Research of Robust Trajectory Tracking Control and Attenuated Chattering: Application on Quadrotor

In this paper, we presented a strategy for accurate trajectory tracking control of a quadrotor with unknown disturbances. To guarantee that the tracking errors of all system state variables converge to zero in finite time and eliminate the chattering phenomenon caused by the switching control action, a control strategy that combines linear predictionmodel of disturbances and fuzzy sliding mode control (SMC) based on logical framework with side conditions (LFSC) was designed. LFSC was applied for both position and attitude tracking of the quadrotor. Firstly, a linear prediction method was devised to minimize the effects of external disturbances. Secondly, a new fuzzy law was implemented to eliminate the chattering phenomenon. In addition, the stabilities of position and attitude were demonstrated by using Lyapunov theory, respectively. Simulation results and comprehensive comparisons demonstrated the superior performance and robustness of the proposed LFSC scheme in the case of external disturbances.


Introduction
e quadrotor, a typical unmanned aerial vehicle consisting of four symmetric propellers, has received much attention recently due to its low cost, easy maintenance, and potential for deployment in difficult environments [1]. e quadrotor design is the preferred choice for aerial robots, as quadrotor vehicles have vertical takeoff and landing capabilities and can hover at low speed [2]. As such, the quadrotor has been applied in many fields, including photography, education, transportation, and agriculture [3,4]. e quadrotor is a highly nonlinear, underactuated, and strongly coupled system; therefore, designing an effective control system for the quadrotor is a challenging task. However, the ability to minimize the effects of external disturbances is a bigger challenge [5].
Over the past several years, a number of advanced control strategies have been proposed to address the quadrotor trajectory tracking control problem [6]. Linear control strategies, including proportional integral derivative [7,8], proportional derivative [9,10], and linear quadratic [11] methods, have been applied successfully to improve stability; however, they are only effective over a small range around the operating point. If the quadrotor is subjected to greater external disturbances as it moves away from its control domain, system stability cannot be guaranteed [12]. Nonlinear control strategies can overcome the drawbacks of linear control methods to achieve good performance, even in harsh environments [13].
Sliding mode control (SMC) [14,15] and backstepping control [16,17] are the two among most widely used nonlinear control methods. Backstepping control is an efficient method for dealing with the trajectory tracking problem of the quadrotor via a nonlinear adaptive controller. However, backstepping control only provides sufficient stability when the disturbances are relatively constant or vary slowly over time. SMC utilizes a high-frequency switching control signal to enforce the system trajectories on the sliding surface, which has been studied for control of different underactuated systems [18][19][20][21]. e main properties of SMC are the proper transient performance and superior robust operation with the presence of model uncertainties and disturbances [22,23]. Zhang et al. [24] adopted the adaptive recursive integral terminal sliding mode control to guarantee the convergence performance of the actual angle and the yaw rate with strong robustness and fast convergence rate. Zhou et al. [25] utilized deep learning method to compensate the uncertainties of the system without requirement of their upper bounds, which makes the designed switching gain much smaller. Song et al. [26] proposed a novel nonsingular fast-terminal sliding mode control method to facilitate the stabilization of nonlinear underactuated systems under disturbances. Gu et al. [27] utilized neural networks to approximate the lumped unknown dynamic model and designed a fast-terminal sliding mode control strategy to achieve the finite-time consensus tracking. Chen et al. [28] proposed a nonsingular terminal sliding mode control algorithm to implement accurate and robust body position trajectory tracking of six-legged robots. Xiong et al. [29] constructed a novel integral sliding mode surface to guarantee the synchronization error convergence to zero in finite time. However, the traditional SMC has the problem of chattering in the control signal which is undesirable [30]. Wang et al. [31] used adaptive integral SMC, backstepping, and terminal SMC to solve the trajectory problem; however, reducing chattering in the control input (u 2 , u 3 , u 4 ) within the context of a hybrid finite-time control strategy is difficult to implement in practice. Mallavalli and Fekih [32] used SMC to solve the fault tolerance problem of the quadrotor; in this case, the gain of the switch function, sgn(s), was a constant, resulting in substantial chattering. Wang et al. [33] proposed a terminal SMC, and Xinghuo Yu and Man Zhihong [34] proposed a fast-terminal SMC; however, the gain of the switch function, sgn(s), was constant.
e traditional SMC has the problem of chattering which makes the SMC method hard to apply in practice. In addition, improving the accuracy and robustness of the control system is very important for the flight of the quadrotor in complex external environment. Considering that chattering phenomenon is caused by the gain of the switch function sgn(s), which is generally designed to be a constant, here we propose to replace the constant with a time-varying function. e motivation of this study is to design a fuzzy SMC strategy based on logical framework with side conditions (LFSC) to allow the quadrotor to achieve an accurate trajectory without chattering. e main contributions are summarized as follows: (1) e proposed LFSC scheme can guarantee that the tracking errors of all system state variables converge to zero in finite time.
(2) e high-frequency chattering phenomenon caused by the switching control action does not appear using the proposed LFSC scheme. (3) Simulation results demonstrate the superior performance and robustness of the proposed LFSC scheme in the case of external disturbances. e rest of this paper is organized as follows: in Section 2, a dynamic model of the quadrotor is presented, and the underactuated problem is solved. In Section 3, the proposed control strategy is described in detail. In Section 4, the simulation results and discussion are provided to show the superiority of the proposed control strategy. In Section 5, conclusions are presented.

