Towards accuracy improvement in solution of inverse kinematic problem in redundant robot: A comparative analysis

Hind Zuhair Khaleel; Amjad J. Humaidi

doi:10.1556/1848.2023.00722

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International Review of Applied Sciences and Engineering

Volume/Issue:: Volume 15: Issue 2

Towards accuracy improvement in solution of inverse kinematic problem in redundant robot: A comparative analysis

Authors:

Hind Zuhair Khaleel

Hind Zuhair Khaleel Control and Systems Engineering Department, University of Technology, Iraq

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hind.z.khaleel@uotechnology.edu.iq https://orcid.org/0000-0002-6020-158X

and

Amjad J. Humaidi

Amjad J. Humaidi Control and Systems Engineering Department, University of Technology, Iraq

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amjad.j.humaidi@uotechnology.edu.iq

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Pages:: 242–251

Online Publication Date:: 11 Jan 2024

Publication Date:: 10 Jun 2024

Article Category:: Research Article

DOI:: https://doi.org/10.1556/1848.2023.00722

Keywords:: redundant robot; inverse kinematics; forward kinematics; GA; PSO; neural network

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Abstract

The redundant manipulators have more DOFs (degree of freedoms) than it requires to perform specified task. The inverse kinematic (IK) of such robots are complex and high nonlinear with multiple solutions and singularities. As such, modern Artificial Intelligence (AI) techniques have been used to address these problems. This study proposed two AI techniques based on Neural Network Genetic Algorithm (NNGA) and Particle Swarm Optimization (PSO) algorithm to solve the inverse kinematics (IK) problem of 3DOF redundant robot arm. Firstly, the forward kinematics for 3 DOF redundant manipulator has been established. Secondly, the proposed schemes based on NNGA and PSO algorithm have been presented and discussed for solving the inverse kinematics of the suggested robot. Thirdly, numerical simulations have been implemented to verify the effectiveness of the proposed methods. Three scenarios based on triangle, circular, and sine-wave trajectories have been used to evaluate the performances of the proposed techniques in terms of accuracy measure. A comparison study in performance has been conducted and the simulated results showed that the PSO algorithm gives 7% improvement compared to NNGA technique for triangle trajectory, while 2% improvement has been achieved by the PSO algorithm for circular and sine-wave trajectories.

Abstract

1 Introduction

The robot is a mechanical device which needs no breaks, and it can operate continuously. Furthermore, the robot outperformed the human workers in terms of speed and accuracy. The forward kinematics (FK) and inverse kinematics (IK) are the essential mathematical tools in controlling and path planning of robotic manipulators. The FK models are used to determine the position of the end effector based on joint angles as input variables. The opposite of FK is the IK, which works to determine the joint angles based on the Cartesian coordinates of the end effector [1–5].

The robot manipulators are classified as redundant and non-redundant robots according to their DOFs and the dimension of working space which is satisfying for specified tasks. The redundant manipulators have more DOFs (degree of freedoms) than it requires to perform a specified task. The 3 DOF manipulators which move in two dimensional plan are examples of redundant robot. In non-redundant robot manipulator, the dimension of workspace is equivalent of the number of DOFs for the robot. The 2DOF robot arm, which has two links and two joints, is an example of non-redundant robot [6, 7].

Redundant robot manipulators have been used in many difficult industrial tasks. An Inverse kinematic (IK) of redundant robot was not easily solved so that many researchers need to interface this robot's mathematical model with Artificial Intelligence (AI) to solve this problem [7–9].

