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Al-Khwarizmi Engineering Journal,Vol. 13, No. 1, P.P. 1

 

Solving the Inverse Kinematic Equations of Elastic Robot Arm 

Utilizing 
 

Hussein M. Al-
*Department of Mechanical 

**College of Engineer

*Email: 

**Email: 

(Received 15

https://doi.org/10.22153/kej

Abstract  
 
 The inverse kinematic equation for a robot is very important to 
of this equation is complex for the rigid robot due to 

structure of robot link. In light robot arms,

than the rigid link robot because the deformation variables (elongation and bending) 

kinematic equation. The finding of an inverse kinematic equation

and both of the end-effector position and deformations variables. In this work, a neural network

solve the problem of inverse kinematic equation. To feed the neural network, experimental data were taken from an 

elastic robot arm for training the network, these data presented by joint angles, deformation variables and end

positions. The results of network training showed a good fit

targets data. In addition, this method for finding the inverse of kinematic equation proved its effectiveness and 

validation when applying the results of neural network 

real light robot’s position.  

 
Keywords: Elastic robot, forward and inverse kinematic equation of elastic robot.  Neural networks.

 

 

1. Introduction  
 

 The forward kinematic equation is a functional 

relationship between the generalized parameters 

(joint displacements) and the end 

positions. By substituting a set of values of 

generalized parameters into the forward kinematic 

equation, the considering ends – effector can be 

obtained. The problem of finding the end

position for a given set of generalized parameters

is called the direct kinematic problem

robot’s end-effector positioning and control, it i

required to find the generalized parameters that 

lead the end – effector to the specified position 

and orientation. This is done by finding the 

inverse of the forward kinematic equation (inverse 

Khwarizmi Engineering Journal,Vol. 13, No. 1, P.P. 13- 25 (2017) 

 

Solving the Inverse Kinematic Equations of Elastic Robot Arm 

Utilizing Neural Network 

-Khafaji*                 Muhsin J. Jweeg** 
Mechanical Engineering / University of Technology 

**College of Engineering/ Al- Nahrain University  

*Email: hussalkhafaji@gamil.com 

**Email: muhsinnah_eng@nahrainuniv.edu.iq 

 
15 March 2016; accepted 14 November 2016) 

https://doi.org/10.22153/kej.2017.11.002 

 
 

robot is very important to the control robot’s motion and position. The solving 

of this equation is complex for the rigid robot due to the dependency of this equation on the joint 

, where the flexibility exists, the solving of this problem is 

deformation variables (elongation and bending) are present

kinematic equation. The finding of an inverse kinematic equation needs to obtain the relation between the joint angles 

effector position and deformations variables. In this work, a neural network has

f inverse kinematic equation. To feed the neural network, experimental data were taken from an 

elastic robot arm for training the network, these data presented by joint angles, deformation variables and end

ng showed a good fit between the output results of the neural network and the 

targets data. In addition, this method for finding the inverse of kinematic equation proved its effectiveness and 

neural network practically in the robot’s operating software for controlling 

orward and inverse kinematic equation of elastic robot.  Neural networks.

The forward kinematic equation is a functional 

relationship between the generalized parameters 

(joint displacements) and the end – effector 

positions. By substituting a set of values of 

forward kinematic 

effector can be 

obtained. The problem of finding the end-effector 

position for a given set of generalized parameters 

called the direct kinematic problem [1]. In 

effector positioning and control, it is 

required to find the generalized parameters that 

effector to the specified position 

done by finding the 

inverse of the forward kinematic equation (inverse 

kinematic problem). The complexity of the 

inverse kinematic problem depends heavily on the 

number of joints and structure of the robot

Flexibility of robot arms will add additional 

in the forward kinematic equation

presented by the deformation variables

magnitude of these variables could 

estimated easily, because their magnitude 

upon external forces, inertia forces, positions and 

orientation of robot arms [3]. 

 The importance of the inverse kinematic 

problems in robot control, and the complications 

of solving it, make many researches deal with this 

subject, and many methods investigated to solve 

the inverse kinematic equation. 

widely applied solution is the clos

     
Al-Khwarizmi 

  Engineering  

Journal 

Solving the Inverse Kinematic Equations of Elastic Robot Arm 

 

control robot’s motion and position. The solving 

joint configuration and 

, the solving of this problem is more complicated 

present in the forward 

to obtain the relation between the joint angles 

has been proposed to 

f inverse kinematic equation. To feed the neural network, experimental data were taken from an 

elastic robot arm for training the network, these data presented by joint angles, deformation variables and end-effector 

between the output results of the neural network and the 

targets data. In addition, this method for finding the inverse of kinematic equation proved its effectiveness and 

ly in the robot’s operating software for controlling the 

orward and inverse kinematic equation of elastic robot.  Neural networks. 

kinematic problem). The complexity of the 

problem depends heavily on the 

number of joints and structure of the robot [2]. 

of robot arms will add additional terms 

in the forward kinematic equation, these terms 

presented by the deformation variables. The 

nitude of these variables could be not 

magnitude depends 

upon external forces, inertia forces, positions and 

of the inverse kinematic 

problems in robot control, and the complications 

of solving it, make many researches deal with this 

subject, and many methods investigated to solve 

 The common and 

widely applied solution is the closed form 



Hussein M. Al-Khafaji                       Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 13- 25(2017) 
 

14 

solution, literatures [3-8] presented different 

methods of closed form. A closed form solution is 

not always a possible way to find the solution, so 

various studies have been proposed using other 

techniques.  Juan   et. Al. [4] suggested the using 

of a Fuzzy reasoning system to position the arm of 

an articulated robot without explicitly solving the 

inverse kinematic equation. Yong et. al. [5] used 

Genetic algorithm to solve the inverse kinematic. 

