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Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880
IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 13
Development of a Two Link Robotic Manipulator
OLAOLUWA D. ALADETOLA1
, KAYODE F. AKINGBADE2
, NAJEEM O. ADELAKUN3
1, 3
Department of Electrical and Electronics Engineering, Federal Polytechnic Ilaro, Ogun State.
2
Department of Electrical and Electronics Engineering, Federal University of Technology, Akure, Ondo
State.
Abstract- Robots play important roles in day to day
activities of human endeavour and can perform
complex tasks speedily and accurately. Robots are
employed to imitate human behaviours and then
apply these behaviours to the skills that lead to
achieving a certain task in solving the challenges
faced by impaired people in society. This robotic arm
was achieved using light-weight steel iron for the
frames, a moderate torque MG995 Towerpro, and
servo motor. Two Atmega328 Arduino
microcontroller was employed to control the motors
through the use of pulse width modulation
technique. Mathematical models were developed for
the mechanism to represent the kinematics involved
at each joint with mathematical variables. Then, the
stability of the system was carried out using a step
input signal being a type zero system.
Indexed Terms- Robot, armature dc servo motors,
Arduino microcontroller, stability.
I. INTRODUCTION
The rapid growth in technology prompt for the latest
matching devices. However, engineers and scientists
are vastly adapting new technologies persistently. The
robots perform significant roles in our lives and can
carry out complex responsibility faster and accurately
when compared to humans. They do not get drained or
perform task emotionally. Robots operate in almost all
human labours mostly in the fields which are
unhealthy or impractical for workers [3]. This fact
causes the workers to have more free time to spend on
skilled Professions which includes the programming,
maintenance, and operation of the robots.
Robotic arms perform simple translational and
rotational motion, which has an end effector that
performs a specific function [4]. A Robot is a
programmable mechanical device that can replace the
functions of the human arm. An automated pick and
place robot arm, which can reach an object in a given
domain or range of space, grip it precisely, and place
it to the desired position. This mechanism is used for
lifting objects and carrying out tasks that require
extreme concentration, expert accuracy and may also
be recursive. It can be employed industrially (e.g.
cranes). According to [12] Presently industrial arms
have risen in its capability and operation through
micro-controllers and programming developments,
enhanced mechanisms, sensing, and drive systems,
which has led to a vast transformation in the robotic
industry [4]. Hence, when designing a robot, factors
such as artificial intelligence, concept and techniques,
and cognitive science are vital to obtaining a viable
design.
II. SYSTEM MODELLING
System modelling involves the use of models to
conceptualize and construct real-life to analyse, study
and simulate the real-life situations or system
virtually. This gives us deeper insight into other
aspects of the system and this helps in determining the
stability and other characteristics of the system. Figure
1 shows that the model has a base upon which other
parts of the manipulator is placed upon. Directly on the
base is the shoulder link, this link is expected to
undergo two different motion patterns first in the 1800
rotation of the base giving the mechanism the aid to
pick and place any desired target within its
circumference of operation [5]. In addition to this
motion, another actuator will be used to perform a
translational motion at an angle which is used to move
the next link, the elbow. At the joint between the
shoulder and the elbow, another actuator is used
making the third that helps to move the elbow link at
an angle translational. Thereafter the end-effector is
put in place to perform the picking and placing of the
target object. The motion of the end-effector is also
done with the aid of actuators and gears [6]. This gives
Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880
IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 14
a brief overview of the whole mechanism and how the
links are connected one to another.
Fig. 1. Two link robotic manipulator system
III. MATHEMATICAL MODELLING OF THE
SYSTEM
The Rotational Double Inverted Pendulum (RDIP)
described here consists of a horizontal base arm
(denoted as Link 1) driven by a servo motor and two
vertical pendulums. The mathematical model of the
RDIP was developed by the use of the Euler-Lagrange
(E-L) function [8].
Fig. 2. Rotational double inverted pendulum
schematic [11].
Ο΄1 is the angle of link 1in the horizontal plane
𝛳̇ 1 is the velocity of link 1in the horizontal plane
π›³Μˆ1 is the acceleration of link 1
Ο΄ 2 is the angle of link 2 in the vertical plane
𝛳̇ 2 is the velocity of link 2 in the vertical plane
π›³Μˆ2 is the acceleration of link 2
Ο΄3 is the angle of link 3 in the horizontal plane
𝛳̇ 3 is the velocity of link 3 in the horizontal plane
π›³Μˆ3 is the acceleration of link 3
J1 is the moment of inertial for link 1
The equation of motion for the system dynamic is
expressed from the Fig.3
Fig. 3. Velocity analysis for link 2
The total kinetic energy of each link in our system is
given by the combination of its moving kinetic term
Km and its rotating kinetic term Kr as
Km=
1
2
mv2
Kr=
1
2
jΞΈi
2Μ‡ (1)
From Fig.3 the total kinetic energy for both links is
expressed as
k=
1
2
jΞΈi
2Μ‡ +
1
2
jΞΈ2
2Μ‡ +
1
2
jΞΈ3
2Μ‡ +
1
2
m2[(L1ΞΈΜ‡1 + i2ΞΈΜ‡2 sin ΞΈ2)2
+
(βˆ’i2ΞΈΜ‡2 sin ΞΈ2)2
] +
1
2
m3[(L1ΞΈΜ‡1 + i2ΞΈΜ‡2 cos ΞΈ2 +i3ΞΈΜ‡3 cos ΞΈ3)2
+
(βˆ’i2ΞΈΜ‡2 sin ΞΈ2 βˆ’ i3ΞΈΜ‡3 sin ΞΈ3)2
] (2)
And the potential energy expressed as
P=m3gl2 cos ΞΈ2+ m3g(L2 cos ΞΈ2+i3 cos ΞΈ3) (3)
From langrange function
If L = K – P (4)
Applying eqn (2) and (3) to eqn (4)
d
dt
βˆ‚L
βˆ‚ΞΈΜ‡ 1
-
βˆ‚L
βˆ‚ΞΈ1
= Ο„-b1ΞΈΜ‡1 (5)
Finding the derivatives from eqn (5)
⊺ = (j1 + L1
2
(m2+m3)) Ο΄1
̈ +l1(m2l2 + m3L2)cos ϴ2ϴ̈ 2+
l1m3l3cos ϴ3ϴ̈ 3 +
b1ΞΈΜ‡1L1(m2l2 + m3L2)Ο΄2
1
Μ‡ sin Ο΄2-L1m3l3Ο΄2
3sinΟ΄3
(6)
Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880
IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 15
d
dt
βˆ‚L
βˆ‚ΞΈΜ‡ 1
-
βˆ‚L
βˆ‚ΞΈ2
= Ο„-b2ΞΈΜ‡2
0=-L1(m2l2+m3l2)cos ΞΈ2 ΞΈ1- (j2+L2
2
m3+l2
2
m2)ΞΈ2 βˆ’
L2m3l3 cos( ΞΈ2 βˆ’ ΞΈ3)ΞΈ3
βˆ’b2ΞΈΜ‡2 βˆ’ L2m3l3Ο΄2
3sin(ΞΈ2-Ο΄3) + (m2l2+
m3L2)g sinΞΈ2 (7)
And
d
dt
βˆ‚L
βˆ‚ΞΈΜ‡ 2
-
βˆ‚L
βˆ‚ΞΈ3
= Ο„-b3ΞΈΜ‡3 (8)
0=-L2m3l3cosΞΈ2ΞΈ1-L2m3l3cos(ΞΈ2 βˆ’ ΞΈ3)ΞΈ2 βˆ’ (j3 +
l2
2
m3)ΞΈ3 βˆ’ b3ΞΈΜ‡3 + L2m3l3ΞΈ3
2Μ‡ sin( ΞΈ2 βˆ’ ΞΈ3)
+m3l3g sinΞΈ3. (9)
IV. THE FORCE AND TORQUE REQUIRED
The mass of the links and the actuators are both
considered in computing the torque at each joint. The
length and the mass of various components are
highlighted below in Figure 4.
