Adaptive Integral-Type Terminal Sliding Mode Control for Unmanned Aerial Vehicle Under Model Uncertainties and External Disturbances

This paper proposes an adaptive integral-type terminal sliding mode approach for the attitude and position tracking control of a quadrotor UAV subject to model uncertainties and external disturbances. First, an integral-type terminal sliding tracker is designed to attain the quadrotor UAV tracking performance in finite time when the upper bound of perturbations and uncertainties are known. Next, an adaptation law is proposed and a modified parameter-tuning integral-type terminal sliding mode tracking control scheme is designed to compensate of the model uncertainties and external disturbances. The stability and finite time convergence of the proposed approach is verified using the Lyapunov theory. Its performance is assessed using a simulation study encompassing various scenarios. Low chattering dynamics, fast convergence rate, and absence of singularities are the main features of the proposed approach.


I. INTRODUCTION
Unmanned Aerial Vehicles (UAVs), especially quadrotors, are widely popular due to their simple and ordinary structure, small size, low maintenance cost and, high hovering accuracy [1], [2]. They have been considered in a wide range of applications including; energy transportation streaks' control [3], organizing requirements of military missions [4], surveillance [5], monitoring [6], [7], search and rescue [8], [9], agriculture [10], [11]. Tracking and stability control of quadrotors, however, is a challenging problem that has motivated various research efforts [12], [13]. This is due to the fact that quadrotors are under-actuated systems with no natural stabilizing elements, that are often operating under unfamiliar and uncertain flight conditions. Thus advanced estimation and control algorithms are required to control the quadcopter.
One of the effective methods to control quadrotors is the Terminal Sliding Mode control (TSMC) which offers both fast reachability of the quadrotor trajectories and high robustness against disturbances [14], [15]. The TSMC can be combined with an adaptive control procedure for the approximation of the upper bound of uncertainties and disturbances The associate editor coordinating the review of this manuscript and approving it for publication was Mou Chen .
to further improve its performance in the presence of uncertainties and disturbances [16], [17].
In [18], a robust adaptive SMC method was proposed for the control of quadrotors in the presence of external disturbances and uncertainties. Though the approach achieved good tracking performance, the finite-time convergence of the attitude and position tracking was not considered in this work. In [19], an appointed-finite-time controller is presented for the attitude tracking control of a quadrotor in the presence of external disturbances. A terminal sliding mode observer was recommended for the estimation of the disturbances at any moment. However, this approach only examined the attitude tracking of the quadrotor and did not consider position tracking. In [20], a robust global SMC approach is proposed for the attitude tracking control of the quadrotor in the presence of external disturbances. An adaptive law is considered to estimate the upper bound of the external disturbances. Position tracking, however, was not investigated in this work. In [21], an adaptive fast SMC is designed for the control of a quadrotor under wind gust conditions. An adaptive law is applied for the reduction of the chattering phenomenon. In [22], a SMC method is presented for control of the quadrotor in the presence of the disturbances. An adaptive fuzzy control approach is adopted to compensate for the VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ disturbances. In [23], a fast TSMC scheme is designed for the hovering control of UAVs in the presence of obstacles. An adaptive control procedure is considered to compensate for these obstacles. Nonetheless, model uncertainties were not considered in these works. In [24], an integral TSMC approach is proposed for the tracking control of the quadrotor in the presence of uncertainties, external disturbances and faults. The proposed approach was combined with an adaptive control technique to elimination the need for any knowledge about the upper bound of disturbances and uncertainties. The model of the quadrotor considered in that paper, however, is the simplified 4DoF model, which is not appropriate in practice. In [25], an adaptive fractional-order fast TSMC approach was proposed for the attitude tracking control of a quadrotor subject to external disturbances and uncertainties. However, the impact of measurement noise on the control method was not examined in the above mentioned paper. Based on the above discussion, this paper proposes an adaptive integral-type terminal sliding mode approach for the attitude and position tracking control of a quadrotor UAV subject to model uncertainties and external disturbances. The main contributions of this paper are as follows: • A robust controller designed for a realistic 6DoF quadrotor UAV with model uncertainties, external disturbances; • A parameter-tuning law to estimate the upper bound of model uncertainties and external disturbances in integral-type terminal sliding mode tracker; • A design that yields finite time tracking, rapid convergence speed and control signals that are free of singularities and exhibit low chattering.
• Dynamic analysis in the presence of noise measurement. The remainder of the paper is organized as follows. Section II introduces the dynamic model of the 6DoF UAV. Section III formulates the control problem and provides some preliminaries. The proposed control approach is derived in section IV. Computer simulations illustrating the performance of the proposed approach are given in section V. Some conclusions are finally drawn in section VI.

