Paths planning of Unmanned Aerial Vehicles (UAVs) in a dynamic environment is considered a challenging task in autonomous flight control design. In this work, an efficient method based on a Multi-Objective Multi-Verse Optimization (MOMVO) algorithm is proposed and successfully applied to solve the path planning problem of quadrotors with moving obstacles. Such a path planning task is formulated as a multicriteria optimization problem under operational constraints. The proposed MOMVO-based planning approach aims to lead the drone to traverse the shortest path from the starting point and the target without collision with moving obstacles. The vehicle moves to the next position from its current one such that the line joining minimizes the total path length and allows aligning its direction towards the goal. To choose the best compromise solution among all the non-dominated Pareto ones obtained for compromise objectives, the modified Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is investigated. A set of homologous metaheuristics such as Multiobjective Salp Swarm Algorithm (MSSA), Multi-Objective Grey Wolf Optimizer (MOGWO), Multi-Objective Particle Swarm Optimization (MOPSO), and Non-Dominated Genetic Algorithm II (NSGAII) is used as a basis for the performance comparison. Demonstrative results and statistical analyses show the superiority and effectiveness of the proposed MOMVO-based planning method. The obtained results are satisfactory and encouraging for future practical implementation of the path planning strategy.

In the last decades, unmanned aerial vehicles have acquired a great potential to complete an autonomous or semi-autonomous mission. A growing number of applications have appeared in real-world environments [

Many researchers have carried out various works to solve the Multi-Objective Path Planning (MOPP) problem for UAVs in a dynamic environment. The authors in [

Although these works have been developed to solve the MOPP problem for a UAV flying in a dynamic environment, most of them converted the multi-objective problem into a single objective problem by using a weighted sum function [

In the above-mentioned studies, the idea of using multi-objective metaheuristics for path planning problem’s formulation and the resolution seems a promising solution. To overcome the limits and inconveniences of the cited methods, particularly in terms of complexity and prohibitive time consuming, a systematic and efficient path planning method is proposed based on an advanced Multi-Objective Multi-Verse Optimization (MOMVO) algorithm. The main contributions of this paper are summarized as follows: 1) a novel strategy of reformulation and solving of a MOPP problem under operational constraints in a dynamic environment is proposed based on the concepts of the multi-criteria multi-verse optimization. Such a planning strategy allows guiding the quadrotor UAV to ensure destination position by calculate the next position in each step time while avoiding all moving obstacles. 2) A modified TOPSIS is employed as a higher-level decision-making approach to choose the best compromise solution among all the non-dominated solutions in the sense of Pareto. 3) A nonparametric statistical analysis method is proposed to compare all reported solvers for the hard path planning problem.

The remainder of this paper is organized as follows. In Section 2, the path planning problem for a quadrotor UAV is formulated as a multi-objective optimization problem under operational constraints. Section 3 presents the proposed multi-objective multi-verse optimizer to solve the formulated path planning problem. Section 4 describes the dynamical model of the studied quadrotor and the PID control design for the position and attitude dynamics stabilization and tracking. In Section 5, demonstrative results and comparisons are carried out and discussed. Section 6 concludes this paper.

The quadrotor passed from the starting point A

In the UAVs’ path planning formalism, the length of the flight path is very important. In this work, the robot determines its next position from its current one and tries to align its direction towards the goal. Consider initially, the UAV placed in the location at a time

