Calculation of the inverse kinematics solution of the 7-DOF redundant robot manipulator by the firefly algorithm and statistical analysis of the results in terms of speed and accuracy

ABSTRACT

In this study, the inverse kinematics solution of a 7-DOF redundant robot manipulator was performed by using firefly algorithm that is a swarm optimization technique. In order to show the power of this technique, a redundant robotic arm which is inadequate inverse kinematic solution by conventional methods has been chosen. Both speed and accuracy are two important factors in robotic studies. For this reason, the comparison of the method used in this study in terms of speed and accuracy has been carried out in depth. The scenario used is as follows: Firstly, the position equations of this manipulator are derived with the DH parameters. Afterward, the position of the end effector is obtained in the work space according to the forward kinematic calculation. Finally, the joint angles that will be directed to the calculated position values with the least error are obtained by the firefly algorithm and the obtained result is compared with other swarm algorithms such as particle swarm optimization and artificial bee colony.

KEYWORDS:

• Firefly algorithm
• swarm intelligence
• inverse kinematics
• redundant robot manipulator
• 7-dof

1. Introduction

Nowadays, swarm algorithms, which called meta-heuristic algorithms, have the most preferred intelligent optimization techniques for difficult problems [1,2]. These algorithms were developed by inspiration from the movements and social behaviours of animals in the nature. For example, particle swarm optimization (PSO) samples the movement of birds [3,4], ABC is similar to the food search behaviour of honey bees [5,6], ACO refers to the food search behaviour of ants [7,8]. Furthermore, they have been used extensively by researchers, because these algorithms have obtained effective results in complex problems with a very high degree of difficulty [9–12]. Even though the individuals of the swarms are actually non-sophisticated, they have the ability to collaborate, even over the most complex problems [13]. The most important feature of the swarm is that the individuals in the flock must be in perfect harmony with each other. Swarm intelligence emerges in this way. The firefly algorithm, which was first proposed by Xin-She Yang in 2008 and is based on the idea that fireflies in tropical regions use their lights to communicate with each other [14], works this way too and it can be applied for solving NP-hard problems [15]. Swarm intelligent optimization techniques have been used extensively by researchers in the robotics [16] field as well as in almost every field such as commerce, science and engineering [17,18]. Inverse kinematics [19,20], trajectory planning [21,22], intelligent robot control [23,24] and the energy efficiency aspect of robot movements [25] are just a few of them. These techniques are often preferred to obtain inverse kinematic solutions, which is a fundamental problem especially in robotics and is an NP-hard problematic [26].

Inverse kinematics [27] is at the base of the robot control and its solution is important. There are many algebraic [28] and numerical [29] methods which were developed by researchers, for the inverse kinematic problem. But the robotic arms used in these developed methods seem to be simple in structure. On the contrary, the robots used today are extremely complicated. So the inverse kinematic equations used for present-day manipulators are also extremely complicated and have a non-linear characteristic that takes a long time to solve. Because there are also a number of solutions for the inverse kinematics problem of these manipulators, it is highly suitable to use intelligent optimization techniques in such problems [30]. There are numerous studies published for the inverse kinematics solution of robot manipulators with the swarm intelligence algorithms. Çavdar et al. [20] have taken a new approach with the change in the search for food sources in the artificial bee colonies and they have successfully obtained the inverse kinematic solution of a six-joint puma robot manipulator. Mahanta et al. [31] have presented the application of soft computing techniques such as ABC, PSO and FA to obtain the inverse kinematics of Kawasaki RS06L 6-DOF robotic manipulator for a pick and place operation. Rokbani et al. [32] proposed a new heuristic approach based on firefly algorithm for inverse kinematic solution of a three-jointed robot arm. El-Sherbini et al. [33] have improved the parameters in the algorithm to increase the ability of the bees to find the best values in the search space. Ayyıldız et al. [34] are comparatively computed for inverse kinematics solution of a four-jointed series robotic using the swarm (PSO, QPSO) and other heuristic (GA, GSA) algorithms. Dereli and Koker [35] have calculated inverse kinematic solution of 7-joint serial manipulators first with the PSO, followed by changing the IW parameters and comparing the results.

In this study, firefly algorithm was examined in terms of accuracy and speed and the results were analysed. A new 7-dof redundant robot manipulator, previously unused in the literature, has been chosen to demonstrate the ability of the algorithm. In the second part of the study, the kinematics analysis of the robot manipulator has been derived, the fitness function has been introduced and the used method has been explained. In section 3, simulation results are analysed comparatively. It has been revealed that contribute to solving of the swarm number is in terms of speed and accuracy.