Aggogeri et al. [10] presented fuzzy logic technique to solve the IK problem to improve the convergence of robot workspace. Shaher et al. [11] utilized two different Genetic Algorithms: the conventional GA and the continuous GA. The IK problem of 3-DOF robot arm has been viewed as an optimization problem. It has been shown that continuous GA outperforms the traditional one in terms of convergence speed. Duka et al. [12] used a feed-forward neural network to find the inverse kinematics of 3-DOF robot by generating the desired trajectories in the Cartesian space. Takatani et al. [13] proposed a neural network structure for solving the inverse kinematics of 3-DOF redundant robot. By hybridization of many neural network structures, the approach achieved good learning performance. Using training data based on postures, endpoints, and cost function, the approach can synthesize the neural network model for solution of inverse kinematics. Habibkhah et al. [14] proposed non-conventional neural network and virtual vector function method to solve the inverse kinematics (IK) of a 3-DOF redundant manipulator robot. Batista et al. [15] used Least Squares (LS), Recursive Least Square (RLS), and dynamic parameter identification algorithm based on Particle Swarm Optimization (PSO) with search space defined by RLS. Kelemen et al. [16] proposed an algorithm that takes into account joint limit avoidance, kinematic singularities avoidance, and an obstacle avoidance when controlling a kinematically redundant manipulator. The potential field approach is employed for end-effector path planning from the starting point to the end point. By using Jacobian-based technique and weight matrices to different tasks, the inverse kinematic model is developed. The other study presented focuses on the development of a low-cost robotic arm with three degrees of freedom designed for various applications, particularly in industry. Some traditional robotics concepts like acquisition, image processing, and object manipulation, were used. Artificial neural networks were utilized for two crucial functions: camera calibration and inverse kinematics resolution. The results from the application of artificial neural networks (ANNs) allowed for the control of the servo motors' trajectory for the robot [17].

In another study, the authors combined the two algorithms GA and NN to provide an inverse kinematics solution for a 3DOF redundant robot. The three-link robot's location and orientation are the inputs. The back-propagation neural network (BPNN) is receiving these inputs. A feed-forward neural network is added to the training of the error backpropagation technique via the BPNN model. This model has outstanding multi-dimensional function mapping capabilities and the capacity to classify arbitrarily complicated patterns. Continuous GA is used to optimize the BPNN weights. The difference between the estimated and desired joint angle outputs is also calculated, as is the Mean Square Error. The fitness function in GA was proposed [18].

The swarm behavior of some animals, such as bird flocks and fish schools, inspired the development of particle swarm optimization (PSO) [19–23]. It has several benefits such as straightforward implementation, global optimal solution with high probability and efficiency, quick convergence, and low computing time. The PSO is useful for solving challenging problems for high complex and nonlinear models [24–27]. Therefore, the PSO algorithm has been proposed as optimization tool for solving the inverse kinematics problem of a three-link redundant robot arm [28]. One can summarize the main contributions by the three main points:

❑Establishing the possible points covering the workspace of redundant robot manipulator based on FK.
❑Proposing optimization-based techniques to solve the IK problem based on NNGA and PSO algorithms
❑Conducting a comparison study between the proposed techniques in terms of accuracy measure.

The organization of this article is put into five sections. The second section presented the forward kinematics of 3-DOF redundant robot arm. The proposed methodology has been discussed in the third section. The fourth section conducted the numerical simulation to verify the effectiveness of the proposed approaches. The fifth section highlighted the main points deduced from the simulated results.

2 The forward kinematics of 3-DOF redundant robot arm

The forward kinematics (FK) of redundant robot is implemented to determine the location and orientation of end-effector for 3-DOF redundant robot manipulator as illustrated in Fig. 2 [18, 28, 29].

The robot manipulator has three arms of lengths $l_{1}$ , $l_{2}$ , and $l_{3}$ with three joint angles designated as $q_{1}$ , $q_{2}$ , and $q_{3}$ . The three joints bind the three links together as indicated shown in Fig. 1.

Fig. 1.

