# Energy-Optimized 3D Path Planning for Unmanned Aerial Vehicles
## 2 Preliminary Studies
### Studies about Drone Energy Consumption  
There are different types of drones defined according to their power supply. The most common drone drives are hybrid and purely electric drives. A summary of these can be seen in Figure 2.  
In the case of hybrid systems, an Energy Management Strategy (EMS) system usually regulates and controls efficient energy consumption. These EMSs can be based on different strategies, such as: rule-based systems; intelligent-based systems (Fuzzy or NN); optimization-based; etc. Classic electric UAVs, which operate only in motorized mode (where there is no recharging of the battery) lack these systems, and energy savings have to be ensured by another approach. This is why energy-efficient path planning was examined and improved in this project. The energy consumption of the drone can be defined in two ways:  
* Practically: by direct measuring of the amps consumed by various parts of the drone. * Theoretically: with calculations of the energy consumption of the various parts. For energy consumption calculations, the following relations must be considered as a minimum. Engine power can be calculated based on forces, torques and moments on the x/y axis. Generally, Newton's Laws are used to calculate forces and moments. See Figure 3 for a better understanding of the equations. In general, the flight lift forces (\(F_{1}\)-\(F_{4}\)) and \(M_{x}\), \(M_{y}\) moments can be calculated as follows:  
\[F_{i}=k_{f}\times\omega_{i}^{2} \tag{1}\]  
\[M_{x}=(F_{3}-F_{4})\times L \tag{2}\]  
\[M_{y}=(F_{1}-F_{2})\times L \tag{3}\]  
where \"\(L\)\" is the distance between two rotors. Figure 2: Classification of drones according to power supply (the energy supply of the actual drone used in the article is colored in red). The operation's conditions are given by the following relations:  
1. **Hovering motion**: Equilibrium conditions for hovering: \(mg=F_{1}+F_{2}+F_{3}+F_{4}\); and all moments = 0; Equation of motion: \[m=F_{1}+F_{2}+F_{3}+F_{4}-mg;m=0\] (4)
2. **Rise/Fall motion**: Conditions for operations: \(mg<F_{1}+F_{2}+F_{3}+F_{4}=\textbf{rise}\); \(mg>F_{1}+F_{2}+F_{3}+F_{4}=\textbf{fall}\); and all moments = 0; Equation of motion: \[m=F_{1}+F_{2}+F_{3}+F_{4}-mg;m>0\] (5)
3. **Yaw motion**: Conditions for hovering \(mg=F_{1}+F_{2}+F_{3}+F_{4}\); and all moments \(\
eq 0\); Equation of motion: \[ \begin{split}\text{(mass * linear acceleration) }m*a=F_{1}+F_{2}+F_{3}+F_{4}-mg;\\ \text{($I_{ZZ}$ * angular acceleration around ``Z'') }I_{ZZ}*a_{Z}=M_{1}+M_{2}+M_{3}+M_{4}\end{split} \] (6)
4. **Pitch/Roll motion**: Conditions for hovering \(mg<F_{1}+F_{2}+F_{3}+F_{4}\); moments \(\
eq 0\); Equation of motion: \[ \begin{split}\text{(mass * linear acceleration) }m*a=F_{1}+F_{2}+F_{3}+F_{4}-mg;\\ \text{($I_{XX}$ * angular acceleration around ``X'') }I_{XX}*a_{X}=(F_{3}-F_{4})\times L\end{split} \] (7)  
For the precise description of the forces and exact modelling of the drone, the rigid-body dynamical relations in a 3D environment must be known. Initially, the reference coordinate system is set up along with the related movements. Afterwards, the following effects on the drone are considered:  
* aerodynamic forces: friction and drag of propellers rotating in air
* secondary aerodynamic effects: blade flapping, ground effect, local flow fields
* inertial counter torques: gravitation, acting at the center, affect the rotation of propellers
* gyroscopic effect: change in the orientation of the drone body and plane rotation of propellers  
The dynamic equation can then be formulated as follows (in essence, this is rigid body dynamics extended to a 3D environment):  
\[ \begin{bmatrix}\overset{\rightarrow}{F}\\ \tau\end{bmatrix} = \begin{bmatrix}m_{3}&0_{3}\\ 0_{3}&\overset{\rightarrow}{I_{3}}\end{bmatrix}\begin{bmatrix}\overset{ \rightarrow}{a}\\ \infty\end{bmatrix} + \begin{bmatrix}\omega\times m\overset{\rightarrow}{v}\\ \omega\times\overset{\rightarrow}{I_{3}\omega}\end{bmatrix}