# Path-Planning for Mobile Robots Using a Novel Variable-Length Differential Evolution Variant

^{1}

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## Abstract

**:**

## 1. Introduction

#### 1.1. A Review of Path-Planning Methods

#### 1.2. Optimization in Path-Planning

#### 1.3. Scope and Contributions of the Proposal

## 2. The Path-Planning Problem

#### 2.1. General Variable-Length-Vector Optimization Problem

#### 2.2. The Path-Planning for Mobile Robots as a Variable-Length-Vector Optimization Problem

#### 2.2.1. The Variable-Length-Vector of Design Variables

#### 2.2.2. The Objective Function

#### 2.2.3. The Constraints

**Obstacles:**Polygons ${P}_{m},m=1,\dots ,{n}_{o}$, defined by an arbitrary number of vertices which describe all the ${n}_{o}$ obstacles in the workspace W.**Path points:**Circles ${C}_{l},l=1,\dots ,s\left(x\right)$ are assigned to each control point in the path. Each circle is centered in the control point and has a diameter of $2r$ that matches the length of the longest diagonal of the 4MWMR in the plane. This shape is selected to allow possible orientation changes of the 4MWMR.**Path edges:**In order to improve the collision detection procedure, rectangular geometries ${R}_{l},l=1,\dots ,s\left(x\right)+1$ are attached to each edge in the path. An edge is the line segment between a pair of consecutive points in the path. The width of each ${R}_{l}$ is equal to the edge length, its height is $2r$ and its orientation matches the inclination $\alpha ,l=1,\dots ,s\left(x\right)+1$ of the edge. The ${R}_{l}$ is centered in the middle of the two points.

## 3. The Novel Variable-Length-Vector Differential Evolution

Algorithm 1: Variable-Length-Vector Differential Evolution (VLV-DE) |

#### 3.1. Initialization

#### 3.2. Evolutionary Cycle

#### 3.3. Normalization

Algorithm 2: Normalize $({x}_{k},{N}_{s})$ |

#### 3.4. Compression

- Selecting a pair of consecutive path points and replacing them with a new single one that preserves information from both. The new variable is obtained as the pair’s mean and can be altered through a variation operator (the polynomial mutation [67]) to enhance the exploitative search.
- Removing the first or the last path point.

#### 3.5. Decompression

- Selecting a pair of consecutive path points to generate a single one based on their information, which is then inserted in the middle of both. This new variable is calculated as the pair’s mean and can be modified by using a variation operator (the polynomial mutation) to improve the exploratory search.
- Including altered versions of the first or the last path points (obtained through the polynomial mutation), at the beginning or the end of the solution, respectively.

Algorithm 3: Compress$({x}_{k},{N}_{s})$ |

Algorithm 4: Decompress$({x}_{k},{N}_{s})$ |

#### 3.6. Evolutionary Operators

#### 3.7. Selection

#### 3.8. Local Search

Algorithm 5: LocalSearch$\left({x}_{i}\right)$ |

Input: ${x}_{i}$ (arbitrary solution)Output: ${x}_{i}^{l}$ (local best solution)1 ${N}_{s}\leftarrow s\left({x}_{i}\right)$; 2 Variate ${N}_{s}$ using Gaussian mutation; 3 ${x}_{i}^{n}\leftarrow $ Normalize(${x}_{i}$,${N}_{s}$); 4 Variate ${x}_{i}^{n}$ using Polynomial Mutation; 5 Evaluate ${x}_{i}^{n}$; 6 Select ${x}_{i}^{l}$ as the best between ${x}_{i}$ and ${x}_{i}^{n}$ |

