# Robust Adaptive Output Feedback Control for a Class of Underactuated Aerial Vehicles with Input and Output Constraints

Authors: Trong-Toan Tran; Tan-No Nguyen; Duc-Duy Nguyen; Viet-Long Nguyen; Nguyen-Vu Truong
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper addresses the problem of nonlinear robust adaptive output feedback controller for a class of underactuated aerial vehicles with input and output constraints. To solve the problem, the modular design strategy is proposed for the control design. By using the neural networks (NNs) to approximate system uncertainties and observers to reconstruct system states, robust adaptive output feedback controllers are developed. By using a combination of saturation functions and barrier functions, input and output constraints are simultaneously dealt with. The design methodology shows that a cascaded system of an input-to-state stable (ISS) subsystem driven by an ultimate bounded (UB) subsystem enjoys ultimate boundedness property. In addition, the tracking error converges to adjustable neighbourhoods of the origin.


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## Body

## 1. Introduction

In recent years, control of underactuated systems has attracted much attention. Unlike the fully actuated system in which the number of independent control inputs is equal to degrees of freedom (DOFs), the underactuated system has fewer independent control inputs than DOFs [1, 2]. Because of the underactuated property, several undesired properties such as higher relative degree and non-minimum phase behavior are manifested. Therefore, powerful techniques developed for fully actuated systems may not be directly applied. For these reasons, control system design for underactuated systems is not an easy task. In fact, based on recent surveys [3–7], control of general underactuated systems is currently a major open problem.In the control design for underactuated systems, all DOFs cannot be controlled; instead, a portion of DOFs is controlled to have desired behaviours while remainders are required to be bounded. This requirement leads the system in the question to a non-minimum phase system [8]. Finding the general solution of non-minimum phase systems is still a quite challenging research topic [9]. In some special cases, underactuated systems with collocated partial feedback linearization, i.e., actuated configuration variables of the underactuated system with noninteracting inputs, can be globally linearized by the use of an invertible change of control as shown in [10]. An alternative approach is the reduction of the order of the system; i.e., the underactuated system is separated into fully actuated subsystems by adding virtual controls, and these subsystems are constructed as a cascaded structure for the control design [11]. However, this approach is only applied in some classes of systems because control of cascaded systems is not solvable in general [12, 13].Constraints appear in most mechanical control systems due to different reasons such as physical restrictions and/or control tasks. There are different kinds of constraints, for example, input constraint, output constraint, and mixed constraints. Optimal control is one of the powerful tools in dealing with constraints [14, 15]. However, the computation is expensive. With the rapid development of control techniques, model predictive control (MPC) has effectiveness to handle constraints [16]. The computation is cheaper in comparison with optimal control because of the receding horizon control. In particular, when constraints have the form of saturations, a combination of the saturation function, barrier function, and direct Lyapunov method is a good choice to seek an analytic solution. The excellent work dealing with bounded control by using the nested saturation function was proposed in [17]. Based on this work, several bounded controllers have been developed [18, 19]. Concerning with output constraints, different approaches have been proposed [20–22]. Among these approaches, the barrier Lyapunov technique is one of the most powerful tools with less computation [23–25].The model of underactuated systems is usually uncertain and affected by disturbances. These factors cause the performance degradation and may destroy the stability of systems. Moreover, when only outputs are available for the feedback, the state feedback control cannot be implemented. In this case, several robust adaptive output feedback control schemata have been proposed to solve the problem [26, 27]. However, these approaches have effectiveness for fully actuated systems. Obstacles arise when systems under consideration are underactuated systems and constraints are taken into account. In this situation, the control system design becomes much more challenging because it is necessary to simultaneously deal with underactuated property, uncertain dynamics, disturbance rejection, state estimation, and constraint satisfaction.Motivated by the above consideration, in this paper, we are interested in designing the robust adaptive output feedback control for a class of underactuated aerial vehicles with input and output constraints. In particular, we focus on the class of underactuated aerial vehicles with a single thrust direction and full torque actuation. To solve the problem, the modular design strategy is proposed for the control system design. Namely, by constructing the underactuated aerial vehicle as a cascaded structure consisting of fully actuated subsystems, the robust adaptive output feedback control can be separately designed for each subsystem. In this setting, the observers are employed to reconstruct system states for the feedback and the radial basis function NNs (RBF NNs) are used to approximate the system uncertainties. To deal with the input and output constraints, a combination of saturation and barrier functions is integrated into the Lyapunov control design. Our design methodology yields the ultimate boundedness of all the states in the closed-loop system, while the tracking error converges to the small neighbourhood of the origin which can be made arbitrary small.The main contributions in this paper include the following aspects: (i) The modular design strategy is proposed to design the robust adaptive output feedback control for the class of underactuated aerial vehicles with taking into account multiple constraints. (ii) The assumption on limitations of orientation angles in the existing studies is removed by dealing with the constraints on orientation angles. It follows that the problem of the gimbal lock is solved, and the vehicle will be never overturned. (iii) The proposed methodology yields the new result for the control cascaded system design. Namely, we show that an ISS subsystem driven by a subsystem enjoying UB property results in the UB property of the overall cascaded subsystem.The rest of the paper is organized as follows. In Section2, the control problem is formulated. Section 3 presents the control design while the stability of the overall system is analysed in Section 4. In Section 5, the simulations are provided to illustrate the effectiveness of the proposed control. The conclusion of this paper is presented in Section 6.


### 1.1. Notation

Throughout this paper,ℝn denotes the Euclidean space with n−dimension. • is the Euclidean norm of vector •. a˜⟶0 denotes that a˜ converges to 0. For integer indices i and j, the symbol σi∈ℝn denotes the vector saturation function with elements σij,j=1,⋯,n. A˜∈ℝn×n denotes the n×n−matrix, and λminA˜ and λmaxA˜ denote the minimum and maximum eigenvalues of matrix A˜, respectively. Given two vector a˜,b˜∈ℝn, then a˜≤b˜ denotes a˜i≤b˜i,i=1,⋯,n.


## 1.1. Notation

Throughout this paper,ℝn denotes the Euclidean space with n−dimension. • is the Euclidean norm of vector •. a˜⟶0 denotes that a˜ converges to 0. For integer indices i and j, the symbol σi∈ℝn denotes the vector saturation function with elements σij,j=1,⋯,n. A˜∈ℝn×n denotes the n×n−matrix, and λminA˜ and λmaxA˜ denote the minimum and maximum eigenvalues of matrix A˜, respectively. Given two vector a˜,b˜∈ℝn, then a˜≤b˜ denotes a˜i≤b˜i,i=1,⋯,n.


## 2. Problem Formulation and Preliminaries

### 2.1. Problem Formulation

The system considered in this paper is the class of underactuated aerial vehicles with a single thrust direction and full torque actuation [2]. In particular, the dynamics of the considered system can be described by the following Euler–Lagrange equations:(1)mq¨=Re3u−mge3+dq,(2)Jηη¨=−Cη,η˙η˙+τ+dη,where q=x,y,zT∈ℝ3 denotes the position of the vehicle in the reference frame; η=ϕ,θ,ψ∈ℝ3 denotes the orientation angles with respect to the roll, pitch, and yaw; u∈ℝ is the total thrust; τ=τϕ,τθ,τψT∈ℝ3 is the torque; m is the mass of the rigid body; g is the acceleration due to gravity; e3=0,0,1T is the unit vector; Jη∈ℝ3×3 is the effective inertial matrix; Cη,η˙∈ℝ3×3 is the Coriolis and centrifugal force matrix; R∈SO3 is the rotation matrix having the form(3)R=CψCθCψSθSψ−SψCθCψSθCψ+SθSϕSϕCθSψSθSϕ+CψCϕSψSθCϕ−CψSϕ−SθCθSϕCθCϕ;and dqanddη∈ℝ3 are unknown disturbances with the known bounds(4)dq≤dq∗;dηdη∗,where dq∗ and dη∗ are the known positive constants.In this paper, we are interested in the input constraints as follows:(5)0<u≤u¯,(6)τ≤τ¯,and we are also interested in the following output constraints:(7)ϕ≤π/2;θ<π/2;ψ≤π,with u¯∈ℝ and τ¯∈ℝ3 being the positive constants.Remark 1.
Constraints (5) and (6) ensure the final values of the control inputs. Constraint (7) ensures that the vehicle will be never overturned [28–30].
We notice that the considered system has four control inputsu,τTT∈ℝ4 and six DOFs qT,ηTT∈ℝ6. Accordingly, it is impossible for such systems to track an arbitrary reference trajectory. Instead, we select four outputs qT,ψT for the tracking purpose as usual in [2, 29]. Therefore, the control problem considered in this paper can be stated as follows.Problem 1.
Consider the class of underactuated vehicles described by (1) and (2). Let qdtT,ψdtT∈ℝ4 be a given smooth time-varying desired trajectory with its time derivatives bounded up to the second order. Under conditions that the velocities q˙ and η˙ are not available for the feedback control, the terms Jη,Cη,η˙, and m are uncertain and the disturbances dq and dτ are bounded; design a robust adaptive output feedback control U=u,τTT for system (1) and (2) such that the following criteria are met:(i)
Constraints (5)–(7) are satisfied(ii)
All the signals in the closed-loop system are bounded(iii)
The outputqT,ψT follows an adjustable neighborhood of its desired trajectory qdT,ψdT
As the problem is formulated, the velocitiesq˙ and η˙ are unavailable for the feedback while the uncertain terms Jη,Cη,η˙, and m, constraints (5)–(7), and disturbances (4) are taken into account. Therefore, direct design of the controls u and τ is not an easy task. To overcome this difficulty, the control strategy based on the modular design is proposed as follows. First, we decouple and construct systems (1) and (2) as the cascaded structure in which position subsystem (1) is driven by orientation subsystem (2). Second, the robust output feedback control is designed for position subsystem (1) such that subsystem (1) is ISS while constraint (5) is satisfied. Third, the robust adaptive output feedback control is designed to achieve the stability of orientation subsystem (2) and simultaneously tackle constraints (6) and (7).Remark 2.
We notice that by dealing with constraint (7), the following two issues are ensured. First, the assumption on limitations of orientation angles ϕ,θ<π/2 in the existing studies [1, 31] is rejected, and the so-called problem gimbal lock is solved. Second, the vehicle will be never overturned.


