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All content in this area was uploaded by Abdel-Nasser Sharkawy on May 09, 2023
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IJRCS
International Journal of Robotics and Control Systems
Vol. 3, No. 2, 2023, pp. 330-353
ISSN 2775-2658
http://pubs2.ascee.org/index.php/ijrcs
http://dx.doi.org/10.31763/ijrcs.v3i2.1017 ijrcs@ascee.org
Forward and Inverse Kinematics Solution of A 3-DOF
Articulated Robotic Manipulator Using Artificial Neural
Network
Abdel-Nasser Sharkawy a, b,1, *, Shawkat Sabah Khairullah c,2
a Mechanical Engineering Department, Faculty of Engineering, South Valley University, Qena 83523, Egypt
b Mechanical Engineering Department, College of Engineering, Fahad Bin Sultan University, Tabuk 47721, Saudi Arabia
c Department of Computer Engineering, College of Engineering, University of Mosul, Mosul, Iraq
1 abdelnassersharkawy@eng.svu.edu.eg; 2 shawkat.sabah@uomosul.edu.iq
* Corresponding Author
1. Introduction
In the research literature review [1]–[3], the positions and orientations of the end-effector in the
forward kinematics problem are determined using the joints’ variables. This process is
straightforward, easy, and not complex. This part is very important and crucial for calculating the error
of the position and/or the orientation to calculate the controller’s qualify [4]. In contrast, inverse
kinematics is opposite to forward kinematics in such a way that the joints’ variables can be calculated
using known positions and orientations of the robot end-effector. This process is complex, difficult,
ARTICLE INFO
ABSTRACT
Article history
Received March 26, 2023
Revised May 01, 2023
Accepted May 09, 2023
In this research paper, the multilayer feedforward neural network
(MLFFNN) is architected and described for solving the forward and
inverse kinematics of the 3-DOF articulated robot. When designing the
MLFFNN network for forward kinematics, the joints' variables are used as
inputs to the network, and the positions and orientations of the robot end-
effector are used as outputs. In the case of inverse kinematics, the
MLFFNN network is designed using only the positions of the robot end-
effector as the inputs, whereas the joints’ variables are the outputs. For both
cases, the training of the proposed multilayer network is accomplished by
Levenberg Marquardt (LM) method. A sinusoidal type of motion using
variable frequencies is commanded to the three joints of the articulated
manipulator, and then the data is collected for the training, testing, and
validation processes. The experimental simulation results demonstrate that
the proposed artificial neural network that is inspired by biological
processes is trained very effectively, as indicated by the calculated mean
squared error (MSE), which is approximately equal to zero. The resulted in
smallest MSE in the case of the forward kinematics is in the
case of the inverse kinematics, is . This proves that the
proposed MLFFNN artificial network is highly reliable and robust in
minimizing error. The proposed method is applied to a 3-DOF manipulator
and could be used in more complex types of robots like 6-DOF or 7-DOF
robots.
Keywords
Artificial Neural Network;
biological;
Forward Kinematics;
Inverse Kinematics;
3-DOF Robot;
Sinusoidal Motion
This is an open-access article under the CC–BY-SA license.
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Abdel-Nasser Sharkawy (Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator
Using Artificial Neural Network)
and computationally expensive [3]. This part is very important for determining the robot’s motion to
reach the desired position and for programming the manipulator for performing tasks.
For the forward kinematics problem, different types of solutions were considered by researchers:
1) Denavit and Hartenberg (DH), 2) product of exponential (POE), 3) dual-quaternion, 4) geometric
approach, and 5) machine learning-based approaches as support vector regression (SVR) and neural
network (NN), as shown in Fig. 1. Denavit and Hartenberg in [5] proposed a new method including
four parameters that are considered necessary for the transformation procedure that occurs between
two joints. All the parameters have been named as the D and H parameters. Furthermore, the four
parameters were considered the operational standard for describing the kinematics of the robot.
