11-19


 

Al-Khwarizmi Engineering Journal,Vol. 11, No. 

 

 
Deflection Analysis of an Elastic Single Link Robotic Manipulator

Rafal M. 
*Department of Automated Manufacturing Engineering

 **Department of Biomedical Engineering

 

 (Received 
 
 
Abstract 

Robotics manipulators with structural flexibility provide an attractive alternative to rigid robotics manipulators for 
many of the new and evolving applications in robotics. In certain applications their use is unavoidable. The increased 
complexity in modeling and control of such manipulators is offset by desirable performance enhancements in some 
respects. In this paperthe single- link flexible robotics manipulator
designed with 0.5 m length , 0.02 m width and with 0.004 m thickness with mass located at the tip. There are four 
subsystems; motion, control, accelerometer and 
robotics manipulator is the base servomotor. It rotates a hub with the link on it and measure the  tip deflection. the 
deflection  was measured for three cases without load,  with 27.5 and with 59.4 gram at the end of the flexible link . 
During each of the above cases I rotated the base servo motor at 
card based on ATMEGA640 microcontroller. 
measured by MMA7631 Accelerometer 
Then the dataI collected collected from accelerometer and plot it using MATLAB software and compared between 
theoretical results obtained from MATLAB program that based on Lagrange equation of motion and experimental 
results and we found the maximum deflection occurred when V=180 deg/sec a

Keywords: Flexible Link Manipulator , Industrial robotics , 
 
 
1. Introduction 

 
Robotic manipulators are used widely

in dangerous, monotonous, and boring
of these robotic manipulators are build and
manner to maximize stiffness and to minimize the 
vibration of the end effectors to achi
position accuracy. The design of 
manipulator is achieved by using heavy material 
and a bulky design. The existing heavy 
manipulators are known to be insufficient in te
of speed and power consumption with respect to 
the operating payload [1]. 

To improve industrial productivity
robotics, it is required to increase the operation 
speed of the link and/or reduce the weight of the 
links. Due to high speed operation and
requirements, a dynamic model that includes the 
joint and / or link flexibilities is needed. 

Khwarizmi Engineering Journal,Vol. 11, No. 3, P.P. 11-19 (2015)

Deflection Analysis of an Elastic Single Link Robotic Manipulator
 

Rafal M. Khalil*               Somer M. Nacy** 
Department of Automated Manufacturing Engineering/ Al-Khwarizmi College of Engineering

Department of Biomedical Engineering/ Al-Khwarizmi College of Engineering/ University of
*E-mail:Rafalalazzawi90@gmail.com     

**E-mail :Somernacy@yahoo.com 

(Received 22 June 2014; accepted 7 May 2015)  

Robotics manipulators with structural flexibility provide an attractive alternative to rigid robotics manipulators for 
many of the new and evolving applications in robotics. In certain applications their use is unavoidable. The increased 

ling and control of such manipulators is offset by desirable performance enhancements in some 
link flexible robotics manipulator was designed and implemented

designed with 0.5 m length , 0.02 m width and with 0.004 m thickness with mass located at the tip. There are four 
motion, control, accelerometer and gyro and a host computer subsystem. The work principle of single

s manipulator is the base servomotor. It rotates a hub with the link on it and measure the  tip deflection. the 
for three cases without load,  with 27.5 and with 59.4 gram at the end of the flexible link . 

rotated the base servo motor at an angular velocity equals to 90 deg./s using control 
ATMEGA640 microcontroller. the deflection  was measured for the three cases and the deflection 

measured by MMA7631 Accelerometer and Gyro . This accelerometer  controlled by  using MEGA Arduino board . 
from accelerometer and plot it using MATLAB software and compared between 

theoretical results obtained from MATLAB program that based on Lagrange equation of motion and experimental 
and we found the maximum deflection occurred when V=180 deg/sec and tip load=59.5 gram

anipulator , Industrial robotics , Robotics Manipulator ,Beam Deflection

widely to help 
boring jobs. Most 

build and in a 
to minimize the 

vibration of the end effectors to achieve good 
position accuracy. The design of high stiffness 

is achieved by using heavy material 
existing heavy and rigid 

manipulators are known to be insufficient in terms 
with respect to 

improve industrial productivity of the 
increase the operation 

reduce the weight of the 
to high speed operation and lightweight 

dynamic model that includes the 
is needed.  

