# An Improved Grey Wolf Optimization with Multi-Strategy Ensemble for Robot Path Planning

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of GWO

#### 2.1. Leadership Hierarchy

#### 2.2. Hunting Mechanism

_{1}and r

_{2}represent random vectors within [0, 1]. Both

**A**and

**C**are coefficient vectors.

**D**denotes the approximate distance between the current wolf position and prey,

**X**

^{t}and

**X**

^{t}

^{+1}stand for the positions of the wolf at tth and t+1th iteration respectively. ${\mathit{X}}_{p}^{t}$ represents the position of the prey at tth iteration.

**C**

_{1},

**C**

_{2}and

**C**

_{3}are random coefficient vectors defined by Equation (4).

**A**

_{1},

**A**

_{2}and

**A**

_{3}are coefficient vectors defined by Equation (3). ${\mathit{X}}_{a}^{t}$, ${\mathit{X}}_{\beta}^{t}$ and ${\mathit{X}}_{\delta}^{t}$ represent the positions of α, β and δ. GWO pseudocode are shown in Algorithm 1, and the location update of wolves are exhibited in Figure 2.

Algorithm 1. The pseudocode of conventional GWO |

1. Generate a population X_{i} (i = 1, 2, …, n) randomly2. Initialize the parameters of GWO (max_iteration, a, A and C)3. Calculate the fitness values and assign α, β and δ4. While (t < max_iteration)5. For each grey wolf6. Update the position of the current grey wolf using Equations (6)–(8)7. End for8. Update a, A and C9. Amend the grey wolves’ positions beyond boundary limits10. Calculate the fitness values of the new positions11. Update the α, β and δ12. t = t + 113. End while14. Return the position of α |

## 3. Development of IGWO

#### 3.1. Modified Position Update Mechanism

_{α}, w

_{β}, and w

_{δ}) calculated by Equations (9) and (10). It is obvious that during the optimization, the corresponding weights of the α, β, and δ are descend sequentially, which is more consistent with the hierarchical system of a wolf pack.

**X**

_{t}

_{+1}and

**X**

_{t}denote the current wolf’s position at t+1th and at tth iteration respectively. ${\mathit{X}}_{rand}^{t}$ represents the randomly selected wolf’s position. rand

_{1}and rand

_{2}are random numbers in [0, 1]. n

_{1}and n

_{2}are adjustment parameters that can bring about a tradeoff of the leader-determined term and the random term. Moreover, n

_{1}and n

_{2}are set to add up to 1, ensuring the convergence of solution. n

_{1}will increase linearly to 1 with the number of iterations, and the corresponding search behavior could be described as follows: While n

_{1}is smaller compared with n

_{2}in the early stage, the random term provides the main guidance of movement for the pack, conducting a full exploration in search space. In the later stage, n

_{1}increases to 1, and the dominating rights returns to the three leaders, enhancing the exploitation capacity of the approach. Compared with Equation (8), the new formula could better simulate the hierarchy of the wolf society, and balance the global and local searches.

#### 3.2. Dynamic Local Optimum Escape Strategy

**is a logical value that determines whether to execute the random walk strategy. K is a fixed threshold. r is a random number. When r is greater than K, the random walk strategy will be executed. Otherwise, the original solution will be maintained.**

_{i}_{3}and rand

_{4}are random numbers inside [0, 1]. σ is a tuning parameter standing for the extent of mutation, which changes dynamically during the search process. a changes according to Equation (5). $ran{d}_{3}\cdot {\mathit{X}}^{t}-ran{d}_{4}\cdot {\mathit{X}}_{rand}^{t}$ is a small random disturbance that can further increase the ability of fleeing local optimum.

#### 3.3. Individual Repositioning Method

**X**

_{new_worst}is the wolf’s position after repositioning, and r

_{1}is a random number within [0.5, 1]. The sum of r

_{1}, r

_{2}and r

_{3}is set to 1 to ensure the convergence of the solutions.

_{1}> r

_{2}> r

_{3}in Equation (23).

## 4. Numerical Optimization Experiments

#### 4.1. Comparison of IGWO with Different GWO Variants

#### 4.1.1. Analysis of Numerical Results

#### 4.1.2. Analysis of Convergence Curves

#### 4.2. Comparison of IGWO with Other Meta-Heuristic Algorithms

#### 4.2.1. Analysis of Numerical Results

#### 4.2.2. Analysis of Convergence Curve

## 5. Application of IGWO in Robot Path Planning

#### 5.1. Environment Models

_{obs}, y

_{obs}) represents the coordinates of the center of the obstacle.

