# UAV Path Planning Based on an Improved Chimp Optimization Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- We established 3D environment models for UAV trajectory planning, covering different terrains or buildings such as plains, mountains, hills, and human engineering.
- (2)
- The path length, flight altitude, and angle loss during the flight of UAVs were considered, which constituted the comprehensive evaluation index of path planning. The cubic spline interpolation method is used to smooth the trajectory of UAVs to solve the problem of low accuracy of interpolation points in B-spline curves [39].
- (3)
- TRS-ChOA: To solve the 3D UAV path planning problem, we propose an enhanced version of the original ChOA based on differential evolution, improved reverse learning, and similarity preference weights.
- (4)
- The optimization performance of TRS-ChOA is verified by the benchmark test function and the CEC2017 complex test function.
- (5)
- Several well-known meta-heuristic methods are compared with the proposed TRS-ChOA in different 3D environments.

## 2. UAV Path Planning Problem Model

#### 2.1. Background

#### 2.2. Environmental Model

_{o}represents the height of the base terrain and h

_{i}represents the height of the i-th mountain. (x

_{oi}, y

_{oi}) represents the center coordinate of the i-th mountain. a

_{i}and b

_{i}are the slopes of the i-th mountain along the x-axis and y-axis directions, respectively. By setting the values of h

_{i}, a

_{i}, and b

_{i}, we can get different mountains or hills, providing a more complex test environment for the UAV. Equation (3) is the expression of environmental height after superimposing the mountain model.

#### 2.3. Track Model

_{i}, and the coordinates of the two nodes are (x

_{i}, y

_{i}, z

_{i}) and (x

_{i}

_{+1}, y

_{i}

_{+1}, z

_{i}

_{+1}), abbreviated as N(i) and N(i + 1). Equation (4) describes the length of the UAV’s actual flight path, Dpath.

_{i}represents the segment vector of the i-th segment trajectory and |s

_{i}| represents its length. Through separate discussions of the three aspects mentioned above, we obtain the loss function for UAV path planning as shown in Equation (7).

_{loss}is the total loss function, and α

_{1}, α

_{2}, and α

_{3}represent the weight. In this paper, we set α

_{1}= 0.4, α

_{2}= 0.4, and α

_{3}= 0.2. After effective processing of C

_{loss}, a flight path composed of line segments is obtained, but this trajectory is only theoretically feasible. We use cubic spline interpolation instead of the B-spline curve method to smooth the trajectory so as to improve the accuracy of interpolation points.

_{0}, y

_{0}), (x

_{1}, y

_{1}), (x

_{2}, y

_{2}), and (x

_{n}, y

_{n}). If linear fitting is used to connect these discrete points, the resulting function is not smooth enough. To smooth the line between two adjacent nodes, we use a cubic function to fit them, as shown in Equation (8).

_{i}, b

_{i}, c

_{i}, and d

_{i}represent the coefficient terms in the curve equation. When fitting a trajectory, it is required that the junction points be continuous and smooth, so the first and second derivatives of a cubic spline curve should also be continuous. For ease of illustration, suppose we need to fit three points P

_{1}(x

_{1}, y

_{1}), P

_{2}(x

_{2}, y

_{2}), and P

_{3}(x

_{3}, y

_{3}), marking the cubic function between P

_{1}and P

_{2}as J and the cubic function between P

_{2}and P

_{3}as K, as shown in Equation (9).

_{1}and P

_{2}are on the curve J, and points P

_{2}and P

_{3}are on the curve K. This relationship is shown in Equation (10).

_{2}are equal. In addition, if the spline is a cubic spline with a free boundary, the second derivatives at the starting point and the ending point are also required to be continuous. Therefore, by calculating the second derivative of Equations (10) and (11), the following is obtained:

_{1}, b

_{1}, c

_{1}, d

_{1}, a

_{2}, b

_{2}, c

_{2}, d

_{2}) of the two cubic splines can be determined through algebraic calculation. However, path-planning involves multiple segments, so matrix equations are often used to solve spline parameters. To calculate the coefficients (a

_{i}, b

_{i}, c

_{i}, d

_{i}) of each spline curve, first calculate the step length h

_{i}between two adjacent nodes, as shown in Equation (12).

_{i}″ = 0) into Equation (13) to obtain the quadratic differential value m

_{i}, thereby deriving the coefficients of the spline curve, as shown in Equation (14).

_{i}≤ x ≤ x

_{i}+ 1 is shown in Equation (15).

_{i}, y

_{i}, z

_{i})} becomes P

_{o}

_{1}= {(t

_{i}, x

_{i})}, P

_{o}

_{2}= {(t

_{i}, y

_{i})}, and P

_{o}

_{3}= {(t

_{i}, z

_{i})}. P

_{o}

_{1}, P

_{o}

_{2}, and P

_{o}

_{3}satisfy the monotonicity of the spline. Set t

_{i}= 1, 2, …, n, do cubic spline interpolation on the three point sets, respectively, and calculate the corresponding (x

_{i}, y

_{i}, z

_{i}), which is the point set after uniform interpolation. By connecting these points, a smooth flight path can be obtained.

## 3. Chimp Optimization Algorithm (ChOA)

#### 3.1. Driving and Chasing the Prey

_{chimp}is the chimp’s position vector. x

_{prey}is the prey’s position vector. t represents the current iteration number. a, m, and c are coefficient vectors. a, m, and c are determined by Equations (18)–(20), respectively.

_{1}and r

_{2}are random vectors within the range of [0, 1]. m is a chaotic vector calculated based on various chaotic maps, which represents the influence of sexual motivation on chimp during hunting.

#### 3.2. Attacking Method

_{Attacker}, d

_{Barrier}, d

_{Chaser}, and d

_{Driver}represent the distance between the four types of chimps and the prey in the current population. x

_{Attacker}, x

_{Barrier}, x

_{Chaser}, and x

_{Driver}are their position vectors relative to the prey. V

_{1}, V

_{2}, V

_{3}, and V

_{4}represent their position update vector. x(t + 1) is the position of the t + 1 generation chimps. a

_{1}~a

_{4}, m

_{1}~m

_{4}, and c

_{1}~c

_{4}are all coefficient vectors. After food satisfaction, the chimps would release hunting responsibilities, meaning they would no longer take on four hunting roles and instead scramble to get food. This chaotic behavior helps prevent the algorithm from falling below its local optimal value. Equation (24) is a mathematical model of this phenomenon.

## 4. Improved Chimp Optimization Algorithm (TRS-ChOA)

#### 4.1. The Differential Evolution

Algorithm 1 (The DE Algorithm) |

1. Generate the initial population x_{i} (i = 1, 2, …, N)2. Evaluate the fitness of each individual in x _{i}3. while (t < T) 4. for i = 1 to N do 5. Select uniform randomly r _{1} ≠ r_{2} ≠ r_{3} ≠ i6. j _{rand} = randint(1, n)7. for j = 1 to d do 8. if randreal _{j} [0, 1) > CR or j == j_{rand} then9. v _{i}(j) = x* (j) + F × (x_{r}_{2} (j) − x_{r}_{3} (j))10. else 11. v _{i}(j) = x_{i}(j)12. end if 13. end for 14. end for 15. Evaluate the offspring v _{i}16. if v _{i} is better than X_{i} then17. Update individual i, x _{i} = v_{i}18. if v _{i} is better than x* then19. Update best individual, x* = v _{i}20. end if 21. end if 22. end while |

_{i}represents the individuals in the population, and d is the dimension of the problem. j

_{rand}represents a random integer between [1, d]. r

_{1}, r

_{2}, and r

_{3}are individuals randomly selected from the population to participate in the mutation process. CR is the crossover probability. Randreal [0, 1) generates a random real number between 0 and 1. x* represents the current best individual. F is the mutation operator, which is generally taken between [0, 2]. x

_{i}(j) is the j-th variable of the i-th individual in the population, and v

_{i}is its offspring. From the pseudocode of the DE algorithm, it can be seen that DE has strong exploration ability but is not good at local exploitation.

_{j}and ub

_{j}represent the lower and upper bounds of the j-th dimension, respectively. x

_{i}(j) represents the j-th dimension of the i-th solution. randreal(0, 1) represents a random number between 0 and 1.

#### 4.2. Improved Reverse Learning

_{1}), refracts at O, and exits along the OQ direction (Angle of refraction θ

_{2}). Equations (27) and (28) can be obtained from the geometric relationship in Figure 2.

_{i}and l

_{i}represent the i-th dimensional vector of the upper and lower bounds, respectively. At the later stage of the iteration, due to the concentrated distribution of particles near high-quality solutions, it may be difficult for the algorithm to find the global optimal solution. Therefore, hyper-parametric ω is introduced. It can adjust adaptively according to different iteration stages to increase the randomness of the solution, so as to improve the ability of the algorithm to escape from local optima. The improvement is shown in Equation (31).

#### 4.3. Similarity Preference Weight

_{11}, x

_{12}, x

_{13}, …, x

_{1n}) and b = (x

_{21}, x

_{22}, x

_{23}, …, x

_{2n}) are shown in Equation (33).

_{1}, λ

_{2}, λ

_{3}, and λ

_{4}represent the current chimp’s preference weights for attacker, driver, barrier, and chaser, respectively. d represents the similarity value calculated from Equation (33).