Besides, the path planning process cannot totally ignore the dynamic characteristics of the UAV. When the UAV moves in the straighter path, the burden of the control system is reducing and the fuel cost of the flight process is decreasing [

where

On the other hand, avoidance of obstacles in a dynamic flight environment is more complex than in a static one. To simplify the characterization of moving obstacles, they can be modeled by spheres of radius

where

When the UAV moved from the actual position

where

The equation of a given sphere with the center’s coordinates

Then, substituting

where

As explained in [

where

Finally, the formulated multi-objective optimization problem for the path planning of the quadrotor UAV according to a given i^{th} waypoint is defined as follows:

where

To handle the inequality constraints of the problem

where ^{th} penalty parameter associated with the j^{th} constraint,

Originally proposed by Mirjalili et al. [

where ^{th} solution,

In

where

To develop a multi-objective version of the MVO metaheuristic for the problem

where

Based on the above motion equations and the archive updating mechanism

The selection of an optimal solution requires in particular a higher-level decision-making approach. The modified TOPSIS method is used to choose with more safety the best compromise solution among all the non-dominated ones in the sense of Pareto. Such a multiple-criteria decision-making approach is implemented as follows [

The basic movements of the quadrotor are realized by varying the speed of each rotor as shown in

The studied quadrotor is presented with its translational

where

The control inputs

where

The proposed control system of the quadrotor UAV is shown in

where

Two cascade loops for decoupling control of all flight dynamics are investigated. An inner loop is set to ensure the attitude and heading’s tracking. And the outer loop is designed for the positions

Solving

To illustrate the performance of the proposed MOMVO-based method, a 3D dynamic environment with moving obstacles is developed under the MATLAB/Simulink software. An interactive Graphical User Interface (GUI) has been implemented for the different simulations. The quadrotor’s 3D trajectory can be viewed by designing an animated quadrotor that receives the simulation data and performs the dynamical responses. Some performance index values, such as path length, flight time, and response plots of the quadrotor along the X, Y, and Z-axis, are presented and discussed. In this study, the quadrotor’s physical parameters are given in Appendix A.

Scenarios | Starting point [km] | Destination point [km] | Center of dynamic obstacles [km] | Dynamic obstacles’ speed [km/s] | |
---|---|---|---|---|---|

[0.0, 0.0, 0.0] | [9.0, 8.0, 0.0] | [5, 5, 2]; [3, 3, 2]; [5, 3, 1]; |
[4, −2, 1]; [2, −2, −2]; [4, 2, 2]; |
||

[1.0, 2.0, 0.0] | [10.0, 10.0, 0.0] | [1, 3, 1]; [3, 5, 1]; [4, 4, 3]; [5, 5, 4]; [7, 3, 4]; [8, 2, 1]; [9, 5, 2] | [2, 1, 1]; [3, −3, 1]; [4, 1, −2]; |
||

[1.0, 2.0, 0.0] | [15.0, 10.0, 0.0] | [2, 3, 1]; [2, 4, 1]; [4, 3, 2]; [5, 3, 3]; [5, 5, 2]; [6, 4, 1]; [7, 7, 2]; [7, 3, 4]; [8, 6, 3]; [10, 8, 2] | [−1, 3, −1]; [−1, 1, 1]; [2, 2, 1]; |
||

[2.0, 4.0, 0.0] | [16.0, 13.0, 0.0] | [1, 3, 1]; [2, 5, 2]; [2, 4, 3]; [2, 7, 1]; [3, 2, 1]; [3, 3, 3]; [4, 1, 2]; [4, 5, 4]; [6, 7, 1]; [7, 2, 2]; [8, 5, 2]; [10, 8, 3] | [1, 2, 1]; [−2, 1, 1]; [3, −1, 3]; |
||

[0.0, 4.0, 0.0] | [16.0, 15.0, 0.0] | [1, 4, 1]; [2, 5, 1]; [2, 2, 1]; [3, 2, 4]; [3, 7, 2]; [4, 2, 1]; [4, 5, 3]; [4, 8, 1]; [5, 3, 4]; [6, 5, 2]; [7, 2, 1]; [7, 4, 5]; [8, 1, 2]; [8, 8, 1]; [9, 5, 2] | [1, 3, 1]; [3, −1, 1]; [4, 1, 3]; |

To compare the performance of the proposed MOMVO-based planning method, others algorithms such as MSSA, MOGWO, MOPSO, and NSGAII are retained. The control parameters of such optimizers are summarized in