Although the presented D and H method is the most concise method compared with all, it contains
some limitations, as described in [6]. For example, Asif and Webb in [7] performed the forward
kinematics of a 6-DOF articulated robot with a spherical wrist by the use of the DH parameters. The
product of exponential and unit dual quaternion was proposed for forward kinematics problems [8],
[9]. In [10], a comparison research work was developed between the POE method and the unit dual
quaternion in determining the forward motion of the 7-DOF KUKA LWR robot. Their experimental
results proved the unit dual quaternion has a higher compactness compared with the product of the
exponential method. In addition, the unit dual quaternion facilitates better comprehension of the
geometrical purpose of the joint axes. The geometric approach has also been proposed for the forward
motion problem. In [11], Kim et al. used the conformal geometric algebra for utilizing three SPS/S to
analyze the forward kinematics of the duplicated motion manipulator. An additional sensor was
required to provide more positional information and to allow the unique solution to be chosen
geometrically from the many found solutions using the geometric approach. SVR [12] is a supervised
machine learning method used widely in regression tasks. NN [13]–[15] has the ability to approximate
any linear/nonlinear function and generalize it in the case of different conditions. SVR was used with
the forward motion of parallel manipulator robots [16]–[19].
Fig. 1. The different types of solutions used for the forward kinematics problem of a robot
For the inverse kinematics problem, also different types of solutions were considered by
researchers, such as 1) closed-form solution method, 2) numerical solution method, 3) evolutionary
computing, and 4) neural networks (NNs), as shown in Fig. 2. Closed-form solution method was used
when the analytic expression or the polynomial has less than 4-DOF [2]. This solution depended on
robot-specific geometry for the formulation of the mathematical model. An example of this approach
was presented in [20], where Lou et al. used the closed-form solution for the inverse kinematics of the
manipulator based on the general spherical joint. Numerical methods were proposed for the kinematics
problem in the inverse mode in case of the resulting polynomial in the solution has more than 4-DOF
[21]. These mentioned methods are less accurate compared to the methods of closed-form [2]. An
Solutions For Forward
Kinematics
Denavit–Hartenberg
Product of Exponential
Dual-Quaternion
Geometric Approach
Machine Learning
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Abdel-Nasser Sharkawy (Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator
Using Artificial Neural Network)
example of these methods was presented in [22], where Elnady developed an iterative technique to
solve the inverse kinematics. Evolutionary computing was proposed in ref. [23], [24] as a method for
inverse kinematics solution. NNs were also developed to solve the problem of inverse kinematics.
Tejomurtula and Kak [25] presented the structured NNs for inverse kinematics. The conventional
backpropagation algorithm was performed for the NN training. This led to some difficulties regarding
accuracy. NNs were also used in ref. [3], [26], [27].
Fig. 2. The different types of solutions used for the inverse kinematics problem of a robot
From the above discussion, further investigation is recommendable for using the NN as a simple
and intelligent technique compared with other methods for forward and particularly inverse
kinematics solutions. Achieving high levels of performance and reliability of the NN is required by
having small values of mean squared error (MSE), which are close to zero and the training error. This
can increase the accuracy and robustness of the application and estimate the forward and inverse
kinematics correctly. In addition, minimizing the input size of the implemented NN is also
recommendable to minimize the complexity and mathematical computations.
The primary contribution of our research paper is proposing an MLFFNN network to solve the
issue of kinematics for the 3-DOF articulated robotic manipulator in both forward and inverse modes.
In forward kinematics used in the design of the network, the positions and orientations for the
manipulator end-effector are outputs, and the joints’ variables for the manipulator are used as the
inputs. For the inverse kinematics case, only the positions of the end-effector are used as the inputs of
the implemented MLFFNN to minimize the size of inputs. The joints’ variables are the outputs. In
each case, the proposed network is trained using LM learning, which provides fast convergence easily.
The main concern during the training on the MLFFNN is obtaining a very small (close to zero) MSE
and training error. The training is executed in MATLAB using collected data considering the
sinusoidal joints’ motion of the manipulator. The testing and the verification of the trained MLFFNN
are presented to investigate its reliability in minimizing the approximation error and its effectiveness
in estimating the forward and inverse kinematics correctly.