### 2.2. Preliminaries

This section briefly recalls the notions of RBF NN, high-gain observer, and saturation function which are used in the control design in the next section.


#### 2.2.1. RBF Neural Networks

The RBF neural networks consist of two layers, in which the hidden layer completes a fixed nonlinear transformation. Then, the output layer linearly combines the outputs of the hidden layer. Therefore, the networks can be simply represented by [32](8)fnnX=WTGX,with the input vector X=x1,x2,⋯,xmT∈ΩX⊂ℝm, weight vector W∈ℝℓ, node number ℓ>1, and basis function GX=g1X,g2X,⋯,gℓXT∈ℝℓ. Here, gjX is the Gaussian function as follows:(9)gjX=exp−‖X−κj‖2hj2,j=1,2,⋯,ℓ,where κj=κj1,κj2,⋯,κjℓT and hj are the center and the width, respectively.It is shown that by universe approximation results that any continuous functionfX defined on a compact set ΩX⊂ℝm can be approximated by WTGX if ℓ is chosen sufficiently large. This is described as(10)fX=W∗TGX+εX,∀X∈ΩX⊂ℝm,where εX is the function approximation error and W∗ is the ideal weigh vector. The following assumption for the function approximation is made.Assumption 1.
(see [32]) Over the compact set ΩX⊂ℝm, the neural network ideal weight vector W∗, and the function approximation error εX of neural networks is bounded by(11)W∗≤Wm,εX≤εm,∀X∈ΩX,where Wm and εm are the positive constants.Lemma 1.
(see [32]) For Gaussian RBF (9), if X^=X−βϑ with constant β>0 and ϑ being a bounded vector, then(12)GX^=GX−βGℓ,where Gℓ is a bounded function vector.


#### 2.2.2. High-Gain Observer

Since only the positions and orientation angles are measurable and the linear and angular velocities are not available, we need to estimate the velocitiesq˙ and η˙ for the output feedback control. The following high-gain observer is used later.Lemma 2.
(see [33]) Assume that the output yt of a system and its first n−1 derivatives are bounded, that is, yk<Yk,k=1,⋯,n−1, with positive constants Yk. Consider the following linear system:(13)ες˙i=ςi+1,i=1,⋯,n−1,ες˙n=ν1ςn−ν2ςn−1−⋯−νn−1ς2−ς1+yt,withε being any small positive constant, where ςi,i=1,⋯,n, are the state variables of the observer. Choosing the parameters ν1,⋯,νn such that the polynomial sn+ν1sn−1+⋯+νn−1s+1 is Hurwitz, then we have the following properties:(i)
ςk+1/εk−yk=−εϑ¯k+1,k=0,1,⋯,n−1, where ϑ¯=ςn+ν1ςn−1+⋯+νn−1ς1 with ϑ¯k denoting the k derivatives of ϑ(ii)
There exist positive constantst∗ and bk only depending on Yk−1k=1,2,⋯,n,ε, and νi,i=1,⋯,n−1, such that we have ϑ¯k≤bk for all t>t∗


#### 2.2.3. Saturation Function

To deal with input constraint (5), the following saturation function is used.Definition 1.
(see [17]). Given a positive constant σ¯, a function σ:ℝ⟶ℝ is said to be a linear saturation with the bound σ¯ if it is a continuous, nondecreasing function satisfying (i) ςσς>0,∀ς≠0; (ii) σς=ς,∀ς≤σ¯; (iii) σς≤σ¯,∀ς∈ℝ; and (iv) ∂σ/∂ς≤1,∀ς∈ℝ.


## 2.1. Problem Formulation

The system considered in this paper is the class of underactuated aerial vehicles with a single thrust direction and full torque actuation [2]. In particular, the dynamics of the considered system can be described by the following Euler–Lagrange equations:(1)mq¨=Re3u−mge3+dq,(2)Jηη¨=−Cη,η˙η˙+τ+dη,where q=x,y,zT∈ℝ3 denotes the position of the vehicle in the reference frame; η=ϕ,θ,ψ∈ℝ3 denotes the orientation angles with respect to the roll, pitch, and yaw; u∈ℝ is the total thrust; τ=τϕ,τθ,τψT∈ℝ3 is the torque; m is the mass of the rigid body; g is the acceleration due to gravity; e3=0,0,1T is the unit vector; Jη∈ℝ3×3 is the effective inertial matrix; Cη,η˙∈ℝ3×3 is the Coriolis and centrifugal force matrix; R∈SO3 is the rotation matrix having the form(3)R=CψCθCψSθSψ−SψCθCψSθCψ+SθSϕSϕCθSψSθSϕ+CψCϕSψSθCϕ−CψSϕ−SθCθSϕCθCϕ;and dqanddη∈ℝ3 are unknown disturbances with the known bounds(4)dq≤dq∗;dηdη∗,where dq∗ and dη∗ are the known positive constants.In this paper, we are interested in the input constraints as follows:(5)0<u≤u¯,(6)τ≤τ¯,and we are also interested in the following output constraints:(7)ϕ≤π/2;θ<π/2;ψ≤π,with u¯∈ℝ and τ¯∈ℝ3 being the positive constants.Remark 1.
Constraints (5) and (6) ensure the final values of the control inputs. Constraint (7) ensures that the vehicle will be never overturned [28–30].
We notice that the considered system has four control inputsu,τTT∈ℝ4 and six DOFs qT,ηTT∈ℝ6. Accordingly, it is impossible for such systems to track an arbitrary reference trajectory. Instead, we select four outputs qT,ψT for the tracking purpose as usual in [2, 29]. Therefore, the control problem considered in this paper can be stated as follows.Problem 1.
Consider the class of underactuated vehicles described by (1) and (2). Let qdtT,ψdtT∈ℝ4 be a given smooth time-varying desired trajectory with its time derivatives bounded up to the second order. Under conditions that the velocities q˙ and η˙ are not available for the feedback control, the terms Jη,Cη,η˙, and m are uncertain and the disturbances dq and dτ are bounded; design a robust adaptive output feedback control U=u,τTT for system (1) and (2) such that the following criteria are met:(i)
Constraints (5)–(7) are satisfied(ii)
All the signals in the closed-loop system are bounded(iii)
The outputqT,ψT follows an adjustable neighborhood of its desired trajectory qdT,ψdT
As the problem is formulated, the velocitiesq˙ and η˙ are unavailable for the feedback while the uncertain terms Jη,Cη,η˙, and m, constraints (5)–(7), and disturbances (4) are taken into account. Therefore, direct design of the controls u and τ is not an easy task. To overcome this difficulty, the control strategy based on the modular design is proposed as follows. First, we decouple and construct systems (1) and (2) as the cascaded structure in which position subsystem (1) is driven by orientation subsystem (2). Second, the robust output feedback control is designed for position subsystem (1) such that subsystem (1) is ISS while constraint (5) is satisfied. Third, the robust adaptive output feedback control is designed to achieve the stability of orientation subsystem (2) and simultaneously tackle constraints (6) and (7).Remark 2.
We notice that by dealing with constraint (7), the following two issues are ensured. First, the assumption on limitations of orientation angles ϕ,θ<π/2 in the existing studies [1, 31] is rejected, and the so-called problem gimbal lock is solved. Second, the vehicle will be never overturned.


## 2.2. Preliminaries

This section briefly recalls the notions of RBF NN, high-gain observer, and saturation function which are used in the control design in the next section.


### 2.2.1. RBF Neural Networks

The RBF neural networks consist of two layers, in which the hidden layer completes a fixed nonlinear transformation. Then, the output layer linearly combines the outputs of the hidden layer. Therefore, the networks can be simply represented by [32](8)fnnX=WTGX,with the input vector X=x1,x2,⋯,xmT∈ΩX⊂ℝm, weight vector W∈ℝℓ, node number ℓ>1, and basis function GX=g1X,g2X,⋯,gℓXT∈ℝℓ. Here, gjX is the Gaussian function as follows:(9)gjX=exp−‖X−κj‖2hj2,j=1,2,⋯,ℓ,where κj=κj1,κj2,⋯,κjℓT and hj are the center and the width, respectively.It is shown that by universe approximation results that any continuous functionfX defined on a compact set ΩX⊂ℝm can be approximated by WTGX if ℓ is chosen sufficiently large. This is described as(10)fX=W∗TGX+εX,∀X∈ΩX⊂ℝm,where εX is the function approximation error and W∗ is the ideal weigh vector. The following assumption for the function approximation is made.Assumption 1.
(see [32]) Over the compact set ΩX⊂ℝm, the neural network ideal weight vector W∗, and the function approximation error εX of neural networks is bounded by(11)W∗≤Wm,εX≤εm,∀X∈ΩX,where Wm and εm are the positive constants.Lemma 1.
(see [32]) For Gaussian RBF (9), if X^=X−βϑ with constant β>0 and ϑ being a bounded vector, then(12)GX^=GX−βGℓ,where Gℓ is a bounded function vector.


