# Robust Algorithm for Calculating the Alignment of Guide Rolls in Slab Continuous Casting Machines
## 3 Angle Conversion Algorithms
### Geometric Approaches
#### 3.2.2 Full Geometric Approach  
The full variant of the geometric approaches does not make additional assumptions about the machine geometry and, therefore, includes the casting radius and different roll radii in its equations. This approach results in two different cases with two slightly different sets of equations. Depending on the geometric circumstances at the current roll pair, the correct case has to be chosen before each step. The underlying geometry of case 1 is shown in Figure 5. The line \(\overline{JH}\) is tangent to both rolls and represents the rulers edge. Correspondingly, \(\rho\) equals the measured angle. The distances \(d_{1}\) and \(d_{2}\) represent the radial components of the current and the next roll position error in the calculation chain. Similar to the simplified approach, the length \(b\) and the angle \(\epsilon\) are independent from the roll positions and can be extracted from the roll plan. Note that \(b\) is a straight line, but the dashed passline is slightly curved. Also defined by the roll plan and independent of the roll positions are the angles \(\alpha_{1}\) and \(\alpha_{2}\) that are formed between a vertical line and a line through the roll's center perpendicular to the passline. Through slightly different values for \(\alpha_{1}\) and \(\alpha_{2}\), the machines curvature is considered. The values of \(r_{1}\) and \(r_{2}\) equal the radii of the two rolls. The angles formed by the radii with  
Figure 4: Geometry of one step of the simplified geometric algorithm. Line \(\overline{DH}\) is tangent to both rolls and represents the ruler’s edge. Due to the simplifying assumptions, the passline through \(\overline{CE}\) is a straight line and both rolls have the same radius. This leads to the four right angles and the fact that the lines \(\overline{BG}\) and \(\overline{DH}\) are parallel. the tangent line are the only right angles in this diagram. The diagram shows the casting radius R, but it is not used in the equations. Mostly, case 1 and case 2 can be distinguished by the position of the short line segment \(s_{1}=\overline{BJ}\). If \(s_{1}\) is outside of the strand (to the left of the passline, when \(d_{1}\geq s_{1}\)), case 1 is used. If it is on the other side (\(d_{1}\leq 0\)), case 2 is used (see Figure 6). In the special case that \(s_{1}\) intersects the passline, the sign of angle \(\gamma\) determines the case. These two special cases are not shown in dedicated figures. Table 1 summarizes the conditions under which each case is applied. With the two sets of equations that will be derived in the following paragraphs, every geometric case can be solved. Up to a point, however, the equations are identical for both cases. First, the angles  
\[\beta_{1}=\alpha_{1}+\rho \tag{5}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Condition**: & \(d_{1}\geq s_{1}\) & \(0<d_{1}<s_{1}\) & \(d_{1}\leq 0\) \\  & \(\gamma\leq 0\) & \(\gamma>0\) & \\ \hline
**Case:** & case 1 & case 1 & case 2 & case 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Choices for the correct case of the full geometric approach. Figure 5: Geometry of one step of the full geometric algorithm (case 1). The roll position errors in this figure are greatly exaggerated. \[\beta_{2}=\alpha_{2}+\rho, \tag{6}\]  
that are part of the right triangles are calculated using the measured angle \(\rho\). With the roll's radii the lengths  
\[a_{1}=\frac{r_{1}}{\sin\beta_{1}} \tag{7}\]  
and  
\[a_{2}=\frac{r_{2}}{\sin\beta_{2}} \tag{8}\]  
are determined. Subsequently, the lengths of the short line sections  
\[s_{1}=a_{1}-r_{1} \tag{9}\]  
and  
\[s_{2}=a_{2}-r_{2} \tag{10}\]  
are calculated. The angle  
\[\gamma=\frac{\pi}{2}-\rho-\alpha_{1} \tag{11}\]  
needs to be determined, when \(0<d_{1}<s_{1}\) in order to determine which case must be used. **Case 1. **: _In case 1, the angle_  
\[\theta=\epsilon+\alpha_{1} \tag{12}\]  
_is calculated first. It is independent from the roll positions.