#### 5.2. Path Smoothing

_{i}, x

_{i}

_{+1}], S(x) is defined as follows:

_{i}, b

_{i}, c

_{i}and d

_{i}are undetermined coefficients. As S(x) has 4n undetermined coefficients, there is a need for at least 4n known conditions to solve the undetermined coefficients, and the specific conditions are given as follows:

_{dD}, y

_{dD}), (x

_{dD}, y

_{dD}), …, (x

_{dD}, y

_{dD}), where the starting and ending coordinates are (x

_{s}, y

_{s}) and (x

_{t}, y

_{t}). Firstly, split the abscissa and ordinate of the above D + 2 points into the sets of {w

_{x}} = {x

_{s}, x

_{d}

_{1}, x

_{d}

_{2},…, x

_{dD}, x

_{t}} and {w

_{y}} = {y

_{s}, y

_{d}

_{1}, y

_{d}

_{2},…, y

_{dD}, y

_{t}}. Then, apply cubic spline interpolation to{w

_{x}} and {w

_{y}} separately and obtain the abscissa and the ordinate of n interpolation points, namely {x

_{1}, x

_{2},…, x

_{n}} and { y

_{1}, y

_{2},…, y

_{n}}. Finally, {(x

_{s}, y

_{s}), (x

_{1}, y

_{1}), (x

_{2}, y

_{2}), …, (x

_{n}, y

_{n}), (x

_{t}, y

_{t})} is the path of the robot after smoothing.

#### 5.3. Construction of Fitness Function

_{s}, y

_{s}) to the end point (x

_{t}, y

_{t}) [63]. Therefore, a fitness function is constructed to measure the performance of a robot obstacle avoidance and path length in this subsection. The mathematical model is defined as follows:

_{l}is the sum of Euclidean distance between the interpolation points and the calculation method is shown in Equation (30). f

_{obs}is a marker variable to evaluate the path obstacle avoidance level, whose initial value is 0. f

_{obs}can be calculated using Equations (31)–(33).

_{i}and y

_{i}respectively represent the x-coordinate and y-coordinate of the i

^{th}interpolation point on a path. (x_obs

_{k}, y_obs

_{k}) is on behalf of the center coordinate of the k

^{th}obstacle. n is the number of interpolation points of the path. n

_{obs}is the total number of obstacles and r

_{k}is the corresponding radius of the obstacle. Equation (31) is to deal with the Euclidean distance between the interpolation point on the candidate path and the center of k

^{th}obstacle. Equation (32) could determine whether the path intersects with the k

^{th}obstacle. If there is a path point entering the k

^{th}obstacle, then ε

_{k}> 0. Otherwise, ε

_{k}= 0. Moreover, if a planned path avoids all the obstacles successfully, f

_{obs}= 0.

#### 5.4. Experimental Environment and Parameter Setting

#### 5.5. Analysis of Path Planning Results

#### 5.5.1. Single Contrast Experiment

#### 5.5.2. Thirty Independent Contrast Experiments

#### 5.6. Contrast Experiment in Complex Environment with Irregular Obstacles

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Adaptive weight coefficients of the leaders. (

**a**) 2-D version of Sphere function; (

**b**) Values of w

_{α}, w

_{β}, and w

_{δ}.

**Figure 22.**Basic environment models. (

**a**) Simple obstacle environment model (case 1); (

**b**) Complex obstacle environment model (case 2).

**Figure 23.**Comparison of three algorithm in case 1, where the yellow square and green five-pointed star respectively represent the starting point and ending point.

**Figure 24.**Comparison of three algorithm in case 2, where the yellow square and green five-pointed star respectively represent the starting point and ending point.

**Figure 27.**Results of the three algorithms in case 1, where the yellow square, green five-pointed star and lines in different colors in the figure represent the starting point, ending point and paths, respectively. (

**a**) Results of RMPSO; (

**b**) Results of MEGWO; (

**c**) Results of IGWO.

**Figure 28.**Results of the three algorithms in case 2, where the yellow square, green five-pointed star and lines in different colors in the figure represent the starting point, ending point and paths, respectively (

**a**) Results of RMPSO; (

**b**) Results of MEGWO; (

**c**) Results of IGWO.

**Figure 33.**Paths planned by four algorithms in complex obstacle environment. (

**a**) Results of RMPSO; (

**b**) Results of MEGWO; (

**c**) Results of mGWO; (

**d**) Results of IGWO.

**Table 1.**Unimodal, multimodal and shifted and rotated multimodal benchmark functions. F * symbolizes the optimal value for each benchmark function.

