# A Metaheuristic Optimization Approach to Solve Inverse Kinematics of Mobile Dual-Arm Robots

^{1}

^{2}

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## Abstract

**:**

^{®}Youbot

^{®}robot. The solution of the inverse kinematics showed precise and accurate results. Although the proposed approach focuses on coordinated manipulation, it can be implemented to solve non-coordinated tasks.

## 1. Introduction

- An approach to solve the inverse kinematics of mobile dual-arm robots is proposed for cooperative manipulation problems.
- A kinematic model for a mobile dual-arm robot based on the KUKA
^{®}(KUKA is a registered trademark of KUKA Aktiengesellschaft Germany) Youbot^{®}(Youbot is a registered trademark of KUKA Aktiengesellschaft Germany) system is described. - An objective function is formulated based on the forward kinematics equations to deal with coordinated manipulation.
- The proposed approach avoids singularity configurations since it does not require using any Jacobian matrix.
- A comparative study is included to compare the performance of the DE, CPABC, SDE, CS, QPSO, IMPSO, and CMA-ES algorithms.

^{®}Youbot

^{®}system. The proposed approach is described in Section 3, where the objective function formulation and the inverse kinematics algorithm for cooperative manipulation tasks are presented. In Section 4, the experimental results for coordinated inverse kinematics and coordinated trajectory tracking tasks are given. A brief analysis of the obtained results and the future research directions are given in Section 5. Finally, conclusions are presented in Section 6.

## 2. Kinematic Analysis of Mobile Dual-Arm Robots

^{®}Youbot

^{®}robot [33]. This system is conformed by two identical manipulators of five DOFs composed of revolute joints and an omnidirectional mobile platform with three DOFs. Each manipulator is composed of revolute joints, and there is a gripper in the end-effector. The technical specification was carefully revised, and the kinematic analysis is given below.

^{®}Youbot

^{®}platform to obtain the matrices in (3). The frames $\left\{{\mathsf{b}}_{i}\right\}$ and $\left\{\mathsf{p}\right\}$ are also included. Based on the provided specifications, the following matrices are established:

^{®}Youbot

^{®}arm to obtain the DH table provided in Table 2. Each link j is represented by a homogeneous matrix ${}^{j-1}{\mathbf{T}}_{j}$, which transforms the frame attached to the link $j-1$ into the frame link j. The matrix ${}^{j-1}{\mathbf{T}}_{j}$ is expressed as

## 3. Description of the Proposed Approach

#### 3.1. Objective Function Formulation

#### 3.2. Inverse Kinematics Based on Metaheuristics Optimization Algorithms

^{®}Youbot

^{®}technical specifications provide the following upper ${\mathbf{q}}_{{i}_{u}}$ and lower ${\mathbf{q}}_{{i}_{u}}$ joint limits.

#### 3.3. Coordinated Trajectory Tracking Algorithm

## 4. Experimental Results

^{®}Youbot

^{®}robot. Moreover, the tests were conducted in simulations and real-world experiments.

^{®}(Intel i7 is a registered trademark of Intel Corporation USA) CPU $3.4$ GHz and 16 GB of RAM. Moreover, the experiments were performed in the Matlab R2021a environment

^{®}(Matlab is a registered trademark of the MathWorks, Inc., USA).

#### 4.1. Simulation Experiments for Coordinated Inverse Kinematics

#### 4.2. Simulation Experiments for Coordinated Trajectory Tracking

#### 4.3. Real-World Experiments for Coordinated Trajectory Tracking

^{®}Youbot

^{®}system; see Figure 1.

^{®}Youbot

^{®}hardware. This component provides a PID algorithm to control the joint positions of each manipulator. Moreover, this component also provides the current state of the joints based on encoder measures, and the current pose is given by odometry. To control the mobile platform pose, we used an adaptive PID scheme [35].

