. The
algorithm ability has improved by combining PSO with
other search techniques [26]. To escape from local minima
and increase the diversity of population, PSO combine with
some evolutionary operators [28] such as crossover,
mutation and selection. Hence, to improve version of PSO
is used in this paper.
III. PROBLEM FORMULATION FOR MULTI ROBOT
PATH PLANNING
The problem formulation for multi-robot path planning is to
determine the next position of the robot from their existing
positions in its workspace by avoiding the collision with
other robots and obstacles (which are static in nature) in its
path to reach at the goal. Multi-robot path planning problem
is formulated by considering the set of principles using the
following assumptions by a uniform treatment.
Assumptions
a. For each robot the Current position (recent position)
and goal position (target position) is known in a given
reference coordinate system.
b. The robot can select any action in a given time from a
fixed set of actions for its motions
99

c. Each robot is performing its action in steps until all
robots reached in their respective target positions.
The following principles have been taken care for satisfying
the given assumptions.
1. For determining the next position from its current
position, the robot tries to align its heading
direction towards the goal.
2. The alignment may cause a collision with the
robots/obstacles (which are static in nature) in the
environment, hence, the robot has to turn it
heading direction left or right by a prescribed
angle to determine its next position
3. If a robot can align itself with a goal without
collision, then, it will move to that determine the
position.
4. If rotating the heading direction left or right
requires the same angle of rotation of the robot
about the z-axis, if it is tied then, broken
randomly.
Consider the initial position of the ith
robot at a time t is
( )(t
i
x , )(t
i
y ), the next position of the same robot at a time
( )tt is ( )( ttxi , )( ttyi
), )(tvi is the velocity of
the robot iR and ( goal
ix , goal
iy ) is the target or goal position
of the robot iR
)1(tyi
Fig. 1: Representation of next position from
current position of the i-th robot.
Fig. 2: Selection of next position
))(),(( ttyttx ii
from the current position
))(),(( tytx ii
to avoid collision with obstacles.
So, the expression for the next
position ))(),(( ttyttx ii
can be derived from the Fig.
1 as follow
)( ttxi = )(t
i
x + t
i
tvi
cos)( (11)
tit
i
vt
i
ytt
i
y sin)()()( (12)
When t =1, the equation (11) and (12) is reduced to
)1(txi = )(t
i
x +
i
tvi
cos)( (13)
it
i
vt
i
yt
i
y sin)()()1( (14)
Consider initially, the robot iR is placed in the location at
( )(t
i
x , )(t
i
y ). We want to find the next position of the
robot ( )1(txi , )1(tyi
), such that the line joining
between{ )(t
i
x , )(t
i
y ),( )1(txi , )1(tyi
)}and{( )1(txi
, )1(tyi
) , ( goal
ix , goal
iy ) } should not touch the obstacle
in the world map is represented in the Fig. 2 and minimizes
the total path length from current position to a goal position
without touching the obstacle by forming constraint. Then
objective function 1
fit that determines the length of the
trajectory for n number of robots,
n
i goal
iyt
i
y
goal
ixt
i
x
t
i
yt
i
yt
i
xt
i
x
fit
1 })2))1((2))1(((
)2)1()((2))1()((({
1 (15)
By putting the value )1(txi and )1(tyi
from the
expressions (13) and (14) into the expression (15), we
obtain,
n
i goal
iyit
i
vt
i
y
goal
ixit
i
vt
i
x
t
i
vfit
1 2)sin)()((
2)cos)()((
)(
1
(16)
The multi-robot path-planning is now represented as an
optimization problem by minimizing the objective function
in Eq.20 with considering the penalty function as the
constraints in the objective function. Minimizing the
objective function in Eq 20 shows that the robot will follow
the shortest distance from the initial point to target point.
The constraints here are two types of penalty. The first
penalty is to avoid collision between teammates (any two
mobile robots), whereas the second penalty is to avoid
collision of a robot with a static obstacle. By combining
these two penalties a linear fuzzy function is developed for
evaluating the obstacle present in the path. So, the objective
function formed based on the fuzzy function is denoted
as jfit .
jOdjd(O
jOdjd(OjOd
jOdjOd
jOdjd(O
jOdjd(O
j
fit
max)()0
max)()
min
)()
min
)(max)(
min
)()
.exp(
min
)()1
(17)
X
Y
))(),(( tytx ii
))(),(( ttyttx ii
Obstacle
X
Y
)(t
i
x
)(t
i
y
)1(txi
))(),(( tytx ii
)(tvi
i
100