The 3DOF redundant robot manipulator

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

The forward kinematics equations for a three-link redundant robot manipulator can be expressed as follows [12],

x_{E} = l_{1} \cos (q_{1}) + l_{2} \cos (q_{1} + q_{2}) + l_{3} \cos (q_{1} + q_{2} + q_{3})

(1)

y_{E} = l_{1} \sin (q_{1}) + l_{2} \sin (q_{1} + q_{2}) + l_{3} \sin (q_{1} + q_{2} + q_{3})

(2)

where (

x_{E}, y_{E})

denotes the location of the three link robot's end-effector and

q_{E}

represents the summation of all three joint angles of the robot (the orientation of end-effector), which can be described by [12, 18, 28].

q_{E} = q_{1} + q_{2} + q_{3}

(3)

Table 1 shows the Denavit-Hartenberg (DH) parameter of the 3DOF redundant robot. The structure 3-DOF redundant manipulator can be described in terms of three variables; namely, $l_{i}$ = length, $α_{i}$ = twist angle, and $d_{i}$ = offset, where, $i$ is the link number from 1 to 3 of the robot. The three joint angles are $q_{1}$ , $q_{2}$ and $q_{3}$ . The limitation for each joint are: $0 < q_{1} < π$ , $- π < q_{2} < 0$ , $- π / 2 < q_{3} < π / 2$ and $0 < q_{E} < 2 π$ .

Table 1.

Link parameter table of 3DOF redundant robot

^{Link number}	$l_{i}$ (m)	$α_{i}$ (rad)	$d_{i}$ (m)	$q_{i}$ (rad)
1	2	0	0	$q_{1}$
2	2	0	0	$q_{2}$
3	2	0	0	$q_{3}$

The circular trajectories equations of position and orientation for the robot's end-effector are illustrated in the following equations [12, 18, 28],

x_{p} = x_{c} + r \cos φ

(4)

y_{p} = y_{c} + r \sin φ

(5)

The desired circular end-effector's position is denoted as (

x_{p}, y_{p}

) and the circle's position center is marked as (

x_{c}, y_{c}

). The circle radius is

r

and

φ

is an angle ranging between 0 and 2π. The desired orientation

q_{d}

is given by

q_{d} = atan 2 {(y}_{p}, x_{p})

(6)

As illustrated in Fig. 2, the square and triangular trajectories of the three link robot are expressed using the parametric Cartesian space trajectory equations [30–32]:

x_{u} = x_{a} + u (x_{b} - x_{a})

(7)

y_{u} = y_{a} + u (y_{b} - y_{a})

(8)

Fig. 2.

The Cartesian trajectory line

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

The beginning point is marked as (

X_{a}

Y_{a}

) while the goal point is expressed as (

X_{b}

Y_{b}

) the u symbol ranged from 0 to 1 and finally, the end-effector's coordinates were denoted as (

X

Y

). The position (

x_{p}

y_{p}

) and orientation

{ph}_{i}

of the sinewave trajectory can be computed as:

x_{p} = 0 : 2 π

(9)

y_{p} = \sin (x_{p})

(10)

{ph}_{i} = atan 2 {(y}_{p}, x_{p})

(11)

3 Methodology

The methodology described the equations of NNGA and PSO algorithms. Then the proposed flowchart is also presented. It involved the implementation of these two algorithms. Next the MSE and the accuracy founded of each algorithm.

3.1 The proposed NNGA algorithm

The NNGA is a combination of the GA algorithm and NN model. The three-link robot's end effector position and orientation are the inputs ( $x_{E}$ , $y_{E}$ , and $q_{E}$ ) which get them from the workspace of the robot arm Eq. (1) and Eq. (2). These data are saved in the MATLAB program. In addition, they are trained using a back-propagation neural network algorithm (BPNN). The BPNN model is defined in the introduction section [18]. The structure of back propagation-based NN (BPNN) consists of three layers as shown in Figure 3. The first layer is the input layer which includes three neurons. The input neurons receive the variables $x_{E}$ , $y_{E}$ , and $q_{E}$ . The hidden layer consists of 10 neurons, while the output layer has one neuron, which is responsible for producing the variable $q_{i}$ as clarified in Figure 3. The weights of the proposed neural structure are optimized by GA to ensure minimization of errors.

Fig. 3.

