**5. Simulation and results**

#### **5.1 Experimental setup**

The main task of this study is to find the optimal value of the joints' variable to ensure the end effector of robots can reach the desired points. The desired point positions of the Scenario 2 and 3 are shown as the **Figure 3**. Research using the ISADE and Pro-ISADE algorithm, which were developed by the authors [13–15], to get simulation results of inverse kinematics problem and then compared it with the results when using PSO, DE and Pro-PSO, Pro-DE algorithms. When solving the IK problem for the 7-DoF serial robot manipulator, the study focused on three main aspects. The first of these is the sensitivity of the solution - in the other word, the amount distance error of end effector is minimum. The second criterion was the execution time. In order to avoid the endless loop, the maximum numbers of generation ð Þ *itermax* were set as 600, 600 and 130 for PSO (Pro-PSO), DE (Pro-DE) and ISADE (Pro-ISADE), respectively. And the final aspect is the searching space of joints' variables. Normally, Normally, almost all studies have been using the Range of Motion (*RoM*) of joints for its boundary space. Our algorithm [15] proposed to use the searching space of current generation is around previous optimal joints' values. In the **Table 2**, the *ubsi*þ<sup>1</sup> *and lbsi*þ<sup>1</sup> are the joints' upper and lower boundary of the current generation.*C*<sup>1</sup> *and C*<sup>2</sup> are weights of personal best and global best, respectively. *w* is the inertia weight. *ρ* is the number of run for each algorithm to choose the best result. Besides, after some trial runs for the algorithms, we noticed that our ISDE algorithm gave much better results than DE and the least was the PSO algorithm. Thus, when setting up the maximum distance error by the fitness value setting for the end effector position, the study set the value of 1*e* � 14 ð Þ *m* ; 1*e* � 15 ð Þ *m* and 1*e* � 17 ð Þ *m* for PSO (Pro-PSO); DE (Pro-DE) and ISADE (Pro-ISADE), respectively or that can be seen in the **Table 2**. In this research, the proposed and other methods were tested in the two different Scenarios. Both the first and second Scenario was coded by Matlab version 2019a and run on the computer equipped with an Intel Core i5-4258U @2.4GHz processor and 8 GB Ram memory.

#### **5.2 Scenario 1 results**

After applying the inverse kinematic problem processing algorithms for a single endpoint, the results are shown in **Figures 4** and **5**. All algorithms are able to handle

#### **Figure 3.**

*Testing scenarios. (a) Scenario 2: 100 random points in workspace; (b) Scenario 3: 100 points on a spiral trajectory.*

**Algorithms**

**29**

PSO Pro-PSO

DE Pro-DE

ISADE Pro-ISADE

**Table 2.** *Optimization*

 *parameters*

 *used in PSO, Pro-PSO, DE Pro-DE, and ISADE, Pro-ISADE.*

 130

130

1e-17 1e-17

*qoi* � *π=*100

*RoM*

600

600

1e-15 1e-15

*qoi* � *π=*100

*RoM*

Scheme 2 in Eq. (10)

 0.9 0.9

*Eq:*ð20)

 \*\* \*\* \*\* \*\*

 \*\* \*\*

 \*\* \*\*

\*\* \*\* \*\* \*\*

*Eq:* 19 ð )

600

600

1e-14 1e-14

*qoi* � *π=*100

 **Max No. of Gen.**

**Max Distance Err. (m) Searching space**

*ubsi*þ**1***lbsi*þ**1**

*RoM*

10

\*\* \*\*

\*\*

\*\*

 1.5 1.5

*w*

¼

*wstart*

þ

*iter itermax* 

∗ *wend*

ð

�

*wstart*

\*\*

 Þ \*\*

0.5

*Use Improved Differential Evolution Algorithms to Handle the Inverse Kinetics Problem…*

