## 2 Materials and Methods  
### Test Station and Experimental Design for Investigating Liquid Slosh  
To investigate the liquid slosh behavior, a test station shown in Figure 2 was build. The station consists of a direct linear drive on which different containers can be conveyed with any desired motion profile. To observe the fluid's behavior, a high-speed camera is coupled to the mechanical setup, whereas the camera is triggered by the motion control system. Hence, the captured frames can be synchronized with the applied motion profile, which simplifies the evaluation. In addition to that, the applied frame rate (FR) is automatically determined by the observed operating speed \(n\). Hence, all experiments supply the same amount of frames for further investigation, independently from the applied operating speed. The specifications of the used hardware are displayed in Table 1.  
The captured high speed frames are evaluated automatically by detecting the region of interest, binarizing the frame, and finally detecting the fluid's surface with the Canny algorithm [21] (see Figure 3).  
To cover a wide range of practical conditions, the experiments are carried out with six container geometries at two different motion profiles and 17 different operating speeds. The geometries differ in their width \(w\). The exact specifications are depicted in Table 2. The stroke, velocity, and acceleration trajectories of the used motion profiles are displayed in Figure 4. Due to the high water content in everyday beverages, water is taken as the fluid to be examined.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Element** & **Specification** & **Remark** \\ \hline High speed camera & Mikrotron EoSens MC3010 & Triggered by GardaSoft CC3320 \\ \hline Direct linear drive & Baumoller LSM 60-0916 & – \\ \hline LED light & Destlight VPad-150 & – \\ \hline Motion control system & Beckhoff TwinCAT 3 & (Programmable logic controller) \\  & Moog MSD & (Converter) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of used hardware elements in the test station.  
Figure 2: Experimental setup to investigate the linear transport unit’s motion’s influence on the liquid slosh behavior for different container geometries. The different containers are clamped into a mechanical socket that is coupled to a linear direct drive. Additionally, a high speed camera and LED lighting are fixed to the mechanical setup to observe the fluid’s surface.  
For creating the parameter identification data, all six geometries are conveyed with the acceleration pulse motion at 17 different operating speeds, whereas every experiment is repeated 10 times, represented by the constant \(K_{1}\). Hence, an amount of 1020 experiments is to be carried out and evaluated. The validation data is generated by applying the quintic polynomial motion profile, whereas each experiment is repeated five times (\(K_{2}=5\)). Therefore, an amount of 510 experiments is to be conducted and evaluated for creating the validation data. The idea behind this strategy is that different system inputs are used for parameter identification and validation.  
Figure 4: Applied motion profiles calculated for a cycle time of \(T_{\text{cycle}}=1\,\text{s}\), displayed over the normalized time \(\tau\): (**a**) acceleration pulse motion is used for parameter identification, and (**b**) polynomial motion is used for generating validation measurements.  
Figure 3: Steps in detecting the fluid’s surface edge for the container’s width \(w\): first, the region of interest is detected; second, the frame is binarized by setting a threshold value; third, the edge is detected by the Canny algorithm. The \((x,\eta)\)-values of the edge are saved.