The BPNN architecture for learning workspace points

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

This neural structure is implemented three times to get the outputs

q_{1}

q_{2}

, and

q_{3}

, which represents the solutions of inverse kinematic (IK). These angular positions are summed together to give the variable

q_{E}

. The number of epoch = 1,000. The training process has utilized 80% of data, while there is 20% of data, which have been used for validating and testing. The number of data depends on particular trajectory. The type of activation functions in the NN structure plays a vital role in performance. To address system complexity and nonlinear issues, nonlinear activation functions are used. Also, some types of activation functions like hyperbolic and linear functions are suitable due to their differentiable nature. As such, the NN structure applied sigmoid activation functions within neurons of hidden layer [18, 32]

f (x) = \frac{1}{1 + \exp (- x)}

(12)

Where

x

is the input and

f (x)

is the sigmoid function. During the forward propagation, the data have also to be summed via the following expressions at the each neuron output of the hidden layer [32],

u_{j} = f (\sum_{i = 1}^{n} v_{i j} x_{i} + b_{j})

(13)

At the output layer, there is only one neuron which works to linearly sum-up all outputs of neurons at the hidden layer,

o = f (\sum_{j = 1}^{m} w_{j} u_{j} + k)

(14)

where,

u_{j}

denotes the output of hidden layer,

n

is the number of input layers,

v_{i j}

represents the input layer weight,

x_{i}

is the value of input variable,

b_{j}

is the bias value of hidden layer,

o

denotes the output of output layer,

n

is the number of hidden layer, and

k

represents the bias value of output layer.

In this study, the Mean Square Error (MSE) between the predicted and desired outputs is used to assess the performance of the proposed method in terms of accuracy [33].

MSE = \frac{1}{n} \sum_{i = 1}^{n} {({desired}_{i} - {predicted}_{i})}^{2}

(15)

where

n

represents the number of compared samples. If MSE < minimum error, the GA has to optimize the weights of BPNN structure. In this work, the NNGA is enhanced by pursuing the following minimization strategy of the objective function,

f (q) = \min (q_{E})

(16)

Figure 4 shows the mechanism of NNGA approach. The inputs to the proposed method are the positions and orientation of end-effector (1^st block) using Eq. (1) and Eq. (2). The back-propagation neural network (BPNN) algorithm is implemented based on received inputs (see 2nd block). The result of BPNN is the rotation of end effector $q_{E}$ (see 3rd block). The fourth stage is to implement the GA for obtaining the optimal value of $q_{E}$ (see 4th block). The forth step is to enhance the result of BPNN method, which may give local minimum in its search, while the GA will help to guide the solution to global minimum and hence the minimization is considerably enhanced. The fifth stage is the last stage which yields the optimal value of orientation (5th block). In this hybrid technique, continuous GA is applied to optimize the orientation $q_{E}$ while training the data in BPNN. The settings of continuous GA's parameters are listed in Table 2.

Fig. 4.

The proposed GA-based BPNN method

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

Table 2.

Setting of continuous GA's parameters

Number of population	30
Number of generation	30
Crossover rate	80%
Mutation rate	15%
Number of variables	$3 (q_{1}, q_{2}, q_{3})$
Lower and upper limits of variable (rad)	$0 < q_{1} < π$ , $- π < q_{2} < 0$ , $- π / 2 < q_{3} < π / 2$

It is worthy to mention that the forward kinematics have been utilized to train ANN for IK.

3.2 The proposed PSO algorithm

The particle swarm optimization (PSO) technique is used to enhance minimization process of objective function in terms of variable q_E. It means that the particle variable of enhanced PSO is q_E, which is limited ranges of the three-link robot arm workspace. Within the optimization process, the orientation q_E of end-effector has two measures: position (

x_{i, j}^{k}

) and velocity (

v_{i, j}^{k}

). They are updated to new values

x_{i, j}^{k + 1}

and

x_{i, j}^{k + 1}

based on the following two equations [17]:

v_{i, j}^{k + 1} = v_{i, j}^{k} + {c_{1} r_{1} (x b e s t}_{i, j}^{k + 1} - x_{i, j}^{k}) + {c_{2} r_{2} (x g b e s t}_{j}^{k + 1} - x_{i, j}^{k})

