# Adaptive Backstepping Control of Quadrotor UAVs with Output Constraints and Input Saturation
## 2 Problem Formulation and Preliminaries  
A quadrotor UAV is a flying machine with six degrees of freedom, consisting of four rotors arranged in a cross structure. As shown in Figure 1, the coordinate system of a quadrotor UAV is divided into an airframe coordinate system \(E_{B}=\{O_{B},X_{B},Y_{B},Z_{B}\}\) and an Earth coordinate system \(E_{E}=\{O_{E},X_{E},Y_{E},Z_{E}\}\). \([x,y,z,\phi,\theta,\psi]\) are the six states of the UAV, where \([x,y,z]\) are the position coordinates of the UAV relative to Earth, \([\phi,\theta,\psi]\) are Euler angles of the UAV's attitude relative to the body. The Euler angles around the \(x\), \(y\) and \(z\) axes are expressed as the transverse roll angle \(\phi\), pitch angle \(\theta\) and yaw angle \(\psi\), respectively. \(F_{1},F_{2},F_{3}\) and \(F_{4}\) represent the four lift forces generated by the four rotors, respectively [37]. The following assumptions are made for the quadrotor UAV dynamics model: (1) The quadrotor UAV is a standard cross structure and symmetrical, while the airframe is rigid. (2) The center of mass and center of gravity of the body structure of the UAV remain the same. (3) When the UAV is flying or aerial hovering, the angular velocity of the UAV's body coordinate system and the earth coordinate system are the same. Then, the dynamics model of the quadrotor UAV is represented as [38]  
\[ \begin{cases}\ddot{x}=\frac{1}{m_{s}}(\cos\phi\sin\theta\cos\psi+\sin\phi\sin \psi)U_{1}-\frac{k_{1}\dot{x}}{m_{s}}\\ \dot{y}=\frac{1}{m_{s}}(\cos\phi\sin\theta\sin\psi-\sin\phi\cos\psi)U_{1}-\frac {k_{2}\dot{y}}{m_{s}}\\ \dot{z}=\frac{1}{m_{s}}(\cos\phi\cos\theta)U_{1}-g-\frac{k_{3}\dot{z}}{m_{s}} \\ \dot{\phi}=\dot{\phi}\dot{\psi}\frac{ly-J_{z}}{J_{x}}+\frac{l}{J_{x}}U_{2}-\frac {k_{4}l}{J_{y}}\dot{\phi}\\ \ddot{\theta}=\dot{\phi}\dot{\psi}\frac{ly_{z}-J_{x}}{J_{y}}+\frac{l}{J_{y}}U_ {3}-\frac{k_{5}l}{J_{y}}\dot{\theta}\\ \ddot{\psi}=\dot{\phi}\dot{\psi}\frac{ly_{x}-J_{y}}{J_{z}}+\frac{1}{J_{z}}U_ {4}-\frac{k_{6}}{J_{z}}\dot{\psi}\end{cases}  \tag{1}\]  
Figure 1: Coordinate systems of the quadrotor UAV. where \(m_{s}\) denotes the total mass of the quadrotor. \(k_{1},k_{2},k_{3},k_{4},k_{5}\) and \(k_{6}\) denote the drag coefficients of each channel, respectively. \(J=diag\{J_{x},J_{y},J_{z}\}\) denote the axial rotational inertia of \(x,y,z\) axes. \(g\) denotes the acceleration of gravity. \(l\) denotes the distance from the center of rotor to the center of the quadrotor. \((U_{1},U_{2},U_{3},U_{4})\) indicate the control inputs of the independent channels. The UAV control system is divided into the position control subsystem and attitude control subsystem. The position and attitude control subsystem is divided into 6 control channels, and each of them is a standard quadratic nonlinear system. To facilitate the design of the controller, we simplify the dynamics model. The UAV has internal uncertainties. We define \(f_{i}(X)\) to represent the unmodeled and uncertain part. In addition, the gyroscope torque is included in the total system perturbation. Considering the input-output constraints, the model uncertainties and external disturbances, a uniform nonlinear state expression for the 6 control channels of the UAV is derived as  
\[ \begin{cases}\dot{x}_{i1}=&x_{i2}\\ \dot{x}_{i2}=&b_{i}u_{i}+f_{i}(X)+w_{i}\end{cases} ^{i}=x,y,z,\phi,\theta,\psi \tag{2}\]  
where \(X= \begin{bmatrix}x_{x1},x_{x2},x_{y1},x_{y2},\dots,x_{y1},x_{y2}\end{bmatrix} ^ {T}= \begin{bmatrix}x,\dot{x},y,\dot{y},z,\dot{z},\phi,\dot{\phi},\theta,\dot{ \theta},\psi,\dot{\psi}\end{bmatrix} ^{T}\). The mass measurement and rotational inertia measurement errors of the UAV are uncertain, so the control gain \(\bar{b}_{i}= \begin{bmatrix}1&\frac{1}{(m_{s}+\Delta m_{s})},\frac{1}{(m_{s}+ \Delta m_{s})},\frac{1}{(m_{s}+\Delta m_{s})},\frac{1}{(j_{x}+\Delta j_{x})}, \frac{1}{(j_{y}+\Delta j_{y})},\frac{1}{(j_{z}+\Delta j_{z})}\end{bmatrix} ^{T}\) is uncertain. The measurement errors are divided into a deterministic part and an uncertain part.