## 4. Results an Discussion

#### 4.1. Details of the Experiment

#### 4.2. Discussion of the Results

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Naranjo, J.E.; Lozada, E.C.; Espín, H.I.; Beltran, C.; García, C.A.; García, M.V. Flexible architecture for transparency of a bilateral tele-operation system implemented in mobile anthropomorphic robots for the oil and gas industry. IFAC-PapersOnLine
**2018**, 51, 239–244. [Google Scholar] [CrossRef] - Wang, Y.; Lang, H.; De Silva, C.W. A hybrid visual servo controller for robust grasping by wheeled mobile robots. IEEE/ASME Trans. Mechatron.
**2009**, 15, 757–769. [Google Scholar] [CrossRef] - Orozco-Rosas, U.; Picos, K.; Montiel, O. Hybrid path planning algorithm based on membrane pseudo-bacterial potential field for autonomous mobile robots. IEEE Access
**2019**, 7, 156787–156803. [Google Scholar] [CrossRef] - Primatesta, S.; Guglieri, G.; Rizzo, A. A risk-aware path planning strategy for uavs in urban environments. J. Intell. Robot. Syst.
**2019**, 95, 629–643. [Google Scholar] [CrossRef] - Alexopoulos, C.; Griffin, P.M. Path planning for a mobile robot. IEEE Trans. Syst. Man Cybern.
**1992**, 22, 318–322. [Google Scholar] [CrossRef] [Green Version] - Niu, H.; Lu, Y.; Savvaris, A.; Tsourdos, A. An energy-efficient path planning algorithm for unmanned surface vehicles. Ocean Eng.
**2018**, 161, 308–321. [Google Scholar] [CrossRef] [Green Version] - Yang, L.; Qi, J.; Song, D.; Xiao, J.; Han, J.; Xia, Y. Survey of robot 3D path planning algorithms. J. Control Sci. Eng.
**2016**. [Google Scholar] [CrossRef] [Green Version] - LaValle, S.M. Rapidly-Exploring Random Trees: A New Tool for Path Planning; Tech. Rep. 98-11; Department of Computer Science, Iowa State University: Ames, IA, USA, 1998. [Google Scholar]
- Kavraki, L.E.; Svestka, P.; Latombe, J.C.; Overmars, M.H. Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Trans. Robot. Autom.
**1996**, 12, 566–580. [Google Scholar] [CrossRef] [Green Version] - Karaman, S.; Frazzoli, E. Incremental sampling-based algorithms for optimal motion planning. Robot. Sci. Syst. VI
**2010**, 104. [Google Scholar] [CrossRef] - Karaman, S.; Frazzoli, E. Sampling-based algorithms for optimal motion planning. Int. J. Robot. Res.
**2011**, 30, 846–894. [Google Scholar] [CrossRef] - Ayanian, N.; Kallem, V.; Kumar, V. Synthesis of feedback controllers for multiple aerial robots with geometric constraints. In Proceedings of the 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, San Francisco, CA, USA, 25–30 September 2011; pp. 3126–3131. [Google Scholar]
- Sethian, J.A. Fast marching methods. SIAM Rev.
**1999**, 41, 199–235. [Google Scholar] [CrossRef] - Barraquand, J.; Langlois, B.; Latombe, J.C. Numerical potential field techniques for robot path planning. IEEE Trans. Syst. Man Cybern.
**1992**, 22, 224–241. [Google Scholar] [CrossRef] - Janet, J.A.; Luo, R.C.; Kay, M.G. The essential visibility graph: An approach to global motion planning for autonomous mobile robots. In Proceedings of the 1995 IEEE International Conference on Robotics and Automation, Nagoya, Japan, 21–27 May 1995; Volume 2, pp. 1958–1963. [Google Scholar]
- Stentz, A. The focussed D* algorithm for real-time replanning. In Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI’95), Montreal, QC, Canada, 20–25 August 1995; Volume 2, pp. 1652–1659. [Google Scholar]
- Britzelmeier, A.; Gerdts, M. A Nonsmooth Newton Method for Linear Model-Predictive Control in Tracking Tasks for a Mobile Robot With Obstacle Avoidance. IEEE Control Syst. Lett.
