RBFNN-Based Singularity-Free Terminal Sliding Mode Control for Uncertain Quadrotor UAVs

In this article, a singularity-free terminal sliding mode (SFTSM) control scheme based on the radial basis function neural network (RBFNN) is proposed for the quadrotor unmanned aerial vehicles (QUAVs) under the presence of inertia uncertainties and external disturbances. Firstly, a singularity-free terminal sliding mode surface (SFTSMS) is constructed to achieve the finite-time convergence without any piecewise continuous function. Then, the adaptive finite-time control is designed with an auxiliary function to avoid the singularity in the error-related inverse matrix. Moreover, the RBFNN and extended state observer (ESO) are introduced to estimate the unknown disturbances, respectively, such that prior knowledge on system model uncertainties is not required for designing attitude controllers. Finally, the attitude and angular velocity errors are finite-time uniformly ultimately bounded (FTUUB), and numerical simulations illustrated the satisfactory performance of the designed control scheme.


Introduction
Due to the advantages of a large range of applications of QUAVs, the attitude control of QUAVs has received extensive attention in many fields. Designing a suitable control law according to the type of flight mission is important for the study of QUAVs attitude control [1]. However, the control scheme proposed in [1] only achieves the asymptotic convergence of the attitude or tracking error, which means that the system states converge to zero within an infinite time. In practical applications, the controller design needs specific requirements for convergence speed and control accuracy.
us, finite-time control (FTC) methods have been proposed to deal with the control problems of different nonlinear systems [2][3][4]. In [2], a parameter update law was constructed to compensate for parameter uncertainty, and an adaptive sliding mode control (SMC) scheme was constructed to ensure the finite-time convergence of the tracking error of QUAV. In [3], combining integral backstepping technology and terminal SMC, a FTC method was designed to guarantee the position and attitude tracking stability of QUAVs, such that the system states are semiglobal practical FTUUB. In [4], a FTC scheme was proposed to guarantee the finite-time convergence of position and attitude states in the QUAV tracking error system.
To solve the dependence of the FTC on a nonlinear system model, multiple control methods have the capability to approximate nonlinear functions, which is used to implement the estimation task of the nonlinear system, i.e., adaptive control [5] and optimal control [6]. In the past years, a method called RBFNN has been widely introduced to approximate the dynamic parameters of nonlinear systems, such that no prior knowledge of model information is required in [7,8]. In [7], a RBFNN method was devolved to estimate the unknown model uncertainties of robot manipulators. In [8], a RBFNN-based SMC scheme was presented to guarantee the asymptotic convergence of the system states under the model uncertainties. Compared with other estimation methods [5,6], RBFNN has faster convergence speed and local approximation capability to avoid local minima problems. us, RBFNN is more suitable for real-time control, such as QUAV attitude control.
Moreover, for complex coupled systems, extended state observer (ESO) as an alternative approach to solve the bounded disturbances is used to decouple the system by treating the coupling terms as a part of the lumped uncertainties. Due to its satisfactory disturbance estimation, ESO-based controllers are widely applied in different practical nonlinear systems, such as rigid spacecrafts [9] and robot manipulators [10]. In [9], a SMC based on backstepping and ESO methods was used to achieve a faster convergence in the rigid spacecraft attitude control system. In [10], an FTC with the model-assisted ESO was used to compensate the bounded uncertainties and guarantee the finite-time convergence of the system states. Although RBFNN and ESO have been successfully applied to a variety of uncertain nonlinear systems [7][8][9][10], it is less used in QUAVs.
Inspired by the above discussions, an RBFNN-based finite-time adaptive attitude tracking controller is designed for the attitude tracking problem of QUAVs with inertial uncertainty and unknown external disturbances, and the main contributions are summarized in the following: (i) Instead of employing any piecewise continuous functions, a SFTSMS is proposed to avoid the singularity directly in the differential of the sliding variable (ii) An auxiliary function is designed to handle a potential singularity resulted from the use of the errorrelated inverse matrix in the attitude controller design (iii) By employing RBFNN and ESO to estimate the unknown dynamics, prior knowledge on system model uncertainties is not required in the controller design, and the tracking errors are FTUUB by the proposed control law e structure of this article is given as follows. e attitude model and necessary preliminaries are given in Section 2. Section 3 shows the detailed design process of SFTSMS. Controller design and rigorous theoretical proofs are depicted in Sections 4 and 5, respectively. Section 6 shows the effective simulations, and the conclusion is given in Section 7.