Fig .4. Mass and length of the robotic arm links
Mass of servo meter used = 52g = 0.052kg
Length of the shoulder = 7.2cm,
Mass of the shoulder link = 0.0201kg
Length of the elbow = 10.4cm,
Mass of the elbow link = 0.402kg
Length of the wrist = 3.2cm,
Mass of the wrist link = 0.015kg
Length of the gripper = 10.5cm,
Mass of the gripper link = 0.0705kg
Mass of the Base = 0.950kg
The torque of each servo motor at no load is obtained
to be 10kgcm at 5V Dc input. This is needed to
calculate the excess torque the motor has to develop in
performing its required task. Moment of inertia is
given as
𝑀 = π‘Žπ‘Ÿπ‘š π‘€π‘’π‘–π‘”β„Žπ‘‘ Γ— Β½ π‘Žπ‘Ÿπ‘š π‘™π‘’π‘›π‘”π‘‘β„Ž
+ π‘šπ‘œπ‘‘π‘œπ‘Ÿ π‘€π‘’π‘–π‘”β„Žπ‘‘ π‘₯ π‘Žπ‘Ÿπ‘š π‘€π‘’π‘–π‘”β„Žπ‘‘
At the gripper, the torque developed by the servo
motor is 10.5 and the excess torque required is
determined to be 9.62987 kg-cm
The torque needed at the wrist calculated to be
0.804125 kg-cm and the excess torque is 9.195875 kg-
cm
At the elbow, the torque needed is determined to be
2.76556 kg-cm and the excess torque 7.23444 kg-cm
The torque at the shoulder is 3.12366 kg-cm and the
excess torque is 6.87634 kg-cm
At the bottom which is the base, the torque required is
3.64666 kg-cm and the excess torque is 6.35334 kg-
cm
The maximum torque that the mechanism developed
at the gripper should be less than the torque at the base.
For the mechanism, the maximum torque at the base is
taken to be 5kg.cm giving allowance of 1.3177kg.cm.
the radius of operation of the mechanism is the sum of
all horizontal link of the mechanism given as 24.3cm.
V. MODELLING THE ARMATURE DC
SERVO MOTORS FOR THE SYSTEM
The armature controlled dc servo motor is used
extensively in the control system for its precision and
stability characteristics [9]. The armature controlled dc
motor modelling and its mathematical representation
is as shown in Figure 5 and equations (9) to (17). It is
assumed that the torque generated by the motor is
proportional to the air-gap flux and also the armature
current.
Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880
IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 16
ef
J,fo
M
+
-
Vb(t)
ia(t)
+
Va(t)
Ra
La Rf
Lf
𝜽
Fig. 5. The DC servo motor circuit diagram [9]
𝑖 π‘Ž(𝑑) is armature current (amperes)
𝑅 π‘Ž is armature resistance
𝑇 π‘š(𝑑) is motor torque (Nm)
𝐾𝑖 is Motor torque constant
𝐿 π‘Ž is armature inductance (Henries)
π‘‰π‘Ž 𝑖𝑠 applied voltage (volts)
π‘Šπ‘š(𝑑) is motor angular velocity (rad/sec)
𝐾𝑒 is motor voltage constant (v/rad/sec)
𝐽 𝑇 is total inertia of motor armature plus load
Ξ± is acceleration (rad/𝑠𝑒𝑐2
)
πœƒ π‘š 𝑖𝑠 shaft angle in radians
The torque developed by the motor is given as
𝑇 π‘š(𝑑)=𝐾 𝑇 𝐼 π‘Ž(𝑑) (10)
Taking the friction force to be negligible, the back emf
𝑉𝑏 is related to the angular velocity by
𝑉𝑏 = 𝐾𝑒 π‘Š = 𝐾𝑒
π‘‘πœƒ π’Ž
𝑑𝑑
(11)
Applying Kirchhoff’s voltage law around the
electrical loop, we have
𝐿 π‘Ž
𝑑𝑖
𝑑𝑑
+𝑅 π‘Ž 𝑖 = Τ‘ π‘Ž 𝐾𝑒
π‘‘πœƒ π’Ž
𝑑𝑑
(12)
Τ‘ π‘Ž = 𝑖 π‘Ž 𝑅 π‘Ž+𝐿 π‘Ž
𝑑𝑖 π‘Ž
𝑑𝑑
+𝐾𝑒 π‘Šπ‘š (13)
Equation (14) is obtained after taking the Laplace
transform of equation (13)
Τ‘ π‘Ž =(
𝐿 π‘Ž
𝑅 π‘Ž
s+1)𝑅 π‘Ž 𝑙 π‘Ž+𝐾𝑒 π‘Šπ‘š (14)
Also it should be recalled that T=𝐾 𝑇 𝑙 π‘Ž= 𝑗 𝑇 𝛼= 𝑗 𝑇 π‘Šπ‘š 𝑠
l π‘Ž=
𝑇
𝐾 𝑇
=
𝑗 𝑇 π‘Š π‘š 𝑠
𝐾 𝑇
(15)
Substitute eqn (15) into eqn (14)
Τ‘ π‘Ž βˆ’ 𝐾𝑒 π‘Š 𝑀=(
𝐿 π‘Ž
𝑅 π‘Ž
s+1)𝑅 π‘Ž
𝑗 𝑇 π‘Š π‘š 𝑠
𝐾 𝑇
(16)
Hence, π‘Šπ‘š becomes
( Τ‘ π‘Žβˆ’πΎ 𝑒 π‘Š 𝑀)𝐾 𝑇
(
𝐿 π‘Ž
𝑅 π‘Ž
s+1) 𝑅 π‘Ž 𝑗 𝑇s
=WM (17)
Fig. 6. The Block Diagram of the DC servomotor.
