Robust adaptive tracking control for unmanned helicopter with constraints

Abstract

In this article, a prescribed performance-based adaptive neural network control scheme is proposed for an uncertain small-scale unmanned helicopter system subject to input saturations and output constraints. The radial basis function neural networks are employed to approximate system uncertainties. A nonlinear disturbance observer is developed to tackle input saturation. Meanwhile, the prescribed performance function is adopted to deal with output constraint. The closed-loop system stability is rigorously proved using Lyapunov synthesis. Finally, simulation results for unmanned helicopter system are presented to demonstrate the effectiveness of developed tracking control scheme using disturbance observer and radial basis function neural network.

Introduction

Unmanned helicopters have received a considerable attention and extensive development in the recent years. ^{1 –3} Especially, the small-scale unmanned helicopter system with complex dynamics is sensitive to model uncertainty and external disturbance. ^{4 –7} To deal with this problem, multifarious robust controller design techniques have been developed. ^{8 –10} Chen et al. ¹¹ and Choi and Yoo ¹² investigated the controller design for the unmanned helicopter using disturbance observer and fuzzy approximator. Zhu and Huo ¹³ proposed a robust nonlinear adaptive backstepping method for the trajectory tracking control of the model-scaled unmanned helicopter with uncertain parameters. Lee and Tsai ¹⁴ and Zhu ¹⁵ proposed a nonlinear adaptive backstepping controller for a class of unmanned helicopters. Yue-Bang et al. ¹⁶ proposed an adaptive backstepping control for simplified unmanned helicopters with unmodeled dynamics and external disturbances. Visual positioning method has been widely used in unmanned helicopter flight control. ^{17 –22} In this article, to facilitate trajectory tracking controller design, a backstepping control strategy is adopted for the small-scale unmanned helicopter system. However, the aforementioned works did not involve the input and constraint problem for unmanned helicopter systems. Thus, in this article, we will develop an adaptive constrained control approach for the unmanned helicopter.

As we all known, neural network (NN) and nonlinear disturbance observer are two of the main solutions to improve the tracking performance and robustness of the closed-loop system. ^{23 –25} Considering the uncertain input delay and time-varying disturbance, Chen et al. ²⁶ studied an NN-based robust control approach for the nonlinear systems. For the nonaffine nonlinear uncertain systems, Zouari et al. ²⁷ developed an adaptive radial basis function NN (RBFNN) feedback control scheme. To guarantee the system with input saturation nonlinearities asymptotically stable, NN was incorporated into the variable structure control design framework in the study by Miao et al. ²⁸ For a class of uncertain nonlinear systems in the presence of input saturations, a nonlinear disturbance observer-based dynamic surface control scheme was proposed by Cui et al. ²⁹ In the study by Chen et al., ³⁰ a recurrent wavelet NN-based nonlinear disturbance observer was designed, and the disturbance estimate was adopted into the controller design. As an effective method to deal with the external disturbance, disturbance observer has been attracted extensive attentions. ^31,32 In this article, we will employ NN and disturbance observer to deal with the model uncertainty and external disturbance of the small unmanned helicopter, respectively.

Many previous works focus on the conventional helicopter flight control, in which the input/output constraints have been rarely considered. If the problem of input saturation is not considered in the controller design process, the tracking performance of closed-loop system will be severely degraded, what’s more, it may cause instability. Fortunately, several research methods on the control constraint problem have been reported in the literature. For example, anti-windup schemes were designed by Tong et al. ³³ and Grimm et al. ³⁴ To efficiently handle the input saturation, a positively invariant sets method was proposed by Cao et al. ³⁵ and Richter. ³⁶ Zhou et al. ³² studied an adaptive control law for the nonlinear systems subject to actuator saturation. Cao and Lin ³⁷ developed an adaptive control algorithm for a class of minimum phase systems with input saturation. Zhong ³⁸ employed an adaptive control method for the flexible spacecraft control system with input saturations. At the same time, there were some control methods proposed for the problem of output constraint. ³⁹ Since the barrier Lyapunov function candidate was originally proposed by Ngo et al., it had been widely used for the nonlinear system control with output constraints. ⁴⁰ By employing the barrier Lyapunov function, Ngo et al. ⁴¹ designed an adaptive neural control law to tackle the problem of control output constraint. Tee et al. ⁴² proposed an asymmetric time-varying barrier Lyapunov function-based controller to ensure constraint requirement for the nonlinear time-varying systems. The barrier Lyapunov function-based backstepping control scheme was proposed by Tee et al., ⁴³ wherein constraints were presented in some or even all of the states. In the study by Tong et al., ⁴⁴ for the constrained uncertain nonlinear systems, an indirect adaptive controller was investigated by employing fuzzy logic system and barrier Lyapunov function. Almost concurrently, by imposing explicitly prescribed bounds on both transient and steady-state tracking error performance, the prescribed performance control was proposed by Li et al. ⁴⁵ Since then, prescribed performance control has attracted considerable research efforts. In the study by Bechlioulis and Rovithakis, ⁴⁶ guaranteed prescribed performance-based adaptive tracking control method was investigated for the strict feedback systems. In the study by Bechlioulis and Rovithakis, ⁴⁷ the prescribed performance was provided to position control for robotic manipulators. ⁴⁸ The prescribed performance control approach was applied for the energy conversion systems in the study by Bechlioulis et al. ⁴⁹ Although there exist some research results about input saturation and output constraint, the control problem for the small-scale unmanned helicopter system need to be further studied considering both input saturation and output constraint. In this article, nonlinear disturbance observer will be introduced to handle the input saturations, and prescribed performance function method will be introduced to handle the output constraints for the unmanned helicopters.

Motivated by above discussion and analysis, we will present an adaptive control law for the small-scale helicopter system subject to model uncertainty, unknown external disturbance, and input/output constraints.

The remainder of this article is organized as follows. In “Problem statement and preliminaries” section, we address the problem statement and present some preliminaries. Following that, “Controller design” section presents the design of adaptive neural tracking controller using the prescribed performance method and nonlinear disturbance observer. In “Simulation results” section, simulation studies of the proposed adaptive tracking controller for the small-scale helicopter system are presented. Some conclusion remarks are addressed in “Conclusion” section.

Problem statement and preliminaries

In this section, it is particularly aimed to briefly review the completed nonlinear dynamic model of unmanned helicopters and introduce some preliminary knowledge about NN and prescribed performance-based control method.

Helicopter model

The considered small-scale unmanned helicopter in this work is Trex-250. Moreover, the external disturbances are introduced by adding some aerodynamical additional forces and torques. The rigid-body dynamic equations of the small-scale unmanned helicopter are described in the form of ¹⁴

$\begin{array}{c}\dot{P}=V\\ \dot{V}=g{e}_3+\frac{1}{m}R\left(\Theta \right)u_T+\Delta F+d_f\\ \dot{\Theta }=H\left(\Theta \right)\Omega \\ I_m\dot{\Omega }=-\Omega \times I_m\Omega -Q_{tr}{e}_2-Q_{mr}{e}_3+G\mathit{\tau }+\Delta M+d_{\mathit{\tau }}\\ \dot{R}\left(\Theta \right)=R\left(\Theta \right)sk\left(\Omega \right)\end{array}$

1

where P = [x, y, z] ^T is the position and V = [u_v, v_v, w_v] ^T is the velocity; Θ = [ϕ, θ, ψ] ^T and Ω = [p, q, r] ^T are the attitude angle and angular rate, respectively. e₃ = [0, 0, 1] ^T , g represents the gravitational acceleration, m ∈ R indicates the total mass, and I_m ∈ R^3×3 denotes the inertial moment matrix. Q_mr and Q_tr, respectively, denote the main rotor anti-torque and tail rotor anti-torque. G∈R^3×3 is the control matrix related to the control moments. ΔF and ΔM, respectively, indicate the force and moment uncertainties, and d_f and d_τ indicate the external disturbances. H(Θ) stands for the transformation matrix which can be represented as

$\begin{array}{l}H\left(\Theta \right)=\left[\begin{array}{ccc}1& S\varphi S\mathit{\theta }/C\mathit{\theta }& C\varphi S\mathit{\theta }/C\mathit{\theta }\\ 0& C\varphi & -S\varphi \\ 0& S\varphi /C\mathit{\theta }& C\varphi /C\mathit{\theta }\end{array}\right]\end{array}$