 The Link flexibility is a consequence of the 
lightweight structure in manipulator arms that are 
modeled and designed to operate at high
speeds with low inertia. Compared 
conventional heavy and bulky robots
weight robot, by introducing 
flexibility on the mechanical system of robots, it 
has a great advantages of 
larger work volume, and lower cost,
payload-to-manipulator-weight ratio, lower 
energy consumption, smaller actuators, better 
transportability, better maneuver ability 
end of these advantages its 
reduced inertia [2]. 

These great disadvantage obtained by 
introducing joint and / or link
robotics mechanical system,
vibration due to low stiffness. If 
cannot be solved, then the 

  
Al-Khwarizmi 
Engineering   

Journal 
(2015) 

Deflection Analysis of an Elastic Single Link Robotic Manipulator 

 
Khwarizmi College of Engineering/ University of Baghdad 

University of Baghdad  

Robotics manipulators with structural flexibility provide an attractive alternative to rigid robotics manipulators for 
many of the new and evolving applications in robotics. In certain applications their use is unavoidable. The increased 

ling and control of such manipulators is offset by desirable performance enhancements in some 
designed and implemented from Perspex and 

designed with 0.5 m length , 0.02 m width and with 0.004 m thickness with mass located at the tip. There are four 
gyro and a host computer subsystem. The work principle of single-link 

s manipulator is the base servomotor. It rotates a hub with the link on it and measure the  tip deflection. the 
for three cases without load,  with 27.5 and with 59.4 gram at the end of the flexible link . 

equals to 90 deg./s using control 
for the three cases and the deflection 

by  using MEGA Arduino board . 
from accelerometer and plot it using MATLAB software and compared between 

theoretical results obtained from MATLAB program that based on Lagrange equation of motion and experimental 
nd tip load=59.5 gram.  

Beam Deflection. 

Link flexibility is a consequence of the 
in manipulator arms that are 

operate at high operation 
speeds with low inertia. Compared between the 
conventional heavy and bulky robots and light 

by introducing joints and /or link 
the mechanical system of robots, it 

advantages of higher operation speed, 
and lower cost, greater 
weight ratio, lower 
smaller actuators, better 

better maneuver ability and at the 
end of these advantages its safer operation due to 

disadvantage obtained by 
and / or link flexibilities to 

system, that system has high 
vibration due to low stiffness. If that problem 

the mechanical system of 



Rafal M. Khalil                                 
 

 

the robot will not been favored in industries
will affect the  repeatability and accuracy of 
end point of manipulator in response to input 
commands.  

To overcome this problem, an accurate 
dynamic model of the manipulator
characterize with joint and/or link 
to be developed. This is a first step 
modeling and designing an efficient control
strategies for these manipulators [3]. 

Before study of flexible manipulator 
construction materials must be focused on
efficient actuation and sensing technologies , and 
simple and effective controller designs. Flexible 
robot manipulators are in use in some extent in 
space applications. This is because of the weight 
resurrection for a spacecraft ,Other potential areas 
of application are manipulation in nuclear and 
other hazardous environment , car
painting, manufacturing of  electronic hardware 
and food industry[4]. 
Cannon and Schmitz [5], studied single
flexible manipulators shown in Fig.1
Lagrange’s equation and the assumed mode 
method for modeling the single link manipulator 
and the vibration is controlled by measuring the 
position and using strain gauges have
to be very useful for achieving good
 

 
Fig. 1. Flexible Link Manipulator.

 
 
Nagarajan and Turcic [6], derived
motion using Lagrange’s equation for elastic 
mechanism systems. 

The elastic links are modeled using the finite 
element method. Both rigid body degrees of 
freedom and the elastic degrees of freedom are 
considered as generalized coordinates in the 
derivation. 