The rest of this paper is divided into the following sections. Section 2 illustrates the forward
kinematics of the 3-DOF articulated robot using DH parameters and the inverse kinematics depending
on the geometrical approach. In Section 3, the collected data considering sinusoidal joints’ motion are
described. The design, analysis, and testing of the proposed network to resolve the kinematics problem
in forward mode are explained in Section 4. In Section 5, the architectural concept, training, testing
and verification of the proposed MLFFNN for solving the problem of inverse kinematics are
Solutions For Inverse Kinematics
Closed-Form Solution Method
Numerical Method
Evolutionary Computing
Neural Networks
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Abdel-Nasser Sharkawy (Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator
Using Artificial Neural Network)
presented. Finally, Section 6 summarizes the work shown in this manuscript, and Section 7 gives some
points for work in the future.
2. Kinematics of a 3-DOF Articulated Manipulator in the Forward and Reverse
Mode
This section discusses the forward kinematics as well as the inverse kinematics of the 3-DOF
articulated (with rotational joints) manipulator. This type of robot is presented in Fig. 3. In Subsection
2.1, parameters of Denavit-Hartenberg (DH) are used to illustrate the concept of forward kinematics,
whereas a geometrical method is used to show the concept of inverse kinematics in Subsection 2.2.
Fig. 3. The 3-DOF articulated robotic manipulator [28], [29]
2.1. Forward Kinematics
The forward kinematics of the 3-DOF manipulator is performed using the DH parameters, which
are considered the standard for robot kinematics description. These parameters are presented in Table
1. In the table, is the link length, is the link twist, is the link offset, and is the joint angle.
Table 1. The DH parameters of the 3-DOF articulated manipulator
Link
1
2
0
0
3
0
0
According to Table 1, the position and orientation of the robot end-effector are represented by a
whole homogeneous transformation matrix, which is obtained as Equation (1) [8].
(1)
where, , , and represent the position of the robot end-effector in , , and directions.
, , ,……. illustrate the orientation of the robot end-effector. ,
, , and .
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Abdel-Nasser Sharkawy (Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator
Using Artificial Neural Network)
2.2. Inverse Kinematics
In this subsection, the inverse kinematics (the joints’ variables , , and ) are determined
using the geometrical approach. This approach solves the joint variable by projecting the robot onto
the plane of and then solving a simple problem of trigonometry.
The joints’ variables using this approach are given as Equations (2), (3), and (4) [29].
(2)
where, is defined only when
.
(3)
where,
and .
(Otherwise, the end-effector's point is outside the work area). This happens because of the constraint
of joint three.
(4)
where, is defined only when
and .
It is clear from equations (2)-(4) the solution of inverse kinematics, depending on the geometrical
approach, is complex and more cumbersome. Furthermore, this is familiar in robotics books that the
inverse kinematics solution is complex. Therefore, an MLFFNN network is used for solving the
kinematics of the 3-DOF articulated manipulator in forward and inverse. MLFFNN has the following
properties and advantages:
1) It is a very simple architecture in comparison to the other different NNs types [13], [30], [31].
2) It has the ability of adaptivity, parallelism, and generalization [32]–[34]. In addition, it can be
linear or nonlinear.
3) It is applied successfully in various types of engineering problems [35]–[40].
However, the MLFFNN network needs a high number of input and target pairs in the training
stage [41], [42]. This disadvantage is taken into consideration in the current research, and we use large
datasets. However, overfitting is avoided, as seen from the results.
In this current research work, the developed MLFFNN is trained depending on the Levenberg-
Marquardt (LM) learning algorithm, which possesses the following qualities:
1) The work can be implemented quickly with the mentioned learning algorithm. It is an
optimization technique with a second order that approximates Newton's Method and has a
solid theoretical foundation as well as quick convergence [43], [44].
2) This algorithm is preferred because it strikes a balance between the gradient descent
algorithm's assured convergence and the quick learning speed of the traditional Newton's
method [43], [45]. In addition, its preference for large datasets and its convergence in fewer
iterations and a short time in comparison with the other learning methods [31].
Using the LM algorithm, the adjustment of the weight , which is applied to the parameter
vector , is calculated using Equation (5) [13], [31],
(5)
where and represent, respectively, the Hessian and the gradient vector of the second-order
function. is the identity matrix which, its dimension is the same dimensions as . is a regularizing
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Abdel-Nasser Sharkawy (Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator
Using Artificial Neural Network)
parameter responsible for forcing the part to be positive definite as well as safely well-
conditioned through the computation.