## 5. Discussion

^{®}Youbot

^{®}system. This system is composed of two 5-DOF manipulators attached to a 3-DOF omnidirectional platform. However, it is important to remark that the proposed scheme is not limited to this system; other manipulator configurations can be used to replace the ones used in this work. Since the forward kinematics is based on the DH model, no modifications to the inverse kinematics algorithm are required.

^{®}Youbot

^{®}system to solve a coordinated trajectory tracking task. Moreover, we used a generic PID algorithm to control each manipulator’s joint position and an adaptive neuron PID to control the mobile platform pose. The reported results showed that the error references were close to zero, which implies that the cooperative task was successful.

## 6. Conclusions

^{®}Youbot

^{®}system, which is composed of two 5-DOF manipulators attached to a 3-DOF omnidirectional platform.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DOFs | Degrees of freedom |

STD | Standard deviation |

DE | Differential evolution |

CPABC | Chaotic and parallelized artificial bee colony |

SDE | Self-adaptive differential evolution |

CS | Cuckoo search |

QPSO | Quantum particle swarm optimization |

IMPSO | Improved particle swarm optimization |

CMA-ES | Covariance matrix adaptation evolution strategy |

## References

- Freddi, A.; Longhi, S.; Monteriù, A.; Ortenzi, D. Redundancy analysis of cooperative dual-arm manipulators. Int. J. Adv. Robot. Syst.
**2016**, 13, 1729881416657754. [Google Scholar] [CrossRef] - Smith, C.; Karayiannidis, Y.; Nalpantidis, L.; Gratal, X.; Qi, P.; Dimarogonas, D.V.; Kragic, D. Dual arm manipulation—A survey. Robot. Auton. Syst.
**2012**, 60, 1340–1353. [Google Scholar] [CrossRef] [Green Version] - Stifter, S. Algebraic methods for computing inverse kinematics. J. Intell. Robot. Syst.
**1994**, 11, 79–89. [Google Scholar] [CrossRef] - Lee, C.; Ziegler, M. Geometric Approach in Solving Inverse Kinematics of PUMA Robots. IEEE Trans. Aerosp. Electron. Syst.
**1984**, AES-20, 695–706. [Google Scholar] [CrossRef] - Siciliano, B.; Lorenzo Sciavicco, L.V. Robotics-Modelling, Planning and Control, 2nd ed.; Advanced Textbooks in Control and Signal Processing; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Ortenzi, D.; Muthusamy, R.; Freddi, A.; Monteriù, A.; Kyrki, V. Dual-arm cooperative manipulation under joint limit constraints. Robot. Auton. Syst.
**2018**, 99, 110–120. [Google Scholar] [CrossRef] [Green Version] - Jamisola, R.S.; Roberts, R.G. A more compact expression of relative Jacobian based on individual manipulator Jacobians. Robot. Auton. Syst.
**2015**, 63, 158–164. [Google Scholar] [CrossRef] - Alkayyali, M.; Tutunji, T.A. PSO-based Algorithm for Inverse Kinematics Solution of Robotic Arm Manipulators. In Proceedings of the 2019 20th International Conference on Research and Education in Mechatronics (REM), Wels, Austria, 23–24 May 2019; pp. 1–6. [Google Scholar]
- Abainia, K.; Ben Ali, Y.M. Bio-inspired Approach for Inverse Kinematics of 6-DOF Robot Manipulator with Obstacle Avoidance. In Proceedings of the 2018 3rd International Conference on Pattern Analysis and Intelligent Systems (PAIS), Tebessa, Algeria, 24–25 October 2018; pp. 1–8. [Google Scholar]
- Umar, A.; Shi, Z.; Wang, W.; Farouk, Z.I.B. A Novel Mutating PSO Based Solution for Inverse Kinematic Analysis of Multi Degree-of-Freedom Robot Manipulators. In Proceedings of the 2019 IEEE International Conference on Artificial Intelligence and Computer Applications (ICAICA), Dalian, China, 29–31 March 2019; pp. 459–463. [Google Scholar]