This closed loop transfer function of the armature dc
controlled motor is obtained from the block diagram
of Figure 6, equations (18) to (21).
π‘Š π‘š
Τ‘ π‘Ž
=
𝐾 𝑇
𝑅 π‘Ž 𝑗 𝑇s[(
𝐿 π‘Ž
𝑅 π‘Ž
)s+1]+𝐾 𝑒 𝐾 𝑇
(18)
π‘Š π‘š
Τ‘ π‘Ž
=
𝐾 𝑇
(𝑅 π‘Ž 𝑗 𝑇
𝐿 π‘Ž
𝑅 π‘Ž
)𝑆2+𝑗 𝑇 𝑅 π‘Ž 𝑆+𝐾 𝑒 𝐾 𝑇
(19)
π‘Š π‘š
Τ‘ π‘Ž
=
𝐾 𝑇
(𝑗 𝑇 𝐿 π‘Ž)𝑆2+𝑗 𝑇 𝑅 π‘Ž 𝑆+𝐾 𝑒 𝐾 𝑇
(20)
πœƒ π‘š
Τ‘ π‘Ž
=
𝐾 𝑇
(𝑗 𝑇 𝐿 π‘Ž)𝑆3+𝑗 𝑇 𝑅 π‘Ž 𝑆2+𝐾 𝑒 𝐾 𝑇
(21)
For prototype design operation of the servo Dc motor,
the following values are assumed for the servo motor
parameters
𝑅 π‘Ž(armature resistance)=1.2
𝐿 π‘Ž (armature inductance)=0.07H
𝑇 π‘š(𝑑)(π‘šπ‘œπ‘‘π‘œπ‘Ÿπ‘‘π‘œπ‘Ÿπ‘žπ‘’π‘’) = 0.7(π‘π‘š)
𝐾𝑖(Motor torque constant)=1N.m/A
Ea (applied voltage) = 5 volts
wm (t) (motor angular velocity) = 70RPM
Ke (motor voltage constant OR motor back – emf
constant) = 0.01(v/rad/sec)
JT (total inertia) = Jmotor + Jload
a(angular acceleration ) = (rad/sec2
To calculate the angular acceleration ∝, is given as
∝ = 0.12217rad/sec2
Therefore to calculate the Jload (moment of inertia of
the load) =
𝑇 π‘™π‘œπ‘Žπ‘‘
∝
. Now considering the motor at the
gripper the torque derived at the previous section is
0.370125 kg.cm.
converting to Newton.neter : 0.370125Γ—
9.81
100
= 0.036309 N.m
Therefore Jload =
0.036309
0.12217
= 0.2972 kg. m2
For this servo dc motor Jm = 0.08 Kg.m2
Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880
IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 17
The JTotal = Jm + Jload
Therefore, that will yield;
JTotal = 0.08 + 0.2972 = 0.3772 kg.m
The transfer function for the first servo motor is
therefore.
π‘Š π‘š
Τ‘ π‘Ž
=
1
(0.3772βˆ—0.07)𝑆2+(0.3772βˆ—1.2)𝑠+(1βˆ—0.01)
π‘Š π‘š
Τ‘ π‘Ž
=
1
(0.02640)𝑆2+0.45264𝑠+0.01
This is the transfer function for the servo motor at the
gripper
At the wrist, torque derived earlier is used to calculate
the total moment on inertia at the joint and the value is
substituted into the general transfer function of a servo
motor. For second motor at the wrist, the torque gotten
is 0.74804 kg.cm, converting it to Newton.meter will
give
0.804125x
9.81
100
= 0.07888Nm
Thereforeπ‘—π‘™π‘œπ‘Žπ‘‘=
0.07888
0.12217
= 0.6457kg.π‘š2
For this servo motor 𝑗 π‘š = 0. 05kg.π‘š2
The π‘—π‘‘π‘œπ‘‘π‘Žπ‘™= 𝑗 π‘š + π‘—π‘™π‘œπ‘Žπ‘‘.Therefore that will yield;
π‘—π‘‘π‘œπ‘‘π‘Žπ‘™=0.05 + 0.6457π‘˜π‘”. π‘š2
= 0.6957kg.π‘š2
The transfer function for the second servo motor is
therefore:
π‘Š π‘š
Τ‘ π‘Ž
=
1
(0.6957βˆ—0.07)𝑆2+(0.6957βˆ—1.2)𝑠+0.01
π‘Š π‘š
Τ‘ π‘Ž
=
1
(0.04869)𝑆2+(0.8348)𝑠+0.01
This is the transfer function for the servo motor at the
wrist
For third motor at the elbow, the torque gotten is
2.76556kg.cm, converting it to Newton.meter will
give
2.76556 x
9.81
100
= 0.2713Nm
Therefore π‘—π‘™π‘œπ‘Žπ‘‘=
0.2713
0.12217
= 2.22068kg.π‘š2
For this servo motor 𝑗 π‘š =0.05kg.π‘š2
π‘—π‘‘π‘œπ‘‘π‘Žπ‘™= 𝑗 π‘š + π‘—π‘™π‘œπ‘Žπ‘‘. Therefore that will yield
π‘—π‘‘π‘œπ‘‘π‘Žπ‘™ =0.05+2.22068kg.π‘š2
= 2.27068kg.π‘š2
The transfer function for this third servo motor is
therefore:
π‘Š π‘š
Τ‘ π‘Ž
=
1
(2.27068βˆ—0.07)𝑆2+(2.27068βˆ—1.2)𝑠+0.01
=
1
(0.158947)𝑆2+2.7248𝑠+0.01
This is the transfer function for the servo motor at the
elbow
For the motor at the shoulder, the torque is
3.12366kg.π‘š2
,converting it to Newton.meter will give
3.12366x
9.81
100
=0.30643Nm
Therefore π‘—π‘™π‘œπ‘Žπ‘‘=
0.30643
0.12217
=2.5082kg.π‘š2
For this servo dc motor 𝑗 π‘š=0.05kg.π‘š2
.
The π‘—π‘‘π‘œπ‘‘π‘Žπ‘™= 𝑗 π‘š + π‘—π‘™π‘œπ‘Žπ‘‘.Therefore that will yield;
π‘—π‘‘π‘œπ‘‘π‘Žπ‘™=0.05 + 2.5082kg.π‘š2
= 2.5582kg.π‘š2
The transfer function for this fourth servo motor is
therefore:
π‘Š π‘š
Τ‘ π‘Ž
=
1
(2.5582βˆ—0.07)𝑆2+(2.5582βˆ—1.2)𝑠+0.01
=
1
(0.179074)𝑆2+3.06984𝑠+0.01
This is the transfer function for the servo motor at the
shoulder The transfer function for this
Fig. 7. Block diagram of the system
VI. SOFTWARE DEVELOPMENT
The electronic design for this mechanism at the
transmitter involves the use of the two Arduino
Atmega 283 microcontroller. It is an 8bit
microcontroller and the first step involves configuring
the register of the Arduino Atmega 283 to work with
the desired pulse width modulation technique used.