Function | Test Function | Dim | Range | F * |
---|---|---|---|---|

F1 | ${f}_{1}={\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}$ | 30 | [−100, 100]^{n} | 0 |

F2 | ${f}_{2}={\displaystyle \sum _{i=1}^{n}\left|{x}_{i}\right|}+{\displaystyle \prod _{i=1}^{n}\left|{x}_{i}\right|}$ | 30 | [−10, 10]^{n} | 0 |

F3 | ${f}_{3}={\displaystyle \sum _{i=1}^{n}\left({\displaystyle \sum _{j=1}^{i}{x}_{j}}\right)}$ | 30 | [−100, 100]^{n} | 0 |

F4 | ${f}_{4}=\underset{n}{{\mathrm{max}}_{i}}\left\{\left|{x}_{i}\right|,1\le i\le n\right\}$ | 30 | [−100, 100]^{n} | 0 |

F5 | ${f}_{5}={\displaystyle \sum _{i=1}^{n}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]}$ | 30 | [−30, 30]^{n} | 0 |

F6 | ${f}_{6}={\displaystyle \sum _{i=1}^{n}i{x}_{i}^{4}+random\left[0,1\right)}$ | 30 | [−1.28, 1.28]^{n} | 0 |

F7 | ${f}_{7}={\displaystyle \sum _{i=1}^{n}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]}$ | 30 | [−5.12, 5.12]^{n} | 0 |

F8 | ${f}_{8}=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{n}{\displaystyle \sum _{j=1}^{n}{x}_{j}}}\right)-\mathrm{exp}\left(\frac{1}{n}\mathrm{cos}\left(2\pi {x}_{j}\right)\right)+20+e$ | 30 | [−32, 32]^{n} | 0 |

F9 | ${f}_{9}=\frac{1}{4000}{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}-{\displaystyle \prod _{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)}+1$ | 30 | [−600, 600]^{n} | 0 |

F10 | $\begin{array}{l}{f}_{10}=\frac{\pi}{n}\left\{10\mathrm{sin}\left(\pi {y}_{1}\right)+{\displaystyle \sum _{i=1}^{n}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {y}_{i+1}\right)\right]+{\displaystyle \sum _{i=1}^{n}u\left({x}_{i},10,100,4\right)}}\right\}\\ {y}_{i}=1+\frac{{x}_{i}+1}{4}\\ u\left({x}_{i},a,k,m\right)=\{\begin{array}{ll}k{\left({x}_{i}-a\right)}^{m}& {x}_{i}a\hfill \\ 0& -a{x}_{i}a\hfill \\ k{\left(-{x}_{i}-a\right)}^{m}& {x}_{i}-a\hfill \end{array}\end{array}$ | 30 | [−50, 50]^{n} | 0 |

F11 | Shifted and Rotated Katsuura Function | 30 | [−100, 100]^{n} | 1200 |

F12 | Shifted and Rotated HappyCat Function | 30 | [−100, 100]^{n} | 1300 |

F13 | Shifted and Rotated HGBat Function | 30 | [−100, 100]^{n} | 1400 |

**Table 2.**Fixed-dimension multimodal and composition benchmark functions. F * symbolizes the optimal value for each benchmark function.

Function | Test Function | Dim | Range | F * |
---|---|---|---|---|

F14 | ${f}_{14}={{\displaystyle \sum _{i=1}^{11}\left[{a}_{i}-\frac{{x}_{i}\left({b}_{i}^{2}+{b}_{i}{x}_{2}\right)}{{b}_{i}{}^{2}+{b}_{i}{x}_{3}+{x}_{4}}\right]}}^{2}$ | 4 | [−5, 5]^{n} | 0.00030 |

F15 | ${f}_{15}={\left({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}{}^{2}+\frac{5}{\pi}{x}_{1}-6\right)}^{2}+10\left(1-\frac{1}{8\pi}\right)\mathrm{cos}\left({x}_{1}\right)+10$ | 2 | [−5, 5]^{n} | 0.398 |

F16 | $\begin{array}{l}{f}_{16}=\left[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\left(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}{}^{2}\right)\right]\\ \times \left[30+{\left(2{x}_{1}-3{x}_{2}\right)}^{2}\times \left(18-32{x}_{1}+12{x}_{1}{}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}{}^{2}\right)\right]\end{array}$ | 2 | [−2, 2]^{n} | 3 |

F17 | ${f}_{17}=-{\displaystyle \sum _{i=1}^{5}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}\right]}^{-1}}$ | 4 | [0, 10]^{n} | −10.1532 |

F18 | Composition Function 1 (N = 5) | 30 | [−100, 100]^{n} | 2300 |

F19 | Composition Function 2 (N = 3) | 30 | [−100, 100]^{n} | 2400 |

F20 | Composition Function 3 (N = 3) | 30 | [−100, 100]^{n} | 2500 |

**Table 3.**Results of GWO variants on unimodal, multimodal and shifted and rotated multimodal benchmark functions. The best values are highlighted in bold.

