The research on Unmanned Aerial Vehicles (UAV) has intensified considerably thanks to the recent growth in the fields of advanced automatic control, artificial intelligence, and miniaturization. In this paper, a Grey Wolf Optimization (GWO) algorithm is proposed and successfully applied to tune all effective parameters of Fast Terminal Sliding Mode (FTSM) controllers for a quadrotor UAV. A full control scheme is first established to deal with the coupled and underactuated dynamics of the drone. Controllers for altitude, attitude, and position dynamics become separately designed and tuned. To work around the repetitive and time-consuming trial-error-based procedures, all FTSM controllers’ parameters for only altitude and attitude dynamics are systematically tuned thanks to the proposed GWO metaheuristic. Such a hard and complex tuning task is formulated as a nonlinear optimization problem under operational constraints. The performance and robustness of the GWO-based control strategy are compared to those based on homologous metaheuristics and standard terminal sliding mode approaches. Numerical simulations are carried out to show the effectiveness and superiority of the proposed GWO-tuned FTSM controllers for the altitude and attitude dynamics’ stabilization and tracking. Nonparametric statistical analyses revealed that the GWO algorithm is more competitive with high performance in terms of fastness, non-premature convergence, and research exploration/ exploitation capabilities.

The quadrotor is one of the most popular architectures of UAV which is widely used in large areas of engineering and civilian applications [

Recently, advanced nonlinear control strategies have been especially proposed for quadrotors UAV. The Terminal Sliding Mode Control (TSMC) and Fast Terminal Sliding Mode Control (FTSMC) approaches are the most powerful and robust used ones [

All the above described TSM and FTSM control methods have shown high performance and robustness improvements in the quadrotors’ stabilization and tracking framework. Unfortunately, they claim the selection and tuning of a large-scale of effective control parameters, i.e., the coefficients of manifolds and sign functions of switching control laws, which make the difficult, non-systematic, and time-consuming procedure of the controllers’ design. Indeed, these effective control parameters are often selected by repetitive trials-errors based methods. To overcome this drawback, various attempts have been proposed in the literature. In [

Tuning the effective parameters of the TSM control approach through optimization methods seems a promising solution for complex and large-scale systems. The metaheuristics theory gives a variety of global optimization algorithms and can be used to solve such a design and tuning problem [

The remainder of this paper is organized as follows. In Section 2, the problem of FTSM parameters’ tuning is stated and then formulated as a constrained optimization problem. A nonlinear dynamical model of the studied quadrotor is established and a full control scheme (attitude, altitude, and position) is given to deal with the coupled and underactuated drone’s dynamics. In Section 3, the proposed GWO algorithm is described and a pseudo-code for its implementation is given. Section 4 presents all simulations and demonstrative results for the proposed GWO-based sliding mode control strategy. Nonparametric statistical analysis based on Friedman and post-hoc tests is investigated to show the superiority and effectiveness of the proposed free-parameters GWO metaheuristic

A quadrotor is an unmanned aerial vehicle that has four motors and detailed with their body-frame

Let a vector

By using the Newton–Euler formalism [

where _{r}_{x}_{y}_{z}_{1}, _{2}, _{3}, and _{4} are the control inputs of the drone given as:

where

Based on the established nonlinear model

Two cascade control loops are investigated to independently drive all flight dynamics of the drone, i.e., an inner control loop to ensure the attitude and heading’s stabilization and/or tracking, and outer loops for the positions

Solving

In _{x}_{y}

In the FTSM control framework, let consider the following model of a given uncertain second-order nonlinear system [

where

Since the task of FTSM control is to design a control law

where

Such a choice of the sliding manifold leads to the following control law of the uncertain system

where

While considering the altitude and attitude dynamics’ models of

where

As depicted in

where

Cost functions of the problem

where

The social hierarchy of the grey wolves is defined by four types of agents such as alpha (

where

where

Finally, the steps of the basic GWO pseudo-code are summarized as follows [

In this section, the proposed GWO is applied to solve the formulated tuning problem