Choi et al. [7] ,”addressed the control
dynamic modeling of a single
manipulator fabricated from composite laminates  
(non- metallic) and compared the results with that 
of aluminum. They have shown that the 
manipulator fabricated from composite laminates 
has superior performance characteristics such as 

                                Al-Khwarizmi Engineering Journal, Vol. 11, No. 

12 
 

industries. This 
accuracy of the 

in response to input 

overcome this problem, an accurate 
manipulator that can 

 flexibility has 
to be developed. This is a first step towards 

designing an efficient controlling 
for these manipulators [3].  

Before study of flexible manipulator the 
must be focused on , 

actuation and sensing technologies , and 
simple and effective controller designs. Flexible 

are in use in some extent in 
space applications. This is because of the weight 
resurrection for a spacecraft ,Other potential areas 

are manipulation in nuclear and 
other hazardous environment , car/vehicle 
painting, manufacturing of  electronic hardware 

, studied single-link 
shown in Fig.1 using the 

Lagrange’s equation and the assumed mode 
method for modeling the single link manipulator 

s controlled by measuring the 
ave been found 

good performance. 

 

Flexible Link Manipulator. [1] 

derived Equations of 
motion using Lagrange’s equation for elastic 

The elastic links are modeled using the finite 
element method. Both rigid body degrees of 

elastic degrees of freedom are 
considered as generalized coordinates in the 

control and the 
dynamic modeling of a single-link flexible 
manipulator fabricated from composite laminates  

the results with that 
of aluminum. They have shown that the 
manipulator fabricated from composite laminates 

characteristics such as 

faster settling time, smaller input torque and 
smaller overshoot relative to the manipulator 
fabricated from aluminum
Discussed the utilization of composite materials in 
the construction of a flexible manipulator to 
provide higher strength and stiffness
ratio and larger structural damping than a metallic 
flexible manipulator . 
 Krishnamurthy et al. [9], 
model for single-link robotic arms fabricated from 
orthotropic composite materials. The equations of 
motion are derived using Hamilton's principle and 
include the coupling between the rigid body 
motion and elastic motion
results presented for aluminum, steel, 
graphite/epoxy, and boron/epoxy indicate that the 
motion-induced vibration is significantly less for 
the composite robotic arms as well as substantial 
savings in energy”. So I made our  flexible link 
manipulator from composite material its most 
preferred because of its light weight and high 
strength and large structural damping
 

 
2. Characteristics of the Physical Arm
 

The schematic of a planar
manipulator is shown in Fig.
inertial coordinate frame, and 
coordinate assigned for a flexible link. 
an d τ represent the hub position, the deflection do 
in the arm, and the torque applied to
respectively. The existing experimental single
link flexible manipulator is a 0.5
flexible structure that can 
plane but it can’t bend the vertical plane. At 
end of the arm the different load was putted and at 
the other end was clamped on a rigid hub
from Teflon material mounted directly on the haft 
of a DC servo motor.  

A torque applied by the DC servo motor  
rotates the arm in a horizontal plane. The other 
end of the arm with payload mass attached is free. 
The beam of the manipulator is made of Perspex

 

 
Fig. 2. A Planar Single-Link Flexible Manipulator.

Khwarizmi Engineering Journal, Vol. 11, No. 3, P.P. 11- 19(2015) 

faster settling time, smaller input torque and 
smaller overshoot relative to the manipulator 

cated from aluminum “. Choi et al.[8] , 
Discussed the utilization of composite materials in 
the construction of a flexible manipulator to 
provide higher strength and stiffness-to-weight 
ratio and larger structural damping than a metallic 

[9], “Present a dynamic 
link robotic arms fabricated from 

orthotropic composite materials. The equations of 
motion are derived using Hamilton's principle and 
include the coupling between the rigid body 

otion and elastic motion. Computer simulated 
results presented for aluminum, steel, 
graphite/epoxy, and boron/epoxy indicate that the 

induced vibration is significantly less for 
the composite robotic arms as well as substantial 

o I made our  flexible link 
manipulator from composite material its most 
preferred because of its light weight and high 
strength and large structural damping.  

