## 2 Materials and Methods
### Backtracking  
In the previous section, the methodology used to determine the orientation of the solar tracker in order to obtain the maximum solar capture was detailed. This section describes the method used to prevent a solar tracker from shading an adjacent tracker. Only two factors needed to be considered for this method: the position of the sun and the orientation of the panels. To study intershading it is convenient to use an auxiliary reference system where the representation plane is perpendicular to the tracker axis. For this purpose, the vector \(\overrightarrow{q}\) was defined as:  
\[\overrightarrow{q}=\overrightarrow{e}\times\overrightarrow{n_{T}} \tag{17}\]  
where \(\overrightarrow{q}\) is a vector perpendicular to the ground normal vector \(\overrightarrow{e}\), as well as to the rotation axis vector \(\overrightarrow{n_{T}}\). The solar vector \(\overrightarrow{s}\) and the plane normal vector in the optimal position \(\overrightarrow{n}\) were named \(\overrightarrow{s^{\prime}}\) and \(\overrightarrow{n^{\prime}}\) in the reference system composed by \(\left\{\overrightarrow{q},\overrightarrow{e},\overrightarrow{n_{T}}\right\}\) (Figure 4). The procedure used to estimate whether a solar panel B would shade an adjacent panel A was to project the shadow of panel B onto the straight line on which the adjacent panel A lay (Figure 5). Since the sun's rays are parallel to each other, the projection of one point was sufficient. First, a vector was defined from the centre of solar panel B in the direction of the solar rays \(\overrightarrow{s^{\prime}}\). Next, the central point of solar panel A was projected onto the centre of panel B, defining \(\overrightarrow{a}\). The examination of this vector enables the determination of whether shading will occur. Figure 4: Reference system indicating the solar trackers, normal vector and solar vector. Figure 5: Vectors in the solar trackers reference system. In order to obtain the vector \(\overrightarrow{a}\), both terms were multiplied by the vector \(\overrightarrow{n^{\prime}}\):  
\[\overrightarrow{d\cdot n^{\prime}}=0+\tau\overrightarrow{s^{\prime}\cdot n^{ \prime}}\ \Rightarrow\ \tau=\ \frac{n^{\prime}{}_{x}d}{s^{\prime}{}_{x}n^{\prime}{}_{x}+s^{\prime}{}_{z} \cdot n^{\prime}{}_{z}} \tag{18}\]  
Substituting the value of \(\tau\) from Equation (19) into Equation (18) results in:  
\[\overrightarrow{d}=\overrightarrow{a}+\frac{n^{\prime}{}_{x}\cdot d}{s^{ \prime}{}_{x}n^{\prime}{}_{x}+s^{\prime}{}_{z}\cdot n^{\prime}{}_{z}} \overrightarrow{s^{\prime}} \tag{19}\]  
Vector \(\overrightarrow{a}\) can be obtained according to the following expression:  
\[\overrightarrow{a}=\left(d-\frac{n^{\prime}{}_{x}\cdot d}{s^{\prime}{}_{x}n^{ \prime}{}_{x}+s^{\prime}{}_{z}\cdot n^{\prime}{}_{z}}s^{\prime}{}_{x} \right)\overrightarrow{p}-\frac{n^{\prime}{}_{x}\cdot d}{s^{\prime}{}_{x} \cdot n^{\prime}{}_{x}+s^{\prime}{}_{z}\cdot n^{\prime}{}_{z}}s^{\prime}{}_{z }\overrightarrow{n_{T}} \tag{20}\]  
As mentioned above, the examination of vector \(\overrightarrow{a}\) enables the determination of whether shading will occur. If the module of the vector is less than the width of the panel, shading will occur. \[\left|\overrightarrow{a}\right|<h \tag{21}\]  
To avoid this situation, backtracking was used to look for a new orientation of the panels that prevented shading. The new orientation was intended to involve a minimum deviation from the optimal capture orientation. The new inclination angle \(\alpha\) was that which caused the shading of a panel to fall on the edge of the adjacent panel (Figure 6). This inclination angle \(\alpha\) was calculated based on a triangle formed by the line that joins the center of the two panels \(d\), the projection of the centre of the panel on the line of the adjacent \(\tau\overrightarrow{s^{\prime}}\) and the distance that corresponds to the module of vector \(\overrightarrow{a}\) in Figure 5, which in Figure 6 is equivalent to the width of the panels, \(h\).