Fig. 8: Electrical design for the system
Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880
IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 18
An external power supply is used to supply the servo
motors and also the Arduino Atmega 283
microcontroller. The receiver is however connected to
the arduino atmega 283 together with the servo
motors. The two DC motors responsible for the
movement of the two wheels of the mechanism is
connected with the motor controller L293D and both
are connected to the Arduino Atmega 283. The system
is being powered by a rechargeable 8v battery.
Fig. 9. Electrical design for the wireless RF remote
control
Fig 8 shows the Arduino Atmega283 microcontroller
connected with the transmitter. The dc battery of 5V
rating is being connected with to power the controller.
The push buttons also are arranged on the Vero board,
thus connected to the microcontroller with cables.
VII. SYSTEM STABILITY
The transfer function for the whole model is gotten
through the MatLab code below which brings about
the test for system stability
Fig...10. MatLab code for acquiring the transfer
function.
The system yields the transfer function below:
=
VIII. RESULTS AND DISCUSSION
The matrices in fig 11 were generated for the state
space form.
Fig. 11. The controllability matrix and the rank of
matrix
Fig 12 shows the rank of the controllability matrix is
not full. A full rank is when the value of the rank is
equal to the value of the row or column of a square
matrix (i.e a n x n matrix having a rank of n). The
controllable matrix for this mechanism is an 8 x 8
matrix and the rank is 4 meaning the rank is not full
hence it is not controllable.
Fig. 12. The observability matrix and the rank of
matrix
Fig 12 shows that the rank of the observability matrix
is full. A full rank is when the value of the rank is equal
to the value of the row or column of the square matrix
(i.e a n x n matrix having a rank of n). The
observability matrix for this system is an 8 x 8 matrix
and the rank is full hence it is controllable.
Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880
IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 19
Fig. 13. The step response plot showing the value of
the step information
Fig 13 shows the step response of the transfer function
of the system is displayed. The rise time of the system
is 1.01e + 03secs and the settling time is 1.84e +
03secs. The system steady state is at 1e + 08secs. This
is considered good for the system.
Fig. 14. The bode plot for the system.
The plot illustrates that the system has a phase margin
of 1730
at a gain cross frequency of 0.75rad/secs. The
point on the magnitude plot in which the curve crosses
0dB is traced down in the phase plot and that the
corresponding point on the phase plot is the phase
margin. Consequently, for the gain margin, the point
on the phase plot that the curve crosses -1800
is traced
up to the magnitude plot and corresponding point on
the magnitude plot curve is the gain margin. On this
plot the phase plot did crosses the -1800
at the phase
crossover frequency of 0.75 rad/secs, the gain margin
at this point is -143dB
Fig. 15. The polar plot for the system.
The polar plot shows that the phase Margin gotten
from the bode plot is 1210
. The gain Margin of the
polar plot is also the same as that of the bode plot and
it is -1430
. This shows that the system is stable.
Fig. 16. The Root locus plot of the System.
Fig 16 shows that the root locus has its zeros on the
left hand side of the plot, this implies that the system
is more stable and its speed up the settling time of the
system response.
A. Establishing the Robustness with Reference
Tracking
Fig 17 shows the initial step plot of the system, the
response time is 544.5 seconds our system is stable
and the robustness is 0.6. The rise time 276 seconds
and the settling time is 2210 seconds.
Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880
IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 20
Fig. 17. Initial System response graph
The system is stable but the rise time is too high. PID
controller is introduced to tune the system to reduce
the rising time and also increase the robustness of the
system.
From Fig 18 the response time is increased the 126
seconds and the robustness of the plant is 0.84seconds.
The rise time was reduced from 276 seconds to 66.1
seconds and the system is stable. Also, the settling
time has reduced. The system is stable and has a
robustness of 0.84. this shows that with the aid of the
PID controller the system is more stable and its rise
time is faster than before and it has good robustness
and performance.
Fig 18. Final system response graph
IX. CONCLUSION
The assistive device will be able to meet most of the
challenges of impaired people in society. The device
was designed to fulfil the required tasks without the
impaired facing any hindrances due to the lack of
muscle activity. The Robotic Arm designed is unique
to impaired people specific conditions of Athetoid
Quadriplegia. As a result, the controllers of this device
are constructed to ease the usage. Assistive robots will
help people with impaired arm functions perform
ordinary tasks, like eating and drinking, or opening a
desk drawer.
The assistive robot construction breakdown involves
the use of a microcontroller and the C – Programming
language for programming, the radio frequency
transmitter, and the receiver together with the servo
motor controls. This device will provide the impaired
an opportunity to partake in the simple everyday tasks.
REFERENCES
[1] Zunt, Dominik. "Who did actually invent the word
"robot" and what does it mean?". The Karel
Čapek website. Retrieved 09-11- 2011.
[2] Nikku, S., β€œIntroduction to Robotics” USA: John
Wiley & Sons, 2011.
[3] Nehmzow, U., ”Mobile Robotics: a practical
introduction”, UK: Springer, 05-04-2003.
[4] Muir, F. P., Neuman, P. C., "Pulsewidth
Modulation Control of Brushless DC Motors for
Robotic Applications", IEEE Transactions on
Industrial Electronics, Issue 3, Vol. 32, pp.222-
229, 1985.
[5] Lin, I. H., Liu, C. Y., Chen, L. C., "Evaluation of
Human-Robot Arm Movement Imitation",
Proceedings of 8th Asian Control Conference
(ASCC), pp.287-292, 2011.
[6] Craig, j.j (2005). The mechanics and control of
mechanical manipulators. Direction of freedom
(DOF): Pearson Education International. pp.333-
357, 2012
[7] Nones, Y. A., β€œHeterogeneous Modelling &
Design of a Robot Arm Control System”, Illinois
Tech Robotics (ITR), pp.1-6, 2003.
[8] Park, I. D., Park, C., Do, H., Choi, T., Kyung, J.,
Design and Analysis of Dual Arm Robot Using
Dynamic Simulation" IEEE 10th International
Conference on Ubiquitous Robots and Ambient
Intelligence (URAI), pp. 681-682, 2013.
[9] RobotWiki. (2014, September 23). Servos.
Retrieved November 5, from psurobotic
http://www.psurobotics.org/wiki/index.php?title=
servos
[10] Scott Schaut (2006). Robots of Westinghouse:
1924-Today. Mansfield Memorial Museum.