### 2.2.2. High-Gain Observer

Since only the positions and orientation angles are measurable and the linear and angular velocities are not available, we need to estimate the velocitiesq˙ and η˙ for the output feedback control. The following high-gain observer is used later.Lemma 2.
(see [33]) Assume that the output yt of a system and its first n−1 derivatives are bounded, that is, yk<Yk,k=1,⋯,n−1, with positive constants Yk. Consider the following linear system:(13)ες˙i=ςi+1,i=1,⋯,n−1,ες˙n=ν1ςn−ν2ςn−1−⋯−νn−1ς2−ς1+yt,withε being any small positive constant, where ςi,i=1,⋯,n, are the state variables of the observer. Choosing the parameters ν1,⋯,νn such that the polynomial sn+ν1sn−1+⋯+νn−1s+1 is Hurwitz, then we have the following properties:(i)
ςk+1/εk−yk=−εϑ¯k+1,k=0,1,⋯,n−1, where ϑ¯=ςn+ν1ςn−1+⋯+νn−1ς1 with ϑ¯k denoting the k derivatives of ϑ(ii)
There exist positive constantst∗ and bk only depending on Yk−1k=1,2,⋯,n,ε, and νi,i=1,⋯,n−1, such that we have ϑ¯k≤bk for all t>t∗


### 2.2.3. Saturation Function

To deal with input constraint (5), the following saturation function is used.Definition 1.
(see [17]). Given a positive constant σ¯, a function σ:ℝ⟶ℝ is said to be a linear saturation with the bound σ¯ if it is a continuous, nondecreasing function satisfying (i) ςσς>0,∀ς≠0; (ii) σς=ς,∀ς≤σ¯; (iii) σς≤σ¯,∀ς∈ℝ; and (iv) ∂σ/∂ς≤1,∀ς∈ℝ.


## 2.2.1. RBF Neural Networks

The RBF neural networks consist of two layers, in which the hidden layer completes a fixed nonlinear transformation. Then, the output layer linearly combines the outputs of the hidden layer. Therefore, the networks can be simply represented by [32](8)fnnX=WTGX,with the input vector X=x1,x2,⋯,xmT∈ΩX⊂ℝm, weight vector W∈ℝℓ, node number ℓ>1, and basis function GX=g1X,g2X,⋯,gℓXT∈ℝℓ. Here, gjX is the Gaussian function as follows:(9)gjX=exp−‖X−κj‖2hj2,j=1,2,⋯,ℓ,where κj=κj1,κj2,⋯,κjℓT and hj are the center and the width, respectively.It is shown that by universe approximation results that any continuous functionfX defined on a compact set ΩX⊂ℝm can be approximated by WTGX if ℓ is chosen sufficiently large. This is described as(10)fX=W∗TGX+εX,∀X∈ΩX⊂ℝm,where εX is the function approximation error and W∗ is the ideal weigh vector. The following assumption for the function approximation is made.Assumption 1.
(see [32]) Over the compact set ΩX⊂ℝm, the neural network ideal weight vector W∗, and the function approximation error εX of neural networks is bounded by(11)W∗≤Wm,εX≤εm,∀X∈ΩX,where Wm and εm are the positive constants.Lemma 1.
(see [32]) For Gaussian RBF (9), if X^=X−βϑ with constant β>0 and ϑ being a bounded vector, then(12)GX^=GX−βGℓ,where Gℓ is a bounded function vector.


## 2.2.2. High-Gain Observer

Since only the positions and orientation angles are measurable and the linear and angular velocities are not available, we need to estimate the velocitiesq˙ and η˙ for the output feedback control. The following high-gain observer is used later.Lemma 2.
(see [33]) Assume that the output yt of a system and its first n−1 derivatives are bounded, that is, yk<Yk,k=1,⋯,n−1, with positive constants Yk. Consider the following linear system:(13)ες˙i=ςi+1,i=1,⋯,n−1,ες˙n=ν1ςn−ν2ςn−1−⋯−νn−1ς2−ς1+yt,withε being any small positive constant, where ςi,i=1,⋯,n, are the state variables of the observer. Choosing the parameters ν1,⋯,νn such that the polynomial sn+ν1sn−1+⋯+νn−1s+1 is Hurwitz, then we have the following properties:(i)
ςk+1/εk−yk=−εϑ¯k+1,k=0,1,⋯,n−1, where ϑ¯=ςn+ν1ςn−1+⋯+νn−1ς1 with ϑ¯k denoting the k derivatives of ϑ(ii)
There exist positive constantst∗ and bk only depending on Yk−1k=1,2,⋯,n,ε, and νi,i=1,⋯,n−1, such that we have ϑ¯k≤bk for all t>t∗


## 2.2.3. Saturation Function

To deal with input constraint (5), the following saturation function is used.Definition 1.
(see [17]). Given a positive constant σ¯, a function σ:ℝ⟶ℝ is said to be a linear saturation with the bound σ¯ if it is a continuous, nondecreasing function satisfying (i) ςσς>0,∀ς≠0; (ii) σς=ς,∀ς≤σ¯; (iii) σς≤σ¯,∀ς∈ℝ; and (iv) ∂σ/∂ς≤1,∀ς∈ℝ.


## 3. Control Design

This section presents the control design for systems (1) and (2) according to the proposed control strategy in the previous section.


### 3.1. Position Control by Bounded Robust Output Feedback

The goal of this section is to decouple subsystem (1) from orientation subsystem (2) and design the robust output feedback control for (1) such that the position q tracks its desired trajectory qd, constraint (5) is satisfied, and subsystem (1) is ISS. To this end, we assume that the uncertain mass m in (1) can be written as(14)m=m^+Δm,where m^ is the known nominal term and Δm is an uncertain term with the known bound as(15)0≤Δm≤Δm∗<m^.Denoting the states as(16)q1=q,q2=q˙,and using the relationship 1/m=1/m^−Δm/mm^ from (14), system (1) can be rewritten as(17)q˙1=q2,q˙2=um^Re3−ge3+dp,where dp=dq−uRe3/mm^ is bounded by(18)dp≤dp∗,with the known bound(19)dp∗=dq∗+Δm∗u¯m^−Δm∗m^.Remark 3.
The upper bounddp∗ is determined by (19) only when the control u is bounded by u¯, i.e., constraint (5) must be satisfied. The satisfaction of this constraint will be guaranteed in the control design later.


#### 3.1.1. Position Observer Design

Because the velocityq2 is not available, we will estimate q2 for the output feedback control by using the following observer:(20)q^˙1=q^2+L1q1−q^1,q^˙2=um^Re3−ge3+L2q1−q^1,where q^1 and q^2 are the estimations of q1 and q2, respectively, and L1andL2∈ℝ3×3 are the positive constant observer gain matrices. By defining the observer error(21)q˜=q˜1T,q˜2TT=q1−q^1T,q2−q^2TTand subtracting (20) from (17), the dynamic equation of the observer error is given by(22)q˜˙=Lq˜+dq˜,where(23)L=−L1I3×3−L20;dq˜=03×1dp.Lemma 3.
For system (22), if L1 and L2 are chosen such that L is Hurwitz, then there exists a time instant To such that the observer error satisfies(24)q˜t≤koq˜0e−ρot,0<t<To,q˜t≤δo,∀t≥To,for some positive constants ko,ρo, and δo, where δo can be made small by adjusting L1 and L2.Proof.
See AppendixA.Remark 4.
It is shown from (24) that the observer error q˜ exponentially decays with the convergent rate ρo. Moreover, the observer error is uniformly ultimately bounded with the ultimate bound δo which can be adjusted by the observer design matrices L1 and L2.


#### 3.1.2. Bounded Robust Output Feedback Control Design

Having position observer (20), we proceed to decouple (1) from (2) and design the control u for position subsystem (1) to solve the tracking problem. To this end, defining the tracking errors(25)p1=q1−qd;p^2=q^2−q˙d,and using the relationship q2=q^2+q˜2, the position tracking error system is obtained as(26)p˙1=p^2+q˜2,p^˙2=um^Re3−ge3−q¨d+L2q˜1,q˜˙=Lq˜+dq˜.We now decouple (26) from (2) by adding and subtracting the right-hand side of p^2−equation in (26) by the virtual control F∈ℝ3 of the following form:(27)F=um^Rde3,where Rd has the same form with R except that the angles ϕ,θ, and ψ are replaced with their desired ϕd,θd, and ψd, respectively. Accordingly, p^2−equation in (26) can be further rewritten as(28)p^˙2=F+ϖ−ge3−q¨d+L2q˜1,with(29)ϖ=uR−Rde3m^.Having (28), we are able to freely design the control F for (26). Furthermore, solving (27) with respect to ϕd,θd, and u [29], we obtain the real control u and the desired orientation angles as(30)u=m^F12+F22+F32,ϕd=arctanSψdF1−CψdF2CψdF1+SψdF22+F32,θd=arctanCψdF1+SψdF2F3,where we use the denotations Sψd=sinψd and Cψd=cosψd.In view of (30), u satisfies constraint (5) if F is bounded with the known bound. For this reason, the control F is designed as(31)F=−σ1p1+p^2−σ2p^2+q¨d+ge3,where σ1andσ2∈ℝ3 are the saturation function vectors in Definition 1.Assumption 2.
The desired trajectoryqd is bounded by(32)q¨d≤ρ,where ρ=ρx,ρy,ρzT∈ℝ3 is a given positive vector.
Under Assumption2, constraint (5) is ensured if the control F satisfies(33)g−σ¯13−σ¯23−ρz>0,σ¯11+σ¯21+ρx2+σ¯12+σ¯22+ρy2+σ¯13+σ¯23+ρz+g2≤u¯2m^2,where σ¯ij is the bound of σij for i=1,2andj=1,2,3.
From (28), (26), and (31) we obtain the closed loop position tracking error system as(34)p˙1=p^2+q˜2,p^˙2=−σ1p1+p^2−σ2p^2+L2q˜1+ϖ,q˜˙=Lq˜+dq˜.Theorem 1.
Consider system (34) under Assumption 2 and Lemma 3. If condition (33) is satisfied, then for any initial condition p0, the following statements hold:(i)
Constraint (5) is ensured.(ii)
Ifϖ,dp˜ is bounded by(35)ϖ≤ϖ∗,dq˜≤dp∗,where dp∗ is defined in (19) and ϖ∗ is a positive constant satisfying(36)0<8ϖ∗≤σ¯1,0<8ϖ∗≤σ¯2,then all the system states are ultimately bounded, i.e., there exist positive definite functions V1p1,p^2,q˜ and W1p1,p^2,q˜, a positive number C1, and a time instant T1>0 such that(37)V˙1<−W1+C1,∀t>T1.
(iii)
System (34) is ISS with the state p1,p^2,q˜ and the input ϖ,dq˜ under restriction (35).Proof.
See AppendixB.Remark 5.
It is shown from (34) that by using observer (20) and p^2−equation for the control design, the disturbance dq is moved from p2−equation to q˜−equation. This allows us to deal with the effect of dp on the tracking performance by adjusting the observer gain L as shown in Lemma 3.Remark 6.
By introducing the virtual controlF for (26), on the one hand, we decouple (26) from (2). On the other hand, we deal with the underactuated property of the considered system. Indeed, in view of (26), the one-dimensional control u∈ℝ1 cannot be used to control the three-dimensional output p^2∈ℝ3.
In view of (34), the tracking errors p1,p^2 are driven by q˜ and ϖ. The observer error q˜ can be made small by adjusting the observer gain matrix L as shown in Lemma 3. The remainder is how to make ϖ small. In view of (29), R⟶Rd implies ϖ⟶0, or we need to have η⟶ηd, where ηd is the desired orientation angles. This goal will be achieved in the next section.


