PATH FINDING FOR A MOBILE ROBOT USING FUZZY … · mobile robot using Fuzzy Logic (FL) and Genetic...

http://www.iaeme.com/IJMET/index.asp 659 [email protected]

International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 8, August 2017, pp. 659–669, Article ID: IJMET_08_08_072

Available online at http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=8

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication Scopus Indexed

PATH FINDING FOR A MOBILE ROBOT USING

FUZZY AND GENETIC ALGORITHMS

Rame Likaj

University of Prishtina / Faculty of Mechanical Engineering,

Prishtina, Kosovo

Xhevahir Bajrami*

University of Prizrena / Faculty of Computer Science,

Prizren, Kosovo. *Correspondence

Ahmet Shala*, Arbnor Pajaziti*

University of Prishtina / Faculty of Mechanical Engineering,

Prishtina, Kosovo. *Correspondence

ABSTRACT

In this paper is presented a new algorithm for global path planning to a goal for a

mobile robot using Fuzzy Logic (FL) and Genetic Algorithm (GA). We have also

shown the detection and avoidance of static and dynamic obstacles in a closed

environment using a single camera to record the state of environment in which the

mobile robot operates. This is done by mapping the environment and optimization

using genetic algorithm. During this process for the mobile robot steps generated by

Fuzzy logic algorithm have been taken, in order to reach the target. Locations of

target and obstacles to find an optimal path are given in a 2-D workplace

environment. Besides, optimization steps of path finding; also we did not use real

sensors for avoiding obstacle. This problem is solved by using virtual sensors to

achieve fast detection and response. For the given mobile robot model, the results of

simulation have been also shown by figures.

Keywords: mobile robot, genetic algorithm, fuzzy logic, camera, virtual sensors,

kinematics, dynamics.

Cite this Article: Rame Likaj, Xhevahir Bajrami*, Ahmet Shala* and Arbnor

Pajaziti*, Path Finding for A Mobile Robot Using Fuzzy and Genetic Algorithms,

International Journal of Mechanical Engineering and Technology 8(8), 2017,

pp. 659–669.

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=8

1. INTRODUCTION

A mobile robot must be able to move in an environment avoiding obstacles in its way and in

the same time must be able to find the shortest path, the smoothest path or the safest path.

Path finding and obstacle avoidance from a mobile robot by using different sensors presents a

Path Finding for A Mobile Robot Using Fuzzy and Genetic Algorithms


problem and there is always room for improvement. In this context several methods of

solutions like different predictable methods [19], global C-space methods [20], potential field

methods and neural networks approaches have been explored. Another method for controlling

mobile robot movement in an environment with static and dynamic obstacles is Genetic

Algorithm, which is proposed at the beginning of 1975 [21]. Fuzzy logic is one of the

algorithms that can be used for path finding in a closed environment as well [22]. In this paper

is presented a method for path finding by making the combination of Genetic Algorithm with

Fuzzy logic outputs and using a camera as a single sensor [23]. In our case to detect position

of the mobile robot we can use several image processing methods taken by a camera sensor.

After detection of the mobile robot position, this position is used as the starting point of the

path, needed for path finding. In the same way, we detect all needed information for the

mobile robot to reach the target.

2. MODELING – KINEMATICS AND DYNAMICS OF MOBILE ROBOT

Kinematics model analysis of vehicle are derived on the basis of known linear velocity of the

vehicle, point B: x�, y�, respectively v� = v , and steering angle of front wheels: ϕ

respectively ∅ [1]. From the non–holonomic constraints follows that the mobile robot moves

in the direction normal to the axis of the driving wheels (rear axle) [2], 4], and it is assumed

that no slipping occurs [6].

From the condition of no slipping towards the rear wheel axis CD at point B (Fig. 1)

follows:

0)cos()sin( =⋅−⋅ θθ BB yx && (1)

From the condition of no-slipping towards the front wheel axis EF at point A (Fig. 1)

follows:

0)cos()sin( =+⋅−+⋅ φθφθ AA yx && (2)

Further follows:

})sin(2)cos(2 θθ ⋅+=⋅+= byybxx BABA (3)

})cos(2)sin(2 θθθθ ⋅⋅+=⋅⋅−= &&&&&& byybxx BABA (4)

⋅+⋅=

=⋅⋅−+⋅−+⋅

=+⋅⋅⋅−+⋅−+⋅⋅⋅−+⋅

)sin()cos(

0)cos(2)cos()sin(

0)cos()cos(2)cos()sin()sin(2)sin(

θθ

φθφθφθ

φθθθφθφθθθφθ

BBB

BB

BB

yxv

byx

bybx

&&

&&&

&&&&

(5)

yB

xB

x

Pv1

Pv2

A

B

E

F

θ

φ

a

Pv

D

C

θ

φ

θ

θ

φ

φ

φ

θ

θ&

mpa

P

a

b

b

θ

φ

v

y

O

Rmin

Figure 1 Position analysis and kinematics model of mobile robot.