Where is positive constant, )jd(O be the distance
between mobile robot and obstacles, max)( jOd is maximum
distance and min)( jOd is the minimum distance with
respect to the obstacle jO . The path is safe and collision
free path, when max)() jj Odd(O and path is unsafe if,
min)() jj Odd(O
Thus, mathematically, the optimization problem for
obstacles can be formulated as follows.
)(
,...2,1
max
2 j
fit
Nj
fit (18)
Thus, the optimization problem can be represented as
follows:
2
1 fit
fitfit (19)
Here, is positive constant. The above optimization
problem is to minimize the Euclidean distance between the
current position and their goal position is presented by the
objective function
1
fit and second objective function is a
constraint to find the safe path.
IV. MULTI-OBJECTIVE OPTIMIZATION PROBLEM
SOLVING USING IGSA
In this section, Multi-robot path planning algorithm has
been proposed using IGSA. The proposed IGSA algorithm
is used to evaluate the next positions of every robot by
presuming the current positions of robots and speeds as the
parameter for optimizing the given multi-objective function.
It determines the optimized path from each state to the goal
state in a dynamic as well as static environment and robot
measures its distance to obstacles with the help of equipping
sensors.
The agents are moving in the search space with the help of
the gravity is considered in the proposed IGSA based path
planning. The outline of the proposed algorithm is discussed
below:
Procedure IPSO ( icurricurr yx __ , , pos-vector)
Begin
Initialize positions of N agents randomly in the population
and parameters such
f
C
i
C
f
C
i
C
2
and
2
,
1
,
1
For d= 1 to no_of _dimension
For iter = 1 to Maxiter do
Begin
Repeat
For each agent i= 1 to m
Begin
a. Evaluate the fitness of each agent
using (19)
b. If (fiti>fit(pbesti) then
i. fit(pbesti)=fiti
ii. pbesti=current position of i
end if
c. If (fit(pbesti)>fit(gbest) then
i. fit(gbesti)=fit(pbesti)
ii. gbest=pbesti
end if
end for
For each agent i= 1 to m
a. Calculate k by (6)
b. Calculate s by (7)
c. Calculate )(tzi
by (8)
d. Update inertia weight by (5)
e. Update the velocity and position by (2)
End for
Until stoping condition is true
End for
End for
Update :
iivicurryicurry
iivicurrxicurrx
sin
cos
Return
End
Pseudo Code for path planning
Input: ( )(t
i
x , )(t
i
y ),( goal
ix , goal
iy ) and )(tiv are the initial
position, goal position and velocity for n robots, ni1
respectively and is the threshold value.
Output: Trajectory of path iTP for each robot iR from
( )(t
i
x , )(t
i
y ) to ( goal
ix , goal
iy )
Begin
For i = 1 to n
)(_ tixicurrx ; )(_ tiyicurry ;
End for
Repeat
For robot 1i to n
Call IPSO ( icurricurr yx __ , , pvector);
// pvector is the position vector denotes updated current
position of the i-th robot //
End for;
For 1i to n
Move-to ( icurricurr yx __ , );
End for;
Until iGicurr_
// icurr_ = ( icurricurr yx __ , ), Gi = ( goal
ix , goal
iy )//
End.
V. COMPUTER SIMULATION
The multi-robot path-planning algorithm is implemented in
a simulated environment. The simulation is conducted in a C
environment on a Pentium processor and the experiment
was performed with 14 robots of circular shape. The radius
of each robot is considered as 6 pixels. Before initiating the
experiment for multi-robot path planning, each robot
starting and goal points are predefined. The experiments
were performed with seven differently shaped obstacles and
with equal velocities for all the robots in a given run of the
program; however, the velocities were adjusted over
different runs of the same program. One of our experimental
world-maps with an initial configuration of the world-map
with 7 obstacles and the starting and the goal positions of 10
circular soft-bots is shown in Fig. 3. The intermediate steps
of movement of the robots are shown in Fig. 4 and Fig. 5.
101