Kuroe et. al. [11, 12] stated the use of neural 

networks as a new method of solving the inverse 

kinematics of a rigid robot manipulator. Previous 

research [13-15] examined the use of artificial 

neural networks to find the inverse kinematic 

equation for articulated robots. While, Shiv et. Al. 

[6] applied neuro-fuzzy to obtain the inverse 

kinematic equation for robot with three degrees of 

freedom.  

Most of researches proposed their solutions for 

the mathematical model of a rigid manipulator. 

With the presence of deflection parameters, the 

problem will be more complicated. Therefore, in 

this work, the proposed neural network with 

multiple layers was suggested as a function 

approximation to find the inverse kinematic 

equation for an elastic robot arm (ERA). The 

multiple layer neural network is useful for 

function approximation because [15]: 

• It is universal approximations. 

• It is efficient approximations. 

• It could be implemented as a learning machine. 

 

 

2. Kinematic Equations of an Elastic 
Robot Arm (with Error) 

 

 The flexibility causes elastic deformations of 

the structural members of the manipulator, 

resulting in large end-effector errors, especially in 

long reach manipulator systems. Hence, as a 

result, the frames defined at the manipulator joints 

are displaced from their expected locations. So, 

the use of kinematic equation of the rigid robot to 

position the manipulator end–effector will place 

the manipulator in a different position than the 

desired one. 

 In Figure 1, the frame Oi-1  is the base frame , 

Oi
I
 is the ideal location and  Oi

a
  is the actual 

location due to the deflection. 

 
 

Fig. 1. Frame displacement due to errors 

 

 

The transformation equation between the 

coordinates   O	�a   and O	�I   consists of two parts, 
rotation and translation. According to the Euler 

angle principle, the rotation part is the result of 

three sets of rotations which are roll, pitch and 

yaw about the axes z, y, x, respectively. The 

sequence of rotation is: ERaI � 	Rot	
z,	d��	Rot
y,	d��	Rot
x,	d��	 …(1) 
 

 That is a rotation of   d�  about the x axis, 
followed by a rotation   d�  about  the y  axis and 
finally a rotation  of  d�  about the  z  axis. 
 

Rot
x,	d�� � �1 0 0 00 cosd� �sind� 00 sind� cosd� 00 0 0 1�	 …(2) 
  

Rot
y,	d�� �	�cosd� 0 �sind� 0	0 1 0 0	sind� 0 cosd� 0	0 0 0 1	�	…(3) 
 

Rot(z, d�� �	�cos d� �sin d� 0 		0sin d� cos d� 0 		00 0 1 		00 0 0 		1�…(4) 
 

ERa   
I � ���� b�! c�" 0�!� b!! c!" 0�"� b"! c"" 00 0 0 1�  	 …(5) 

 

Where: a�� � cos d� cos d� a!� � sin d� cos d� a"� � �sin d� b�!#cos d� sin d� sin d� �sin d� cos d� b!! � sin d� sin d� sin d�$ cos d� cos d� b"! � cos d� sin d� 



Hussein M. Al-Khafaji                       Al
 c�" � cos d� sin d� cos d�$ sin d� sin d� c!" � sin d� sin d� cos d�� cos d� sin d� c"" � cos d� cos d� 
 

The equation …(5) represents the rotation part of 

the transformation matrix. The translation matrix 

is: 

ETa		I	 � �
1 0 0 δx0 1 0 δy0 0 1 δz0 0 0 1 �	

  

The transformation between coordinates   

and O	�I  becomes: 
 

Ea  
I � ETa  I ERa  I  

E'		(	 � �
��� b�! c�" δx�!� b!! c!" δy�"� b"! c"" δz0 0 0 1 �		 

 Here, the δx, δy, δz, d�, d� and d�
generalized error parameters. Usually

generalized parameters are introduced on the basis 

of different approximations, such as assumed 

modes, finite elements, or Ritz-Kantorovich 

expansions, with different implications on the 

model complexity and accuracy [9]. 

The references [17-19] have supposed that the 

generalized deflection parameters are small, so a 

first order approximation can be applied to their 

trigonometric functions and product, and higher 

order equals zero. sin d� = d�   , sin d� = d�      sin d� = d� cos d� =cos d�   =cos d� =1 
Based on these assumptions, the matrix 

…(8) will be equal to: 

E)*+, � �
1 �d� d� δxd� 1 �d� δy�d� d� 1 δz0 0 0 1 � 

 

 In this work, the values of deflection’s 

are taken in consideration. 