Rame Likaj, Xhevahir Bajrami, Ahmet Shala and Arbnor Pajaziti


The non–holonomic constraint states that the vehicle can only move in the direction

normal to the axis of the driving wheels (rear axle) i.e., the vehicle platform satisfies the

conditions of pure rolling and non-slipping in the matrix form:

0)( =⋅ ssC & (6)

Where, [ ]TBB yxs θ= and [ ]TBB yxs θ&&&& = - represents general variables, and finally:

The solution of equation (1) actually represents the inverse kinematics of mobile robot.

From Figure 1 is performed calculation of the following expressions:

)tan()tan(

φθθφ b

vbvvB =⇒⋅== &&

(7)

})sin()cos( θθ ⋅=⋅= vyvx BB&&

(8)

From the equation (7) is calculated angle :

( )BB xya && /tan=θ (9)

And, from equation (6) is calculated angle ϕ

( )vba /tan θφ &⋅= (10)

In Figure 2 is shown the implementation of MATLAB model of the inverse kinematics for

mobile robot.

Figure 2 MATLAB Model-solution for inverse kinematic of mobile robot.

Figure 3 Coordinates of point B in X axle-direction. Figure 4 Coordinates of point B in Y

axle-direction.

Figure 5 Velocity vx of point B. Figure 6 Velocity vy of point B.

0 0.5 1 1.5 2 2.5 3 3.50

10

20

30

40

Time (Seconds)

x(t

)

0 0.5 1 1.5 2 2.5 3 3.5

-10

-5

0

5

10

Time (Seconds)

y(t

)

0 0.5 1 1.5 2 2.5 3 3.5

8

10

12

14

16

Time (Seconds)

vx

(t)

0 0.5 1 1.5 2 2.5 3 3.5-15

-10

-5

0

5

10

15

Time (Seconds)

vy

(t)



Figure 7 The angleθ between mobile robot and x – axle. Figure 8 Steering angleφ of front wheels

The following Figures (Figure 3 - Figure 8) shows the simulation results of mobile robot

model, for the position of mobile robot in X and Y axis during its movement as well as the

angle of mobile robot platform and steering angle of the front wheels.

2.1. Dynamic modeling of the mobile robot

The Lagrange formulations are used to derive the dynamic equations of the vehicle:

τ=∂

∂−

∂

∂

q

L

q

L

dt

d

& (10)

After the calculation of Lagrange function (kinetic and potential energy), the dynamic

equations of the vehicle can be expressed in matrix form:

τ=+⋅ ),()( qqHqqD &&& (11)

Where:

=

=

=

d

sh

M

F

H

HqqH

DD

DDqD τ

2

1

2221

1211),()( &

(12)

22

2

1

2

1θ&⋅⋅+⋅⋅= PPpa

pak JvmE

(13)

Where:

pam - Mobile robot platform mass, perm - general mass, a2 - width of mobile robot platform

)4(2

1 21

2bamJ paP +⋅= - moment of inertia for mobile robot platform

=

∂

∂

∂

∂=

=

∂

∂

∂

∂==++=

∂

∂

∂

∂=

2

)tan(4

)(tan12

2

22

2

21122

11

Rm

LD

b

Rm

v

LDDmmm

v

L

vD

rr

rrperrrpa

φφ

φφ

φ

&&

&

(14)

+⋅

⋅⋅+⋅⋅=

+⋅

⋅+⋅⋅⋅=

)1)((tan2

)tan(

)1)((tan4

)tan(2

22

22

222

1

φφφ

φφφφ

&

&&

vb

RmvmH

b

RmvmH

rrper

rrper

(15)

The final vehicle model - Lagrange equations which describe the movement of the vehicle

are defined by:

0 0.5 1 1.5 2 2.5 3 3.5

-1

-0.5

0

0.5

1

Time (Seconds)

teta

(t)

0 0.5 1 1.5 2 2.5 3 3.5-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time (Seconds)

Fi(

t)



+

⋅

++

φφ

φφ

&&

&v

Rm

b

Rm

b

Rmmmm

rrrr

rrperrrpa

2)tan(

4

)tan(4

)(tan12

22

22

=

+⋅

⋅⋅+⋅⋅

+⋅

⋅+⋅⋅⋅

+d

sh

rrper

rrper

M

F

vb

Rmvm

b

Rmvm

)1)((tan2

)tan(

)1)((tan4

)tan(2

22

2

222

φφφ

φφφφ

&

&&

(16)

Where 1τ denotes the nominal driving force acting on the rear axle and 2τ denotes torque

for steering wheels. In order to be able to follow the sinusoid trajectory, the vehicle nominal

driving force )(tFsh (acting on the rear axle CD), and its torque for steering wheels )(tMd

(nominally to front wheels E, F) are obtained by equation (16).

