## 4 Mathematical Model
### Motor  
The motor model is based on the direct current (DC) motor fundamentals and simple electrical model [5]. Figure 3 depicts the equivalent circuit of the simple DC motor model, where the propeller torque acts as a load, and the basic equations are as shown in Equation (8) and (9). Where \(V_{a}\) is the applied voltage; \(L_{m}\) and \(R_{m}\) are the motor inductance and resistance, respectively; \(I_{m}\) is the current consumption; \(e_{b}\) is the back electro motive force (EMF); \(e_{a}\) is the voltage drop due to an armature reaction; \(\tau_{m}\) is motor torque; \(\omega_{m}\) is motor speed in radian per second; and, \(B_{m}\) is the viscous friction coefficient. Although the construction of the model is simple, it contains all of the basic electrical and mechanical elements required in order to represent motor dynamics. This is appropriate when it is necessary to examine the overall behavior characteristics as simple as possible with adequate fidelity, as in this study. \[V_{a}=L_{m}\cdot\frac{dI_{m}}{dt}+I_{m}\cdot R_{m}+e_{b}+e_{a} \tag{8}\]\[\tau_{m}=J_{rotor}\cdot\frac{d\omega_{m}}{dt}+B_{m}\cdot\omega_{m}+Q_{prop} \tag{9}\]  
Equations (8) and (9), respectively, reflect the electrical and mechanical characteristics of the motor, and, if there is no change in speed of the motor, the differential terms are eliminable as Equations (10) and (11). The input and output power of the motor are calculated while using Equations (12) and (13), and the efficiency is calculated using Equation (14). \[V_{a}=I_{m}\cdot R_{m}+e_{b}+e_{a} \tag{10}\]  
\[\tau_{m}=B_{m}\cdot\omega_{m}+Q_{prop} \tag{11}\]  
\[P_{in}=V_{a}\cdot I_{m} \tag{12}\]  
\[P_{out}=\tau_{m}\cdot\omega_{m} \tag{13}\]  
\[\eta_{m}=\frac{P_{out}}{P_{in}}=\frac{\tau_{m}\cdot\omega_{m}}{V_{a}\cdot I_{m}} \tag{14}\]  
The relationship between mechanical energy and electrical energy can be expressed using motor constants. Equations (15) and (16) show that the back EMF and motor speed, motor torque, and current consumption are linearly proportional, where \(K_{b}\) is a back EMF constant and \(K_{t}\) is a torque constant. The mechanical power output of the motor and the power by back EMF have the same values, according to the law of energy conservation. This can be expressed as Equation (17), and applying the Equations (15) and (16) here, it can show that the motor torque constant and the back EMF constant also have the same value. The speed constant, \(K_{v}\), which is provided as a key performance index from most multicopter motor manufacturers, represents the motor speed per unit voltage in the no-load condition (\(\omega_{m_{0}}\)), as shown in Equation (18). This corresponds to the inverse of the back EMF constant. Thus, each motor constant is combined as one, as in Equation (19). \[e_{b}=K_{b}\cdot\omega_{m} \tag{15}\]  
\[\tau_{m}=K_{t}\cdot I_{m} \tag{16}\]  
\[e_{b}\cdot I_{m}=\tau_{m}\cdot\omega_{m} \tag{17}\]  
\[\tau_{m}=J_{rotor}\cdot\frac{d\omega_{m}}{dt}+B_{m}\cdot\omega_{m}+Q_{prop} \tag{9}\]0  
\[\tau_{m}=J_{rotor}\cdot\frac{d\omega_{m}}{dt}+B_{m}\cdot\omega_{m}+Q_{prop} \tag{9}\]1 Equivalent circuit of DC motor. \[\tau_{m}=J_{rotor}\cdot\frac{d\omega_{m}}{dt}+B_{m}\cdot\omega_{m}+Q_{prop} \tag{9}\]2  
A motor size constant, \(K_{m}\), another important reference index when selecting a motor, is expressed as the ratio of the motor torque and the square root of resistive power loss and it can be calculated from the torque constant and the motor resistance, as Equation (20). The motor size constant is proportional to the weight of the motor and it is also useful for estimating the viscous friction coefficient. This is because the viscous friction coefficient also tends to be proportional to the size of the motor. Although the viscous friction coefficient is not usually provided as a product specification of the motor, the value can be estimated from the no-load current. From the Equation (11), if there is no external load, the viscous friction can be regarded as the only load applied to the motor.