# Research on Amphibious Multi-Rotor UAV Out-of-Water Control Based on ADRC
## 4 Controller Design  
Amphibious multi-rotor UAVs are subject to hydrodynamic and other factors when moving underwater, which affects the stability of the controller. The disturbing factors become more complicated when the out-of-water movement is performed. To improve the anti-interference of the amphibious multi-rotor UAV, a cascade ADRC position and attitude controller is designed in this paper in combination with the ADRC algorithm [42], and the control block diagram is shown in Figure 4. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Speed (m/s)** & **X (N)** & **Y (N)** & **Z (N)** & **K (N+m)** & **M (N+m)** & **N (N+m)** \\ \hline
0.1 & \(-0.00034626\) & \(-0.00157796\) & 0.39501 & 0.000368329 & \(-0.00058171\) & 0.00063545 \\
0.2 & \(-0.00353503\) & \(-0.0244471\) & 1.54798 & 0.00116013 & \(-0.00236987\) & 0.00309158 \\
0.3 & \(0.000714899\) & \(-0.0484334\) & 3.46933 & 0.00341651 & \(-0.00486215\) & 0.00592401 \\
0.4 & \(0.1179082\) & \(-0.109463\) & 6.13351 & 0.0052411 & \(-0.00950867\) & 0.00688129 \\
0.5 & \(0.0392365\) & \(-0.183024\) & 9.55917 & 0.0065315 & \(-0.0149868\) & 0.0164673 \\
0.6 & \(0.0541794\) & \(-0.268715\) & 13.7434 & 0.01118391 & \(-0.0206249\) & 0.0269714 \\
0.7 & \(0.0988873\) & \(-0.344484\) & 18.6816 & 0.0181143 & \(-0.0289125\) & 0.0298976 \\
0.8 & \(0.134385\) & \(-0.397333\) & 24.3859 & 0.0283891 & \(-0.0361443\) & 0.0440083 \\
0.9 & \(0.126005\) & \(-0.610409\) & 30.7818 & 0.0251821 & \(-0.0526489\) & 0.0636737 \\
1.0 & \(0.157797\) & \(-0.749914\) & 37.9765 & 0.0347552 & \(-0.0656979\) & 0.0860169 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Viscous hydrodynamic moment test table. In the outgoing process, it is assumed that \(d_{f}\) is the amount of variation due to buoyancy changes, \(d_{k}\) is the error between the actual hydrodynamic forces experienced and the estimated hydrodynamic forces, and \(d_{l}\) is the amount of other external disturbances experienced. Then, the external force (moment) applied in Equation (5) can be expressed as Equation (14):  
\[\left\{ \begin{array}{l}X=\tau_{pX}+{\tau^{\prime}}_{X}+\tau_{CX}+d_{fX}+d_{ kX}+d_{lX}\\ Y=\tau_{pY}+{\tau^{\prime}}_{Y}+\tau_{CY}+d_{fY}+d_{kY}+d_{lY}\\ Z=\tau_{pZ}+{\tau^{\prime}}_{Z}+\tau_{CZ}+d_{fZ}+d_{kZ}+d_{lZ}\\ K=\tau_{pK}+{\tau^{\prime}}_{K}+\tau_{CK}+d_{fK}+d_{kk}+d_{lK}\\ M=\tau_{pM}+{\tau^{\prime}}_{M}+\tau_{CM}+d_{fM}+d_{kM}+d_{lM}\\ N=\tau_{pN}+{\tau^{\prime}}_{N}+\tau_{CN}+d_{fN}+d_{kN}+d_{lN}\end{array} \right. \tag{14}\]  
The inner loop control includes pitch angle control, traverse angle control, and yaw angle control, assuming that \([\phi_{d},\theta_{d},\psi_{d}]^{T}\) is the given attitude angle and \([\phi,\theta,\psi]^{T}\) is the actual attitude angle obtained by feedback. The ADRC attitude controller design is introduced as an example of yaw angle control. Since the motion of the amphibious multi-rotor UAV is a small-angle motion in an equilibrium attitude, the angular velocity \([p,q,r]^{T}\) and attitude angle \([\phi,\theta,\psi]^{T}\) can be approximated by the quasi-integral relationship. The yaw angle control model is shown in Equation (15). \[\ddot{\psi}=\frac{\dot{\phi}\dot{\theta}(I_{x}-I_{y})}{I_{z}}\tau_{pN}+d_{fN}+ d_{kN}+d_{lN} \tag{15}\]  
The ADRC arrangement transition process obtains the value of a given yaw angle \(\psi_{d}\) from the outer ring. \[\left\{ \begin{array}{l}X_{11}(k+1)=X_{11}(k)+T_{1}\cdot X_{12}(k)\\ X_{12}(k+1)=X_{12}(k)+T_{1}\cdot fst(\epsilon_{1},X_{12}(k),r_{1},h_{1})\\ e_{1}=X_{11}(k)-\psi_{d}(k)\end{array} \right. \tag{16}\]  
In Equation (16), \(T_{1}\) is the sampling time; \(r_{1}\) is the factor that determines the tracking speed; \(h_{1}\) is the filtering factor, and \(fst(\cdot)\) is the fast synthesis function of the second-order discrete system. After the expansion state observer and the nonlinear error feedback, the final control quantity is generated as in Equation (17). \[\left\{