The final stage of world maps, where all the robots reached
in their predefine goal respectively is shown in Fig. 6.
The experiment has been conducted using a central version
of the algorithm using the fitness function (19) for deciding
the next position of the robot. In our experiment, the
following parameters are used in the simulation and
Khepera II environment:
Table 1: Parameter used in the Simulation and Khepera
i
C
1
0.5
f
C
1
0.5
i
C
2
2.0
f
C
2
2.0
Maxiter 30
Wini 0.4
2.0
A. Average total trajectory path Distance (ATTPD)
Consider a trajectory path from the predefine starting point
kS to the goal point kG generated by the program for
robot kR in the th
j run is kjTP . If 1kTP , 2kTP ,….., kjTP are
the trajectory paths generated over th
j runs, then the
average total trajectory path traversed (ATTPT) by a robot
kR is given by jTP
j
r
ir /
1
and the average trajectory path
distance for this robot is evaluated by measuring the
difference between ATPT and the ideal shortest path
between kS to kG .If the ideal trajectory path for robot
kR obtained geometrically is realkTP , then the average
path distance is given by
jPTP
j
r
irrealk /
1
.
Therefore, for n robots in the workspace the average total
path distance (ATPD) is
n
i
j
r
irrealk jPTP
1 1
)/(
B. Average Uncovered trajectory Target Distance
(AUTTD)
Given a goal position kG and the current position kC of a
robot on a 2-dimensional workspace, where kG of kC are 2-
dimensional vectors, the uncovered trajectory distance for
the robot k is kk CG , where . denotes Euclidean
norm. For n robots, uncovered trajectory target distance
(UTTD) is UTTD =
n
i
kk CG
1
. For k runs of the
program, we evaluate the average of UTTDs, and call it the
average uncovered/untraveled trajectory target distance
(AUTTD). Fig. 15 shows that by decreasing the velocity,
AUTTD takes longer time to converge and gradually
terminated with iteration. Again, it is noted that larger the
velocity of the robot, the faster falloff in the AUTTD. Fig.
16 shows that, larger the number of robots, slower the
convergence rate. Slower the convergence causes the delay
in falloff in AUTTD.
The performance analysis was undertaken in the simulation
environment and the ATPT was ploted for n robots, called
average total trajectory path travelled (ATTPT)by varying
no of robots 1 to 5 presented in the Fig. 17 and generate
paths using three algorithms, including real-coded DE,PSO
and IPSO. It is noteworthy from the Fig.16 that IPSO
possess least ATTPT in comparison to the algorithms
irrespective of the number of robots.
The performance analysis has been performed in terms of
AUTTD over the number of steps in Fig. 18. It provides
graphs between AUTTD verse no stages required during the
planning of path using three algorithms with number of
obstacles=7 and no of robots = 5. It is apparent from Fig.18
that AUTTD returns the smallest value for IGSA
irrespective of the number of planning steps.
The performance of the result has been analyzed by plotting
the average total trajectory path deviation(ATTPD) with the
number of robots as variable in Fig 19. This path is
generated by three different Evolutionary algorithms such as
PSO, DE, IPSO. Fig.19 shows the result of ATTPD
computation, when the number of robots varies between 1 to
5. Here, we observed that IPSO performs better than the
other two algorithms as ATTPD is smallest for IPSO in
comparison to other two algorithms irrespective of the
number of robots.
Again, the performance analysis was undertaken by
comparing the running time over the maximum number of
iterations using three algorithms. Fig. 20 provides the time
required for robots to reach in their respective goal position
by three different algorithms and it shows that IPSO takes
less time for robots to reach in destination.
Finally, the performance has analyzed in the terms of
number of stepts to gaoal with the maximum number of
iterations in Fig. 21. This figure indicates the steps to
travelled in IPSO is less then other metaherustic algorithm
with irrespective of the robots.
The comaprion result of simulation and experimental result
for two number of robots has elaborated in Table 2. This
indicates that IPSO takes less error in than other algorithms.
Fig.3: Initial world map with 7 obstacles and 5 robots
Parameters Values
102