 
Characteristics of the Physical Arm 

The schematic of a planar single-Link flexible 
manipulator is shown in Fig. 2, (X0,Y 0) is an 
inertial coordinate frame, and ( X 1 ,Y 1)is the 
coordinate assigned for a flexible link. θ,Ψ(x ,t), 

represent the hub position, the deflection do 
the torque applied to the hub, 

The existing experimental single-
link flexible manipulator is a 0.5 m long, a 
flexible structure that can bend in the horizontal 

the vertical plane. At the 
the different load was putted and at 

clamped on a rigid hub made 
mounted directly on the haft 

torque applied by the DC servo motor  
rotates the arm in a horizontal plane. The other 
end of the arm with payload mass attached is free. 

manipulator is made of Perspex. 

 

Link Flexible Manipulator. 



Rafal M. Khalil                                 Al-Khwarizmi Engineering Journal, Vol. 11, No. 3, P.P. 11- 19(2015) 
 

13 
 

 

3. Mechanical System Modeling 
 

The equations of motion of this system 
involving a rotary flexible link manipulator, 
involves modeling the rigid rotational base and 
the flexible link together as rigid bodies. a 
simplification of the partial differential equation 
describe the motion of a the flexible link, a single 
degree of freedom approximation in this system is 
used. At the first start with the derivation of the 
dynamic model  of the system by computing 
various rotational moments of inertia terms. The 
rotational inertia for a flexible link is given  by:-    

J���� = �� m����L�                                            …(1)                 
Where: L is the total flexible link length , and mlink 
is the total mass of the flexible link, For a single 
degree of freedom of this system, the natural 
frequency is related with torsional stiffness of the 
link and rotational inertia in the following 
manner:- 

ω��
����������                                                       … (2)           
Where: �� n was found experimentally and Kstiff is 
the lateral stiffness constant of the link ,defined 
as:- ������ = ��                                                   … (3)    
Where: F is the force applied at the tip of the link 
and   is the tip deflection.  = � !"� #$                                                          … (4)  
And substitutes equation (4) into equation (3) 
yields the flexural stiffness of cantilever beam :- ������ = � #$!"                                                   … (5) 

Where I is the moment of  area of the link and 
E is the young modules of elasticity of the link. In 
addition, any frictional damping effects between 
the flexible link and the rotary base was 
neglected. Next, the generalized dynamic equation 
of the system was driven  for the tip and base 
using The Lagrange’s energy equations of motion 
in terms of a set of generalized variables phi θ and 
alpha α ,where α is the angle of tip deflection and 
θ is the base rotation given in the following:- %%� & %'%(′) − %'%( + %,%( = -(                                 … (6) 
Where: P is the total potential energy of the system 
and T is the total kinetic energy of the system, and 
Qi is the i

th generalized force within the ith degree 
of freedom. the virtual forces that applied onto the 
generalized coordinates obtained from Qθand Qα, 
be: -(./                                                                … (7) -0 = 0                                                            … (8) 

The dynamic equations was driven for the 
mechanical subsystem from:  

2.. = − 4567889:;5< = + �9:;5< /                                  … (9) =.. = −������ & �9:;5< + �9:;5<) = + �9:;5< /     … (10) 
Next, rewriting equations (9) and (10) into a state 
space form  that gives the following equation[10]: 

>2?=?2@=@
A =

BC
CC
D00 00 1 00 10 − 4567889:;5< 0 00 −������ & �9:;5< + �9:;5<) 0 0FG

GG
H >2=2?=? A + BC

CC
D 00�9:;5<�9:;5<FG

GG
H /                                                                                             

                                                                              …(11) 
 
 
4. Mathematical Design 
 
Table 1, 
Physical parameters of the system. 