2. Materials and methods

2.1. Kinematic analysis of a 7-dof redundant robot

The robot arm that we give as the name SUNGUR 370 and is shown in Figure 1, is used in this study is a redundant manipulator, which is composed of a set of links connected to each other by revolute seven joints. This newly designed robot arm has a complex structure besides its ability to work smoothly and avoid obstacles successfully [36].

Figure 1. The structure of SUNGUR 370.

DH parameters [37,38] have been used for kinematic analysis of 7-jointed robot arm.

In Table 1, i shows the joint sequence. The lengths are given in metres and the angles are given in degree. (1) $\begin{array}{rl}{}_{i-1}^{\phantom{i-}i}T& =\left[\begin{array}{cccc}cos{\theta }_i& -cos{\alpha }_{i}sin{\theta }_i& sin{\alpha }_{i}sin{\theta }_i& a_{i}cos{\theta }_i\\ sin{\theta }_i& cos{\alpha }_{i}cos{\theta }_i& -cos{\theta }_{i}sin{\alpha }_i& a_{i}sin{\theta }_i\\ 0& sin{\alpha }_i& cos{\alpha }_i& d_i\\ 0& 0& 0& 1\end{array}\right],\end{array}$(1) (2) $\begin{array}{rl}{A}_{\text{End}-\text{Effector}}& ={}_0^{7}T={}_0^{1}T.{}_1^{2}T.{}_2^{3}T.{}_3^{4}T.{}_4^{5}T.{}_5^{6}T.{}_6^{7}T=\left[\begin{array}{cccc}{n}_x& s_x& a_x& P_x\\ n_y& s_y& a_y& P_y\\ n_z& s_z& a_z& P_z\\ 0& 0& 0& 1\end{array}\right],\end{array}$(2) where _iTⁱ⁺¹ is the transfer matrix of link i. ₀T⁷ matrix produces a Cartesian coordinate for any seven joint angles. In Equation (2), P_x, P_y, and P_z denote the elements of the position vector whereas n_x, n_y, n_z, s_x, s_y, s_z, a_x, a_y, a_z denote the rotational elements of the transformation matrix. In this study, only the position vector will be used to calculate the position error. The position vector equation is as follows (where s and c denote the sine and cosine functions): (3) $\begin{array}{rl}{P}_x& =\left(c{\theta }_{1}c{\theta }_{2}c{\theta }_{3}c{\theta }_4-s{\theta }_{1}s{\theta }_{3}s{\theta }_4-c{\theta }_{1}s{\theta }_{2}s{\theta }_4\right)\\ & \times \left(c{\theta }_{5}c{\theta }_6{l}_{7}c{\theta }_7-c{\theta }_{5}s{\theta }_6{l}_{7}s{\theta }_7-s{\theta }_5{d}_7+c{\theta }_5{l}_{6}c{\theta }_6+l_{5}c{\theta }_5\right)\\ & +\left(-c{\theta }_{1}c{\theta }_{2}s{\theta }_3-s{\theta }_{1}c{\theta }_3\right)\left(s{\theta }_{5}c{\theta }_6{l}_{7}c{\theta }_7-s{\theta }_{5}s{\theta }_6{l}_{7}s{\theta }_7+c{\theta }_5{d}_7+s{\theta }_{5}c{\theta }_6{l}_6+l_{5}s{\theta }_5\right)\\ & +\left(c{\theta }_{1}c{\theta }_{2}c{\theta }_{3}s{\theta }_4-s{\theta }_{1}s{\theta }_{3}s{\theta }_4+c{\theta }_{1}c{\theta }_{4}s{\theta }_2\right)\left(-s{\theta }_6{l}_{7}c{\theta }_7-c{\theta }_6{l}_{7}s{\theta }_7-l_{6}s{\theta }_6\right)\\ & +c{\theta }_{1}c{\theta }_2\left(c{\theta }_{3}c{\theta }_4{l}_4+l_{3}c{\theta }_3\right)-s{\theta }_1\left(s{\theta }_{3}c{\theta }_4{l}_4+l_{3}s{\theta }_3\right)\\ & -c{\theta }_{1}s{\theta }_2{l}_{4}s{\theta }_4+c{\theta }_{1}c{\theta }_2{l}_2,\end{array}$(3) (4) $\begin{array}{rl}{P}_y& =s{\theta }_{1}c{\theta }_{2}c{\theta }_{3}c{\theta }_4+c{\theta }_{1}s{\theta }_{3}c{\theta }_4-s{\theta }_{1}s{\theta }_{2}s{\theta }_4)\\ & \times \left(c{\theta }_{5}c{\theta }_6{l}_{7}c{\theta }_7-c{\theta }_{5}s{\theta }_6{l}_{7}s{\theta }_7-s{\theta }_5{d}_7+c{\theta }_{5}c{\theta }_6{l}_6+l_{5}c{\theta }_5\right)\\ & +\left(-s{\theta }_{1}c{\theta }_{2}s{\theta }_3+c{\theta }_{1}c{\theta }_3\right)\left(s{\theta }_{5}c{\theta }_6{l}_{7}c{\theta }_7-s{\theta }_{5}s{\theta }_6{l}_{7}s{\theta }_7+c{\theta }_5{d}_7+s{\theta }_{5}c{\theta }_6{l}_6+l_{5}s{\theta }_5\right)\\ & +\left(s{\theta }_{1}c{\theta }_{2}c{\theta }_{3}s{\theta }_4+c{\theta }_{1}s{\theta }_{3}s{\theta }_4+s{\theta }_{1}s{\theta }_{2}c{\theta }_4\right)\left(-s{\theta }_6{l}_{7}c{\theta }_7-c{\theta }_6{l}_{7}s{\theta }_7-l_{6}s{\theta }_6\right)\\ & +s{\theta }_{1}c{\theta }_2\left(c{\theta }_{3}c{\theta }_4{l}_4+l_{3}c{\theta }_3\right)+c{\theta }_1\left(s{\theta }_{3}c{\theta }_4{l}_4+l_{3}s{\theta }_3\right)\\ & -s{\theta }_{1}s{\theta }_{2}s{\theta }_4{l}_4+s{\theta }_{1}c{\theta }_2{l}_2,\end{array}$(4) (5) $\begin{array}{rl}{P}_z& =\left(-s{\theta }_{2}c{\theta }_{3}c{\theta }_4-c{\theta }_{2}s{\theta }_4\right)\left(c{\theta }_{5}c{\theta }_6{l}_{7}c{\theta }_7-c{\theta }_{5}s{\theta }_6{l}_{7}s{\theta }_7-s{\theta }_5{d}_7+c{\theta }_{5}c{\theta }_6{l}_6+l_{5}c{\theta }_5\right)\\ & +s{\theta }_{2}s{\theta }_3\left(s{\theta }_{5}c{\theta }_6{l}_{7}c{\theta }_7-s{\theta }_{5}s{\theta }_6{l}_{7}s{\theta }_7+c{\theta }_5{d}_7+s{\theta }_{5}c{\theta }_6{l}_6+s{\theta }_5{l}_5\right)\\ & +\left(-s{\theta }_{2}c{\theta }_{3}s{\theta }_4+c{\theta }_{2}c{\theta }_4\right)\left(-s{\theta }_6{l}_{7}c{\theta }_7-c{\theta }_6{l}_{7}s{\theta }_7-s{\theta }_6{l}_6\right)\\ & -s{\theta }_2\left(c{\theta }_{3}c{\theta }_4{l}_4+l_{3}c{\theta }_3\right)-c{\theta }_{2}s{\theta }_4{l}_4-s{\theta }_2{l}_2+l_1.\end{array}$(5)