Figure 9 Nominal driving forces acting on the rear axle. Figure 10 Steering wheels nominal torque.

3. DETECTION OF MOBILE ROBOT AND OBSTACLES

In our case, since the mobile robot must operate in order to be detected by camera, a selected

color from the constructed environment is used [3]. In each edge of the mobile robot a unique

color is located as a single point, which can be detected from camera (Fig. 11), and based on

these points we don’t detect only our mobile robot, but also we use it as a scale factor between

image taken from camera and real environment. By using color detection to take information,

the target point is also detected. The detection of obstacles is made based on shapes

(polygons). After detection of the mobile robot and obstacles, a matrix is generated by using

MATLAB, where each element has the information whether there is detection of an obstacle

or not [13].

Figure 11 Image taken from camera. Figure 12 Plotted mobile robot and virtual sensors on the

generated matrix.



In this matrix, we added three virtual distance sensors based on mobile robot size, which

are used to detect obstacles and generate information’s from these sensors, which will be used

as Fuzzy logic inputs (Fig. 12) [13].

3.1. Convert Pixel Distance to Metric Distance

The distance computed in the previous stage does not represent the real distance in meters to

the wall, but only the image distances specified by pixels. If one knows the optics of the

camera-mirror system, it is however possible to recover the metric distance (Figure 13).

Figure 13 Camera-mirror system.

Let d be the distance in meters on the ground floor and ρ the distance in pixels at the

image plane. Then, we have: )tan()( θθ hd =

(17)

For most omnidirectional cameras, the relation between θ and ρ can be approximated by a

first order Taylor expansion, that is: ραθ )/1(≈

(18)

Where, α depends on the mirror shape and the camera-mirror distance. Then, the final

relation becomes: ( )αρρ /tan)( hd =

(19)

In our case, we use α = 95 pixel, while the height h must be calibrated in meter, in order to

get the correct distance measurements. A reference distance measured on the ground floor is

used and compared with the results which have been obtained by equation (19).

4. FUZZY LOGIC FOR OBSTACLE AVOIDANCE

Fuzzy logic algorithm used to control mobile robot direction to the target, takes four inputs

which are left, middle and right sensor states, where each sensor state detects whether there is

or not an obstacle, and the fourth input is angle from the starting point to the target [11], [16].

Combination of six rules which are created on Fuzzy logic, provide different outputs based on

the inputs given by the actual state of mobile robot and obstacles [12], [13]. Fuzzy logic

algorithm is used in this work, for path finding from starting point to the target [5], [7]. In the

case when one of the mobile robot sensors is activated then a steps counter starts in

background. When counter reaches the limit the mobile robot should move toward the target.

Thus, we must repeat the same process until the mobile robot reaches the target.



Figure 14 Active Sensor. Figure 15 New direction of mobile robot after counter

reaches the limit.

In Fig. 14 the middle sensor of mobile robot is activated, after fuzzy logic receives the

information from this sensor, and the direction of mobile robot is changed based on rules, in

order to avoid the obstacle [16]. In Fig. 15 is presented the new direction of mobile robot

toward target, which is calculated by:

==

=<

=>

≠

−+

=

00

00

000

,0arctan2

),(atan2

22

yandxifundefined

yandxif

yandxif

yify

xyx

yx

π

(20)

Where: � = ��

Figure 16 Controller design in Fuzzy Logic Control (FLC).

Figure 17 Membership functions (a) sensors activity and (b) angle to the target from middle sensor of

mobile robot [13].

In Figure 17(a) are presented membership functions of mobile robot sensors which might

be taken as inputs for two fuzzy values, active or passive. Based on the rules of fuzzy logic

the mobile robot has three options for choosing the right direction.

ifsensorleftisactiveandsensormiddleisnotactiveand

sensorrightisnotactivatethendirectionisright



While for the optimization of mobile robot path in the Fig. 17(b) is presented membership

function which takes the input only for the case when middle sensor of mobile robot is active

and by knowing the target angle it helps us to choose the direction.

ifsensorleftisnotactiveandsensormiddleisactiveand

sensorrightianotactiveandangletothetargetisright

Thendirectionisright

The simulation results show that the application of FLC (Fuzzy logic controller)

significantly reduces the errors of front wheels steering angle (φ ) as well as error of angle

between mobile robot platform and X axle (θ).

