Quadrotor Stabilization and Tracking Using Nonlinear Surface Sliding Mode Control and Observer

: We propose a control method wherein the estimated angles converge to the desired value for quadrotor attitude stabilization and position tracking. To improve the performance of a quadrotor system, the unmeasured states of the quadrotor are estimated using a sliding mode observer (SMO). We set up a quadrotor dynamic model and augment the quadrotor dynamics by an SMO. We also derive the control inputs by sliding mode control (SMC) and calculate the desired angle of the quadrotor to reach the target position with the control inputs. For fast convergence speed and increased robustness of tracking performance, a nonlinear sliding surface is applied to SMC. The angle of the quadrotor converges to the desired value through the operation of SMC with a nonlinear sliding surface. The target tracking performance is improved by adaptively switching the deceleration curve of the sliding mode surface with a nonlinear curve. Using a tracking system based on a nonlinear surface sliding mode control (NSMC) and SMO, the quadrotor reaches the target position with a decreased settling time. The performance and effectiveness of the proposed system are proved through simulation results.


Introduction
Recently, unmanned aerial vehicles (UAVs) have attracted considerable research attention. A quadrotor that is one of the most widely used UAVs in various industrial fields is classified as a rotorcraft since it takes off vertically using four motors. Many researchers have studied quadrotors for various purposes, such as swarming, parcel delivery, exploration, and vehicles [1][2][3][4][5][6]. In various studies, the basic technology for quadrotor stabilization and movement is attitude control based on proportional-integral-derivative (PID) or sliding mode control (SMC) as general controllers. PID, which has the advantage of simple control structure, is a widely used maneuvering algorithm in controller design [7][8][9][10][11][12]. However, PID has a disadvantage against disturbances and is limited for precision control. Meanwhile, SMC shows strong, robust performance in the presence of disturbances and model uncertainties [13][14][15][16][17][18][19].
A quadrotor system is sensitive to parameter uncertainties and external disturbances. To solve this problem, we propose an augmented system that adopts correction variables ( i z ). When the state converges to the sliding surface, the trajectory becomes less sensitive to disturbances during the sliding phase. The stability of the control system is proved by following the Lyapunov stability theorem. However, in spite of the robust characteristic to disturbances or model uncertainties, sliding mode control has drawbacks of chattering phenomena and time delay for attitude stabilization. Even if the state converges to the sliding surface, a discontinuous control value with high frequency switching action is induced so that it causes unnecessary noises and switching stress in the system. As a solution, we propose a nonlinear sliding surface in SMC to reduce the error between the desired value and the measured value. Moreover, the proposed nonlinear sliding surface reduces the response time of the controller. The control input is derived to satisfy the stability condition of the Lyapunov function with the sliding surface.
Sliding mode observer (SMO) is widely used to improve the state estimation performance [20][21][22][23][24]. In SMO, we estimate the unmeasured state and use a saturation function to set up the correction variable with the estimated and the measured data. To improve tracking performance, SMC has been applied in various controller design problems [25]. A quadrotor is a nonlinear system with four independent inputs and six coordinate outputs. A novel SMC based on a nonlinear disturbance observer was proposed in [26] wherein the disturbance estimation was included to attenuate the mismatched uncertainties. The control algorithm in [27] was a hybrid of SMC and backstepping to stabilize the attitude and was widely adapted in various control applications. To achieve sensorless control of a permanent magnet synchronous motor, a sliding mode observer was proposed in [28] according to the back electromotive force model. By estimating the capacitor voltages from measurement of the load current, an adaptive gain second-order sliding mode observer for a multi-cell power converter system was designed in [29]. The capacitor voltages were estimated under the input sequences using the observability concept.
In this paper, we propose an SMC-and SMO-based control method for the position tracking and attitude stabilization of a quadrotor system. We set up the quadrotor dynamic model and design the augmented quadrotor dynamics with correction variables obtained by SMO. In general, there are two approaches to derive the quadrotor motion dynamics: Newton-Euler and Euler-Lagrange methods. Newton-Euler formalism gives physical insights through the derivation. Euler-Lagrange formalism provides the linkage between the classical framework and the Lagrangian or Hamiltonian method [30]. In this paper, the dynamics of the quadrotor is formulated based on Newton-Euler approach. To derive the appropriate control input ( ) u t by SMC, we use the augmented quadrotor dynamics and the angles estimated by SMO. In SMC, the desired angle   This paper is structured as follows. In Section 2, we obtain the quadrotor dynamic model. In Section 3, the SMO estimates the unmeasured angle and augmented dynamics of the quadrotor. In Section 4, a nonlinear sliding surface is designed to reduce the influence of the deceleration curve. In Section 5, the desired angle   SMO. In Section 6, we prove the proposed method by simulation. Finally, in Section 7, we present our conclusion.