All reported algorithms are independently run 20 times on a PC with i7 Core 2 Duo-2.67 GHz CPU and 6.00 GB RAM. The termination criterion is set as a maximum number of iteration reached _{Gen}_{pop}

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

ABC [ |
Limit of abandonments 32 | |

CSA [ |
Awareness probability 0.2, flight length 1. | |

HSA [ |
Pitch rate 0.1, fret width damping ratio 0.995. | |

WCA [ |
Number of rivers 4, maximum distance 1e −16. | |

SFLA [ |
Memeplex size 10, number of off-springs 3, step size 2. | |

PSO-In [ |
Max and min inertia factors (0.9; 0.4), social and cognitive coefficients (0.5; 0.3) | |

FA [ |
Light absorption and attraction 1, mutation coefficient 0.2, dumping ratio 0.98. |

WCA | HSA | SFLA | CSA | GWO | PSO-In | FA | ABC | ||
---|---|---|---|---|---|---|---|---|---|

Best | 2.521 | 2.398 | 2.383 | 12.919 | 2.3406 | 2.5472 | 2.487 | 2.245 | |

Mean | 2.587 | 2.635 | 2.630 | 23.651 | 2.4135 | 2.6726 | 2.520 | 2.392 | |

Worst | 4.974 | 12.700 | 7.907 | 129.35 | 3.3795 | 5.65 | 2.832 | 4.532 | |

STD | 0.279 | 1.069 | 0.954 | 33.413 | 0.1473 | 0.481 | 0.063 | 0.402 | |

ET (s) | 625.32 | 745.29 | 312.38 | 387.37 | 157.35 | 428.37 | 453.26 | 394.54 | |

Best | 2.045 | 1.838 | 1.922 | 5.3609 | 1.7545 | 2.6187 | 2.177 | 1.752 | |

Mean | 2.069 | 2.067 | 2.482 | 35.34 | 1.9772 | 3.3831 | 2.291 | 2.330 | |

Worst | 2.501 | 3.673 | 8.170 | 173.13 | 14.442 | 19.591 | 6.131 | 14.380 | |

STD | 0.077 | 0.450 | 1.364 | 54.709 | 1.2732 | 2.5627 | 0.431 | 1.420 | |

ET (s) | 829.19 | 766.26 | 315.27 | 482.26 | 199.36 | 613.26 | 511.45 | 514.32 | |

Best | 4.221 | 4.147 | 4.227 | 4.7521 | 4.0882 | 4.2623 | 4.158 | 4.179 | |

Mean | 4.240 | 4.232 | 4.686 | 36.178 | 4.1861 | 4.8124 | 4.170 | 4.469 | |

Worst | 4.598 | 4.345 | 15.71 | 3147.3 | 5.6978 | 16.243 | 4.410 | 13.040 | |

STD | 0.055 | 0.062 | 1.993 | 314.25 | 0.2231 | 2.2454 | 0.033 | 1.239 | |

ET (s) | 543.64 | 528.36 | 392.26 | 298.48 | 150.35 | 407.36 | 349.36 | 455.34 | |

Best | 1.011 | 0.935 | 1.074 | 1.7898 | 0.9302 | 1.0054 | 0.951 | 0.983 | |

Mean | 1.035 | 0.973 | 1.097 | 308.01 | 0.9791 | 0.9837 | 1.059 | 1.160 | |

Worst | 1.772 | 1.352 | 1.368 | 4274.9 | 2.3031 | 3.3248 | 4.582 | 2.617 | |

STD | 0.082 | 0.093 | 0.050 | 1.094 | 0.1647 | 0.4928 | 0.535 | 0.326 | |

ET (s) | 973.33 | 892.21 | 391.27 | 367.32 | 230.16 | 688.35 | 426.39 | 584.51 | |