Description of the Quadrotor Model.
A schematic diagram of a quadrotor UAV is shown in Figure 1, the earthfixed coordinate system is defined as the E-frame (O e , X e , Y e , Z e ), and the body-fixed system is defined as the . e main frame of the quadrotor is assumed to be a rigid body. e four propellers are installed in two vertical directions: propellers 1 and 3 rotate in the counterclockwise direction, while propellers 2 and 4 rotate in the clockwise direction to generate a lift force and balance the yaw torque as needed. Changing all four rotor speeds by the same amount changes the lift force, thus affecting the altitude of the quadrotor. Pitch rotation can be obtained by varying the speeds of propellers 1 and 3 in opposite directions. Roll rotation can be generated in a similar way by changing the speeds of propellers 2 and 4. e quadrotor has six degrees of freedom, including translational motions and three rotational motions, with only four independent inputs generated by increasing or decreasing the speeds of the four propellers [35]. e thrusts generated by the four rotors are denoted by f i (i � 1, 2, 3, 4), respectively. Figure 1, two reference frames are defined to describe the quadrotor kinematics model: the E-frame (O e , X e , Y e , Z e ) and the B-frame

Kinematic Model. As shown in
, this paper assumes that the origin O b of the body coordinate system is located at the center of the quadrotor; X b and Y b point toward rotors 1 and 2, respectively. en, according to the right-hand rule, Z b points upwards. e E-frame (O e , X e , Y e , Z e ) is used to define the absolute position of the quadrotor according to X � [x, y, z] T and Euler angles η � [ϕ, θ, ψ] T , where ϕ, θ, and ψ denote the roll angle, pitch, and yaw, respectively. V � [μ, υ, ω] T and Ω � [p, q, r] T denote the linear and angular velocities of the quadrotor, respectively. In this context, the quadrotor can be modelled by [36] where rotation matrices are given by and where s * and c * denote sin( * ) and cos( * ), respectively. Equations (1)-(3) are used to calculate the actual position and attitude of the quadrotor.

Dynamic
Model. e dynamic model is built in consideration of model uncertainties and disturbances. It must be noted that ground and gyro effects were not taken into account because the purpose of this research was to design a control system for the model; therefore, the model was kept as simple as possible, with only the main effects being taken into account [37].

Assumption 1.
e main frame of the quadrotor is symmetrical and rigid.

Assumption 2.
e origin O b of the quadrotor's body coordinate system is located at the center of mass of the quadrotor.