Table 1. D-H parameters for robot manipulator.

i	ai(m)	αi (°)	di(m)	Θi (°)(Range)
1	0	−90	l1 = 0.5	−180 < Θ1 < 180
2	l2 = 0.2	90	0	−90 < Θ2 < 30
3	l3 = 0.25	−90	0	−90 < Θ3 < 120
4	l4 = 0.3	90	0	−90 < Θ4 < 90
5	l5 = 0.2	−90	0	−90 < Θ5 < 90
6	l6 = 0.2	0	0	−90 < Θ6 < 90
7	l7 = 0.1	0	d7 = 0.05	−30 < Q7 < 90

2.2. Model of position error and fitness function

The aim of this study is to obtain the joint angles that will move the end effector to the nearest point to the predetermined position. The Euclid formula which given Equation (6), is used to find the position error that appears in Figure 2. This equality is also a fitness function which will be used in this study. (6) $PositionError=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2}.$(6) In order to test the operation of the firefly algorithm, the robot arm is directed to a predetermined position.

Figure 2. Representing the position error.

In this study, the defined joints are as follows: Θ₁= 45°, Θ₂= 0°, Θ₃= 45°, Θ₄= 0°, Θ₅= 45°, Θ₆= 0°, Θ₇= 0°. The position of the robot arm which should go through these joints is calculated by firefly algorithm. When the robot arm receives these joints, the end effector will point to a certain position (Figure 3). The aim of this work is to find the new joints with firefly algorithm, which will position the end element in this determined position.