- Wang, M.; Luo, J.; Yuan, J.; Walter, U. Coordinated trajectory planning of dual-arm space robot using constrained particle swarm optimization. Acta Astronaut.
**2018**, 146, 259–272. [Google Scholar] [CrossRef] - Tam, B.; Linh, T.; Nguyen, T.; Nguyen, T.; Hasegawa, H.; Watanabe, D. DE-based Algorithm for Solving the Inverse Kinematics on a Robotic Arm Manipulators. J. Phys. Conf. Ser.
**2021**, 1922, 012008. [Google Scholar] [CrossRef] - Nguyen, T.T.; Nguyen, V.H.; Nguyen, X.H. Comparing the Results of Applying DE, PSO and Proposed Pro DE, Pro PSO Algorithms for Inverse Kinematics Problem of a 5-DOF Scara Robot. In Proceedings of the 2020 International Conference on Advanced Mechatronic Systems (ICAMechS), Hanoi, Vietnam, 10–13 December 2020; pp. 45–49. [Google Scholar]
- Nizar, I.I. Investigation of Inverse kinematics Solution for a Human-like Aerial Manipulator Based on The Metaheuristic Algorithms. In Proceedings of the 2019 International Seminar on Electron Devices Design and Production (SED), Prague, Czech Republic, 23–24 April 2019; pp. 1–13. [Google Scholar]
- Kumar, A.; Banga, V.K.; Kumar, D.; Yingthawornsuk, T. Kinematics Solution using Metaheuristic Algorithms. In Proceedings of the 2019 15th International Conference on Signal-Image Technology and Internet-Based Systems (SITIS), Sorrento-Naples, Italy, 26–29 November 2019; pp. 505–510. [Google Scholar]
- Dereli, S.; Koker, R. Simulation based calculation of the inverse kinematics solution of 7-DOF robot manipulator using artificial bee colony algorithm. SN Appl. Sci.
**2020**, 2, 27. [Google Scholar] [CrossRef] [Green Version] - Abdor-Sierra, J.A.; Merchán-Cruz, E.A.; Rodríguez-Cañizo, R.G. A comparative analysis of metaheuristic algorithms for solving the inverse kinematics of robot manipulators. Results Eng.
**2022**, 16, 100597. [Google Scholar] [CrossRef] - Yiyang, L.; Xi, J.; Hongfei, B.; Zhining, W.; Liangliang, S. A General Robot Inverse Kinematics Solution Method Based on Improved PSO Algorithm. IEEE Access
**2021**, 9, 32341–32350. [Google Scholar] [CrossRef] - Nguyen, T.; Bui, T.; Pham, H. Using proposed optimization algorithm for solving inverse kinematics of human upper limb applying in rehabilitation robotic. Artif. Intell. Rev.
**2022**, 55, 679–705. [Google Scholar] [CrossRef] - Dereli, S.; Koker, R. A meta-heuristic proposal for inverse kinematics solution of 7-DOF serial robotic manipulator: Quantum behaved particle swarm algorithm. Artif. Intell. Rev.
**2020**, 53, 949–964. [Google Scholar] [CrossRef] - Zhang, L.; Xiao, N. A novel artificial bee colony algorithm for inverse kinematics calculation of 7-DOF serial manipulators. Soft Comput.
**2019**, 23, 3269–3277. [Google Scholar] [CrossRef] - Larsen, L.; Kim, J. Path planning of cooperating industrial robots using evolutionary algorithms. Robot.-Comput.-Integr. Manuf.
**2021**, 67, 102053. [Google Scholar] [CrossRef] - Liu, F.; Huang, H.; Li, B.; Xi, F. A parallel learning particle swarm optimizer for inverse kinematics of robotic manipulator. Int. J. Intell. Syst.
**2021**, 36, 6101–6132. [Google Scholar] [CrossRef] - Šegota, S.B.; Anđelić, N.; Lorencin, I.; Saga, M.; Car, Z. Path planning optimization of six-degree-of-freedom robotic manipulators using evolutionary algorithms. Int. J. Adv. Robot. Syst.
**2020**, 17, 1729881420908076. [Google Scholar] - Li, C.; Dong, H.; Li, X.; Zhang, W.; Liu, X.; Yao, L.; Sun, H. Inverse Kinematics Study for Intelligent Agriculture Robot Development via Differential Evolution Algorithm. In Proceedings of the 2021 International Conference on Computer, Control and Robotics (ICCCR), Shanghai, China, 8–10 January 2021; pp. 37–41. [Google Scholar]
- Karahan, O.; Karci, H.; Tangel, A. Optimal trajectory generation in joint space for 6R industrial serial robots using cuckoo search algorithm. Intell. Serv. Robot.