2

Here, sk(Ω) indicates skew-symmetric matrix for the angular rate vector, and R(Θ) denotes the rotation matrix from body-fixed coordinate frame to inertial coordinate frame, which are derived as

$\begin{array}{l}R\left(\Theta \right)=\left[\begin{array}{ccc}{R}_{1,1}& R_{2,1}& R_{3,1}\\ R_{1,2}& R_{2,2}& R_{3,2}\\ R_{1,3}& R_{2,3}& R_{3,3}\end{array}\right]\end{array}$

3

$\begin{array}{l}{R}_{1,1}=C\mathit{\theta }C\psi \\ R_{2,1}=S\varphi S\mathit{\theta }C\psi -C\varphi S\psi \\ R_{3,1}=C\varphi S\mathit{\theta }C\psi +S\varphi S\psi \\ R_{1,2}=C\mathit{\theta }S\psi \\ R_{2,2}=S\varphi S\mathit{\theta }S\psi +C\varphi C\psi \\ R_{3,2}=C\varphi S\mathit{\theta }S\psi -S\varphi C\psi \\ R_{1,3}=-S\mathit{\theta }\\ R_{2,3}=S\varphi C\mathit{\theta }\\ R_{3,3}=C\varphi C\mathit{\theta }\end{array}$

$\begin{array}{l}sk\left(\Omega \right)=\left[\begin{array}{ccc}0& -r& q\\ r& 0& -p\\ -q& p& 0\end{array}\right]\end{array}$

4

where S(⋅) and C(⋅) indicate trigonometric functions sin(⋅) and cos(⋅), respectively. u_T ∈ R³ and τ ∈ R³ denote acting forces and control torques on the helicopter, respectively. Furthermore, considering the input saturation, the acting forces can be described by

$\begin{array}{l}{u}_{T_i}=\text{sat}\left(v_{T_i}\right)=\{\begin{array}{cc}\text{sgn}\left(v_{T_i}\right){\overline{u}}_{T_i}& \left|v_{T_i}\right|\ge {\overline{u}}_{T_i}\\ v_{T_i}& \left|v_{T_i}\right|<{\overline{u}}_{T_i}\end{array}\end{array}$

5

where ${\overline{u}}_{T_i}$ is the bound of $u_{T_i}$ , i = 1,2,3, and $v_T=[v_{T_1},v_{T_2},v_{T_3}{]}^T$ is the designed control vector. In view of the saturation, the actual control moment can be expressed as

$\begin{array}{l}{\mathit{\tau }}_i=\text{sat}\left(v_{{\mathit{\tau }}_i}\right)=\{\begin{array}{cc}\text{sgn}\left(v_{{\mathit{\tau }}_i}\right){\overline{\mathit{\tau }}}_i& \left|v_{{\mathit{\tau }}_i}\right|\ge {\overline{\mathit{\tau }}}_i\\ v_{{\mathit{\tau }}_i}& \left|v_{{\mathit{\tau }}_i}\right|<{\overline{\mathit{\tau }}}_i\end{array}\end{array}$

6

where ${\overline{\mathit{\tau }}}_i$ is the bound of τ_i, i = 1,2,3, and $v_{\tau }=[v_{{\mathit{\tau }}_1},v_{{\mathit{\tau }}_2},v_{{\mathit{\tau }}_3}{]}^T$ is the designed control vector.

Neural networks

In recent literature of robust adaptive control, due to the inherent approximation capabilities of RBFNNs, they are mostly adopted as approximator for the unknown nonlinearities. Suppose that Δ(ω) : R^q → R represents an unknown and smooth nonlinear function, which can be approximated on the compact set Ω ⊆ R^p by a class of linearly parameterized RBFNNs ²⁶

$\begin{array}{l}\Delta (\mathit{\omega })={\widehat{W}}^{T}S(\mathit{\omega })+\epsilon (\mathit{\omega })\end{array}$

7

where ω = [ω₁, ω₂, …, ω_q] ^T ∈ R^q means the approximator input vector, S(ω) = [s₁, s₂, …, s_p] ^T ∈ R^p is the basis function, $\widehat{W}\in R^p$ is a weight vector of the RBFNN, and $\widehat{W}$ indicates the estimate of W*. ε means approximation error which satisfies $\left|\epsilon \right|\le \overline{\epsilon }$ and $\overline{\epsilon }$ represents the bound of unknown parameter. Assume that the optimal approximation can be expressed as ²³

$\begin{array}{l}\Delta (\mathit{\omega })=W^{*T}S(\mathit{\omega })+{\epsilon }^{*}(\mathit{\omega })\end{array}$

8

where ε* represents the approximation error for special case: $\widehat{W}=W^{*}$ . Meanwhile, W* denotes the ideal weights in the output layer, which can be defined by

$\begin{array}{l}{W}^{*}=\text{arg}\underset{\widehat{W}\in \Omega }{{\displaystyle \text{min}}}\left[\underset{\mathit{\omega }\in {\Xi }_{\mathit{\omega }}}{{\displaystyle \text{sup}}}\left|\Delta (\mathit{\omega })-{\widehat{W}}^{T}S(\mathit{\omega })\right|\right]\end{array}$

9

where $\Omega =\{\widehat{W}:\parallel \widehat{W}\parallel \le M\}$ represents a valid field of estimate parameter $\widehat{W}$ , M > 0. Ξ_ω indicates an allowable set of state ω. And the Gaussian function can be written as ²⁹

$\begin{array}{l}{s}_i(\mathit{\omega })=\text{exp}\left[-\frac{{\left(\mathit{\omega }-c_i\right)}^T\left(\mathit{\omega }-c_i\right)}{b_i^2}\right],i=1,2,\dots ,p\end{array}$

10

where b_i ∈ R and c_i ∈ R^m are the width and center of the neural cell, respectively.

Prescribed performance function method

We know that one of the control objective is to deal with the output constraint, and the prescribed performance function method will be introduced in this article. Here, we recall some preliminary results and definitions which are necessary in the following design and analysis.

Definition 1

A smooth bounded function ρ(t) : R⁺ + {0} → R⁺ can be called as a performance function, when ρ(t) is decreasing, $\left|\mathit{\delta }\left(0\right)\right|<\rho \left(0\right)$ and $\underset{t\to \infty }{{\displaystyle \text{lim}}}\rho \left(t\right)={\rho }_{\infty }>0$ . For all t ≥ 0, the prescribed performance will be attained if the following condition holds ⁴⁵

$\begin{array}{l}-\underset{\_}{\alpha }\rho \left(t\right)<\mathit{\delta }\left(t\right)<\rho \left(t\right),\mathit{\delta }\left(0\right)\ge 0\\ -\rho \left(t\right)<\mathit{\delta }\left(t\right)<\overline{\alpha }\rho \left(t\right),\mathit{\delta }\left(0\right)\le 0\end{array}$

11

where $0\le \underset{\_}{\alpha }\le 1$ , $0\le \overline{\alpha }\le 1$ , and ρ(t) is a performance function associated δ(t).

To achieve the tracking performance (11), the error transformation method is introduced to transform constrained tracking errors into unconstrained signals, and the error transformation function is defined as ⁴⁶

$\begin{array}{l}\mathit{\delta }\left(t\right)=\rho \left(t\right)S\left(\varpi \left(t\right)\right)\end{array}$

12

where ϖ is the new transformed error. The transformation function S(ϖ) is rigorously increasing, and it satisfies ⁴⁷

$\begin{array}{c}\{\begin{array}{cc}\underset{\varpi \to -\infty }{\text{lim}}S\left(\varpi \right)=-\underset{\_}{\alpha },& \mathit{\delta }\left(0\right)\ge 0\\ \underset{\varpi \to -\infty }{\text{lim}}S\left(\varpi \right)=-1,& \mathit{\delta }\left(0\right)\le 0\end{array}\\ \{\begin{array}{cc}\underset{\varpi \to \infty }{\text{lim}}S\left(\varpi \right)=1,& \mathit{\delta }\left(0\right)\ge 0\\ \underset{\varpi \to \infty }{\text{lim}}S\left(\varpi \right)=\overline{\alpha },& \mathit{\delta }\left(0\right)\le 0\end{array}\end{array}$

13

To proceed with the design of the robust adaptive control for the small-scale helicopter system (1), we make the following assumptions.

Assumption 1

The desired trajectories P_d(t) and ψ_d(t) are the known bounded sufficiently smooth functions of time, with bounded and continuous first derivative.

Assumption 2

All state vectors are available for measurement.