Figure 3. Determined position of the end effector for this study.

2.3. Firefly algorithm

Firefly Algorithm that was developed by Yang in 2008 [39], is a meta-heuristic and swarm-based optimization method used today to solve the most difficult optimization problems in many areas [40]. At the same time, the FA is part of the group of statistical algorithms since it randomly searches for the solution [41,42].

Firefly Algorithm is based on the social behaviour of fireflies in tropical climate regions. Fireflies produce rhythmic flashing lights to both raise fear on their enemies and to attract other fireflies regardless of their gender to them [41,42]. The less bright fireflies move toward the more brilliant fireflies. So, bright fireflies are always more attractive than others. Therefore, brightness and light intensity are two important variables in this algorithm [43,44]. However, the attractiveness varies according to the position of the firefly. This leads to another variant in the name of the light absorption coefficient [45,46]. To achieve optimal solutions in firefly algorithm: The fitness function of a given optimization problem is related to the intensity of the flashing light or light that helps to go to bright and more attractive places in the firefly's drive [47,48].

Flowchart of FA Algorithm is shown Figure 4. The following equation is used for the light intensity which has a decreasing structure as the distance (r) increases (γ; light absorption coefficient, range of values 0.1 and 1): (7) $I=I_0{e}^{-\gamma r^2}.$(7) The attractiveness of a firefly is proportional to the light intensity and is found in a formula similar to light intensity, as follow (range of values 0.1 and 1): (8) $\beta ={\beta }_0{e}^{-\gamma r^2},$(8) where β₀ is the attractiveness at r = 0. The distance between any two fireflies (x_i and x_j) is expressed as the Euclidean relationship, as follows: (9) $r_{ij}=x_i-x_j=\sqrt{\sum _{k=1}^d{\left(x_i-x_j\right)}^2},$(9) where d is the number of dimension. The movement of firefly i towards the more attractive firefly j is expressed in the following way: (10) $x_i=x_i+{\beta }_0{e}^{-\gamma r^2}\left(x_j-x_i\right)+\alpha {\epsilon }_i.$(10)

Figure 4. Flowchart of firefly algorithm.

This Equation (10) consists of three terms. The first term is the current position of the firefly, the second term is the more attractive firefly, and the last term (α) is a series of varying randomly (α: range of values 0.1 and 1).

Firefly algorithm, which is among the intelligent optimization techniques that are known for producing effective solutions to NP type problems, is widely used in many areas.

3. Simulation results

The main subject of this study is that the optimal angle values of joints are found for the end effector of the robot arm to reach the desired point. The target position of the robot arm can be seen in Figure 3. Simulations which are number of firefly, minimum position error, computation time and comparison with other techniques have been performed at the Matlab environment. The initial population of the algorithm has been randomly selected at the determined intervals of the joints. In addition, the algorithms have been run ten times and the minimum position error between these values has been considered the best solution. FA parameters are used as follows: α: 0.3, γ: 0.9 and β: 0.9.

Although one of the most important features of the swarm algorithms is the better solution as the number of particles increases, this solving time also increases. In this study, inverse kinematic calculations were performed with 30, 40, 50, 60, 70, 80 and 90 fireflies and the results are shown in Table 2.

Table 2. Particle number, computation time and position error simulation results.

Firefly number	Solution time (s)	Position error (cm)
30	0,1825	8,77e−03
40	0,5678	5,90e−04
50	0,8788	1.23e−05
60	1,8641	8.38e−07
70	5,8822	5.14e−08
80	6,0668	4.17e−09
90	8,8297	5.59e−11

As a result of the tests made, the joint angles appearing in Table 3 indicate that the end effector of the robot manipulator reaches the desired point with minimum error. It is clear that the value of joints is different in all tests due to the infinite number of solutions of redundant manipulators.

Table 3. The angle of the joints obtained by the number of particles.

Manual	45	0	45	0	45	0	0
	Estimated final angle of joints
Firefly number	θ1	θ2	θ3	θ4	θ5	θ6	θ7
30	105,218	5,890	−28,611	−37,947	60,792	39,698	25,477
40	38,135	29,674	78,172	−6,108	−3,328	−49,456	59,064
50	138,874	−5,254	−48,312	−19,639	7,576	66,989	26,008
60	161,233	1,565	−49,229	7,320	−47,714	−38,932	55,175
70	95,716	26,773	−6,055	−33,4	26,509	−59,975	79,581
80	156,924	29,087	−82,614	−38,518	25,683	17,636	43,932
90	63,975	−15,475	46,035	−1,071	0,336	40,476	67,894

The obtained joint angles reach the end effector of the robot manipulator to a certain point in the work space. The x, y, z coordinates of this point are shown in Table 4.

