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

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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

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**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