Model Description and Preliminaries
2.1. Quadrotor Attitude Dynamics. As depicted in Figure 1, the dynamics of the QUAV is represented as follows [11]: where q 0 ∈ R and q v � [q 1 , q 2 , q 3 ] T ∈ R 3×1 denote the scalar and vector elements of the unit quaternion Q � [q 0 , q T v ] T , respectively, and I 3 ∈ R 3×3 is the identity matrix, and the skew-symmetric matrix A × ∈ R 3×3 is given by e reference attitude vector is defined as Q r � [q r 0 , q r 1 , q r 2 , q r 3 ] T ∈ R 4×1 , and _ Q r and € Q r are bounded. e attitude tracking error e q is and the dynamics model is expressed as where ω ∈ R 3×1 denotes the angular velocity, J ∈ R 3×3 is a positive definite inertia matrix, u � [u 1 , u 2 , u 3 ] T ∈ R 3×1 represents the generalized control torque produced by rotating propellers, and d � [d 1 , d 2 , d 3 ] T ∈ R 3×1 denotes the unknown but bounded continuous external disturbances.
Defining J � J 0 + ΔJ, (4) is expressed as where J 0 and ΔJ are denoted as the nominal and the unknown inertia matrix, respectively. e angular velocity tracking error is where ω r is given by From (5)- (7), the tracking error model is expressed as Figure 1: e schematic of quadrotor UAVs. 2 Computational Intelligence and Neuroscience Remark 1. Unit quaternion is able to represent the attitude uniquely because the equilibrium states correspond to a unique physical equilibrium orientation [12] in the QUAV.

RBFNN.
In practical systems, neural networks (NNs) are online estimation techniques for unknown nonlinear uncertainties. Due to the approximation characteristics and faster learning convergence, RBFNN is widely used in the estimation of nonlinear functions in the field of control. is section will introduce the structure of RBFNN. RBFNN consists of three parts: input layer, output layer, and hidden layer. As shown in Figure 2 is the weight of the mth network node, y represents the output vector, and ϕ(x) � [ϕ 1 , ϕ 2 , . . . , ϕ m ] T is the basis function, which can approximate nonlinear uncertainties with high precision through the linear combination of Gaussian functions, which is given by the following [7]: where μ k is the center of the RBF and a k means the scaling parameter of the network node m. us, the output vector y is expressed as Considering that RBFNN has good nonlinear approximation ability [13], the approximate system model of the nonlinear function F is where ε represents the estimation error.

SFTSMS. A SFTSMS is constructed as
where 1 and a 2 are positive odd integers, and the term sig r (x) is given by (14) is expressed as Due to the facts that 1 − a > 0, the singularity will not occur in (16).
When S � 0, (14) is rewritten as According to 2 − a > 1, the following equation is obtained as From (18), the equivalent equation of (14) is obtained as follows [12]: From (8) and (19), one has Computational Intelligence and Neuroscience 3 To illustrate e q is finite-time convergent, a Lyapunov function is chosen as From (20), the time derivative of V 1 is where According to Lemma 1 and (8), e q and e ω converge to the equilibrium within a finite-time T s satisfying where e q (0) represents the initial states of e q (t). is completes the proof.

Auxiliary Function Design.
Substituting (8) into (16) yields where Due to the existence of the term of F 0 in the expression of Y − 1 , it may cause the potential singularity issue when Θ � 0. Consequently, an auxiliary function φ is constructed to solve the singularity caused by Y − 1 in the controller design.
us, (24) is expressed as From the definition of S and Θ, it has From (26), the auxiliary function φ is defined as and (25) can be rewritten as where F denotes the lumped uncertainties presented by

ESO.
e bounded external disturbances D in (28) are estimated by the ESO. Considering the disturbances D as an extended state, system (28) is rewritten as where h(t) represents the derivative of D.
With the unknown nonlinear uncertainties F and the disturbances D estimated by the RBFNN and ESO, respectively, the finite-time control law u is given by where K 3 > 0, W � diag(W 1 , W 2 , W 3 ), and W i is used to estimate W * i with i � 1, 2, 3. e update law of W i is given by where δ i > 0 and ϖ i > 0 with i � 1, 2, 3.

