## 2 Tracking Error Model of Azimuth Axis Tilt Influence  
### Azimuth Axis Tilt Error  
An SDC system mainly consists of a column, DATD, parabolic concentrator, and Stirling heat engine. The column is the front-end mechanism in the SDC system which is fixed on the horizontal ground by bolts. The DATD is installed at the top of the column, and the parabolic concentrator is fixed with the truss through the u-shaped body and installed on the DATD. Ideally, the azimuth axis (that is, the center of the column) is perpendicular to the horizontal plane, the elevation axis is perpendicular to the azimuth axis, and the concentrator focal axis is perpendicular to the elevation axis. The angle between the focal axis and the azimuth axis varies with the position of the sun. However, after long-term operation, the foundation is prone to settlement and the column inclines, leading to the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Researchers** & **Research Objects** & **Main Contribution** \\ \hline [PERSON] et al. [21] & Fresnel lens solar concentrator & Uses polar-axis tracking \\ [PERSON] et al. [22] & Solar parabolic dish and cavity receiver & Proposes a low-cost solar \\ [PERSON] et al. [23] & An optimized solar dish & The non-ideal optical factors \\  & collector & considered are relatively \\ [PERSON] et al. [24] & Heliostat & Proposes a tracking error \\  & & correction model to improve \\  & & the tracking accuracy \\ [PERSON] et al. [25] & Dual-axis tracking systems & Proposes a novel \\  & & methodology for optimizing \\  & & the overall performance \\ [PERSON] et al. [26] & Large telescopes & Establishes a new model of a \\  & & telescope pointing error \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Previous researchers’ related work (in part). non-orthogonal relationship between the azimuth axis and the horizontal plane, which leads to the tracking error of the SDC system. To represent the tilt of the azimuth axis of an SDC system and the position of the sun, the coordinate system _O-xyz_, as shown in Figure 2, is established. Its center \(O\) is located at the intersection of the horizontal ground and the central axis of the column, with the _y_-axis pointing to due south, _x_-axis pointing to due east, and _z_-axis pointing to zenith. In this study, the sun azimuth angle \(x\) is positive to the west and negative to the east, the sun elevation angle \(B\) is positive to the water intake plane upward, and \(B\) = 90\({}^{\circ}\) when pointing to the zenith. Ideally, the initial unit vector of azimuth axis is N\({}_{\mathrm{e}}\) = [0,0,\(-\)1], the initial unit vector of elevation axis is N\({}_{\mathrm{e}}\) = [0,1,0], and the initial unit vector of focal axis is N\({}_{\mathrm{f}}\) = [\(-\)1,0,0]. After settlement of the foundation, the azimuth axis vector tilts. In order to describe the deviation error between the azimuth axis and its ideal position, the parameters of the radial angle \(\theta_{1}\) and tangential angle \(\phi_{1}\) are introduced to represent the inclination angle and direction of the azimuth axis, respectively. Based on the rigid body motion theory [27], the column tilt is equivalent to the azimuth axis vector rotating by a certain radial angle and tangential angle, in turn. The azimuth axis produces the tilt error, which can be considered as the ideal azimuth axis vector Na rotating by an angle \(\theta_{1}\) around the _x_-axis and then rotating by an angle \(\phi_{1}\) around the _z_-axis. The azimuth axis vector N\({}_{\mathrm{a1}}\), after introducing the tilt error, is shown in Equation (1), where Rot(x, \(\theta_{1}\)) and Rot(z, \(\phi_{1}\)), respectively, represent the rotation matrix around the _x_- and _z_-axes, which are shown in Equations (2) and (3), respectively:  
\[\mathrm{N}_{\mathrm{a1}}=\mathrm{N}_{\mathrm{a}}\cdot\mathrm{Rot}(x,\theta_{1} )\cdot\mathrm{Rot}(x,\phi_{1})=[\mathrm{sin}\theta_{1}\mathrm{sin}\phi_{1}, \mathrm{sin}\theta_{1}\mathrm{cos}\phi_{1},-\mathrm{cos}\theta_{1}] \tag{1}\]  
\[\mathrm{Rot}(x,\theta_{1})=\left[