ISSN: 1693-6930
TELKOMNIKA
Vol. 10, No. 4, December 2012 : 645 – 654 648
A. generate the initial population, Initial population is generated randomly in order to obtain the initial solution. Population itself
consists of a chromosome that represents the desired solution. B. evaluation of solutions,
This process will evaluate each population by calculating the value of fitness function until criteria are met. Generation that has the best fitness value is expected the desired optimal
solution. C. forming a new generation.
In shaping a new generation used of the three operators, that are reproductionselection operator, crossover, and mutation.
3. Research Method
There are several step to design Genetic Algorithm of SMC for Manipulator Robot, such as: manipulator modeling, SMC design for manipulator, and optimizing SMC by genetic
algoritm.
3.1 Model of manipulator.
There are two steps to model a manipulator robot, which are: kinematics modeling, and dynamics modeling. Robot kinematics is analytical study of robot arm movement to the
coordinate framework of silentmoving reference regardless of force causing the movement. Kinematics model represent the relation of end effectors in three dimension space with variable
of joint in the joint space. Robot dynamics is mathematical formulation which depicts dynamic behavior of manipulator considered force causing the movement.
By using lagrange-euler method, is obtained inverse dynamic equations for each joints expressing joint torque to accelerations with DC motors actuator [8] by following
equations 10-11.
10 cos
sin 3
2 3
3 cos
1 1
1 2
1 2
2 2
2 2
1 1
1 1
2 2
2 2
2 2
1 1
L m
L L
L L
L m
L
n F
l m
n n
J l
m n
θ θ
θ θ
θ θ
θ τ
+ −
+ =
11 cos
2 1
cos sin
3 1
3 3
2 2
2 2
2 2
2 2
1 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
L L
m L
L L
L m
gl m
n n
F l
m n
n J
l m
n
θ θ
θ θ
θ θ
τ
+ +
+ +
=
where
τ
1
and
τ
2
are the torque of joint 1 and joint 2, m
1
and m
2
are mass for each link, l
1
and l
2
are length of each lengths, J
m1
and J
m2
are inertias of motors, F
m1
and F
m2
are viscous coefficients of motors,
θ
L1
and
θ
L2
are the joints angle of movement, and n
1
and n
2
are gear ratio for each joint.
The type DC motor is armature-controlled. The output of DC motor is controlled by armature voltage, whereas field current kept in constant. Figure 2 is the schematic of DC motor.
Figure 2. Schematic of DC motor
TELKOMNIKA ISSN: 1693-6930
Genetic Algorithm of Sliding Mode Control Design for Manipulator Robot Ahmad Riyad Firdaus 649
Since the torque developed at the motor shaft increas linearly with the armature current, independent of speed and angular position, then the torque can be written by the following
equation 12.
a a
i K
=
τ
12 Whereas, armature voltage
b a
a a
a a
e dt
di L
R i
V +
+ =
13 where
n K
e
L m
m b
b
θ θ
θ =
= dan
thus,
− =
L a
b a
a a
nR K
R V
K
θ τ
14 By substituting the equation 10, 11 and 14, is obtained:
1 1
1 1
1 a
V B
H D
+ =
θ
15
2 2
2 2
2 2
a L
V B
G H
D +
+ =
θ
16 where,
1 1
1 2
1 2
2 2
2 2
1 1
1 1
1 1
1 1
1 1
1 2
2 2
2 2
2 1
1
cos sin
3 2
3 3
cos
a a
L L
L L
L m
a b
a m
L
R K
B l
m n
n F
R n
K K
H n
J l
m n
D
= +
+ −
= +
=
θ θ
θ θ
θ θ
2 2
2 2
2 2
2 2
2 1
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
cos 2
1 cos
sin 3
1 3
3
a a
L L
L L
L m
a b
a m
R K
B gl
m n
G l
m n
n F
R n
K K
H n
J l
m n
D
= −
= −
+ −
= +
=
θ θ
θ θ
θ
If select the state
2 4
2 3
1 2
1 1
; ;
;
L L
L L
x x
x x
θ θ
θ θ
= =
= =
, the control input are
2 2
1 1
;
a a
V u
V u
= =
and desired output are
2 2
1 1
;
L L
y y
θ θ
= =
, thus, the nonlinear state equation of manipulator 2- DOF can be written by the following equation 17.
+
+ =
− −
− −
2 1
2 1
2 1
1 1
2 2
1 2
4 1
1 1
2
4 3
2 1
u u
B D
B D
G H
D x
H D
x
x x
x x
17
x y
= 1
1
18
3.2 SMC for Manipulator