0.5

 �

*ρ*

**Mut. rate**

 **Cross. rate**

*C***1** *C***2**

*w*

 *Fi*

*DOI: http://dx.doi.org/10.5772/intechopen.97138*

½

**max** *iter* *Use Improved Differential Evolution Algorithms to Handle the Inverse Kinetics Problem… DOI: http://dx.doi.org/10.5772/intechopen.97138*


**Table 2.**

*Optimization parameters used in PSO, Pro-PSO, DE Pro-DE, and ISADE, Pro-ISADE.*

**5. Simulation and results**

*Robotics Software Design and Engineering*

The main task of this study is to find the optimal value of the joints' variable to ensure the end effector of robots can reach the desired points. The desired point positions of the Scenario 2 and 3 are shown as the **Figure 3**. Research using the ISADE and Pro-ISADE algorithm, which were developed by the authors [13–15], to get simulation results of inverse kinematics problem and then compared it with the results when using PSO, DE and Pro-PSO, Pro-DE algorithms. When solving the IK problem for the 7-DoF serial robot manipulator, the study focused on three main aspects. The first of these is the sensitivity of the solution - in the other word, the amount distance error of end effector is minimum. The second criterion was the execution time. In order to avoid the endless loop, the maximum numbers of generation ð Þ *itermax* were set as 600, 600 and 130 for PSO (Pro-PSO), DE (Pro-DE) and ISADE (Pro-ISADE), respectively. And the final aspect is the searching space of joints' variables. Normally, Normally, almost all studies have been using the Range of Motion (*RoM*) of joints for its boundary space. Our algorithm [15] proposed to use the searching space of current generation is around previous optimal joints' values. In the **Table 2**, the *ubsi*þ<sup>1</sup> *and lbsi*þ<sup>1</sup> are the joints' upper and lower boundary of the current generation.*C*<sup>1</sup> *and C*<sup>2</sup> are weights of personal best and global best, respectively. *w* is the inertia weight. *ρ* is the number of run for each algorithm to choose the best result. Besides, after some trial runs for the algorithms, we noticed that our ISDE algorithm gave much better results than DE and the least was the PSO algorithm. Thus, when setting up the maximum distance error by the fitness value setting for the end effector position, the study set the value of 1*e* � 14 ð Þ *m* ; 1*e* � 15 ð Þ *m* and 1*e* � 17 ð Þ *m* for PSO (Pro-PSO); DE (Pro-DE) and ISADE (Pro-ISADE), respectively or that can be seen in the **Table 2**. In this research, the proposed and other methods were tested in the two different Scenarios. Both the first and second Scenario was coded by Matlab version 2019a and run on the computer equipped with an Intel Core i5-4258U @2.4GHz processor and 8 GB Ram memory.

After applying the inverse kinematic problem processing algorithms for a single endpoint, the results are shown in **Figures 4** and **5**. All algorithms are able to handle

*Testing scenarios. (a) Scenario 2: 100 random points in workspace; (b) Scenario 3: 100 points on a spiral*

**5.1 Experimental setup**

**5.2 Scenario 1 results**

**Figure 3.**

*trajectory.*

**28**

processing speed of the ISADE algorithm is the best, followed by the DE algorithm and finally with the PSO algorithm. In **Table 3** the study of selecting stop conditions for algorithms is the maximum number of iterations of 85 rounds. After 10 runs, the best results are shown in the table. The ISADE algorithm gives the best processing results in terms of both quality and speed. The endpoint deviation can reach 2.8422e-13 (m) in 0.049 (s) time. For the PSO algorithm, it can handle the reverse kinematic problem for the end point with an accuracy of 2.6815e-4 in a period of 0.0941 (s). and, 5.7514e-10 (m) and 0.0715 (s) are the accuracy of end effector and

*Use Improved Differential Evolution Algorithms to Handle the Inverse Kinetics Problem…*

