## 3 Design and Implementation of the ATD  
### Mechanical Structure  
To construct ATD, to be equipped on a typical multirotor UAV, we use ducted fan as the main actuator unit, which is lightweight, compact, and able to generate relatively high thrust. The generated thrust from ATD acts on the multirotor to allow it to translate and generate the force in horizontal direction. At least three forces applied in a plane are required for positioning a body in that plane. The structure of ATD is shown in Figure 2a. Three ducted fans are placed every 120 degrees and set as Y-configurations, so that a well-balanced force can be generated by the combination of any of two thrusts of ducted fans. To avoid exhaust from the ducted fan from being affected by the downwash of the multirotor's propellers, the ducted fans are mounted outside the propeller radius of the multirotor. ATD comprises three ducted fans, power module, PCA9685-I2C to PWM interface, ESC, and a CPU board (LattePanda Alpha 864). The ducted fan is 50 mm in diameter and can generate 0.95 kg of maximum thrust. The CPU board is used for processing feedback control. The distance \(L\) from ducted fan can be set depending on the size of multirotor platform. The specifications of the device are summarized in Table 1.  
To control ATD, for any feasible control force \(F\), the output thrust of three ducted fans should be calculated. According to the arrangement of ducted fans, the model of ATD is shown in Figure 2b. The coordination is defined as the same as the multirotor body frame. \(F_{1},\ F_{2}\), and \(F_{3}\) show the output thrusts of the ducted fans. The blocked region in Figure 2b is the feasible control force volume which is mixed by thrust of three ducted fans. The region can be separated into \(S_{1}\), \(S_{2}\), and \(S_{3}\) and given as follows. \[S_{1}:=\{F\in\Re^{2}\mid-kF_{x}\leq F_{y},\ kF_{x}-F_{max}\geq F_{y},\ -kF_{x}-F_{max}\geq F_{y},\ kF_{x}\leq F_{y}\} \tag{1}\]  
\[S_{2}:=\{F\in\Re^{2}\mid kF_{x}\geq F_{y},\ -\frac{k}{2}F_{max}\leq F_{x}\leq 0, \ kF_{x}+F_{max}\leq F_{y}\} \tag{2}\]  
\[S_{3}:=\{F\in\Re^{2}\mid-kF_{x}\geq F_{y},\ 0\leq F_{x}\leq\frac{k}{2}F_{max},\ -kF_{x}+F_{max}\leq F_{y}\} \tag{3}\] \begin{table}
\begin{tabular}{l c} \hline \hline Gross weight & 0.69 kg without battery \\ \hline Size & 0.7 m (W) \(\times\) 0.6 m (D) \(\times\) 0.12 m (H) \\ \hline Number of ducted fan & 3 \\ \hline Ducted fan size/motor & \(\phi\)50 mm/4300 KV \\ \hline Max thrust & 950 g for each ducted fan \\ \hline L & 0.4 m \\ \multicolumn{2}{c}{L is adjustable depending on multirotor size.} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of ATD. Figure 2: (**a**) Overview of ATD and its defined parameters; and (**b**) the model of ATD. The blocked region shows feasible control force volume and parameters corresponding to (**a**). The coordination is defined as the same as the body frame of multirotor. In Equations (1)-(3), \(\pm k\) represents the slope of \(F_{1}\) and \(F_{2}\) based on the coordination, and the value is \(\pm\frac{\sqrt{3}}{3}\). \(F_{max}\) shows the maximum output thrust of ducted fan. Thus, for any feasible force \(F\), output thrust of ducted fans \([F_{1}\;\;F_{2}\;\;F_{3}]^{T}\) can be given by as follows. \[A=\left\{ \begin{array}{cc}\left[\begin{array}{cc}k&1\\ k&-1\\ 0&0\end{array}\right]&(F\in S_{1})\\ \\ \left[\begin{array}{cc}2k&0\\ 0&0\\ -k&1\end{array}\right]&(F\in S_{1})\\ \\ \left[\begin{array}{cc}0&0\\ 2k&0\\ k&1\end{array}\right]&(F\in S_{3})\\ \end{array} \right.,\qquad\left[ \begin{array}{c}F_{1}\\ F_{2}\\ F_{3}\end{array} \right]=A\left[ \begin{array}{c}F_{x}\\ F_{y}\end{array} \right]\\ \\ \left[ \begin{array}{cc}0&0\\ 2k&0\\ k&1\end{array} \right]&(F\in S_{3})\\ \end{array}\right. \tag{4}\]  
The input thrust of ducted fan \(u_{i}\) with its output thrust \(F_{i}\) is \(u_{i}\propto F_{i}\), and we define the equation as follows.