MixedGWO | GWOCS | LearnGWO | mGWO | RW_GWO | IGWO | ||
---|---|---|---|---|---|---|---|

F1 | Mean | 3.83 × 10^{−30} | 1.60 × 10^{−59} | 1.3498 × 10^{−123} | 7.04 × 10^{−60} | 1.46 × 10^{−22} | 0 |

Std | 3.94 × 10^{−29} | 2.27 × 10^{−58} | 2.8395 × 10^{−122} | 1.47 × 10^{−58} | 9.50 × 10^{−22} | 0 | |

R/T | 5/+ | 4/+ | 2/+ | 3/+ | 6/+ | 1 | |

F2 | Mean | 1.09 × 10^{−18} | 4.00 × 10^{−35} | 6.09 × 10^{−69} | 3.02 × 10^{−38} | 1.04 × 10^{−11} | 0 |

Std | 1.02 × 10^{−17} | 2.50 × 10^{−34} | 4.57 × 10^{−68} | 7.56 × 10^{−37} | 1.96 × 10^{−11} | 0 | |

R/T | 5/+ | 4/+ | 2/+ | 3/+ | 6/+ | 1 | |

F3 | Mean | 8.28 × 10^{−5} | 3.57 × 10^{−15} | 2.83 × 10^{−89} | 3.35 × 10^{+3} | 1.02 × 10^{−9} | 0 |

Std | 0.0011 | 9.19 × 10^{−14} | 7.08 × 10^{−88} | 1.23 × 10^{+4} | 1.41 × 10^{−8} | 0 | |

R/T | 5/+ | 3/+ | 2/+ | 6/+ | 4/+ | 1 | |

F4 | Mean | 7.28 × 10^{−6} | 1.01 × 10^{−14} | 2.91 × 10^{−51} | 3.44 × 10^{−10} | 5.46 × 10^{−8} | 0 |

Std | 6.54 × 10^{−5} | 1.03 × 10^{−13} | 3.76 × 10^{−50} | 9.61 × 10^{−9} | 1.37 × 10^{−6} | 0 | |

R/T | 6/+ | 3/+ | 2/+ | 4/+ | 5/+ | 1 | |

F5 | Mean | 28.8373 | 26.9401 | 28.5497 | 28.6775 | 26.5379 | 27.5219 |

Std | 0.1036 | 3.7477 | 1.9255 | 0.7313 | 2.9594 | 3.7676 | |

R/T | 6/+ | 2/− | 4/+ | 5/+ | 1/− | 3 | |

F6 | Mean | 0.0025 | 8.88 × 10^{−4} | 8.10 × 10^{−5} | 4.16 × 10^{−4} | 9.20 × 10^{−4} | 8.95 × 10^{−5} |

Std | 0.0051 | 0.0018 | 3.21 × 10^{−4} | 0.0021 | 2.70 × 10^{−3} | 2.50 × 10^{−4} | |

R/T | 6/+ | 4/+ | 1/− | 3/+ | 5/+ | 2 | |

F7 | Mean | 19.3405 | 1.3586 | 0 | 7.0415 | 1.4371 | 0 |

Std | 50.5218 | 19.1616 | 0 | 144.5174 | 17.6327 | 0 | |

R/T | 5/+ | 2/+ | 1/≈ | 4/+ | 3/+ | 1 | |

F8 | Mean | 1.58 × 10^{14} | 1.55 × 10^{−14} | 4.56 × 10^{−15} | 4.56 × 10^{−15} | 6.18 × 10^{−12} | 8.88 × 10^{−16} |

Std | 2.87 × 10^{−14} | 1.58 × 10^{−14} | 3.49 × 10^{−15} | 6.12 × 10^{−15} | 1.19 × 10^{−11} | 0 | |

R/T | 5/+ | 4/+ | 2/+ | 3/+ | 6/+ | 1 | |

F9 | Mean | 0.0021 | 5.75 × 10^{−4} | 1.11 × 10^{−4} | 0.0021 | 3.26 × 10^{−4} | 0 |