### 3.2. Orientation Control by Bounded Robust Adaptive Output Feedback

The goal of this section is the design of the controlτ for orientation subsystem (2) to solve the orientation tracking problem, i.e., η tracks its desired ηd and constraints (6) and (7) are simultaneously tackled. To this end, we denote the states as(38)η1=η,η2=η˙,and rewrite system (2) in the form(39)η˙1=η2,Jη1η˙2=−Cη1,η2η2+τ+dτ.


#### 3.2.1. High-Gain Observer for Orientation Subsystems

Because the angular velocityη2 in (39) is not available, we will estimate η2 by using the following high-gain observer:(40)εχ˙1=χ2,εχ˙2=−ν1χ2−χ1+η1t,where χ1andχ2∈ℝ3 are the state vectors of observer (40) and ν1 is chosen such that s2+ν1s+1 is Hurwitz. Then, the estimations η^1 and η^2 of η1 and η2, respectively, are reconstructed as(41)η^1=χ1,η^2=1εχ2.Lemma 4.
(see [33]). For orientation subsystem (2), high-gain observer (40) provides the estimations η^1,η^2 of η1,η2, and the estimation errors are bounded by(42)η1−η^1≤εϑ1,η2−η^2≤εϑ2,for some positive constants ϑ1 and ϑ2.


#### 3.2.2. Bounded Robust Adaptive Output Feedback Control Design

Having high-gain observer (40), we proceed to design the control τ for orientation subsystem (39). In the control design, constraints (6) and (7) are simultaneously dealt with by the use of the saturation and barrier functions. The RBF NNs are used to approximate the unknown terms Jη1 and Cη1,η2 in (39). The control design is based on the backstepping method. For more detail, the design schema consists of following two steps:Step 1
Define the tracking errors as(43)ζ1=η1−ηd,ζ2=η2−α,ζ^2=η^2−α,where α is the virtual control. Let ζ˜2 be the error between ζ2 and ζ^2, involving (42); ζ˜2 is bounded by(44)ζ˜2=ζ2−ζ^2=η˙−η^˙≤εϑ2,and dynamic equation of ζ1 is(45)ζ˙1=η˙1−η˙d=ζ2+α−η˙d.To deal with output constraint (7), let us consider the barrier Lyapunov function candidate [24] as follows:(46)V2=∑i=1312logbi2bi2−ζ1i2,where bi for i=1,2,3 is defined as(47)b1=π2−ϕ¯d,b2=π2−θ¯d,b3=π−ψ¯d,with ψ¯d being the bound of ψd. The time derivative of (46) along (45) is given by(48)V˙2=∑13ζ1iζ˙1ibi2−ζ1i2=ζ1TBζ1ζ˙1,where(49)Bζ1=1b12−ζ1120001b22−ζ1220001b32−ζ132.Substituting (45) into (48) yields(50)V˙2=ζ1TBζ1α−η˙d+ζ1TBζ1ζ2.Choosing(51)α=−A0+Bζ1−1A1ζ1+η˙d,where A0andA1∈ℝ3×3 are positive matrices, and substituting (51) into (50), yield(52)V˙2=−ζ1TBζ1A0ζ1−ζ1TA1ζ1+ζ1TBζ1ζ2.Using Lemma2 in [34], in the set ζ1i<bi,i=1,2,3, we have(53)logbi2bi2−ζ1i2<ζ1i2bi2−ζ1i2,and the first term in (52) satisfies(54)−ζ1TBζ1A0ζ1<−2λminA0∑i=1312logbi2bi2−ζ1i2<−2λminA0V2.Substituting (54) into (52) yields(55)V˙2<−2λminA0V2−ζ1TA1ζ1+ζ1TBζ1ζ2.The first two terms in (55) are negative; the last term will be cancelled in the next step.Step 2
Taking the time derivativeζ2 in (43) along (2) yields(56)Jη1ζ˙2=Jη1η˙2−Jη1α˙=−Cη1,η2ζ2−Cη1,η2α+τ+dτ−Jη1α˙.Consider the Lyapunov function candidate as(57)V3=V2+12ζ2TJη1ζ2,and its time derivative along (56) is given by(58)V˙3=V˙2+12ζ2TJ˙η1ζ2+ζTJη1ζ˙2=V˙2+12ζ2TJ˙η1−2Cη1,η2ζ2+ζ2T−Cη1,η2α+τ+dτ−Jη1α˙.Thus, the real controlτ appears in (58). We need to design the control τ to make V˙3≤0. To this end, we notice that the term ζ2TJ˙η1−2Cη1,η2ζ2=0 according to the skew symmetric property. The unknown terms Cη1,η2α and Jη1α˙ are estimated by using the RBF NNs approximation function from (10) as(59)Cη1,η2α+Jη1α˙=W∗TΦZ+εZ,where Z=η1T,η2TT is the input vector of the NNs and W∗ is the ideal unknown constant weight matrix, which has the following form:(60)W∗=w1∗T01×ℓ01×ℓ01×ℓw2∗T01×ℓ01×ℓ01×ℓw3∗TT∈ℝ3ℓ×3,with wi∗∈ℝℓ,i=1,2,3, and W∗ is bounded by(61)W∗≤Wm,where Wm is a positive constant; ℓ is the node number of the corresponding NN; and ΦZ∈ℝ3ℓ is the basis function vector of the following form:(62)ΦZ=Φ1TZ,Φ2TZ,Φ3TZT∈ℝ3ℓ,with ΦiZ∈ℝℓ; ϵZ is the error vector with the known bound as(63)ϵZ≤ϵ∗,where ϵ∗ is a positive constant.It was mentioned that the optimal weightW∗ is unavailable for the approximation purpose. Furthermore, the state η2 is also unavailable for the feedback and hence cannot be the input of the NNs. Therefore, approximation (59) cannot be implemented. To deal with this situation, consider the relation between η2 and η^2 in Lemma 4; we use the estimated state η^2 as the input of the NNs for the approximation of the left-hand side of (59). The possibility of the use of the estimated state η^2 is shown in Appendix C. Accordingly, the input of the NNs is Z^=η1T,η^2TT, and hence, the basis function vector is ΦZ^.Next, we need to deal with input constraint (6). To do this, the control τ is designed in the following form:(64)τi=τ¯i,ifτ0i>τ¯i,τoi,ifτ0i≤τ¯i,−τ¯i,ifτ0i<−τ¯i,for i=ϕ,θ,ψ, with τo=τoϕ,τoθ,τoψT being the free control to be designed. Denoting the control input error as(65)Δτ=τ−τoand substituting (55), (59), and (65) into (58), we obtain(66)V˙3=−2λminA0V2−ζ1TA1ζ1+ζ1TBζ1ζ2+ζ2T−W∗TΦZ−εZ+τo+Δτ+dτ.In view of (66), Δτ≠0 when the saturation appears. We need to reject Δτ. This can be accomplished by using the following auxiliary system [35]:(67)υ˙=−Kυυ−ζ^2TΔτ+Δτ2/k0υ2υ+Δτ,υ≥ϱ,0,υ<ϱ,where υ∈ℝ3 is the auxiliary vector; Kυ is a positive definite matrix; k0 is a positive number; and ϱ is a positive constant that should be chosen as an appropriate value in accordance with the requirement of the tracking performance.Thus, we have transformed the controlτ under input constraint (6) to the free control τo by adding the auxiliary variable υ. Consequently, in order to obtain the control τo based on the Lyapunov method, we consider the following Lyapunov function candidate:(68)V4=V3+12∑i=13w˜iTΓi−1w˜i+12υTυ,where w˜i=wi∗−w^i, for i=1,2,3, is the estimation error and w^i is the estimation of wi∗. The goal now is to select the control τo to make the time derivative V˙4 negative. The process of selecting τo is detailed in Appendix C. At this point, in order to state the result of this section, we propose the control τo and the update law w^i in advance as follows:(69)τo=−Bζ1ζ1−A2ζ^2+W^TΦZ^+υ,(70)w^˙i=−ΓiΦiZ^ζ^2i+μiw^i,i=1,2,3,where A2 is a positive matrix and μi is a positive constant.In control (69), the first term is proposed to cancel the second term in (55). The second term is used for the stabilization of ζ2 and dealing with dτ and ϵ. The third term is the estimation of (59). The last term is used to tackle the appearance of the input saturation.Theorem 2.
For orientation subsystem (2), under high-gain observers (40) and (41), controllers (64), (69), and (67), and update law (70), if the initial conditions are such that ζ10<b, where b is defined in (47), then(i)
All the system states are ultimately bounded, i.e., there exist positive numbersκ and C4 such that(71)V˙4≤−κV4+C4.(ii)
Constraints (6) and (7) are satisfied; in particular, the tracking error ζ1 is bounded by(72)ζ1t≤b1−e2V4t,∀t>0.(iii)
The tracking errorζt converges to a small neighbourhood of the origin which can be made arbitrarily small.Proof.
See AppendixC.Remark 7.
It is noticed that the second term ofV3 in (57) is the function of the state ζ2 instead of ζ^2. This allows us to know the behavior of the state ζ2 even when ζ2 is unavailable for the feedback.Remark 8.
In order to deal with constraint (7), we use the Euler angles in the representation of the attitude. This makes the control design become much easier because the use of the unit quaternion may lead to inconsistent behavior, i.e., the unwinding phenomenon [2]. Although the use of Euler angles has drawback which is the so-called gimbal lock [36], i.e., the rotation matrix is singular when ϕ=π/2 or θ=π/2, this drawback is tackled by the satisfaction of constraint (7).