As mentioned above, in this Scenario 2, algorithms was used to resolve inverse kinematics problem for 100 randomly chosen points within the workspace of the robot. When processed at each point, the end effector started at the same initial position of [0 0 0 0 0 0 0] for 7 serial joints values. Because the end effector points all come from the same starting point to go to each of the 100 points, the study only used the ISADE algorithm and compares with the results from PSO and DE algorithms without using the Pro-ISADE algorithm as well as Pro-DE and Pro-PSO. The 100 randomly selected points were shown in the **Figure 3a**. Results when applying ISADE and the other algorithm were presented in the **Figure 6**. As shown in the Figure, all algorithms have solved problem well. In particular, with the ISADE algorithm, although the fitness value in experimental setup required 1000 and 100 times higher than the required by applying the PSO and DE algorithms, respectively, it was not only guaranteed required precision but also showed faster processing speed and fewer iterations compared to the 2 other algorithms. Specifically, as shown in **Figure 6b** and **c** and especially **Table 4**, the average execution time when using ISADE to solve IK of each points was around 0.0685 second, while this value of the PSO and DE algorithm were on average 0.2307 (s) and 0.0978 (s) respectively. The main reason for this, as seen in **Figure 6b** and **Table 4**, was the number of generations to reach the optimal values much higher in PSO algorithm and slightly higher in DE algorithm, compared to in ISADE algorithm. Specifically, the PSO algorithm needed an average of 413.24 and the DE algorithm needed average of 124.45 loops to find a solution, while the ISADE algorithm used an average of 85.63 loops. Another remarkable thing is although there was not much difference in the number of iterations to solve the problem between the two algorithms DE and ISADE, but the ISADE algorithm still gave a processing speed of 1.42 times higher than DE algorithm though required 100 times more accuracy for the ISADE algorithm. This demonstrated the very high efficiency of the ISADE algorithm when it was applied to handle inverse kinematics problem for this robot. In short, in the optimization study for randomly chosen points in working space, the ISADE algorithm presented the best algorithm to

resolve the IK requirement in term of accuracy, iteration and execution time.

In Scenario 2, the end effector moved through the 100 points located on a specific trajectory that was defined in Eq. (22) and shown in **Figure 3b**. The main difference between Scenario 2 and Scenario 3 is that, instead of after solving each IK problem for each point, the end effector goes back to the original point to continue processing for the next points like in Scenario 2, in Scenario 3 the end effector starts from previous point in order to calculate for the next point. Stemming from this feature, the searching space of joints' variable also starts previous optimal joints' values. However, depending on the searching space we have 2 smaller cases such as:

execution time for DE algorithm.

*DOI: http://dx.doi.org/10.5772/intechopen.97138*

**5.3 Scenario 2 results**

**5.4 Scenario 3 results**

**31**

**Figure 4.** *End effector distance error vs. generations in Scenario 1.*

#### **Figure 5.**

*End effector distance error vs. time in Scenario 1.*


#### **Table 3.**

*Comparison of ISADE with other algorithms.*

the inverse kinetics problem, but the best results have been obtained with the ISADE algorithm as shown in **Table 3**.

**Figures 4** and **5** show convergence speed of algorithms corresponding to the number of iterations and processing time, respectively. The results show that the *Use Improved Differential Evolution Algorithms to Handle the Inverse Kinetics Problem… DOI: http://dx.doi.org/10.5772/intechopen.97138*

processing speed of the ISADE algorithm is the best, followed by the DE algorithm and finally with the PSO algorithm. In **Table 3** the study of selecting stop conditions for algorithms is the maximum number of iterations of 85 rounds. After 10 runs, the best results are shown in the table. The ISADE algorithm gives the best processing results in terms of both quality and speed. The endpoint deviation can reach 2.8422e-13 (m) in 0.049 (s) time. For the PSO algorithm, it can handle the reverse kinematic problem for the end point with an accuracy of 2.6815e-4 in a period of 0.0941 (s). and, 5.7514e-10 (m) and 0.0715 (s) are the accuracy of end effector and execution time for DE algorithm.