ISBN 0-9785844-1-4. Received 09-11- 2010
Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880
IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 21
[11] Bo Li (2013), Rotational Double Inverted
Pendulum, M.Sc. Thesis, School of Engineering,
University of Dayton, Dayton, Ohio, U.S.
[12] Ranwaka, R., Perera, T. R., Adhuran, J.,
Samarakoon, C., & Rajakaruna, R. (2013), 111-
Microcontroller Based Robot Arm with Three –
Dimensional Reach.

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Development of a two link robotic manipulator

  • 1. Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880 IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 13 Development of a Two Link Robotic Manipulator OLAOLUWA D. ALADETOLA1 , KAYODE F. AKINGBADE2 , NAJEEM O. ADELAKUN3 1, 3 Department of Electrical and Electronics Engineering, Federal Polytechnic Ilaro, Ogun State. 2 Department of Electrical and Electronics Engineering, Federal University of Technology, Akure, Ondo State. Abstract- Robots play important roles in day to day activities of human endeavour and can perform complex tasks speedily and accurately. Robots are employed to imitate human behaviours and then apply these behaviours to the skills that lead to achieving a certain task in solving the challenges faced by impaired people in society. This robotic arm was achieved using light-weight steel iron for the frames, a moderate torque MG995 Towerpro, and servo motor. Two Atmega328 Arduino microcontroller was employed to control the motors through the use of pulse width modulation technique. Mathematical models were developed for the mechanism to represent the kinematics involved at each joint with mathematical variables. Then, the stability of the system was carried out using a step input signal being a type zero system. Indexed Terms- Robot, armature dc servo motors, Arduino microcontroller, stability. I. INTRODUCTION The rapid growth in technology prompt for the latest matching devices. However, engineers and scientists are vastly adapting new technologies persistently. The robots perform significant roles in our lives and can carry out complex responsibility faster and accurately when compared to humans. They do not get drained or perform task emotionally. Robots operate in almost all human labours mostly in the fields which are unhealthy or impractical for workers [3]. This fact causes the workers to have more free time to spend on skilled Professions which includes the programming, maintenance, and operation of the robots. Robotic arms perform simple translational and rotational motion, which has an end effector that performs a specific function [4]. A Robot is a programmable mechanical device that can replace the functions of the human arm. An automated pick and place robot arm, which can reach an object in a given domain or range of space, grip it precisely, and place it to the desired position. This mechanism is used for lifting objects and carrying out tasks that require extreme concentration, expert accuracy and may also be recursive. It can be employed industrially (e.g. cranes). According to [12] Presently industrial arms have risen in its capability and operation through micro-controllers and programming developments, enhanced mechanisms, sensing, and drive systems, which has led to a vast transformation in the robotic industry [4]. Hence, when designing a robot, factors such as artificial intelligence, concept and techniques, and cognitive science are vital to obtaining a viable design. II. SYSTEM MODELLING System modelling involves the use of models to conceptualize and construct real-life to analyse, study and simulate the real-life situations or system virtually. This gives us deeper insight into other aspects of the system and this helps in determining the stability and other characteristics of the system. Figure 1 shows that the model has a base upon which other parts of the manipulator is placed upon. Directly on the base is the shoulder link, this link is expected to undergo two different motion patterns first in the 1800 rotation of the base giving the mechanism the aid to pick and place any desired target within its circumference of operation [5]. In addition to this motion, another actuator will be used to perform a translational motion at an angle which is used to move the next link, the elbow. At the joint between the shoulder and the elbow, another actuator is used making the third that helps to move the elbow link at an angle translational. Thereafter the end-effector is put in place to perform the picking and placing of the target object. The motion of the end-effector is also done with the aid of actuators and gears [6]. This gives
  • 2. Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880 IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 14 a brief overview of the whole mechanism and how the links are connected one to another. Fig. 1. Two link robotic manipulator system III. MATHEMATICAL MODELLING OF THE SYSTEM The Rotational Double Inverted Pendulum (RDIP) described here consists of a horizontal base arm (denoted as Link 1) driven by a servo motor and two vertical pendulums. The mathematical model of the RDIP was developed by the use of the Euler-Lagrange (E-L) function [8]. Fig. 2. Rotational double inverted pendulum schematic [11]. Ο΄1 is the angle of link 1in the horizontal plane 𝛳̇ 1 is the velocity of link 1in the horizontal plane π›³Μˆ1 is the acceleration of link 1 Ο΄ 2 is the angle of link 2 in the vertical plane 𝛳̇ 2 is the velocity of link 2 in the vertical plane π›³Μˆ2 is the acceleration of link 2 Ο΄3 is the angle of link 3 in the horizontal plane 𝛳̇ 3 is the velocity of link 3 in the horizontal