Std | 0.0409 | 0.0112 | 0.0033 | 0.0632 | 0.0096 | 0 | |

R/T | 5/+ | 4/+ | 2/≈ | 6/≈ | 3/+ | 1 | |

F10 | Mean | 0.5419 | 0.0396 | 0.6079 | 0.0619 | 0.0012 | 0.1822 |

Std | 0.8062 | 0.1137 | 0.8432 | 0.1819 | 0.009 | 0.5448 | |

R/T | 5/+ | 2/− | 6/+ | 3/− | 1/− | 4 | |

F11 | Mean | 1203.1 | 1202.5 | 1203.2 | 1202.0 | 1200.9 | 1201.5 |

Std | 1.4643 | 7.837 | 2.5871 | 1.8016 | 5.2961 | 2.1432 | |

R/T | 4/+ | 6/+ | 5/+ | 3/+ | 1/− | 2 | |

F12 | Mean | 1302.1 | 1300.7 | 1304.4 | 1304.0 | 1300.6 | 1300.5 |

Std | 6.7783 | 3.1098 | 2.5492 | 4.0168 | 0.5621 | 0.0695 | |

R/T | 4/+ | 3/+ | 6/+ | 5/+ | 2/+ | 1 | |

F13 | Mean | 1435.3 | 1410.8 | 1499.5 | 1472.9 | 1400.8 | 1400.4 |

Std | 107.6842 | 50.4025 | 124.8757 | 126.7876 | 1.7689 | 1.7390 | |

R/T | 4/+ | 3/+ | 6/+ | 5/+ | 2/+ | 1 |

**Table 4.**Results of GWO variants on fixed-dimension multimodal and composition benchmarks. The best values are highlighted in bold.

MixedGWO | GWOCS | LearnGWO | mGWO | RW_GWO | IGWO | ||
---|---|---|---|---|---|---|---|

F14 | Mean | 0.0099 | 3.78 × 10^{−4} | 5.70 × 10^{−3} | 8.73 × 10^{−4} | 0.0023 | 3.33 × 10^{−4} |

Std | 0.0073 | 0.0013 | 4.37 × 10^{−2} | 0.0047 | 0.0329 | 2.58 × 10^{−4} | |

R/T | 6/+ | 2/+ | 4/+ | 3/+ | 5/+ | 1 | |

F15 | Mean | 0.4576 | 0.398 | 0.4003 | 0.398 | 0.398 | 0.398 |

Std | 0 | 8.74 × 10^{−7} | 0.0224 | 3.97 × 10^{−8} | 2.14 × 10^{−6} | 1.03 × 10^{−8} | |

R/T | 6/+ | 3/+ | 5/+ | 2/+ | 4/+ | 1 | |

F16 | Mean | 3 | 3 | 3.0093 | 3 | 3 | 3 |

Std | 0 | 3.69 × 10^{−5} | 0.273 | 1.06 × 10^{−12} | 6.53 × 10^{−5} | 3.56 × 10^{−5} | |

R/T | 1/− | 4/+ | 6/+ | 2/− | 5/+ | 3 | |

F17 | Mean | −10.4028 | −9.873 | −4.8074 | −9.1627 | −9.481 | −10.057 |

Std | 2.83 × 10^{−4} | 7.1031 | 3.8401 | 12.3134 | 11.3324 | 5.7099 | |

R/T | 1− | 3/+ | 6/+ | 5/+ | 4/+ | 2 | |

F18 | Mean | 2662.4 | 2642.7 | 2745.2 | 2673.7 | 2626.2 | 2500 |

Std | 115.9621 | 69.9378 | 210.9216 | 190.9163 | 23.952 | 0 | |

R/T | 5/+ | 3/+ | 4/+ | 6/+ | 2/+ | 1 | |

F19 | Mean | 2600.2 | 2600 | 2600 | 2600.4 | 2600 | 2600 |

Std | 0.3285 | 0.0604 | 2.1052 × 10^{−6} | 1.4886 | 0.09 | 0 | |

R/T | 5/+ | 3/+ | 2/+ | 6/+ | 4/+ | 1 | |

F20 | Mean | 2704.5 | 2700.4 | 2700 | 2700.9 | 2712.5 | 2700 |

Std | 38.3336 | 13.1324 | 6.4311 × 10^{−13} | 27.4906 | 30.3025 | 4.5475 × 10^{−13} | |

R/T | 5/+ | 3/+ | 2/+ | 4/+ | 6/+ | 1 |

Result | MixedGWO | GWOCS | LearnGWO | mGWO | RW_GWO | IGWO |
---|---|---|---|---|---|---|