#### **5.3 Scenario 2 results**

As mentioned above, in this Scenario 2, algorithms was used to resolve inverse kinematics problem for 100 randomly chosen points within the workspace of the robot. When processed at each point, the end effector started at the same initial position of [0 0 0 0 0 0 0] for 7 serial joints values. Because the end effector points all come from the same starting point to go to each of the 100 points, the study only used the ISADE algorithm and compares with the results from PSO and DE algorithms without using the Pro-ISADE algorithm as well as Pro-DE and Pro-PSO.

The 100 randomly selected points were shown in the **Figure 3a**. Results when applying ISADE and the other algorithm were presented in the **Figure 6**. As shown in the Figure, all algorithms have solved problem well. In particular, with the ISADE algorithm, although the fitness value in experimental setup required 1000 and 100 times higher than the required by applying the PSO and DE algorithms, respectively, it was not only guaranteed required precision but also showed faster processing speed and fewer iterations compared to the 2 other algorithms. Specifically, as shown in **Figure 6b** and **c** and especially **Table 4**, the average execution time when using ISADE to solve IK of each points was around 0.0685 second, while this value of the PSO and DE algorithm were on average 0.2307 (s) and 0.0978 (s) respectively. The main reason for this, as seen in **Figure 6b** and **Table 4**, was the number of generations to reach the optimal values much higher in PSO algorithm and slightly higher in DE algorithm, compared to in ISADE algorithm. Specifically, the PSO algorithm needed an average of 413.24 and the DE algorithm needed average of 124.45 loops to find a solution, while the ISADE algorithm used an average of 85.63 loops. Another remarkable thing is although there was not much difference in the number of iterations to solve the problem between the two algorithms DE and ISADE, but the ISADE algorithm still gave a processing speed of 1.42 times higher than DE algorithm though required 100 times more accuracy for the ISADE algorithm. This demonstrated the very high efficiency of the ISADE algorithm when it was applied to handle inverse kinematics problem for this robot. In short, in the optimization study for randomly chosen points in working space, the ISADE algorithm presented the best algorithm to resolve the IK requirement in term of accuracy, iteration and execution time.

#### **5.4 Scenario 3 results**

In Scenario 2, the end effector moved through the 100 points located on a specific trajectory that was defined in Eq. (22) and shown in **Figure 3b**. The main difference between Scenario 2 and Scenario 3 is that, instead of after solving each IK problem for each point, the end effector goes back to the original point to continue processing for the next points like in Scenario 2, in Scenario 3 the end effector starts from previous point in order to calculate for the next point. Stemming from this feature, the searching space of joints' variable also starts previous optimal joints' values. However, depending on the searching space we have 2 smaller cases such as:

the inverse kinetics problem, but the best results have been obtained with the

PSO 85 2.6815e-04 0.0941 DE 85 5.7514e-10 0.0715 ISADE 85 2.8422e-13 0.0490

**Figures 4** and **5** show convergence speed of algorithms corresponding to the number of iterations and processing time, respectively. The results show that the

**Max. Iteration Position error (m) Calculation time (s)**

ISADE algorithm as shown in **Table 3**.

*Comparison of ISADE with other algorithms.*

*End effector distance error vs. time in Scenario 1.*

**Figure 4.**

**Figure 5.**

**Table 3.**

**30**

*End effector distance error vs. generations in Scenario 1.*

*Robotics Software Design and Engineering*

• Scenario 3.2: Searching spaces for joints' variables are around the previous optimal joints' values. The study compared the results when using Pro-ISADE

Fitness value 1e-14 1e-15 1e-17 Avg. error 7.3016e-13 2.2938e-13 2.1644e-14 STD 2.0415e-13 5.991e-14 6.2125e-15 Avg. iteration 413.24 124.45 85.63 Avg. execution time 0.2307 0.0978 0.0685