Parameter Values 
l* h * b (Arm dim.) 0.5 * 0.02 *0.004m 
E (Modules of 
elasticity ) 

2.1 Gpa 

M (link mass) 0.041 kg 
Mt (Tip payload) 0, 0.0595,0.0274 kg 
Jb(inertia of the base) 0.0229 kg.m2 
T (Motor torque const.) 7,15.5 kg-cm 
• T ( Torque) = 1.520 N-m 
• Kg (motor gear ration ) = 2:1 
• Vf= (feed voltage)=6 , 12 V 

 
The MATLAB program built to find the tip 

deflection based on Lagrange equation [10] of 
motion as mention above and derived using my 
parameter shown in table 1.1 .the ordinary 
differential equation (eq.11)  solved using Laplace 
inverse to find the tip deflection. 

 
 

5. Experimental Work 
 
5.1. Design and Implementation of Single 

Link Flexible Manipulator 
 

The single link flexible manipulator has been 
designed and manufactured in the Al-Khwarizmi 
engineering college workshop; the basic 
components that have been used to form the 
flexible manipulator illustrated as following:  
(1) High torque servo motor (Changeable 
Velocities), (2) Teflon Hub with radius 
(0.0615m), with a  width of (0.09 m), (3) Rotating 
base two layer with radius (0.2m) , with a  space 
between them equal (0.05 m) and thickness equal 



Rafal M. Khalil                                 Al-Khwarizmi Engineering Journal, Vol. 11, No. 3, P.P. 11- 19(2015) 
 

14 
 

 

(0.005 m). (4) Two aluminum fixed plate, (5) 
Perspex link with length (0.520 m) , with width of 
(0.042 m) and with thickness (0.004 m). (6) 
Power supply to control the velocity of  servo 
motors,(7) MMA 7631 Accelerometer + Gyro 
sensor to measure the tip deflection of the flexible 
link, (8) Arduino mega 2560 board to control the 
accelerometer + gyro, (9) Servo motor control 
card to control the angle and velocity of the base 
servo motor . The manufacturing of the flexible 
link has been done by the following steps: (a) the 
hub was turning to cylindrical shape with 6.15 
mm radius. (b) the Perspex plate was cutting using 
laser beam cutting to the dimension mention 
previously . (c) the aluminum plate shaped to fix 
the Perspex link with the Teflon hub . (d) the 
Teflon hub with the flexible link mounted above 
the rotating base and the base servo motor inside 
the rotating base . (e) Put the load at the end the 
flexible link and stick the accelerometer on it. The 
Single link flexible manipulator shown in figures 
below: 
 

 
 

Fig. 3. Hub with rotating base. 
 

The side , front and top view was drawn using 
AutoCAD software as shown in Figure 4,5,and 6: 

 

 
 

Fig. 4. Flexible link manipulator. 
 

 

 

Fig. 5. Flexible link manipulator (Side View ). 

 

 
 
 
 
 

Fig. 5. Flexible link manipulator (Side View ). 
 
 

 
 
 
Fig. 6. (a) Flexible link manipulator (front view), (b) 
Aluminum fix plate. 

 
 

Fig. 7. Flexible link manipulator (top view). 
 
 
 
 

 

Hub 

Perspex Link 

Load 
Rotating 

Base 

Base 
servo 
motor 

(A) (B) 



Rafal M. Khalil                                 Al-Khwarizmi Engineering Journal, Vol. 11, No. 3, P.P. 11- 19(2015) 
 

15 
 

 

5.2. Control Part 
 
The single link flexible manipulator rotate 

horizontally using high torque servo motor and 
this is motor was controlled by servo motor 
control card based on ATMEGA640. The  
velocity and the angle of the servo motor 
controlled through graphical user interface GUI of 

servo motion software built in with this card.The  
MMA7631 Accelerometer + Gyro  sensor , its 
block diagram shown in ( Fig. 8).stick at the tip of 
the link and this is sensor controlled by Arduino 
Mega 2560 board. The program built using 
Arduino language and this is language based on 
C++language . 

 
Fig. 8. Simplified accelerometer functional block diagram. 

 

6. Results and Discussion 
 

6.1. Theoretical Results 
 

The theoretical results obtained from 
MATLAB software.. The MATLAB program was 
built based on Lagrange equation. The tip load 
was changed and the motor velocity also changed 
and measured the deflection,   as shown in the 
results below: 

 

 
 

Fig. 9. Tip deflection at tip load=0 gram. 
 