plane π›³Μˆ3 is the acceleration of link 3 J1 is the moment of inertial for link 1 The equation of motion for the system dynamic is expressed from the Fig.3 Fig. 3. Velocity analysis for link 2 The total kinetic energy of each link in our system is given by the combination of its moving kinetic term Km and its rotating kinetic term Kr as Km= 1 2 mv2 Kr= 1 2 jΞΈi 2Μ‡ (1) From Fig.3 the total kinetic energy for both links is expressed as k= 1 2 jΞΈi 2Μ‡ + 1 2 jΞΈ2 2Μ‡ + 1 2 jΞΈ3 2Μ‡ + 1 2 m2[(L1ΞΈΜ‡1 + i2ΞΈΜ‡2 sin ΞΈ2)2 + (βˆ’i2ΞΈΜ‡2 sin ΞΈ2)2 ] + 1 2 m3[(L1ΞΈΜ‡1 + i2ΞΈΜ‡2 cos ΞΈ2 +i3ΞΈΜ‡3 cos ΞΈ3)2 + (βˆ’i2ΞΈΜ‡2 sin ΞΈ2 βˆ’ i3ΞΈΜ‡3 sin ΞΈ3)2 ] (2) And the potential energy expressed as P=m3gl2 cos ΞΈ2+ m3g(L2 cos ΞΈ2+i3 cos ΞΈ3) (3) From langrange function If L = K – P (4) Applying eqn (2) and (3) to eqn (4) d dt βˆ‚L βˆ‚ΞΈΜ‡ 1 - βˆ‚L βˆ‚ΞΈ1 = Ο„-b1ΞΈΜ‡1 (5) Finding the derivatives from eqn (5) ⊺ = (j1 + L1 2 (m2+m3)) Ο΄1 ̈ +l1(m2l2 + m3L2)cos Ο΄2ϴ̈ 2+ l1m3l3cos Ο΄3ϴ̈ 3 + b1ΞΈΜ‡1L1(m2l2 + m3L2)Ο΄2 1 Μ‡ sin Ο΄2-L1m3l3Ο΄2 3sinΟ΄3 (6)
  • 3. Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880 IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 15 d dt βˆ‚L βˆ‚ΞΈΜ‡ 1 - βˆ‚L βˆ‚ΞΈ2 = Ο„-b2ΞΈΜ‡2 0=-L1(m2l2+m3l2)cos ΞΈ2 ΞΈ1- (j2+L2 2 m3+l2 2 m2)ΞΈ2 βˆ’ L2m3l3 cos( ΞΈ2 βˆ’ ΞΈ3)ΞΈ3 βˆ’b2ΞΈΜ‡2 βˆ’ L2m3l3Ο΄2 3sin(ΞΈ2-Ο΄3) + (m2l2+ m3L2)g sinΞΈ2 (7) And d dt βˆ‚L βˆ‚ΞΈΜ‡ 2 - βˆ‚L βˆ‚ΞΈ3 = Ο„-b3ΞΈΜ‡3 (8) 0=-L2m3l3cosΞΈ2ΞΈ1-L2m3l3cos(ΞΈ2 βˆ’ ΞΈ3)ΞΈ2 βˆ’ (j3 + l2 2 m3)ΞΈ3 βˆ’ b3ΞΈΜ‡3 + L2m3l3ΞΈ3 2Μ‡ sin( ΞΈ2 βˆ’ ΞΈ3) +m3l3g sinΞΈ3. (9) IV. THE FORCE AND TORQUE REQUIRED The mass of the links and the actuators are both considered in computing the torque at each joint. The length and the mass of various components are highlighted below in Figure 4. Fig .4. Mass and length of the robotic arm links Mass of servo meter used = 52g = 0.052kg Length of the shoulder = 7.2cm, Mass of the shoulder link = 0.0201kg Length of the elbow = 10.4cm, Mass of the elbow link = 0.402kg Length of the wrist = 3.2cm, Mass of the wrist link = 0.015kg Length of the gripper = 10.5cm, Mass of the gripper link = 0.0705kg Mass of the Base = 0.950kg The torque of each servo motor at no load is obtained to be 10kgcm at 5V Dc input. This is needed to calculate the excess torque the motor has to develop in performing its required task. Moment of inertia is given as 𝑀 = π‘Žπ‘Ÿπ‘š π‘€π‘’π‘–π‘”β„Žπ‘‘ Γ— Β½ π‘Žπ‘Ÿπ‘š π‘™π‘’π‘›π‘”π‘‘β„Ž + π‘šπ‘œπ‘‘π‘œπ‘Ÿ π‘€π‘’π‘–π‘”β„Žπ‘‘ π‘₯ π‘Žπ‘Ÿπ‘š π‘€π‘’π‘–π‘”β„Žπ‘‘ At the gripper, the torque developed by the servo motor is 10.5 and the excess torque required is determined to be 9.62987 kg-cm The torque needed at the wrist calculated to be 0.804125 kg-cm and the excess torque is 9.195875 kg- cm At the elbow, the torque needed is determined to be 2.76556 kg-cm and the excess torque 7.23444 kg-cm The torque at the shoulder is 3.12366 kg-cm and the excess torque is 6.87634 kg-cm At the bottom which is the base, the torque required is 3.64666 kg-cm and the excess torque is 6.35334 kg- cm The maximum torque that the mechanism developed at the gripper should be less than the torque at the base. For the mechanism, the maximum torque at the base is taken to be 5kg.cm giving allowance of 1.3177kg.cm. the radius of operation of the mechanism is the sum of all horizontal link of the mechanism given as 24.3cm. V. MODELLING THE ARMATURE DC SERVO MOTORS FOR THE SYSTEM The armature controlled dc servo motor is used extensively in the control system for its precision and stability characteristics [9]. The armature controlled dc motor modelling and its mathematical representation is as shown in Figure 5 and equations (9) to (17). It is assumed that the torque generated by the motor is proportional to the air-gap flux and also the armature current.
  • 4. Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880 IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 16 ef J,fo M + - Vb(t) ia(t) + Va(t) Ra La Rf Lf 𝜽 Fig. 5. The DC servo motor circuit diagram [9] 𝑖 π‘Ž(𝑑) is armature current (amperes) 𝑅 π‘Ž is armature resistance 𝑇 π‘š(𝑑) is motor torque (Nm) 𝐾𝑖 is Motor torque constant 𝐿 π‘Ž is armature inductance (Henries) π‘‰π‘Ž 𝑖𝑠 applied voltage (volts) π‘Šπ‘š(𝑑) is motor angular velocity (rad/sec) 𝐾𝑒 is motor voltage constant (v/rad/sec) 𝐽 𝑇 is total inertia of motor armature plus load Ξ± is acceleration (rad/𝑠𝑒𝑐2 ) πœƒ π‘š 𝑖𝑠 shaft angle in radians The torque developed by the motor is given as 𝑇 π‘š(𝑑)=𝐾 𝑇 𝐼 π‘Ž(𝑑) (10) Taking the friction force to be negligible, the back emf 𝑉𝑏 is related to the angular velocity by 𝑉𝑏 = 𝐾𝑒 π‘Š = 𝐾𝑒 π‘‘πœƒ π’Ž 𝑑𝑑 (11) Applying Kirchhoff’s voltage law around the electrical loop, we have 𝐿 π‘Ž 𝑑𝑖 𝑑𝑑 +𝑅 π‘Ž 𝑖 = Τ‘ π‘Ž 𝐾𝑒 π‘‘πœƒ π’Ž 𝑑𝑑 (12) Τ‘ π‘Ž = 𝑖 π‘Ž 𝑅 π‘Ž+𝐿 π‘Ž 𝑑𝑖 π‘Ž 𝑑𝑑 +𝐾𝑒 π‘Šπ‘š (13) Equation (14) is obtained after taking the Laplace transform of equation (13) Τ‘ π‘Ž =( 𝐿 π‘Ž 𝑅 π‘Ž s+1)𝑅 π‘Ž 𝑙 π‘Ž+𝐾𝑒 π‘Šπ‘š (14) Also it should be recalled that T=𝐾 𝑇 𝑙 π‘Ž= 𝑗 𝑇 𝛼= 𝑗 𝑇 π‘Šπ‘š 𝑠 l π‘Ž= 𝑇 𝐾 𝑇 = 𝑗 𝑇 π‘Š π‘š 𝑠 𝐾 𝑇 (15) Substitute eqn (15) into eqn (14) Τ‘ π‘Ž βˆ’ 𝐾𝑒 π‘Š 𝑀=( 𝐿 π‘Ž 𝑅 π‘Ž s+1)𝑅 π‘Ž 𝑗 𝑇 π‘Š π‘š 𝑠 𝐾 𝑇 (16) Hence, π‘Šπ‘š becomes ( Τ‘ π‘Žβˆ’πΎ 𝑒 π‘Š 𝑀)𝐾 𝑇 ( 𝐿 π‘Ž 𝑅 π‘Ž s+1) 𝑅 π‘Ž 𝑗 𝑇s =WM (17) Fig. 6. The Block Diagram of the DC servomotor. This closed loop transfer function of the armature dc controlled motor is obtained from the block diagram of Figure 6, equations (18) to (21). π‘Š π‘š Τ‘ π‘Ž = 𝐾 𝑇 𝑅 π‘Ž 𝑗 𝑇s[( 𝐿 π‘Ž 𝑅 π‘Ž )s+1]+𝐾 𝑒 𝐾 𝑇 (18) π‘Š π‘š Τ‘ π‘Ž = 𝐾 𝑇 (𝑅 π‘Ž 𝑗 𝑇 𝐿 π‘Ž 𝑅 π‘Ž )𝑆2+𝑗 𝑇 𝑅 π‘Ž 𝑆+𝐾 𝑒 𝐾 𝑇 (19) π‘Š π‘š Τ‘ π‘Ž = 𝐾 𝑇 (𝑗 𝑇 𝐿 π‘Ž)𝑆2+𝑗 𝑇 𝑅 π‘Ž 𝑆+𝐾 𝑒 𝐾 𝑇 (20) πœƒ π‘š Τ‘ π‘Ž = 𝐾 𝑇 (𝑗 𝑇 𝐿 π‘Ž)𝑆3+𝑗 𝑇 𝑅 π‘Ž 𝑆2+𝐾 𝑒 𝐾 𝑇 (21) For prototype design operation of the servo