Fig.4: Intermediate state of the world map during execution using
IPSO for 5 robots and 7 obstacles after 9 steps
Fig.5 Intermediate stage of the world map during execution of
IPSO for 5 robots and 7 obstacles after 17 steps
Fig.6. Five robots reached in their respective pre-defined goal.
Fig. 7 : All robots reached in their respective pre-defined
goal by PSO
Fig. 8 : All robots reached in their respective pre-defined
goal by DE
VI. EXPERIMENT ON KHEPERA II ROBOT
Khepera II (Fig. 9) is a miniature robot (diameter of 8cm)
equipped with 8 built-in infrared range and light sensors,
and 2 relatively accurate encoders for the two motors. The
range sensors are positioned at fixed angles and have
limited range detection capabilities. The sensors are
numbered clockwise from the leftmost sensor 0 to sensor
7 and its internal structure (Fig. 10). Sensor values are
numerical ranging from 0 (for distance > 5 cm) to 1023
(approximately 2 cm).The on board microprocessor has a
flash memory size of 256KB, and the CPU of 8 MHz.
Khepera can be used on a desk, connected to a
workstation through a wired serial link. This
configuration allows an optional experimental
configuration with everything at hand: the robot, the
environment and the host computer. The Khepera II
network and its accessories are presented in the Fig. 11
for the conduct of experiments.
Fig.9. The Khepera II Robot.
Fig.11. Position of sensors and and internal structure of
Khepera II
103

Fig.11 Khepera network and its accessories
IPSO is implemented in the Khepera-II robot with
considering two robots and compared with a different
evolutionary computing algorithm is demonstrated in the
Fig.13. It shows better convergence in comparing to the
other meta-heuristic algorithm presented in the Fig. 13.
Fig.12: khepera environment setup for multi-robot path planning
Fig 13. Snapshot of Intermediate stage of the multi - robot path
planning using IGSA in Khepera environment.
The initial world map for conducting the experiment in the
Khepera II is presented in the Fig. 11 to 8 obstacles of
different shape and predefine initial state and goal is marked
on the map, where different meta heuristic algorithm is
applied. Fig. 12 shows the intermediate moment of the robot
in the trajectory path towards the goal by respective robot
using IGSA. Finally, different matter-heuristic algorithm is
applied in Khepera environment and result of the trajectory
path is presented in the Fig. 13.
Fig 14. Optimal path representation of different algorithm for
multi-robot path planning in khepera environment is represented
by different color code
Fig. 15. Average Uncovered Trajectory Distance vs.
Number of stages with variable velocity for fixed number
of obstacles=7
Fig.16. Average Uncovered trajectory Distance vs. Number
of stages with variable number of robots for fixed number of
obstacles=7(constant)
104

Fig..17. Average total trajectory path Traversed vs. Number
of robots with variable number of obstacles for fixed
velocity
Fig. 18. Average untraveled Trajectory target distance vs.
Number of steps to goal in different algorithms
Fig. 20 : Run Time Vs Maximum Iteration for different
algorithms
Fig. 21 : Number of steps to Goal Vs Maximum Iteration
for different algorithms
Fig.19 :Average total trajectory path deviation vs No. of
robots for algorithm with fixed no. of
obstacles=7
Table 2: Comparison of Simulation result with experimental Result for Two number robots in different algorithm
VII. CONCLUSION AND FUTURE WORKS
An improved gravitational search algorithm was proposed
for trajectory path planning of multi-robots in order to
find collision free smoothness optimal path from
predefine start position to end position for each robot in
the environment. The results obtained from the
experimental work performs better compared with
proposed algorithm. Comparisons of the performances
among different techniques have been carried out. From
the simulation and khepera-II environment, it is observed
that the IPSO technique is best over other technique for
navigation of multi- mobile robot. However, in this paper,
both the environment and obstacles are static relative to
the robots; where as other robots are dynamic for priority
Algorithm Path travelled(in pixel)
In
simulation
In Experimental
model
% error in simulation
and experimental
IPSO 430 478 10.04%
PSO 580 659 12%
DE 610 709 14%
105

robots. In future, work will be carried out using dynamic
obstacles other than robots such as running vehicle,
animals and onboard camera during the mult-robot path
planning.
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