Figure 1 shows the coordinate for

flexible links, the kinematic equation covering the 

positions of this links with presence of flexibility 

is: 

T)*+,=A�-.q�/Ea 1  I 1 A!�.q!/Ea 2  I2 ⋯ Ann1�.qn/
 

Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 1

15 

represents the rotation part of 

the transformation matrix. The translation matrix 

	 	 							…(6) 
The transformation between coordinates   O	�a     

        …(7) 

        …(8) 

� are called 
generalized error parameters. Usually, these 

generalized parameters are introduced on the basis 

such as assumed 

Kantorovich 

expansions, with different implications on the 

have supposed that the 

generalized deflection parameters are small, so a 

first order approximation can be applied to their 

trigonometric functions and product, and higher 

the matrix Ea  
I

 in         

�   … (9) 
deflection’s parameters 

shows the coordinate for serial of 

the kinematic equation covering the 

links with presence of flexibility 

. /Ea n  I n …(10) 

An
n1�

  is the homogenous transformation matrix in 

Denavit- Hartenber (DH) equation

 

A221�
� �cos 32 � sin32 cos 42 sin 32sin 32 cos 32 cos 42 � cos 30 sin42 cos0 0
                                                                     

 

Where:  32	, 42	, ai 	and		d2 represent the joint parameter 
defined as in Figure 2. 

By using equation (10), the transformation 

relationship between the two frames 

can be found. 

 

 

Fig. 2. The Denavit-Hartenberg notatio

 

 T)*+, may be expressed in 
form as a nonlinear function of q and 

 T)*+, � 5
q, 7�					 
                       

Where: T)*+,  is the set of Cartesian coordinates 
describing the position and orientation of the 

manipulator’s end- effector with respect to the 

inertial frame. 

 	q � 8q� q! ⋯ ⋯ ⋯ ⋯
 7 � 87� 7	! ⋯ ⋯ ⋯ ⋯
    

Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 13- 25(2017) 

ansformation matrix in 

(DH) equation (11): 

32 sin 42 ai cos 3232 sin 42 ai 	sin 32cos 42 d20 1 � 
                                                                     … (11) 

represent the joint parameter 

, the transformation 

relationship between the two frames O	�a   and O�1�  

 

Hartenberg notatio. 

may be expressed in a simple compact 

form as a nonlinear function of q and 7 vectors: 
     …(12) 

is the set of Cartesian coordinates 

describing the position and orientation of the 

effector with respect to the 

⋯ q9:;		 			…(13) 
⋯ 7	<:=		   …(14) 



Hussein M. Al-Khafaji                       Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 13- 25(2017) 
 

16 

7 � 8δx> δy> δz? d�> d�> d�>:= 
 …(15) 

 

Where: q  is the vector in joint space. 7  is a dimensional link deflection vector space, 
spanned to be the link deflections of the 

manipulator arm. 

n   is the number of joints. @  generally equals to 6 x m. 
m   is the number of flexible  links. 

The elements of the vector 7 cannot easily specify 
their values, since these elastic coordinates are 
dependent on the rigid coordinate q and on the 

applied forces F. 

For more generally: 

 7 � A
F, q�		   …(16) 
 

 By substituting equation   …(16) into equation 

(12), T)*+, will be: 
 	T)*+, � 5
q, E
F, q��		   …(17) 
 

 As mentioned in the previous section, the two 

degrees of freedom of the arm robot manipulator 

used in this study are confined to move within the 

vertical plane, and so is the deformation of each 

link. Hence, there is no rotation about the axes x 

and y, also there is no transformation in the z 

direction (the parameters δz, d� and d� are equal 
zero). Figure 3 shows a single flexible link 

coordinate system, the O21� represents the base 
local coordinate system, and the frame O	2(  is the 
local coordinate system assigned to link i in its 

undeformed position, while the O	2a   is the actual 
local coordinate system of link i , when it is under 

deformation.  

 The homogenous transformation matrix for 

this single link was derived as follow: 

• According to the motion of the links within the 
vertical plane, the homogenous transformation 

matrix  E2a  (  between O	2a  and  O	2I  becomes: 
 

E2'		2(	 � �
cosd� �sind� 0 			δxsin d� cosd� 0 		δy0 0 1 			00 0 0 		1 �  …(18) 

 

• The homogenous transformation matrix 

between the O	�I and O�1� is presented by the 
matrix A��1� in equation (9). 

 

• The homogeneous transformation matrix for 
this a single link is found from the relations   

(12) and (18). T2C21�' � A221�'	E2'		2(	 					            …(19) 
 

T2C21� � �
T�� T�! T�" T�ET!� T!! T!" T!ET"� T"! T"" T"E0 0 0 1 �     …(20) 

Where: F2 � FGH 32																		H2 � sin32					F42 � cos 42		 H42 � sin42 FI2 � cosI�2			 HI2 � sinI�2 T�� � F2FI2 � F42H2HI2 T!�												 � s�cd� 	$ sd�cα�c�		T"� � H42HI2 T�! � �F2HI2 � F42H2FI2 T!! � �H2HI2 $ F2FI2F42 T"! � H42FI2		 T�" � H42H2 T!" � �H42F2 T"" � F42 T�E � F28KL2 $ �2: � F42H2KM2 T!E � H28KL2 $ �2: $ KM2F42F2 T"E � H42KM2 $ d2 
 

 
 

Fig. 3. The coordinates system of a single flexible 

link under deformation. 

