## 2 Methodology
### Service Life in Tracking Systems  
The increased service life of solar trackers means that the system must operate in reliable conditions without failures for an extended period. For the present research, the two-axis solar tracking system in azimuth-elevation configuration is considered a two-degree-of-freedom robotic system in an open kinematic chain. Hence, the common failures in industrial robotic systems occur in the joint mechanisms, which generally limits the service life [32]. Failures depend on characteristics such as transmission type, tooth geometry, and environmental conditions. Typically, expected life is based on transmission errors due to increased backlash and bearing deterioration, intensifying the stresses due to friction. Consequently, the lifetime analysis of the trackers will be carried out based on the wear and failure of the robot joints and processing hardware, where the critical devices in the tracker are the axis, mainly the gearboxes, where their service life (\(L_{h}\)) in the robotic joints is determined by speed and torque. Each axis includes the motor driver, rotational position sensor, DC motor, gearbox, mechanical transmission, link, and housing. According to [32], it can be calculated as:  
\[L_{h}\ =\ K\cdot\frac{N_{o}}{N_{m}}\bigg{(}\frac{T_{o}}{T_{m}}\bigg{)}^{\frac{10}{ \gamma}} \tag{3}\]  
where \(K\) is the rated service life at rated torque and at rated output speed, \(N_{o}\) is the rated output speed, \(N_{m}\) is the average output speed, \(T_{o}\) is the rated output torque, and \(T_{m}\) is the average output torque. The service life and the rated output torque and speed are parameters defined by the reducer model in the datasheet. In addition, the path of each axis must be segmented based on the defined maximum tracking error as presented in [29] due to the tracking strategy will be step by step, obtaining the steps \(\beta_{i}\) for \(i=1,\ldots,n_{\beta}\) and \(\gamma_{j}\) for \(j=1,\ldots,n_{\gamma}\), for the azimuth and elevation axis, respectively.  
Therefore, the solar path can be expressed in terms of the required time to move from one step to the next on each axis, defined as an operation time (\(\text{t}_{\text{op}}\)) when the axis is moving and as idle time (\(\text{t}_{\text{td}}\)) when the axis is waiting or stopped [29]. Figure 1 shows the segmented path of the azimuth and elevation axis where each movement is independent, and they do not necessarily arrive at the position at the same time. In other words, the operation time period is independent due to the solar path, where \(\text{t}_{\text{op},i}\
eq\text{t}_{\text{op},i}\). Therefore, the wear that exists on each of the axes depends on the operating time, which affects the lifetime. Then, based on [32], the average output speed and average output torque for the azimuth path are defined by the following expressions:  
\[|N_{m}|_{avg}\ =\ \frac{\sum_{i=1}^{n_{\beta}}\text{t}_{\text{op},i}\ |N_{m,i}|}{\sum_{i=1}^{n_{\beta}}\text{t}_{\text{op},i}} \tag{4}\]  
\[|T_{m}|_{avg}\ =\ \Bigg{(}\frac{\sum_{i=1}^{n_{\beta}}\text{t}_{\text{op},i}\ |N_{m,i}|\ |T_{m,i}|^{c}}{\sum_{i=1}^{n_{\beta}}\text{t}_{\text{op},i}\ |N_{m,i}|}\Bigg{)}^{\frac{1}{c}} \tag{5}\]  
where \(c\) is equal to \(\frac{10}{3}\). In the case of the elevation path, the Equations (4)-(6) are modified for the number of elevation steps for \(j\ =\ 1,\ldots,n_{\gamma}\) and with the mechanical transmission parameters. Consequently, the Equation (3) is modify as:  
\[L_{h}\ =\ \frac{\lambda_{h}}{|N_{m}|_{avg}}\bigg{(}\frac{1}{|T_{m}|_{avg}} \bigg{)}^{c} \tag{6}\]  
where \(\lambda_{h}\ =\ K\cdot N_{o}\cdot T_{o}\).  
Figure 1: Schematic of the solar path segmentation in the time domain.