#### 4.4. TRS-ChOA Pseudocode

Algorithm 2 (TRS-ChOA Algorithm) |

1. Generate the initial population x_{i} (i = 1, 2, …, N)2. Initialize f, m, a and c |

3. Divide chimps randomly into independent groups 4. Calculate the fitness of each chimp 5. x _{Attacker} = the best search agent6. x _{Chaser} = the second-best search agent7. x _{Barrier} = the third-best search agent8. x _{Driver} = the fourth-best search agent |

9. while (t < T) 10. for i = 1 to N do 11. Extract the chimp’s group 12. Use its group strategy to update f, m, c, a and d |

13. Select uniform randomly r_{1} ≠ r_{2} ≠ i14. Update α by the Equation (25), j _{rand} = randint(1,n), p = randreal(0,1), μ = randreal(0,1)15. for j = 1 to d do 16. if p ≤ α then 17. if randreal _{j}[0,1) ≤ CR or j == j_{rand} then18. v _{i}(j) = randchoice{x_{Attacker} (j), x_{Chaser} (j), x_{Barrier} (j), x_{Driver} (j)} + F × (x_{r}_{1}(j) – x_{r}_{2}(j))19. else 20. v _{i}(j) = Chaotic_value21. end if 22. else if p > α then 23. if randreal _{j} [0,1) ≤ 0.5 then24. v _{i}(j)= x_{Attacker} (j) – a × d25. else 26. Update the position of the current search agent using the Equation (34) 27. end if 28. end if 29. end for 30. end for 31. Calculate the reverse position of each chimp by Equation (31) 32. Update high-quality individuals by Equation (32) 33. Ranking chimp individuals by fitness value |

34. Update x_{Attacker}, x_{Driver}, x_{Barrier}, x_{Chaser}35. t = t + 1 36. end while 37. return x _{Attacker} |

#### 4.5. Time Complexity Analysis of TRS-ChOA

- (1)
- The time complexity after combining with differential evolution is represented as O (N × d), so the time complexity of the algorithm becomes O (N × d × T + N × d) = O (N × d × T) after it is introduced;
- (2)
- The time complexity of using improved reverse learning to update the position of the population is O (N × d × T), However, this is a juxtaposed loop, so the time complexity of the algorithm is O (N × d × T + N × d × T) = O (N × d × T).
- (3)
- Assuming that the time required to introduce the similarity preference weight is t, then the time complexity of the algorithm is O (N × d × T + t) = O (N × d × T)

## 5. TRS-ChOA Optimized Performance Test

#### 5.1. Benchmark Function Test

_{1}–f

_{6}are single module functions, which are mainly used to study the convergence speed and accuracy of the algorithm. f

_{7}–f

_{13}are multi-modal functions used to evaluate the algorithm’s exploration ability and its ability to avoid local optima. In the experiment, the dimensions of the test functions are d = 30/500/1000 to verify the ability of the algorithm to handle low- and high-dimensional problems. To verify the effectiveness of the three improvement strategies proposed in Section 4, the ChOA that integrates the differential evolution algorithm, improves reverse learning, and similarity preference weights are recorded as TChOA, RChOA, and SchOA, respectively, and they are tested together with ChOA and TRS-ChOA. This is an ablation study. To ensure the fairness of the experiment, basic parameters were set uniformly during the experiment: population size N = 30, maximum number of iterations T = 500. After each algorithm runs 50 times independently, the mean value and standard deviation of the results are recorded. The specific experimental results are shown in Table 3.

_{1}, f

_{2}, f

_{3}, f

_{4}, f

_{9}, and f

_{11}, while ChOA does not obtain the global optima on any of the 13 functions. On the multi-modal function f

_{9}, RChOA and SChOA can find theoretical optima at 30 and 500 dimensions, and have higher accuracy than ChOA at 1000 dimensions, indicating that reverse learning strategies and similarity preference weights can help the algorithm escape local optima. TChOA performs better than ChOA on functions f

_{1}, f

_{2}, f

_{3}, f

_{4}, f

_{5}, f

_{6}, and f

_{10}. Most of these functions are single-modal functions, which indicates that The differential evolution can improve the optimization ability of the algorithm on single-mode problems but is not effective in dealing with multi-mode problems. Although TRS-ChOA cannot find the theoretical optimal value for all functions, its optimization results can be close to the theoretical extreme value. For example, on function f

_{8}, TRS-ChOA can converge to −12,567.28, which is the closest theoretical value among all algorithms. It can be concluded that TRS-ChOA exhibits stronger search performance than ChOA in both low-dimensional and high-dimensional functions and is an optimization algorithm with better stability and robustness.

_{1}–f

_{13}. The dimensions of the functions are d = 500, the population size is N = 30, and the maximum iteration number is T = 500. The specific results are shown in Figure 3. From the convergence curves of functions f

_{1}, f

_{2}, f

_{3}, f

_{4}, f

_{9}, f

_{10}, and f

_{11}, it can be seen that compared with the other four algorithms, TRS-ChOA has a faster convergence speed, while ChOA has the slowest optimization speed, indicating that the three improvement strategies enhance the performance of ChOA to varying degrees. From the convergence curves of functions f

_{5}, f

_{7}, f

_{8}, and f

_{12}, it can be found that TChOA, RChOA, SChOA, and TRS-ChOA have higher optimization accuracy than ChOA while guaranteeing the optimization speed, which indicates that the three improvement strategies increase the population diversity of ChOA and improve the ability of the algorithm to escape from local optima. According to the above analysis, TRS-ChOA has a faster convergence rate in the optimization process of benchmark functions and can escape in time to improve the accuracy of optimization when falling into local extremes. In short, it is a better algorithm for solving global optimization problems.

#### 5.2. Wilcoxon Rank-Sum Test

_{6}, the p value of RChOA is 1.12 × 10

^{−13}, indicating that there is a significant difference between RChOA and TRS-ChOA, and RChOA demonstrates stronger optimization performance on f

_{6}than TRS-ChOA. NaN indicates that the difference is not significant, i.e., the optimization performance of the two algorithms is equivalent. In most cases, the value of p is less than 5%, indicating that the optimization performance of TRS-ChOA on benchmark functions is significantly superior to the other four algorithms.