*Use Improved Differential Evolution Algorithms to Handle the Inverse Kinetics Problem…*

The results were presented in the **Figure 7** and **Table 5**. Similar to the Scenario 2, although the experimental installation required the ISADE (and Pro-ISADE) algorithm to be 100 and 1000 times more accurate than the algorithm DE (Pro-DE) and PSO (Pro PSSO), respectively, all of 6 algorithms gave appropriated solutions for all the points in the trajectory. It can be seen that, in both cases 3.1 and 3.2 the ISADE and Pro-ISADE algorithms showed the best ability to resolve the inverse kinematics problems in all 3 aspects: accuracy, execution time and number of generations. More specifically, in Scenario 3.1, when searching space for joints' variables were *RoMs*, the average achieved accuracies for ISADE was around 2.0748e-14 (m) that is much better than the values of 7.5404e-13 (m) and 2.2260e-13 (m) corresponding for PSO and DE algorithms. Although the ISADE algorithm was set to a fitness value to achieve such higher accuracy, the execution time of the algorithm was still below the time of PSO and DE algorithm. These average execution time values were 0.0679 (s); 0.0845 (s) and 0.3478 (s) second for ISADE, DE and Pro algorithm, respectively. The above results can be partly explained based on the number of necessary iterations that each algorithm was needed to find the optimal values of joints variables. From **Figure 6c**, it showed that, when solving the IK problem for almost points in the spiral trajectory, the ISADE method used the least number of iterations. The **Table 5** presented more clearly, on the average each point in the trajectory the ISADE needed 85.19 generations to find the optimal values, these

**PSO DE ISADE**

algorithm with when using Pro-PSO and Pro-DE algorithms.

**Table 4.**

**33**

*Comparative results in case 2.*

*DOI: http://dx.doi.org/10.5772/intechopen.97138*

means number for DE and PSO algorithm are 125.44 and 391.1

including distance error, execution time and number of generations.

In Scenario 3.2, the searching space for joints' variables were around previous optimal values that were set up as in the **Table 2**. Similar to the Scenario 3.1, all of the comparison parameters gotten from using Pro-ISADE algorithm were better than that values from Pro-DE and Pro-PSO algorithms. These parameters are described in the as well as **Table 5**. In order to comparison between Scenario 3.1 with Scenario 3.2, all average parameters was shown in the **Table 5**. From all comparison, the proposed ISADE or Pro-ISADE were always proved the best solution to solve the inverse kinematics requirements for the manipulator robot. Moreover, **Table 5** also showed that, the Pro-ISADE had better performance compared to ISADE. By using Pro-ISADE algorithm, it reduced all of parameters

Another very important result gotten from Scenario 3.2 is the quality of joints' values. **Figure 8** show the joints' value in two cases of using ISADE in Scenario 3.1 and using Pro-ISADE in Scenario 3.2. It is clear that the joints' value in the Scenario 3.1 were change dramatically. On the contrary, the values of joints in Scenario 3.2 changed continuously and slowly. The quality of joints variable values as **Figure 9b**,

**Figure 6.** *Results for Scenario 2. (a) Distance error. (b) Execution time. (c) Number of generations.*

• Scenario 3.1: Searching spaces for joints' variables are *RoMs*. Then, like the Scenario 2, the study compared the results when using the ISADE algorithm with the results when using the PSO and DE algorithms.


*Use Improved Differential Evolution Algorithms to Handle the Inverse Kinetics Problem… DOI: http://dx.doi.org/10.5772/intechopen.97138*

#### **Table 4.**

*Comparative results in case 2.*

• Scenario 3.2: Searching spaces for joints' variables are around the previous optimal joints' values. The study compared the results when using Pro-ISADE algorithm with when using Pro-PSO and Pro-DE algorithms.