**2020**, 4, 886–891. [Google Scholar] [CrossRef] - Liu, L.; Wang, D.; Peng, Z. Path following of marine surface vehicles with dynamical uncertainty and time-varying ocean disturbances. Neurocomputing
**2016**, 173, 799–808. [Google Scholar] [CrossRef] - Song, Q.; Zhao, Q.; Wang, S.; Liu, Q.; Chen, X. Dynamic path planning for unmanned vehicles based on fuzzy logic and improved ant colony optimization. IEEE Access
**2020**, 8, 62107–62115. [Google Scholar] [CrossRef] - Singh, N.H.; Thongam, K. Neural network-based approaches for mobile robot navigation in static and moving obstacles environments. Intell. Serv. Robot.
**2019**, 12, 55–67. [Google Scholar] [CrossRef] - Wang, J.; Chi, W.; Li, C.; Wang, C.; Meng, M.Q.H. Neural RRT*: Learning-based optimal path planning. IEEE Trans. Autom. Sci. Eng.
**2020**, 17, 1748–1758. [Google Scholar] [CrossRef] - Amer, M.; Namaane, A.; M’sirdi, N. Optimization of hybrid renewable energy systems (HRES) using PSO for cost reduction. Energy Procedia
**2013**, 42, 318–327. [Google Scholar] [CrossRef] [Green Version] - Han, M.; Wang, M.H.; Fan, J.C. Trajectory optimization based on improved differential evolution algorithm. Control Decis.
**2012**, 27, 247–251. [Google Scholar] - Huang, J.; Qingyun, W. Robust optimization of hub-and-spoke airline network design based on multi-objective genetic algorithm. J. Transp. Syst. Eng. Inf. Technol.
**2009**, 9, 86–92. [Google Scholar] [CrossRef] - Flores-Caballero, G.; Rodríguez-Molina, A.; Aldape-Pérez, M.; Villarreal-Cervantes, M.G. Optimized Path-Planning in Continuous Spaces for Unmanned Aerial Vehicles Using Meta-Heuristics. IEEE Access
**2020**, 8, 176774–176788. [Google Scholar] [CrossRef] - Karami, A.H.; Hasanzadeh, M. An adaptive genetic algorithm for robot motion planning in 2D complex environments. Comput. Electr. Eng.
**2015**, 43, 317–329. [Google Scholar] [CrossRef] - Nazarahari, M.; Khanmirza, E.; Doostie, S. Multi-objective multi-robot path planning in continuous environment using an enhanced genetic algorithm. Expert Syst. Appl.
**2019**, 115, 106–120. [Google Scholar] [CrossRef] - Jones, D.F.; Mirrazavi, S.K.; Tamiz, M. Multi-objective meta-heuristics: An overview of the current state-of-the-art. Eur. J. Oper. Res.
**2002**, 137, 1–9. [Google Scholar] [CrossRef] - Li, H.; Deb, K. Challenges for evolutionary multiobjective optimization algorithms in solving variable-length problems. In Proceedings of the 2017 IEEE Congress on Evolutionary Computation (CEC), San Sebastian, Spain, 5–8 June 2017; pp. 2217–2224. [Google Scholar]
- Meza, G.R.; Ferragud, X.B.; Saez, J.S.; Durá, J.M.H. Controller Tuning with Evolutionary Multiobjective Optimization: A Holistic Multiobjective Optimization Design Procedure; Springer International Publishing: Cham, Switzerland, 2016; Volume 85. [Google Scholar]
- Onal, C.D.; Tolley, M.T.; Wood, R.J.; Rus, D. Origami-inspired printed robots. IEEE/ASME Trans. Mechatron.
**2014**, 20, 2214–2221. [Google Scholar] [CrossRef] [Green Version] - Rabadi, G.; Moraga, R.J.; Al-Salem, A. Heuristics for the unrelated parallel machine scheduling problem with setup times. J. Intell. Manuf.
**2006**, 17, 85–97. [Google Scholar] [CrossRef] - Dewang, H.S.; Mohanty, P.K.; Kundu, S. A robust path planning for mobile robot using smart particle swarm optimization. Procedia Comput. Sci.
**2018**, 133, 290–297. [Google Scholar] [CrossRef] - Lamini, C.; Benhlima, S.; Elbekri, A. Genetic algorithm based approach for autonomous mobile robot path planning. Procedia Comput. Sci.
**2018**, 127, 180–189. [Google Scholar] [CrossRef] - Martinez-Soltero, E.G.; Hernandez-Barragan, J. Robot navigation based on differential evolution. IFAC-PapersOnLine
**2018**, 51, 350–354. [Google Scholar] [CrossRef] - Tang, B.; Zhu, Z.; Luo, J. Hybridizing particle swarm optimization and differential evolution for the mobile robot global path planning. Int. J. Adv. Robot. Syst.
**2016**, 13, 86. [Google Scholar] [CrossRef] [Green Version] - Oleiwi, B.K.; Roth, H.; Kazem, B.I. Multi objective optimization of path and trajectory planning for non-holonomic mobile robot using enhanced genetic algorithm. In Proceedings of the 8th International Conference on Neural Networks and Artificial Intelligence (ICNNAI 2014), Brest, Belarus, 3–6 June 2014; pp. 50–62. [Google Scholar]
- Niederberger, C.; Radovic, D.; Gross, M. Generic path planning for real-time applications. In Proceedings of the Computer Graphics International, Crete, Greece, 19 June 2004; pp. 299–306. [Google Scholar]