 

 

As shown in Figure 4, the total kinematic 

equation of flexible robot of two degrees of 

freedom is:  

T)*+, � T�CN'	T!C�'             …(21) 
 



Hussein M. Al-Khafaji                       Al
 

The transformation matrix of robot in general 

form is: 

T)*+, � �
OP GP QP RPOS GS QS RSOT GT QT RT0 0 0 1 �          

Where: 

 

 

Fig. 4. the coordinates systems of two elastic links and the angles of joints

nU � c�c!8cd!sd�cα� $ cα�cd�cα!sd!�	s�s!8sd�cd! $ cd�c$ c!s�8cd�cd! � sd�sd$ c�s!8cd!cd�cα�� sd�cα�sd!cα!:� c�8sα�sd!sα!:	
                                                        

 nV � c!8sα�sd�cd! $ sα�cd�cα!sd!:$ s!8sα�cd�cd!� sα�sd�sd!cα!: $ cα	 	 																																																				 	oW � s�s!8cα�sd�cα!cd! $ cα�cd�sd!� c�c!8cd�sd! $ sd�cd$ c�s!8sd�sd! � cd�cα$ s�c!8cα�cd�cd!� cα�sd�sd!cα!:$ s�sα�sα!cd!																																																																										

Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 1

17 

robot in general 

           …(22) 

nW � 	c�c!8cd�cd! � sd�sd!cα!� c�s!8cd�cα!sd$ s�s!8cα�cα!sd� cα�cd�cd!:� c!s�8cd!sd�c$ cα�cd�cα!sd!$ s�sα�sα!sd!	
     

the coordinates systems of two elastic links and the angles of joints.

!:cα!sd!:sd!cα!:
                                                         …(24) 

α�sd!sα!																																																				…(25) 	 	 			…
25�	!:cd!cα!:α!cd!:
													…	
26�	

oU � c�c!8cd�cα�cd!cα! � sd�c� s�c!8sd�cd!c� c�s!8cd�cα�sd$ sd�cα�cα!cd!$ s�s!8sd�sd! �� c�8sα�sα!cd!																																																													oV � c!8sα�cd�cd!cα! � sα�sd$ s!8sα�cd�sd!� sα�sd�cα!cd!																																																																								tW � c�s!cd�sα! � s�s!cα�sd�s$ c�c!sd�sα! $$ s�sα�cα!																																																																														tU � s�s!cd�sα! $ c�s!sd�cα�s$ s�c!sd�sα! �� c�sα�cα! 
                                                                  

 tV � s!sα�sd�sα! � c!sα�cd�sα
                                                            

Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 13- 25(2017) 

!:sd! $ sd�cd!:sd�sd!: cα�!:	
    …(23) 

cα�sd!:cα! $ cd�sd!:sd!!:� cd�cα!cd!:!:																																																																												…	
27�	
sd�sd!:!!: $ cα�sα!cd!																																																																											…	
28�	sα!$ s�c!cα�cd�sα!

																																																																												…	
29�	
sα!� c!cd�cα�sα!

                                                                  … (30) 

α! $ cα�cα! 
                                                            … (31) 

 



Hussein M. Al-Khafaji                       Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 13- 25(2017) 
 

18 

 

RP � F�F!8FI�
KL! $ �!� � HI�KM!F4!:� F�H!8FI�F4!KM!$ HI�
KL! $ �!�:� H�F!8F4�HI�
KL! $ �!�$ F4�FI�KM!F4!:$ H�H!8F4�HI�F4!KM!� F4�FI�
KL! $ �!�:$ H4�H�H4!KM! $ H4�H�I!$ F�
KL� $ ��� � F4�H�KM� 
… (32) 

 pU � s�c!8cd�
δx! $ a!� � sd�δy!cα!:� s�s!8sd�
δx! $ a!�$ cd�cα!δy!:$ c�c!8sd�cα�
δx! $ a!�$ cd�cα�δy!cα!:$ c�s!8cd�cα�
δx! $ a!�� sd�cα�cα!δy!: $ c�8cα�δy�� sα�sα!δy! � sα�d!:$ s�8δx� $ a�:																																																																														…	
33�		pV � c!8sα�sd�
δx! $ a!� $ sα�cd�δy!cα!:$ s!8sα�cd�
δx! $ a!�� sα�sd�cα!δy!: $ cα�sα!δy!$ cα�d! $ sα�δy� $ d�																																																																														…	
34�		c� � cos θ�				cd� � cos dϕ�			cα� � cosα�	s� � sin θ�	 sd� � sindϕ�	 								sα� � sin α�				 The	 degree	 of	 freedom	 of	 the	 robot	 arm	manipulator	which	is	used	in	this	work,	is	two	degrees	 of	 freedom,	 and	 confined	 to	 move	within	 the	 vertical	 plane	 with	 the	 kinematic	parameters	listed	in	the	Table	
1�.		Table	Table	Table	Table	1111,,,,				The	parameters	of	two	linksThe	parameters	of	two	linksThe	parameters	of	two	linksThe	parameters	of	two	links....				JointsJointsJointsJoints				 pq				 ddddq				 rq				 aaaai	i	i	i					1111				 θ�	 0	 0	 a1		2222				 ps				 0000				 0000				 aaaa2222									 			 By	substituting	the	values	of	parameters	of	the	manipulator	from	the	Table	
1�	in	to	the	equations	 
23�	 to	 
34�	 and	 with	 triangular	operations,	these	equations	will	be	as	follow:		nW � 	8c�!:8cd�!: � 8s�!:8sd�!:																					…
35�		nU � c�c!8sd�!: $	s�s!8sd�1!: $ c!s�8cd�1!:$ c�s!8cd�!:																																																																														…	
36�	