**2022**, 15, 627–648. [Google Scholar] [CrossRef] - Lopez-Franco, C.; Hernandez-Barragan, J.; Alanis, A.Y.; Arana-Daniel, N. A soft computing approach for inverse kinematics of robot manipulators. Eng. Appl. Artif. Intell.
**2018**, 74, 104–120. [Google Scholar] [CrossRef] - López-Franco, C.; Hernández-Barragán, J.; Alanis, A.Y.; Arana-Daniel, N.; López-Franco, M. Inverse kinematics of mobile manipulators based on differential evolution. Int. J. Adv. Robot. Syst.
**2018**, 15, 1729881417752738. [Google Scholar] [CrossRef] [Green Version] - Hernández-Barragán, J.; López-Franco, C.; Alanis, A.Y.; Arana-Daniel, N.; López-Franco, M. Dual-arm cooperative manipulation based on differential evolution. Int. J. Adv. Robot. Syst.
**2019**, 16, 1729881418825188. [Google Scholar] [CrossRef] - Hernandez-Barragan, J.; Lopez-Franco, C.; Arana-Daniel, N.; Alanis, A.Y. Inverse kinematics for cooperative mobile manipulators based on self-adaptive differential evolution. PeerJ Comput. Sci.
**2021**, 7, e419. [Google Scholar] [CrossRef] [PubMed] - Lopez-Franco, C.; Diaz, D.; Hernandez-Barragan, J.; Arana-Daniel, N.; Lopez-Franco, M. A Metaheuristic Optimization Approach for Trajectory Tracking of Robot Manipulators. Mathematics
**2022**, 10, 1051. [Google Scholar] [CrossRef] - Li, Z.; Yang, C.; Su, C.; Deng, J.; Zhang, W. Vision-Based Model Predictive Control for Steering of a Nonholonomic Mobile Robot. IEEE Trans. Control. Syst. Technol.
**2016**, 24, 553–564. [Google Scholar] [CrossRef] - Bischoff, R.; Huggenberger, U.; Prassler, E. KUKA youBot - a mobile manipulator for research and education. In Proceedings of the 2011 IEEE International Conference on Robotics and Automation, Shanghai, China, 9–13 May 2011; pp. 1–4. [Google Scholar]
- Spong, M.W.; Vidyasagar, M. Robot Dynamics and Control; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Hernandez-Barragan, J.; Rios, J.D.; Alanis, A.Y.; Lopez-Franco, C.; Gomez-Avila, J.; Arana-Daniel, N. Adaptive Single Neuron Anti-Windup PID Controller Based on the Extended Kalman Filter Algorithm. Electronics
**2020**, 9, 636. [Google Scholar] [CrossRef] - Simon, D. Evolutionary Optimization Algorithms; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Sun, Y.; Nelson, B.J. Biological Cell Injection Using an Autonomous MicroRobotic System. T Int. J. Robot. Res.
**2002**, 21, 861–868. [Google Scholar] [CrossRef] - Gai, S.N.; Sun, R.; Chen, S.J.; Ji, S. 6-DOF Robotic Obstacle Avoidance Path Planning Based on Artificial Potential Field Method. In Proceedings of the 2019 16th International Conference on Ubiquitous Robots (UR), Jeju, Korea, 24–27 June 2019; pp. 165–168. [Google Scholar]
- Khan, A.H.; Li, S.; Cao, X. Tracking control of redundant manipulator under active remote center-of-motion constraints: An RNN-based metaheuristic approach. Sci. China Inf. Sci.
**2021**, 64, 132203. [Google Scholar] [CrossRef] - Wang, J.; Zhang, Y. Recurrent neural networks for real-time computation of inverse kinematics of redundant manipulators. In Machine Intelligence: Quo Vadis? World Scientific: Singapore, 2004; pp. 299–319. [Google Scholar]
- Xie, Z.; Jin, L.; Luo, X. Kinematics-Based Motion-Force Control for Redundant Manipulators with Quaternion Control. IEEE Trans. Autom. Sci. Eng.
**2022**, 1–14. [Google Scholar] [CrossRef] - Fabris, F.; Krohling, R.A. A co-evolutionary differential evolution algorithm for solving min–max optimization problems implemented on GPU using C-CUDA. Expert Syst. Appl.
**2012**, 39, 10324–10333. [Google Scholar] [CrossRef]