+/≈/− | 18/0/2 | 18/0/2 | 17/2/1 | 17/1/2 | 17/0/3 | ~ |

Mean rank | 4.7 | 3.25 | 3.5 | 4.05 | 3.75 | 1.5 |

Overall rank | 6 | 2 | 4 | 5 | 3 | 1 |

GWO | SCA | PSO | WOA | ABC | TSA | MVO | IGWO | ||
---|---|---|---|---|---|---|---|---|---|

F1 | Mean | 2.07 × 10^{−59} | 1.25 × 10^{−2} | 9.43 × 10^{−9} | 6.59 × 10^{−150} | 0.7887 | 6.02 × 10^{−6} | 0.2963 | 0 |

Std | 1.33 × 10^{−58} | 1.07 × 10^{−1} | 1.88 × 10^{−7} | 1.77 × 10^{−148} | 2.1909 | 1.59 × 10^{−5} | 0.3969 | 0 | |

R/T | 3/+ | 6/+ | 4/+ | 2/+ | 8/+ | 5/+ | 7/+ | 1 | |

F2 | Mean | 1.83 × 10^{−34} | 1.20 × 10^{−5} | 6.0004 | 1.27 × 10^{−102} | 0.0363 | 0.0208 | 0.4606 | 0 |

Std | 1.78 × 10^{−33} | 1.26 × 10^{−5} | 38.9865 | 2.18 × 10^{−101} | 0.056 | 0.0305 | 1.1093 | 0 | |

R/T | 3/+ | 4/+ | 8/+ | 2/+ | 6/+ | 5/+ | 7/+ | 1 | |

F3 | Mean | 1.20 × 10^{−12} | 6.57 × 10^{3} | 16.3022 | 1.76 × 10^{4} | 3.41 × 10^{4} | 2.67 × 10^{4} | 45.3638 | 0 |

Std | 3.44 × 10^{−11} | 4.86 × 10^{3} | 46.3105 | 4.31 × 10^{4} | 2.04 × 10^{4} | 1.49 × 10^{4} | 120.8618 | 0 | |

R/T | 2/+ | 5/+ | 3/+ | 6/+ | 8/+ | 7/+ | 4/+ | 1 | |

F4 | Mean | 2.25 × 10^{−14} | 17.9 | 0.6072 | 36.9902 | 53.3308 | 26.7721 | 1.0175 | 0 |

Std | 1.07 × 10^{−13} | 9.85 | 0.6718 | 113.8156 | 24.6552 | 17.5272 | 1.7441 | 0 | |

R/T | 2/+ | 5/+ | 3/+ | 7/+ | 8/+ | 6/+ | 4/+ | 1 | |

F5 | Mean | 26.7285 | 579.7010 | 56.6336 | 27.2767 | 1.72 × 10^{4} | 125.1067 | 425.0734 | 27.6068 |

Std | 3.1903 | 1.41 × 10^{4} | 233.9869 | 3.1856 | 5.29 × 10^{4} | 232.5186 | 3.53 × 10^{3} | 3.7676 | |

R/T | 1/− | 6/+ | 4/+ | 2/− | 8/+ | 5/+ | 7/+ | 3 | |

F6 | Mean | 8.28 × 10^{−4} | 5.09 × 10^{−2} | 4.8865 | 0.0021 | 0.2127 | 0.2946 | 0.0222 | 9.13 × 10^{−5} |

Std | 0.0016 | 7.87 × 10^{−2} | 26.4596 | 0.0104 | 0.1851 | 0.2579 | 0.0465 | 2.49 × 10^{−4} | |

R/T | 2/+ | 5/+ | 8/+ | 3/+ | 6/+ | 7/+ | 4/+ | 1 | |

F7 | Mean | 0.7531 | 10.4 | 86.7988 | 8.53 × 10^{−15} | 224.9901 | 107.1537 | 111.7582 | 0 |

Std | 10.908 | 16.4 | 123.4636 | 1.21 × 10^{−13} | 93.4056 | 53.0099 | 198.3001 | 0 | |

R/T | 3/+ | 4/+ | 5/+ | 2/+ | 8/+ | 6/+ | 7/+ | 1 | |

F8 | Mean | 1.67 × 10^{−14} | 16.1 | 4.24 × 10^{−05} | 4.97 × 10^{−15} | 1.569 | 1.3307 | 0.9705 | 8.88 × 10^{−16} |

Std | 1.55 × 10^{−14} | 8.48 | 2.64 × 10^{−04} | 1.36 × 10^{−14} | 2.5771 | 4.6003 | 3.2009 | 0 | |

