## 4 Mathematical Model
### Battery  
The main challenge with the battery model is describing the voltage drop due to internal resistance. Assuming that the effects of the disturbance in flight are small enough to be neglected and that the output of all motors remains stable, the drain on the battery can also be seen as a constant steady discharge. In this paper, the battery model refers to a simple model with only one internal resistance, as shown in Figure 10.  
Figure 10: Simple battery model.  
Figure 9: ESC continuous current vs. on-state resistance.  
The terminal voltage, \(V_{t}\), of the battery is calculated, as shown in Equation (31), where \(V_{oc}\) is the open-circuit voltage of the battery, \(I_{load}\) is the load current, and \(R_{butt}\) is the internal resistance of the battery. The load current is the total current consumption of the entire UAV system, as shown in Equation (32). From Equations (31) and (32), the equation for the terminal voltage is derived, as shown in Equation (33). The open-circuit voltage of the LiPo battery depends on the state of charge (\(SoC\)) and outputs per cell of approximately 4.2 V in the fully charged condition and approximately 3.4 V in the fully discharged condition. The nominal voltage is based on the open-circuit voltage of the half-charged condition (\(SoC=0.5\)), which is approximately 3.7 V per cell. The energy capacity of the battery is given by multiplying the electric charge capacity by the nominal voltage, and the total energy capacity is calculated, as shown in Equation (34).  
\[V_{t}=V_{oc}-I_{load}\cdot R_{butt} \tag{31}\]  
\[I_{load}=\frac{P_{sys}}{V_{t}} \tag{32}\]  
\[V_{t}=\frac{1}{2}\cdot(V_{oc}+\sqrt{V_{oc}^{2}-4\cdot P_{sys}\cdot R_{butt}}) \tag{33}\]  
\[E_{butt}=N_{butt.pack}\cdot N_{butt.cell}\cdot(3.7V\cdot e_{butt.cell}) \tag{34}\]  
The LiPo battery comprises a series/parallel connection of multiple single cells, as illustrated in Figure 1. Thus, the total internal resistance of the battery set (\(R_{butt.set}\)) can be calculated while using Equation (35), where \(N_{butt.cell}\) is the number of serial-connected cells that are contained in a single battery pack, and \(N_{butt.Pack}\) is the number of battery packs connected in parallel. The internal resistance of a single LiPo battery cell (\(R_{butt.cell}\)) depends on the electric charge capacity.  
\[R_{butt.set}=\frac{N_{butt.cell}}{N_{butt.pack}}\cdot R_{butt.cell} \tag{35}\]  
In this study, experiments were conducted on single-cell LiPo battery samples that are shown in Figure 11 in order to obtain estimation models of the battery open-circuit voltage and the internal resistance according to SoC. The maximum discharge rate of the sample cells that are mainly used in multicopter UAVs is approximately 25 C.  
Figure 11: Single cell LiPo battery battery samples.  
Figure 12 shows the average tendency of open circuit voltages according to SoC measured in a discharge experiment on single cell LiPo battery samples. Equation (36) shows the model estimated from the hysteresis curve.  
\[V_{\alpha c}=1.7\cdot SoC^{3}-2.1\cdot SoC^{2}+1.2\cdot SoC+3.4 \tag{36}\]  
The measurements for the internal resistance were performed while using a DC load method relative to the nominal voltage at room temperature (24 \({}^{\circ}\)C). Other major factors that affect the change in internal resistance are the operating temperature of the battery, the deterioration of performance due to aging, and theSoC; however, only the normal condition was considered in this study. Figure 13 shows the tendency of internal resistance of the sample battery cells to the capacity. Equation (37) shows the model of the internal resistance estimated from this tendency.  
\[R_{batt.cell}=21.0\cdot(e_{batt.cell})^{-0.8056} \tag{37}\]  
Figure 12: Average open-circuit voltage curve of the sample LiPo battery cells.  
Figure 13: Internal resistance of the Single cell LiPo battery samples according to the capacity.