## 3.1. Position Control by Bounded Robust Output Feedback

The goal of this section is to decouple subsystem (1) from orientation subsystem (2) and design the robust output feedback control for (1) such that the position q tracks its desired trajectory qd, constraint (5) is satisfied, and subsystem (1) is ISS. To this end, we assume that the uncertain mass m in (1) can be written as(14)m=m^+Δm,where m^ is the known nominal term and Δm is an uncertain term with the known bound as(15)0≤Δm≤Δm∗<m^.Denoting the states as(16)q1=q,q2=q˙,and using the relationship 1/m=1/m^−Δm/mm^ from (14), system (1) can be rewritten as(17)q˙1=q2,q˙2=um^Re3−ge3+dp,where dp=dq−uRe3/mm^ is bounded by(18)dp≤dp∗,with the known bound(19)dp∗=dq∗+Δm∗u¯m^−Δm∗m^.Remark 3.
The upper bounddp∗ is determined by (19) only when the control u is bounded by u¯, i.e., constraint (5) must be satisfied. The satisfaction of this constraint will be guaranteed in the control design later.


### 3.1.1. Position Observer Design

Because the velocityq2 is not available, we will estimate q2 for the output feedback control by using the following observer:(20)q^˙1=q^2+L1q1−q^1,q^˙2=um^Re3−ge3+L2q1−q^1,where q^1 and q^2 are the estimations of q1 and q2, respectively, and L1andL2∈ℝ3×3 are the positive constant observer gain matrices. By defining the observer error(21)q˜=q˜1T,q˜2TT=q1−q^1T,q2−q^2TTand subtracting (20) from (17), the dynamic equation of the observer error is given by(22)q˜˙=Lq˜+dq˜,where(23)L=−L1I3×3−L20;dq˜=03×1dp.Lemma 3.
For system (22), if L1 and L2 are chosen such that L is Hurwitz, then there exists a time instant To such that the observer error satisfies(24)q˜t≤koq˜0e−ρot,0<t<To,q˜t≤δo,∀t≥To,for some positive constants ko,ρo, and δo, where δo can be made small by adjusting L1 and L2.Proof.
See AppendixA.Remark 4.
It is shown from (24) that the observer error q˜ exponentially decays with the convergent rate ρo. Moreover, the observer error is uniformly ultimately bounded with the ultimate bound δo which can be adjusted by the observer design matrices L1 and L2.


### 3.1.2. Bounded Robust Output Feedback Control Design

Having position observer (20), we proceed to decouple (1) from (2) and design the control u for position subsystem (1) to solve the tracking problem. To this end, defining the tracking errors(25)p1=q1−qd;p^2=q^2−q˙d,and using the relationship q2=q^2+q˜2, the position tracking error system is obtained as(26)p˙1=p^2+q˜2,p^˙2=um^Re3−ge3−q¨d+L2q˜1,q˜˙=Lq˜+dq˜.We now decouple (26) from (2) by adding and subtracting the right-hand side of p^2−equation in (26) by the virtual control F∈ℝ3 of the following form:(27)F=um^Rde3,where Rd has the same form with R except that the angles ϕ,θ, and ψ are replaced with their desired ϕd,θd, and ψd, respectively. Accordingly, p^2−equation in (26) can be further rewritten as(28)p^˙2=F+ϖ−ge3−q¨d+L2q˜1,with(29)ϖ=uR−Rde3m^.Having (28), we are able to freely design the control F for (26). Furthermore, solving (27) with respect to ϕd,θd, and u [29], we obtain the real control u and the desired orientation angles as(30)u=m^F12+F22+F32,ϕd=arctanSψdF1−CψdF2CψdF1+SψdF22+F32,θd=arctanCψdF1+SψdF2F3,where we use the denotations Sψd=sinψd and Cψd=cosψd.In view of (30), u satisfies constraint (5) if F is bounded with the known bound. For this reason, the control F is designed as(31)F=−σ1p1+p^2−σ2p^2+q¨d+ge3,where σ1andσ2∈ℝ3 are the saturation function vectors in Definition 1.Assumption 2.
The desired trajectoryqd is bounded by(32)q¨d≤ρ,where ρ=ρx,ρy,ρzT∈ℝ3 is a given positive vector.
Under Assumption2, constraint (5) is ensured if the control F satisfies(33)g−σ¯13−σ¯23−ρz>0,σ¯11+σ¯21+ρx2+σ¯12+σ¯22+ρy2+σ¯13+σ¯23+ρz+g2≤u¯2m^2,where σ¯ij is the bound of σij for i=1,2andj=1,2,3.
From (28), (26), and (31) we obtain the closed loop position tracking error system as(34)p˙1=p^2+q˜2,p^˙2=−σ1p1+p^2−σ2p^2+L2q˜1+ϖ,q˜˙=Lq˜+dq˜.Theorem 1.
Consider system (34) under Assumption 2 and Lemma 3. If condition (33) is satisfied, then for any initial condition p0, the following statements hold:(i)
Constraint (5) is ensured.(ii)
Ifϖ,dp˜ is bounded by(35)ϖ≤ϖ∗,dq˜≤dp∗,where dp∗ is defined in (19) and ϖ∗ is a positive constant satisfying(36)0<8ϖ∗≤σ¯1,0<8ϖ∗≤σ¯2,then all the system states are ultimately bounded, i.e., there exist positive definite functions V1p1,p^2,q˜ and W1p1,p^2,q˜, a positive number C1, and a time instant T1>0 such that(37)V˙1<−W1+C1,∀t>T1.
(iii)
System (34) is ISS with the state p1,p^2,q˜ and the input ϖ,dq˜ under restriction (35).Proof.
See AppendixB.Remark 5.
It is shown from (34) that by using observer (20) and p^2−equation for the control design, the disturbance dq is moved from p2−equation to q˜−equation. This allows us to deal with the effect of dp on the tracking performance by adjusting the observer gain L as shown in Lemma 3.Remark 6.
By introducing the virtual controlF for (26), on the one hand, we decouple (26) from (2). On the other hand, we deal with the underactuated property of the considered system. Indeed, in view of (26), the one-dimensional control u∈ℝ1 cannot be used to control the three-dimensional output p^2∈ℝ3.
In view of (34), the tracking errors p1,p^2 are driven by q˜ and ϖ. The observer error q˜ can be made small by adjusting the observer gain matrix L as shown in Lemma 3. The remainder is how to make ϖ small. In view of (29), R⟶Rd implies ϖ⟶0, or we need to have η⟶ηd, where ηd is the desired orientation angles. This goal will be achieved in the next section.


## 3.1.1. Position Observer Design

Because the velocityq2 is not available, we will estimate q2 for the output feedback control by using the following observer:(20)q^˙1=q^2+L1q1−q^1,q^˙2=um^Re3−ge3+L2q1−q^1,where q^1 and q^2 are the estimations of q1 and q2, respectively, and L1andL2∈ℝ3×3 are the positive constant observer gain matrices. By defining the observer error(21)q˜=q˜1T,q˜2TT=q1−q^1T,q2−q^2TTand subtracting (20) from (17), the dynamic equation of the observer error is given by(22)q˜˙=Lq˜+dq˜,where(23)L=−L1I3×3−L20;dq˜=03×1dp.Lemma 3.
For system (22), if L1 and L2 are chosen such that L is Hurwitz, then there exists a time instant To such that the observer error satisfies(24)q˜t≤koq˜0e−ρot,0<t<To,q˜t≤δo,∀t≥To,for some positive constants ko,ρo, and δo, where δo can be made small by adjusting L1 and L2.Proof.
See AppendixA.Remark 4.
It is shown from (24) that the observer error q˜ exponentially decays with the convergent rate ρo. Moreover, the observer error is uniformly ultimately bounded with the ultimate bound δo which can be adjusted by the observer design matrices L1 and L2.