Assumption 3

The roll angle ϕ satisfies inequality constraints −π/2 < ϕ < π/2, and the pitch angle θ satisfies inequality constraints −π/2 < θ < π/2. ¹⁴

Assumption 4

The uncertain dynamics ΔF and ΔM is bounded

$\begin{array}{l}\parallel \Delta F\parallel \le \overline{\Delta }F,\parallel \Delta M\parallel \le \overline{\Delta }M\end{array}$

14

where $\overline{\Delta }F>0$ and $\overline{\Delta }M>0$ are unknown constants.

Assumption 5

The external disturbance d_f and d_τ are slowly varying variables, and $\parallel {\dot{d}}_f\parallel \le {\vartheta }_1$ and $\parallel {\dot{d}}_{\mathit{\tau }}\parallel \le {\vartheta }_2$ with ϑ₁ > 0 and ϑ₂ > 0.

Assumption 6

The time derivations of saturation errors $\Delta {\dot{v}}_T$ and $\Delta {\dot{v}}_{\mathit{\tau }}$ are bounded, and $\parallel \Delta {\dot{v}}_T\parallel \le {\vartheta }_T$ and $\parallel \Delta {\dot{v}}_{\mathit{\tau }}\parallel \le {\vartheta }_{\mathit{\tau }}$ , where ϑ_T and ϑ_τ indicate the upper bounds, Δv_T = v_T − sat(v_T), and Δv_τ = v_τ − sat(v_τ).

Remark 1

For the practical unmanned helicopter systems, it is well known that the time derivations of Δv_T and Δv_τ are always bounded when actuators are determined.

Lemma 1

For any bounded initial conditions, suppose that there exist a C¹ positive definite and continuous Lyapunov function V(x) and two class K functions π₁, π₂ : Rⁿ → R, satisfying ${\pi }_1\left(\parallel x\parallel \right)\le V\left(x\right)\le {\pi }_2\left(\parallel x\parallel \right)$ . ³⁰ If the inequality holds $\dot{V}\left(x\right)\le -c_{1}V\left(x\right)+c_2$ , for the positive constants c₁ and c₂, then the solution x(t) will be uniformly bounded.

The objective of this article is to design a robust adaptive controller subject to input and output constraint for the small-scale helicopter system such that it can track desired trajectories P_d(t) and ψ_d(t).

Controller design

In this section, the prescribed performance-based robust adaptive control scheme will be developed for the uncertain helicopter system. As mentioned earlier, the disturbance observer and prescribed performance function method are adopted to deal with input saturation and output constraint, respectively. Firstly, choose one appropriate prescribed performance function and design the virtual control based on the new error transformation form in the initial step. Then, the control strategy is to find an immediate control in each step, in which the RBFNN weight update law and disturbance observer are considered. The detailed design procedure can be described in the following steps.

Step 1: The position tracking error is defined as

$\begin{array}{l}{\mathit{\delta }}_P=P-P_d\end{array}$

15

where P_d ∈ R³ is the reference position trajectory. A performance function ρ_i(t) is chosen as ⁴⁶

$\begin{array}{l}{\rho }_i\left(t\right)=\left({\rho }_{i0}-{\rho }_{i\infty }\right)e^{-lt}+{\rho }_{i\infty },i=1,2,3\end{array}$

16

where ρ_i∞, ρ_i0, and l are positive constants. The decreasing rate e^−lt of ρ_i(t) denotes the desired convergence speed of δ_Pi. The constant ρ_i∞ represents the maximum allowable amplitude of tracking error at the steady state. Therefore, by choosing appropriate performance function ρ _i (t) and design parameters, the bounds of system output trajectory can be assigned.

An error transformation is defined as

$\begin{array}{c}{\varpi }_{Pi}=S^{-1}\left(\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i},{\overline{\alpha }}_i,{\underset{\_}{\alpha }}_i\right)\\ =\frac{1}{2}\text{ln}\frac{{\underset{\_}{a}}_i+\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}}{{\overline{\alpha }}_i-\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}}i=1,2,3\end{array}$

17

where ${\overline{\alpha }}_i$ and ${\underset{\_}{\alpha }}_i$ are the designed positive constants, which indicate the upper and lower bounds of the tracking errors. The time derivative of ϖ_Pi becomes

$\begin{array}{c}{\dot{\varpi }}_{Pi}=\frac{\mathit{\partial }{\varpi }_{Pi}}{\mathit{\partial }\left(\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)}\frac{1}{{\rho }_i}{\dot{\mathit{\delta }}}_{Pi}-\frac{\mathit{\partial }{\varpi }_{Pi}}{\mathit{\partial }\left(\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)}\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i^2}{\dot{\rho }}_i\\ =\frac{1}{2}\frac{{\overline{\alpha }}_i-\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}}{{\underset{\text{-}}{a}}_i+\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}}\left(\frac{1}{{\overline{\alpha }}_i-\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}}+\frac{\left({\underset{\text{-}}{a}}_i+\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)}{{\left({\overline{\alpha }}_i-\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)}^2}\right)\frac{1}{{\rho }_i}{\dot{\mathit{\delta }}}_{Pi}\\ -\frac{1}{2}\frac{{\overline{\alpha }}_i-\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}}{{\underset{\text{-}}{a}}_i+\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}}\left(\frac{1}{{\overline{\alpha }}_i-\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}}+\frac{\left({\underset{\text{-}}{a}}_i+\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)}{{\left({\overline{\alpha }}_i-\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)}^2}\right)\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i^2}{\dot{\rho }}_i\\ =\frac{1}{2}\frac{{\overline{\alpha }}_i+{\underset{\text{-}}{a}}_i}{\left({\underset{\text{-}}{a}}_i+\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)\left({\overline{\alpha }}_i-\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)}\frac{1}{{\rho }_i}\left(V_i-{\dot{P}}_{di}\right)\\ -\frac{1}{2}\frac{{\overline{\alpha }}_i+{\underset{\text{-}}{a}}_i}{\left({\underset{\text{-}}{a}}_i+\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)\left({\overline{\alpha }}_i-\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)}\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i^2}{\dot{\rho }}_i\end{array}$

18

In order to simplify the analysis, we define M_Pi and N_Pi as follows

$\begin{array}{l}{M}_{Pi}=\frac{{\overline{\alpha }}_i+{\underset{\_}{a}}_i}{2{\rho }_i\left({\underset{\_}{a}}_i+\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)\left({\overline{\alpha }}_i-\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i}\right)}>0\\ N_{Pi}=\frac{{\mathit{\delta }}_{Pi}}{{\rho }_i^2}{\dot{\rho }}_i\end{array}$

19

Considering equations (18) and (19), we have

$\begin{array}{l}{\dot{\varpi }}_P=M_P\left(V-{\dot{P}}_d-N_P\right)\end{array}$

20

where M_P = diag{M_P1, M_P2, M_P3} and N_P = [N_P1, N_P2, N_P3] ^T .

${\dot{P}}_d$ is the time derivative of reference trajectory P_d. The immediate control is designed as

$\begin{array}{l}{V}_d={\dot{P}}_d-M_P^{-1}{K}_P{\varpi }_P+N_P\end{array}$

21

where K_P ∈ R^3×3 is the constant positive definite matrices. Then, the time derivative of ϖ_P becomes

$\begin{array}{l}{\dot{\varpi }}_P=M_P\left(V-V_d\right)-K_P{\varpi }_P\end{array}$

22

Choose the Lyapunov function candidate

$\begin{array}{l}{V}_P=\frac{1}{2}{\varpi }_P^T{\varpi }_P\end{array}$

23

Substituting equation (22) into equation (23), we obtain

$\begin{array}{l}{\dot{V}}_P={\varpi }_P^T{\dot{\varpi }}_P=-{\varpi }_P^T{K}_P{\varpi }_P+{\varpi }_P^T{M}_P\left(V-V_d\right)\end{array}$

24

• Step 2: Define the velocity tracking error item δ_V ∈ R³ as

$\begin{array}{l}{\mathit{\delta }}_V=V-V_d\end{array}$

25

Let γ = R³(Θ) ∈ R^3×1, where R³(Θ) is the third column vector of the rotation matrix R(Θ). According to the definition of R(Θ), we know ∥γ∥ = 1. Define v_T = −T_mre₃, where $e_3={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^T$ , and T_mr ∈ R will be defined in equation (29). According to the definition of input saturation (5), the difference Δu_T between u_T and v_T can be defined as Δu_T = u_T − v_T. Then, we have $u_T=-T_{mr}{e}_3+\Delta u_T$ . Considering the translational dynamic and differentiating δ_V with respect to time yields

