# Generic Component-Based Mission-Centric Energy Model for Micro-Scale Unmanned Aerial Vehicles
## 3 Model
### Power Function of Components  
To compute the energy consumption of the quadcopter in a certain time frame of a mission, we assign each component \(C_{i}\) of the quadcopter a power function \(P_{C_{i}}(\vec{p}):\mathcal{P}\rightarrow\mathbb{R}\), which maps the state of each component described by the parameter vector to an output power expressed as a real number. This power function uses the whole parameter vector \(\vec{p}\) of the quadcopter as input, even though it may only depend on individual parameters \(\vec{p}_{i}\). The power function of the quadcopter is defined as the sum of power functions for all components. This reflects the law of conservation of energy:  
\[P(\vec{p})=\sum_{Q}P_{C_{i}}(\vec{p}) \tag{2}\]  
To calculate the power of the quadcopter for a specific mission, the mission description function \(M(t)\) can be used to transform the mission time frame to the parameter vector space. Afterwards, the power function of the quadcopter can be used to transform the parameter vector at each point in mission time to a power.  
\[P(M(t)):\mathcal{T}\rightarrow\mathbb{R} \tag{3}\]  
**Definition 5** (Energy Function).: _To estimate the energy needed for a manoeuvre in the time frame \(T=[t_{start},t_{end}]\subset\mathcal{T}\) within the mission, we calculate the integral of the concatenated function, according to the physical relation of power and energy \(E(t)=P(t)\)._  
\[E(T)=\int_{t_{start}}^{t_{end}}P(M(t))\,dt,\]  
The power functions of the components contain the free parameters of this model. To parametrise the energy model for a quadcopter the power functions for each of its components must be modelled using an estimated power function \(g_{C_{i}}(\vec{p})\) and an error function \(e_{C_{i}}(\vec{p})\):  
\[P_{C_{i}}(\vec{p})=g_{C_{i}}(\vec{p})\pm e_{C_{i}}(\vec{p}) \tag{4}\]  
A correctly modelled power function introduces a bounded error function \(0\leq e_{C_{i}}(\vec{p})\leq e_{C_{i}}\), \(\forall\vec{p}\in\mathcal{P}\).  
We consider three special types of power functions as generic templates for the components in a quadcopter: _Constant Power Functions_, _Discrete Power Functions_ and _Continuous Power Functions_.  
The most simplistic form of a power function of a component is the _Constant Power Function_. The characteristic of this function is that no \(p_{i}\) has an influence on the drawn power. Consequently, the power of the component is not dependent on the state \(\vec{p}\) and therefore mission. The _Constant Power Function_ is defined as  
\[g(\vec{p})=k,k\in\mathbb{R},\forall\vec{p}\in\mathcal{P} \tag{5}\]  
This type of power function can be applied to components which do not fluctuate in their power usage. Example for such components are static sensors and micro controllers with a constant power draw.  
A second type of power function is the _Discrete Power Function_. This function enables a selection of the used power function based on the input parameter vector \(\vec{p}\), according to  
\[g(\vec{p})=\begin{cases}P_{1}(\vec{p})&,\vec{p}\in\mathcal{P}_{1}\\ P_{2}(\vec{p})&,\vec{p}\in\mathcal{P}_{2}\\...&\\ P_{l}(\vec{p})&,\vec{p}\in\mathcal{P}_{l}\end{cases} \tag{6}\]  
All \(\mathcal{P}_{i}\) need to define non-overlapping non-empty subsets of \(\mathcal{P}\), where each \(\mathcal{P}_{i}\) represents a dedicated activation condition. This function type can be used to model components with discrete activation states like LEDs, additional motors or manipulators to interact with the environment.  
The third type of power functions are _Continuous Power Functions_. We consider multivariate polynomials of degree \(o\), which enable the representation of physical phenomena like inertia or drag with reasonable precision:  
\[g(p)=\sum_{i=0}^{n}\sum_{j=0}^{o}a_{j}p_{i}^{j} \tag{7}\]  
In general, not all parameters \(a_{j}\) are necessary. Based on expert knowledge of the technical implementation of a component, a simplified polynomial may be used.