## 3.1.2. Bounded Robust Output Feedback Control Design

Having position observer (20), we proceed to decouple (1) from (2) and design the control u for position subsystem (1) to solve the tracking problem. To this end, defining the tracking errors(25)p1=q1−qd;p^2=q^2−q˙d,and using the relationship q2=q^2+q˜2, the position tracking error system is obtained as(26)p˙1=p^2+q˜2,p^˙2=um^Re3−ge3−q¨d+L2q˜1,q˜˙=Lq˜+dq˜.We now decouple (26) from (2) by adding and subtracting the right-hand side of p^2−equation in (26) by the virtual control F∈ℝ3 of the following form:(27)F=um^Rde3,where Rd has the same form with R except that the angles ϕ,θ, and ψ are replaced with their desired ϕd,θd, and ψd, respectively. Accordingly, p^2−equation in (26) can be further rewritten as(28)p^˙2=F+ϖ−ge3−q¨d+L2q˜1,with(29)ϖ=uR−Rde3m^.Having (28), we are able to freely design the control F for (26). Furthermore, solving (27) with respect to ϕd,θd, and u [29], we obtain the real control u and the desired orientation angles as(30)u=m^F12+F22+F32,ϕd=arctanSψdF1−CψdF2CψdF1+SψdF22+F32,θd=arctanCψdF1+SψdF2F3,where we use the denotations Sψd=sinψd and Cψd=cosψd.In view of (30), u satisfies constraint (5) if F is bounded with the known bound. For this reason, the control F is designed as(31)F=−σ1p1+p^2−σ2p^2+q¨d+ge3,where σ1andσ2∈ℝ3 are the saturation function vectors in Definition 1.Assumption 2.
The desired trajectoryqd is bounded by(32)q¨d≤ρ,where ρ=ρx,ρy,ρzT∈ℝ3 is a given positive vector.
Under Assumption2, constraint (5) is ensured if the control F satisfies(33)g−σ¯13−σ¯23−ρz>0,σ¯11+σ¯21+ρx2+σ¯12+σ¯22+ρy2+σ¯13+σ¯23+ρz+g2≤u¯2m^2,where σ¯ij is the bound of σij for i=1,2andj=1,2,3.
From (28), (26), and (31) we obtain the closed loop position tracking error system as(34)p˙1=p^2+q˜2,p^˙2=−σ1p1+p^2−σ2p^2+L2q˜1+ϖ,q˜˙=Lq˜+dq˜.Theorem 1.
Consider system (34) under Assumption 2 and Lemma 3. If condition (33) is satisfied, then for any initial condition p0, the following statements hold:(i)
Constraint (5) is ensured.(ii)
Ifϖ,dp˜ is bounded by(35)ϖ≤ϖ∗,dq˜≤dp∗,where dp∗ is defined in (19) and ϖ∗ is a positive constant satisfying(36)0<8ϖ∗≤σ¯1,0<8ϖ∗≤σ¯2,then all the system states are ultimately bounded, i.e., there exist positive definite functions V1p1,p^2,q˜ and W1p1,p^2,q˜, a positive number C1, and a time instant T1>0 such that(37)V˙1<−W1+C1,∀t>T1.
(iii)
System (34) is ISS with the state p1,p^2,q˜ and the input ϖ,dq˜ under restriction (35).Proof.
See AppendixB.Remark 5.
It is shown from (34) that by using observer (20) and p^2−equation for the control design, the disturbance dq is moved from p2−equation to q˜−equation. This allows us to deal with the effect of dp on the tracking performance by adjusting the observer gain L as shown in Lemma 3.Remark 6.
By introducing the virtual controlF for (26), on the one hand, we decouple (26) from (2). On the other hand, we deal with the underactuated property of the considered system. Indeed, in view of (26), the one-dimensional control u∈ℝ1 cannot be used to control the three-dimensional output p^2∈ℝ3.
In view of (34), the tracking errors p1,p^2 are driven by q˜ and ϖ. The observer error q˜ can be made small by adjusting the observer gain matrix L as shown in Lemma 3. The remainder is how to make ϖ small. In view of (29), R⟶Rd implies ϖ⟶0, or we need to have η⟶ηd, where ηd is the desired orientation angles. This goal will be achieved in the next section.


## 3.2. Orientation Control by Bounded Robust Adaptive Output Feedback

The goal of this section is the design of the controlτ for orientation subsystem (2) to solve the orientation tracking problem, i.e., η tracks its desired ηd and constraints (6) and (7) are simultaneously tackled. To this end, we denote the states as(38)η1=η,η2=η˙,and rewrite system (2) in the form(39)η˙1=η2,Jη1η˙2=−Cη1,η2η2+τ+dτ.


### 3.2.1. High-Gain Observer for Orientation Subsystems

Because the angular velocityη2 in (39) is not available, we will estimate η2 by using the following high-gain observer:(40)εχ˙1=χ2,εχ˙2=−ν1χ2−χ1+η1t,where χ1andχ2∈ℝ3 are the state vectors of observer (40) and ν1 is chosen such that s2+ν1s+1 is Hurwitz. Then, the estimations η^1 and η^2 of η1 and η2, respectively, are reconstructed as(41)η^1=χ1,η^2=1εχ2.Lemma 4.
(see [33]). For orientation subsystem (2), high-gain observer (40) provides the estimations η^1,η^2 of η1,η2, and the estimation errors are bounded by(42)η1−η^1≤εϑ1,η2−η^2≤εϑ2,for some positive constants ϑ1 and ϑ2.


### 3.2.2. Bounded Robust Adaptive Output Feedback Control Design

Having high-gain observer (40), we proceed to design the control τ for orientation subsystem (39). In the control design, constraints (6) and (7) are simultaneously dealt with by the use of the saturation and barrier functions. The RBF NNs are used to approximate the unknown terms Jη1 and Cη1,η2 in (39). The control design is based on the backstepping method. For more detail, the design schema consists of following two steps:Step 1
Define the tracking errors as(43)ζ1=η1−ηd,ζ2=η2−α,ζ^2=η^2−α,where α is the virtual control. Let ζ˜2 be the error between ζ2 and ζ^2, involving (42); ζ˜2 is bounded by(44)ζ˜2=ζ2−ζ^2=η˙−η^˙≤εϑ2,and dynamic equation of ζ1 is(45)ζ˙1=η˙1−η˙d=ζ2+α−η˙d.To deal with output constraint (7), let us consider the barrier Lyapunov function candidate [24] as follows:(46)V2=∑i=1312logbi2bi2−ζ1i2,where bi for i=1,2,3 is defined as(47)b1=π2−ϕ¯d,b2=π2−θ¯d,b3=π−ψ¯d,with ψ¯d being the bound of ψd. The time derivative of (46) along (45) is given by(48)V˙2=∑13ζ1iζ˙1ibi2−ζ1i2=ζ1TBζ1ζ˙1,where(49)Bζ1=1b12−ζ1120001b22−ζ1220001b32−ζ132.Substituting (45) into (48) yields(50)V˙2=ζ1TBζ1α−η˙d+ζ1TBζ1ζ2.Choosing(51)α=−A0+Bζ1−1A1ζ1+η˙d,where A0andA1∈ℝ3×3 are positive matrices, and substituting (51) into (50), yield(52)V˙2=−ζ1TBζ1A0ζ1−ζ1TA1ζ1+ζ1TBζ1ζ2.Using Lemma2 in [34], in the set ζ1i<bi,i=1,2,3, we have(53)logbi2bi2−ζ1i2<ζ1i2bi2−ζ1i2,and the first term in (52) satisfies(54)−ζ1TBζ1A0ζ1<−2λminA0∑i=1312logbi2bi2−ζ1i2<−2λminA0V2.Substituting (54) into (52) yields(55)V˙2<−2λminA0V2−ζ1TA1ζ1+ζ1TBζ1ζ2.The first two terms in (55) are negative; the last term will be cancelled in the next step.Step 2
Taking the time derivativeζ2 in (43) along (2) yields(56)Jη1ζ˙2=Jη1η˙2−Jη1α˙=−Cη1,η2ζ2−Cη1,η2α+τ+dτ−Jη1α˙.Consider the Lyapunov function candidate as(57)V3=V2+12ζ2TJη1ζ2,and its time derivative along (56) is given by(58)V˙3=V˙2+12ζ2TJ˙η1ζ2+ζTJη1ζ˙2=V˙2+12ζ2TJ˙η1−2Cη1,η2ζ2+ζ2T−Cη1,η2α+τ+dτ−Jη1α˙.Thus, the real controlτ appears in (58). We need to design the control τ to make V˙3≤0. To this end, we notice that the term ζ2TJ˙η1−2Cη1,η2ζ2=0 according to the skew symmetric property. The unknown terms Cη1,η2α and Jη1α˙ are estimated by using the RBF NNs approximation function from (10) as(59)Cη1,η2α+Jη1α˙=W∗TΦZ+εZ,where Z=η1T,η2TT is the input vector of the NNs and W∗ is the ideal unknown constant weight matrix, which has the following form:(60)W∗=w1∗T01×ℓ01×ℓ01×ℓw2∗T01×ℓ01×ℓ01×ℓw3∗TT∈ℝ3ℓ×3,with wi∗∈ℝℓ,i=1,2,3, and W∗ is bounded by(61)W∗≤Wm,where Wm is a positive constant; ℓ is the node number of the corresponding NN; and ΦZ∈ℝ3ℓ is the basis function vector of the following form:(62)ΦZ=Φ1TZ,Φ2TZ,Φ3TZT∈ℝ3ℓ,with ΦiZ∈ℝℓ; ϵZ is the error vector with the known bound as(63)ϵZ≤ϵ∗,where ϵ∗ is a positive constant.It was mentioned that the optimal weightW∗ is unavailable for the approximation purpose. Furthermore, the state η2 is also unavailable for the feedback and hence cannot be the input of the NNs. Therefore, approximation (59) cannot be implemented. To deal with this situation, consider the relation between η2 and η^2 in Lemma 4; we use the estimated state η^2 as the input of the NNs for the approximation of the left-hand side of (59). The possibility of the use of the estimated state η^2 is shown in Appendix C. Accordingly, the input of the NNs is Z^=η1T,η^2TT, and hence, the basis function vector is ΦZ^.Next, we need to deal with input constraint (6). To do this, the control τ is designed in the following form:(64)τi=τ¯i,ifτ0i>τ¯i,τoi,ifτ0i≤τ¯i,−τ¯i,ifτ0i<−τ¯i,for i=ϕ,θ,ψ, with τo=τoϕ,τoθ,τoψT being the free control to be designed. Denoting the control input error as(65)Δτ=τ−τoand substituting (55), (59), and (65) into (58), we obtain(66)V˙3=−2λminA0V2−ζ1TA1ζ1+ζ1TBζ1ζ2+ζ2T−W∗TΦZ−εZ+τo+Δτ+dτ.In view of (66), Δτ≠0 when the saturation appears. We need to reject Δτ. This can be accomplished by using the following auxiliary system [35]:(67)υ˙=−Kυυ−ζ^2TΔτ+Δτ2/k0υ2υ+Δτ,υ≥ϱ,0,υ<ϱ,where υ∈ℝ3 is the auxiliary vector; Kυ is a positive definite matrix; k0 is a positive number; and ϱ is a positive constant that should be chosen as an appropriate value in accordance with the requirement of the tracking performance.Thus, we have transformed the controlτ under input constraint (6) to the free control τo by adding the auxiliary variable υ. Consequently, in order to obtain the control τo based on the Lyapunov method, we consider the following Lyapunov function candidate:(68)V4=V3+12∑i=13w˜iTΓi−1w˜i+12υTυ,where w˜i=wi∗−w^i, for i=1,2,3, is the estimation error and w^i is the estimation of wi∗. The goal now is to select the control τo to make the time derivative V˙4 negative. The process of selecting τo is detailed in Appendix C. At this point, in order to state the result of this section, we propose the control τo and the update law w^i in advance as follows:(69)τo=−Bζ1ζ1−A2ζ^2+W^TΦZ^+υ,(70)w^˙i=−ΓiΦiZ^ζ^2i+μiw^i,i=1,2,3,where A2 is a positive matrix and μi is a positive constant.In control (69), the first term is proposed to cancel the second term in (55). The second term is used for the stabilization of ζ2 and dealing with dτ and ϵ. The third term is the estimation of (59). The last term is used to tackle the appearance of the input saturation.Theorem 2.
For orientation subsystem (2), under high-gain observers (40) and (41), controllers (64), (69), and (67), and update law (70), if the initial conditions are such that ζ10<b, where b is defined in (47), then(i)
All the system states are ultimately bounded, i.e., there exist positive numbersκ and C4 such that(71)V˙4≤−κV4+C4.(ii)
Constraints (6) and (7) are satisfied; in particular, the tracking error ζ1 is bounded by(72)ζ1t≤b1−e2V4t,∀t>0.(iii)
The tracking errorζt converges to a small neighbourhood of the origin which can be made arbitrarily small.Proof.
See AppendixC.Remark 7.
It is noticed that the second term ofV3 in (57) is the function of the state ζ2 instead of ζ^2. This allows us to know the behavior of the state ζ2 even when ζ2 is unavailable for the feedback.Remark 8.
In order to deal with constraint (7), we use the Euler angles in the representation of the attitude. This makes the control design become much easier because the use of the unit quaternion may lead to inconsistent behavior, i.e., the unwinding phenomenon [2]. Although the use of Euler angles has drawback which is the so-called gimbal lock [36], i.e., the rotation matrix is singular when ϕ=π/2 or θ=π/2, this drawback is tackled by the satisfaction of constraint (7).