nV � 0																																																																…
37�		oW � s�s!8sd�1!: � c�c!8sd�!: � c�s!8cd�1!: �										s�c!8cd�!:																																																…
38�	oU � c�!8cd�!: � s�!8sd�1!::	 																		…
39�	oV � 0																																																																…
40�	tW � 0	 																																																														…
41�	tU � 0																																																																	…
42�	tV � 1																																																																	…
43�	pW � c�c!8cd�
δx! $ a!� � sd�δy!:� c�s!8cd�δy!$ sd�
δx! $ a!�:� s�c!8sd�
δx! $ a!�$ cd�δy!: $ s�s!8sd�δy!� cd�
δx! $ a!�: $ c�
δx�$ a�� � s�δy�																																																																											…	
44�	pU � s�c!8cd�
δx! $ a!� � sd�δy!:� s�s!8sd�
δx! $ a!�$ cd�δy!: $ c�c!8sd�
δx!$ a!� $ cd�δy!:$ c�s!8cd�
δx! $ a!�� sd�δy!: $ c�δy� $ s�8δx�$ a�:																																																																												…	
45�	pV � 0	 																																																															…
46�	Where:	c�! � cos
θ� $ θ!�	s�! � sin
θ� $ θ!�	cd�! � cos
dϕ� $ dϕ!�		sd�! � sin
dϕ� $ dϕ!�	cd�1! � cos
dϕ� � dϕ!�	sd�1! � sin
dϕ� � dϕ!�		
 Equations (44) to (46) represent the end-

effector position with presence of flexibility of 

robot, which will be called, the flexible kinematic 

equation. The inverse of these equations will be 

found by using the neural network. 

 
 

3. Experimental Work 
a. The rig of Experiment 
 

 In this work, the test rig was represented by 

two degrees’ manipulator (two elastic links) 

confined to move in a vertical plane. The two 

links were designed to have properties gathering 

the lightness approximate to the flexible beam and 

stiffness approximate to the rigid beam to reduce 

the vibration effects; the two links are hallow 

rectangular beams made from the aluminum type 

(6061-T6) with length 57 cm and 62 cm for first 

and second arm, respectively. In addition, the 

manipulator has a measurement system (bonded 



Hussein M. Al-Khafaji                       Al
 

strain gauges on each arm) and data acquisition 

system to measure the deflection variable

the rotation of the tip of the beam 

experimentally, the measuring of the deflections 

at two points along the beam was done

points where the strain gauge No.3 and at the tip, 

and by applying these values of the deflections 

into the following equation [20] :  tanI� �	∆xS∆P    
Where:  δy	 represent the deflection. ∆δy	 represents the difference between the two 
deflections. ∆L  represents the distance between the two 
points where the deflections measured. 

 The relations between the deflections and the 

voltages of strain gauges were estimated 

experimentally by curve fitting between deflection

and voltage as shown in Figure 5. 

 
For Link-1: 

 
The relation between the tip deflection and the

voltage of the strain gauge No.1 is: KMy2z � 0.0026	{ $ 8 | 101}  
The relation between the tip deflection and the 

voltage of the strain gauge No.2 is: KMy2z � 0.001	{ $ 3 | 101}  
The relation between the deflection at the 

point 31cm from the center of the shaft of the first 

joint and the voltage of the strain gauge No.3 is:KMCy	"�	~> � 0.0008	{ � 2 | 101} 
 
The Link-2: 

 
The relation between the tip deflection and the 

voltage of the strain gauge No.1 is: KMy2z � 0.0016	{ $ 6 | 101}	  
The relation between the tip deflection 

and the voltage of the strain gauge No.2 is:KMy2z � 0.0022	{ $ 3 | 101}		 	
The relation between the deflection at the 

point 36 cm from the center of the shaft of the 

second joint and the voltage of the strain gauge 

No.3 is: KMCy	"�	~> � 0.0022	{ � 3 | 101}	KL was experimentally found by extracting the 
strain from two sensors at the upper and lower 

surface at the same position of the strain gauge 

No.1 and No. 2. The difference value represents 

the extension of the beam in the longitudinal 

direction, as represented by the relationKL � 7. �                  
 

Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 1

19 

strain gauges on each arm) and data acquisition 

deflection variable, To find 

the rotation of the tip of the beam dϕ 

experimentally, the measuring of the deflections 

at two points along the beam was done at the 

points where the strain gauge No.3 and at the tip, 

values of the deflections 

   … (47) 

the difference between the two 

represents the distance between the two 

 

The relations between the deflections and the 

voltages of strain gauges were estimated 

between deflection 

The relation between the tip deflection and the 

   …(48) 
The relation between the tip deflection and the 

 
    …(49) 

The relation between the deflection at the 

from the center of the shaft of the first 

joint and the voltage of the strain gauge No.3 is: 

    …(50) 

The relation between the tip deflection and the 

  …(51) 
The relation between the tip deflection 

and the voltage of the strain gauge No.2 is: 	 					…(52) 
The relation between the deflection at the 

point 36 cm from the center of the shaft of the 

and the voltage of the strain gauge 

	 				…(53)	
was experimentally found by extracting the 

strain from two sensors at the upper and lower 

surface at the same position of the strain gauge 

difference value represents 

the extension of the beam in the longitudinal 

direction, as represented by the relation [20]: 

           …(54) 

Figure 6 shows the two degrees robot arm 

manipulator and the strain gages.	

