## 8 Conclusions  
In this paper, the facet installation measurement method and facet alignment method of the mirror unit of solar parabolic dish concentrator were presented and corresponding mathematical models were developed. The methods were verified by numerical simulation and an indoor experiment consisting of a metal plane facet alignment. The main conclusions of this study are as follows:  
* The proposed mirror facet installation measurement method using photogrammetry only needs to measure the spatial coordinates of three feature points on the mirror surface (forming a triangle distribution) and can determine the facet alignment parameters of the mirror unit (axis vector and vertex position vector). The whole process is not only simple and convenient but does not cause optical pollution of the reflection mirror.
* The three-rotation alignment method (adjusting three support bolts in turn) and two-rotation alignment method (adjusting two of the three support bolts in turn) for aligning the mirror facet were presented. The advantage of these methods is that the adjustment value and adjustment order of each bolt can be determined prior to the alignment of the mirror facet, which could provide quantitative adjusting information to an operator. Theoretically, according to adjusting information just single aligning the mirror unit can eliminate the axis vector error of mirror unit, which can avoid the need for repeated adjustments, so could reduce labor costs.
* Using a numeral simulation for a case where the initial error range is 2.95-61.19 mrad before alignment, values for the error angle \(\theta\) of axis vector range were found to be 0.001-0.08 mrad and 0.003-0.45 mrad by adopting the three-rotation and two-rotation alignment method, respectively. These results show that both methods are valid and have high theoretical accuracy.
* In the facet alignment experiments using a metal plane, the error angle \(\theta\) decreased from 33.09 to 6.16 mrad and 37.59 to 4.69 mrad using three-rotation alignment methods and the error angles \(\theta\) decreased from 26.75 to 6.26 mrad and 46.11 to 2.05 mrad using two-rotation alignment methods. The results have thus shown that the two methods can be used for mirror facet alignment in the solar concentrator.
* The position error of the vertex can be greatly reduced using facet alignment methods presented in this paper but cannot be completely eliminated. For engineering applications, the real-timedisplay system of the facet installation measurement and facet alignment could be adopted to improve alignment accuracy (vertex position error could be effectively controlled) and efficiency of mirror units.
* In this paper, the presented methods for measuring facet alignment of the mirror unit are not subject to the geometry of the reflector mirror and therefore have extensive applicability. They could be applied to dish concentrator, solar tower, and trough concentrator and furthermore could provide a reference for the measurement and adjustment of rigid body facets in other engineering fields.  
Conceptualization, J.Y.; Methodology, J.Y., Y.P. and D.N.; Software, J.Y.; Validation, J.Y. and D.N.; Formal Analysis, J.Y.; Investigation, J.Y.; Data Curation, D.N.; Writing-Review & Editing, J.Y. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the National Natural Science Foundation of People's Republic of China (No. 51641504), Hunan Province Natural Science Foundation of People's Republic of China (No. 2019 JJ50202, 2019 JJ40085), and Scientific Research Project of Hunan Education Department (No. 19C0794, 18B565).  
The authors specially acknowledge the editors and referees who made important comments to improve this article.  
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.