**Figure 1.**Mobile dual-arm system KUKA

^{®}Youbot

^{®}. It is composed of two 5-DOF manipulators and a 3-DOF mobile platform.

**Figure 3.**Coordinate frames assignment for the KUKA

^{®}Youbot

^{®}mobile platform based on the technical specifications manual.

**Figure 4.**Coordinate frames’ assignment for the KUKA

^{®}Youbot

^{®}manipulator based on the technical specifications manual.

**Figure 8.**Coordinated trajectory tracking comparative results. The compared algorithms are CMA-ES and SDE. The “Desired” label means the desired end-effector trajectory.

**Figure 10.**Joint motion comparative results for coordinated trapezoidal tracking. The compared algorithms are CMA-ES and CS.

**Figure 11.**Motion results for the mobile platform. The “Reference” label indicates the optimal pose value provided by the CMA-ES algorithm, and the “Measured” label is the current odometry value.

**Figure 12.**Motion result for manipulator 1. The “Reference” label indicates the optimal joint value provided by the CMA-ES algorithm, and the “Measured” label is the current measurement.

**Figure 13.**Motion result for manipulator 2. The “Reference” label indicates the optimal joint value provided by the CMA-ES algorithm, and the “Measured” label is the current measurement.

**Figure 14.**Coordinated trajectory tracking results. The “Reference” label represents the trajectory achieved by the CMA-ES algorithm, and the “Measured” label is the current end-effector position.

Reference | Metaheuristics | Robotic System | Application |
---|---|---|---|

[16] | ABC | 7-DOF manipulator | IK solutions |

[17] | PSO, DE, QPSO, IMPSO | 6-DOF and 7-DOF manipulators | IK solutions |

[18] | IMPSO | 6-DOF manipulator | IK solutions |

[19] | ISADE | 7-DOF manipulator | IK solutions |

[20] | QPSO | 7-DOF manipulator | IK solutions |

[21] | CPABC | 7-DOF manipulator | IK solutions |

[22] | EA | 14-DOF dual-arm | Path planning |

[23] | PLPSO | 6-DOF manipulator | IK solutions |

[24] | GA, DE | 6-DOF manipulator | Path planning |

[25] | DE | 7-DOF manipulator | IK solutions |

[26] | CS | 6-DOF manipulator | Path planning |

Link | a (m) | $\mathit{\alpha}\phantom{\rule{0.222222em}{0ex}}$(rad) | d (m) | $\mathit{\theta}$ (rad) |
---|---|---|---|---|

