# Adaptive Backstepping Control of Quadrotor UAVs with Output Constraints and Input Saturation
## 4 Simulation Results  
In this section, the proposed BLF-based ANNBC is applied to the quadrotor UAV to verify the anti-interference and tracking performance under constrained conditions. The superiority of the present method is verified through comparative simulation experiments with the backstepping dynamic surface control (BDSC) [43]. The controller design gives a uniform form of control law for each control channel of the UAV, and the parameters and constraints are different from each control channel. Considering the dynamics of equation (1), the parameters of the quadrotor UAV in [37] are listed in Table 1.  
The desired position-attitude trajectory of the UAV are as follows  
\[ \begin{split}\left[x_{d}(t),y_{d}(t),z_{d}(t)\right]^{T}& =\left[t,2\cos(0.5t),\sin(t)\right]^{T}\\ \psi_{d}(t)&=2\sin(t)\end{split}  \tag{47}\]  
The initial conditions for the quadrotor UAV are given as  
\[ \begin{split}\left[x(t_{0}),y(t_{0}),z(t_{0}),\theta(t_{0}),\phi( t_{0}),\psi(t_{0})\right]^{T}&=\left[0,0,2,0,0,0\right]^{T}\end{split}  \tag{48}\]  
To make the comparison experiments more convincing, the parameters of the two control methods are identical. The relevant parameters of the control law (31), (22) and the adaptive law (32), (33) are given as \(k_{x1}=1\), \(k_{x2}=5\), \(k_{y1}=1\), \(k_{y2}=5\), \(k_{z1}=1\), \(k_{z2}=5\), \(k_{\theta 1}=k_{\theta 2}=10\), \(k_{\varphi 1}=k_{\varphi 2}=10\). \(\beta_{i}=1\), \(\gamma_{i}=0.1\), \(\rho_{i}=1\), \(\upsilon_{i}=0.1\), \(\varsigma_{i}=0.01\), \(\sigma_{i}=1\), \(\Theta_{i}(t_{0})=0\), \(\vartheta_{i}(t_{0})=0\)  
Assume that the following uncertainties and external disturbances are [37]  
\[ \begin{split}\left[\Delta_{tt}(t),\Delta_{Jx}(t),\Delta_{Jy}(t), \Delta_{Jz}(t)\right]^{T}&=\left[0.2,0.001\times\sin(t),0.001 \times\cos(t),0.001(\sin(t)+1)\right]^{T},\\ \left[\Delta_{H1},\Delta_{J2},\Delta_{J3},\Delta_{H4},\Delta_{J5}, \Delta_{J6}\right]&=0.5\times\mathrm{rand}(1),\\ \left[w_{x}(t),w_{y}(t),w_{z}(t)\right]^{T}&=\left[0.3\times\sin(t),0.3\times\cos(t),0.1\times\sin(0.5t)\right]^{T},\\ \left[w_{\theta}(t),w_{\theta}(t),w_{\theta}(t)\right]^{T}& =\left[0.5\times\cos(0.1t),0.3\times\sin(0.01t),1-0.5\times\sin(0.5t) \right]^{T}.\end{split}  \tag{49}\]  
In this paper, we use RBF neural network to approximate the unknown part, and the selected structure is 2-5-1, which means two inputs, five hidden layer nodes, and one \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Units** \\ \hline \(m_{s}\) & 0.468 & Kg \\ \(l\) & 0.3 & m \\ \(J_{xx}\) & \(y_{y},J_{z}\) & 0.0023 & N\(\cdot\)m \\ \(k_{1}\),\(k_{2}\),\(k_{3}\) & 0.01 & N\(\cdot\)s/m \\ \(k_{4}\),\(k_{5}\),\(k_{6}\) & 0.012 & N\(\cdot\)s/rad \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modeled parameters of the UAV. The position-attitude tracking errors of the UAV control system using the ANNBC are shown in Figures 7 and 8, respectively. The position control subsystem has a large tracking error at the beginning. By introducing RBF neural networks and robust terms, the position-attitude tracking errors \(\left[e_{x},e_{y},e_{z},e_{\phi},e_{\theta},e_{\psi}\right]\) within the range of \(\left[0.012,\,0.057\right]\), \(\left[0.021,\,0.103\right]\), \(\left[-0.072,\,0.045\right]\), \(\left[-0.013,\,0.031\right]\), \(\left[-0.015,\,0.018\right]\) and \(\left[-0.033,\,0.002\right]\), respectively. The state of the attitude control channel needs to be constrained. It can be found that with ANNBC, the tracking error of the attitude control channel is always in the range of \(\left[-0.51,0.51\right]\), satisfying the set constraint. In the presence of external disturbances and internal uncertainties, our proposed ANNBC can achieve output constraints at all moments. While from Figures 5 and 6, since there is no anti-interference and output constraints design, small perturbation leads to a large tracking error. Output constraints are broken and the dynamic properties of the system are affected.