- Yang, K.; Jung, D.; Sukkarieh, S. Continuous curvature path-smoothing algorithm using cubic B zier spiral curves for non-holonomic robots. Adv. Robot.
**2013**, 27, 247–258. [Google Scholar] [CrossRef] - Son, T.D.; Ahn, H.S.; Moore, K.L. Iterative learning control in optimal tracking problems with specified data points. Automatica
**2013**, 49, 1465–1472. [Google Scholar] [CrossRef] - Davoodi, M.; Panahi, F.; Mohades, A.; Hashemi, S.N. Clear and smooth path planning. Appl. Soft Comput.
**2015**, 32, 568–579. [Google Scholar] [CrossRef] - Shwail, S.H.; Karim, A.; Turner, S. Probabilistic multi robot path planning in dynamic environments: A comparison between A* and DFS. Int. J. Comput. Appl.
**2013**, 82. [Google Scholar] - Tu, J.; Yang, S.X. Genetic algorithm based path planning for a mobile robot. In Proceedings of the 2003 IEEE International Conference on Robotics and Automation (ICRA 2003), Taipei, Taiwan, 14–19 September 2003; Volume 1, pp. 1221–1226. [Google Scholar]
- Riquelme, J.; Ridao, M.; Camacho, E.; Toro, M. Using genetic algorithms with variable-length individuals for planning two-manipulators motion. In Artificial Neural Nets and Genetic Algorithms; Springer: Vienna, Austria, 1998; pp. 26–30. [Google Scholar]
- Limbourg, P.; Kochs, H.D. Preventive maintenance scheduling by variable dimension evolutionary algorithms. Int. J. Press. Vessels Pip.
**2006**, 83, 262–269. [Google Scholar] [CrossRef] - Chuanjiao, S.; Wei, Z.; Yuanqing, W. Scheduling combination and headway optimization of bus rapid transit. J. Transp. Syst. Eng. Inf. Technol.
**2008**, 8, 61–67. [Google Scholar] - Ahn, C.W.; Ramakrishna, R.S. A genetic algorithm for shortest path routing problem and the sizing of populations. IEEE Trans. Evol. Comput.
**2002**, 6, 566–579. [Google Scholar] - Qiongbing, Z.; Lixin, D. A new crossover mechanism for genetic algorithms with variable-length chromosomes for path optimization problems. Expert Syst. Appl.
**2016**, 60, 183–189. [Google Scholar] [CrossRef] - Chen, Y.; Mahalec, V.; Chen, Y.; Liu, X.; He, R.; Sun, K. Reconfiguration of satellite orbit for cooperative observation using variable-size multi-objective differential evolution. Eur. J. Oper. Res.
**2015**, 242, 10–20. [Google Scholar] [CrossRef] - Das, S.; Abraham, A.; Konar, A. Automatic clustering using an improved differential evolution algorithm. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum.
**2007**, 38, 218–237. [Google Scholar] [CrossRef] - Yuan, H.; He, J. Evolutionary design of operational amplifier using variable-length differential evolution algorithm. In Proceedings of the 2010 International Conference on Computer Application and System Modeling (ICCASM 2010), Taiyuan, China, 22–24 October 2010; Volume 4, p. V4-610. [Google Scholar]
- Pereira, C.M.; Lapa, C.M.; Mol, A.C.; Da Luz, A.F. A particle swarm optimization (PSO) approach for non-periodic preventive maintenance scheduling programming. Prog. Nucl. Energy
**2010**, 52, 710–714. [Google Scholar] [CrossRef] - Wang, B.; Sun, Y.; Xue, B.; Zhang, M. Evolving deep convolutional neural networks by variable-length particle swarm optimization for image classification. In Proceedings of the 2018 IEEE Congress on Evolutionary Computation (IEEE CEC 2018), Rio de Janeiro, Brazil, 8–13 July 2018; pp. 1–8. [Google Scholar]
- Xue, B.; Ma, X.; Gu, J.; Li, Y. An improved extreme learning machine based on variable-length particle swarm optimization. In Proceedings of the 2013 IEEE International Conference on Robotics and Biomimetics (ROBIO), Shenzhen, China, 12–14 December 2013; pp. 1030–1035. [Google Scholar]
- Wilt, C.M.; Thayer, J.T.; Ruml, W. A comparison of greedy search algorithms. In Proceedings of the Third Annual Symposium on Combinatorial Search, Atlanta, GA, USA, 8–10 July 2010; pp. 130–136. [Google Scholar]