Figure 18 left Error of front wheels steering. Figure 19 right Error for angle theta between mobile

angle with FLC robot platform and x-axle with FLC.

Figure 20 Error of front wheels steering angle without applying Fuzzy logic.

5. GENETIC ALGORITHM

Using a static value for steps counter as mentioned in previous section, the path which is

generated by Fuzzy logic in most cases is not the optimal path, which means mobile robot has

taken some extra steps, which are not needed in order to go to the target. The generated path

in this way can be optimized by implementing Genetic Algorithm over steps counter, in order

to find the array of best possible steps counter for each case when one of the mobile robot

sensors is active. The first step of Genetic Algorithm implementation it to create an initial

population with a predefined population size. The population contains the numbers of

individuals [10]. Each individual represents a value for steps counter which will help us in

choosing the right time to re-calculate direction toward target. The initial population of n size

can be expressed as:

Initial Population = <p., p/, …, p0> (21)

0 0.5 1 1.5 2 2.5 3 3.5

-1

-0.5

0

0.5

1

x 10-3

Time (Seconds)

automjeti

Efi

0 0.5 1 1.5 2 2.5 3 3.5

-8

-6

-4

-2

0

2

4

6

8

x 10-4

Time (Seconds)

automjeti

Ete

ta



Each value p0 represents an Integer value between 1 and the maximum of steps, which

can be taken in a diagonal from point 0,0 to the point x, y where x, yare equal to image size.

Next step is fitness function, which represents an important part of Genetic Algorithm [8].

Appropriate selection of the fitness function will lead the search towards the optimal solution.

Since we already have a path which is generated by Fuzzy logic, where steps counter has a

default value, we can assume that the steps taken within that path are most optimal but less

possible. For each iteration which is performed by Genetic Algorithm a new path is generated

[9], for the cases where the steps counter value is equal to p0 value and in the same time the

smallest value of all steps related to that generated path now is considered as best possible

path. With the same logic, this process continues until Genetic Algorithm finds the optimized

path for actual state of mobile robot, target and obstacles position. This approach can be

presented as:

=

<

csreturn

asos

osasif

(22)

Where is: as = actual steps count

os =optimized steps count

sc =steps counter for recalculating direction toward target

The optimal path, in our case, is the shortest path between the starting and ending point.

Based on this, the fitness function is responsible for finding this path.

6. EXPERIMENTAL RESULTS

The environment shown at Figure 21, respectively Figure 22 is used as simulation-

environment. Based on the generated map, the mobile robot should move from the starting

position, which is its own actual position to the target. The angle from mobile robot position

toward target is pre-calculated using formula shown in (21). After starting the simulation,

Fuzzy logic takes care for mobile robot to avoid the obstacles during its path to the target. A

mobile robot position is plotted for each step and at the end we have visualized the mobile

robot’s path to reach the target [16].

Figure 21 Mobile robot and generated path Figure 22 Mobile robot and optimized path to the target

to the target.

From in the Fig. 21 it can be seen the total of 235 steps needed for achieving the goal,

same as the processing time needed for the path finding. In this GUI the Turn steps that

represent steps counter are also shown, which will help us in choosing the right time to re-

calculate the direction towards the target, which by default is 10. This path is shown in Fig. 21



and is generated by using only the Fuzzy logic, without Genetic Algorithm, but it results in

some steps which are not necessary, so path needs to be optimized [16]. In Fig. 21 we can see

the difference between path finding using only Fuzzy Logic vs path finding using Fuzzy

Logic and Genetic Algorithm. The simulations show that by adding Genetic Algorithm to

Fuzzy Logic, the optimization process is improved in terms of path length. There is a full

match between target and mobile robot position as shown in Fig. 21 and Fig. 22, but for the

case where the length of the path is taken into consideration, better results are obtained as is

shown in Fig. 22.

7. CONCLUSIONS

Knowing that the generation of the map can be performed by using several kinds of sensors

which might be very expensive, for the detection of obstacles in this paper we used only a

simple camera and image processing. In this way we were able to achieve better or almost

same results by using inexpensive devices. In this study we presented the idea for using

Genetic Algorithm approach in order to optimize the mobile robot path finding problem. We

proposed a simplified fitness function that utilizes the path length and our preliminary

experiments showed that the proposed approach is quite effective and efficient.

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