## 3.2.1. High-Gain Observer for Orientation Subsystems

Because the angular velocityη2 in (39) is not available, we will estimate η2 by using the following high-gain observer:(40)εχ˙1=χ2,εχ˙2=−ν1χ2−χ1+η1t,where χ1andχ2∈ℝ3 are the state vectors of observer (40) and ν1 is chosen such that s2+ν1s+1 is Hurwitz. Then, the estimations η^1 and η^2 of η1 and η2, respectively, are reconstructed as(41)η^1=χ1,η^2=1εχ2.Lemma 4.
(see [33]). For orientation subsystem (2), high-gain observer (40) provides the estimations η^1,η^2 of η1,η2, and the estimation errors are bounded by(42)η1−η^1≤εϑ1,η2−η^2≤εϑ2,for some positive constants ϑ1 and ϑ2.


## 3.2.2. Bounded Robust Adaptive Output Feedback Control Design

Having high-gain observer (40), we proceed to design the control τ for orientation subsystem (39). In the control design, constraints (6) and (7) are simultaneously dealt with by the use of the saturation and barrier functions. The RBF NNs are used to approximate the unknown terms Jη1 and Cη1,η2 in (39). The control design is based on the backstepping method. For more detail, the design schema consists of following two steps:Step 1
Define the tracking errors as(43)ζ1=η1−ηd,ζ2=η2−α,ζ^2=η^2−α,where α is the virtual control. Let ζ˜2 be the error between ζ2 and ζ^2, involving (42); ζ˜2 is bounded by(44)ζ˜2=ζ2−ζ^2=η˙−η^˙≤εϑ2,and dynamic equation of ζ1 is(45)ζ˙1=η˙1−η˙d=ζ2+α−η˙d.To deal with output constraint (7), let us consider the barrier Lyapunov function candidate [24] as follows:(46)V2=∑i=1312logbi2bi2−ζ1i2,where bi for i=1,2,3 is defined as(47)b1=π2−ϕ¯d,b2=π2−θ¯d,b3=π−ψ¯d,with ψ¯d being the bound of ψd. The time derivative of (46) along (45) is given by(48)V˙2=∑13ζ1iζ˙1ibi2−ζ1i2=ζ1TBζ1ζ˙1,where(49)Bζ1=1b12−ζ1120001b22−ζ1220001b32−ζ132.Substituting (45) into (48) yields(50)V˙2=ζ1TBζ1α−η˙d+ζ1TBζ1ζ2.Choosing(51)α=−A0+Bζ1−1A1ζ1+η˙d,where A0andA1∈ℝ3×3 are positive matrices, and substituting (51) into (50), yield(52)V˙2=−ζ1TBζ1A0ζ1−ζ1TA1ζ1+ζ1TBζ1ζ2.Using Lemma2 in [34], in the set ζ1i<bi,i=1,2,3, we have(53)logbi2bi2−ζ1i2<ζ1i2bi2−ζ1i2,and the first term in (52) satisfies(54)−ζ1TBζ1A0ζ1<−2λminA0∑i=1312logbi2bi2−ζ1i2<−2λminA0V2.Substituting (54) into (52) yields(55)V˙2<−2λminA0V2−ζ1TA1ζ1+ζ1TBζ1ζ2.The first two terms in (55) are negative; the last term will be cancelled in the next step.Step 2
Taking the time derivativeζ2 in (43) along (2) yields(56)Jη1ζ˙2=Jη1η˙2−Jη1α˙=−Cη1,η2ζ2−Cη1,η2α+τ+dτ−Jη1α˙.Consider the Lyapunov function candidate as(57)V3=V2+12ζ2TJη1ζ2,and its time derivative along (56) is given by(58)V˙3=V˙2+12ζ2TJ˙η1ζ2+ζTJη1ζ˙2=V˙2+12ζ2TJ˙η1−2Cη1,η2ζ2+ζ2T−Cη1,η2α+τ+dτ−Jη1α˙.Thus, the real controlτ appears in (58). We need to design the control τ to make V˙3≤0. To this end, we notice that the term ζ2TJ˙η1−2Cη1,η2ζ2=0 according to the skew symmetric property. The unknown terms Cη1,η2α and Jη1α˙ are estimated by using the RBF NNs approximation function from (10) as(59)Cη1,η2α+Jη1α˙=W∗TΦZ+εZ,where Z=η1T,η2TT is the input vector of the NNs and W∗ is the ideal unknown constant weight matrix, which has the following form:(60)W∗=w1∗T01×ℓ01×ℓ01×ℓw2∗T01×ℓ01×ℓ01×ℓw3∗TT∈ℝ3ℓ×3,with wi∗∈ℝℓ,i=1,2,3, and W∗ is bounded by(61)W∗≤Wm,where Wm is a positive constant; ℓ is the node number of the corresponding NN; and ΦZ∈ℝ3ℓ is the basis function vector of the following form:(62)ΦZ=Φ1TZ,Φ2TZ,Φ3TZT∈ℝ3ℓ,with ΦiZ∈ℝℓ; ϵZ is the error vector with the known bound as(63)ϵZ≤ϵ∗,where ϵ∗ is a positive constant.It was mentioned that the optimal weightW∗ is unavailable for the approximation purpose. Furthermore, the state η2 is also unavailable for the feedback and hence cannot be the input of the NNs. Therefore, approximation (59) cannot be implemented. To deal with this situation, consider the relation between η2 and η^2 in Lemma 4; we use the estimated state η^2 as the input of the NNs for the approximation of the left-hand side of (59). The possibility of the use of the estimated state η^2 is shown in Appendix C. Accordingly, the input of the NNs is Z^=η1T,η^2TT, and hence, the basis function vector is ΦZ^.Next, we need to deal with input constraint (6). To do this, the control τ is designed in the following form:(64)τi=τ¯i,ifτ0i>τ¯i,τoi,ifτ0i≤τ¯i,−τ¯i,ifτ0i<−τ¯i,for i=ϕ,θ,ψ, with τo=τoϕ,τoθ,τoψT being the free control to be designed. Denoting the control input error as(65)Δτ=τ−τoand substituting (55), (59), and (65) into (58), we obtain(66)V˙3=−2λminA0V2−ζ1TA1ζ1+ζ1TBζ1ζ2+ζ2T−W∗TΦZ−εZ+τo+Δτ+dτ.In view of (66), Δτ≠0 when the saturation appears. We need to reject Δτ. This can be accomplished by using the following auxiliary system [35]:(67)υ˙=−Kυυ−ζ^2TΔτ+Δτ2/k0υ2υ+Δτ,υ≥ϱ,0,υ<ϱ,where υ∈ℝ3 is the auxiliary vector; Kυ is a positive definite matrix; k0 is a positive number; and ϱ is a positive constant that should be chosen as an appropriate value in accordance with the requirement of the tracking performance.Thus, we have transformed the controlτ under input constraint (6) to the free control τo by adding the auxiliary variable υ. Consequently, in order to obtain the control τo based on the Lyapunov method, we consider the following Lyapunov function candidate:(68)V4=V3+12∑i=13w˜iTΓi−1w˜i+12υTυ,where w˜i=wi∗−w^i, for i=1,2,3, is the estimation error and w^i is the estimation of wi∗. The goal now is to select the control τo to make the time derivative V˙4 negative. The process of selecting τo is detailed in Appendix C. At this point, in order to state the result of this section, we propose the control τo and the update law w^i in advance as follows:(69)τo=−Bζ1ζ1−A2ζ^2+W^TΦZ^+υ,(70)w^˙i=−ΓiΦiZ^ζ^2i+μiw^i,i=1,2,3,where A2 is a positive matrix and μi is a positive constant.In control (69), the first term is proposed to cancel the second term in (55). The second term is used for the stabilization of ζ2 and dealing with dτ and ϵ. The third term is the estimation of (59). The last term is used to tackle the appearance of the input saturation.Theorem 2.
For orientation subsystem (2), under high-gain observers (40) and (41), controllers (64), (69), and (67), and update law (70), if the initial conditions are such that ζ10<b, where b is defined in (47), then(i)
All the system states are ultimately bounded, i.e., there exist positive numbersκ and C4 such that(71)V˙4≤−κV4+C4.(ii)
Constraints (6) and (7) are satisfied; in particular, the tracking error ζ1 is bounded by(72)ζ1t≤b1−e2V4t,∀t>0.(iii)
The tracking errorζt converges to a small neighbourhood of the origin which can be made arbitrarily small.Proof.
See AppendixC.Remark 7.
It is noticed that the second term ofV3 in (57) is the function of the state ζ2 instead of ζ^2. This allows us to know the behavior of the state ζ2 even when ζ2 is unavailable for the feedback.Remark 8.
In order to deal with constraint (7), we use the Euler angles in the representation of the attitude. This makes the control design become much easier because the use of the unit quaternion may lead to inconsistent behavior, i.e., the unwinding phenomenon [2]. Although the use of Euler angles has drawback which is the so-called gimbal lock [36], i.e., the rotation matrix is singular when ϕ=π/2 or θ=π/2, this drawback is tackled by the satisfaction of constraint (7).