The next section discusses in detail the generated data that have been used for the analysis,
testing, and validation of the MLFFNN network.
3. Generated Data
This section describes the data that have been utilized for validating and testing the proposed
implemented MLFFNN network to solve the forward and inverse kinematics. A sinusoidal motion is
chosen to be commanded to the three joints of the articulated robotic manipulator. This motion is
given by Equation (6).
(6)
where is the joint 1, 2, and 3, is the frequency of the sinusoidal type of motion, and it is variable
and linearly increasing.
This type of kinematic is similar to the motion presented in our previous work [46]. The range of
each joint motion is . The position and the orientation of the robot end-effector is
calculated according to Equation (1) considering the following parameters: ,
, and . The number of the collected samples is 30395. All these data are presented
from Fig. 4, Fig. 5, and Fig. 6.
Fig. 4. The joints’ variables , , and in radians
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Abdel-Nasser Sharkawy (Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator
Using Artificial Neural Network)
Fig. 5. The position of the robot end-effector , , and in meters
Fig. 6. The orientations of the robot’s end-effector , , ,……. in radians
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Using Artificial Neural Network)
4. Forward Kinematics Solution Using MLFFNN
This section presents designing, training, testing, and verifying the proposed MLFFNN to solve
the forward kinematics of a 3-DOF articulated manipulator.
4.1. Network Design
The proposed MLFFNN is designed using three layers. The first layer is the input layer which
contains the three inputs, which are the joints’ variables , , and . The hyperbolic tangent,
denoted by tanh, serves as the activation function for the second layer, which is the hidden, non-linear
layer. It also contains hidden neurons. The output layer is the third layer. This layer is linear and
estimates the values of the position and the orientations for the targeted robot (,
, ,
,
,
…….
), which are twelve outputs. These estimated outputs are compared with the actual
positions and orientations (which are presented in Fig. 5 and Fig. 6). The main and followed criteria
during the design of the MLFFNN-network is implementing a simple NN can achieve high levels of
performance which can be represented by small mean squared error and training error. In other
meaning, these values should be close to zero value. The proposed MLFFNN is presented in Fig. 7.
Fig. 7. The MLFFNN was used as a solution for the kinematics of the 3-DOF articulated manipulator in a
forward mode. The inputs are the joints’ variables, and the outputs are the positions and the orientations of
the end-effector for the robot. The hidden layer includes 40 hidden neurons. The figure is drawn using an
online program available at https://app.diagrams.net/
The intended end-effector positions and orientations of the robot are used only for the training of
the designed MLFFNN. In addition, the error resulting from training must be small as possible and
close to zero. The training process is described in the following subsection.
Input Layer
Hidden Layer
Output Layer
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Abdel-Nasser Sharkawy (Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator
Using Artificial Neural Network)
Fig. 8. The following steps for the train and the test processes of the MLFFNN in solving the forward
kinematics. This graphical chart was sketched by the available online program at https://app.diagrams.net/
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Abdel-Nasser Sharkawy (Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator
Using Artificial Neural Network)
4.2. Designed Network Training
The collected data that are presented in section 3 are used for analyzing the behavior of the
proposed MLFFNN network. These data are divided into three groups as follows:
1) 80% of these data (24315 samples) are used for the training,
2) 10% (3040 samples) are used for validation, and
3) The last 10% (3040 samples) are used for testing.
This division in data occurs in MATLAB randomly. In addition, this is very important to be able
to test and validate the trained NN by different types of data used for the training procedure. Therefore,
we can be sure that the trained NN works effectively and can estimate the output value correctly. The
training is occurring in MATLAB using LM learning. The steps and methodology that followed during
the training and test processes of the designed MLFFNN network are presented in Fig. 8.