 

Fig. 9 shows the variation of tip deflection 
with time for two velocity (90 and 180) 
degree/sec. and for tip attached mass equals zero 

gram. It shows that a time passes, the deflection 
fluctuates between a values. For v=90 degree/sec. 
blue curve, Maximum value was found to be (4.6 
mm). For v=180 degree/sec. red curve, Maximum 
value was found to be (9.9 mm).This figure also 
shows that the maximum tip deflection values 
decreases as the velocity of the servo motor 
decrease from 180 to 90 degrees/sec. 

 

 
 

Fig. 10. Tip deflection at tip load=27.4 gram. 
 
 

Fig. 10 shows the variation of tip deflection 
with time for two velocity (90 and 180) 
degree/sec. and for tip attached mass equals 27.4 
gram. It shows that a time passes, the deflection 
fluctuates between a values. For v=90 degree/sec. 



Rafal M. Khalil                                 Al-Khwarizmi Engineering Journal, Vol. 11, No. 3, P.P. 11- 19(2015) 
 

16 
 

 

blue curve, Maximum value was found to be (6.1 
mm). For v=180 degree/sec .red curve, Maximum 
value was found to be (1.31cm).This figure also 
shows that the maximum tip deflection values 
decreases as the velocity of the servo motor 
decrease from 180 to 90 degrees/sec. 

 

 
 

Fig. 11. Tip deflection at tip load=59.5 gram. 
 
 

Fig. 11 shows the variation of tip deflection 
with time for two velocity (90 and 
180)degree/sec. and for tip attached mass equals 
59.5 gram. It shows that a time passes, the 
deflection fluctuates between a values. For v=90 
degree/sec. blue curve, Maximum value was 
found to be (7.8 mm). For v=180 degree/sec. red 
curve, Maximum value was found to be 
(1.67cm).This figure also shows that the 
maximum tip deflection values decreases as the 
velocity of the servo motor decrease from 180 to 
90 degrees/sec. 

 
 

6.2. Experimental Results 
 

The experimental results were obtained from 
the single link flexible manipulator and the tip 
deflection measured using accelerometer stick at 
the end of the link .The results were taken when 
the base servo motor operated at velocity equal 
(90,180) degrees/sec and for 120 degrees using 
servo motor control card GUI and for three 
different loads. The experimental results were 
shown below: 

 
 

 
 

Fig. 12. Experimental tip deflection at tip load=0 
gram. 
 
 

Fig. 12 shows the variation of tip deflection 
with time for two velocity (90 and 180) 
degree/sec. and for tip attached mass equals zero 
gram. It shows that a time passes, the deflection 
fluctuates between a values. For v=90 degree/sec. 
blue curve, Maximum value was found to be (7 
mm). For v=180 degree/sec. red curve, Maximum 
value was found to be (1.2cm).This figure also 
shows that the maximum tip deflection values 
decreases as the velocity of the servo motor 
decrease from 180 to 90 degrees/sec. 

 

 
 
Fig. 13. Experimental tip deflection at tip load=27.4 
gram. 
 
 

Fig. 13 shows the variation of tip deflection 
with time for two velocity (90 and 180) 
degree/sec. and for tip attached mass equals 27.4 
gram. It shows that a time passes, the deflection 
fluctuates between a values. For v=90 degree/sec. 
blue curve, Maximum value was found to be (9.4 
mm). For v=180 degree/sec. red curve, Maximum 
value was found to be (1.72cm).This figure also 
shows that the maximum tip deflection values 



Rafal M. Khalil                                 Al-Khwarizmi Engineering Journal, Vol. 11, No. 3, P.P. 11- 19(2015) 
 

17 
 

 

decreases as the velocity of the servo motor 
decrease from 180 to 90 degrees/sec. 

 

 
 

Fig. 14.  Experimental tip deflection at tip load=59.5 
gram. 
 