Substituting (36) into (46) yields
where us, (47) can be transformed into the following form: Due to Lemma 2 and (48), S can converge into Δ S satisfying within a finite-time bounded by where S 0 is the initial value of S. From (14), one has S j � sig e q j + Λ 1 2 − a sig 2− a e ω j + Λ 2 sig e q j � η j , where j � 1, 2, 3. e terms (Λ 1 /2 − a)sig 2− a (e ω j + Λ 2 sig(e q j )) and sig(e qj ) are both positive or negative. Considering only positive case and Lemma 3, the following relationship exists without loss of generality: 6 Computational Intelligence and Neuroscience Multiplying Λ 1 and adding sig(e q j ) on both sides of (52), it has sig e q j + Λ 1 e ω j + Λ 2 sig e q j 2− a 2 − a ≥ sig e q j + Λ 1 e ω j + Λ 2 sig e q j + Λ 1 (a − 1) 2 − a . (53) Substituting (53) into (51) yields sig e q j + Λ 1 e ω j + Λ 1 Λ 2 sig e q j ≤ ΔS, where ΔS � ΔS + (Λ 1 (1 − a)/2 − a). From (54), the attitude tracking errors e q and angular velocity e ω are finite-time stable. us, e q and e ω converge into small regions: within a finite-time, respectively. From the above discussion, the convergence time T of closed-loop system states in (8) is bounded, which satisfies Consequently, the sliding variable S and the tracking errors e q and e ω are FTUUB.
is completes the proof.

Remark 2.
e convergence time is determined by the parameters Λ 1 , Λ 2 , K 1 , K 2 , and ι 0 . From (56), we can find that when the parameters Λ 1 , Λ 2 , K 1 , and K 2 are selected to be larger, the convergence speed will become faster while it will cause some chattering issues.
us, the choice of Λ 1 , Λ 2 , K 1 , and K 2 should be set appropriately to keep a balance between convergence speed and chattering.

Simulations Results
In this section, numerical simulations are displayed to show the attitude tracking performance. To indicate the superiority of the designed control scheme, two different control schemes are given for comparison, i.e., SFTSM controller [20] and adaptive linear sliding mode (ALSM) controller [21].
To intuitively display the physical meaning of the reference attitude trajectories, the reference attitude quaternion [q r 0 , q r v1 , q r v2 , q r v3 ] is expressed as where [ϕ d , θ d , ψ d ] are the desired Euler angles. e parameters in (8)  In SFTSM, the parameters in (14) are set as Λ 1 � 0.1, Λ 2 � 0.4, and a � 5/7. e parameters in (36) are chosen as K 1 � 0.5, K 2 � 1.1, K 3 � 4, and c � 0.1. e number of RBFNN nodes is 10. e corresponding parameters in (9) and adaptive update law (37) are chosen as μ i ∈ (− 2, 2), In ALSM, the linear sliding variable S is given by where Λ 0 � 7.5, and the controller is in which the updating law of θ 0 is given by where c 0 � 1, and the other parameters in the ALSM are set the same as those of SFTSM.
To illustrate the better tracking performance and transient convergence performance in SFTSM, the fixed initial Euler angles are selected with ϕ d � θ d � ψ d � 10°as the reference trajectories. Figures 3 and 4 show the time response of control inputs in SFTSM and ALSM, respectively. It is concluded that the maximum amplitude of the control input is 10 N · m. e comparative results of the Euler angle tracking performance are shown in Figures 5-7. From Figure 5, the convergence time of ϕ in SFTSM is 0.5 s, and the convergence time of ϕ in ALSM is 1.5 s. Due to the same analysis in Figures 6 and 7, SFTSM can achieve faster convergence rate and better transient performance of Euler angles than ALSM.
For the purpose of showing the superiority of SFTSM, the corresponding attitude quaternion tracking performance in ALSM is presented in Figures 8-11. e convergence time of SFTSM is faster almost 1 s than ALSM, which means SFTSM can realize the better transient performance than ALSM. Based on the mentioned analysis, quaternion-based tracking performance in SFTSM is still outperformed than Computational Intelligence and Neuroscience

Conclusion
In this study, a finite-time convergent RBFNN-based adaptive controller has been constructed to resolve a tracking problem of quadrotor UAVs. Firstly, a SFTSMS is proposed to realize the finite-time convergence of the tracking errors, which can directly avoid the potential singularity problem without requiring any piecewise continuous functions. Besides, an auxiliary function is proposed to purposely prevent the hidden singularity issue caused by the error-related matrix in the controller design. en, a finitetime attitude controller is designed to guarantee that the system states were FTUUB. With the presented control scheme by RBFNN and ESO, prior knowledge about the unknown nonlinear uncertainties and external disturbances is not required. Finally, comparative simulations have shown the effectiveness of the designed control scheme.

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

Disclosure
Preprint is submitted to Computational Intelligence and Neuroscience.  Computational Intelligence and Neuroscience 9