II. MODELING OF THE 6DoF QUADROTOR SYSTEM
Quadrotors are Vertical Take-Off and landing vehicles which have four control inputs. Their control is typically realized via the angular speed of the four propellers. These latter can generate thrust, pitching, rolling and yawing moments as shown in FIGURE 1 [26]. Quadrotors are under-actuated systems. This is due to the fact that six states are being controlled by four control inputs.
The dynamics of a quadrotor with 6Dof can be represented by [27], [28]: where, the variables x, y, z represent the position coordinates of the quadrotor and φ, θ, ψ are the attitude of the quadrotor. m is the mass and d is the distance between the rotation axes and the center of the quadrotor. The terms K fdx , K fdy , K fdz are the drag coefficients and K fax , K fay , K faz denote the aerodynamic fiction factors. Also, C D and J r are the drag factors and motor inertia, respectively. The expressions I x , I y , I z denote the inertia to the axes x, y, z, respectively. u i (i = z, φ, θ, ψ) are the control inputs.¯ = w 1 − w 2 + w 3 − w 4 , where w 1 , w 2 , w 3 , w 4 are the angular velocities with the following laws: where K p is the lift power factor. Remark 1: In real applications, it is essential to control the quadrotor in six degrees-of-freedom. However, quadrotors are under-actuated systems with four independent control inputs acting autonomously. To circumvent this problem, an auxiliary input control is defined as: Thus, system (1 can be re-written as:

III. PROBLEM FORMULATION AND PRELIMINARIES
Considering x(t) = φ,φ, θ,θ, ψ,ψ, x,ẋ, y,ẏ, z,ż T as the state vector and u(t) = [u 2 , u 3 , u 4 , u 1 ] T as the input vector, the dynamics of the 6DOF quadrotor (4) can be represented using the following state space model: where f (x(t)) and g(x(t)) are defined by: with a 1 = represents system uncertainties and external disturbances.
The main control objective is to design a sliding mode-based position and attitude tracking controller for the 6DoF quadrotor subject to model uncertainties and external disturbances that: 1) guarantees finite time convergence, 2) is free of singularities, 3) exhibit low chattering.

Lemma 1:
The Lyapunov function V (t) is a continuous positive-definite function which satisfies the following differential inequality [30]: where m and n are positive constants. η = a b is a constant parameter, and a, b are odd positive integers that satisfy 0 < a < b. Then, the function V (t) converges to zero in the finite time t s : where t 0 is the initial time.
The output of the quadrotor UAV system is considered as y(t) = [φ, θ, ψ, x, y, z] T . The tracking error signals are expressed by: Define the integral-type TSMC surfaces as: (19) s y (t) = e y 2 (t) + η y e y1 (t) + κ y t 0 e y1 (τ ) q y /p y dτ, (20) where η i and κ i , i = 2, 3, 4, x, y, 1 are positive scalars, and q i and p i are two odd positive integers with q i < p i . If the initial errors are equal to zero, then the tracking problem is assumed to be the error remaining on switching surfaces s i (t) = 0, i = 2, 3, 4, x, y, 1 for t ≥ 0. If the error states reach the sliding manifold s i (t) = 0, i = 2, 3, 4, x, y, 1, they stay on it while sliding to e(t) = 0 andė(t) = 0. Differentiating the above sliding surfaces with respect to time, one finds: Substituting Eqs. (5)-(7 and (10)-(15) in the above equations, yields: Now, the equivalent controller u eq i , i = 2, 3, 4, x, y, 1 can be obtained from equationṡ i (t) = 0, (i = 2, 3, 4, x, y, 1) as: The total control input is defined as where u T i 's are obtained based on the adaptive integral-type TSMC method described in the next section.

IV. PROPOSED CONTROL APPROACH
In what follows, an integral-type TSMC-based finite time tracking control approach is first derived with the assumption that the upper bounds of uncertainties and disturbances are known. Then, an adaptive integral-type TSMC approach is synthesized for the case when the upper bounds of perturbations and model uncertainties are assumed to be unknown.

A. FINITE-TIME INTERGRAL-TYPE SMC DESIGN
The integral-type TSMC controller is defined as: u T y = − m u 1 (λ y s y (t) + γ y s y (t) α y sgn(s y (t)) + δ y sgn(s y (t))), The finite-time convergence of the integral-type terminal sliding surfaces (16)-(21) is investigated using the following theorem: Theorem 1: Consider the nonlinear 6DoF quadrotor UAV described using (5) with model uncertainties and external disturbances which satisfy Assumption 1. Let the error signal between the desired and actual outputs defined by (10)- (15). Then, the sliding surfaces (16)- (21) converge to the origin in finite time, if we consider the control inputs (40) with (34)-(39) as equivalent controls and (41)-(46) as the finite-time integral-type TSMC. Hence, the attitude and position tracking control of the quadrotor is satisfied.