Best | 0.097 | 0.093 | 0.097 | 1.666 | 0.0939 | 0.1449 | 0.109 | 0.096 | |

Mean | 0.099 | 0.103 | 0.102 | 12.646 | 0.1072 | 0.1617 | 0.113 | 0.110 | |

Worst | 0.145 | 0.186 | 0.143 | 66.782 | 1.0132 | 0.2564 | 0.148 | 0.255 | |

STD | 0.005 | 0.018 | 0.012 | 22.666 | 0.0923 | 0.0293 | 0.006 | 0.027 | |

ET (s) | 984.52 | 994.11 | 587.68 | 723.53 | 383.28 | 847.73 | 635.21 | 613.85 |

The time-domain performances of the controlled dynamics are shown in

From these demonstrative results, one can observe the superiority of the proposed control approach to reduce the undesirable chattering phenomenon in comparison with the standard TSMC approach. The control amplitude of each flight dynamic is moderated and further reduced. Large transient oscillations and amplitudes are recorded for the reported FTSMC and TSMC cases.

Algorithms | Unit step response | |||
---|---|---|---|---|

Rise time | Settling time | Overshoot (%) | Steady-state error | |

0.5833 | 0.8622 | 0.0039 | 0.0378 | |

0.5689 | 1.2805 | 1.3994 | 0.0043 | |

0.5569 | 1.4924 | 3.4234 | 0.0068 | |

0.9529 | 1.5413 | 0.0053 | 0.0229 | |

0.5591 | 1.4418 | 4.8454 | 0.0122 | |

0.5533 | 7.7761 | 10.1420 | 0.0368 | |

0.5473 | 1.2895 | 6.3926 | 0.0014 | |

0.5377 | 1.3256 | 6.1483 | 0.0052 |

Algorithms | Unit step response | |||
---|---|---|---|---|

Rise time | Settling time | Overshoot (%) | Steady-state error | |

0.3355 | 9.0926 | 28.7371 | 0.0014 | |

0.2826 | 1.4277 | 67.5484 | 1.242e −04 | |

0.3754 | 1.3220 | 26.5182 | 6.781e −04 | |

0.2164 | 9.8061 | 4.4428 | 0.0032 | |

0.3126 | 2.9209 | 4.6088 | 5.997e −04 | |

0.2347 | 9.4247 | 11.9934 | 5.179e −04 | |

0.3161 | 9.9768 | 15.2130 | 9.929e −04 | |

0.4603 | 1.2503 | 38.1779 | 0.0126 |

Algorithms | Unit step response | |||
---|---|---|---|---|

Rise time | Settling time | Overshoot (%) | Steady-state error | |

0.6517 | 9.9878 | 20.3243 | 0.0105 | |

0.6183 | 1.3882 | 26.3044 | 8.184e −04 | |

0.7027 | 1.4033 | 14.8132 | 4.276e −05 | |

0.6044 | 8.3874 | 5.3774 | 0.0138 | |

0.7006 | 9.9751 | 2.0786 | 0.0057 | |

0.5035 | 1.9252 | 40.2486 | 5.029e −05 | |

0.3962 | 1.4412 | 59.9361 | 0.0039 | |

0.5636 | 9.8719 | 33.4547 | 0.0056 |

Algorithms | Unit step response | |||
---|---|---|---|---|

Rise time | Settling time | Overshoot (%) | Steady-state error | |

0.8242 | 1.1367 | 0.5399 | 0.0027 | |

0.6537 | 0.9046 | 1.0184 | 8.184e −04 | |

0.7092 | 1.0367 | 1.3303 | 0.0049 | |

0.7176 | 0.9570 | 0.7762 | 0.0011 | |

0.5075 | 1.5670 | 1.1222 | 1.856e −04 | |

0.4868 | 0.6403 | 1.4360 | 2.019e −04 | |

0.5527 | 0.7370 | 1.6817 | 0.0037 | |

0.5969 | 1.1032 | 0.3840 | 0.0017 |

Nonparametric statistical comparison of the proposed metaheuristics is carried out based on the Friedman and pair-wise