After executing many experiments and trials by using different initialization of weights and
different hidden neurons number, the preferred settings of parameters that enable the MLFFNN's high
performance are obtained. These parameters are the best number of hidden neurons, the number of the
used epochs or iterations, the very small MSE value, and the training time. All these parameters are
presented in Table 2. Some other resulting parameters are presented in the Appendix. The intended
high performance is achieving the lowest MSE as well as the training error. The time of training is not
a very important issue because the training happens offline, and the main aim is to have a very well-
trained MLFFNN network that can estimate the outputs efficiently with an error close to zero. The
MSE value is calculated using Equation (7),
(7)
where is the actual output that is used for the training process, and is the estimated
output by the designed NN. is the number of samples.
Table 2. Obtained best parameters which lead to the high performance of the designed MLFFNN for
forward kinematics solution
Parameter
Value
Number of hidden neurons
40
Number of epochs (iterations)
1000
Lowest MSE
Training time
1 hour, 33 minutes, and 1 second
The MSE and the regression obtained from the training are shown in Fig. 9 and Fig. 10. The
resulting error histogram is presented in the Appendix. As presented in Fig. 9, the obtained MSE is a
very small value and approximately zero. In addition, the resulting regression is 1, as shown in Fig.
10. These results illustrate that the required positions and orientations for the targeted robot end
effector converge/coincide with the corresponding estimated ones by the MLFFNN network. In other
meaning, the training error between them is approximately zero. The simulation results demonstrate
that the analyzed MLFFNN network is learned and trained efficiently, which can estimate the
positions and the orientations of the robot’s end-effector. The process of testing and the verification
of the trained method is presented in the following subsection. This process of very important to show
the effectiveness and reliability of the method.
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Abdel-Nasser Sharkawy (Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator
Using Artificial Neural Network)
Fig. 9. The gained lowest MSE from training the designed MLFFNN for forward kinematics
Fig. 10. The obtained regression from training the designed MLFFNN for forward kinematics
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Using Artificial Neural Network)
4.3. Trained Network Verification and Testing
All samples (30395) are used to verify and test the trained MLFFNN. Results and comparisons
of the robot's desired effector's positions and orientations, and the corresponding estimated ones by
the trained network are presented from Fig. 11 to Fig. 14. Furthermore, the average, the maximum,
the minimum, and the standard deviation (std) of the approximation error between the desired and
estimated end-effector positions are shown in Table 3.
From Fig. 11, Fig. 12, Fig. 13 and Table 3, it is clear that desired positions of the robot end-
effector in , , and directions and the corresponding estimated ones by the NN are coincide.
The approximation error between them is a very small value and approximately zero. In Fig. 14, the
desired orientations of the robot also coincide with the corresponding approximated ones by the neural
network. This also means that the approximation error is a very small value and about the zero value.
These results prove that the trained MLFFNN is highly reliable in minimizing errors and is trained
very well. Furthermore, it is able to solve the forward kinematics (finding positions and orientations
of end-effector) of a 3-DOF articulated manipulator in a correct way.
Table 3. The resulting parameters values: average, maximum, minimum, and standard deviation (std) of the
approximation error between the desired and estimated end-effector positions
Parameter
Position in
direction
Position
in
direction
Position in
direction
Approximation of absolute error between
the desired and estimated positions of the
end-effector
Average
1.4058e-04
2.2879e-04
9.8047e-05
Maximum
7.9550e-04
0.0036
7.1878e-04
Minimum
4.5048e-09
2.6214e-09
6.1091e-10
Standard
deviation (std)
1.1052e-04
2.3766e-04
8.4661e-05
Fig. 11. The comparison between the desired and estimated position of the robot’s end-effector in
direction. (a)
and , (b)
Fig. 12. The simulation comparative results between the desired and estimated position of the robot’s end-
effector in direction. (a)
and
, (b)
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Abdel-Nasser Sharkawy (Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator
Using Artificial Neural Network)
Fig. 13. The simulation comparative results between the desired and estimated position of the robot’s end-
effector in direction. (a)
and , (b)
Fig. 14. The simulation comparative results between the desired orientations of the robot’s end-effector and
corresponding actual ones by the trained MLFFNN-network
It should be noted that the trained MLFFNN takes 1.379 seconds in MATLAB to do all the
calculations and give the results. This time is very short, and if the method is applied
experimentally/practically to any 3-DOF robot, it cannot affect the continuity of the robot’ motion.
