# Robust Algorithm for Calculating the Alignment of Guide Rolls in Slab Continuous Casting Machines
## 4 Results and Discussion
### Performance Under Error Influence
This result is due to the zero-mean compensation and an effect called \"calculation error accumulation\". The roll position errors lead to differences between the measured tangential angles and the ideal tangential angles. The mean of this difference is not exactly equal to zero because only about 100 values were sampled from the random distribution and because the relationship between the roll positions and tangential angles is non-linear. The compensation method interprets this as a constant error component in the angle measurement and removes it. However, because there was no measurement error in the first place, this acts like an addition of a small constant error component to the angle measurements. Even though this added error is small, the performance deteriorates significantly due to calculation error accumulation. Calculation error accumulation is indicated by a calculation error that rises nearly linearly over the length of the machine. The results from the geometric algorithms show this behavior in Figure 12. A constant measurement error in combination with the chain-like structure of the geometric algorithms are responsible here. At each step in the calculation  
Figure 12: Results of Experiment 1 with activated zero-mean compensation. The calculation errors are much worse compared to the baseline performance because the zero-mean compensation is detrimental in this scenario. Starting at roll 57, the horizontal section of the plant is recognizable in all diagrams. That is expected, since the tangential error components have no influence on the tangential angles in the horizontal section of the plant when the radial components are zero. Still, some calculation error accumulation is visible for the geometric algorithms. It is not affected by error accumulation and is able to minimize the computation error in the horizontal section. Since the radial components of the roll errors are actually zero in this experiment, the third term of the loss function has a positive impact. In Experiment 3, the influence of a constant angle measurement error of 0.01\({}^{\circ}\) is shown. Since the angle measurement error is purely constant in this experiment, the zero-mean compensation corrects it entirely. This leads to the algorithms performing at their baseline performance which is not shown again. Rather, to demonstrate its effect, the compensation was temporarily deactivated for this experiment. The results are shown in Figure 15 and demonstrate the detrimental effect of error accumulation and the necessity of the compensation. Both geometric algorithms show effectively the same result, which is an almost linearly rising calculation error. Towards the end of the machine, this error accumulates to a value close to \(-6\,\mathrm{mm}\), which is unacceptable. This gave impetus to the development of the optimization-based algorithm in the first place. Figure 14: Results of Experiment 2 with zero-mean compensation. Figure 15: Result of Experiment 3 without zero-mean compensation. Without the compensation, the geometric algorithms suffer heavily from the influence of a constant error in the angle measurements. The optimization-based algorithm shows a much better performance, even without the zero-mean compensation. This is due to the third term in the loss function that tries to minimize the roll position. If \(w_{\text{error}}\) would be set to zero, the result of the optimization-based algorithm would look like the result of the geometric algorithms. It is worth noting that between rolls 90 and 102, a small amount of the effect also affects the optimization-based algorithm. This experiment demonstrates the importance of the zero-mean compensation. It shows how susceptible the geometric algorithms are to constant components in the angle measurement errors. It also becomes clear that the additional term in the loss function of the optimization-based algorithm improves the robustness against this type of error significantly.