$\begin{array}{l}{\dot{\mathit{\delta }}}_V=g{e}_3-\frac{1}{m}\gamma T_{mr}-{\dot{V}}_d+\Delta F+d_f+\frac{1}{m}\gamma \Delta u_T\end{array}$

26

Since ΔF is unknown, using the RBFNN to approximate ΔF, we obtain

$\begin{array}{l}\Delta F=p_V^{-1}{W}_V^{*T}S\left(V\right)+{\epsilon }_V^{*}\end{array}$

27

where p_V > 0 is a design parameter. According to the definition of γ, we know ∥γ∥ = 1. Thus, define ${\xi }_V={\epsilon }_V^{*}+d_f+\frac{1}{m}\gamma \Delta u_T$ . As we know, the external disturbance ξ_V represents the time-dependent signal, and it can be seen as largely affected by the exogenous factors. Thus, ${\dot{\xi }}_V$ is bounded. Substituting equation (27) into equation (26) results in

$\begin{array}{l}{\dot{\mathit{\delta }}}_V=g{e}_3-\frac{1}{m}\gamma T_{mr}-{\dot{V}}_d+p_V^{-1}{W}_V^{*T}S\left(V\right)+{\xi }_V\end{array}$

28

The desired direction γ _d and magnitude T_mr of the thrust vector can be designed as

$\begin{array}{c}{\gamma }_d=\{\begin{array}{cc}\frac{T_{mr0}}{\parallel T_{mr0}\parallel }& T_{mr0}\ne 0\\ 0& T_{mr0}=0\end{array}\\ T_{mr}=m\parallel T_{mr0}\parallel \\ T_{mr0}=g{e}_3-{\dot{V}}_d+K_V{\mathit{\delta }}_V+M_P{\varpi }_P+p_V^{-1}{\widehat{W}}_V^T{S}_V+{\widehat{\xi }}_V\end{array}$

29

where $K_V=K_V^T>0,K_V\in R^{3\times 3}$ is the designed matrix. ${\widehat{W}}_V$ and ${\widehat{\xi }}_V$ are the estimated values of $W_V^{*}$ and ξ_V, respectively. ${\tilde{W}}_V$ and ${\tilde{\xi }}_V$ represent the estimated errors, which are defined as ${\tilde{W}}_V=W_V^{*}-{\widehat{W}}_V$ and ${\tilde{\xi }}_V={\xi }_V-{\widehat{\xi }}_V$ . Invoking equation (29), the control input can be defined as

$\begin{array}{l}{v}_T=-T_{mr}{e}_3\end{array}$

30

The updating law of the RBFNN can be chosen as

$\begin{array}{l}{\dot{\widehat{W}}}_V={\Gamma }_V\left(p_V^{-1}{\mathit{\delta }}_{V}S\left(V\right)-{\mathit{\sigma }}_V{\Gamma }_V^{-1}{\widehat{W}}_V\right)\end{array}$

31

where σ_V > 0 is a designed parameter, and Γ_V ∈ R^3×3 denotes the symmetric positive definite matrix. Meanwhile, the disturbance observer is chosen as follows

$\begin{array}{l}{\widehat{\xi }}_V={\mathit{\omega }}_V+p_{V}V\\ {\dot{\mathit{\omega }}}_V=-p_V{\mathit{\omega }}_V+{\mathit{\delta }}_V\\ -p_V\left(g{e}_3-\frac{1}{m}\gamma T_{mr}+p_V^{-1}{\widehat{W}}_V^T{S}_V+p_{V}V\right)\end{array}$

32

where ω_V is the auxiliary variable, and ${\widehat{\xi }}_V$ is the disturbance estimate. Then, we have

$\begin{array}{l}{\dot{\widehat{\xi }}}_V={\dot{\mathit{\omega }}}_V+p_V\dot{V}\\ =-p_V{\mathit{\omega }}_V+{\mathit{\delta }}_V+p_V\dot{V}\\ -p_V\left(g{e}_3-\frac{1}{m}\gamma T_{mr}+p_V^{-1}{\widehat{W}}_V^T{S}_V+p_{V}V\right)\\ =-p_V{\mathit{\omega }}_V+{\mathit{\delta }}_V\\ -p_V\left(g{e}_3-\frac{1}{m}\gamma T_{mr}+p_V^{-1}{\widehat{W}}_V^T{S}_V+p_{V}V\right)\\ +p_V\left(g{e}_3-\frac{1}{m}\gamma T_{mr}+p_V^{-1}{W}_V^{*T}{S}_V+{\xi }_V\right)\\ =-p_V\left({\mathit{\omega }}_V+p_{V}V\right)+{\tilde{W}}_V^T{S}_V+p_V{\xi }_V+{\mathit{\delta }}_V\\ =p_V{\tilde{\xi }}_V+{\tilde{W}}_V^T{S}_V+{\mathit{\delta }}_V\end{array}$

33

Considering equations (28) and (29), the time derivative of δ_V can be rewritten as

$\begin{array}{l}{\dot{\mathit{\delta }}}_V=-K_V{\mathit{\delta }}_V+p_V^{-1}{\tilde{W}}_V^T{S}_V+{\tilde{\xi }}_V\\ -M_P{\varpi }_P-\frac{1}{m}{\mathit{\delta }}_{\gamma }{T}_{mr}\end{array}$

34

Choose the Lyapunov function candidate as

$\begin{array}{l}{V}_V=\frac{1}{2}{\mathit{\delta }}_V^T{\mathit{\delta }}_V+\frac{1}{2}tr\left({\tilde{W}}_V^T{\Gamma }_V^{-1}{\tilde{W}}_V\right)+\frac{1}{2}{\tilde{\xi }}_V^T{\tilde{\xi }}_V\end{array}$

35

Let δ_γ = γ − γ_d and δ_γ = [δ_γ1, δ_γ2, δ_γ3] ^T . According to equation (29), we know that δ_γ3 = 0. Considering equations (24) and (34), we have

$\begin{array}{l}{\dot{V}}_V=-{\mathit{\delta }}_V^T{K}_V{\mathit{\delta }}_V-{\mathit{\delta }}_V^T{M}_P{\varpi }_P-\frac{1}{m}{T}_{mr}{\mathit{\delta }}_V^T{\mathit{\delta }}_{\gamma }\hfill \\ +p_V^{-1}{\mathit{\delta }}_V^T{\tilde{W}}_V^T{S}_V+{\mathit{\delta }}_V^T{\tilde{\xi }}_V\hfill \\ -tr\left({\tilde{W}}_V^T{\Gamma }_V^{-1}{\dot{\widehat{W}}}_V\right)-{\tilde{\xi }}_V^T{\dot{\widehat{\xi }}}_V+{\tilde{\xi }}_V^T{\dot{\xi }}_V\hfill \\ =-{\mathit{\delta }}_V^T{K}_V{\mathit{\delta }}_V-{\mathit{\delta }}_V^T{M}_P{\varpi }_P-\frac{1}{m}{T}_{mr}{\mathit{\delta }}_V^T{\mathit{\delta }}_{\gamma }\hfill \\ +{\mathit{\delta }}_V^T{\tilde{\xi }}_V+tr\left({\sigma }_V{\tilde{W}}_V^T{\Gamma }_V^{-1}{\widehat{W}}_V\right)\hfill \\ -{\tilde{\xi }}_V^T{\dot{\widehat{\xi }}}_V+{\tilde{\xi }}_V^T{\dot{\xi }}_V\hfill \\ =-{\mathit{\delta }}_V^T{K}_V{\mathit{\delta }}_V-{\mathit{\delta }}_V^T{M}_P{\varpi }_P-\frac{1}{m}{T}_{mr}{\mathit{\delta }}_{V\mathrm{1,2}}^T{\mathit{\delta }}_{\zeta }\hfill \\ +tr\left({\sigma }_V{\tilde{W}}_V^T{\Gamma }_V^{-1}{\widehat{W}}_V\right)\hfill \\ -p_V{\tilde{\xi }}_V^T{\tilde{\xi }}_V-{\tilde{\xi }}_V^T{\tilde{W}}_V^T{S}_V+{\tilde{\xi }}_V^T{\dot{\xi }}_V\hfill \end{array}$