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Using Artificial Neural Network)
5. Inverse Kinematics Solution Using MLFFNN
This section shows in detail the designing, training, testing, and verifying of the proposed
implemented MLFFNN to resolve the 3-DOF articulated robotic manipulator's inverse kinematics.
Fig. 15 shows his design of MLFFNN.
Fig. 15. The designed MLFFNN is used to solve the kinematics of the 3-DOF articulated manipulator in
reverse mode. The inputs are the positions of the end-effector, and the outputs are the joints’ variables. The
hidden layer includes 120 hidden neurons. The figure is drawn using the online program available at
https://app.diagrams.net/
5.1. Network Design
The proposed MLFFNN network is designed in such a way that using three layers of neurons are
interconnected with each other. The first layer is called the input layer, which contains the three inputs,
which are only the positions of the robot’s end-effector (, , and ). The second layer, which
contains hidden neurons, is called the hidden layer, which is considered non-linear in its operation,
and the activation function for this layer is the hyperbolic tangent which is represented by tanh. The
third and last layer is the output layer. This layer is linear and estimates the joints’ variables (, ,
and ), which are three outputs. These estimated outputs are compared with the actual (desired)
joints’ variables (which are presented in Fig. 4). This proposed MLFFNN is presented in Fig. 15.
The desired joints’ variables are utilized for training the designed MLFFNN network. The error
from the training phase must be very small and close to the zero value. In detail, training for the
implemented MLFFNN is described in the following subsection (5.2. Designed Network Training).
Input Layer
Hidden Layer
Output Layer
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Abdel-Nasser Sharkawy (Forward and Inverse Kinematics Solution of A 3-DOF Articulated Robotic Manipulator
Using Artificial Neural Network)
5.2. Designed Network Training
The same procedure that is presented in Subsection 4.2 is followed here. The collected data
mentioned in Section 3 is divided as follows:
1) 80% of the collected data (24315 samples) are used for the training,
2) 10% (3040 samples) are used for validation, and
3) The last 10% (3040 samples) are used for testing.
The most suitable parameters which lead to high levels of performance for the designed
MLFFNN network are the best number of hidden numbers, the number of epochs or iterations, the
very small MSE, and the training time. These parameters are presented in Table 4. As mentioned in
Section 4, the training time is not very important. Some other parameters are presented in the
Appendix.
Table 4. The obtained best settings that were obtained led to the high performance of the designed
MLFFNN for kinematics solution in an inverse method
Parameter
Value
Number of hidden neurons
120
Number of epochs (iterations)
1000
Lowest MSE
Training time
44 minutes and 16 seconds
The resulting MSE and regression from the training are shown in Fig. 16 and Fig. 17. The
obtained error histogram is presented in the Appendix. As shown from the figures, the MSE is very
small and approximately zero. The regression is equal to 1. This means that the desired joints’
variables are coinciding/converging with corresponding estimated ones by the implemented NN.
Therefore, the training error is very small. This indicates that the designed MLFFNN-neural network
is trained efficiently and it is ready to solve inverse kinematics correctly. Testing and the verification
of the trained NN should be investigated, as presented in the following section. This process is very
crucial for showing the effectiveness of the trained method.
Fig. 16. The obtained lowest MSE from training the designed MLFFNN for inverse kinematics
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Using Artificial Neural Network)
Fig. 17. The obtained regression from training the designed MLFFNN for inverse kinematics
5.3. Trained Network Verification and Testing
In this step, all samples (30395) are used to verify and test the trained MLFFNN. The results and
the comparisons between the targeted joints’ variables and the corresponding computed ones by the
trained MLFFNN-neural network are presented in Fig. 18, Fig. 19, and Fig. 20. Furthermore, Table 5
presents the average, maximum, minimum, and standard deviation of the approximation absolute error
between the desired and estimated joints’ variables.