Optimizers | Parameters | |
---|---|---|

MSSA [ |
Without control parameters (free-parameters algorithm) | |

MOGWO [ |
Grid inflation 0.1, grids per each dimension 10, leader selection pressure 4, and repository member selection pressure 2 | |

MOPSO [ |
Cognitive and social accelerations 2, grid inflation 0.1, leader selection pressure 2, and grids per each dimension 7 | |

NSGAII [ |
Crossover percentage 0.7, mutation percentage 0.4, and mutation rate 0.02 | |

MOMVO | Lower and upper wormhole existence probabilities |

To have a fair and reliable comparison, the common parameters such as the maximum number of iterations and the population size are set as 100 and 50, respectively. For statistical comparison purposes, all algorithms are independently executed 10 times and compared in the sense of the solutions’ quality. In each step time, the quadrotor calculates the next position by solving the formulated multi-objective optimization problem

These results show the repartition topology of the set of optimal non-dominated Pareto solutions on the compromised surface. A higher-level decision-making approach,

To highlight the diversity and coverage of the obtained non-dominated solutions, various metrics such as Maximum Spread (MS) [

MSSA | MOMVO | MOGWO | NSGAII | MOPSO | |
---|---|---|---|---|---|

Best | 18.4022 | 20.8615 | 11.8244 | 17.9264 | 20.1983 |

Mean | 18.3603 | 20.4966 | 11.0232 | 06.5371 | 20.0943 |

Worst | 18.3295 | 19.9721 | 10.0944 | 02.4744 | 19.9385 |

STD | 00.0227 | 00.2899 | 00.5777 | 06.0948 | 00.0910 |

MSSA | MOMVO | MOGWO | NSGAII | MOPSO | |
---|---|---|---|---|---|

Best | 0.000247 | 0.1074 | 0.8655 | 2.52e-09 | 0.0917 |

Mean | 0.00020 | 0.1017 | 0.7521 | 5.18e-10 | 0.0883 |

Worst | 0.00011 | 0.0820 | 0.5420 | 0.0000 | 0.0811 |

STD | 4.003e-05 | 0.0079 | 0.1177 | 2.52e-09 | 0.0031 |

Best | Mean | Worst | STD | |
---|---|---|---|---|

C (MOMVO, MSSA) | 0.1000 | 0.0200 | 0.0100 | 0.0521 |

C (MSSA, MOMVO) | 0.4221 | 0.3233 | 0.2400 | 0.0752 |

C (MOMVO,MOGWO) | 1.0000 | 0.9900 | 0.9800 | 0.0012 |

C (MOGWO, MOMVO) | 0.0600 | 0.0400 | 0.0000 | 0.0314 |

C (MOMVO, NSGAII) | 0.2100 | 0.1500 | 0.0100 | 0.0546 |

C (NSGAII, MOMVO) | 1.0000 | 0.9700 | 0.8400 | 0.0642 |

C (MOMVO, MOPSO) | 0.0800 | 0.0400 | 0.0000 | 0.0253 |

C (MOPSO, MOMVO) | 0.0200 | 0.0067 | 0.0000 | 0.0103 |

To analyze the statistical performance of the MOMVO-based planning method, a comparative study with MSSA, MOGWO, NSGAII, and MOPSO algorithms is performed on three performance criteria, such as path length, elapsed time, and capacity to avoid the moving obstacles as shown in

Scenarios | MSSA | MOMVO | MOGWO | NSGAII | MOPSO | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

PL^{1} |
ET^{2} |
PL | ET | PL | ET | PL | ET | PL | ET | ||

1 | Best | 13.054 | 482.27 | 12.301 | 480.22 | 14.243 | 789.4 | 12.192 | 1110.2 | 17.191 | 757.1 |

Mean | 13.066 | 496.08 | 12.427 | 493.66 | 14.861 | 854.5 | 12.326 | 1265.4 | 19.857 | 821.5 | |