#### 5.3. CEC2017 Function Test

## 6. UAV Path Planning Test

#### 6.1. Parameter Settings

#### 6.2. Simulation Experiment and Results

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Fan, B.; Li, Y.; Zhang, R.; Fu, Q. Review on the technological development and application of UAV systems. Chin. J. Electron.
**2020**, 29, 199–207. [Google Scholar] [CrossRef] - Aggarwal, S.; Kumar, N. Path planning techniques for unmanned aerial vehicles: A review, solutions, and challenges. Comput. Commun.
**2020**, 149, 270–299. [Google Scholar] [CrossRef] - Ren, H.; Zhao, Y.; Xiao, W.; Hu, Z. A review of UAV monitoring in mining areas: Current status and future perspectives. Int. J. Coal Sci. Technol.
**2019**, 6, 320–333. [Google Scholar] [CrossRef] [Green Version] - Zhao, Y.; Zheng, Z.; Liu, Y. Survey on computational-intelligence-based UAV path planning. Knowl.-Based Syst.
**2018**, 158, 54–64. [Google Scholar] [CrossRef] - Shin, J.-J.; Bang, H. UAV path planning under dynamic threats using an improved PSO algorithm. Int. J. Aerosp. Eng.
**2020**, 2020, 8820284. [Google Scholar] [CrossRef] - Liu, W.; Zheng, Z.; Cai, K.-Y. Bi-level programming based real-time path planning for unmanned aerial vehicles. Knowl.-Based Syst.
**2013**, 44, 34–47. [Google Scholar] [CrossRef] - Lu, N.; Zhou, Y.; Shi, C.; Cheng, N.; Cai, L.; Li, B. Planning while flying: A measurement-aided dynamic planning of drone small cells. IEEE Internet Things J.
**2018**, 6, 2693–2705. [Google Scholar] [CrossRef] - Lluvia, I.; Lazkano, E.; Ansuategi, A. Active mapping and robot exploration: A survey. Sensors
**2021**, 21, 2445. [Google Scholar] [CrossRef] - Munoz, P.; Rodriguez-Moreno, M. Improving efficiency in any-angle path-planning algorithms. In Proceedings of the 2012 6th IEEE International Conference Intelligent Systems, Sofia, Bulgaria, 6–8 September 2012; IEEE: Piscataway, NJ, USA, 2012; pp. 213–218. [Google Scholar]
- Choset, H.; Lynch, K.M.; Hutchinson, S.; Kantor, G.A.; Burgard, W. Principles of Robot Motion: Theory, Algorithms, and Implementations; MIT Press: Cambridge, MA, USA, 2005. [Google Scholar]
- Kim, J.; Kim, S.; Choo, Y. Stealth path planning for a high speed torpedo-shaped autonomous underwater vehicle to approach a target ship. Cyber-Phys. Syst.
**2018**, 4, 1–16. [Google Scholar] [CrossRef] - Pettie, S. A new approach to all-pairs shortest paths on real-weighted graphs. Theor. Comput. Sci.
**2004**, 312, 47–74. [Google Scholar] [CrossRef] [Green Version] - Nash, A.; Daniel, K.; Koenig, S.; Felner, A. Theta*: Any-angle path planning on grids. AAAI
**2007**, 7, 1177–1183. [Google Scholar] - Kim, J. Fast Path Planning of Autonomous Vehicles in 3D Environments. Appl. Sci.
**2022**, 12, 4014. [Google Scholar] [CrossRef] - Fulcher, J. Computational intelligence: An introduction. In Computational Intelligence: A Compendium; Springer: Berlin/Heidelberg, Germany, 2008; pp. 3–78. [Google Scholar]
- Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the ICNN′95-International Conference on Neural Networks, Perth, WA, Australia, 27 November–1 December 1995; IEEE: Piscataway, NJ, USA, 1995; pp. 1942–1948. [Google Scholar]
- Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef] [Green Version] - Yang, X.-S.; Deb, S. Cuckoo search via Lévy flights. In Proceedings of the 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC), Coimbatore, India, 9–11 December 2009; IEEE: Piscataway, NJ, USA, 2009; pp. 210–214. [Google Scholar]
- Xue, J.; Shen, B. A novel swarm intelligence optimization approach: Sparrow search algorithm. Syst. Sci. Control Eng.
**2020**, 8, 22–34. [Google Scholar] [CrossRef] - Mirjalili, S.; Lewis, A. The whale optimization algorithm. Adv. Eng. Softw.
**2016**, 95, 51–67. [Google Scholar] [CrossRef] - Mirjalili, S. The ant lion optimizer. Adv. Eng. Softw.
**2015**, 83, 80–98. [Google Scholar] [CrossRef] - Kaur, S.; Awasthi, L.K.; Sangal, A.; Dhiman, G. Tunicate Swarm Algorithm: A new bio-inspired based metaheuristic paradigm for global optimization. Eng. Appl. Artif. Intell.
**2020**, 90, 103541. [Google Scholar] [CrossRef] - Poudel, S.; Arafat, M.Y.; Moh, S. Bio-Inspired Optimization-Based Path Planning Algorithms in Unmanned Aerial Vehicles: A Survey. Sensors
**2023**, 23, 3051. [Google Scholar] [CrossRef] - Roberge, V.; Tarbouchi, M.; Labonté, G. Comparison of parallel genetic algorithm and particle swarm optimization for real-time UAV path planning. IEEE Trans. Ind. Inform.
**2012**, 9, 132–141. [Google Scholar] [CrossRef] - Wen, X.; Ruan, Y.; Li, Y.; Xia, H.; Zhang, R.; Wang, C.; Liu, W.; Jiang, X. Improved genetic algorithm based 3-D deployment of UAVs. J. Commun. Netw.
**2022**, 24, 223–231. [Google Scholar] [CrossRef] - Guan, Y.; Gao, M.; Bai, Y. Double-ant colony based UAV path planning algorithm. In Proceedings of the 2019 11th International Conference on Machine Learning and Computing, Zhuhai, China, 22–24 February 2019; pp. 258–262. [Google Scholar]
- Chai, X.; Zheng, Z.; Xiao, J.; Yan, L.; Qu, B.; Wen, P.; Wang, H.; Zhou, Y.; Sun, H. Multi-strategy fusion differential evolution algorithm for UAV path planning in complex environment. Aerosp. Sci. Technol.
**2022**, 121, 107287. [Google Scholar] [CrossRef] - Zhang, R.; Li, S.; Ding, Y.; Qin, X.; Xia, Q. UAV Path Planning Algorithm Based on Improved Harris Hawks Optimization. Sensors
**2022**, 22, 5232. [Google Scholar] [CrossRef] [PubMed] - Ji, Y.; Zhao, X.; Hao, J. A novel UAV path planning algorithm based on double-dynamic biogeography-based learning particle swarm optimization. Mob. Inf. Syst.
**2022**, 2022, 8519708. [Google Scholar] [CrossRef] - Qu, C.; Gai, W.; Zhang, J.; Zhong, M. A novel hybrid grey wolf optimizer algorithm for unmanned aerial vehicle (UAV) path planning. Knowl.-Based Syst.
**2020**, 194, 105530. [Google Scholar] [CrossRef] - Qu, C.; Gai, W.; Zhong, M.; Zhang, J. A novel reinforcement learning based grey wolf optimizer algorithm for unmanned aerial vehicles (UAVs) path planning. Appl. Soft Comput.
**2020**, 89, 106099. [Google Scholar] [CrossRef] - Yu, X.; Li, C.; Zhou, J. A constrained differential evolution algorithm to solve UAV path planning in disaster scenarios. Knowl.-Based Syst.
**2020**, 204, 106209. [Google Scholar] [CrossRef] - Jiang, W.; Lyu, Y.; Li, Y.; Guo, Y.; Zhang, W. UAV path planning and collision avoidance in 3D environments based on POMPD and improved grey wolf optimizer. Aerosp. Sci. Technol.
**2022**, 121, 107314. [Google Scholar] [CrossRef] - Du, N.; Zhou, Y.; Deng, W.; Luo, Q. Improved chimp optimization algorithm for three-dimensional path planning problem. Multimed. Tools Appl.
**2022**, 81, 27397–27422. [Google Scholar] [CrossRef] - Khishe, M.; Mosavi, M.R. Chimp optimization algorithm. Expert Syst. Appl.
**2020**, 149, 113338. [Google Scholar] [CrossRef] - Kaur, M.; Kaur, R.; Singh, N.; Dhiman, G. Schoa: A newly fusion of sine and cosine with chimp optimization algorithm for hls of datapaths in digital filters and engineering applications. Eng. Comput.
**2021**, 38, 975–1003. [Google Scholar] [CrossRef] - Hu, T.; Khishe, M.; Mohammadi, M.; Parvizi, G.-R.; Karim, S.H.T.; Rashid, T.A. Real-time COVID-19 diagnosis from X-Ray images using deep CNN and extreme learning machines stabilized by chimp optimization algorithm. Biomed. Signal Process Control
**2021**, 68, 102764. [Google Scholar] [CrossRef] [PubMed] - Houssein, E.H.; Emam, M.M.; Ali, A.A. An efficient multilevel thresholding segmentation method for thermography breast cancer imaging based on improved chimp optimization algorithm. Expert Syst. Appl.
**2021**, 185, 115651. [Google Scholar] [CrossRef] - Thompson, S.E.; Patel, R.V. Formulation of joint trajectories for industrial robots using B-splines. IEEE Trans. Ind. Electron.
**1987**, IE-34, 192–199. [Google Scholar] [CrossRef] - Tisdale, J.; Kim, Z.; Hedrick, J.K. Autonomous UAV path planning and estimation. IEEE Robot. Autom. Mag.
**2009**, 16, 35–42. [Google Scholar] [CrossRef] - Lv, Z.; Yang, L.; He, Y.; Liu, Z.; Han, Z. 3D environment modeling with height dimension reduction and path planning for UAV. In Proceedings of the 2017 9th International Conference on Modelling, Identification and Control (ICMIC), Kunming, China, 10–12 July 2017; IEEE: Piscataway, NJ, USA, 2017; pp. 734–739. [Google Scholar]
- Hussain, A.; Muhammad, Y.S.; Nauman Sajid, M.; Hussain, I.; Mohamd Shoukry, A.; Gani, S. Genetic algorithm for traveling salesman problem with modified cycle crossover operator. Comput. Intell. Neurosci.
**2017**, 2017, 7430125. [Google Scholar] [CrossRef] - Ergezer, H.; Leblebicioğlu, K. 3D path planning for multiple UAVs for maximum information collection. J. Intell. Robot. Syst.
**2014**, 73, 737–762. [Google Scholar] [CrossRef] - Besada-Portas, E.; de la Torre, L.; de la Cruz, J.M.; de Andrés-Toro, B. Evolutionary trajectory planner for multiple UAVs in realistic scenarios. IEEE Trans. Robot.
**2010**, 26, 619–634. [Google Scholar] [CrossRef] - McKinley, S.; Levine, M. Cubic spline interpolation. Coll. Redw.
**1998**, 45, 1049–1060. [Google Scholar] - Brest, J.; Zumer, V.; Maucec, M.S. Self-adaptive differential evolution algorithm in constrained real-parameter optimization. In Proceedings of the 2006 IEEE international conference on evolutionary computation, Vancouver, BC, Canada, 16–21 July 2006; IEEE: Piscataway, NJ, USA, 2006; pp. 215–222. [Google Scholar]
- Gong, W.; Cai, Z.; Ling, C.X. DE/BBO: A hybrid differential evolution with biogeography-based optimization for global numerical optimization. Soft Comput.
**2010**, 15, 645–665. [Google Scholar] [CrossRef] [Green Version] - Wu, G.; Mallipeddi, R.; Suganthan, P.N. Problem Definitions and Evaluation Criteria for the CEC 2017 Competition on Constrained Real-Parameter Optimization; Technical Report; National University of Defense Technology: Changsha, China; Kyungpook National University: Daegu, Republic of Korea; Nanyang Technological University: Singapore, 2017. [Google Scholar]
- Suganthan, P.N.; Hansen, N.; Liang, J.J.; Deb, K.; Auger, A. Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL Rep.
**2005**, 2005005, 2005. [Google Scholar] - Derrac, J.; García, S.; Molina, D.; Herrera, F. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol. Comput.
**2011**, 1, 3–18. [Google Scholar] [CrossRef]

**Figure 3.**Convergence curve of TRS − ChOA and other algorithms on the benchmark test function. (

**a**) Convergence curve of f

_{1}. (

**b**) Convergence curve of f

_{2}. (

**c**) Convergence curve of f

_{3}. (

**d**) Convergence curve of f

_{4}. (

**e**) Convergence curve of f

_{5}. (

**f**) Convergence curve of f

_{6}. (

**g**) Convergence curve of f

_{7}. (

**h**) Convergence curve of f

_{8}. (

**i**) Convergence curve of f

_{9}. (

**j**) Convergence curve of f

_{10}. (

**k**) Convergence curve of f

_{11}. (

**l**) Convergence curve of f

_{12}. (

**m**) Convergence curve of f

_{13}.