Assumption 3.
e aerodynamic parameters of the rotors and propellers are the same. Assumption 4. Ground and gyro effects can be ignored (in this case, compared to the brushless motor, the propeller is very light; thus, the moment of inertia due to the propeller is ignored here).
According to Newton's laws of motion and Euler's formula, a simplified dynamic model of the quadrotor is given below [38,39] where m is the mass of the quadrotor, g is the acceleration of gravity, K i (i � 1, 2, 3, 4, 5, 6) are the drag coefficients for the system, and I x , I y and I z are the principal moments of inertia. When the quadrotor is flying at low speed indoors, the control inputs u 1 , u 2 , u 3 , and u 4 represent the lift torque, roll torque, pitch torque, and yaw torque of the quadrotor, respectively. u 1 , u 2 , u 3 , and u 4 are as follows: Equation (5) can be rearranged in matrix form as follows: where l is the linear distance from the center of the rotor to the center of gravity.
where b is the thrust coefficient, which depends on the blade rotor characteristics; α is the force to moment scaling factor [38], and w i is the angular speed of the ith propeller of the quadrotor. From equation (4), the quadrotor dynamics presented in E-frame in the presence of external disturbances is given by where d i (i � 1, 2, 3, 4, 5, 6) are the unknown disturbances that contain system uncertainties and other unknowns. e control objective is to design the input control so that the quadrotor tracks the time-varying desired trajectory

Mathematical Problems in Engineering
However, in the dynamic model of the quadrotor from equation (8), there are only four control inputs, but six outputs x, y, z, ϕ, θ, ψ to control. To deal with the underactuated problem, we consider three virtual control inputs (v 1 , v 2 , v 3 ), as follows [35]: Applying the three virtual control inputs to (8), the dynamic model can be rewritten as By setting the desired yaw angle ψ d and using equation (9), the input control u 1 , roll angle ϕ d , and pitch angle θ d are given by erefore, the trajectory tracking control objective can be described as follows: given the desired trajectory [x d , y d , z d , ψ d ] T , the idea is to design the control laws v 1 , v 2 , v 3 and u 2 , u 3 , u 4 , such that the tracking errors converge to zero asymptotically.

Controller Design and Stability Analysis
In this section, the proposed controller was divided into an inner loop (attitude) controller and an outer loop (position) controller. For the inner and outer loops, a novel fuzzy sliding mode controller based on LFSC was first developed. e proposed controller guaranteed that the reference position (x d , y d , z d ) and attitude ψ d could be accurately tracked. Using equation (10), the reference attitude (ϕ d , θ d ) could also be accurately tracked. Finally, the entire closedloop system could quickly track the reference signals. e overall control structure of the quadrotor is shown in Figure 2.

Outer Loop Controller Design.
e position could be extracted from equation (10) as follows: where Defined the position tracking error as where e X � [x e , y e , z e ] T . en, the LFSC manifold was given by where c X and C were positive constants. Parameter c X was related to the rate of approaching the sliding mode surface, the larger the parameter c X , the faster the approaching rate, while the greater the overshoot. Parameter C is related to retain the system states on the sliding mode surface. s X � [s x , s y , s z ] T . en, the derivative of s X with respect to time was Consider the tracking error (13) and the LFSC manifold (14). e virtual control law for the outer loop was as follows: where ξ was a positive constant. Parameter ξ was related to retain the control system stability.
were the parameters obtained by the fuzzy controller. To reduce the impact of external disturbances and attenuate chattering, d o at the next moment must be predicted.
where t was time and T was the sampling interval of time.
e Lyapunov function was chosen as follows: en, the derivative of V 1 with respect to time was given by Substituting the virtual control law (16) into (20), we obtained the derivative of V 1 : To guarantee V 1 ≤ 0, the appropriate parameters of H 1 (t) must be selected, such that H 1 (t) was sufficiently large to balance the error in the prediction of the disturbances, expressed as e combination of (21) and (22) implied that the LFSC manifold designed in (14) was feasible. To ensure that (22) is true, a fuzzy controller could be constructed to obtain H 1 (t).
e fuzzy controller in this paper has two parameters, s x , s x · , as the input; H 1 (t) was the only output. e fuzzy rules are shown in Table 1. e rules of the controllers are expressed in Table 1, for all possible combinations. Based on the Mathematical Problems in Engineering control experience of the quadrotor, the fuzzy rules were established according to s x and s x · to adjust parameter H 1 (t). e membership function of the fuzzy controller is shown in Figure 3.
From Table 1 e proof process of V 2 · ≤ 0 and V 3 · ≤ 0 were similar to that of V 1 · ≤ 0. A block diagram of the LFSC method is shown in Figure 4.