 

Fig. 5. the distribution of strain gauges and applied 

load for the two links. 

 	

 

Fig. 6. the two degrees robot arm manipulator and 

the strain gages. 

Link-2 

Link-1 

Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 13- 25(2017) 

6 shows the two degrees robot arm 

manipulator and the strain gages.	

	
the distribution of strain gauges and applied 

	
the two degrees robot arm manipulator and 



Hussein M. Al-Khafaji                       Al
 

b. Collection of Data 
 

 The experimental data for feeding the neural 

networks for training include joint’s angle, 

deformation variables and corresponding end

effector positions. To collect data, a control mode 

was designed to operate the robot 

sequence of motions, which include moving the 

first links in a range of angles (0 to 30

and the second link ( -35
o
 to 60

o
 step 5

the control mode is activated, the first link

at angle 0
o
, and the second link is still a while at 

the start angle, then moves 5
o
 to the next step

After completion the range of angles, 

arm returns back to the start angle, and the first 

joint moves 5
o
 to next step and so on. At each step 

(position), the deformation variable 

effector position were measured 

automatically. This procedure was repeated for 

four values of load (0,1,2, and 3) kg hang

the end of the link two. The number 

for each parameter was 560 reading, which will be 

used for training neural network. Figure

the samples of positions during this test.

 

 

Fig. 7. samples of positions during experimental 

work. 

 
 

 Some samples of the results of the deformation

parameters, which were read in above 

shown in the figures (figure 8 to figure 

figures represent the change of values for 

and d�	at the tip with angles of the two links
These results were used as inputs to equations 

(44) and   (45) to calculate the end effector in x 

and y coordinate that would be with the measured 

generalized parameters as input values to the 

neural network. 
 

Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 1

20 

for feeding the neural 

include joint’s angle, 

corresponding end-

a control mode 

was designed to operate the robot in specific 

ch include moving the 

30
o
 step 5

o
 ) 

step 5
o
). When 

, the first link is still 

, and the second link is still a while at 

to the next step. 

, the second 

returns back to the start angle, and the first 

. At each step 

 and end-

 and saved 

. This procedure was repeated for 

,1,2, and 3) kg hanging at 

 of readings 

which will be 

Figure 7 shows 

of positions during this test. 

 

samples of positions during experimental 

deformation 

, which were read in above test, are 

figure 13), these 

represent the change of values for δ�, δ� 
at the tip with angles of the two links. 

These results were used as inputs to equations 

calculate the end effector in x 

and y coordinate that would be with the measured 

generalized parameters as input values to the 

 

Fig.8 δx Vs. θ2 at θ1=4 Deg. link
 

 

Fig. 9. δy Vs. θ2 at θ1=0 Deg.  link
 

 

Fig.  10.  dϕ Vs. θ2 at θ1=0 Deg.  link

 

 

Fig. 11. δx Vs. θ2 Deg. Link

 

Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 13- 25(2017) 

 

θ1=4 Deg. link-1. 

 

θ1=0 Deg.  link-1. 

 

θ1=0 Deg.  link-1. 

 

θ2 Deg. Link-2. 



Hussein M. Al-Khafaji                       Al
 

 

Fig. 12. δy  Vs. θ2 Deg. Link-2
 

 

Fig. 13  dϕ  Vs. θ2 Deg. Link-2
 

 

4. Neural Network for Inverse K
Equation of Elastic Robot 

 

 As mentioned in previous section, there is 

difficulty to solve the problem for finding the 

inverse of the kinematic equation with presence

the deformation parameters. So, to overcome this 

difficulty, the neural network was used 

inverse of the kinematic equations (4

These equations relate the vector of operating

(task) space coordinate to the joints and 

deformations vector, these equations could be 

presented in compact form: � � 5
�, 7�                                                   
Where:  � � 8RP, RS:  ,          RT � 0                         � � 83�,3!:																																																	7 � �KL1,KM1,I�1,KL2,KM2,I�2�									
 

 The inverse kinematic problem of rigid robot is 

defined as giving end effector positions (

the joint variable vector (��, in the elastic robot
beside the vector ( � ), there will 
deformations vector 7 that should be given to get 
(��. So, the inverse equation will be:  

Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 1

21 

 

2. 

 

2. 

Kinematic 

section, there is a 

difficulty to solve the problem for finding the 

inverse of the kinematic equation with presence of 

So, to overcome this 

used to find the 

(44) to (46). 

the vector of operating 

(task) space coordinate to the joints and 

deformations vector, these equations could be 

                                                   …(55) 

                         …(56) 									…(57) � 									…(58) 
The inverse kinematic problem of rigid robot is 

d effector positions (�) to get 
elastic robot. 

there will be the 

should be given to get 

� � 51�
�, 7�                                              
 The proposed neural network was used to 

model the equation (59), which

kinematic inverse neural network.

many neural networks were trained. The structure 

of each network was with (one 

two hidden layers-output layer). Each network 

had a different number of neurons in the hidden 

layers. The numbers was chosen randomly. The 

selection of neurons number was specified by trial 

and error. The input layer received the eight 

variables (δy1, δx1, dϕ1, δy2, δ
py). The output layer gave the angles (

network result, as shown in Figure 1

 

 

Fig. 14. the structure of kinematic inverse neural 

network. 