**Figure 4.**UAV trajectory planning environment model. (

**a**) Environment 1. (

**b**) Environment 2. (

**c**) Environment 3.

**Figure 5.**UAV-simulated flight path. (

**a**) Environment 1 route, (

**b**) environment 2 route, (

**c**) environment 3 route.

**Figure 6.**UAV simulated flight path (vertical view). (

**a**) Environment 1 route (vertical view), (

**b**) environment 2 route (vertical view), (

**c**) environment 3 route (vertical view).

**Figure 7.**Convergence curve of the trajectory loss function. (

**a**) Convergence curve of the loss function in environment 1. (

**b**) Convergence curve of the loss function in environment 2. (

**c**) Convergence curve of the loss function in environment 3.

Algorithm | Parameters Setting | Reference |
---|---|---|

GWO | r_{1} ∈ [0, 1], r_{2} ∈ [0, 1] | [17] |

SSA | proportion of discoverers: 20% proportion of scouter: 10% alert threshold: 0.7 | [19] |

WOA | b = 1, r_{1} ∈ [0, 1], r_{2} ∈ [0, 1],l ∈ [−1, 1], p ∈ [0, 1] | [20] |

ALO | w = 1, t ≤ 0.1 T w = 2, t > 0.1 T w = 3, t > 0.5 T w = 4, t > 0.75 T w = 5, t > 0.9 T w = 6, t > 0.95 T | [21] |

ChOA | r_{1} ∈ [0, 1], r_{2} ∈ [0, 1], m = chaos (3,1,1) | [35] |

TRS-ChOA | F ∈ [0, 1], CR = 0.1, k ∈ [0, 1], σ = 2.5 | Section 4 of this article |

Fun No. | Name | Range | Dim | Optimal Value | Function Type |
---|---|---|---|---|---|

f_{1} | Sphere Function | [−100, 100] | 30, 500, 1000 | 0 | Single-modal |

f_{2} | Schwefel’s problem 2.22 | [−10, 10] | 30, 500, 1000 | 0 | Single-modal |

f_{3} | Schwefel’s problem 1.2 | [−100, 100] | 30, 500, 1000 | 0 | Single-modal |

f_{4} | Schwefel’s problem 2.21 | [−100, 100] | 30, 500, 1000 | 0 | Single-modal |

f_{5} | Generalized Rosenbrock’s Function | [−30, 30] | 30, 500, 1000 | 0 | Single-modal |

f_{6} | Step Function | [−100, 100] | 30, 500, 1000 | 0 | Single-modal |

f_{7} | Quartic Function | [−1.28, 1.28] | 30, 500, 1000 | 0 | Single-modal |

f_{8} | Generalized Schwefel’s problem 2.26 | [−500, 500] | 30, 500, 1000 | 12,569.5 | Multi-modal |

f_{9} | Generalized Rastrigin’s Function. | [−5.12, 5.12] | 30, 500, 1000 | 0 | Multi-modal |

f_{10} | Ackley’sFunction | [−32, 32] | 30, 500, 1000 | 0 | Multi-modal |

f_{11} | Generalized Criewank’s Function | [−600, 600] | 30, 500, 1000 | 0 | Multi-modal |

f_{12} | Generalized Penalized Function 1 | [−50, 50] | 30, 500, 1000 | 0 | Fixed multi-modal |

f_{13} | Generalized Penalized Function 2 | [−50, 50] | 30, 500, 1000 | 0 | Fixed multi-modal |

Fun No. | Dim | ChOA | TChOA | RChOA | SChOA | TRS-ChOA | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | ||

f_{1} | d = 30 | 1.69 × 10^{−21} | 4.76 × 10^{−21} | 1.70 × 10^{−215} | 4.95 × 10^{−214} | 1.46 × 10^{−243} | 1.53 × 10^{−240} | 1.76 × 10^{−300} | 8.66 × 10^{−298} | 0 | 0 |

d = 500 | 2.03 × 10^{−22} | 3.61 × 10^{−22} | 5.69 × 10^{−168} | 6.42 × 10^{−174} | 3.58 × 10^{−233} | 1.76 × 10^{−230} | 2.20 × 10^{−301} | 6.33 × 10^{−300} | 0 | 0 | |

d = 1000 | 7.84 × 10^{−16} | 4.38 × 10^{−17} | 3.77 × 10^{−164} | 4.29 × 10^{−150} | 1.25 × 10^{−200} | 9.23 × 10^{−203} | 6.53 × 10^{−276} | 4.85 × 10^{−280} | 0 | 0 | |

f_{2} | d = 30 | 2.13 × 10^{−15} | 1.41 × 10^{−15} | 1.13 × 10^{−78} | 5.74 × 10^{−77} | 3.38 × 10^{−112} | 3.14 × 10^{−103} | 4.49 × 10^{−326} | 2.47 × 10^{−341} | 0 | 0 |

d = 500 | 3.74 × 10^{−15} | 1.89 × 10^{−14} | 4.81 × 10^{−82} | 5.11 × 10^{−85} | 6.52 × 10^{−105} | 4.32 × 10^{−105} | 7.55 × 10^{−317} | 4.35 × 10^{−377} | 0 | 0 | |

d = 1000 | 6.51 × 10^{−15} | 7.03 × 10^{-−9} | 7.64 × 10^{−77} | 3.58 × 10^{−77} | 2.78 × 10^{−67} | 7.53 × 10^{−69} | 4.89 × 10^{−300} | 3.14 × 10^{−305} | 0 | 0 | |

f_{3} | d = 30 | 5.15 × 10^{1} | 2.90 × 10^{1} | 3.50 × 10^{−167} | 1.22 × 10^{−165} | 4.33 × 10^{−214} | 8.64 × 10^{−213} | 0 | 0 | 0 | 0 |

d = 500 | 2.82 × 10^{1} | 7.26 × 10^{1} | 2.21 × 10^{−154} | 1.97 × 10^{−148} | 4.67 × 10^{−212} | 6.38 × 10^{−213} | 6.47 × 10^{−303} | 4.54 × 10^{−302} | 0 | 0 | |

d = 1000 | 4.15 × 10^{1} | 3.91 × 10^{1} | 5.30 × 10^{−144} | 1.79 × 10^{−139} | 3.55 × 10^{−210} | 7.24 × 10^{−218} | 6.52 × 10^{−170} | 6.09 × 10^{−175} | 0 | 0 | |

f_{4} | d = 30 | 4.92 × 10^{−1} | 2.34 × 10^{−1} | 1.36 × 10^{−81} | 1.93 × 10^{−80} | 3.72 × 10^{−141} | 3.69 × 10^{−151} | 3.36 × 10^{−184} | 2.36 × 10^{−185} | 0 | 0 |

d = 500 | 4.57 × 10^{−1} | 1.99 × 10^{−1} | 2.14 × 10^{−56} | 4.31 × 10^{−67} | 3.11 × 10^{−110} | 2.76 × 10^{−133} | 3.52 × 10^{−217} | 8.12 × 10^{−142} | 0 | 0 | |

d = 1000 | 8.44 × 10^{0} | 7.34 × 10^{0} | 3.42 × 10^{−40} | 2.66 × 10^{−42} | 1.52 × 10^{−86} | 3.74 × 10^{−76} | 4.16 × 10^{−182} | 3.81 × 10^{−163} | 0 | 0 | |

f_{5} | d = 30 | 2.90 × 10^{1} | 4.25 × 10^{1} | 1.33 × 10^{1} | 1.04 × 10^{1} | 2.82 × 10^{−2} | 1.76 × 10^{−1} | 2.88 × 10^{1} | 1.29 × 10^{1} | 2.64 × 10^{−4} | 8.62 × 10^{−4} |

d = 500 | 4.36 × 10^{1} | 1.69 × 10^{1} | 4.22 × 10^{2} | 1.67 × 10^{1} | 4.66 × 10^{−1} | 1.80 × 10^{−1} | 2.69 × 10^{1} | 3.58 × 10^{0} | 2.15 × 10^{−5} | 8.33 × 10^{−5} | |

d = 1000 | 8.92 × 10^{2} | 4.53 × 10^{2} | 1.73 × 10^{2} | 6.51 × 10^{1} | 2.06 × 10^{1} | 4.69 × 10^{0} | 1.18 × 10^{2} | 2.39 × 10^{1} | 3.05 × 10^{−5} | 6.69 × 10^{−4} | |

f_{6} | d = 30 | 3.53 × 10^{1} | 3.05 × 10^{0} | 8.24 × 10^{−4} | 5.56 × 10^{−4} | 3.62 × 10^{−5} | 1.51 × 10^{−5} | 1.33 × 10^{0} | 7.39 × 10^{−1} | 1.37 × 10^{−4} | 6.12 × 10^{−4} |

d = 500 | 4.31 × 10^{0} | 4.81 × 10^{0} | 7.96 × 10^{−2} | 4.81 × 10^{−2} | 3.69 × 10^{−5} | 1.71 × 10^{−4} | 1.57 × 10^{0} | 6.63 × 10^{−1} | 1.47 × 10^{−2} | 5.77 × 10^{−3} | |