Table 4. Positions of the end effector in work space.

	Px (cm)	Py (cm)	Pz (cm)
Target position	−24,7487	100,9619	50	–	–
Firefly number	Estimated positions			Position error (cm)	Computation time (s)
30	−24,2647	100,2518	49,8330	8,77e−03	0,1825
40	−24,7363	100,9702	49,9915	5,90e−04	0,5678
50	−24,7466	100,9619	49,9949	1,23e−05	0,8788
60	−24,7444	100,9641	50,0021	8,38e−07	1,8641
70	−24,7624	100,9565	50,0079	5,14e−08	5,8821
80	−24,7447	100,9637	49,9965	4,17e−09	6,0668
90	−24,7412	100,9625	50,0042	5,59e−11	8,8297

According to swarm numbers, the position errors calculated using the joint angles obtained with the firefly technique is shown in Figure 5. It is obvious that the increase in the number of firefly has helped the solution to be in the best condition.

Figure 5. Solving contribution of swarm numbers.

Although the algorithms were terminated at 1000 iterations, the solution could not be improved after a certain value. This value is considered to be the best solution for the algorithm and is also shown in Figures 5 and 6. The increase in the number of swarm makes it possible to obtain a better solution, but it is moving up to the top of the computation time (Figure 6).

Figure 6. The graphic of computation time and position error.

It has been emphasized earlier in this work that the inverse kinematic solution set of a 7-DOF redundant manipulator is an infinite number of elements. This situation is obvious in Figure 7. Here, the final positions of the manipulator have been revealed in the RoboAnalyzer [49] interface through the joint angles obtained with each firefly swarm. When the literature search is done in the inverse kinematic solution, it is used in artificial bee colony and particle swarm optimization as well as firefly algorithm.

Figure 7. Final position of the 7-DOF manipulator according to the number of firefly.

In this study, the joint angles obtained with these algorithms are shown in Table 5. Since the firefly algorithm works based on a swarm optimization technique, it is of course important to compare it with other swarm algorithms. The comparison was made according to the execution time and the number of iterations to reach the solution and it was illustrated in two different graphs. Each algorithm was run 10 times and the smallest position error was obtained as the best solution.

Table 5. Comparative results of other techniques.

	Actual Position	PSO	ABC	Firefly
PX (cm)	−24.7487	−24,6665	−24.7149	−24.7402
PY (cm)	100.9619	100,3896	101.8105	100.9723
PZ (cm)	50	49,6557	50.0104	49.9989
Error	–	6,72e−03	5.45e−04	6.53e−05
Computation time (s)	–	0.4498	0,4441	0,9204
Parameters		Particle number: 300	Population: 100	Firefly number: 50
		c1: 1.7	Limit: 350	Gamma (γ): 0.9
		c2: 1.7	Onlooker:100	Alpha (α): 0.3
		Initial: Randomly	Initial: Randomly	Beta (β): 0.9
				Initial: Randomly

Figure 8 shows the computation times of all three algorithms. ABC and PSO techniques completed 500 iterations around 0.7 and 0.9 s, but the best solution reached 0.45 s. However, the firefly algorithm has the longest run time in terms of completing both 500 iterations and reaching the best solution. These times are 11 and 0.9 s.

Figure 8. The graphic of computation time [50].

Figure 9 illustrates the position error comparisons of the algorithms. In terms of both the minimum position error value and the number of iterations that the solution reaches, the technique with the worst solution among these three algorithms is PSO. However, from this point of view, the best solution values were obtained by the firefly algorithm.

Figure 9. The graphic of position error (cm) and iteration [50].

4. Conclusion

In this study, the inverse kinematics solution, that is the most difficult problem group, is calculated by using firefly algorithm which is one of the recent popular optimization techniques. Simulation results show that the firefly algorithm has been yield very successful data. As the size of the swarm grew, both the quality of the solution and the calculation time has increased. Therefore, it seems that swarm size is an important parameter in swarm-based optimization algorithms. So, this algorithm can be used in very sensitive work in which it is more important that the work is done rather than the importance of the solution time and it can also be used for tasks that need to be done in a short period of time. It is obvious that the firefly algorithm produces better solutions between 10 times and 10,000 times than other swarm algorithms in the tests made. When compared with other techniques in terms of computation time, it can be said that the solution is reached in relatively short time. As a result, the firefly algorithm which can produce effective results in the most complex problems has proved successful in the inverse kinematic solution of a redundant robot arm.

Disclosure statement

No potential conflict of interest was reported by the authors.