Table 5. The resulting parameters values: average, maximum, minimum, and standard deviation (std) of the
approximation absolute error between the desired and estimated joints’ variables
Parameter
Theta 1
Theta 2
Theta 3
Approximation of absolute error between the
desired and estimated joints’ variables
Average
7.9624e-04
7.3061e-04
6.5449e-04
Maximum
0.0043
0.0044
0.0032
Minimum
2.8322e-08
4.7931e-08
1.5800e-08
Standard
deviation (std)
6.2181e-04
6.1428e-04
5.2187e-04
It is clear from the presented figures (Fig. 18, Fig. 19, and Fig. 20) and Table 5 that the desired
joints’ variables (Theta1, Theta 2, and Theta 3) coincide with the corresponding estimated ones by the
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NN. The approximation error between them is a very small value and approximately zero value. This
proves that the trained MLFFNN is highly reliable in minimizing error and is trained very well.
Fig. 18. The comparison between the desired and estimated joint variable Theta 1. (a) and , (b)
Fig. 19. The comparison between the desired and estimated joint variable Theta 2. (a) and , (b)
Fig. 20. The comparison between the desired and estimated joint variable Theta 3. (a) and , (b)
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Using Artificial Neural Network)
It should be noted that the trained MLFFNN takes 0.587 seconds in MATLAB to do all the
calculations and give the results. This time is very short, and if the method is applied
experimentally/practically to any 3-DOF robot, it cannot affect the continuity of the robot’ motion.
The time in the case of the inverse kinematics is less than the time in the forward kinematics due to
the fact that MLFFNN-network estimates 12 outputs in the forward kinematics case, and it estimates
only 3 outputs in the other case.
Remark: To solve the inverse kinematics, we created another MLFFNN and used both the
positions and orientations of the robot end-effector as its inputs. The results are better compared with
the presented one in this section (Section 5). However, the presented MLFFNN in this section (Section
5) gives very good results because the MSE and training/approximation error are both very low values
that are close to zero. In addition, we prefer it since the inputs are smaller than when using the end-
effector locations and orientations. The complexity of the presented MLFFNN is also lower.
Limitation of the proposed method: The proposed method is investigated using a limited range
of the joints’ motion of the manipulator as the range of each joint motion is . Therefore,
applying the method using the entire experiment space of the robot joints should be investigated and
considered in future work. The proposed method is used with a 3-DOF articulated robot. For
investigating the generalization of the proposed network, it should be applied to other different types
of robots and more complex robots like 6-DOF robots or 7-DOF robots.
In the current work, only sinusoidal motion is recommended for the robot joints. Other different
types of joint motion should also be considered and used. The proposed method is validated only using
simulated data, and no experimental data or real robot experiments are conducted to validate the
effectiveness of the proposed method in real-world scenarios. This happens because we do not have a
real robot at the time of doing this paper. However, in future work, the experimental validation should
be considered and investigated.
The accuracy of the proposed MLFFNN for solving the problem of a 3-DOF manipulator
working in the inverse kinematics is compared with other previous NNs-based approaches such as the
ones presented by Koker et al. [47], Daya et al. [48], Duka [3], and Jiménez-López et al. [49]. With
Duka [3], a feedforward NN was used for solving the inverse kinematics of a 3-DOF planar
manipulator. The NN was designed using six inputs which were the three end-effector positions and
the three orientations. With Koker et al. [47], a backpropagation NN with a sigmoidal activation
function was proposed as a foundation for the inverse motion problem for the 3-DOF manipulator
robot. In [48], Daya et al. developed an NN-based approach for the inverse kinematics solution of a
3-DOF manipulator.
In their design, six inputs were used, which were the three positions and the three orientations of
the end-effector. Jiménez-López et al. [49] implemented a NN for the inverse kinematic solution of a
3-DOF manipulator. In their approach, the NN was trained using Bayesian regularization
backpropagation. In addition, the limited range of the joints’ motion of the manipulator was
considered. The comparison includes the resulting MSE, which is the main parameter to show the
accuracy of each method. The comparison is shown in Fig. 21. As shown in Fig. 21, the proposed
method has the resulting smallest MSE value compared with other previous related approaches.
Hence, the results prove that the proposed method is the most accurate, and the resulting
approximation error is the smallest.