d = 1000 | 6.37 × 10^{1} | 4.22 × 10^{1} | 1.67 × 10^{−3} | 4.10 × 10^{−2} | 9.04 × 10^{−4} | 2.58 × 10^{−2} | 7.33 × 10^{1} | 1.09 × 10^{0} | 1.55 × 10^{−2} | 7.04 × 10^{−2} | |

f_{7} | d = 30 | 1.82 × 10^{−3} | 6.88 × 10^{−4} | 2.55 × 10^{−2} | 3.41 × 10^{−2} | 2.13 × 10^{−3} | 1.46 × 10^{−3} | 5.38 × 10^{−4} | 1.91 × 10^{−4} | 4.77 × 10^{−7} | 9.28 × 10^{−8} |

d = 500 | 2.03 × 10^{−3} | 5.16 × 10^{−4} | 2.18 × 10^{−2} | 3.63 × 10^{−2} | 4.30 × 10^{−4} | 3.87 × 10^{−4} | 4.01 × 10^{−3} | 3.19 × 10^{−3} | 4.35 × 10^{−6} | 7.81 × 10^{−8} | |

d = 1000 | 1.66 × 10^{−2} | 6.09 × 10^{−2} | 8.25 × 10^{−1} | 4.52 × 10^{−1} | 1.94 × 10^{−3} | 8.33 × 10^{−2} | 6.13 × 10^{−3} | 5.46 × 10^{−1} | 5.72 × 10^{−6} | 3.20 × 10^{−5} | |

f_{8} | d = 30 | −5734.36 | 8.95 × 10^{−9} | −5498.63 | 3.21 × 10^{2} | −10340.06 | 2.21 × 10^{3} | −8334.31 | 6.26 × 10^{2} | −12,567.28 | 2.53 × 10^{−10} |

d = 500 | −5529.71 | 6.34 × 10^{−8} | −5736.44 | 2.73 × 10^{2} | −10649.63 | 2.08 × 10^{2} | −8221.50 | 5.90 × 10^{2} | −12,496.63 | 4.10 × 10^{−10} | |

d = 1000 | −6017.87 | 3.65 × 10^{−5} | −4396.07 | 8.60 × 10^{3} | −8774.59 | 6.77 × 10^{2} | −8005.19 | 2.53 × 10^{3} | −12,195.10 | 2.74 × 10^{−9} | |

f_{9} | d = 30 | 1.37 × 10^{1} | 6.11 × 10^{2} | 8.07 × 10^{1} | 2.88 × 10^{0} | 0 | 0 | 0 | 0 | 0 | 0 |

d = 500 | 1.63 × 10^{1} | 6.40 × 10^{2} | 7.46 × 10^{1} | 3.00 × 10^{0} | 0 | 0 | 0 | 0 | 0 | 0 | |

d = 1000 | 2.70 × 10^{2} | 8.39 × 10^{0} | 3.98 × 10^{2} | 5.14 × 10^{0} | 7.57 × 10^{−279} | 9.41 × 10^{−278} | 7.03 × 10^{−131} | 3.72 × 10^{−110} | 0 | 0 | |

f_{10} | d = 30 | 2.00 × 10^{1} | 9.03 × 10^{−14} | 5.89 × 10^{0} | 3.86 × 10^{−1} | 4.34 × 10^{−15} | 3.97 × 10^{−11} | 1.60 × 10^{−12} | 7.11 × 10^{−16} | 8.88 × 10^{−14} | 0 |

d = 500 | 3.12 × 10^{1} | 6.51 × 10^{−13} | 5.53 × 10^{0} | 4.02 × 10^{−1} | 4.22 × 10^{−12} | 2.67 × 10^{−8} | 3.07 × 10^{−11} | 2.97 × 10^{−15} | 6.93 × 10^{−14} | 0 | |

d = 1000 | 4.73 × 10^{1} | 7.73 × 10^{−4} | 8.14 × 10^{1} | 7.50 × 10^{−2} | 5.96 × 10^{−13} | 7.51 × 10^{−8} | 8.46 × 10^{−7} | 5.29 × 10^{−10} | 5.00 × 10^{−13} | 1.37 × 10^{−12} | |

f_{11} | d = 30 | 4.42 × 10^{−2} | 3.97 × 10^{−12} | 4.45 × 10^{−1} | 3.24 × 10^{−2} | 1.61 × 10^{−216} | 1.62 × 10^{−223} | 0 | 0 | 0 | 0 |

d = 500 | 4.90 × 10^{−3} | 3.62 × 10^{−11} | 4.73 × 10^{−2} | 1.88 × 10^{−2} | 0 | 0 | 0 | 0 | 0 | 0 | |

d = 1000 | 2.26 × 10^{−3} | 4.74 × 10^{−11} | 5.30 × 10^{−2} | 4.09 × 10^{−2} | 2.73 × 10^{−10} | 2.11 × 10^{−10} | 6.16 × 10^{−14} | 8.40 × 10^{−15} | 3.71 × 10^{−13} | 6.11 × 10^{−16} | |

f_{12} | d = 30 | 4.68 × 10^{−1} | 1.60 × 10^{−11} | 1.41 × 10^{1} | 3.17 × 10^{0} | 9.29 × 10^{−2} | 7.76 × 10^{−2} | 9.02 × 10^{−3} | 5.12 × 10^{−2} | 6.07 × 10^{−6} | 8.36 × 10^{−13} |

d = 500 | 3.90 × 10^{−1} | 4.25 × 10^{−10} | 2.68 × 10^{1} | 5.33 × 10^{0} | 6.75 × 10^{−2} | 5.77 × 10^{−1} | 4.57 × 10^{−2} | 8.16 × 10^{−2} | 9.12 × 10^{−6} | 6.31 × 10^{−12} | |

d = 1000 | 9.57 × 10^{0} | 7.75 × 10^{−8} | 4.69 × 10^{2} | 8.01 × 10^{1} | 5.10 × 10^{−2} | 9.69 × 10^{−2} | 3.88 × 10^{−3} | 1.90 × 10^{−2} | 5.17 × 10^{−6} | 6.21 × 10^{−12} | |

f_{13} | d = 30 | 2.71 × 10^{0} | 1.66 × 10^{−13} | 2.25 × 10^{1} | 3.37 × 10^{1} | 5.48 × 10^{−1} | 2.82 × 10^{−6} | 2.85 × 10^{−1} | 7.45 × 10^{−9} | 1.93 × 10^{−4} | 5.65 × 10^{−17} |

d = 500 | 6.24 × 10^{0} | 3.57 × 10^{−12} | 3.06 × 10^{1} | 5.65 × 10^{1} | 3.98 × 10^{−1} | 2.17 × 10^{−5} | 1.74 × 10^{0} | 8.61 × 10^{−5} | 2.01 × 10^{−4} | 4.85 × 10^{−15} | |

d = 1000 | 4.82 × 10^{0} | 3.07 × 10^{−10} | 6.44 × 10^{1} | 6.00 × 10^{1} | 4.32 × 10^{−1} | 2.73 × 10^{−1} | 9.38 × 10^{0} | 1.00 × 10^{−3} | 2.13 × 10^{−4} | 9.32 × 10^{−14} |

Fun No. | ChOA (P_{1}) | TChOA (P_{2}) | RChOA (P_{3}) | SChOA (P_{4}) |
---|---|---|---|---|