(17)

x_{i, j}^{k + 1} = x_{i, j}^{k} + v_{i, j}^{k + 1}

(18)

Where i and j represent the ith and jth components of joint's position,

r_{1}, r_{2}

are random values ranged between 0 and 1. The symbols

c_{1}, c_{2}

present the cognition and social coefficients whose values range between 0 and 2, respectively, and k is the number of iterations. The next step in the algorithm is based on best particle individual (

{x b e s t}_{i, j}

) and global of best particle (

{x g b e s t}_{j}

). This algorithm enhanced the performance of precious work [20] by proposing the following minimization problem of the objective function defined by:

Cost = \min (q_{E})

(19)

The mechanism of the proposed method based on PSO algorithm can be described by Figure 5. The mechanism consists of three stages. The first stage is the input to the PSO algorithm (1st block). The PSO algorithm receives these inputs for optimization purpose (2nd block); the mechanism results in optimal value of $q_{E}$ (3rd block). According to Eq. (19), the PSO algorithm works to minimize the fitness function based on orientation variable ( $q_{E}$ ) of the robot to reach the best value. The parameters of PSO algorithm are listed in Table 3.

Fig. 5.

The proposed PSO algorithm block diagram

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

Table 3.

The parameters of PSO

Parameter Description	Value
Number of iteration	100
Values of cognitive parameter $c_{1}$ and social parameter $c_{2}$	1.9
Values of constant ( $r_{1}, r_{2}$ )	0.95
Number of Variables	$3 (q_{1}, q_{2}, q_{3})$
Lower and upper limits of variable (rad)	$0 < q_{1} < π$ $- π < q_{2} < 0$ $- π / 2 < q_{3} < π / 2$

The flowchart of proposed schemes is shown in Figure 6. The flowchart begins with the random generation of positions and orientations of the 3DOF robot redundant arm based on forward kinematics. These points are chosen to lie within the robot's workspace. Then the proposed methods, NNGA and PSO algorithm, are applied. Furthermore, the output from each two approaches is the best values of orientation $q_{E}$ , which represents the sum of all joint angular position; that is, $q_{E} = q_{1} + q_{2} + q_{3}$ . The PSO depends on generated input $q_{E}$ from Eq. (1)–Eq. (3), which is ranged between 0 and $2 π$ and the optimization is based on cost function Eq. (17). Accordingly, the three suggested trajectories (triangle, circular, and sine-wave) have been produced based on these best values using enhanced NNGA and PSO. For each trajectory, the accuracy of these proposed techniques is assessed in terms of Mean Square of Error (MSE). Finally, the best positions and orientations have been determined accordingly.

Fig. 6.

The flowchart of proposed techniques

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

4 Simulation results and discussion

The MATLAB programming software has been used to verify the proposed methods to solve the IK problem of 3-DOF redundant arm. The dimension of each length link is 2 m as indicted in Table (1). Eq. (1) and Eq. (2) are applied to perform forward kinematics (FK). There are 1,000 positions ( $x, y$ ), and orientation $(q_{E})$ of end effector's positions are randomly generated. These orientations ranged between 0 and 2π rad for the suggested robot. The robot's link colors start from the first green link and end with the third red link. The middle link color is black. The positions are marked as blue colors. The 3DOF redundant robot moves within its workspace as shown in Figure 7.

Fig. 7.

The 3DOF redundant arm workspace

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

The triangle trajectory has been generated based on proposed NNGA and PSO algorithms as illustrated in Figure 8. The start point of the triangle trajectory is the same as the end trajectory. The trajectory ends at coordinates x = 2.0100, y = 1.0110 based on NNGA algorithm (blue line), it ends at x = 2.000 and y = 1.000 based on PSO algorithm (black line). The black-colored trajectory, which belongs to PSO algorithm, is more accurate than the trajectory, which belongs to NNGA algorithm, as shown in Figs 8 and 9.

