## 2 Modular Multifunctional Approach Advantages
### The Amplitude of the Payload Design Space
The \(n\) payload modules are independent. This means that they can operate independently and do not need to be connected to fulfill a single payload function. The thickness of the payload modules \(t_{payl}\) is between 6.5 mm and 18 mm, while for the bus modules, the thickness \(t_{sys}\) is fixed to 6.5 mm. Thus, the bus modules can either be stored in the internal volume or on the external faces. There are \(k\) smart tiles occupied by the bus subsystems, where \(k\) varies between 0 and 6. Figure 3: Example of the payload design space amplitude \(A\) as a function of the number of payload configurations \(P\). From elementary combinatorics, the total number \(P\) of possible payload configurations is:  
\[P=h! \begin{pmatrix}n\\ h\end{pmatrix} =\frac{n!}{(n-h)!} \tag{10}\]  
The number of possible payload configurations \(P\) must be correlated with the amplitude of the payload design space \(A\), i.e., it must be decided which is the function \(f\) at Equation (8). To respect the condition that \(A(1)=0\) (Equation (9)), it is assumed that \(f(P)=P-1\), and thus \(A=P-1\):  
\[A=P-1=\frac{n!}{(n-h)!}-1 \tag{11}\]  
It is useful to divide the amplitude \(A\) by the total volume of the payload \(V_{payl}\). The specific amplitude of the payload design space \(A^{*}\) is thus obtained:  
\[A^{*}=\frac{A}{V_{payl}}=\frac{1}{V_{payl}}\left(\frac{n!}{(n-h)!}-1\right) \tag{12}\]  
The study case considers a variable payload module thickness \(t_{payl}\). For each value of the thickness, the total number of payload modules \(n\), from which the \(h<n\) modules are chosen, is taken as equal to the maximum number of modules that fit in the internal volume:  
\[n=\text{floor}\left(\frac{l_{cube}}{t_{payl}}\right) \tag{13}\]  
The number of payload modules \(h\) stored in the CubeSat internal volume is computed accounting for the volume occupied by the \(6-k\) bus modules:  
\[h=\text{floor}\left(\frac{l_{cube}-t_{sys}(6-k)}{t_{payl}}\right) \tag{14}\]  
The payload volume \(V_{payl}\) is obtained from the number of modules \(h\):  
\[V_{payl}=h\,\cdot\,l_{cube}^{2}\,\cdot\,t_{payl} \tag{15}\]  
Finally, the total volume \(V_{CS}\) of the CubeSat is evaluated adding the fixed internal volume \(V_{int}\) with the volume occupied by the smart tiles \(V_{tiles}\):  
\[V_{CS}=V_{int}+V_{tiles}=l_{cube}^{3}+k\,l_{cube}^{2}\,t_{sys} \tag{16}\]  
Figure 4: CubeSat schematization for the study case. The volumetric efficiency of the CubeSat \(\gamma_{CS}\) is also evaluated, using its definition at Equation (2). The equations above refer to the 1U form factor and can be easily adapted for the other form factors. The results for the form factors 1U, 2U, 3U, and 6U are shown in Figure 5. The figure summarizes the advantages of the smart tiles approach applied to form factors 1U, 2U, 3U, and 6U. Figure 5a shows the total volumetric increments as a function of the total CubeSat volume \(V_{CS}\), for different numbers of tiles \(k\). The maximum volume gain \(\delta_{CS}=\gamma_{CS}-1\) for the 1U case is 63.93% is obtained with six tiles. For the 2U, the same number of tiles leads to a lower value of 22.41%, for the 3U an even lower value of 13.52% and for the 6U just 8.12%. In general, the higher the internal volume of the CubeSat is, the lower the volume gain is for a fixed number of tiles \(k\), as shown in the box on the left in the figure (Equation (2)). These results suggest that the maximum advantage in terms of volumetric efficiency can be reached in the 1U case. Nevertheless, remarkable volume gains can be obtained for the 2U, 3U, and 6U cases, but this requires major efforts. For bigger CubeSats, more complex payloads can be carried and thus more types of tiles may be needed. In addition, more tiles would need to be built and tested. Figure 5b represents the specific payload design space amplitude \(A^{*}\) as a function of the total spacecraft volume \(V_{CS}\) for various payload modules thicknesses \(t_{payl}\). It can obviously be concluded that bigger form factors have a wider payload design space.