f_{1} | 8.01 × 10^{−14} | 8.01 × 10^{−14} | 8.01 × 10^{−14} | 8.01 × 10^{−14} |

f_{2} | 1.83 × 10^{−15} | 1.83 × 10^{−15} | 1.83 × 10^{−15} | 1.83 × 10^{−15} |

f_{2} | 3.16 × 10^{−13} | 3.16 × 10^{−13} | 3.16 × 10^{−13} | 3.16 × 10^{−13} |

f_{2} | 3.02 × 10^{−16} | 3.16 × 10^{−13} | 3.16 × 10^{−13} | 3.16 × 10^{−13} |

f_{3} | 2.03 × 10^{−7} | 4.73 × 10^{−12} | 5.80 × 10^{−15} | 1.01 × 10^{−17} |

f_{4} | 1.86 × 10^{−12} | 8.66 × 10^{−14} | 2.64 × 10^{−15} | 1.83 × 10^{−17} |

f_{5} | 3.09 × 10^{−9} | 1.71 × 10^{−10} | 1.83 × 10^{−11} | 1.83 × 10^{−11} |

f_{6} | 7.77 × 10^{−13} | NaN | 1.12 × 10^{−13} | 1.83 × 10^{−13} |

f_{7} | 1.64 × 10^{−14} | 9.53 × 10^{−17} | 7.07 × 10^{−18} | 1.01 × 10^{−17} |

f_{8} | 2.65 × 10^{−18} | 7.08 × 10^{−12} | 0.91 × 10^{−6} | 9.54 × 10^{−18} |

f_{9} | 3.31 × 10^{−20} | 3.31 × 10^{−20} | NaN | 8.97 × 10^{−7} |

f_{10} | 3.31 × 10^{−20} | 3.43 × 10^{−13} | 6.45 × 10^{−15} | NaN |

f_{11} | 3.31 × 10^{−20} | 3.31 × 10^{−20} | 3.27 × 10^{−6} | NaN |

f_{12} | 7.10 × 10^{−9} | 1.46 × 10^{−12} | 7.06 × 10^{−18} | 7.79 × 10^{−12} |

f_{13} | 1.58 × 10^{−12} | 1.62 × 10^{−13} | 4.44 × 10^{−15} | 2.38 × 10^{−10} |

+/=/― | 13/0/0 | 12/1/0 | 11/1/1 | 11/2/0 |

Fun No. | Dim | Function Type | Range | Optimal Value |
---|---|---|---|---|

CEC01 | 10, 50 | UF (Uni-modal Function) | [−100, 100] | 100 |

CEC02 | 10, 50 | UF | [−100, 100] | 200 |

CEC03 | 10, 50 | SMF (Simple Multimodal Functions) | [−100, 100] | 300 |

CEC04 | 10, 50 | SMF | [−100, 100] | 400 |

CEC05 | 10, 50 | SMF | [−100, 100] | 500 |

CEC06 | 10, 50 | SMF | [−100, 100] | 600 |

CEC07 | 10, 50 | SMF | [−100, 100] | 700 |

CEC08 | 10, 50 | SMF | [−100, 100] | 800 |

CEC09 | 10, 50 | SMF | [−100, 100] | 900 |

CEC10 | 10, 50 | HF (Hybrid Function) | [−100, 100] | 1000 |

CEC11 | 10, 50 | HF | [−100, 100] | 1100 |

CEC12 | 10, 50 | HF | [−100, 100] | 1200 |

CEC13 | 10, 50 | HF | [−100, 100] | 1300 |

CEC14 | 10, 50 | HF | [−100, 100] | 1400 |

CEC15 | 10, 50 | HF | [−100, 100] | 1500 |

CEC16 | 10, 50 | HF | [−100, 100] | 1600 |

CEC17 | 10, 50 | HF | [−100, 100] | 1700 |

CEC18 | 10, 50 | HF | [−100, 100] | 1800 |

CEC19 | 10, 50 | HF | [−100, 100] | 1900 |

CEC20 | 10, 50 | CF (Composition Function) | [−100, 100] | 2000 |

CEC21 | 10, 50 | CF | [−100, 100] | 2100 |

CEC22 | 10, 50 | CF | [−100, 100] | 2200 |

CEC23 | 10, 50 | CF | [−100, 100] | 2300 |

CEC24 | 10, 50 | CF | [−100, 100] | 2400 |

CEC25 | 10, 50 | CF | [−100, 100] | 2500 |

CEC26 | 10, 50 | CF | [−100, 100] | 2600 |

CEC27 | 10, 50 | CF | [−100, 100] | 2700 |

CEC28 | 10, 50 | CF | [−100, 100] | 2800 |

CEC29 | 10, 50 | CF | [−100, 100] | 2900 |

Fun No. | Dim | ChOA | SSA | GWO | WOA | TRS-ChOA | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | ||

CEC1 | d = 10 | 2.32 × 10^{2} | 7.39 × 10^{1} | 8.47 × 10^{2} | 2.15 × 10^{2} | 1.36 × 10^{2} | 3.13 × 10^{1} | 5.78 × 10^{2} | 2.62 × 10^{0} | 1.14 × 10^{2} | 1.59 × 10^{0} |

d = 50 | 4.52 × 10^{2} | 2.31 × 10^{2} | 3.62 × 10^{3} | 4.22 × 10^{2} | 2.37 × 10^{2} | 1.16 × 10^{2} | 7.40 × 10^{2} | 1.72 × 10^{2} | 1.32 × 10^{2} | 6.21 × 10^{1} | |

CEC2 | d = 10 | 9.70 × 10^{2} | 5.46 × 10^{0} | 6.53 × 10^{2} | 1.59 × 10^{1} | 3.35 × 10^{2} | 4.61 × 10^{1} | 4.41 × 10^{2} | 3.84 × 10^{−1} | 3.51 × 10^{2} | 4.32 × 10^{0} |

d = 50 | 6.94 × 10^{2} | 2.40 × 10^{2} | 5.39 × 10^{2} | 3.11 × 10^{2} | 4.02 × 10^{2} | 4.11 × 10^{2} | 5.37 × 10^{2} | 3.78 × 10^{1} | 3.77 × 10^{2} | 1.60 × 10^{0} | |

CEC3 | d = 10 | 4.80 × 10^{2} | 1.13 × 10^{2} | 4.85 × 10^{2} | 1.97 × 10^{1} | 3.32 × 10^{2} | 4.44 × 10^{2} | 1.00 × 10^{3} | 6.93 × 10^{2} | 3.28 × 10^{2} | 6.55 × 10^{−1} |

d = 50 | 6.40 × 10^{2} | 3.27 × 10^{2} | 2.66 × 10^{3} | 5.64 × 10^{2} | 5.49 × 10^{2} | 3.13 × 10^{1} | 7.46 × 10^{2} | 2.86 × 10^{2} | 4.09 × 10^{2} | 9.85 × 10^{0} | |

CEC4 | d = 10 | 4.51 × 10^{2} | 4.75 × 10^{−2} | 4.22 × 10^{2} | 9.53 × 10^{−2} | 3.69 × 10^{2} | 1.35 × 10^{1} | 4.36 × 10^{2} | 4.30 × 10^{−2} | 4.02 × 10^{2} | 4.28 × 10^{−2} |

d = 50 | 4.89 × 10^{2} | 5.77 × 10^{−1} | 6.35 × 10^{2} | 4.91 × 10^{−1} | 4.62 × 10^{2} | 4.62 × 10^{1} | 5.76 × 10^{2} | 2.73 × 10^{1} | 4.55 × 10^{2} | 4.58 × 10^{−1} | |

CEC5 | d = 10 | 6.44 × 10^{2} | 1.14 × 10^{0} | 5.14 × 10^{2} | 3.24 × 10^{0} | 5.75 × 10^{2} | 1.33 × 10^{1} | 6.44 × 10^{2} | 1.26 × 10^{0} | 5.26 × 10^{2} | 6.82 × 10^{−1} |

d = 50 | 6.97 × 10^{2} | 4.03 × 10^{0} | 5.96 × 10^{2} | 1.95 × 10^{1} | 5.20 × 10^{2} | 4.67 × 10^{−2} | 5.76 × 10^{2} | 2.73 × 10^{1} | 5.19 × 10^{2} | 2.73 × 10^{−2} | |

CEC6 | d = 10 | 6.37 × 10^{2} | 2.99 × 10^{1} | 6.47 × 10^{2} | 6.21 × 10^{1} | 6.01 × 10^{2} | 1.35 × 10^{−1} | 6.72 × 10^{2} | 6.72 × 10^{−2} | 6.22 × 10^{2} | 1.24 × 10^{0} |

d = 50 | 6.84 × 10^{2} | 3.62 × 10^{1} | 6.72 × 10^{2} | 2.51 × 10^{1} | 6.49 × 10^{2} | 5.23 × 10^{−1} | 6.94 × 10^{2} | 5.26 × 10^{0} | 6.43 × 10^{2} | 7.30 × 10^{0} | |

CEC7 | d = 10 | 7.66 × 10^{2} | 5.39 × 10^{1} | 8.06 × 10^{2} | 3.67 × 10^{0} | 7.15 × 10^{2} | 6.59 × 10^{−3} | 8.21 × 10^{2} | 3.00 × 10^{1} | 7.04 × 10^{2} | 6.39 × 10^{−1} |

d = 50 | 8.04 × 10^{2} | 1.41 × 10^{1} | 2.93 × 10^{3} | 4.80 × 10^{1} | 8.01 × 10^{2} | 3.57 × 10^{0} | 1.53 × 10^{3} | 5.12 × 10^{1} | 7.87 × 10^{2} | 0 | |

CEC8 | d = 10 | 2.41 × 10^{3} | 9.41 × 10^{1} | 1.19 × 10^{3} | 9.13 × 10^{1} | 9.36 × 10^{2} | 5.43 × 10^{1} | 9.77 × 10^{2} | 1.84 × 10^{2} | 9.04 × 10^{2} | 4.32 × 10^{1} |

d = 50 | 1.47 × 10^{3} | 6.27 × 10^{2} | 1.65 × 10^{3} | 2.41 × 10^{2} | 1.79 × 10^{3} | 1.75 × 10^{2} | 1.01 × 10^{3} | 3.17 × 10^{2} | 1.23 × 10^{3} | 8.02 × 10^{1} | |

CEC9 | d = 10 | 1.91 × 10^{3} | 3.72 × 10^{2} | 3.37 × 10^{3} | 8.44 × 10^{1} | 2.85 × 10^{3} | 9.62 × 10^{−1} | 1.38 × 10^{3} | 3.16 × 10^{2} | 9.44 × 10^{3} | 7.06 × 10^{−3} |

d = 50 | 3.34 × 10^{3} | 8.00 × 10^{2} | 4.17 × 10^{3} | 1.94 × 10^{2} | 1.45 × 10^{3} | 1.16 × 10^{1} | 3.93 × 10^{3} | 2.40 × 10^{2} | 9.65 × 10^{3} | 1.00 × 10^{1} | |

CEC10 | d = 10 | 5.34 × 10^{2} | 7.01 × 10^{2} | 2.63 × 10^{3} | 2.61 × 10^{2} | 9.10 × 10^{2} | 8.21 × 10^{0} | 5.20 × 10^{3} | 7.46 × 10^{0} | 1.03 × 10^{3} | 3.13 × 10^{0} |

d = 50 | 7.99 × 10^{2} | 8.96 × 10^{2} | 9.13 × 10^{2} | 1.36 × 10^{2} | 1.81 × 10^{3} | 1.96 × 10^{2} | 3.18 × 10^{3} | 2.17 × 10^{2} | 1.08 × 10^{3} | 4.32 × 10^{0} | |

