# Experimental Design of a Mobile Landing Platform to Assist Aerial Surveys in Fluvial Environments
## 3 Materials and Methods
### Motor and Rudder Servo Sizing  
To select the necessary hardware for the vessel an analysis of the forces acting on the body and rudder of the vessel was performed. This analysis will help define the requirements related to engine power and the rudder servo minimum torque. For this analysis we will first set a suitable workspace for notation. The Strider V1.0 was considered as a single rigid solid [20], with an inertial reference frame attached to the Earth and a body-fixed reference frame. The inertial frame was defined with the standard notation of SNAME (Society of Naval Architects and Marine Engineers), following the NED convention, \(x,\ y\) and \(z\) pointing North, East and Down respectively (see Figure 4). Angular motion is defined in terms of the Euler angles being \(\mathbf{\Theta}^{E}=[\phi\ \theta\ \psi]^{T}\ [rad]\) the roll, pitch and yaw angles respectively as shown in Figure 4. We will also define \(\boldsymbol{\omega}^{B}=[p\ q\ r]^{T}\ [rad\ s^{-1}]\) and \(\boldsymbol{\tau}^{B}=[K\ N\ M]^{T}\ [N\ m]\) as the angular velocities and the torques about the \(x_{0},\ y_{0}\) and \(z_{0}\) directions with respect to the body frame. Concerning linear motion, \([x_{0}\ y_{0}\ z_{0}][m]\) stands for the linear position of the body-fixed frame with respect to the body-fixed frame. While \(\mathbf{V}^{B}=[u\ v\ w]^{T}\ [m\ s^{-1}]\) stands for the linear velocity of the vessel and \(\mathbf{F}^{B}=[X\ Y\ Z]^{T}\ [N]\) for the forces and torques applied to the body, both referenced in the  
Figure 4: Standard notation and sign conventions for ship motion description on the Strider V1.0. Figure 3: Material used in the structure construction: (**a**) Final assembly, (**b**) Structure without the polyurethane foam base, (**c**) The PVC materials used. body-fixed frame. Motions on the \(x_{0}\), \(y_{0}\) and \(z_{0}\) direction are usually called surge, sway, and heave, respectively. We start the analysis by using [PERSON]'s second law, to describe translational motion:  
\[m_{v}\dot{\textbf{V}}^{B}=\textbf{F}^{B} \tag{1}\]  
where \(m_{v}\left[Kg\right]\) is the mass of the vessel. From this expression we can then divide the \(\textbf{F}^{B}\) force vector into its different components to analyze the forces applied to the body of the vessel. This description of the force vector is really simple but sufficient for motor sizing. \[\textbf{F}^{B}= \begin{bmatrix}X\\ Y\\ Z\end{bmatrix} = \begin{bmatrix}F_{AirDrag_{x}}+F_{WaterDrag_{x}}+F_{Thrust}\\ F_{AirDrag_{y}}+F_{WaterDrag_{y}}\\ f_{b}+W\end{bmatrix} +F_{Other} \tag{2}\]  
where \(F_{AirDrag_{x}}\left[N\right]\) and \(F_{WaterDrag_{y}}\left[N\right]\) are air and water drag horizontal forces derived from movement through both fluids and \(F_{Thrust}\left[N\right]\) is the thrust produced by the propeller. \(f_{b}\left[N\right]\) is the buoyancy force and \(F_{Other}\left[N\right]\) are other forces not considered under normal operation such as exogenous forces or forces not relevant or considered for motor sizing. Since the only force opposing the thrust force of the vessel, disregarding \(F_{Other}\) for motor sizing, are \(F_{AirDrag_{x}}\) and \(F_{WaterDrag_{y}}\), we will characterize them to obtain the necessary motor power to move the vessel. Now, for the calculation of air and water resistance, a fluid dynamics approach was used. Reynolds number \(Re\left[+\right]\) was calculated using Equation (3) to characterize drag forces in both fluids. \[Re=\frac{\rho\cdot u\cdot L}{\mu} \tag{3}\]  
where \(\rho\left[Kg\;m^{-3}\right]\) and \(\mu\) are the density and dynamic viscosity of the studied fluid, \(u\) is the surge speed as described by Figure 4 and \(L\left[m\right]\) is the characteristic length of the drag surfaces.