Quadrotor Dynamic Modeling
The quadrotor, which can take off and land vertically, is composed of four motors. ,   rotate in a counterclockwise direction. A quadrotor can change its direction rapidly by its two pairs of motors [31]. We can acquire different positions and attitudes of the quadrotor by adjusting the velocity of the four propellers. Figure 1 also shows the earth-fixed frame and rigid body model of a quadrotor. In this paper, the dynamic model of the quadrotor is represented based on Newton-Euler equations. The Newton-Euler equations demonstrate the combined translational and rotational dynamics of a rigid body frame. The angles   , ,    , which represent roll, pitch, and yaw, respectively, are obtained from the rotation of the quadrotor's body frame in the x -, y -, and z -axes.
Here, , , x y z I is the body inertia, r J is the rotor inertia,  is the overall rotor speed, l is the lever length, g is gravity, and m is the mass of the quadrotor. , , x y and z are the movement distance of the quadrotor in the direction of , , x y and z axes, respectively.
As shown in Equation (1), the attitude and altitude of the quadrotor are decided by the gyroscopic effect resulted from the rigid body rotation in space and the rotor speed. The control inputs for quadrotor attitude are represented by 1 2 , , u u and 3 .
u The other control input, for altitude, is represented by 4 . u To consider the unknown disturbance in real environment, the dynamic model of Equation (1) where i  is the rotational velocity of the i-th motor, t is the thrust factor, and d is the drag factor. We can obtain the whole motor velocity as represented in Equation (2). The dynamic model in Equation (1) can be rewritten in the state-space form of The state vector is defined as follows: x In Equation (1), the mechanical structure can be represented as In Equation (4), 1 c through 8 c are parameters comprising body inertia, lever length, and rotor inertia. , x y u u are the control inputs of the quadrotor using roll, pitch, and yaw. From Equations (1) and (3), the dynamics of the quadrotor can be obtained as Equation (5).

Sliding Mode Observer-Based Augmented Dynamics
The purpose of an SMO is to estimate the states of a quadrotor system. Sliding mode observer works using a switching function to minimize the error between the real quadrotor's state and the observer output [32][33][34]. In this section, we estimate the quadrotor rotational angle and angular velocity and design the augmented quadrotor dynamics by SMO. The super-twisting higher order sliding mode method is adopted as a state observer design for model uncertainties [34]. Using the super-twisting observer, we estimate the state values in finite time under the presence of disturbances. For the stabilization and tracking control of the quadrotor, the sliding mode controller is used in the observer design.
We propose an observer based on differentiation of the state variables 1 x through 12 x of the quadrotor attitude dynamics as Equation (8). 1ˆ( cos cos ) The integral value and correction variables are applied to the roll, pitch, and yaw angles for quadrotor stabilization. Based on SMO with an augmented quadrotor model, the control procedure of the quadrotor is shown in Figure 2.

Nonlinear Sliding Surface-Based SMC Design
For the case of a conventional SMC, a linear sliding surface can be designed using the state error and differential value of the error. In this section, we set up a nonlinear sliding surface for the controller to have a fast transient response. The general linear sliding surface is designed as follows: , ( 1,3,5,7,9,11).
In Equation ( Figure 3 shows the system trajectory to reach the linear sliding surface. First, the system states move along the acceleration curve to reach the sliding surface and then switch to the deceleration curve. With the deceleration process, the settling time of the state error increases [35]. To reduce the effect of the deceleration curve, we propose a nonlinear sliding surface that compensates the deceleration curve with a sigmoid function. As a first step in designing the nonlinear sliding surface, we set up a new axis-based linear sliding surface as , , where i   and i   are the maximum and minimum values of the sigmoid function, respectively, and 0 i a  is a scaling constant that determines the slope value. As the value of i a increases, the shape of sliding surface becomes the step function. According to different values of i a , the nonlinear sliding surface will be adjusted to ensure the reduced reaching time. Figure 4 shows the proposed nonlinear sliding surface constructed by following the sigmoid function in the i i p   plane. We represent the sigmoid function of Equation (11)  ( ( , )).