CEC11 | d = 10 | 1.00 × 10^{3} | 2.85 × 10^{1} | 1.39 × 10^{3} | 3.59 × 10^{1} | 1.13 × 10^{3} | 3.85 × 10^{1} | 1.28 × 10^{3} | 7.62 × 10^{0} | 1.11 × 10^{3} | 2.25 × 10^{0} |

d = 50 | 1.41 × 10^{4} | 4.28 × 10^{2} | 2.77 × 10^{3} | 2.99 × 10^{1} | 4.31 × 10^{3} | 3.38 × 10^{2} | 7.19 × 10^{3} | 2.85 × 10^{2} | 1.30 × 10^{3} | 7.36 × 10^{0} | |

CEC12 | d = 10 | 3.45 × 10^{3} | 2.09 × 10^{1} | 2.07 × 10^{3} | 8.00 × 10^{0} | 2.96 × 10^{3} | 4.45 × 10^{1} | 1.84 × 10^{3} | 4.51 × 10^{2} | 1.54 × 10^{3} | 7.31 × 10^{−1} |

d = 50 | 4.06 × 10^{3} | 5.23 × 10^{1} | 2.62 × 10^{3} | 1.36 × 10^{2} | 1.49 × 10^{3} | 5.27 × 10^{2} | 1.78 × 10^{3} | 2.81 × 10^{2} | 2.19 × 10^{3} | 4.85 × 10^{0} | |

CEC13 | d = 10 | 1.30 × 10^{3} | 3.60 × 10^{1} | 1.42 × 10^{3} | 7.62 × 10^{0} | 1.33 × 10^{3} | 1.57 × 10^{1} | 6.55 × 10^{3} | 1.98 × 10^{3} | 1.35 × 10^{3} | 1.43 × 10^{1} |

d = 50 | 1.01 × 10^{4} | 1.44 × 10^{2} | 2.58 × 10^{3} | 1.01 × 10^{2} | 1.34 × 10^{3} | 1.83 × 10^{2} | 7.01 × 10^{3} | 7.46 × 10^{2} | 1.28 × 10^{3} | 1.20 × 10^{2} | |

CEC14 | d = 10 | 3.79 × 10^{3} | 2.16 × 10^{0} | 1.71 × 10^{3} | 1.56 × 10^{2} | 1.64 × 10^{3} | 2.27 × 10^{2} | 8.84 × 10^{3} | 2.48 × 10^{2} | 1.63 × 10^{3} | 5.57 × 10^{0} |

d = 50 | 2.95 × 10^{3} | 7.02 × 10^{1} | 3.26 × 10^{3} | 3.71 × 10^{2} | 1.89 × 10^{3} | 1.23 × 10^{2} | 1.07 × 10^{4} | 5.05 × 10^{2} | 1.57 × 10^{3} | 8.23 × 10^{0} | |

CEC15 | d = 10 | 2.44 × 10^{3} | 8.15 × 10^{2} | 2.25 × 10^{3} | 5.61 × 10^{2} | 5.49 × 10^{3} | 7.61 × 10^{1} | 1.74 × 10^{3} | 3.46 × 10^{1} | 1.68 × 10^{3} | 2.17 × 10^{1} |

d = 50 | 3.76 × 10^{3} | 1.54 × 10^{2} | 3.77 × 10^{3} | 9.99 × 10^{1} | 6.25 × 10^{3} | 8.09 × 10^{0} | 1.99 × 10^{3} | 2.27 × 10^{2} | 1.39 × 10^{3} | 4.44 × 10^{0} | |

CEC16 | d = 10 | 1.85 × 10^{3} | 5.07 × 10^{1} | 2.50 × 10^{3} | 5.57 × 10^{1} | 1.71 × 10^{3} | 7.33 × 10^{−1} | 1.77 × 10^{3} | 1.65 × 10^{1} | 1.66 × 10^{3} | 6.59 × 10^{−1} |

d = 50 | 1.90 × 10^{3} | 4.39 × 10^{1} | 3.98 × 10^{3} | 4.35 × 10^{1} | 1.61 × 10^{3} | 4.82 × 10^{0} | 5.28 × 10^{3} | 6.92 × 10^{2} | 1.60 × 10^{3} | 4.78 × 10^{0} | |

CEC17 | d = 10 | 2.02 × 10^{3} | 3.18 × 10^{1} | 2.03 × 10^{3} | 5.70 × 10^{1} | 1.73 × 10^{3} | 1.17 × 10^{0} | 1.79 × 10^{3} | 6.83 × 10^{0} | 1.72 × 10^{3} | 1.08 × 10^{0} |

d = 50 | 1.87 × 10^{3} | 4.67 × 10^{1} | 3.77 × 10^{3} | 9.92 × 10^{1} | 3.13 × 10^{3} | 5.60 × 10^{2} | 4.46 × 10^{3} | 4.30 × 10^{2} | 1.83 × 10^{3} | 6.85 × 10^{1} | |

CEC18 | d = 10 | 3.23 × 10^{3} | 1.78 × 10^{1} | 1.90 × 10^{3} | 1.06 × 10^{2} | 1.86 × 10^{3} | 3.27 × 10^{2} | 1.54 × 10^{3} | 2.12 × 10^{2} | 1.88 × 10^{3} | 1.25 × 10^{1} |

d = 50 | 1.45 × 10^{3} | 5.14 × 10^{2} | 3.26 × 10^{3} | 2.29 × 10^{1} | 2.39 × 10^{3} | 4.38 × 10^{2} | 2.36 × 10^{3} | 7.03 × 10^{2} | 2.06 × 10^{3} | 1.73 × 10^{1} | |

CEC19 | d = 10 | 1.46 × 10^{3} | 3.69 × 10^{2} | 4.24 × 10^{3} | 4.50 × 10^{2} | 3.12 × 10^{3} | 1.05 × 10^{2} | 3.18 × 10^{3} | 2.37 × 10^{2} | 2.27 × 10^{3} | 1.01 × 10^{2} |

d = 50 | 3.36 × 10^{3} | 4.05 × 10^{2} | 5.38 × 10^{3} | 7.82 × 10^{1} | 2.76 × 10^{3} | 9.47 × 10^{0} | 3.37 × 10^{3} | 9.15 × 10^{1} | 2.69 × 10^{3} | 5.05 × 10^{0} | |

CEC20 | d = 10 | 4.06 × 10^{3} | 2.70 × 10^{0} | 2.41 × 10^{3} | 2.55 × 10^{2} | 2.63 × 10^{3} | 2.94 × 10^{1} | 1.53 × 10^{4} | 2.40 × 10^{3} | 2.28 × 10^{3} | 2.31 × 10^{0} |

d = 50 | 3.00 × 10^{3} | 7.18 × 10^{1} | 5.75 × 10^{3} | 9.57 × 10^{0} | 4.79 × 10^{3} | 2.70 × 10^{2} | 5.18 × 10^{3} | 2.17 × 10^{2} | 2.89 × 10^{3} | 8.86 × 10^{0} | |

CEC21 | d = 10 | 2.33 × 10^{3} | 1.28 × 10^{2} | 2.37 × 10^{3} | 4.09 × 10^{2} | 2.10 × 10^{3} | 3.12 × 10^{−1} | 2.18 × 10^{3} | 9.06 × 10^{0} | 2.03 × 10^{3} | 5.31 × 10^{0} |

d = 50 | 2.94 × 10^{3} | 5.05 × 10^{1} | 2.63 × 10^{3} | 5.33 × 10^{2} | 2.64 × 10^{3} | 1.50 × 10^{2} | 2.88 × 10^{3} | 3.54 × 10^{1} | 2.15 × 10^{3} | 3.51 × 10^{1} | |

CEC22 | d = 10 | 2.41 × 10^{3} | 6.22 × 10^{1} | 2.88 × 10^{3} | 2.68 × 10^{1} | 2.29 × 10^{3} | 2.84 × 10^{1} | 2.32 × 10^{3} | 2.49 × 10^{1} | 2.18 × 10^{3} | 1.97 × 10^{1} |

d = 50 | 8.29 × 10^{3} | 8.13 × 10^{2} | 1.09 × 10^{4} | 3.87 × 10^{2} | 2.31 × 10^{3} | 7.52 × 10^{1} | 1.60 × 10^{4} | 3.83 × 10^{2} | 2.28 × 10^{3} | 5.36 × 10^{1} | |

CEC23 | d = 10 | 2.67 × 10^{3} | 4.28 × 10^{1} | 2.59 × 10^{3} | 3.06 × 10^{2} | 2.63 × 10^{3} | 0 | 2.69 × 10^{3} | 7.35 × 10^{0} | 2.37 × 10^{3} | 4.28 × 10^{1} |

d = 50 | 4.13 × 10^{3} | 1.59 × 10^{2} | 2.90 × 10^{3} | 5.17 × 10^{2} | 2.64 × 10^{3} | 4.48 × 10^{−1} | 3.64 × 10^{3} | 1.25 × 10^{2} | 2.29 × 10^{3} | 4.11 × 10^{1} | |