Dc motor, the following values are assumed for the servo motor parameters 𝑅 π‘Ž(armature resistance)=1.2 𝐿 π‘Ž (armature inductance)=0.07H 𝑇 π‘š(𝑑)(π‘šπ‘œπ‘‘π‘œπ‘Ÿπ‘‘π‘œπ‘Ÿπ‘žπ‘’π‘’) = 0.7(π‘π‘š) 𝐾𝑖(Motor torque constant)=1N.m/A Ea (applied voltage) = 5 volts wm (t) (motor angular velocity) = 70RPM Ke (motor voltage constant OR motor back – emf constant) = 0.01(v/rad/sec) JT (total inertia) = Jmotor + Jload a(angular acceleration ) = (rad/sec2 To calculate the angular acceleration ∝, is given as ∝ = 0.12217rad/sec2 Therefore to calculate the Jload (moment of inertia of the load) = 𝑇 π‘™π‘œπ‘Žπ‘‘ ∝ . Now considering the motor at the gripper the torque derived at the previous section is 0.370125 kg.cm. converting to Newton.neter : 0.370125Γ— 9.81 100 = 0.036309 N.m Therefore Jload = 0.036309 0.12217 = 0.2972 kg. m2 For this servo dc motor Jm = 0.08 Kg.m2
  • 5. Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880 IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 17 The JTotal = Jm + Jload Therefore, that will yield; JTotal = 0.08 + 0.2972 = 0.3772 kg.m The transfer function for the first servo motor is therefore. π‘Š π‘š Τ‘ π‘Ž = 1 (0.3772βˆ—0.07)𝑆2+(0.3772βˆ—1.2)𝑠+(1βˆ—0.01) π‘Š π‘š Τ‘ π‘Ž = 1 (0.02640)𝑆2+0.45264𝑠+0.01 This is the transfer function for the servo motor at the gripper At the wrist, torque derived earlier is used to calculate the total moment on inertia at the joint and the value is substituted into the general transfer function of a servo motor. For second motor at the wrist, the torque gotten is 0.74804 kg.cm, converting it to Newton.meter will give 0.804125x 9.81 100 = 0.07888Nm Thereforeπ‘—π‘™π‘œπ‘Žπ‘‘= 0.07888 0.12217 = 0.6457kg.π‘š2 For this servo motor 𝑗 π‘š = 0. 05kg.π‘š2 The π‘—π‘‘π‘œπ‘‘π‘Žπ‘™= 𝑗 π‘š + π‘—π‘™π‘œπ‘Žπ‘‘.Therefore that will yield; π‘—π‘‘π‘œπ‘‘π‘Žπ‘™=0.05 + 0.6457π‘˜π‘”. π‘š2 = 0.6957kg.π‘š2 The transfer function for the second servo motor is therefore: π‘Š π‘š Τ‘ π‘Ž = 1 (0.6957βˆ—0.07)𝑆2+(0.6957βˆ—1.2)𝑠+0.01 π‘Š π‘š Τ‘ π‘Ž = 1 (0.04869)𝑆2+(0.8348)𝑠+0.01 This is the transfer function for the servo motor at the wrist For third motor at the elbow, the torque gotten is 2.76556kg.cm, converting it to Newton.meter will give 2.76556 x 9.81 100 = 0.2713Nm Therefore π‘—π‘™π‘œπ‘Žπ‘‘= 0.2713 0.12217 = 2.22068kg.π‘š2 For this servo motor 𝑗 π‘š =0.05kg.π‘š2 π‘—π‘‘π‘œπ‘‘π‘Žπ‘™= 𝑗 π‘š + π‘—π‘™π‘œπ‘Žπ‘‘. Therefore that will yield π‘—π‘‘π‘œπ‘‘π‘Žπ‘™ =0.05+2.22068kg.π‘š2 = 2.27068kg.π‘š2 The transfer function for this third servo motor is therefore: π‘Š π‘š Τ‘ π‘Ž = 1 (2.27068βˆ—0.07)𝑆2+(2.27068βˆ—1.2)𝑠+0.01 = 1 (0.158947)𝑆2+2.7248𝑠+0.01 This is the transfer function for the servo motor at the elbow For the motor at the shoulder, the torque is 3.12366kg.π‘š2 ,converting it to Newton.meter will give 3.12366x 9.81 100 =0.30643Nm Therefore π‘—π‘™π‘œπ‘Žπ‘‘= 0.30643 0.12217 =2.5082kg.π‘š2 For this servo dc motor 𝑗 π‘š=0.05kg.π‘š2 . The π‘—π‘‘π‘œπ‘‘π‘Žπ‘™= 𝑗 π‘š + π‘—π‘™π‘œπ‘Žπ‘‘.Therefore that will yield; π‘—π‘‘π‘œπ‘‘π‘Žπ‘™=0.05 + 2.5082kg.π‘š2 = 2.5582kg.π‘š2 The transfer function for this fourth servo motor is therefore: π‘Š π‘š Τ‘ π‘Ž = 1 (2.5582βˆ—0.07)𝑆2+(2.5582βˆ—1.2)𝑠+0.01 = 1 (0.179074)𝑆2+3.06984𝑠+0.01 This is the transfer function for the servo motor at the shoulder The transfer function for this Fig. 7. Block diagram of the system VI. SOFTWARE DEVELOPMENT The electronic design for this mechanism at the transmitter involves the use of the two Arduino Atmega 283 microcontroller. It is an 8bit microcontroller and the first step involves configuring the register of the Arduino Atmega 283 to work with the desired pulse width modulation technique used. Fig. 8: Electrical design for the system
  • 6. Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880 IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 18 An external power supply is used to supply the servo motors and also the Arduino Atmega 283 microcontroller. The receiver is however connected to the arduino atmega 283 together with the servo motors. The two DC motors responsible for the movement of the two wheels of the mechanism is connected with the motor controller L293D and both are connected to the Arduino Atmega 283. The system is being powered by a rechargeable 8v battery. Fig. 9. Electrical design for the wireless RF remote control Fig 8 shows the Arduino Atmega283 microcontroller connected with the transmitter. The dc battery of 5V rating is being connected with to power the controller. The push buttons also are arranged on the Vero board, thus connected to the microcontroller with cables. VII. SYSTEM STABILITY The transfer function for the whole model is gotten through the MatLab code below which brings about the test for system stability Fig...10. MatLab code for acquiring the transfer function. The system yields the transfer function below: = VIII. RESULTS AND DISCUSSION The matrices in fig 11 were generated for the state space form. Fig. 11. The controllability matrix and the rank of matrix Fig 12 shows the rank of the controllability matrix is not full. A full rank is when the value of the rank is equal to the value of the row or column of a square matrix (i.e a n x n matrix having a rank of n). The controllable matrix for this mechanism is an 8 x 8 matrix and the rank is 4 meaning the rank is not full hence it is not controllable. Fig. 12. The observability matrix and the rank of matrix Fig 12 shows that the rank of the observability matrix is full. A full rank is when the value of the rank is equal to the value of the row or column of the square matrix (i.e a n x n matrix having a rank of n). The observability matrix for this system is an 8 x 8 matrix and the rank is full hence it is controllable.
