# Robust Algorithm for Calculating the Alignment of Guide Rolls in Slab Continuous Casting Machines
## 4 Results and Discussion
### Performance Under Error Influence  
To evaluate the performance of the algorithms under non-ideal conditions, sets of roll positions with random position errors, including radial and tangential components, are generated. The radial components represent the target solution, which the algorithm will have to reproduce based on tangential angles. These are calculated from the roll positions using the same Equations (21)-(24) as in the forward function of the optimization-based algorithm. After that, artificial errors are applied to the angle values to simulate measurement errors. Two types of measurement errors can be applied: constant and random. Constant errors represent systematic measurement errors that can occur in the real world, for example, with a slightly tilted or miscalibrated angle sensor. They are applied by adding a constant value to every tangential angle. Random errors occur in the real world as measurement noise or mechanical vibrations. For these, a random distribution of errors is added to the angles. The underlying concept for random sets of values (position errors or measurement errors) is a normal distribution (see Figure 11). In this paper, the amplitude of a random distribution is defined as two times the standard deviation of that distribution, such that 95.4% of randomly sampled values lie within that amplitude. Each algorithm will be tested in five different experiments. Table 2 provides an overview of the experimental conditions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{Error Amplitudes} \\  & **Roll Position** & **Roll Position** & **Angle Meas.** & **Angle Meas.** \\
**Nr.** & **(Random, Radial)** & **(Random, Tangential)** & **mm (Constant)** & **(Random)** \\ \hline
1 & **0.5** mm & 0 mm & 0\({}^{\circ}\) & 0\({}^{\circ}\) \\
2 & 0 mm & **0.5** mm & 0\({}^{\circ}\) & 0\({}^{\circ}\) \\
3 & 0 mm & 0 mm & **0.01\({}^{\circ}\)** & 0\({}^{\circ}\) \\
4 & 0 mm & 0 mm & 0\({}^{\circ}\) & **0.01\({}^{\circ}\)** \\
5 & **0.5** mm & **0.5** mm & **0.01\({}^{\circ}\)** & **0.01\({}^{\circ}\)** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental conditions to test the algorithms. Figure 11: (**a**) Normal distribution with an amplitude (two times standard deviation) of 0.5. A total of 95.4% of sampled values will fall within this amplitude. (**b**) Example of 100 values sampled from that distribution. The first four experiments reveal the susceptibility of the algorithms to each of the four types of errors. In the fifth experiment, the algorithms are tested under the influence of all error types at once, which is the most realistic case. The result of the fifth experiment will determine the best-performing algorithm. To keep the chapter to a reasonable size, only one error amplitude is tested in each experiment. The amplitudes represent plausible real-world values. An amplitude of 0.5 mm was chosen for the roll position errors according to the maximum tolerated errors by CCM operators. The amplitude of 0.01\({}^{\circ}\) was chosen for the angle measurement errors based on the targeted accuracy of SCM manufacturers and rough estimations of the measurement accuracy needed to achieve the desired position calculation accuracy. These error amplitudes are well suited to reveal the strengths and weaknesses of the algorithms, as will be demonstrated by the experimental results. Every experiment is run 100 times for each algorithm in order to capture the behavior under different random error distributions. All 100 runs are shown in the same diagram to give an impression of the average, best and worst case performances. In Experiment 1, the influence of a radial component in the roll position errors is investigated. It is expected that the results of this experiment are very similar to the baseline performance, since there are no measurement errors and the algorithms are designed to reproduce the radial components. However, as can be seen in Figure 12, the performance is much worse than the algorithm's baselines.