CEC24 | d = 10 | 2.64 × 10^{3} | 3.22 × 10^{1} | 2.73 × 10^{3} | 6.85 × 10^{1} | 2.29 × 10^{3} | 2.38 × 10^{0} | 2.73 × 10^{3} | 8.76 × 10^{0} | 2.46 × 10^{3} | 2.02 × 10^{0} |

d = 50 | 3.77 × 10^{3} | 2.65 × 10^{1} | 2.99 × 10^{3} | 8.36 × 10^{1} | 2.68 × 10^{3} | 6.69 × 10^{−1} | 3.50 × 10^{3} | 3.29 × 10^{2} | 2.43 × 10^{3} | 1.99 × 10^{−1} | |

CEC25 | d = 10 | 3.36 × 10^{3} | 2.45 × 10^{2} | 1.03 × 10^{4} | 4.42 × 10^{2} | 2.90 × 10^{3} | 1.53 × 10^{2} | 4.45 × 10^{3} | 5.55 × 10^{2} | 2.64 × 10^{3} | 1.45 × 10^{2} |

d = 50 | 6.63 × 10^{3} | 2.52 × 10^{1} | 9.13 × 10^{3} | 3.57 × 10^{2} | 3.18 × 10^{3} | 3.41 × 10^{2} | 6.29 × 10^{3} | 8.12 × 10^{1} | 3.02 × 10^{3} | 2.21 × 10^{1} | |

CEC26 | d = 10 | 4.69 × 10^{3} | 1.58 × 10^{3} | 3.89 × 10^{3} | 2.29 × 10^{2} | 4.45 × 10^{3} | 1.91 × 10^{2} | 4.84 × 10^{3} | 6.35 × 10^{0} | 3.50 × 10^{3} | 4.75 × 10^{0} |

d = 50 | 3.08 × 10^{3} | 9.14 × 10^{2} | 4.03 × 10^{3} | 1.57 × 10^{2} | 4.62 × 10^{3} | 2.16 × 10^{2} | 3.98 × 10^{3} | 9.50 × 10^{2} | 2.94 × 10^{3} | 1.38 × 10^{2} | |

CEC27 | d = 10 | 4.30 × 10^{3} | 3.28 × 10^{2} | 2.29 × 10^{3} | 1.11 × 10^{2} | 3.18 × 10^{3} | 4.14 × 10^{1} | 5.52 × 10^{3} | 3.08 × 10^{2} | 3.18 × 10^{3} | 2.20 × 10^{2} |

d = 50 | 3.01 × 10^{3} | 5.24 × 10^{1} | 3.53 × 10^{3} | 3.34 × 10^{2} | 2.89 × 10^{3} | 1.54 × 10^{2} | 7.85 × 10^{3} | 1.25 × 10^{2} | 3.04 × 10^{3} | 4.00 × 10^{1} | |

CEC28 | d = 10 | 3.44 × 10^{3} | 4.37 × 10^{2} | 4.88 × 10^{3} | 1.20 × 10^{2} | 3.19 × 10^{3} | 2.40 × 10^{2} | 6.07 × 10^{3} | 1.37 × 10^{2} | 2.99 × 10^{3} | 7.49 × 10^{1} |

d = 50 | 5.96 × 10^{3} | 2.27 × 10^{0} | 6.79 × 10^{3} | 6.17 × 10^{1} | 3.91 × 10^{3} | 2.87 × 10^{2} | 9.38 × 10^{3} | 3.08 × 10^{2} | 3.37 × 10^{3} | 1.56 × 10^{0} | |

CEC29 | d = 10 | 2.82 × 10^{3} | 8.13 × 10^{2} | 3.11 × 10^{3} | 3.18 × 10^{2} | 2.27 × 10^{3} | 1.50 × 10^{2} | 4.58 × 10^{3} | 9.71 × 10^{1} | 3.10 × 10^{3} | 7.58 × 10^{1} |

d = 50 | 2.76 × 10^{3} | 3.03 × 10^{2} | 4.35 × 10^{3} | 6.07 × 10^{2} | 1.14 × 10^{4} | 3.62 × 10^{3} | 6.61 × 10^{3} | 7.49 × 10^{2} | 3.18 × 10^{3} | 2.74 × 10^{2} |

3D Environment | Threat Areas | Parameters | Value | |||||
---|---|---|---|---|---|---|---|---|

Model 1 | mountains | Central coordinate | [80,25] | [70,80] | [175,45] | [140,125] | [60,150] | [120,175] |

height | 40 | 40 | 50 | 40 | 40 | 45 | ||

Slope in the X direction | 40 | 15 | 35 | 15 | 20 | 35 | ||

Slope in the Y direction | 40 | 15 | 60 | 15 | 20 | 20 | ||

buildings | Central coordinate | [35,120] | [105,115] | |||||

height | 40 | 40 | ||||||

Apothem | 15 | 15 | ||||||

Side length | 30 | 15 | ||||||

Radar areas | Central coordinate | [45,50] | [120,75] | [175,165] | ||||

height | 40 | 40 | 40 | |||||

radius | 10 | 15 | 12 | |||||

Model 2 | mountains | Central coordinate | [100,160] | [170,40] | [105,50] | |||

height | 60 | 70 | 80 | |||||

Slope in the X direction | 40 | 20 | 45 | |||||

Slope in the Y direction | 40 | 20 | 20 | |||||

buildings | Central coordinate | [30,125] | [50,40] | |||||

height | 40 | 30 | ||||||

Apothem | 24 | 10 | ||||||

Side length | 34.87 | 20 | ||||||

Radar areas | Central coordinate | [160,150] | ||||||

height | 50 | |||||||

radius | 20 | |||||||

Model 3 | mountains | Central coordinate | [50,60] | [130,100] | [135,90] | [170,150] | [170,50] | |

height | 40 | 40 | 50 | 40 | 40 | |||

Slope in the X direction | 40 | 15 | 60 | 15 | 20 | |||

Slope in the Y direction | 40 | 15 | 35 | 15 | 20 | |||

buildings | Central coordinate | [110,40] | ||||||

height | 60 | |||||||

Apothem | 17.3 | |||||||

Side length | 34.6 | |||||||

Radar areas | Central coordinate | [50,170] | [170,195] | |||||

height | 30 | 50 | ||||||

radius | 25 | 10 |

3D Model | Path Length | Longest | Shortest | Mean | |
---|---|---|---|---|---|

Algorithm | |||||

Environment 1 | WOA | 410.32 | 284.73 | 390.27 | |

ALO | 399.76 | 363.45 | 374.18 | ||

SSA | 341.25 | 295.07 | 318.08 | ||

ChOA | 292.08 | 259.62 | 283.24 | ||

GWO | 406.91 | 372.94 | 381.06 | ||

TRS-ChOA | 286.40 | 251.79 | 263.29 | ||

Environment 2 | WOA | 354.61 | 293.55 | 318.46 | |

ALO | 317.42 | 261.00 | 284.72 | ||

SSA | 320.08 | 294.18 | 315.65 | ||

ChOA | 327.10 | 281.38 | 304.01 | ||

GWO | 321.94 | 300.25 | 309.19 | ||

TRS-ChOA | 297.36 | 274.82 | 279.56 | ||

Environment 3 | WOA | 352.47 | 312.29 | 338.97 | |

ALO | 354.12 | 347.53 | 350.68 | ||

SSA | 297.74 | 263.80 | 285.88 | ||

ChOA | 333.26 | 304.91 | 317.93 | ||

GWO | 324.59 | 299.47 | 313.93 | ||

TRS-ChOA | 284.06 | 262.56 | 267.71 |

3D Model | Fitness Value | Optimal | Worst | Mean | |
---|---|---|---|---|---|

Algorithm | |||||

Environment 1 | WOA | 464.13 | 487.51 | 477.19 | |

ALO | 300.00 | 364.74 | 311.42 | ||

SSA | 287.85 | 329.99 | 300.61 | ||

ChOA | 463.11 | 490.07 | 476.08 | ||

GWO | 198.65 | 347.61 | 224.93 | ||

TRS-ChOA | 113.60 | 152.79 | 115.37 | ||

Environment 2 | WOA | 392.64 | 985.06 | 937.40 | |

ALO | 435.29 | 908.31 | 850.31 | ||

SSA | 224.57 | 678.34 | 343.06 | ||

ChOA | 339.15 | 852.38 | 771.56 | ||

GWO | 455.74 | 890.00 | 860.67 | ||

TRS-ChOA | 119.48 | 481.90 | 125.04 | ||

Environment 3 | WOA | 314.60 | 869.75 | 573.51 | |

ALO | 428.00 | 589.91 | 560.29 | ||

SSA | 365.77 | 714.70 | 401.57 | ||

ChOA | 316.49 | 462.39 | 385.48 | ||

GWO | 293.68 | 613.44 | 442.83 | ||

TRS-ChOA | 250.84 | 419.56 | 268.16 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, Q.; He, Q.; Zhang, D.
UAV Path Planning Based on an Improved Chimp Optimization Algorithm. *Axioms* **2023**, *12*, 702.
https://doi.org/10.3390/axioms12070702

**AMA Style**

Chen Q, He Q, Zhang D.
UAV Path Planning Based on an Improved Chimp Optimization Algorithm. *Axioms*. 2023; 12(7):702.
https://doi.org/10.3390/axioms12070702

**Chicago/Turabian Style**

Chen, Qinglong, Qing He, and Damin Zhang.
2023. "UAV Path Planning Based on an Improved Chimp Optimization Algorithm" *Axioms* 12, no. 7: 702.
https://doi.org/10.3390/axioms12070702