Integral Adaptive Sliding Mode Control for Quadcopter UAV Under Variable Payload and Disturbance

The quadcopter unmanned aerial vehicle (UAV) system is considered a good platform for control scheme design as it is highly nonlinear with coupled dynamics and an under-actuated system. Considering these challenges, this manuscript aimed to propose a control algorithm to control the quadcopter system in the presence of uncertainty and disturbance influences. The integral adaptive sliding mode control scheme has been proposed to control the system. The proposed control scheme is composed of the outer loop controller to control the position of the quadcopter, while the inner loop controls the attitude of the quadcopter. The proposed control law has three major terms, firstly the equivalent control which is developed based on the Lyapunov approach to handle most of the uncertainty and disturbance, secondly the adaptive switching gain, which is achieving fast adaptation against uncertainty, finally the switching function which has been approximated by a tangent hyperbolic function to reduce the unwanted chattering phenomena. The proposed control scheme and its performance have been investigated via a MATLAB/Simulink. The results prove that the implemented control scheme is robust even in the presence of uncertainties and disturbance and the quadcopter tracks the predefined trajectories with limited chattering influence.

The quadcopter is a complex system due to high nonlin-29 earity in the dynamics, unbounded dynamics and uncertain-30 ties [3]. Furthermore, the quadcopter often operates in a harsh 31 environment where external disturbance plays a significant 32 role. Therefore, all these challenges must be taken into con-33 sideration in the control design stage. 34 Several advanced control techniques have been developed 35 to control the quadcopter systems in order to handle these 36 challenges. For example, sliding mode control [4], [5], [6], 37 backstepping control [7], [8], feedback linearization [9], [10], 38 adaptive SMC control [11], [12], [13], and high-order sliding 39 mode control [14]. 40 The SMC is classified among the robust nonlinear con-41 trol algorithm, which provides better performance against 42 Section III discusses the proposed control scheme. Section IV 98 evaluates the proposed control scheme via the MATLAB/ 99 Simulink platform, where the simulation results are presented 100 and discussed. Finally, section V concludes the article.

102
In this section, a brief description of the quadcopter has been 103 introduced, then the quadcopter kinematics and the dynamics 104 equations have been presented. 105 A. QUADCOPTER SYSTEM DESCRIPTION 106 The quadcopter system composes of four rotors which 107 attached to the mainframe in symmetric cross shape [26], 108 [27], as illustrated in Figure 1. To obtain the kinematics equations of the quadcopter, the 111 coordinates space can be given as follows. Therefore, the translational motion equations of the quad-118 copter can be written as follows. Similarly, the quadcopter rotational motion equations can 125 be obtained as follows.
whereη denotes the angular velocity in the E-frame, and ω 128 denotes the angular velocity in the B-frame. T is the transfer 129 matrix [29], and it's given as follows.
The quadcopter dynamic equations in 6-DOF (space) are 133 presented as follows.

169
The control objective is to develop an integral adaptive 170 SMC control law to stabilize the error dynamics of the 171 quadcopter attitude. The error attitude dynamics are given as 172 follows:  Step 1: is to write the dynamics of the errors as in (12).
Consequently, the integral SMC surface for attitude angles 184 individually is chosen as follows. 188 where, s φ , s θ and s ψ represent the integral SMC surfaces.

191
Step 3: is to implement the sliding mode condition as in the 192 below (15).
where C 1 and C 2 are positive constants. to.

206
Step 4: is to select the control input u 2 , u 3 , u 4 as follows.
Step 5: is to calculate the estimated φ , θ , ψ based on 220 selected Lyapunov functions as follows.
As per (22) the adaption laws are calculated as follows.
Step 6: to reduce the chattering, the sign(s) is approxi-242 mated by the hyperbolic function tangent function (tanh(s)). 243 Therefore, the switching function in (24) will be replaced by 244 (tanh(s)), where its formula is given as follows. Step 7: the switching gains are designed as in (26) to sense 249 and adapt the changes in the uncertainties quickly. There are 250 three parameters ϕ i , α i and δ i (t) can be used and selected to 251 VOLUME 10, 2022 achieve fast adaptation, which contributes to the chattering 252 reduction.
The parameter ϕ i can be tuned as follows: where, ϕ i up is controlling the increment rate and ϕ i down is 258 controlling the decrement rate. As it can be observed, the overall dynamics of the quad-261 copter (attitude and position) as given in (8) (8), where, µ x , µ y and µ z are the position disturbances. 275 u 1 = u 2 x + u 2 y +(u z + g) 2 (30) 276 and, where u x , u y and u z denote the virtual controls generated 280 to control the quadcopter position through (u 1 , φ d and θ d ).