1 | 0.033 | $\pi /2$ | 0.147 | ${\theta}_{1}$ |

2 | 0.155 | 0 | 0 | ${\theta}_{2}$ |

3 | 0.135 | 0 | 0 | ${\theta}_{3}$ |

4 | 0 | $\pi /2$ | 0 | ${\theta}_{4}$ |

5 | 0 | 0 | 0.2175 | ${\theta}_{5}$ |

Reference | Algorithm | Parameter Settings |
---|---|---|

[28] | DE | DE/rand/1/bin, $F=0.6$, ${C}_{R}=0.9$ |

[21] | CPABC | Limits $L=40$, ${M}_{R}=0.8$, ${S}_{F}=0.6$ |

[30] | SDE | $F=0.5$, ${C}_{R}=0.8$ |

[26] | CS | Discover rate $P=0.25$ |

[20] | QPSO | ${\beta}_{0}=0.5$, ${\beta}_{1}=1.0$ |

[17] | IMPSO | ${w}_{i}=\left\{0.2,0.4,0.6,0.8,1.0,1.2,1.4,1.6,1.8,2.0\right\}$, ${C}_{1}=1.4962$, ${C}_{2}=1.4962$ |

[27] | CMAES | Standard $(\mu ,\lambda )$-CMA-ES, $\lambda =30$ |

DE | CPABC | SDE | CS | QPSO | IMPSO | CMA-ES | |
---|---|---|---|---|---|---|---|

Mean | 5.136 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.022031 | 5.440 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.00013183 | 0.0008852 | 0.0018518 | $\mathbf{7.965}\times {10}^{-15}$ |

STD | 2.360 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.097928 | 1.193 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 6.929 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.0037455 | 0.0084942 | $\mathbf{1.509}\times {10}^{-14}$ |

Best | 7.488 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 1.717 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ | 3.683 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 2.513 $\times {10}^{-5}$ | $\mathbf{6.245}\times {10}^{-17}$ | 8.370 $\times {10}^{-17}$ | 6.973 $\times {10}^{-16}$ |

Worst | 0.00016689 | 0.58487 | 6.887$\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.0003229 | 0.022734 | 0.055906 | $\mathbf{1.03}\times {10}^{-13}$ |

DE | CPABC | SDE | CS | QPSO | IMPSO | CMA-ES | |
---|---|---|---|---|---|---|---|

Mean | 1.036 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.00051808 | 2.768 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.0003566 | 0.0002991 | 0.0062474 | $\mathbf{3.803}\times {10}^{-13}$ |

STD | 2.287 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.0018515 | 5.428 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.00021011 | 0.00099298 | 0.035237 | $\mathbf{1.469}\times {10}^{-12}$ |

Best | 1.257 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 7.467 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ | 1.049 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 9.012 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.875 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | $\mathbf{1.746}\times {10}^{-16}$ | 4.616$\times {10}^{-16}$ |

Worst | 0.00010999 | 0.011715 | 3.109 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.0010176 | 0.0062429 | 0.24645 | $\mathbf{8.107}\times {10}^{-12}$ |

DE | CPABC | SDE | CS | QPSO | IMPSO | CMA-ES | |
---|---|---|---|---|---|---|---|

Mean | 8.119 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.0036393 | 2.754 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.00026886 | 0.0002542 | 0.0034882 | $\mathbf{8.839}\times {10}^{-13}$ |

STD | 1.843 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.021661 | 5.134 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.00016798 | 0.001258 | 0.020727 | $\mathbf{2.685}\times {10}^{-12}$ |

Best | 5.514 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 7.992 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ | 2.743 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 3.107 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.257 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | $\mathbf{2.310}\times {10}^{-16}$ | 1.994 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ |

Worst | 0.00011067 | 0.15316 | 2.599 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.00074317 | 0.0085917 | 0.14618 | $\mathbf{1.523}\times {10}^{-11}$ |

DE | CPABC | SDE | CS | QPSO | IMPSO | CMA-ES | |
---|---|---|---|---|---|---|---|

Mean | 4.213 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.02832 | 3.229 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.00043625 | 0.023282 | 0.040247 | $\mathbf{8.710}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-14}$ |

STD | 5.632 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.087143 | 1.264 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.00028418 | 0.015985 | 0.09733 | $\mathbf{2.608}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ |

Best | 8.997 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 1.023 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-15}$ | 1.919 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 5.754 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 5.658 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | $\mathbf{7.850}\times {10}^{-17}$ | 8.01 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-16}$ |