Fig. 14 shows the variation of tip deflection 
with time for two velocity (90 and 180) 
degree/sec. and for tip attached mass equals 59.5 
gram. It shows that a time passes, the deflection 
fluctuates between a values. For v=90 degree/sec. 
blue curve, Maximum value was found to be (1.3 
cm). For v=180 degree/sec. red curve, Maximum 
value was found to be (1.96cm).This figure also 
shows that the maximum tip deflection values 
decreases as the velocity of the servo motor 
decrease from 180 to 90 degrees/sec. 

 
 

 

 

 

Table 2, 
Theoretical &Experimental tip deflection result at different load. 

Motor Velocity 
(deg/s) 

Tip Load 
(gram) 

 Theoretical Max. 
Tip deflection (mm) 

Experimental Max. 
Tip deflection(mm) 

Percentage Difference 

V=90 Zero 4.6 7 34 % 
V=180 9.9 12 17.5% 
V=90 27.4 6.1 9.4 35 % 
V=180 13.1 17.2 23.8 % 
V=90 59.5 7.8 13 40% 
V=180 16.7 19.6  
 

7. Conclusions 
 
The concluding remarks withdrawn from the 

obtained results as following: 
• Load: maximum tip deflection of the flexible 

link increases as the tip attached mass 
increases from 0, 27.4 and 59.5 gram as shown 
in result obtained previously . 

• Velocity: the deflection of the flexible link 
increases as the velocity of the base servo 
motor angle increases from [90-180] 
degrees/sec as shown in (Table:1).  

• Damping effect: the damping effect is clear 
after approximately 5 second and obvious in 
the experimental results only  because we 
neglect the damping effect at the theoretical 
part. 

 
 

8. References 
 
[1] Aarts R.G.K.M. and Jonker J.B.," Dynamic 

simulation of planar flexible   link 
manipulators using adaptive modal 

integration", Multibody System Dynamics, 
Vol.7, pp. 31–50, 2002. 

[2] Shaheed M.H.  andTokhi M.O., "Dynamic 
modeling of a single-link flexible   
manipulator: parametric and non-parametric 
approaches" Robotica, Vol.20, pp. 93–109, 
2002. 

[3] Sciavicco, L., and Siciliano, B., "Modeling 
and Control of Robot Manipulators", 
Springer London, 2004. 

[4] Book ,M.O. Tokhi and A.K.M. 
Azad,"Flexible Robot manipulator 
,Modeling, Simulation and Control" , The 
Institution of Engineering and Technology,  
series 68 ,pp. 19-20,2008. 

[5] Cannon  R.H. and Schmitz E., "Initial 
experiments on end-point control of a 
flexible one-link robot", The International 
Journal of Robotics Research Vol.3, No.3, 
pp.62–75, 1984. 

[6] Nagarajan S. and Turcic D.A., "Lagrangian 
formulation of the equations of motion for 
the elastic mechanisms with mutual 
dependencebetween rigid body and elastic 
motions", Part 1: Element level equations, 



Rafal M. Khalil                                 Al-Khwarizmi Engineering Journal, Vol. 11, No. 3, P.P. 11- 19(2015) 
 

18 
 

 

and Part II: System equations, ASME Journal 
of Dynamic Systems, Measurement, and 
Control, Vol.112, No. 2, pp.203–214, and 
pp.215–224, 1990. 

[7] Choi S.B., Thompson  B.S. and Gandhi 
M.V., "Modeling and control of a single-link 
flexible manipulator featuring a graphite–
epoxycomposite arm", Proceedings of the 
IEEE International Conference on Robotics 
and Automation, Vol.2, pp.1450–1455, 1990. 

[8] Choi, S.B., Gandhi, M.V. and Thompson,B. 
S., "An experimental investigation of an 
articulating robotic manipulator with a 
graphite epoxy arm". Journal of Robotic 
Systems, Vol.5, pp.73–79, 1988. 

[9] Krishnamurthy K.,  Chandrashekhara K. and 
Roy S., "A study of single-link robots 
fabricated from orthotropic composite 
materials", Computers and Structures, 
Vol.136, No.1, pp.139–146, 1990. 