36

where δ_ζ will be defined in equation (45), δ_V1,2 = [δ_V1, δ_V2] ^T indicates the first and second lines of δ_V, and δ_ς will be designed in next step. Define ϑ₁ = ϑ_T + ϑ_df. Considering the following facts:

$\begin{array}{l}{\tilde{\xi }}_V^T{\dot{\xi }}_V\le \frac{1}{2}{\tilde{\xi }}_V^T{\tilde{\xi }}_V+\frac{1}{2}{\dot{\xi }}_V^T{\dot{\xi }}_V\le \frac{1}{2}{\tilde{\xi }}_V^T{\tilde{\xi }}_V+\frac{1}{2}{\vartheta }_1^2\end{array}$

37

$-2{\tilde{\xi }}_V{\tilde{W}}_V^T{S}_V\le 2{\mu }_V\parallel {\tilde{\xi }}_V\parallel \parallel {\tilde{W}}_V\parallel \le {\mu }_V{\gamma }_V{\parallel {\tilde{\xi }}_V\parallel }^2+\frac{{\mu }_V}{{\gamma }_V}{\parallel {\tilde{W}}_V\parallel }^2$

38

$\begin{array}{l}2{\sigma }_V{\tilde{W}}_V^T{\Gamma }_V^{-1}{\widehat{W}}_V={\sigma }_V{\tilde{W}}_V^T{\Gamma }_V^{-1}{\tilde{W}}_V+{\sigma }_V{\widehat{W}}_V^T{\Gamma }_V^{-1}{\widehat{W}}_V\\ -{\sigma }_V{W}_V^{*T}{\Gamma }_V^{-1}{W}_V^{*}\\ \ge {\sigma }_V{\lambda }_{\text{max}}^{-1}\left({\Gamma }_V\right){\parallel {\tilde{W}}_V\parallel }^2-{\sigma }_V{\lambda }_{\text{max}}^{-1}\left({\Gamma }_V\right){\parallel W_V^{*}\parallel }^2\end{array}$

39

where μ_V > 0 and γ_V > 0 are the design parameters. Invoking equations (38) and (39), equation (36) can be rewritten as

$\begin{array}{l}{\dot{V}}_V\le -{\mathit{\delta }}_V^T{K}_V{\mathit{\delta }}_V-{\mathit{\delta }}_V^T{M}_P{\varpi }_P-\frac{1}{m}{T}_{mr}{\mathit{\delta }}_{V1,2}^T{\mathit{\delta }}_{\zeta }\\ -\left(p_V-{\mu }_V{\gamma }_V-\frac{1}{2}\right){\tilde{\xi }}_V^2\\ -\left({\sigma }_V{\lambda }_{\text{max}}^{-1}\left({\Gamma }_V\right)-\frac{{\mu }_V}{{\gamma }_V}\right){\parallel {\tilde{W}}_V\parallel }^2\\ +{\sigma }_V{\lambda }_{\text{max}}^{-1}\left({\Gamma }_V\right){\parallel W_V^{*}\parallel }^2+\frac{1}{2}{\vartheta }_1^2\end{array}$

40

• Step 3: Define a new orientation variable ζ ∈ R² as

$\begin{array}{l}\zeta ={\left[{\gamma }_1,{\gamma }_2\right]}^T\end{array}$

41

According to the rotation matrix dynamic (3), we have

$\begin{array}{l}{\dot{\gamma }}_1={\dot{R}}_{3,1}=R_{1,1}q-R_{2,1}p\\ {\dot{\gamma }}_2={\dot{R}}_{3,2}=R_{1,2}q-R_{2,2}p\end{array}$

42

Then, the orientation dynamics can be obtained as

$\begin{array}{l}\dot{\zeta }=Z\left(\Theta \right)\left[\begin{array}{c}p\\ q\end{array}\right]\end{array}$

43

where

$\begin{array}{l}Z\left(\Theta \right)=\left[\begin{array}{cc}-R_{2,1},& R_{1,1}\\ -R_{2,2},& R_{1,2}\end{array}\right]\\ =\left[\begin{array}{cc}-(S\varphi S\theta C\psi -C\varphi S\psi ),& C\theta C\psi \\ -(S\varphi S\theta S\psi +C\varphi C\psi ),& C\theta S\psi \end{array}\right]\end{array}$

44

According to the definition of Z(Θ) and assumption 3, we can obtain |Z(Θ)| ≠ 0, where |Z(Θ)| indicates the polynomial of Z(Θ). Thus, |Z(Θ)| is reversible, and Z⁻¹(Θ) denotes the inverse matrix of Z(Θ).

Using the ideal vector γ _d , define ζ_d = [γ_d1, γ_d2] ^T , γ_dk, k = 1, 2 indicates the kth of γ _k . Define the third error δ_ζ ∈ R² as

$\begin{array}{l}{\mathit{\delta }}_{\zeta }=\zeta -{\zeta }_d\end{array}$

45

Choose the angular velocity virtual control as

$\begin{array}{c}\left[\begin{array}{c}{p}_d\\ q_d\end{array}\right]=Z^{-1}\left(\Theta \right)\left({\dot{\zeta }}_d-K_{\zeta }{\mathit{\delta }}_{\zeta }\right)\end{array}$

46

where K_ζ ∈ R^2×2 is the design matrix. Define δ_Ω1,2 = [p, q] ^T − [p_d,q_d] ^T . The reduced orientation error dynamics can be obtained as

$\begin{array}{c}{\dot{\delta }}_{\zeta }=-K_{\zeta }{\mathit{\delta }}_{\zeta }+Z\left(\Theta \right){\mathit{\delta }}_{\Omega 1,2}\end{array}$

47

Choose the Lyapunov function candidate as

$\begin{array}{l}{V}_{\zeta }=\frac{1}{2}{\mathit{\delta }}_{\zeta }^T{\mathit{\delta }}_{\zeta }\end{array}$

48

Using equations (45) and (47), one obtains

$\begin{array}{l}{\dot{V}}_{\zeta }=-{\mathit{\delta }}_{\zeta }^T{K}_{\zeta }{\mathit{\delta }}_{\zeta }+{\delta }_{\varsigma }^{T}Z\left(\Theta \right){\mathit{\delta }}_{\Omega 1,2}\end{array}$

49

• Step 4: According to equation (1), the yaw dynamics is obtained

$\begin{array}{l}\dot{\psi }=H_3\left(\Theta \right)\Omega \end{array}$

50

where H₃(Θ) ∈ R^1×3 indicates the third row of the matrix H(Θ). Define the yaw dynamics error δ_ψ ∈ R as

$\begin{array}{l}{\mathit{\delta }}_{\psi }=\psi -{\psi }_d\end{array}$

51

where ψ _d is the reference yaw signal. Invoking equations (50) and (51), we have

$\begin{array}{c}{\dot{\delta }}_{\psi }=\dot{\psi }-{\dot{\psi }}_d=-{\dot{\psi }}_d+H_3\left(\Theta \right)\Omega \\ =-{\dot{\psi }}_d+\frac{S\varphi }{C\theta }q+\frac{C\varphi }{C\theta }r\end{array}$

52

Similar to equation (17), we define the error transformation as follows

$\begin{array}{l}{\varpi }_{\psi }=S^{-1}\left(\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }},{\overline{\beta }}_{\psi },{\underset{\_}{\beta }}_{\psi }\right)=\frac{1}{2}\text{ln}\frac{\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}+{\underset{\_}{\beta }}_{\psi }}{{\overline{\beta }}_{\psi }-\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}}\end{array}$

53

where ${\overline{\beta }}_{\psi }>0$ and ${\underset{\_}{\beta }}_{\psi }>0$ are the bounds of ρ_ψ. Invoking equation (52), we have

