## 4 Energy Consumption Modeling
### Subsystem Model
The wind has a direct effect on the air-induced velocity of the propeller. Applying the blade unit theory and the principles of the air momentum theory to account for the effect of air-induced velocity on the propeller thrust and torque can improve the accuracy of the flight energy consumption model. In order to determine the air-induced velocity, it is necessary to calculate it using the principles of the air momentum theory. As depicted in Figure 4, Point 1 represents the airflow inlet where the relative velocity, i.e., the total air velocity of the FPTLIR, can be obtained by summing up both horizontal and vertical components of air velocities. The airflow is accelerated perpendicular to the plane of propeller rotation, resulting in distinct induced velocities at Point 2 (propeller rotation plane) and Point 3 (airflow outlet). Applying the conservation laws for momentum and kinetic energy, it can be seen that the induced velocity at Point 3 is twice as much as that at Point 2. Through this analysis, a relationship between thrust, airflow mass flow, and induced velocity at the propeller rotation plane can be expressed by Equation (16). \[\dot{m}=\rho\pi{R_{2}}^{2}\sqrt{{v_{x}}^{2}+\left({v_{z}}+{v_{in}}\right)^{2}} \tag{16}\] \[F_{\Gamma}=\dot{m}{v_{o}}=2\dot{m}{v_{in}}=2\rho\pi{R_{2}}^{2}{v _{in}}\sqrt{{v_{x}}^{2}+\left({v_{z}}+{v_{in}}\right)^{2}}\]  
where \(\dot{m}\) is the mass of air flow, \(v_{in}\) is the induced velocity at point 2, and \(v_{o}\) is the induced velocity at point 3. The induced velocity at point 2 can be determined by solving expression (17) as presented as follows. \[v_{in}^{4}+2v_{z}v_{in}^{3}+v^{2}v_{in}^{2}=\left({F_{\Gamma}}/{2\rho\pi{R_{2 }}^{2}}\right)^{2} \tag{17}\]  
Brushless DC rotor motors are used as the end-effector to drive the flight system of the FPTLIR, and improvement of the accuracy of the energy consumption model can be achieved by incorporating their dynamics into the modelling process. As the rotor motor is controlled electronically with a fast response time, transient effects are not considered. The equivalent circuit diagram of the rotor motor is illustrated in Figure 4, and its mathematical representation follows a three-parameter model [41; 42]. By combining this with a simplified motor model, Equation (18) can be used to calculate main road currents and voltages applied to the rotor motor. \[ \begin{array}{l}I_{t}=F_{\text{Q}}K_{V}+I_{0}\\ U_{t}=\frac{\omega_{V}}{K_{V}}+F_{\text{Q}}K_{V}R_{m}\end{array}  \tag{18}\]  
where \(I_{t}\) is the dry circuit current input to the rotor motor, \(U_{t}\) is the dry circuit voltage, \(F_{\text{Q}}\) is the load torque of the propeller, \(K_{V}\) is the velocity constant of the rotor motor, \(I_{0}\) is the current required by the rotor motor when the load on the rotor motor is zero, and other losses of the rotor motor such as friction losses and magnetic losses are included in the no-load current \(I_{0}\) of the rotor motor. \(\omega\) is the angular velocity of the rotor motor and the paddle, and \(R_{m}\) is the resistance of the rotor motor winding. The relationship between the path and the voltage and current of the end effector can be described by the above derivation.