# Research on Amphibious Multi-Rotor UAV Out-of-Water Control Based on ADRC
## 2 Dynamic and Kinematic Modeling  
To facilitate the study of the dynamic and kinematic model of the amphibious multi-rotor UAV, this paper establishes a coordinate system as shown in Figure 2. \(E-\xi\eta\zeta\) is the coordinate system consisting of a fixed coordinate system established on the horizontal plane, and \(o-xyz\) is an airframe motion coordinate system fixed on the amphibious multi-rotor UAV. First, to facilitate the analysis of the motion and forces on the amphibious multi-rotor UAV, the following assumptions are made for the amphibious multi-rotor UAV under study. (1) The amphibious multi-rotor aircraft can be regarded as a rigid body with uniformly distributed mass and uniformly symmetric left and right. (2) The center and center of gravity of the amphibious multi-rotor aircraft body coincide with the origin of the body coordinate system. (3) There is a constant acceleration of gravity, ignoring the impact of the rotor on the surrounding airflow when the vehicle is in the air. Figure 1: Water and air amphibious multi-rotor UAV physical picture. Figure 2: Coordinate system. (4) The flow velocity of the water where the UAV is located is zero and the fluid is incompressible. Second, before modeling, the symbols of physical quantities involved in this paper, including linear and angular velocities in the motion coordinate system, forces, and moments, and positions, linear velocities, and Euler angles in the fixed coordinate system, are defined as shown in Table 1.  
In a fixed coordinate system, according to the momentum theorem, the moment of momentum of the aircraft's center of mass can be expressed as  
\[m\frac{dv_{EG}}{dt}=\tau \tag{1}\]  
where \(\tau\) is the total force of the external forces on the amphibious multi-rotor UAV in a fixed coordinate system. In addition, if the center of gravity of the amphibious multi-rotor UAV does not coincide with the origin of the coordinate system, according to the moment of momentum theorem, the moment of momentum of the rigid body to the point can be expressed as  
\[L_{H}=r_{G}\times H \tag{2}\]  
where, \(H\) is the momentum of the center of gravity \(G\), \(H=mv_{EO}+mw\times r_{G}\). According to the momentum moment theorem of rigid body, the total momentum moment of the amphibious multi-rotor UAV can be expressed as  
\[ \begin{array}{ll}L&=L_{G}+L_{H}\\ &=I_{G}w+r_{G}\times(mv_{EO}+mw\times r_{G})\\ &=Iw+mv_{EO}\times r_{G}\end{array}  \tag{3}\]  
Thus, according to the Newtonian Euler equation, it follows that  
\[\frac{dL}{dt}=N \tag{4}\]  
where, \(N\) represents the total torque of UAV in the motion coordinate system. \begin{table}
\begin{tabular}{c c c c} \hline
**Name** & **Physical Meaning** & **Name** & **Physical Meaning** \\ \hline \(r=[\xi,\eta,\zeta]^{T}\) & Position vectors in the coordinate system & \(m\) & Quality \\ \(E-\xi\eta\zeta\) & & & \\ \(\Omega=[\phi,\theta,\psi]^{T}\) & Pose vectors in coordinate system & \(v_{EG}\) & Velocity of the center of gravity in \(E-\xi\eta\zeta\) coordinate system \\ \hline \(V=[u,v,w]^{T}\) & The velocity vector at the origin \(o\) of the coordinate system \(o-xyz\) & \(v_{EO}\) & The velocity at \(o\) in \(E-\xi\eta\zeta\) coordinate system \\ \hline \(W=[p,q,r]^{T}\) & Angular velocity vector of the vehicle in the coordinate system \(o-xyz\) & \(L\) & Total moment of momentum in \(o-xyz\) coordinate system \\ \hline \(F=[X,Y,Z,K,M,N]^{T}\) & The combined external forces and moments acting on the vehicle in the coordinate system \(o-xyz\) & \(L_{H}\) & The moment of momentum of the center of gravity with respect to \(o\) \\ \hline \(r_{G}\) & The center of gravity of amphibious vehicle in coordinate system \(o-xyz\) & \(L_{G}\) & The moment of momentum of the amphibious multi-rotor aircraft with respect to \(o\) \\ \hline \(l_{G}=[l_{x},l_{y},l_{z}]^{T}\) & The inertia matrix for a coordinate system whose origin is not at the center & & \\  & of gravity & & \\ \hline \end{tabular}
\end{table} 
Table 1: Amphibious multi-rotor vehicle state physical quantities.