$\begin{array}{c}{\dot{\varpi }}_{\psi }=\frac{\mathit{\partial }{\varpi }_{\psi }}{\mathit{\partial }\left(\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)}\frac{1}{{\rho }_{\psi }}{\dot{\delta }}_{\psi }-\frac{\mathit{\partial }{\varpi }_{\psi }}{\mathit{\partial }\left(\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)}\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }^2}{\dot{\rho }}_{\psi }\\ =\frac{1}{2}\frac{{\overline{\beta }}_{\psi }-\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}}{{\underset{̱}{\beta }}_{\psi }+\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}}\left(\frac{1}{{\overline{\beta }}_{\psi }-\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}}+\frac{\left({\overline{\beta }}_{\psi }+\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)}{{\left({\underset{̱}{\beta }}_{\psi }-\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)}^2}\right)\frac{1}{{\rho }_{\psi }}{\dot{\delta }}_{\psi }\\ -\frac{1}{2}\frac{{\overline{\beta }}_{\psi }-\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}}{{\underset{̱}{\beta }}_{\psi }+\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}}\left(\frac{1}{{\overline{\beta }}_{\psi }-\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}}+\frac{\left({\overline{\beta }}_{\psi }+\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)}{{\left({\underset{̱}{\beta }}_{\psi }-\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)}^2}\right)\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }^2}{\dot{\rho }}_{\psi }\\ =\frac{{\overline{\beta }}_{\psi }+{\underset{̱}{\beta }}_{\psi }}{2\left({\overline{\beta }}_{\psi }-\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)\left({\underset{̱}{\beta }}_{\psi }+\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)}\frac{1}{{\rho }_{\psi }}\left(\begin{array}{c}\frac{S\varphi }{C\theta }q+\frac{C\varphi }{C\theta }r-{\dot{\psi }}_d\end{array}\right)\\ -\frac{{\overline{\beta }}_{\psi }+{\underset{̱}{\beta }}_{\psi }}{2\left({\overline{\beta }}_{\psi }-\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)\left({\underset{̱}{\beta }}_{\psi }+\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)}\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }^2}{\dot{\rho }}_{\psi }\end{array}$

54

In order to simplify the analysis, we define M_ψ and N_ψ as follows

$\begin{array}{l}{M}_{\psi }=\frac{{\overline{\beta }}_{\psi }+{\underset{\_}{\beta }}_{\psi }}{2\left({\overline{\beta }}_{\psi }-\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)\left({\underset{\_}{\beta }}_{\psi }+\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }}\right)}\frac{1}{{\rho }_{\psi }}>0\\ N_{\psi }=\frac{{\mathit{\delta }}_{\psi }}{{\rho }_{\psi }^2}{\dot{\rho }}_{\psi }\end{array}$

55

Designed the immediate control as

$\begin{array}{l}{r}_d=\frac{C\theta }{C\varphi }\left(-M_{\psi }^{-1}{K}_{\psi }{\varpi }_{\psi }-\frac{S\varphi }{C\theta }q+{\dot{\psi }}_d+N_{\psi }\right)\end{array}$

56

where K_ψ > 0 is the designed constant. Choose the Lyapunov function candidate as

$\begin{array}{l}{V}_{\psi }=\frac{1}{2}{\varpi }_{\psi }^2\end{array}$

57

The time derivative of V_ψ can be obtained as

$\begin{array}{l}{\dot{V}}_{\psi }={\varpi }_{\psi }{\dot{\varpi }}_{\psi }=-{\varpi }_{\psi }{K}_{\psi }{\varpi }_{\psi }+{\varpi }_{\psi }{M}_{\psi }\frac{C\varphi }{C\theta }\left(r-r_d\right)\end{array}$

58

• Step 5: According to steps 3 and 4, we have obtained Ω_d = [p_d, q_d, r_d] ^T . Define the angular velocity error δ_Ω ∈ R³ as

$\begin{array}{l}{\mathit{\delta }}_{\Omega }=\Omega -{\Omega }_d\end{array}$

59

Then we have

$\begin{array}{l}{\dot{\delta }}_{\Omega }=\dot{\Omega }-{\dot{\Omega }}_d\\ =-\Omega \times I_m\Omega -Q_{tr}{e}_2-Q_{mr}{e}_3\\ +G\mathit{\tau }+\Delta M+d_{\tau }-{\dot{\Omega }}_d\end{array}$

60

Since Δ_M is unknown, using the RBFNN to approximate Δ_M, we obtain

$\begin{array}{l}\Delta M=p_{\Omega }^{-1}{W}_{\Omega }^{*T}S\left(\Omega \right)+{\epsilon }_{\Omega }^{*}\end{array}$

61

where p_Ω > 0 is the design parameter. According to the definition of input saturation (6), the difference Δu_τ between v_τ and τ can be defined as Δu_τ = τ − v_τ. Define ${\xi }_{\Omega }={\epsilon }_{\Omega }^{*}+d_{\tau }+\Delta u_{\tau }$ . Substituting equation (61) into equation (60) results in

$\begin{array}{l}{\dot{\delta }}_{\Omega }=-\Omega \times I_m\Omega -Q_{tr}{e}_2-Q_{mr}{e}_3+G{v}_{\tau }-{\dot{\Omega }}_d\\ +p_{\Omega }^{-1}{W}_{\Omega }^{*T}S\left(\Omega \right)+{\xi }_{\Omega }\end{array}$

62

Invoking equation (62), one can easily obtain the following ideal moment control

$\begin{array}{l}{v}_{\tau }=G^{-1}\left(\begin{array}{c}{\dot{\Omega }}_d-K_{\Omega }{\mathit{\delta }}_{\Omega }+\Omega \times I_m\Omega +Q_{tr}{e}_2\\ +Q_{mr}{e}_3-p_{\Omega }^{-1}{\widehat{W}}_{\Omega }^{T}S\left(\Omega \right)-{\widehat{\xi }}_{\Omega }\end{array}\right)\end{array}$

63

where K_Ω ∈ R^3×3 is the designed matrix. ${\widehat{W}}_{\Omega }$ and ${\widehat{\xi }}_{\Omega }$ are the estimated values of $W_{\Omega }^{*}$ and ξ_Ω, respectively. ${\tilde{W}}_{\Omega }$ and ${\tilde{\xi }}_{\Omega }$ represent the estimated errors, which are defined as ${\tilde{W}}_{\Omega }=W_{\Omega }^{*}-{\widehat{W}}_{\Omega }$ and ${\tilde{\xi }}_{\Omega }={\xi }_{\Omega }-{\widehat{\xi }}_{\Omega }$ . The updating law of the RBFNN can be obtained as

$\begin{array}{l}{\dot{\widehat{W}}}_{\Omega }={\Gamma }_{\Omega }\left(p_{\Omega }^{-1}{\mathit{\delta }}_{\Omega }S\left(\Omega \right)-{\sigma }_{\Omega }{\Gamma }_{\Omega }^{-1}{\widehat{W}}_{\Omega }\right)\end{array}$

64

where σ_Ω > 0, σ_Ω ∈ R is a designed parameter, and Γ_Ω ∈ R^3×3 represents the symmetric positive definite matrix. The disturbance observer is designed as follows

$\begin{array}{l}{\widehat{\xi }}_{\Omega }={\mathit{\omega }}_{\Omega }+p_{\Omega }\Omega \\ {\dot{\omega }}_{\Omega }=-p_{\Omega }{\mathit{\omega }}_{\Omega }+{\mathit{\delta }}_{\Omega }\\ -p_{\Omega }\left(\begin{array}{c}-\Omega \times I_m\Omega -Q_{tr}{e}_2-Q_{mr}{e}_3\\ +G{v}_{\tau }+p_{\Omega }^{-1}{\widehat{W}}_{\Omega }^{T}S+p_{\Omega }\Omega \end{array}\right)\end{array}$

65

where ω_Ω is the auxiliary variable, and ${\widehat{\xi }}_{\Omega }$ is the disturbance estimate. Then, we have

$\begin{array}{c}{\dot{\widehat{\xi }}}_{\Omega }={\dot{\omega }}_{\Omega }+p_{\Omega }\dot{\Omega }=-p_{\Omega }{\mathit{\omega }}_{\Omega }+{\mathit{\delta }}_{\Omega }+p_{\Omega }\dot{\Omega }\\ -p_{\Omega }\left(\begin{array}{c}-\Omega \times I_m\Omega -Q_{tr}{e}_2-Q_{mr}{e}_3\\ +G{v}_{\tau }+p_{\Omega }^{-1}{\widehat{W}}_{\Omega }^{T}S+p_{\Omega }\Omega \end{array}\right)\\ =-p_{\Omega }{\mathit{\omega }}_{\Omega }+{\mathit{\delta }}_{\Omega }\\ -p_{\Omega }\left(\begin{array}{c}-\Omega \times I_m\Omega -Q_{tr}{e}_2-Q_{mr}{e}_3\\ +G{v}_{\tau }+p_{\Omega }^{-1}{\widehat{W}}_{\Omega }^{T}S+p_{\Omega }\Omega \end{array}\right)\\ +p_{\Omega }\left(\begin{array}{c}-\Omega \times I_m\Omega -Q_{tr}{e}_2-Q_{mr}{e}_3\\ +G{v}_{\tau }+p_{\Omega }^{-1}{W}_{\Omega }^{*T}S+{\xi }_{\Omega }\end{array}\right)\\ =-p_{\Omega }\left({\mathit{\omega }}_{\Omega }+p_{\Omega }\Omega \right)+{\tilde{W}}_{\Omega }^{T}S+p_{\Omega }{\xi }_{\Omega }+{\mathit{\delta }}_{\Omega }\\ =p_{\Omega }{\tilde{\xi }}_{\Omega }+{\tilde{W}}_{\Omega }^{T}S+{\mathit{\delta }}_{\Omega }\end{array}$