Fig. 8.

The triangle trajectories based on NNGA and PSO algorithms

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

Fig. 9.

The circular trajectories based on NNGA and PSO algorithms

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

The circular trajectories are simulated based on the proposed NNGA and PSO algorithms. The red-colored trajectory belongs to NNGA, while the black-colored trajectory belongs to PSO algorithm. The start and end points of a circular trajectory are the same. The trajectory ends at coordinates = 2.0034 and y = 2.0013 based on NNGA algorithm (red line), while it ends x = 4.0000, y = 2.0000 based on PSO algorithm (black line).

The sine-wave trajectories are simulated based on the proposed NNGA and PSO algorithms as shown in Figure 10. In case of NNGA method, the start of orientation $φ_{i}$ for red-colored trajectory begins with 0.4636 rad and it ends at the same point. In case of PSO method, the start of orientation $φ_{i}$ for black-colored trajectory begins with 0.4636 rad and it ends with 0.4586 rad. It is clear from the figure that the trajectory based on PSO algorithm (black-colored trajectory) outperforms the trajectory based on GA-based NN method in terms of accuracy.

Fig. 10.

The sine-wave trajectories of NNGA and PSO algorithms

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

In these simulated scenarios, 52 points of end effector positions have been taken for each circular and triangle trajectories the simulated number. On the one hand, 15 orientations have been used for end effector orientation in case of sine-wave trajectory. The MSE has been used as accuracy indicator of the proposed NNGA and PSO algorithms for each trajectory. According to Table 4, the accuracy based on PSO algorithm is better than that based on NNGA method for all suggested trajectories.

Table 4.

Evaluation of MSE based on proposed methods

Trajectory	Algorithm	MSE
Triangle trajectory	proposed NNGA	$2.3725 \times 10^{- 2}$
Triangle trajectory	proposed PSO	$0.8491 \times 10^{- 5}$
Circular trajectory	proposed NNGA	$2.9534 \times 10^{- 5}$
Circular trajectory	proposed PSO	$1.6928 \times 10^{- 5}$
Sin-wave trajectory	proposed NNGA	$1.0735 \times 10^{- 5}$
Sin-wave trajectory	proposed PSO	$0.0903 \times 10^{- 5}$

Figure (11) shows the spectrum of accuracy values of the proposed methods for three trajectories as follows:

❑In case of triangular trajectory, the accuracy based on PSO algorithm equals 0.92%, which is better than the accuracy of the NNGA method which is equal to 0.855%. This indicates that an improvement ratio of 7% is obtained.
❑In case of circular trajectory, the accuracy based on PSO algorithm equals 0.9%, which is better than the accuracy of the NNGA method which is equal to 0.88%. This indicates that an improvement ratio of 2% is obtained.
❑In case of circular trajectory, the accuracy based on PSO algorithm equals 0.96%, which is better than the accuracy of the NNGA method which is equal to 0.94%. This indicates that an improvement ratio of 2% is obtained.

Fig. 11.

The Accuracy results in comparison between proposed NNGA and proposed PSO algorithms

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

The main difference between previous works [18, 20] and this work is that objective functions (Eq. (14) for GA and Eq. (17) for PSO) have been proposed in this paper. These proposed equations solved inverse kinematic joint angle ( ${TH}_{E}$ ) of 3DOF redundant robot. In addition, they enhanced the circular trajectory of 3DOF redundant robot as seen in Table 5. In this Table, the results show the position (x, y) of the proposed work in the circle trajectory, which is more accurate than position (x, y) of the related work. Furthermore, MSE of the proposed work is too small and near zero, as seen in Table 5.

Table 5.

