## 2 Support-Adjust Structure of Mirror Unit  
A 38 kW dish-Stirling solar power system (XEM-Dish System [4; 5]) was designed and built by our team and Xiangtan Electric Manufacturing Group (in China) and is shown in Figure 1a. The dish concentrator (diameter 17.70 m and focal length 9.49 m) was made up of 164 mirror units in the XEM-Dish System. Each mirror unit was fixed onto the connection plate of the space truss structure using three or four bolts and the bolts were adjusted to align the mirror (Figure 1c). An image of the system and the assembly model of the support-adjust structures at the back of the mirror units are shown in Figure 1c,d, respectively. Each support-adjust structure consists of a bolt, ball-hinge gasket set (including a convex and concave spherical gasket, forming the ball hinge pair), and nut (Figure 1d). This is the conventional structure for supporting and adjusting the mirror unit in a dish concentrator and has been widely used [21]. As shown in Figure 1d, the upper end of each bolt had ball-shaped design paired with a spherical concave surface (Figure 1b) on the back of the mirror unit forming the ball-hinge pair. The center of the ball-hinge pair is referred to as the ball hinge center. The ball-hinge gasket set and nuts were installed on the upper and lower sides of the bolt and all the nuts were tightened to fix the bolt and ball hinge center. Once bolts 1, 2, and 3 were fixed, installation of the mirror unit was complete.  
In the O-xyz coordinate system of the dish concentrator (Figure 2), the origin O is the vertex of the parabolic reflecting mirror and the \(z\)-axis points to the focal point F on the parabolic reflecting mirror. **A** (i.e., **OA**), **B**, and **C** are position vectors of the ball hinge center with coordinates (A, B, and C) on the back of the mirror unit corresponding to the ball hinge center of bolts 1, 2, and 3. The facet alignment parameter of the mirror unit was described by axis vector **N\({}_{0}\)** and vertex position vector **Q** (vertex point Q). In ideal conditions, i.e., when the mirror unit has no installation error, and for a parabolic reflecting mirror surface, Q is coincident with the origin O, such that **Q** = [0, 0, 0] and **N\({}_{0}\)** is coincident with the \(z\)-axis such that **N\({}_{0}\)** = [0, 0, 1.0]. If the reflecting mirror surface is flat, Q is the centroid of the mirror's plane and **N\({}_{0}\)** is the normal vector, which will be used in Section 6.2. In general, facet alignment error that exists after initial installation of mirror unit is called facet 1 in this paper. In this condition, the axis vector of the mirror unit has changed from **N\({}_{0}\)** to **N\({}_{t}\)**, point Q has moved to point Q\({}_{t}\) (with position vector **Q\({}_{t}\)**), and the ball hinge center has moved to the point A\({}_{t}\)\(\sim\)C\({}_{t}\), as shown in Figure 2.  
The optical performance of a dish concentrator is adversely affected by installation error, which should thus be reduced or eliminated. As an example, the facet alignment process was carried out as follows. Bolt 1 was first adjusted by unscrewing the nuts on either sides of the bolt while bolt 2 and bolt 3 remained fixed and bolt 1 was then moved up or down a certain distance \(d_{\rm A}\) to make the mirror unit rotate around the axis B\({}_{t}\)C\({}_{t}\) changing the mirror facet alignment. Next, the nuts were tightened on both sides of bolt 1 and the steps were performed to adjust bolts 2 and 3 in turn to achieve facet alignment of the mirror unit. The mirror unit thus underwent three rotational motions with the axis of rotation consisting of two ball-hinge center points. In order to achieve efficient and accurate alignment, the facet alignment parameters was measured (vertex position vector **Q\({}_{t}\)** and axis vector **N\({}_{t}\)**) and the adjustment values \(d_{\rm A}\), \(d_{\rm B}\), and \(d_{\rm C}\) of bolt 1, 2, and 3 were determined as well as the adjustment order, respectively. These aspects are the highlights of this study.