The results were presented in the **Figure 7** and **Table 5**. Similar to the Scenario 2, although the experimental installation required the ISADE (and Pro-ISADE) algorithm to be 100 and 1000 times more accurate than the algorithm DE (Pro-DE) and PSO (Pro PSSO), respectively, all of 6 algorithms gave appropriated solutions for all the points in the trajectory. It can be seen that, in both cases 3.1 and 3.2 the ISADE and Pro-ISADE algorithms showed the best ability to resolve the inverse kinematics problems in all 3 aspects: accuracy, execution time and number of generations. More specifically, in Scenario 3.1, when searching space for joints' variables were *RoMs*, the average achieved accuracies for ISADE was around 2.0748e-14 (m) that is much better than the values of 7.5404e-13 (m) and 2.2260e-13 (m) corresponding for PSO and DE algorithms. Although the ISADE algorithm was set to a fitness value to achieve such higher accuracy, the execution time of the algorithm was still below the time of PSO and DE algorithm. These average execution time values were 0.0679 (s); 0.0845 (s) and 0.3478 (s) second for ISADE, DE and Pro algorithm, respectively. The above results can be partly explained based on the number of necessary iterations that each algorithm was needed to find the optimal values of joints variables. From **Figure 6c**, it showed that, when solving the IK problem for almost points in the spiral trajectory, the ISADE method used the least number of iterations. The **Table 5** presented more clearly, on the average each point in the trajectory the ISADE needed 85.19 generations to find the optimal values, these means number for DE and PSO algorithm are 125.44 and 391.1

In Scenario 3.2, the searching space for joints' variables were around previous optimal values that were set up as in the **Table 2**. Similar to the Scenario 3.1, all of the comparison parameters gotten from using Pro-ISADE algorithm were better than that values from Pro-DE and Pro-PSO algorithms. These parameters are described in the as well as **Table 5**. In order to comparison between Scenario 3.1 with Scenario 3.2, all average parameters was shown in the **Table 5**. From all comparison, the proposed ISADE or Pro-ISADE were always proved the best solution to solve the inverse kinematics requirements for the manipulator robot. Moreover, **Table 5** also showed that, the Pro-ISADE had better performance compared to ISADE. By using Pro-ISADE algorithm, it reduced all of parameters including distance error, execution time and number of generations.

Another very important result gotten from Scenario 3.2 is the quality of joints' values. **Figure 8** show the joints' value in two cases of using ISADE in Scenario 3.1 and using Pro-ISADE in Scenario 3.2. It is clear that the joints' value in the Scenario 3.1 were change dramatically. On the contrary, the values of joints in Scenario 3.2 changed continuously and slowly. The quality of joints variable values as **Figure 9b**,

• Scenario 3.1: Searching spaces for joints' variables are *RoMs*. Then, like the Scenario 2, the study compared the results when using the ISADE algorithm

with the results when using the PSO and DE algorithms.

*Results for Scenario 2. (a) Distance error. (b) Execution time. (c) Number of generations.*

**Figure 6.**

*Robotics Software Design and Engineering*

**32**

**PSO Pro-PSO DE Pro-DE ISADE Pro-ISADE**

Fitness value 1e-9 Not applied 1e-10 Not applied 1e-12 Not applied Avg. error (m) 2.4151e-09 Not applied 6.9655e-10 Not applied 6.8362e-11 Not applied STD (m) 5.8117e-10 Not applied 2.0075e-10 Not applied 2.3796e-11 Not applied Avg. iteration 357.91 Not applied 76.54 Not applied 64.34 Not applied Avg. execution time (s) 0.2931 Not applied 0.1115 Not applied 0.0455 Not applied

*Use Improved Differential Evolution Algorithms to Handle the Inverse Kinetics Problem…*