Comparison with related works

References and algorithm	Circular trajectory (x, y) (m)	Objective function	MSE (m)
[18] NNGA (2018)	(2.098, 1.9708)	$\min (\frac{1}{1 + MSE})$	$0.0229$
Proposed NNGA	(2.0034, 2.0013)	$\min ({TH}_{E})$	$2.9534 \times 10^{- 5}$
[20] PSO (2019)	(4.0131, 2.0059)	$\min (\| \| θ_{T} - θ_{i} \| \|)$	$0.0001$
Proposed PSO	(4.000, 2.000)	$\min ({TH}_{E})$	$1.6928 \times 10^{- 8}$

The behavior of fitness function with respect to generation in case of NNGA method is shown in Figure 12. According to the figure, the best fitness value based on proposed NNGA is equal to $q_{E}$ = 0.0317382 rad, while the mean value of fitness function is equal to $q_{E}$ = 0.031743 rad.

Fig. 12.

The behavior of fitness value with respect to iteration (NNGA method)

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

Figure 13 shows the behavior of fitness function with respect to generation in case of PSO algorithm. According to the figure, the best fitness value based on proposed PSO algorithm is equal to $q_{E}$ = 0.0309428 rad.

Fig. 13.

The behavior of fitness value with respect to iteration (PSO algorithm)

Citation: International Review of Applied Sciences and Engineering 15, 2; 10.1556/1848.2023.00722

5 Conclusion

The inverse kinematics of redundant manipulators is complex and its solution is a challenging problem. This study proposed NNGA and PSO algorithm to solve the inverse kinematics problem of 3DOF redundant robot arm. The performance of the proposed techniques has been evaluated in terms of accuracy. The MSE has been used as accuracy indicator for the solution based on these approaches. In the sense of solution accuracy, a comparison study has been made between the proposed methods based on numerical simulation. Three trajectories have been tested using the proposed methods. According to simulated results, it has been shown that the accuracies due to PSO algorithm is better than that based on NNGA approach for all suggested trajectories. The improvement in accuracy has been determined and it is found that a 7% improvement in accuracy is obtained using PSO algorithm in case of triangular trajectory, while 2% improvement has been recorded in case of circular and sine-wave trajectories. In addition, this study has been compared to previous works in the literature. It has been shown that the proposed solution schemes outperform the other techniques.

This study can be extended for future work by suggesting other modern optimization techniques or using embedded system design to implement the proposed algorithms to reduce the calculation efforts [34–45].

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International Review of Applied Sciences and Engineering

Print ISSN:: 2062-0810

Online ISSN:: 2063-4269

Editorial Board

Mohammad Nazir AHMAD, Institute of Visual Informatics, Universiti Kebangsaan Malaysia, Malaysia

Murat BAKIROV, Center for Materials and Lifetime Management Ltd., Moscow, Russia

Nicolae BALC, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Umberto BERARDI, Toronto Metropolitan University, Toronto, Canada

Ildikó BODNÁR, University of Debrecen, Debrecen, Hungary

Sándor BODZÁS, University of Debrecen, Debrecen, Hungary

Fatih Mehmet BOTSALI, Selçuk University, Konya, Turkey

Samuel BRUNNER, Empa Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland

István BUDAI, University of Debrecen, Debrecen, Hungary

Constantin BUNGAU, University of Oradea, Oradea, Romania

Shanshan CAI, Huazhong University of Science and Technology, Wuhan, China

Michele De CARLI, University of Padua, Padua, Italy

Robert CERNY, Czech Technical University in Prague, Prague, Czech Republic

Erdem CUCE, Recep Tayyip Erdogan University, Rize, Turkey

György CSOMÓS, University of Debrecen, Debrecen, Hungary

Tamás CSOKNYAI, Budapest University of Technology and Economics, Budapest, Hungary

Anna FORMICA, IASI National Research Council, Rome, Italy

Alexandru GACSADI, University of Oradea, Oradea, Romania

Eugen Ioan GERGELY, University of Oradea, Oradea, Romania

Janez GRUM, University of Ljubljana, Ljubljana, Slovenia

Géza HUSI, University of Debrecen, Debrecen, Hungary

Ghaleb A. HUSSEINI, American University of Sharjah, Sharjah, United Arab Emirates