Worst | 13.306 | 516.63 | 13.251 | 506.63 | 15.251 | 958.8 | 13.241 | 1421.5 | 22.502 | 886.7 | |

STD | 0.1083 | 5.5645 | 0.2544 | 4.0314 | 0.3514 | 5.142 | 0.641 | 5.874 | 0.554 | 4.544 | |

2 | Best | 12.621 | 520.12 | 12.451 | 501.24 | 14.892 | 834.5 | 12.351 | 1187.2 | 18.524 | 798.4 |

Mean | 12.462 | 556.23 | 12.384 | 524.24 | 15.214 | 863.2 | 12.841 | 1354.2 | 20.241 | 854.2 | |

Worst | 13.762 | 620.63 | 13.484 | 589.92 | 15.458 | 987.1 | 13.541 | 1465.5 | 22.741 | 932.1 | |

STD | 0.2145 | 5.741 | 0.2014 | 4.1345 | 0.3741 | 5.142 | 0.667 | 5.984 | 0.574 | 4.651 | |

3 | Best | 16.201 | 825.12 | 16.121 | 817.21 | 17.451 | 1115.7 | 15.942 | 7545.3 | 25.874 | 2624.8 |

Mean | 17.149 | 855.41 | 16.443 | 848.05 | 18.899 | 1570.6 | 16.512 | 7646.5 | 27.317 | 2909.3 | |

Worst | 17.354 | 945.12 | 17.207 | 895.45 | 19.542 | 1618.2 | 16.774 | 8068.5 | 28.651 | 3187.4 | |

STD | 0.2451 | 5.781 | 0.2214 | 4.245 | 0.392 | 5.413 | 0.6754 | 6.2654 | 0.6224 | 4.988 | |

4 | Best | 16.673 | 923.5 | 16.654 | 891.2 | 19.214 | 1704.7 | 16.741 | 7998.5 | 27.874 | 3478.1 |

Mean | 16.715 | 1064.5 | 16.682 | 917.64 | 20.415 | 1845.8 | 16.784 | 8142.2 | 28.145 | 3584.2 | |

Worst | 20.854 | 1123.5 | 19.177 | 1013.2 | 21.214 | 1991.4 | 17.514 | 8534.6 | 29.941 | 3782.1 | |

STD | 0.3641 | 5.804 | 0.3146 | 5.5243 | 0.4201 | 5.6103 | 0.6774 | 6.4541 | 0.6412 | 5.2341 | |

5 | Best | 20.121 | 1105.4 | 19.471 | 1089.6 | 21.754 | 2154.2 | 19.104 | 9120.1 | 30.987 | 4212.2 |

Mean | 21.286 | 1188.19 | 20.275 | 1129.07 | 22.730 | 2263.7 | 19.115 | 9293.3 | 33.281 | 4422.4 | |

Worst | 21.764 | 1272.3 | 20.941 | 1262.4 | 23.147 | 2549.7 | 20.321 | 9752.2 | 35.102 | 4949.7 | |

STD | 0.4587 | 5.8715 | 0.4031 | 5.6441 | 0.5342 | 5.4924 | 0.6871 | 6.6934 | 0.6733 | 5.441 |

^{1}Path Length (Km), ^{2}Elapsed Time (s).

To find out which algorithms differ from the others, Fisher’s LSD post-hoc test is applied [

Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 | Scenario 5 | Ranks’ sum | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Score | Rank | Score | Rank | Score | Rank | Score | Rank | Score | Rank | ||

MSSA | 13.066 | 3 | 12.462 | 2 | 17.149 | 3 | 16.715 | 2 | 21.286 | 3 | 13 |

MOMVO | 12.427 | 2 | 12.384 | 1 | 16.443 | 1 | 16.682 | 1 | 20.275 | 2 | 7 |

MOGWO | 14.861 | 4 | 15.214 | 4 | 18.899 | 4 | 20.415 | 4 | 22.730 | 4 | 20 |

NSGA-II | 12.326 | 1 | 12.841 | 3 | 16.512 | 2 | 16.784 | 3 | 19.115 | 1 | 10 |

MOPSO | 19.857 | 5 | 20.241 | 5 | 27.317 | 5 | 28.145 | 5 | 33.281 | 5 | 25 |

Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 | Scenario 5 | Ranks’ sum | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Score | Rank | Score | Rank | Score | Rank | Score | Rank | Score | Rank | ||

