## 2 Modular Multifunctional Approach Advantages
### The Amplitude of the Payload Design Space  
The amplitude of the payload design space \(A\) is a function of the number of payload configurations \(P\) through a function \(f\). \(P\) itself is in turn a function \(g\) of the spacecraft bus configuration:  
\[A=f(P)=f(g(\text{spacecraft bus configuration})) \tag{8}\]  
Some elementary properties of the amplitude \(A\) of the payload design space are:  
1. If there is just one possible payload configuration, the amplitude of the relative design space is zero: \[A(1)=0\] (9) According to the chosen payload design philosophy, this condition is undesired, because there is no choice in the payload design and only one kind of space mission can be realized;
2. \(f\) is a strictly increasing monotonic function of the number of configurations \(P\). More precisely, \(f\) is assumed to be piecewise linear. This is due to the fact that configurations produced in different portions of the CubeSat volume produce different magnitudes of increments in design space amplitude. For example, an increment in the configurations produced in the internal volume of the CubeSat produces a certain increment of the design space amplitude. If the same increment in configurations is introduced in the tuna can volume [2] (p. 11), the relative increment in the design space amplitude must be inferior to the increment in the CubeSat internal volume, due to the fact that the area of the tuna can cross section is inferior to the area of the CubeSat internal cross-section. Moreover, the tuna can have a circular cross-section, while the CubeSat internal cross-section is square. The possible behaviors of the design space amplitude \(A\) are shown in Figure 3, as a function of the number of payload configurations \(P\). The limit to the number of possible payload configurations, \(P_{max}\), corresponds to the maximum amplitude of the payload design space, \(A_{max}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**s** & \(\delta_{CS}\) \\
**[-]** & **[\%]** \\ \hline
1 & 4.71 \\
2 & 9.89 \\
3 & 15.60 \\
4 & 21.94 \\
5 & 29.02 \\
6 & 36.97 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: CubeSat volumetric increments \(\delta_{CS}\) for an increasing number \(s\) of tiles with \(\delta=4.71\%\). To evaluate the effectiveness of the smart tiles approach with respect to the amplitude of the design space \(A\), a simple study case is considered. The following simplifying hypotheses, shown in Figure 4, are introduced:  
1. Different locations for the bus systems are considered. In the traditional solution, they are placed only in the internal volume of the satellite (except for small components like solar cells), while with a full smart tiles approach, they only occupy the external faces of the cube. Intermediate solutions are also considered, i.e., with part of the bus occupying the internal volume (for bulky components like reaction wheels) and part on the external faces (for thinner components like solar cells);
2. The payload is hosted in the internal volume of the CubeSat only. This happens if the payload thickness is greater than 6.5 mm and cannot fit on the external faces (CubeSat mechanical requirement 3.2.3, [2] (p. 9)). Usually, the external faces are dedicated to the solar cells and to the electrical power system (EPS) and the payload is commonly located in the inside of the spacecraft. Possible reasons leading to choose an internal payload may be the payload delicacy or the protection from a thermal or radiation environment;
3. The payload is composed of a set of modules with an equal, prismatic shape. The modules section is coincident with the cross-section of the internal volume of the CubeSat (in Figure 4, the payload modules are represented with the yellow boxes). This configuration is common for CubeSats payloads ([18] (p. 11), [19] (p. 4), [20] (p. 3)) and could represent a set of printed circuit boards (PCBs);
4. A set of \(n\) payload modules is available. Due to internal volume restrictions, only \(h<n\) modules can fit in the payload bay;
5.