IAE | ITAE | ISE | ITSE | MSE | ||||||
---|---|---|---|---|---|---|---|---|---|---|

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

2.587 | 4 | 2.069 | 3 | 4.240 | 4 | 1.035 | 4 | 0.099 | 1 | |

2.635 | 6 | 2.067 | 2 | 4.232 | 3 | 0.973 | 1 | 0.103 | 3 | |

2.630 | 5 | 2.482 | 5 | 4.686 | 6 | 1.097 | 6 | 0.102 | 2 | |

23.651 | 8 | 35.340 | 8 | 36.178 | 8 | 308.01 | 8 | 12.646 | 8 | |

2.413 | 2 | 1.977 | 1 | 4.186 | 2 | 0.979 | 2 | 0.107 | 4 | |

2.672 | 7 | 3.383 | 7 | 4.812 | 7 | 0.983 | 3 | 0.161 | 7 | |

2.520 | 3 | 2.291 | 4 | 4.170 | 1 | 1.059 | 5 | 0.113 | 6 | |

2.392 | 1 | 2.330 | 6 | 4.469 | 5 | 1.160 | 7 | 0.110 | 5 |

Based on the results of _{0.995} = 2.763 value of the t-distribution with 28 degrees of freedom.

WCA | HSA | SFLA | CSA | GWO | PSO-In | FA | ABC | |
---|---|---|---|---|---|---|---|---|

Average rank | 3.2 | 3 | 4.8 | 8 | 2.2 | 6.2 | 3.8 | 4.8 |

Summation | 16 | 15 | 25 | 40 | 11 | 31 | 19 | 24 |

Squared rank sum. | 58 | 59 | 126 | 320 | 29 | 205 | 87 | 136 |

F-score | 7.200 | |||||||

F-statistics at 99% | 3.36 |

Absolute difference of the rank’s summation | HSA | SFLA | CSA | GWO | PSO-In | FA | ABC |
---|---|---|---|---|---|---|---|

1 | 9 | 24 | 5 | 15 | 3 | 8 | |

– | 10 | 25 | 4 | 16 | 4 | 9 | |

– | – | 15 | 4 | 6 | 6 | 1 | |

– | – | – | 29 | 9 | 21 | 16 | |

– | – | – | – | 20 | 8 | 13 | |

– | – | – | – | – | 12 | 7 | |

– | – | – | – | – | – | 5 |

From these results, it can be clearly deduced that the GWO metaheuristic has the same performance in solving the optimization problem

In this work, a systematic and intelligent tuning method of all effective parameters of fast sliding mode controllers is proposed and successfully applied for a quadrotor UAV. The gains of sliding manifolds and switching functions of the FTSM control laws are selected thanks to the proposed free-parameters GWO algorithm. For such a hard and large-scale tuning problem, the tedious and time-consuming trials-errors based methods are no longer used, and the design time is further reduced. A full control scheme for the studied quadrotor UAV is first given to deal with the underactuated and coupled flight dynamics. Only the dynamics of altitude and attitude are considered for the optimization-based tuning process using the proposed GWO algorithm compared to other homologous methods. Demonstrative results in terms of optimization capabilities and time-domain performance are carried out to show the effectiveness of the proposed GWO-based tuning method. In comparison with the reported optimizers, the mean values of STD and elapsed time obtained for the proposed GWO algorithm are minimal and equal to 0.38012 and 224.1 s, respectively. This finding further encourages the use of such a free-parameters metaheuristic in real-world and online optimization scenarios. Regarding the chattering attenuation, the proposed GWO-tuned controllers succeeded to damp and cancel severe oscillations on control signals with the amplitude of 1.6 N/m for the no optimized FTSMC technique and 0.8 N/m for the classical TSMC approach. The performance metrics in terms of rising and settling times as well as the first overshoots are further improved by a reduction to 50% of their values in the cases without GWO-based tuning, i.e., classical TSMC, FTSMC, and NTSMC approaches. The superiority of the proposed GWO-tuned FTSMC in terms of stabilization and tracking is clearly shown. Nonparametric statistical analyses, i.e., using the Friedman and post-hoc tests, show that the proposed free-parameters GWO metaheuristic outperforms the reported algorithms retained as comparison tools.