Worst | 2.941 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.39983 | 8.697 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.0014147 | 0.064859 | 0.4156 | $\mathbf{1.488}\times {10}^{-12}$ |

**Table 8.**Position error results for sinusoidal trajectory. The best results are highlighted in bold.

DE | SDE | CS | CMA-ES | |
---|---|---|---|---|

Mean | 5.104 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 5.5523 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.0099478 | $\mathbf{1.1142}\times {10}^{-8}$ |

STD | 7.5244 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.94 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.0055781 | $\mathbf{2.9257}\times {10}^{-8}$ |

Min | 6.0195 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.3216 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.0028493 | $\mathbf{2.6832}\times {10}^{-13}$ |

Max | 0.00087922 | 0.00026586 | 0.036553 | $\mathbf{3.7112}\times {10}^{-7}$ |

DE | SDE | CS | CMA-ES | |
---|---|---|---|---|

Mean | 4.0829 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 5.2769 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.009863 | $\mathbf{3.4473}\times {10}^{-9}$ |

STD | 2.9189 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.8536 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.0046131 | $\mathbf{6.085}\times {10}^{-9}$ |

Min | 3.7308 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 6.9321 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.0025768 | $\mathbf{3.7374}\times {10}^{-15}$ |

Max | 0.00021541 | 0.0002404 | 0.027738 | $\mathbf{4.8502}\times {10}^{-8}$ |

**Table 10.**Position error results for trapezoidal trajectory. The best results are highlighted in bold.

DE | SDE | CS | CMA-ES | |
---|---|---|---|---|

Mean | 4.1641 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 4.6393 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.010247 | $\mathbf{7.3893}\times {10}^{-9}$ |

STD | 3.4045 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.4973 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.0047796 | $\mathbf{2.0364}\times {10}^{-8}$ |

Min | 4.1663 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 5.7924 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.0015987 | $\mathbf{3.4801}\times {10}^{-16}$ |

Max | 0.00024229 | 0.00019833 | 0.030391 | $\mathbf{1.2199}\times {10}^{-7}$ |

**Table 11.**Position error results for rose curve trajectory. The best results are highlighted in bold.

DE | SDE | CS | CMA-ES | |
---|---|---|---|---|

Mean | 3.5829 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 4.2613 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.0092417 | $\mathbf{2.3824}\times {10}^{-9}$ |

STD | 2.8703 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.9939 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 0.0048608 | $\mathbf{3.9706}\times {10}^{-9}$ |

Min | 5.3653 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 4.1256 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.0014107 | $\mathbf{3.6486}\times {10}^{-12}$ |

Max | 0.0001964 | 0.00018327 | 0.031768 | $\mathbf{2.4175}\times {10}^{-8}$ |

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## Share and Cite

**MDPI and ACS Style**

Hernandez-Barragan, J.; Martinez-Soltero, G.; Rios, J.D.; Lopez-Franco, C.; Alanis, A.Y.
A Metaheuristic Optimization Approach to Solve Inverse Kinematics of Mobile Dual-Arm Robots. *Mathematics* **2022**, *10*, 4135.
https://doi.org/10.3390/math10214135

**AMA Style**

Hernandez-Barragan J, Martinez-Soltero G, Rios JD, Lopez-Franco C, Alanis AY.
A Metaheuristic Optimization Approach to Solve Inverse Kinematics of Mobile Dual-Arm Robots. *Mathematics*. 2022; 10(21):4135.
https://doi.org/10.3390/math10214135

**Chicago/Turabian Style**

Hernandez-Barragan, Jesus, Gabriel Martinez-Soltero, Jorge D. Rios, Carlos Lopez-Franco, and Alma Y. Alanis.
2022. "A Metaheuristic Optimization Approach to Solve Inverse Kinematics of Mobile Dual-Arm Robots" *Mathematics* 10, no. 21: 4135.
https://doi.org/10.3390/math10214135