R/T | 3/+ | 8/+ | 4/+ | 2/+ | 7/+ | 6/+ | 5/+ | 1 | |

F9 | Mean | 0.0036 | 0.335 | 0.0094 | 0.0036 | 0.8348 | 0.2145 | 0.554 | 0 |

Std | 0.0299 | 0.335 | 0.0443 | 0.0699 | 0.4122 | 0.3981 | 0.5881 | 0 | |

R/T | 2/+ | 6/+ | 4/+ | 3/≈ | 8/+ | 5/+ | 7/+ | 1 | |

F10 | Mean | 0.0427 | 4.1166 | 0.0173 | 0.0069 | 6.3958 × 10^{3} | 0.6925 | 1.2145 | 0.1822 |

Std | 0.1214 | 68.7625 | 0.2116 | 0.0234 | 6.5686 × 10^{4} | 2.1067 | 4.9150 | 0.5448 | |

R/T | 3/− | 7/+ | 2/− | 1/− | 8/+ | 5/+ | 6/+ | 4 | |

F11 | Mean | 1202.3 | 1203.1 | 1200.4 | 1202.3 | 1203.3 | 1201.5 | 1200.7 | 1201.5 |

Std | 6.2334 | 1.2522 | 1.6106 | 3.9568 | 2.4321 | 2.8913 | 1.8405 | 1.1522 | |

R/T | 6/+ | 7/+ | 1/− | 5/+ | 8/+ | 4/+ | 2/− | 3 | |

F12 | Mean | 1300.6 | 1303.8 | 1300.5 | 1300.6 | 1300.6 | 1300.6 | 1300.7 | 1300.5 |

Std | 2.0869 | 1.7531 | 0.6553 | 0.7098 | 0.4312 | 0.3043 | 0.8459 | 0.0700 | |

R/T | 6/+ | 8/+ | 2/+ | 5/+ | 4/+ | 3/+ | 7/+ | 1 | |

F13 | Mean | 1405.5 | 1472.1 | 1400.7 | 1403.0 | 1400.8 | 1400.3 | 1400.7 | 1400.6 |

Std | 44.3370 | 74.1032 | 8.0383 | 26.9729 | 0.2715 | 0.2175 | 1.9733 | 1.9105 | |

R/T | 7/+ | 8/+ | 4/+ | 6/+ | 5/+ | 1/− | 3/+ | 2 |

GWO | SCA | PSO | WOA | ABC | TSA | MVO | IGWO | ||
---|---|---|---|---|---|---|---|---|---|

F14 | mean | 0.0014 | 1.00 × 10^{−3} | 0.0057 | 6.09 × 10^{−4} | 6.22 × 10^{−4} | 3.83 × 10^{−4} | 0.0034 | 3.33 × 10^{−4} |

sd | 0.0195 | 4.23 × 10^{−4} | 0.0335 | 0.0015 | 4.51 × 10^{−4} | 2.51 × 10^{−4} | 0.0366 | 2.58 × 10^{−4} | |

R/T | 6/+ | 5/+ | 8/+ | 3/+ | 4/+ | 2/+ | 7/+ | 1 | |

F15 | mean | 0.398 | 0.399 | 0.398 | 0.398 | 0.398 | 0.398 | 0.398 | 0.398 |

sd | 9.97 × 10^{−5} | 1.45 × 10^{−3} | 0 | 2.43 × 10^{−6} | 0 | 0 | 1.16 × 10^{−6} | 5.71 × 10^{−08} | |

R/T | 5/+ | 6/+ | 1/− | 4/+ | 1/− | 1/− | 3/+ | 2 | |

F16 | mean | 3 | 3 | 3 | 3 | 3 | 3 | 5.7 | 3 |

sd | 2.95 × 10^{−05} | 1.37 × 10^{−05} | 4.97 × 10^{−15} | 1.50 × 10^{−04} | 1.54 × 10^{−15} | 3.29 × 10^{−15} | 79.6386 | 1.07 × 10^{−06} | |

R/T | 6/+ | 5/+ | 3/− | 7/+ | 1/− | 2/− | 8/+ | 4 | |

F17 | mean | −9.8160 | −3.45 | −7.9464 | −8.5398 | −10.1526 | −10.1532 | −7.6246 | −10.0786 |

sd | 6.9026 | 12.5768 | 15.1201 | 13.6539 | 0.0175 | 1.46 × 10^{−14} | 15.1507 | 2.3102 | |