Inner Loop Controller Design.
e attitude could be extracted from equation (10) as follows: where J � (I x , I y , e LFSC manifold was given by where c η and C 1 were positive constants. e meaning of c η and C 1 were similar to c X and C respectively. en, the derivative of s η with respect to time was as follows: Consider the tracking error (25) and the LFSC manifold (26). In this case, control law u was designed as where ξ 1 was a positive constant, the meaning of ξ 1 was similar to ξ. en, the meaning of H 4 (t), H 5 (t), H 6 (t) and d 4 (t), d 5 (t), d 6 (t) were similar to H 1 (t) and d 1 (t) in Section 3.1, respectively.
e Lyapunov function was as follows: Substitute the control law (28) into (30). en, the derivative of V 4 was given by To guarantee V 4 ≤ 0, appropriate parameters of H 4 (t) must be selected, such that |H 4 (t)| was sufficient to balance the error in the prediction of interference, which could be expressed as e combination of (31) and (32) implied that the sliding manifold design described in (26) was feasible. e proof process of equation (32) was similar to equation (22) in Section 3.1, in which s ϕ s ϕ · ≤ 0, and the Lyapunov function V 4 ≤ 0 could be guaranteed. Based on the Lyapunov method, the system was asymptotically stable: e proof process of V 5 · ≤ 0 and V 6 · ≤ 0 were similar to that of V 4 · ≤ 0.

Results and Discussion
In this section, several trajectory tracking simulation experiments were performed in the MATLAB R2016b/Simulink, which was equipped in a computer consisting of a 2.60 GHz CPU with 8 GB of RAM and a 256 GB solid-state disk drive. e control performance obtained by the proposed LFSC scheme was compared to SMC [40] and fuzzy SMC [41] schemes, to demonstrate the superiority of the proposed LFSC strategy. e parameters of the quadrotor used in the simulation studies are shown in Table 2.
e external disturbances considered in all of the simulation studies, to validate the robustness of the proposed LFSC control strategy, were time-varying. Case 1. In this case, the desired trajectory of the position and yaw angle was given by e initial position and yaw angle of the quadrotor [x 0 , y 0 , z 0 , ψ 0 ] were [1.9, 2.9, 0.5, 0.001].
e Gaussian function and white noise functions which were imposed on the quadrotor were given by M 1 cos(t) (i � 4, 5, 6).
In order to achieve appropriate control performance, appropriate parameters c X , ξ, C, c η , ξ 1 , and C 1 were designed according to reference [38]. e parameters for the proposed LFSC controllers were shown in Table 3.
Simulation results are shown in Figures 5-11. To demonstrate the superiority of the proposed LFSC scheme, simulation experiments of traditional SMC and fuzzy SMC were conducted, more details of the SMC and the fuzzy SMC methods for a quadrotor UAV have been introduced in [40,41], respectively. e trajectory tracking results in 3D space are shown in Figure 5. e position tracking errors (e x , e y , e z ) are shown in Figure 6. e following results can be observed from Figure 6: (1) It could be seen that starting from an initial position far from the desired trajectory, the proposed LFSC method successfully forces state variables (x, y, z) to their desired trajectory. (2) e proposed LFSC method successfully forces position tracking errors (e x , e y , e z ) to converge to zero in finite time, while the SMC and fuzzy SMC methods force position tracking errors (e x , e y , e z ) to converge to small bounded fields around zero.
(3) Moreover, when disturbed by the wind gust, time is about 15 s, the performance of the proposed LFSC controller was much better than that achieved by SMC or fuzzy SMC controllers in terms of settle time, overshoot, and robustness. It was clearly seen that the proposed LFSC method was able to make the quadrotor follow the desired position trajectory with strong robustness and the highest accuracy. Due to the linear prediction and the fuzzy controller, disturbances were well compensated. e attitude tracking errors (e ϕ , e θ , e ψ ) are shown in Figure 7. e following results can be observed from Figure 7: (1) e proposed LFSC method successfully forces attitude tracking errors (e ϕ , e θ , e ψ ) to converge to zero in finite time, while the SMC and fuzzy SMC methods force attitude tracking errors (e ϕ , e θ , e ψ ) to converge to small bounded fields around zero. (2) e performance of the proposed LFSC method was much better than that achieved by the SMC or fuzzy SMC method in terms of tracking accuracy and robustness. It could be seen that the timevariant disturbances were not well compensated with traditional SMC or fuzzy SMC method. By introducing the linear prediction and new fuzzy controller, the LFSC method achieved good attitude tracking performance. e response curves of virtual control input v 1 , v 2 , and v 3 under the three control methods are displayed in Figures 8-10, respectively. By observing these figures, the high-frequency chattering phenomenon caused by the switching control action are shown in Figures 8 and 9. While in Figure 10, chattering was considerably reduced by adopting the proposed LFSC control law. In Figure 10, the changes in three LFSC virtual control inputs v 1 , v 2 , v 3 were shown, whereby the virtual control input v 3 was around 19.6 N, which was equal to the gravity force of the quadrotor. In addition, three virtual control inputs v 1 , v 2 , v 3 showed huge fluctuations around 15 s, which is caused by the wind gust.
In Figure 11, the changes in four control inputs (u 1 , u 2 , u 3 , u 4 ) are shown, whereby the control input u 1 was 19.71 N, which was slightly greater than the gravity force of the quadrotor. Compared with the results given in [40,41], the amplitudes of the proposed LFSC controllers (u 1 , u 2 , u 3 , u 4 ) were greatly decreased in the presence of external disturbances.
Case 2. In order to further evaluate the performance of the proposed LFSC control strategy, the disturbances were simulated by white noise functions with standard deviation of 2. In this case, the desired trajectory of the position and yaw angle was given by  Figure 13. When the disturbance d i is introduced at t � 15 s, the null steady-state error could not be reached with both SMC method and fuzzy SMC method. In contrast, the proposed LFSC control strategy is able to force the quadrotor follow the desired trajectory with the highest accuracy and null steady-state error. Due to the linear prediction and the fuzzy controller, disturbances were well compensated. e attitude tracking errors (e ϕ , e θ , e ψ ) are shown in Figure 14. It could be seen that the time-variant disturbances were not well compensated with both SMC method and fuzzy SMC method when the time was up to 15 seconds. By introducing the linear prediction and the fuzzy controller, the LFSC method achieves good attitude tracking performance.  Mathematical Problems in Engineering 9 x (m ) y (m )     Figures 17 and 18, due to the linear prediction and the fuzzy controller, the chattering problems was considerably reduced by adopting the proposed LFSC control law.

