## 5 Conclusions  
This manuscript presents the implementation and deployment of an optical system for automatic in-line dimensional inspection of revolution warm workpieces. The circular runout of warm-forged revolution workpieces is critical as a severe misalignment between the punch press and the forming matrix axes disables the posterior machining, resulting in a scrapped part. The system splits the inspection in two steps: (1) the dimensional assessment planning, performed only once by the metrologist, off-line the production, and (2) the in-line automated dimensional inspection. The developed system automatically assesses, in less than 60 s, the circular runout of the workpiece, whose temperature nears 600 \({}^{\circ}\)C. Our prior-to-operation test results show that the measurements of the developed system for warm workpieces (27 \({}^{\circ}\)C \(\leq\) T \(\leq\) 560 \({}^{\circ}\)C) deviate less than 0.1 with respect to the standard CMM measurements of the cooled-down workpieces, for workpieces as long as 160 mm. In addition, the temperature-vs.-runout analysis shows no correlation between these two variables at such level of uncertainty. The measuring system repeatability and reproducibility (R&R) has been validated with an ANOVA test. This assessment of dimensions in warm workpieces fills a gap in processes in which the advantage of early detection of an inherent planning, design, or manufacturing error compensates for the disadvantage of precision loss due to the cooling of the workpiece.  
Our system has been deployed by an automotive part manufacturer in a warm forming production line of stub axles, working as early detection of dimensional misfits. This early detection reduces the time needed to detect a defective part from 20 min to 1 min. Since the forge is a highly repetitive manufacturing process, when a defective part is found, all the pieces between the last correct part and the defective one are systematically scrapped. Thus, considering the production cadence, the number of parts that are scrapped each time a defective part is detected has been reduced from 400 to 20 (95%).  
Future work concerns: (1) A warm-workpiece assessment method for data which are highly sensitive to cooling effects, which accounts for the heat loss effects on the workpiece geometry. (2) The integration of a robot arm for automatic placement of forged workpieces, to increase the measurement efficiency. (3) Metrological certification of the equipment by an ENAC accredited laboratory.  
[PERSON]-[PERSON] and [PERSON] conceptualized and implemented the Computational Geometry and Dimensional Inspection methodology. [PERSON] and [PERSON] designed, implemented and executed the system calibration and the Computer Vision system for data acquisition. [PERSON] and [PERSON] designed the methodology and experiments, and supervised the the Dimensional Inspection of warm-forge parts. [PERSON]-[PERSON] and [PERSON] supervised the Stochastic Computational Geometry aspects and applications of this research. All the authors contributed in the writing of this manuscript.  
This research was partially supported by the Basque Government under the grant Basque Industry 4.0.  
We want to thank our colleagues from Sariki Metrologia S.A. whose knowledge of dimensional inspection has been very valuable during this research.  
The authors declare no conflict of interest.