66

Choose the Lyapunov function candidate as

$\begin{array}{l}{V}_{\Omega }=\frac{1}{2}{\mathit{\delta }}_{\Omega }^T{\mathit{\delta }}_{\Omega }+\frac{1}{2}tr\left({\tilde{W}}_{\Omega }^T{\Gamma }_{\Omega }^{-1}{\tilde{W}}_{\Omega }\right)+\frac{1}{2}{\tilde{\xi }}_{\Omega }^T{\tilde{\xi }}_{\Omega }\end{array}$

67

The time derivative of V_Ω can be obtained as

$\begin{array}{l}{\dot{V}}_{\Omega }=-{\mathit{\delta }}_{\Omega }^T{K}_{\Omega }{\mathit{\delta }}_{\Omega }+{\mathit{\delta }}_{\Omega }^T{p}_{\Omega }^{-1}{\tilde{W}}_{\Omega }{S}_{\Omega }+{\mathit{\delta }}_{\Omega }^T{\tilde{\xi }}_{\Omega }\hfill \\ -tr\left({\tilde{W}}_{\Omega }^T{\Gamma }_{\Omega }^{-1}{\dot{\widehat{W}}}_{\Omega }\right)-{\tilde{\xi }}_{\Omega }{\dot{\widehat{\xi }}}_{\Omega }+{\mathit{\delta }}_{\gamma }{\mathit{\delta }}_{\Omega }+{\tilde{\xi }}_{\Omega }^T{\dot{\xi }}_{\Omega }\hfill \\ =-{\mathit{\delta }}_{\Omega }^T{K}_{\Omega }{\mathit{\delta }}_{\Omega }+tr\left({\tilde{W}}_{\Omega }^T{\Gamma }_{\Omega }{\widehat{W}}_{\Omega }\right)+{\mathit{\delta }}_{\Omega }^T{\tilde{\xi }}_{\Omega }\hfill \\ -{\tilde{\xi }}_{\Omega }{\dot{\widehat{\xi }}}_{\Omega }+{\mathit{\delta }}_{\gamma }{\mathit{\delta }}_{\Omega }+{\tilde{\xi }}_{\Omega }^T{\dot{\xi }}_{\Omega }\hfill \\ =-{\mathit{\delta }}_{\Omega }^T{K}_{\Omega }{\mathit{\delta }}_{\Omega }+tr\left({\tilde{W}}_{\Omega }^T{\Gamma }_{\Omega }{\widehat{W}}_{\Omega }\right)\hfill \\ -{\tilde{\xi }}_{\Omega }^T{p}_{\Omega }{\tilde{\xi }}_{\Omega }-{\tilde{\xi }}_{\Omega }^T{\tilde{W}}_{\Omega }^T{S}_{\Omega }+{\tilde{\xi }}_{\Omega }^T{\dot{\xi }}_{\Omega }\hfill \end{array}$

68

Define ϑ₂ = ϑ_τ + ϑ_dτ. Considering the following facts

$\begin{array}{l}{\tilde{\xi }}_{\Omega }^T{\dot{\xi }}_{\Omega }\le \frac{1}{2}{\tilde{\xi }}_{\Omega }^T{\tilde{\xi }}_{\Omega }+\frac{1}{2}{\dot{\xi }}_{\Omega }^T{\dot{\xi }}_{\Omega }\le \frac{1}{2}{\tilde{\xi }}_{\Omega }^T{\tilde{\xi }}_{\Omega }+\frac{1}{2}{\vartheta }_2^2\end{array}$

69

$2{\tilde{\xi }}_{\Omega }{\tilde{W}}_{\Omega }^T{S}_{\Omega }\le 2{\mu }_{\Omega }\parallel {\tilde{\xi }}_{\Omega }\parallel \parallel {\tilde{W}}_{\Omega }\parallel \le {\mu }_{\Omega }{\gamma }_{\Omega }{\parallel {\tilde{\xi }}_{\Omega }\parallel }^2+\frac{{\mu }_{\Omega }}{{\gamma }_{\Omega }}{\parallel {\tilde{W}}_{\Omega }\parallel }^2$

70

$\begin{array}{l}2{\sigma }_{\Omega }{\tilde{W}}_{\Omega }^T{\Gamma }_{\Omega }^{-1}{\widehat{W}}_{\Omega }={\sigma }_{\Omega }{\tilde{W}}_{\Omega }^T{\Gamma }_{\Omega }^{-1}{\tilde{W}}_{\Omega }+{\sigma }_{\Omega }{\widehat{W}}_{\Omega }^T{\Gamma }_{\Omega }^{-1}{\widehat{W}}_{\Omega }\\ -{\sigma }_{\Omega }{W}_{\Omega }^{*T}{\Gamma }_{\Omega }^{-1}{W}_{\Omega }^{*}\\ \ge {\sigma }_{\Omega }{\lambda }_{\text{max}}^{-1}\left({\Gamma }_{\Omega }\right){\parallel {\tilde{W}}_{\Omega }\parallel }^2-{\sigma }_{\Omega }{\lambda }_{\text{max}}^{-1}\left({\Gamma }_{\Omega }\right){\parallel W_{\Omega }^{*}\parallel }^2\end{array}$

71

where μ_Ω > 0 and γ_Ω > 0 are the design parameters. Invoking equations (69) to (71), equation (68) can be rewritten as

$\begin{array}{l}{\dot{V}}_{\Omega }\le -{\mathit{\delta }}_{\Omega }^T{K}_{\Omega }{\mathit{\delta }}_{\Omega }-{\mathit{\delta }}_{\Omega }^T{\mathit{\delta }}_{Z,\psi }-\left(p_{\Omega }-{\mu }_{\Omega }{\gamma }_{\Omega }-\frac{1}{2}\right){\tilde{\xi }}_{\Omega }^T{\tilde{\xi }}_{\Omega }\\ -\left({\sigma }_{\Omega }{\lambda }_{\text{max}}^{-1}\left({\Gamma }_{\Omega }\right)-\frac{{\mu }_{\Omega }}{{\gamma }_{\Omega }}\right){\parallel {\tilde{W}}_{\Omega }\parallel }^2\\ +{\sigma }_{\Omega }{\lambda }_{\text{max}}^{-1}\left({\Gamma }_{\Omega }\right){\parallel W_{\Omega }^{*}\parallel }^2+\frac{1}{2}{\vartheta }_2^2\end{array}$

72

Theorem 1

Consider the Multiple-Input Multiple-Output (MIMO) nonlinear unmanned helicopter system (1) in the presence of input and output constraints. The adaptive laws of NNs are designed as equations (32) and (65). Disturbance observers are designed as equations (33) and (66). The adaptive prescribed performance control laws are designed as equations (21), (29), (46), (56), and (63). Then the closed-loop system is globally stable. Furthermore, by choosing appropriate design parameters, the trajectory tracking errors converge to an arbitrarily small neighborhood of the origin.