  • 7. Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880 IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 19 Fig. 13. The step response plot showing the value of the step information Fig 13 shows the step response of the transfer function of the system is displayed. The rise time of the system is 1.01e + 03secs and the settling time is 1.84e + 03secs. The system steady state is at 1e + 08secs. This is considered good for the system. Fig. 14. The bode plot for the system. The plot illustrates that the system has a phase margin of 1730 at a gain cross frequency of 0.75rad/secs. The point on the magnitude plot in which the curve crosses 0dB is traced down in the phase plot and that the corresponding point on the phase plot is the phase margin. Consequently, for the gain margin, the point on the phase plot that the curve crosses -1800 is traced up to the magnitude plot and corresponding point on the magnitude plot curve is the gain margin. On this plot the phase plot did crosses the -1800 at the phase crossover frequency of 0.75 rad/secs, the gain margin at this point is -143dB Fig. 15. The polar plot for the system. The polar plot shows that the phase Margin gotten from the bode plot is 1210 . The gain Margin of the polar plot is also the same as that of the bode plot and it is -1430 . This shows that the system is stable. Fig. 16. The Root locus plot of the System. Fig 16 shows that the root locus has its zeros on the left hand side of the plot, this implies that the system is more stable and its speed up the settling time of the system response. A. Establishing the Robustness with Reference Tracking Fig 17 shows the initial step plot of the system, the response time is 544.5 seconds our system is stable and the robustness is 0.6. The rise time 276 seconds and the settling time is 2210 seconds.
  • 8. Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880 IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 20 Fig. 17. Initial System response graph The system is stable but the rise time is too high. PID controller is introduced to tune the system to reduce the rising time and also increase the robustness of the system. From Fig 18 the response time is increased the 126 seconds and the robustness of the plant is 0.84seconds. The rise time was reduced from 276 seconds to 66.1 seconds and the system is stable. Also, the settling time has reduced. The system is stable and has a robustness of 0.84. this shows that with the aid of the PID controller the system is more stable and its rise time is faster than before and it has good robustness and performance. Fig 18. Final system response graph IX. CONCLUSION The assistive device will be able to meet most of the challenges of impaired people in society. The device was designed to fulfil the required tasks without the impaired facing any hindrances due to the lack of muscle activity. The Robotic Arm designed is unique to impaired people specific conditions of Athetoid Quadriplegia. As a result, the controllers of this device are constructed to ease the usage. Assistive robots will help people with impaired arm functions perform ordinary tasks, like eating and drinking, or opening a desk drawer. The assistive robot construction breakdown involves the use of a microcontroller and the C – Programming language for programming, the radio frequency transmitter, and the receiver together with the servo motor controls. This device will provide the impaired an opportunity to partake in the simple everyday tasks. REFERENCES [1] Zunt, Dominik. "Who did actually invent the word "robot" and what does it mean?". The Karel Čapek website. Retrieved 09-11- 2011. [2] Nikku, S., β€œIntroduction to Robotics” USA: John Wiley & Sons, 2011. [3] Nehmzow, U., ”Mobile Robotics: a practical introduction”, UK: Springer, 05-04-2003. [4] Muir, F. P., Neuman, P. C., "Pulsewidth Modulation Control of Brushless DC Motors for Robotic Applications", IEEE Transactions on Industrial Electronics, Issue 3, Vol. 32, pp.222- 229, 1985. [5] Lin, I. H., Liu, C. Y., Chen, L. C., "Evaluation of Human-Robot Arm Movement Imitation", Proceedings of 8th Asian Control Conference (ASCC), pp.287-292, 2011. [6] Craig, j.j (2005). The mechanics and control of mechanical manipulators. Direction of freedom (DOF): Pearson Education International. pp.333- 357, 2012 [7] Nones, Y. A., β€œHeterogeneous Modelling & Design of a Robot Arm Control System”, Illinois Tech Robotics (ITR), pp.1-6, 2003. [8] Park, I. D., Park, C., Do, H., Choi, T., Kyung, J., Design and Analysis of Dual Arm Robot Using Dynamic Simulation" IEEE 10th International Conference on Ubiquitous Robots and Ambient Intelligence (URAI), pp. 681-682, 2013. [9] RobotWiki. (2014, September 23). Servos. Retrieved November 5, from psurobotic http://www.psurobotics.org/wiki/index.php?title= servos [10] Scott Schaut (2006). Robots of Westinghouse: 1924-Today. Mansfield Memorial Museum. ISBN 0-9785844-1-4. Received 09-11- 2010
  • 9. Β© FEB 2020 | IRE Journals | Volume 3 Issue 8 | ISSN: 2456-8880 IRE 1701909 ICONIC RESEARCH AND ENGINEERING JOURNALS 21 [11] Bo Li (2013), Rotational Double Inverted Pendulum, M.Sc. Thesis, School of Engineering, University of Dayton, Dayton, Ohio, U.S. [12] Ranwaka, R., Perera, T. R., Adhuran, J., Samarakoon, C., & Rajakaruna, R. (2013), 111- Microcontroller Based Robot Arm with Three – Dimensional Reach.