Fitness value *1e-14* 1e-14 *1e-15 1e-17* 1e-12 Avg. error (m) *7.4140e-13* 7.4650e-13 *2.2260e-13* 2.2950e-13 *2.0748e-14* 2.0103e-14 STD (m) *1.9574e-13* 1.9736e-13 *6.5615e-14* 6.1330e-14 *1.0414e-14* 9.8913e-15 Avg. iteration *429.950* 407.8800 *125.4400* 114.2700 *85.1900* 75.2300 Avg. execution time (s) *0.3604* 0.2576 *0.1015* 0.0845 *0.0679* 0.0554

*Joint variables' results. (a) Joint variables' values in Scenario 3.1 using ISADE algorithm. (b) Joint variables'*

Scenario 1

**Table 5.**

**Figure 8.**

**35**

*values in Scenario 3.2 using Pro-ISADE algorithm.*

Scenario 3.1 (Italic values) and Scenario 3.2

*DOI: http://dx.doi.org/10.5772/intechopen.97138*

*Comparative results between all cases.*

*Italics were used to differentiate the results of Scenario 3.1 and 3.2.*

**Figure 7.** *Results for Scenario 3.1. (a) Distance error. (b) Execution time. (c) Number of generations.*

that received by using Pro-ISADE, will ensure feasibility in the next stages of calculation and design for the robot. These values, along with the values of speed, acceleration, as well as the weight parameters of the stages, will be used in the dynamic problem as well as in future control.

*Use Improved Differential Evolution Algorithms to Handle the Inverse Kinetics Problem… DOI: http://dx.doi.org/10.5772/intechopen.97138*


#### **Table 5.**

*Comparative results between all cases.*

#### **Figure 8.**

*Joint variables' results. (a) Joint variables' values in Scenario 3.1 using ISADE algorithm. (b) Joint variables' values in Scenario 3.2 using Pro-ISADE algorithm.*

that received by using Pro-ISADE, will ensure feasibility in the next stages of calculation and design for the robot. These values, along with the values of speed, acceleration, as well as the weight parameters of the stages, will be used in the

*Results for Scenario 3.1. (a) Distance error. (b) Execution time. (c) Number of generations.*

dynamic problem as well as in future control.

*Robotics Software Design and Engineering*

**Figure 7.**

**34**

different robot models. The Table presents: the used algorithm for the IK calculation, selected manipulators for the test, the algorithms that are used to comparison. For example, El-Sherbiny et al. [11] used the Adaptive Neuro Fuzzy Inference System (ANFIS) algorithm to calculate the IK problem of a 5 DOF robot, and then compared results with GA algorithm. Both algorithms could get the appropriate solutions, but ANFIS algorithm proved to be the best one. The comparison also shows that a number of studies [12, 22, 23], using optimal algorithms such as PSO, ABC, Q-PSO … handle the inverse kinetic requirements for the model of 7 degrees of freedom. All the used algorithms have proven the ability to handle the problem, but it is not difficult to see that most of these studies have the lower accuracy and processing speed than the ISADE as well as the Pro-ISADE algorithm proposed in

*Use Improved Differential Evolution Algorithms to Handle the Inverse Kinetics Problem…*

**Results of Used algorithm**

4-DOF PSO GA

Rokbani et al. [20] 3-DOF 10 Firefly 60 Firefly

5-DOF Adaptive Neuro

Shi and Xie [21] 6-DOF Adaboost NN —

Serkan Dereli [12] 7-DOF Q-PSO PSO; ABC; Firefly

Serkan Dereli [23] 7-DOF firefly PSO, ABC

*Italics were used to differentiate the results of Scenario 3.1 and 3.2.*

*Comparison with some other studies.*

Our study 7-DOF ISADE, Pro-ISADE PSO, DE, Pro-PSO, Pro-DE

Fuzzy Inference System (ANFIS)

> 0.00267 — 0.3 —

7-DOF Random IW-PSO Global–Local Best

**Results of Compared algorithm**

1.27e17 1.78e18 Position error (m) 1.21e03 7.15e3 Execution time (s)