[10] M. O. Tokhi and A. K. M. Azad,”Modelling 
of a single-link flexible manipulator system: 
theoretical and practical investigations”, 
Robotica , V.14,No.1,pp.91-102,1996. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 



 )2015( 11- 19، ���� 3، ا��
د11
ا���ارز�
 ا���
	�� ا�����	� �                                                          
	�لر�ل ���د 
 

19 
 

 

 
 

 &���% ا$��اف ذراع رو��ت ��ن ا��دي ا�����
 

%��' 

 داوود                *ر)% ���)� ���	**  
� ا������ ا��������� ا�)'ارز$#/ "!  ھ����اد/ +*�� ا�(�-. �/$01  * 

� ا�34 ا��0�2 ��� ا�)'ارز$# / #"!  ھ����اد/ +*�� ا�(�-. �/$01 ** 
 Rafalalazzawi90@gmail.com :ا�;7:� ا7�8�9و#5

 Somernacy@yahoo.com :ا�;7:� ا7�8�9و#5
  

 

 
 ا��*��

 
� ذات ا�(�8*�� ا��A57 ھ# .�:@ ا+?7 <0=�ان �D00ر5 ةاط7اف ا�7و.'ا�0ت ا���H$�� ذات ا�(�8*�� ا��*;� ا  .Iط7اف �D0وان �7و.'�0ت ا��� J$ �:�/*�

�:�K0ت ا�H�;40ل ا�7و.'�0ت ةا��K$ #> ا��.وا���4'رة M/. #>4�$ 7N$ 9 7$ا A57ام ا9ط7اف ا���(��� ا�����  وا��82   ھPاA و :0H�; @.0Hت ا�H/� #> ز:0دة
 J�!2� 0دة:Q� ه ا9ط7افP).0)=ادا .Pھ #>S2;راع $��'ع $ ا ا�Pا ا�Pوھ #D0�U 7و.'ت� �Vذراع $7ن وا P�Nم .����  و��'H5 W8;�.4'ل J $0دة ا�;7:0

� ا�Pراع$�7  ٠.٠٠٤ ^$�7 و�� 7D٠.٠٢ض ، 7�$  ٠.٥:5(0 #> A�;?$ �� `A:'50 ، $� و1'د +�*�a5ا �� ا�72+� ا�?50': :ھ#ھ�0ك ار./$'a�$ A و �$'a�$
�و  ا�?A:'50 ا�!�74ة$'a�$ :'50?ا� AK$ا��8;�'�7 و.7ا�� :7ان $;�أ �D@ ذراع ا�7و.'ت ا���D0# ا��7ن ھ' ��و. D0H*� '>7�!و:7   ة$72ك ا��وھ' :H'م .�

�D0Hو1'د  ةا� ��Dم و1'د وزن <# ا�74ف و�D ��D 90تV ثe` #> 0س ا7259افH:راع وPي :�2@ <# ط7ف ا�Pا7259اف ا� W�H5راع ا��7ن وPو$/(0 ا�
�0� $J ھPه ا�902ت H5'م .�72:^ $72ك ا� ٥٩.٥و  ٢٧.٤V @+ ��Dراع ا��7ن وP7ام <# ط7ف ا�jD7� ��D ة�D0H*� '>7�!��ار  �دورا�5 H�.٩٠ A1در/ 
 A�50`��ام .04"� �82  .�72+0ت ا�!�7<'  .'�0ط(���0� )  k*D @�/� )ATMEGA640 ا�0س $�82 ا��# اV @+ ��D راعPا7259اف <# ط7ف ا� W�H5و

4� V!0س �*!AD7 وا��وران  $J ھPه ا�902ت��ام �  ا�!�74ة D*�و) MMA7631(.'ا(��I. A� '�:ارد �V'�)Mega2560 ( J$ ا�;��0ت ��K. م'H5  `
k*D #�;$ بe�0��0. m$057. 0ء�. J$ 0)�*D 0�*�V #ا�� �:7aا�� m=0ا��� J�. A50رHب وا��e�0ا�� m$057;. 0)��ا�0س $/0د9ت  V!0س ا�!AD7 وا��وران ور

0)�*D 0�*�V #ا�� �� وا���m=0 ا�/�*�+72*� m59+7ا  .