Attitude and Altitude Control
The purpose of SMC is to make the measurement angle and tracking error converge to the desired value in finite time using a proper control input i u . In this section, we use SMC to stabilize the quadrotor attitude and to control the altitude tracking with the SMO-based quadrotor attitude dynamics of Section 3. We use the Lyapunov function to satisfy the stability condition [36]. The Lyapunov stability function for attitude is confirmed as follows: where 2 x  is designed in Equation (8) where 1 1 ( )   is the sigmoid function . In Equations (14) and (15), we use the saturation function to replace the discontinuous signum function.

Simulation Results
In this section, to verify the effectiveness of the proposed controller design method, simulations using SMC and SMO-based NSMC methods in the quadrotor system were demonstrated. In our simulation, the control system of a quadrotor is modeled using MATLAB/Simulink with ODE 45 as a solver. The disturbance model of the simulation is supposed to follow white Gaussian noise. Table 1 shows quadrotor dynamic system parameters used in the simulation. Moreover, we set the control parameters respectively. First, we confirm the estimation performance to prove the improvement of the quadrotor control system by SMO. For the quadrotor attitude control, Figure 5 shows the stabilization of the quadrotor angle   ,   , which is initially slanted by 17 degrees. In Figure 5, the dotted line is a stabilization result by conventional SMC and the solid line is a stabilization result by SMO-based nonlinear surface sliding mode control (NSMC). We can confirm that the performance of NSMC combined with SMO is better than that of conventional SMC. The settling time for the angle   ,   of a quadrotor is reduced with the use of SMO-based NSMC. Figure 6 shows that the quadrotor reaches the target position   7, 7,5 m from the initial position   3, 3, 0 m, following the reference trajectory. In Figure 6, the SMO-based control shows stable tracking performance following the reference trajectory and has a small perturbation around the turning points. Figure 7 shows the trajectory projected onto the x y  plane. To compare tracking performance, we simulate the trajectory shown in Figure 6 using the SMO-based NSMC and conventional SMC without an observer. The shows robustness against the effects of measurement noise. However, the conventional SMC trajectory shows chattering caused by the measurement noise. We can confirm that the trajectory driven by the SMO-based NSMC reaches the target point with less fluctuation.     Figure 8 shows the tracking performance of a quadrotor. The trajectory of the quadrotor with disturbances and the simulation results along the x-, y-, and z-axes are shown in this figure. While the conventional SMC shows low performance with slow convergence speed and large steady state error due to disturbance, the proposed NSMC shows more robust performance against disturbance since the proposed augmented dynamics with correction variables represents the quadrotor system precisely and NSMC reduces the delay of reaching time. This robustness against disturbance is possible owing to the state estimation precision of SMO. We can confirm the tracking performance on the x-, y-, and z-axes in Figure 8, wherein the nonlinear-sliding-surface-based SMC reduces bounces, settling time, and rising time more than conventional SMC. Moreover, chattering appears in the steady state of conventional SMC owing to the absence of SMO. We can confirm that the performance of NSMC is better than that of conventional SMC.

Conclusions
In this paper, we proposed a nonlinear-sliding-surface-based SMC for tracking and stabilization with an SMO. We set up the conventional quadrotor dynamic model based on Newton-Euler formalism. To improve the performance of the quadrotor system, we remodeled the augmented quadrotor dynamics and estimated the unmeasured states and their derivatives through the SMO. To improve the SMC control performance, we set up a nonlinear sliding surface to reduce the convergence time with a modified deceleration curve. We derived the proper control inputs 1 2 3 4 ( , , , , , and ) x y u u u u u u using SMC and calculated the desired angle   , d d

 
from the control input , .
x y u Moreover, the augmented quadrotor dynamics with the estimated states and the correction variables using SMO was derived to improve the representation of quadrotor system. We confirmed the improved performance in attitude control by SMO, which estimated the angles of the quadrotor and augmented its dynamics. Moreover, we proved the performance of a tracking system driven by NSMC.