7.70e06 3.96e04 Position error (m) 0.0196 0.1753 Execution time (s)

GA

5.426e03 7.64e04 Position error (m) 0.0308 83.1239 Execution time (s)

IW-PSO 6.20e03 3.64e03 Position error (m) 1.6 1.2 Execution time (s)

6.69347e-11 1.4547e-3 Position error (m) 0.2195 0.4806 Execution time (s)

6.53e05 5.45e04 Position error (m) 0,9204 0,4441 Execution time (s)

2.0103e-14 *7.4140e-13* Position error (m) 0.0554 *0.3604* Execution time (s)

**Average of**

this study.

Ayyıldız and Çetinkaya [7]

El-Sherbiny et al.

Dereli and Köker

[11]

[22]

**Table 6.**

**37**

**Research Robot**

**arm**

*DOI: http://dx.doi.org/10.5772/intechopen.97138*

**Figure 9.** *Results for Scenario 3.2. (a) Distance error. (b) Execution time. (c) Number of generations.*

In short, after comparing the results of Scenario 3.1 and 3.2, it is possible to conclude that the ISADE algorithm and Pro-ISADE are the best solutions to solve the IK problem for the robot in all aspects: endpoint accuracy, execution time and number of generation. The Pro-ISADE algorithm not only guarantees the above parameters, it also ensures the quality of the joints' variables to serve the next computational and design stages.

**Table 5** summarizes results of the average error, the standard deviation of error (STD), the average iteration and the average execution time of all Scenarios. As in the table, the algorithms of ISADE and Pro-ISADE got the better results than the other algorithms.

As mentioned at the beginning of this article, intelligent optimization techniques have been using more and more popular in difficult and complex tasks including the IK problem for redundant manipulator robots. **Table 6** shows some studies used meta-heuristic optimization algorithms to resolve the inverse kinematics task for

*Use Improved Differential Evolution Algorithms to Handle the Inverse Kinetics Problem… DOI: http://dx.doi.org/10.5772/intechopen.97138*

different robot models. The Table presents: the used algorithm for the IK calculation, selected manipulators for the test, the algorithms that are used to comparison. For example, El-Sherbiny et al. [11] used the Adaptive Neuro Fuzzy Inference System (ANFIS) algorithm to calculate the IK problem of a 5 DOF robot, and then compared results with GA algorithm. Both algorithms could get the appropriate solutions, but ANFIS algorithm proved to be the best one. The comparison also shows that a number of studies [12, 22, 23], using optimal algorithms such as PSO, ABC, Q-PSO … handle the inverse kinetic requirements for the model of 7 degrees of freedom. All the used algorithms have proven the ability to handle the problem, but it is not difficult to see that most of these studies have the lower accuracy and processing speed than the ISADE as well as the Pro-ISADE algorithm proposed in this study.


#### **Table 6.**

*Comparison with some other studies.*

In short, after comparing the results of Scenario 3.1 and 3.2, it is possible to conclude that the ISADE algorithm and Pro-ISADE are the best solutions to solve the IK problem for the robot in all aspects: endpoint accuracy, execution time and number of generation. The Pro-ISADE algorithm not only guarantees the above parameters, it also ensures the quality of the joints' variables to serve the next

*Results for Scenario 3.2. (a) Distance error. (b) Execution time. (c) Number of generations.*

**Table 5** summarizes results of the average error, the standard deviation of error (STD), the average iteration and the average execution time of all Scenarios. As in the table, the algorithms of ISADE and Pro-ISADE got the better results than the

As mentioned at the beginning of this article, intelligent optimization techniques have been using more and more popular in difficult and complex tasks including the IK problem for redundant manipulator robots. **Table 6** shows some studies used meta-heuristic optimization algorithms to resolve the inverse kinematics task for

computational and design stages.

*Robotics Software Design and Engineering*

other algorithms.

**36**

**Figure 9.**