 

 

The transfer function (τ) used in the hidden 
layers for each network was the hyperbolic 

tangent function (due to the bipolar value for the 

output of the function) as shown in equatioτ
x� � tanh
L� � *��1*���*���*��� 
 

Where  � is the slope parameter.
For the output layer was pure line. The back 

propagation training algorithm with one variable 

learning rate was used to train these neural 

networks. 

 The input variables (δy1, δx1, d
dϕ2)   were imported from the results of the test in 
the part of experimental work and by using these 

variables with corresponding angles (

in equations (44) and (46) to calculate the last two 

variables (px and py). Hence, these variables were 

used to train the networks. 

The networks that used were simulated by 

using the scientific and engineering package 

Matlab. 

 To find the optimum neural network, the tested 

networks output was compared with respect to the 

Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 13- 25(2017) 

                                              …(59) 

The proposed neural network was used to 

which is called the 

kinematic inverse neural network. In this work, 

many neural networks were trained. The structure 

one input layer and 

output layer). Each network 

had a different number of neurons in the hidden 

layers. The numbers was chosen randomly. The 

selection of neurons number was specified by trial 

received the eight 

δy2, δx2, dϕ2, px, and 
py). The output layer gave the angles (θ1, θ2) as a 

Figure 14. 

 

the structure of kinematic inverse neural 

used in the hidden 

layers for each network was the hyperbolic 

tangent function (due to the bipolar value for the 

as shown in equation [21]: 

      …(60) 

is the slope parameter. 

the output layer was pure line. The back 

propagation training algorithm with one variable 

learning rate was used to train these neural 

δ δx1, dϕ1, δy2, δx2, 
2)   were imported from the results of the test in 

experimental work and by using these 

variables with corresponding angles (θ1 and  θ2 )  
to calculate the last two 

variables (px and py). Hence, these variables were 

were simulated by 

using the scientific and engineering package 

To find the optimum neural network, the tested 

compared with respect to the 



Hussein M. Al-Khafaji                       Al
 

angles that were used in equations (44) 

get px and py. The best fit obtaine

network has the following structure, as shown in

Figure 14: 

1- One input layer (receives the eight variables).
2- Two hidden layers: 

a- n=15 neurons in the first hidden layer.
b- k=13 neurons in second hidden layer.

3- One output layer (gives two outputs).
The Network training parameters that used in this 

network are listed as follow: 

 

 Training parameters  Value 

 Epochs   5000 

 Goal    1×10-8

 Minimum gradient  1×10-10

 Learning rate   1×10-5

 Learning increment  0.9 

 

 

5. Results and Discussion  
 

 Figures 15 and 16 show the comparison 

between the optimum neural network

outputs and the target data for (θ1 and 
respectively. These figures reveal a good fitting 

between the network results and the target angles

The roots square errors of result networ

(0.999742) and (0.999804) for the first and the 

second angles, respectively. 

 

 

Fig. 15. The comparison between θ1 target and 
ONN. 

 

Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 1

22 

) and (45) to 

get px and py. The best fit obtained with the 

network has the following structure, as shown in 

One input layer (receives the eight variables). 

hidden layer. 

hidden layer. 

One output layer (gives two outputs). 

Network training parameters that used in this 

  

  

8  

10  

5  

 

show the comparison 

between the optimum neural network (ONN) 

outputs and the target data for (θ1 and θ2), 
respectively. These figures reveal a good fitting 

between the network results and the target angles. 

of result network were 

(0.999742) and (0.999804) for the first and the 

 

θ1 target and θ1 

 

Fig. 16 the comparison between θ
ONN. 
 

 

Figures 17 and 18 depict the comparison 

between the px and py, that resulted from the 

equations (44) and (45) with the neural network. 

Each figure has two curves; the first one is from 

applying the target angles, and the second one is 

resulted from applying the ONN.

 The Figures from 15 to 18 show a good fitting 

between the neural network results and the target 

angles, so that the results of this 

were used in the control system of the elastic 

robot to control the motion and position

 

    

Fig. 17 the comparison between px target and px 

ONN. 

 

 

Fig. 18. the comparison between py target and py 

ONN. 

Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 13- 25(2017) 

 

the comparison between θ2 target and θ2 

depict the comparison 

 resulted from the 

the neural network. 

Each figure has two curves; the first one is from 

applying the target angles, and the second one is 

NN. 

show a good fitting 

network results and the target 

the results of this neural network 

control system of the elastic 

robot to control the motion and position.  

 

the comparison between px target and px 

 

the comparison between py target and py 



Hussein M. Al-Khafaji                       Al
 

As mentioned above, there are many networks 

with different structures that were experimented 

to find the best one. At the beginning, the 

networks were with one hidden layer trained, but 

the results were far from the target in spite of 

increasing the number of neurons. Therefore

using networks with two hidden layers, the result 

began to approach from the target. Figure

22 illustrate the results for some of the networks 

that were experimented. The way that these 

networks were named depended upon how many 

neurons were used in each layer, so the name 

(network 12-40) means the first layer has 12 

neurons, and the second layer has 40 neurons.