R/T | 4/+ | 8/+ | 7/+ | 5/+ | 2/− | 1/− | 6/+ | 3 | |

F18 | mean | 2641.6 | 2713.4 | 2616.4 | 2668.3 | 2617.4 | 2615.3 | 2623.7 | 2500 |

sd | 68.8862 | 142.2616 | 12.1211 | 264.5586 | 4.1657 | 0.0414 | 30.339 | 0 | |

R/T | 6/+ | 8/+ | 3/+ | 7/+ | 4/+ | 2/+ | 5/+ | 1 | |

F19 | mean | 2600 | 2607.1 | 2624.4 | 2608.7 | 2638.5 | 2633.5 | 2636.5 | 2600 |

sd | 0.0727 | 40.0755 | 43.338 | 35.6168 | 20.2054 | 8.637 | 35.5488 | 0 | |

R/T | 2/+ | 3/+ | 5/+ | 4/+ | 8/+ | 6/+ | 7/+ | 1 | |

F20 | mean | 2712.3 | 2739.6 | 2718.1 | 2722.9 | 2741.4 | 2719.9 | 2708.1 | 2700 |

sd | 31.8856 | 50.1478 | 26.4854 | 112.5271 | 42.1794 | 13.3521 | 11.7235 | 4.55 × 10^{−13} | |

R/T | 3/+ | 7/+ | 4/+ | 6/+ | 8/+ | 5/+ | 2/+ | 1 |

Result | GWO | SCA | PSO | WOA | ABC | TSA | MVO | IGWO |
---|---|---|---|---|---|---|---|---|

+/≈/− | 18/0/2 | 20/0/0 | 16/0/4 | 17/1/2 | 17/0/3 | 16/0/4 | 19/0/1 | ~ |

Mean rank | 3.75 | 6.05 | 4.15 | 4.1 | 6 | 4.2 | 5.4 | 1.7 |

Overall rank | 2 | 8 | 4 | 3 | 7 | 5 | 6 | 1 |

Simple Environment (Case 1) | Complex Environment (Case 2) | |
---|---|---|

Obstacles | 3 | 9 |

Starting point | (0,0) | (0, 0) |

Ending point | (4,6) | (10, 10) |

The shortest length | 7.21 | 14.14 |

Simple Environment (Case 1) | Complex Environment (Case 2) | |
---|---|---|

Population size | 30 | 30 |

Path points | 2 | 2 |

Interpolation points | 100 | 100 |

Iterations | 100 | 100 |

Mean | Best | Worst | Unsafe Path | Success Rate | |
---|---|---|---|---|---|

RMPSO | 7.8486 | 7.8308 | 7.8989 | 3 | 90% |

MEGWO | 7.9557 | 7.7764 | 8.3557 | 3 | 90% |

IGWO | 7.6529 | 7.5669 | 7.6981 | 0 | 100% |

Mean | Best | Worst | Unsafe Paths | Success Rate | |
---|---|---|---|---|---|

RMPSO | 15.5894 | 14.9284 | 17.5323 | 4 | 86.67% |

MEGWO | 16.1491 | 14.5662 | 17.6817 | 3 | 90% |

IGWO | 14.7052 | 14.5691 | 14.8698 | 0 | 100% |

**Table 13.**Experimental results in complex obstacle environment, and the best values are highlighted in bold.

Iteration | Path Length | Success Rate | |
---|---|---|---|

RMPSO | 96 | 36.289 | 86.67% |

MEGWO | 93 | 40.1963 | 83.33% |

mGWO | 101 | 39.3321 | 80% |

IGWO | 72 | 31.8779 | 90% |

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**MDPI and ACS Style**

Dong, L.; Yuan, X.; Yan, B.; Song, Y.; Xu, Q.; Yang, X.
An Improved Grey Wolf Optimization with Multi-Strategy Ensemble for Robot Path Planning. *Sensors* **2022**, *22*, 6843.
https://doi.org/10.3390/s22186843

**AMA Style**

Dong L, Yuan X, Yan B, Song Y, Xu Q, Yang X.
An Improved Grey Wolf Optimization with Multi-Strategy Ensemble for Robot Path Planning. *Sensors*. 2022; 22(18):6843.
https://doi.org/10.3390/s22186843

**Chicago/Turabian Style**

Dong, Lin, Xianfeng Yuan, Bingshuo Yan, Yong Song, Qingyang Xu, and Xiongyan Yang.
2022. "An Improved Grey Wolf Optimization with Multi-Strategy Ensemble for Robot Path Planning" *Sensors* 22, no. 18: 6843.
https://doi.org/10.3390/s22186843