## 4. Stability Analysis

This section investigates the stability of the overall closed-loop system under the controllers designed in the previous section. The overall error system consists of the closed-loop position error subsystem from (34) as follows:(73)p˙1=p^2+q˜2,p^˙2=−σ1p1+p^2−σ2p^2+L2q˜1+ϖ,q˜˙=Lq˜+dq˜.Also, it has the closed-loop orientation error subsystem from (40), (41), (45), (56), (64), (69), and (70) as follows:(74)ζ˙1=ζ2+α−Bζ1−1A1ζ1,Jη1ζ˙2=−Cη1,η2ζ2−W∗TΦZ+W^TΦZ^−Bζ1ζ1−A2ζ^2+υ+Δτ+εZ+dτ,υ˙=−Kυυ−ζ^2Δτ+Δτ2/k0υ2υ+Δτ,υ≥ϱ,0,υ<ϱ.Systems (73) and (74) have the cascaded structure in which (73) is driven by (74). The result proposed in [11, 37] cannot be directly applied because of constraint (7) and disturbances. In the following, we show that the controllers designed in the previous section achieve the ultimate boundedness property of the overall systems (73) and (74).Theorem 3.
Consider the closed-loop overall tracking error system described by (73) and (74) under Assumption 2. If the initial condition is such that ζ10<b, then the considered system enjoys the ultimate boundedness property.Proof.
The proof consists of two stages: (i) the solution of systems (73) and (74) does not have finite escape time, and (ii) all the system states are ultimately bounded. That is,(i)
Completeness of the solution: since the solution of system (74) is ultimately bounded from Theorem 2, the remainder is to show that the solution of system (73) is bounded for all the bounded time, or in other words, will not approach infinity in the finite time. To do this, considering the time derivative V˙1 from (B.10) by noticing(75)2p^2q˜1≤p^22+q12;2ϖσ1≤ϖ2+σ12;2ϖp^2≤ϖ2+p^22;2q˜dq˜≤q˜2+dq˜2,thenV˙1 in (B.10) becomes(76)V˙1≤γ1σ12+γ2p^22+γ3q˜2+γ4ϖ2+γ5dq˜2,whereγ1=1/2;γ2=λmaxL2+1/2;γ3=λmaxL2/2+λmaxPo;γ4=1; and γ5=λmaxPo. Furthermore, from (18), (23), and (29), ϖ and dq˜ are bounded by(77)ϖ≤2u¯/m^;dq˜≤dp∗.Inequality (76) can be further written as(78)V˙1≤γ6V1+γ7,whereγ6=maxi=1,2,32γi and γ7=γ42u¯/m^+γ5dp∗/γ6. Taking the integral (78) over the time 0,t yields(79)∫0tdV1V1+γ7≤∫0tγ6dν⇒lnV1t+γ7V10+γ7≤γ6t,or(80)V1t≤V10+γ7eγ6t−γ7.Inequality (80) shows that for the finite time t, V1t will not approach infinity, and hence, the solution p1,p^2,q˜ does not escape in the finite time.(ii)
Ultimate boundedness property of the overall system: we need to prove that the time derivative of the total Lyapunov functionV˙1+V˙4 is negative outside some compact set. It is noticed that V˙4 satisfies (71) from Theorem 2. The remainder is V˙1. We shall show that V˙1 satisfies (37) in Theorem 1, or in other words, we need to verify condition (36) of Theorem 1. To this end, by direct computation from (29), we have(81)ϖ≤kϖζ1,where kϖ=32u¯/m^>0. From (72) of Theorem 2, the solution ζ1t is bounded by(82)ζ1t≤b1−e−2V4t.
To evaluate the behavior ofV4, solving (71) with respect to V4, we obtain(83)V4t≤V40e−κt+C4κ1−e−κt.
Inequality (83) shows that V4t exponentially decays and the size of the final value is C4/κ which can be made arbitrarily small. Accordingly, from (82) and (83), there exists a time instant Tϖ such that(84)ζ1t≤b1−e−2V4Tϖ≤ϖ∗kϖ,∀t>Tϖ,with ϖ∗ being the constant in (36). Involving (81), we have(85)ϖ≤kϖζ1t≤ϖ∗,∀t>Tϖ.
Condition (85) implies the satisfaction of condition (36) in Theorem 1. According to Theorem 1, there exists a time instant T2>maxT1,Tϖ such that inequality (37) in Theorem 1 is achieved or we have(86)V˙1≤−W1+C1,∀t>T2.
Now let us consider the Lyapunov function candidate(87)Vc=V1+V4,from (86) and (71); the time derivative V˙c along closed-loop system (73) and (74) is given by(88)V˙c=V˙1+V˙2≤−W1+C1−κV4+C4,∀t>T2,or(89)V˙c≤−Wc+Cc,∀t>T2,where Wc=W1+κV4 and Cc=C1+C4.
The foregoing inequality (89) shows that the solution of the overall closed loop system is ultimately bounded with the ultimate bound Cc which can be made arbitrarily small by making C1 and C4 small. The proof is completed.Remark 9.
It is noticed from Theorem3 that cascaded systems (73) and (74), in which ISS system (73) with the bounded input is driven by system (74) with ultimate boundedness property, enjoy ultimate boundedness property. This fact is a novelty and gives us a flexible tool in the control design for the cascaded system. In comparison with the existing studies [11, 37], this fact allows us not only to deal with constraints, disturbances, and uncertainties of the system but also take observers for state estimation into account.
Based on Theorem3, our final result is stated in the following theorem.Theorem 4.
Consider the class of underactuated vehicles described by (1) and (2) with constraints (5)–(7), the velocities estimated by observers (20) and (40), respectively, the controls u and τ obtained by (30), (31), (64), (69), and (67), and update laws described by (70). Let design parameters be chosen according to Theorems 1 and 2. Then, for every initial condition such that ζ10<b, Problem 1 is solved, i.e., constraints (5)–(7) are ensured, all the signals in the closed loop system are ultimately bounded, and the tracking error converges to adjustable neighbourhoods of the origin.Proof.
Constraint (5) is satisfied from Theorem 1, and constraints (6) and (7) are satisfied from Theorem 2. All the signals in the closed loop are ultimately bounded, and the tracking error converges to the adjustable neighbourhoods of the origin from Theorem 3.


## 5. Simulation Results

This section presents the numerical simulation results of the proposed controller. The system in the simulation is taken from the model of a quadrotor [1, 38]. Matrices Jη and Cη,η˙ are given in Appendix D. The nominal parameters are m^=1kg; J^=dia0.125/4,0.125/4,0.125/4kgm2; and g=9.81 m/s2. We assume that Δm=0.05m^; J=J^+0.1J^; and dq and dη have random values in −0.02;0.02 and −0.01;0.01, respectively. The upper bounds of constraints (5) and (6) are u¯=14N and τ¯=0.05;0.05;0.05TNm. The desired trajectory is xdt=0.5cos0.5tm, ydt=0.5sin0.5tm, zdt=−0.5tm, and ψdt=0.2e−trad. The initial conditions are q0=0,0,4Tm; q˙0=0.5,0.5,0Tm/s; and η0=0.1,0.1,0.1Trad. The computation of the controller is as follows.For the position subsystem, the controlu is computed from (30) and (31). We set σ¯11=σ¯21=1, σ¯12=σ¯22=1, and σ¯13=σ¯23=1.8 for (31) and L1=L2=diag20,20,20 for the position observer (20). Thus, condition (33) is satisfied.For the orientation subsystem, we setν1=2andε=0.005 for observer (40); A2=diag3,3,3 for (69); ko=1,Kv=diag1,1,1,andϱ=0.05 for (67). The update laws are designed based on (70). The neural network has 27 nodes, the centers κi of the basis function vector are evenly spaced in the range of −2,2×−2,2×−1,1, the widths are hi=1 for all setting, and Γi=diag1,⋯,1∈ℝ27×27andμi=0.01,i=1,⋯,3 for (70).For the comparison purpose, we carry out the simulation with the full-state feedback control proposed in [1]. The position and linear velocities are shown in Figures 1 and 2. The orientation angles and angular velocities are shown in Figures 3 and 4. It can be seen in Figures 2 and 4 that the estimated translational and angular velocities of the proposed control converge to the desired velocities. Furthermore, our proposed control can reject the velocity errors in comparison with the controller proposed in [1]. The control inputs are shown in Figures 5 and 6. It is seen that the constraints on input are protected. Finally, the trajectory is shown in Figure 7.Figure 1
The positionsx,y, and z.Figure 2
The translational velocitiesx˙,y˙, and z˙.Figure 3
The orientation anglesϕ,θ, and ψ.Figure 4
The angular velocitiesϕ˙ and θ..Figure 5
The total thrustu and generalized torque τϕ.Figure 6
The generalized torquesτθ and τψFigure 7
The trajectory.


## 6. Conclusion

This paper studies the robust adaptive output feedback control for the class of underactuated aerial vehicles with multiple constraints. By constructing the system as the cascaded structure, the modular design strategy is proposed to separately design the controller for each subsystem. By integrating observers, saturations, and barrier functions and NN approximation into the control system design, the controller has ability to simultaneously deal with system uncertainties, unavailable velocities, and constraints. The overall system enjoys the ultimate boundedness property. The simulation on the quadrotor helicopter shows that the control objectives are completed.



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*Source: 7193136-2020-05-16.xml*