Nikolay IVANOV, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

Antal JÁRAI, Eötvös Loránd University, Budapest, Hungary

Gudni JÓHANNESSON, The National Energy Authority of Iceland, Reykjavik, Iceland

László KAJTÁR, Budapest University of Technology and Economics, Budapest, Hungary

Ferenc KALMÁR, University of Debrecen, Debrecen, Hungary

Tünde KALMÁR, University of Debrecen, Debrecen, Hungary

Milos KALOUSEK, Brno University of Technology, Brno, Czech Republik

Jan KOCI, Czech Technical University in Prague, Prague, Czech Republic

Vaclav KOCI, Czech Technical University in Prague, Prague, Czech Republic

Imre KOCSIS, University of Debrecen, Debrecen, Hungary

Imre KOVÁCS, University of Debrecen, Debrecen, Hungary

Angela Daniela LA ROSA, Norwegian University of Science and Technology, Trondheim, Norway

Éva LOVRA, Univeqrsity of Debrecen, Debrecen, Hungary

Elena LUCCHI, Eurac Research, Institute for Renewable Energy, Bolzano, Italy

Tamás MANKOVITS, University of Debrecen, Debrecen, Hungary

Igor MEDVED, Slovak Technical University in Bratislava, Bratislava, Slovakia

Ligia MOGA, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Marco MOLINARI, Royal Institute of Technology, Stockholm, Sweden

Henrieta MORAVCIKOVA, Slovak Academy of Sciences, Bratislava, Slovakia

Phalguni MUKHOPHADYAYA, University of Victoria, Victoria, Canada

Balázs NAGY, Budapest University of Technology and Economics, Budapest, Hungary

Husam S. NAJM, Rutgers University, New Brunswick, USA

Jozsef NYERS, Subotica Tech College of Applied Sciences, Subotica, Serbia

Bjarne W. OLESEN, Technical University of Denmark, Lyngby, Denmark

Stefan ONIGA, North University of Baia Mare, Baia Mare, Romania

Joaquim Norberto PIRES, Universidade de Coimbra, Coimbra, Portugal

László POKORÁDI, Óbuda University, Budapest, Hungary

Roman RABENSEIFER, Slovak University of Technology in Bratislava, Bratislava, Slovak Republik

Mohammad H. A. SALAH, Hashemite University, Zarqua, Jordan

Dietrich SCHMIDT, Fraunhofer Institute for Wind Energy and Energy System Technology IWES, Kassel, Germany

Lorand SZABÓ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Csaba SZÁSZ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Ioan SZÁVA, Transylvania University of Brasov, Brasov, Romania

Péter SZEMES, University of Debrecen, Debrecen, Hungary

Edit SZŰCS, University of Debrecen, Debrecen, Hungary

Radu TARCA, University of Oradea, Oradea, Romania

Zsolt TIBA, University of Debrecen, Debrecen, Hungary

László TÓTH, University of Debrecen, Debrecen, Hungary

László TÖRÖK, University of Debrecen, Debrecen, Hungary

Anton TRNIK, Constantine the Philosopher University in Nitra, Nitra, Slovakia

Ibrahim UZMAY, Erciyes University, Kayseri, Turkey

Andrea VALLATI, Sapienza University, Rome, Italy

Tibor VESSELÉNYI, University of Oradea, Oradea, Romania

Nalinaksh S. VYAS, Indian Institute of Technology, Kanpur, India

Deborah WHITE, The University of Adelaide, Adelaide, Australia

International Review of Applied Sciences and Engineering
Address of the institute: Faculty of Engineering, University of Debrecen
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2023
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International Review of Applied Sciences and Engineering
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International Review of Applied Sciences and Engineering
Language	English
Size	A4
Year of Foundation	2010
Volumes per Year	1
Issues per Year	3
Founder	Debreceni Egyetem
Founder's Address	H-4032 Debrecen, Hungary Egyetem tér 1
Publisher	Akadémiai Kiadó
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ISSN	2062-0810 (Print)
ISSN	2063-4269 (Online)