- Alakshendra, V.; Chiddarwar, S.S. Adaptive robust control of Mecanum-wheeled mobile robot with uncertainties. Nonlinear Dyn.
**2017**, 87, 2147–2169. [Google Scholar] [CrossRef] - Sánchez, P.; Casado, R.; Bermúdez, A. Real-Time Collision-Free Navigation of Multiple UAVs Based on Bounding Boxes. Electronics
**2020**, 9, 1632. [Google Scholar] [CrossRef] - Lin, M.C.; Manocha, D.; Cohen, J.; Gottschalk, S. Collision detection: Algorithms and applications. In Algorithms for Robotic Motion and Manipulation; A K Peters/CRC Press: New York, NY, USA, 1997; pp. 129–142. [Google Scholar]
- van der Spuy, R. Collisions Between Polygons. In AdvancED Game Design with Flash; Apress: Berkeley, CA, USA, 2010; pp. 223–303. [Google Scholar] [CrossRef]
- Madavan, N.K. Multiobjective optimization using a Pareto differential evolution approach. In Proceedings of the 2002 Congress on Evolutionary Computation. CEC’02 (Cat. No. 02TH8600), Honolulu, HI, USA, 12–17 May 2002; Volume 2, pp. 1145–1150. [Google Scholar]
- Chakraborty, U.K. Advances in Differential Evolution; Springer: Berlin/Heidelberg, Germany, 2008; Volume 143. [Google Scholar]
- Price, K.V. Differential evolution. In Handbook of Optimization; Springer: Berlin/Heidelberg, Germany, 2013; pp. 187–214. [Google Scholar]
- Mezura-Montes, E.; Velázquez-Reyes, J.; Coello Coello, C.A. A comparative study of differential evolution variants for global optimization. In Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation (GECCO ’06), Seattle, WA, USA, 8–12 July 2006; pp. 485–492. [Google Scholar]
- Thayer, J.T.; Ruml, W. Faster than Weighted A*: An Optimistic Approach to Bounded Suboptimal Search. In Proceedings of the Eighteenth International Conference on Automated Planning and Scheduling (ICAPS 2008), Sydney, Australia, 14–18 September 2008; pp. 355–362. [Google Scholar]
- Seet, B.C.; Liu, G.; Lee, B.S.; Foh, C.H.; Wong, K.J.; Lee, K.K. A-STAR: A mobile ad hoc routing strategy for metropolis vehicular communications. In Proceedings of the International Conference on Research in Networking, Athens, Greece, 9–14 May 2004; pp. 989–999. [Google Scholar]
- Fogel, D.B.; Atmar, J.W. Comparing genetic operators with Gaussian mutations in simulated evolutionary processes using linear systems. Biol. Cybern.
**1990**, 63, 111–114. [Google Scholar] [CrossRef] - Zeng, G.Q.; Chen, J.; Li, L.M.; Chen, M.R.; Wu, L.; Dai, Y.X.; Zheng, C.W. An improved multi-objective population-based extremal optimization algorithm with polynomial mutation. Inf. Sci.
**2016**, 330, 49–73. [Google Scholar] [CrossRef] - Goldberg, D.E. Genetic Algorithms; Pearson Education: Tamil Nadu, India, 2006. [Google Scholar]
- Montero, E.; Riff, M.C. Effective collaborative strategies to setup tuners. Soft Comput.
**2020**, 24, 5019–5041. [Google Scholar] [CrossRef] - Nasir, J.; Islam, F.; Malik, U.; Ayaz, Y.; Hasan, O.; Khan, M.; Muhammad, M.S. RRT*-SMART: A rapid convergence implementation of RRT. Int. J. Adv. Robot. Syst.
**2013**, 10, 299. [Google Scholar] [CrossRef] [Green Version] - Broderick, J.A.; Tilbury, D.M.; Atkins, E.M. Characterizing energy usage of a commercially available ground robot: Method and results. J. Field Robot.
**2014**, 31, 441–454. [Google Scholar] [CrossRef] [Green Version] - Opara, K.R.; Arabas, J. Differential Evolution: A survey of theoretical analyses. Swarm Evol. Comput.
**2019**, 44, 546–558. [Google Scholar] [CrossRef] - Hasegawa, T.; Terasaki, H. Collision avoidance: Divide-and-conquer approach by space characterization and intermediate goals. IEEE Trans. Syst. Man Cybern.
**1988**, 18, 337–347. [Google Scholar] [CrossRef] - Tasoulis, D.K.; Pavlidis, N.G.; Plagianakos, V.P.; Vrahatis, M.N. Parallel differential evolution. In Proceedings of the 2004 Congress on Evolutionary Computation (IEEE CEC 2004), Portland, OR, USA, 19–23 June 2004; Volume 2, pp. 2023–2029. [Google Scholar]