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Using Artificial Neural Network)
Fig. 21. The comparison between the proposed method for solving the inverse kinematics of a 3-DOF
manipulator and other previous related approaches
6. Conclusion
In this research, the kinematics of a 3-DOF articulated robotic manipulator in both forward and
inverse modes are solved using the concept of the MLFFNN-network algorithm. For Forward
kinematics, the MLFFNN is designed using the joints’ variables as its inputs. In addition, its outputs
are the positions and orientations of the end-effector of the robot. For the kinematics in the inverse
case, only the positions of the end-effector are used as the inputs of the MLFFNN. The outputs are the
joints’ variables. For both cases, the designed MLFFNN network is trained using the LM learning
algorithm using data collected from the sinusoidal joint motion of the manipulator. The resulting
training errors and the MSE from the training stage have relatively very small values and close to zero
values. The simulation results demonstrate that the proposed neural network is trained in an effective
way. The trained MLFFNN is tested and verified, and the results prove that the approximation error
between the actual output and the corresponding predicted one by the NN is equal to a very small
value. Therefore, the MLFFNN is highly reliable in minimizing errors. Furthermore, it is efficient to
solve the kinematics of the robot in forward and inverse modes correctly. Our proposed research
approach is compared with other previously published methods. The experimental simulation result
reveals that our proposed neural network-based method has the highest levels of accuracy compared
with others.
0.000121
6.26E-06
0.00054387
0.00020037
9.07E-07
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
The Lowest MSE Value
The Method
Comparison Between the NNs Based Approaches
Koker et al. Approach Daya et al. Approach
Duka Approach Jiménez-López et al. Approach
The proposed method
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Using Artificial Neural Network)
7. Future Work
Different aspects of future work can be discussed in this section. Firstly, as the range of each joint
motion of the used robot is limited in the current work and is , the future work can
consider the whole workspace of the robot joints. Secondly, the use of the NN in solving the forward
and inverse kinematics of a complex robot, such as a 6-DOF or a 7-DOF manipulator, can also be
considered. Thirdly, different types of NNs-based architectures inspired by biological concepts can be
used and compared, such as WaveNet, RNN, and RBF. Deep learning methods can also be
investigated and compared. Fourthly, we would synthesize the hardware implementation for the
proposed MLFFNN network on file programmable gate array (FPGA) technology and use different
fault-tolerant techniques. FPGA has many advantages, as follows:
1) It can easily change its functionality after designing the product.
2) It does not require a larger board area, and it is more energy efficient than other equivalent
discrete circuits.
3) It can carry out several operations on data simultaneously.
Finally, investigating the current work experimentally with a real robot is recommended.
Abbreviations
Abbreviation
Meaning
DH
Denavit–Hartenberg
POE
Product of Exponential
SVR
Support Vector Regression
NN
Neural Network
DOF
Degree of Freedom
LWR
Light Weight Robot
MLFFNN
Multilayer Feedforward Neural Network
LM
Levenberg–Marquardt
MSE
Mean Squared Error
Tanh
Hyperbolic Tangent
FPGA
File Programmable Gate Array
Supplementary Materials: The generated datasets during and/or analyzed during the current study are available
from the corresponding author upon reasonable request.
Author Contribution: All authors contributed to this paper. Most of the presented work is done by Abdel-
Nasser Sharkawy. All authors read and approved the final paper.
Funding: This research received no external funding
Acknowledgment: It is not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix
Part 1: Some parameters resulted from the training process of the MLFFNN that is used for solving
the forward and inverse kinematics problem. These parameters are shown in Fig. A1.
Part 2: The obtained error histogram from the MLFFNN training in the forward and inverse
kinematics cases. These histograms are presented in Fig. A2. As shown from these histograms, the
error, which is the difference between the actual output and the estimated one by NN, is very close
to zero. This supports that the proposed NN is trained in a very effective way.
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(a) In the forward kinematics case
(b) In the inverse kinematics case
Fig. A1. Some parameters result from the training of the MLFFNN (a) The case that is used for the forward
kinematics problem (b) The case that is used for the inverse kinematics problem
(a) The error histogram resulted in the case of the
forward kinematics.
(b) The error histogram resulted in the case of
inverse kinematics.
Fig. A2. The resulting error histogram from the training stage
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