B. FINITE-TIME INTERGRAL-TYPE SMC DESIGN
In real applications, it is impossible to determine the upper bound of the model uncertanities and external disturbances d(t). To solve this problem, an estimation of the positive constant δ i , i.e.δ i (t) is considered.
Assume the estimation error as: 2, 3, 4, x, y, 1).  By taking the time-derivative ofδ i (t), one finds: Now, the adaptation laws are defined as: where l i are positive constants. Subsequently, the adaptive integral-type TSMC controller is designed as: ).
(70) VOLUME 9, 2021  with µ i , ν i > 0, 0 < β i < 1 and ξ i = c i l i where c i , l i are two odd numbers which satisfy the following condition 0 < c i < l i .
In the following Theorem, the convergence of the sliding surfaces (16)-(21) is examined based on the adaptive integral-type sliding mode control scheme.
The block diagram of the proposed control approach is illustrated in FIGURE 2.

V. SIMULATION RESULTS
The performance of the proposed approach is illustrated using the quadrotor UAV which parameters are illustrated in Table 1. In addition, model uncertainties and external disturbances are considered as d i = 0.1e −t+1 , i = 1, 3, 5, 7, 9 and d i = 0.25 sin (πt) + 2, i = 2, 4, 6, 8, 10, 12, respectively. Besides, the desired values with respect to the pitch, roll and yaw are selected as: φ = π 3 sin( π 3 t + 2), θ = π 6 sin( π 6 t + 2) and φ = π 4 sin( π 4 t + 2), respectively. Also, the desired values for x, y, z are chosen as 0.5, 0.5, 1, respectively. The initial condition of quadrotor states and adaption laws are considered as x i (0) = 0.1(∀i = 1, . . . , 12) and δ i (0) = 0.1(∀i = 2, 3, 4, x, y, 1). The controller parameters are illustrated in Table 2. These latter were obtained by trial and error. The finite-time value of t s φ is calculated based on the inequality (9. The required values in (9 are considered as Three scenarios are considered in the performance analysis.     The time histories of the tracking errors are depicted in FIGURE 4. As it can be shown from these figures, the system states track the desired references appropriately and the tracking error signals reach zero in finite time. The time histories of the switching surfaces are displayed in FIGURE 5. The trajectories of the control inputs are depicted in FIGURE 6. It is observed from these figures that both sliding surfaces and control signals converge to the origin in finite time when using the proposed sliding mode control signal. Note also the appropriate amplitude and low chattering dynamics of the control inputs.    and the tracking errors reach zero properly. The time histories of the terminal sliding surfaces and adaptive controller VOLUME 9, 2021 signals are given in FIGURE 9 and FIGURE 10, respectively. Furthermore, the time responses of the adaptive parameters are exhibited in FIGURE 11. As can be seen from these plots, the trajectories of the terminal sliding surfaces and control inputs converge to zero. One can observe that the obtained control signals do not exhibit any chattering and hence, the responses are chattering-free.

C. PERFORMANCE ANALYSIS IN THE PRESENCE OF MEASUREMENT NOISE
This scenario investigates the impact of measurement noise on the quadrotor system. A zero mean white noise with noise power 0.0001 and sample time 0.01 as shown in FIGURE 12 is added to the measurement. The time histories of the state tracking, sliding surfaces and control inputs in this case are illustrated in FIGURE 13, FIGURE 14 and FIGURE 15, respectively when using the integral-type TSMC. The same variables are displayed in FIGURE 16, FIGURE 17 and FIGURE 18 respectively, when using the adaptive integral-type TSMC. As one can observe, besides low chattering due to the noise, the simulation results are similar to the ones obtained without measurement noise.

VI. CONCLUSION
This paper proposed an adaptive terminal integral-type terminal sliding control procedure for the trajectory and position tracking of a quadrotor UAV subject to model uncertainties and external disturbances. A terminal integral-type sliding mode tracking control approach was recommended to achieve the trajectory tracking performance of this system with known bound for uncertainties and disturbances. Then, a modified adaptive integral-type terminal sliding mode tracker was proposed for the quadrotor UAVs with uncertainties and disturbances with unknown bounds. The simulation results showed that the proposed approach is free of singularities, exhibits low chattering dynamics and has fast convergence rate. The practical implementation of the proposed approach and its extension to the control of time-delayed quadrotor UAVs will be the focus of our future research.