# Robust Algorithm for Calculating the Alignment of Guide Rolls in Slab Continuous Casting Machines
## 5 Conclusions  
Continuously monitoring and improving the alignment condition of a CCM is a critical task to ensure a consistent and high-quality production of steel. Measuring the alignment is still a challenge today and the angle measurement data from SCMs often contain systematic and stochastic errors. The contribution of this paper is the development of an algorithm that robustly and accurately converts the angle measurements into the Cartesian coordinates of the rolls with the desired accuracy of 0.1 mm. Three different approaches were developed, tested and compared. The first two approaches are purely geometric and rely on trigonometry, while the third approach is based on the gradient descent optimization algorithm. Under ideal conditions, all algorithms show very good performance. The simplified geometric algorithm yields a computational error of around 0.02 mm, while the full geometric and the optimization-based algorithm produce effectively no computational error. Five different experiments were conducted to investigate the susceptibility of the algorithms to the four error types that may occur in the real world: radial and tangential components in roll position errors as well as constant and random components in angle measurement errors. The experiments showed that the influence of roll position errors (radial and tangential) on the accuracy of the algorithms is small and remains in an acceptable range. Constant angle measurement errors on the other hand lead to large error accumulation within the geometric approaches. A constant angle measurement error as small as 0.01\({}^{\circ}\) leads to computation errors of almost 6 mm. However, utilizing a zero-mean compensation method effectively mitigates this effect. Therefore, random angle measurement errors remain as the dominant challenge for the algorithms, since these cannot be compensated or neglected. Under the influence of random angle measurements with an amplitude of 0.01\({}^{\circ}\), the algorithms yield 90 percentile computational errors of 0.23 mm, 0.21 mm and 0.08 mm, respectively. The fifth experiment, which applies all four types of errors simultaneously, represents the most realistic and representative case. The results showed that the optimization-based algorithm performs, on average, about three times better than the geometric algorithms: 0.34 mm, 0.32 mm and 0.10 mm 90 percentile computational error, respectively. It is the only algorithm that reaches the desired accuracy. However, it has to be noted that the performance of the optimization-based algorithm relies on the weight factor \(w_{\text{error}}\), which depends on the amplitude of random measurement errors. This amplitude must be determined experimentally and the weight factor must be tuned for any given SCM and CCM in order to yield the shown results. Nevertheless, the gains in calculation accuracy over the geometric algorithms justify the effort. Figure 19: Comparison of computation times for of each algorithm. The next step in the continuation of this research is a test of the algorithms on an actual CCM. To evaluate the results of the algorithms, the actual alignment of the plant would need to be determined using a laser scanner or total station prior to the measurement with the SCM. Additionally, further improving the angle measurement accuracy of SCM would be beneficial because the experiments showed that an accuracy of 0.01\({}^{\circ}\) is just about enough to reach the targeted position calculation accuracy with the optimization-based algorithm. Conceptualization, R.R. and M.J.; methodology, R.R. ; software, R.R. ; writing--original draft preparation, R.R. ; writing--review and editing, N.A. and M.J.; visualization, R.R. ; supervision, M.J.; project administration, N.A. and M.J.; funding acquisition, N.A. and M.J.; All authors have read and agreed to the published version of the manuscript. This research was funded by the Central Innovation Programme for small and medium-sized enterprises (SMEs) of the German Federal Ministry for Economic Affairs and Climate Action.