Conclusions
In this paper, an LFSC scheme was presented to guarantee that the tracking errors of all system state variables converge to zero in finite time and eliminate the chattering phenomenon caused by the switching control action, in the presence of external disturbances and uncertainties. Firstly, a linear prediction method was devised to minimize the effects of external disturbances. Secondly, a new fuzzy law was implemented to eliminate the chattering phenomenon. In addition, the stabilities of position and attitude were demonstrated by Lyapunov theory, respectively. Finally, several quadrotor trajectory tracking simulation examples were presented. e control performances obtained using  traditional SMC and fuzzy SMC schemes were compared to demonstrate the superior performance of the proposed LFSC scheme.
e main conclusions are summarized as follows.
(1) e proposed LFSC scheme can guarantee that the tracking errors of all system state variables converge to zero in finite time.
(2) e high-frequency chattering phenomenon caused by the switching control action does not appear using the proposed LFSC scheme. (3) Simulation results demonstrate the superior performance and robustness of the proposed LFSC scheme in the case of external disturbances. (4) Compared with [42,43], simulations demonstrate the accuracy and superiority of the proposed LFSC method.
e simplicity of the approach, and the use of continuous control signals, makes it readily applicable to a real quadrotor. Advantages of the proposed LFSC method are accuracy, robustness, state variables converge to zero in finite time, and no chattering phenomenon. erefore, the quadrotor with the LFSC method can be applied to the emergency mission, disaster relief mission, and special military mission. Further work will focus on utilizing the deep learning method to compensate the uncertainties of the system without requirement of their upper bounds and utilizing software ANSYS to research disturbances caused by complex external environments.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.