MSSA | 496.08 | 2 | 556.23 | 2 | 855.41 | 2 | 1064.5 | 2 | 1188.1 | 2 | 10 |

MOMVO | 493.66 | 1 | 524.24 | 1 | 848.05 | 1 | 917.64 | 1 | 1129.0 | 1 | 5 |

MOGWO | 854.5 | 4 | 863.2 | 4 | 1570.6 | 3 | 1845.8 | 3 | 2263.7 | 3 | 17 |

NSGA-II | 1265.4 | 5 | 1354.2 | 5 | 7646.5 | 5 | 8142.2 | 5 | 9293.3 | 5 | 25 |

MOPSO | 821.5 | 3 | 854.2 | 3 | 2909.3 | 4 | 3584.2 | 4 | 4422.4 | 4 | 18 |

MOMVO | MOGWO | NSGA-II | MOPSO | |
---|---|---|---|---|

MSSA | 3 | |||

MOMVO | – | 3 | ||

MOGWO | – | 0 | ||

NSGA-II | – | – | – |

MOMVO | MOGWO | NSGA-II | MOPSO | |
---|---|---|---|---|

MSSA | ||||

MOMVO | – | |||

MOGWO | – | – | 1 | |

NSGA-II | – | – | – |

By visualizing the simulations of the 3D trajectory of the quadrotor, all proposed algorithms succeed in completing the flight mission still avoiding all the moving obstacles. The simulation results of the proposed MOMVO-based method are shown in

By visualizing these figures, we can notice that the proposed algorithm MOMVO gives the most direct path, which guarantees high efficiency in flight missions. The MOPSO algorithm generates a trajectory with many fluctuations along the Z-axis. From these results, the quadrotor has started the mission after a time delay which is due to the calculation time of the next point. A minimum execution time for an algorithm ensures the high efficiency of collision avoidance with the dynamic obstacles and causes a minimum flight time.

In this paper, a multi-objective multi-verse optimizer-based method has been proposed and successfully applied to solve the path planning problem of quadrotors UAV in a 3D dynamic environment. The path planning problem was formulated as a multi-objective optimization problem under operational constraints. The proposed planning approach aims to lead the drone to traverse a short and fast path in a dynamic environment without collision with the moving obstacles. An interactive graphical interface was developed under MATLAB/Simulink software environment to implement the proposed MOMVO-based path planning strategy. The demonstrative results and nonparametric statistical analyses in the sense of Friedman and the post-hoc tests show that the proposed MOMVO-based method is efficient and powerful compared to other reported algorithms. In comparison with MSSA, MOGWO, MOPSO, and NSGA-II optimizers, the main advantages of the proposed multi-verse algorithm are the remarkable simplicity of software implementation as well as the reduced number of its control parameters. The exploration/exploitation capabilities are superior to those of the other reported algorithms. Besides, the paired comparisons for two different optimization criteria showed that the MOMVO algorithm outperforms all the reported optimizers.

Future works deal with the real-world prototyping and experimentation of the proposed MOMVO-based planning approach using a real model of quadrotor available in our laboratory. The Parrot AR. Drone 2.0 kit associated with MATLAB/Simulink software will be used for the experimentations.

Symbol | Description | Value/unit |
---|---|---|

Lift coefficient | ||

Drag coefficient | ||

Mass | ||

Arm length | ||

Motor inertia | ||

Quadrotor inertia | ||

Aerodynamic friction coefficients | ||

Translational drag coefficients | ||

Acceleration of the gravity |