281
Particularly, as shown in Figure 2, the outer loop controller

289
Step 1: Subsequently, the error dynamics of the quadcopter 290 position subsystem are written as follows.
Step 2: Similarly, as in the inner loop controller design 295 section, the integral sliding surface for the outer loop con-296 troller is written as follows. where, s x , s y and s z represent the sliding surfaces for x, y, 301 and z dynamics, respectively. While k 1 x , k 1 y , k 1 z > 0 and 302 k 2 x , k 2 y , k 2 z > 0 are the outer loop integral adaptive SMC 303 control gains.

304
Step 3: is to apply the sliding mode condition as in (15).
s y = u y + µ y −ÿ d + k 1 yė y + k 2 y e y 311 s z = u z + µ z −z d + k 1 zė z + k 2 z e z (36) 312 Step 4: is to select the control input u x , u y , u z as follows: 313 Therefore, from (36) and (37), we get  Step 5: is to calculate the estimated x , y , z based on 329 selected Lyapunov functions as follows.
where,˜ x ,˜ y ,˜ z are the uncertainty errors, and calculated The adaption laws are calculated as follows.
Thus, the control laws will be as follows.
To reduce the chattering, steps 6 and 7 are implemented as in 351 the previous section.

353
The proposed control scheme has been simulated and eval-  Table 3 and Table 4 for the 356 inner and the outer loop controllers, respectively.

358
To evaluate the performance of the proposed control scheme, 359 three different scenarios have been investigated. In the first 360 case, the quadcopter has been commanded to reach the 361 desired position. In the second case, the quadcopter has 362 been commanded to track a predefined reference trajectory. 363 Finally, the proposed control scheme has been evaluated, 364 considering both the external disturbance and the quadcopter 365 mass uncertainty, where the quadcopter is following the pre-366 defined track. In this scenario, the desired setpoint in space is defined 370 as (x d , y d , z d ), and the quadcopter UAV has been com-371 manded to reach that point. The proposed control scheme 372 is designed to achieve the assigned task precisely and 373 fastly as shown in Figure 3 and Figure 4 for the position 374 VOLUME 10, 2022   and attitude, respectively. Figure 5 and Figure 6 presented

388
The simulation results in this scenario prove that the 389 proposed controller is robust against the reference tracking 390 modification. In the previous scenario, the performance was 391 evaluated for the quadcopter to reach the reference setpoint, 392 whereas, in this scenario, the proposed controller is evaluated 393   to follow a full predefined trajectory tracking in space. Figure 11 and Figure 12 visualize the error signals for the 395 position and attitude, respectively. The control efforts that 396 enabled the quadcopter to achieve this task successfully are 397 shown in Figure 13.   applications such as spraying agricultural pesticides where 404 the disturbances and uncertainties are challenging. In this 405 case the uncertainty has been represented by a variable pay-406 load where it is assumed as an added mass (such as liquid 407 pesticides) 0.18 kg to the quadcopter nominal mass 0.65 kg; 408 the added mass is about 30%. As shown in Figure 14 the 409 added mass discharges linearly until the liquid pesticides 410 finish and the quadcopter mass returns to the nominal value. 411 Meanwhile, the disturbances have been considered as well as 412 depicted in Figure 15.      Figure 17 and the attitude as in Figure 18). The 417 assigned task has been achieved successfully, and the con-418 troller performance can be observed as in Figure 19, and      contributed to chattering reduction. Based on the proposed 447 integral SMC, it is suggested to proceed further to identify 448 the unknown dynamics, which is also important for UAV 449 control. Currently, intelligent algorithms are widely used in 450 online estimation, e.g., neuroadaptive control for complicated 451 underactuated systems with simultaneous output and veloc-452 ity constraints; adaptive fuzzy control for a class of MIMO 453 underactuated systems with plant uncertainties and actuator 454 dead-zones. These topics are the subject of future research. 455 order sliding mode control of induction motors with core loss, '  Skudai Johor, in 1989, the master's degree in 586 control system engineering from The University 587 of Sheffield, U.K., in 1993, and the Ph.D. degree 588 in electronic instrumentation engineering from 589 Sheffield Hallam University, U.K., in 1996. He is 590 currently a Professor with the Department of 591 Control and Mechatronics Engineering, Faculty 592 of Electrical Engineering, UTM. His research interests include system 593 identification and estimation, signal processing, process tomography for 594 industrial process, process control instrumentation, sensors and actuators, 595 hydraulic, and pneumatic systems.