**Figure 3.**Multimodality in the path-planning problem: (

**a**) a path with minimum length and a minimum number of control points and (

**b**) the same path using more control points.

**Figure 4.**Exploration in the path-planning problem: (

**a**) a path with minimum length using two control points and (

**b**) a longer path using a single control point.

Test Case | Obstacles | Size (m, m) | ${\mathit{p}}_{\mathit{s}}$ (m, m) | ${\mathit{p}}_{\mathit{t}}$ (m, m) | Complexity |
---|---|---|---|---|---|

(A) | 32 | (100, 100) | (5, 5) | (87.5, 87.5) | 0/1E8 |

(B) | 2 | (100, 100) | (5, 5) | (97.5, 97.5) | 0/1E8 |

(C) | 17 | (100, 100) | (5, 5) | (97.5, 97.5) | 0/1E8 |

**Table 2.**Differential Evolution for Variable-Length-Vector (VLV-DE) results for test cases (A), (B) and (C).

Test Case (A) | Test Case (B) | Test Case (c) | |
---|---|---|---|

Mean $s\left(x\right)$ | 9 | 13.2666 | 13.9666 |

STD | 1.2317 | 1.2015 | 1.5196 |

C.I. | [8.61788, 9.3821] | [12.8939, 13.6394] | [13.4952, 14.4380] |

Mean J $\left(\mathrm{m}\right)$ | 2627.0570 | 3453.4441 | 4095.0317 |

STD $\left(\mathrm{m}\right)$ | 45.9620 | 11.4728 | 48.2714 |

C.I. $\left(\mathrm{m}\right)$ | [2612.7988, 2641.3152] | [3449.8850, 3457.0032] | [4080.0571, 4110.0064] |

Best $s\left(x\right)$ | 9 | 15 | 13 |

Best J $\left(\mathrm{m}\right)$ | 2548.8696 | 3415.7757 | 3989.3064 |

Worst $s\left(x\right)$ | 11 | 13 | 17 |

Worst J $\left(\mathrm{m}\right)$ | 2731.2721 | 3471.7303 | 4194.8301 |

Test Case | $\mathit{s}\left(\mathit{x}\right)$ | J (m) |
---|---|---|

(A) | 166 | 3314.1421 |

(B) | 219 | 4374.1421 |

(C) | 232 | 4634.1421 |

Test Case | $\mathit{s}\left(\mathit{x}\right)$ | J (m) |
---|---|---|

(A) | 8 | 3334.5370 |

(B) | 12 | 3562.2011 |

(C) | 15 | 4559.8692 |

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**MDPI and ACS Style**

Rodríguez-Molina, A.; Solís-Romero, J.; Villarreal-Cervantes, M.G.; Serrano-Pérez, O.; Flores-Caballero, G.
Path-Planning for Mobile Robots Using a Novel Variable-Length Differential Evolution Variant. *Mathematics* **2021**, *9*, 357.
https://doi.org/10.3390/math9040357

**AMA Style**

Rodríguez-Molina A, Solís-Romero J, Villarreal-Cervantes MG, Serrano-Pérez O, Flores-Caballero G.
Path-Planning for Mobile Robots Using a Novel Variable-Length Differential Evolution Variant. *Mathematics*. 2021; 9(4):357.
https://doi.org/10.3390/math9040357

**Chicago/Turabian Style**

Rodríguez-Molina, Alejandro, José Solís-Romero, Miguel Gabriel Villarreal-Cervantes, Omar Serrano-Pérez, and Geovanni Flores-Caballero.
2021. "Path-Planning for Mobile Robots Using a Novel Variable-Length Differential Evolution Variant" *Mathematics* 9, no. 4: 357.
https://doi.org/10.3390/math9040357