 

 

Fig.19.  θ1 Target Vs.θ1 neural network two layers
 

 

Fig. 20.  θ1 target Vs. θ1 neural network two layers
 

 

Fig. 21. θ2 target Vs. θ2 neural network two layers

Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 1

23 

As mentioned above, there are many networks 

were experimented 

to find the best one. At the beginning, the 

with one hidden layer trained, but 

the results were far from the target in spite of 

Therefore, by 

using networks with two hidden layers, the result 

Figures 19 to 

for some of the networks 

that were experimented. The way that these 

networks were named depended upon how many 

neurons were used in each layer, so the name 

40) means the first layer has 12 

neurons, and the second layer has 40 neurons. 

 

1 neural network two layers. 

 

1 neural network two layers. 

 

2 neural network two layers. 

 

Fig. 22. θ2 target Vs. θ2 neural network two layers
  

 

6. Conclusion 
  

The first advantage of using neural network is 

to overcome the difficulties of kinematic inverse 

equation of the flexible robot, and the 

advantage is that it takes the values of training 

from the experimental work by operating the 

robot itself, and measures the variables with 

presence of more than one effect during the 

operation (like joint’s stiffens, joint’s gear box 

backlash, noise, etc..). These effects will 

in consideration to establish the network 

automatically (during learning to find

approximation), Therefore, the result

when used in the control system of robot will

overcome the problem of these effects, but may be 

when using the other methods

algebraic method, the inverse kinematic equation 

will be solved mathematically and

consideration these effects. The proposed neural 

network gave a good fit with the target. The roots 

square errors of results network were 

and (0.999804) for the first and the second angles, 

respectively. And it is found from the training and 

testing of many networks that, not conditionally, 

the increase of number of neurons in the hidden 

layers will give the best result. Also, this neural 

network was used in the operating software of 

ERA robot  and showed a good performance to 

position the end effector of the elastic 

 

 

7. References 
 

[1] Slotine, H. Asada and J. J. E., Robot 
Analysis and Control, New York: John Wiley 

and Sons, 1985.  

[2] Manseur, Rachid, Robot Modeling and 
KIneamtics, India : Firwall Media, 2007. 

[3] Bingul, Serdar Kucuk and Zafer, "The 
inverse kinematics solutions of industrial 

Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 13- 25(2017) 

 

2 neural network two layers. 

The first advantage of using neural network is 

to overcome the difficulties of kinematic inverse 

equation of the flexible robot, and the second 

the values of training 

from the experimental work by operating the 

the variables with 

presence of more than one effect during the 

operation (like joint’s stiffens, joint’s gear box 

hese effects will be taken 

in consideration to establish the network 

automatically (during learning to find the function 

he result of network 

when used in the control system of robot will 

overcome the problem of these effects, but may be 

the other methods, especially the 

the inverse kinematic equation 

l be solved mathematically and will not take in 

The proposed neural 

network gave a good fit with the target. The roots 

of results network were  (0.999742) 

and (0.999804) for the first and the second angles, 

tively. And it is found from the training and 

testing of many networks that, not conditionally, 

the increase of number of neurons in the hidden 

layers will give the best result. Also, this neural 

network was used in the operating software of 

a good performance to 

elastic robot. 

Slotine, H. Asada and J. J. E., Robot 

Analysis and Control, New York: John Wiley 

Manseur, Rachid, Robot Modeling and 

KIneamtics, India : Firwall Media, 2007.  

Bingul, Serdar Kucuk and Zafer, "The 

inverse kinematics solutions of industrial 



Hussein M. Al-Khafaji                       Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 13- 25(2017) 
 

24 

 

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[5] Michael Wenz and Heinz Wörn, "Solving the 
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[7] E. Sariyildiz and H. Temeltas, "Solution of 
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[8] Emre Sariyidiz, Eray Cakiray and Hakan 
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[9] Juan Martinez, John Bowles and Patrick 
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Learning and Cybernetics, 12-15 July, 2009.  

[11] Yasuaki Kuroe, Yasuhiro Nakai and 
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[12] Yasuaki Kuroe, Yasuhiro Nakai and 
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[13] Jolly Shah, S. S. Rattan, B.C. Nakra , 
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[14] Panchanand Jha, BB Biswal, "A Neural 
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[15] Adrian- Vasile Duka , "Neural Network 
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[16] Shiv Manjaree , Vijyant Agarwal and B.C. 
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6 , no. 4, pp. 557-562, 2013.  

[17] U. Y. X. a. T. Y. Hiroshi, "Modeling and 
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Robot," in Workshop on Intelligent Robots 

and Systems IROS, 1991.  

[18] C. S. D. P. D. J. H. a. J. F. Mavroidis, "A 
systematic Error Analysis of Robotic 

Manipulators: Application to High 

Performance Medical Robot," in 

International Conference in Robotic and 

Automation, Albuquerque, 1997.  

[19] J. Zhao-Hui, "Compensability of End-
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Arms," IEEE, pp. 486-491, 1992.  

[20] Stephen P. Timoshenko and James M. Gere, 
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