Proof

For considering the convergence of closed-loop state tracking errors and disturbance estimate errors, the Lyapunov function candidate for closed-loop control system can be chosen as

$\begin{array}{l}{V}_{\Sigma }=V_P+V_V+V_{\varsigma }+V_{\psi }+V_{\Omega }\end{array}$

73

Differentiating V_Σ and considering equations (24), (36), (49), (58), and (72), we obtain

$\begin{array}{l}{\dot{V}}_{\Sigma }\le -{\varpi }_P^T{K}_P{\varpi }_P-{\mathit{\delta }}_V^T{K}_V{\mathit{\delta }}_V-{\mathit{\delta }}_{\zeta }^T{\Gamma }_{\zeta }{\mathit{\delta }}_{\zeta }\\ -{\mathit{\delta }}_{\psi }^T{K}_{\psi }{\mathit{\delta }}_{\psi }-{\mathit{\delta }}_{\Omega }^T{K}_{\Omega }{\mathit{\delta }}_{\Omega }-\left(p_V-{\mu }_V{\gamma }_V-\frac{1}{2}\right){\tilde{\xi }}_V^T{\tilde{\xi }}_V\\ -\left(p_{\Omega }-{\mu }_{\Omega }{\gamma }_{\Omega }-\frac{1}{2}\right){\tilde{\xi }}_{\Omega }^T{\tilde{\xi }}_{\Omega }-\left({\sigma }_V{\lambda }_{\text{max}}^{-1}\left({\Gamma }_V\right)-\frac{{\mu }_V}{{\gamma }_V}\right){\parallel {\tilde{W}}_V\parallel }^2\\ -\left({\sigma }_{\Omega }{\lambda }_{\text{max}}^{-1}\left({\Gamma }_{\Omega }\right)-\frac{{\mu }_{\Omega }}{{\gamma }_{\Omega }}\right){\parallel {\tilde{W}}_{\Omega }\parallel }^2+{\sigma }_V{\lambda }_{\text{max}}^{-1}\left({\Gamma }_V\right){\parallel W_V^{*}\parallel }^2\\ +{\sigma }_{\Omega }{\lambda }_{\text{max}}^{-1}\left({\Gamma }_{\Omega }\right){\parallel W_{\Omega }^{*}\parallel }^2+\frac{1}{2}{\vartheta }_1^2+\frac{1}{2}{\vartheta }_2^2\end{array}$

74

where ρ and C are given by

$\begin{array}{l}\rho :=\text{min}\left\{\begin{array}{c}{\lambda }_{\text{min}}\left(K_P\right),{\lambda }_{\text{min}}\left(K_V\right),\\ {\lambda }_{\text{min}}\left(K_{\varsigma }\right),K_{\psi },{\lambda }_{\text{min}}\left(K_{\Omega }\right),\\ p_V-{\mu }_V{\gamma }_V-\frac{1}{2},\\ p_{\Omega }-{\mu }_{\Omega }{\gamma }_{\Omega }-\frac{1}{2},\\ {\sigma }_V{\lambda }_{\text{max}}^{-1}\left({\Gamma }_V\right)-\frac{{\mu }_V}{{\gamma }_V},\\ {\sigma }_{\Omega }{\lambda }_{\text{max}}^{-1}\left({\Gamma }_{\Omega }\right)-\frac{{\mu }_{\Omega }}{{\gamma }_{\Omega }}\end{array}\right\}\end{array}$

75

$\begin{array}{l}C:={\sigma }_V{\lambda }_{\text{max}}^{-1}\left({\Gamma }_V\right){\parallel W_V^{*}\parallel }^2+{\sigma }_{\Omega }{\lambda }_{\text{max}}^{-1}\left({\Gamma }_{\Omega }\right){\parallel W_{\Omega }^{*}\parallel }^2\\ +\frac{1}{2}{\vartheta }_1^2+\frac{1}{2}{\vartheta }_2^2\end{array}$

76

Furthermore, the corresponding design parameters K_P, K_V, K_ς, K_ψ, K_Ω, p_V, μ_V, γ_V, p_Ω, μ_Ω, γ_Ω, σ_V, Γ_V, σ_Ω, and Γ_Ω are chosen such that

$\begin{array}{l}{p}_V-{\mu }_V{\gamma }_V-\frac{1}{2}>0\\ p_{\Omega }-{\mu }_{\Omega }{\gamma }_{\Omega }-\frac{1}{2}>0\\ {\sigma }_V{\lambda }_{\text{max}}^{-1}\left({\Gamma }_V\right)-\frac{{\mu }_V}{{\gamma }_V}>0\\ {\sigma }_{\Omega }{\lambda }_{\text{max}}^{-1}\left({\Gamma }_{\Omega }\right)-\frac{{\mu }_{\Omega }}{{\gamma }_{\Omega }}>0\end{array}$

77

According to equations (74) to (76), we have

$\begin{array}{l}0\le V_{\Sigma }\le \frac{C}{\rho }+\left[V_{\Sigma }\left(0\right)-\frac{C}{\rho }\right]e^{-\rho t}\end{array}$

78

From equation (78) and lemma 1, we know that the tracking errors δ_P, δ_V, δ_ς, δ_ψ, and δ_Ω; approximation errors ${\tilde{W}}_V\text{and}{\tilde{W}}_{\Omega }$ ; and estimation errors ${\tilde{\xi }}_V$ and ${\tilde{\xi }}_{\Omega }$ are bounded. Moreover, according to the definition of the transformed performance (12), we know that the boundedness of transformed errors ϖ_P and ϖ_ψ can guarantee the boundedness of tracking errors δ_P and δ_ψ, respectively. This concludes the proof.

Simulation results

Now, we consider an uncertain nonlinear unmanned helicopter system, and the main parameters are given by Chen et al., ²⁰ which is shown in Table 1.

Table 1.

Parameters for the small scale unmanned helicopter.

Symbol	Description	Value
m(kg)	The mass of helicopter	8.2
Im (kg·m2)	The moment of inertia	diag{0.18, 0.34, 0.28}
g (m·s−2)	The acceleration of gravity	9.8
Qtr (N·m)	The anti-torques of tail rotor	0.0002
Qmr (N·m)	The anti-torques of main rotor	0.002

In order to verify the effectiveness of the proposed control method, two typical flight trajectories are proposed for the unmanned helicopter system. The reference trajectory is given by $P_d\left(t\right)={\left[\text{sin}\left(t\right),\text{cos}\left(t\right)\mathrm{,1}\right]}^T\left(m\right)$ , and the desired heading angle is given by ψ _d (t) = 0(rad). Suppose that initial values of position and heading angle are set to be P(0) = [0, 0.9, 0.9] ^T (m) and ψ(0) = 0(rad). In order to evaluate the robustness of the proposed controller against the model uncertainties and the external disturbance, there is ΔM = −0.1 (−Ω × I_mΩ) uncertainty on aerodynamic moment coefficients. According to theorem 1, the adaptive prescribed performance control schemes are proposed as equations (21), (29), (46), (56), and (63); the disturbance observers of each subsystem are designed as equations (32) and (65); and the NN adaptive laws are chosen as equations (31) and (64). The designed parameters are chosen as K_P = diag{1, 1, 1}, K_V = diag{4, 4, 4}, K_ψ = 100, K_ς = diag{5, 5}, K_Ω = diag{10, 10, 10}, p_V = 1.5, p_Ω = 1.5, Γ_V = diag{10, 10, 10}, Γ_Ω = diag{10, 10, 10}, σV = 0.05, σ_Ω = 0.05, $\underset{\_}{\alpha }=0.3$ , $\overline{\alpha }=0.3$ , and ρ = diag{2, 2, 2}. Choose the input saturation value as ${\overline{u}}_T=50$ and $\overline{\mathit{\tau }}=50$ .

Under the developed control scheme, the simulation results for unmanned helicopter are shown in Figures 1 to 8. Figures 1 to 3 show the position tracking results, where the solid lines indicate the real trajectories and the dashed lines indicate the desired trajectories. Obviously, the tracking performance of system is satisfactory; meanwhile, the tracking errors always be maintained between the prescribed performance bounds. Figures 4 to 6 show the velocities, attitude angles, and angle velocities of the unmanned helicopter, respectively. We observe that the states remain bounded. Figures 7 and 8 show the control moment and control thrust that do not exceed the limitation of inputs.

Figure 1.

Tracking response of x.

Figure 2.

Tracking response of y.

Figure 3.

Tracking response of z.

Figure 4.

The response of velocity V.

Figure 5.

The response of angle Θ.

Figure 6.

The response of angular velocity Ω.

Figure 7.

Control input moment τ.

Figure 8.

Control input thrust T.

It can be shown from these simulation results of the numerical example that the closed-loop system for the nonlinear helicopter system with input saturation and parametric uncertainties is asymptotically stable. Thus, the proposed robust control method is valid.

Conclusion

A robust adaptive neural control scheme has been presented to achieve trajectory tracking for a class of small-scaled unmanned helicopter in this article. Based on the adaptive backstepping control framework, nonlinear disturbance observer has been developed to compensate the input saturation and prescribed performance function method has been used to deal with the output constraint. Finally, simulation results have been given to illustrate tracking performance of the proposed robust adaptive control approach. The future research is going to consider solving the constrained control problem for unmanned helicopter with various external disturbances.