## 6 Experimental Verification  
The developed flow rate control system was verified using experiments with the tilting-ladle-type automatic pouring machine depicted in Figure 1b. In the experiments, the developed flow rate control was compared with a conventional feed-forward flow rate control. The feed-forward flow rate control was constructed using the inverse of the pouring process model [3]. In the practical use of a tilting-ladle-type automatic pouring machine, it is difficult to precisely measure the ladle tilt angle at which the molten metal can be initially poured. This is because the starting condition for pouring the molten metal is influenced by the surface tension and density variations in the molten metal. Therefore, we assumed that majority of the disturbance in the flow rate control was caused by a difference between the tilt angles at the beginning of pouring the liquid in the experiments and the controller design. Figure 9: Reference trajectories in flow rate control. (**a**) Reference trajectory of liquid height at pouring mouth, which is added to the flow rate control. (**b**) Trajectory of flow rate, which is derived by substituting the reference trajectory of liquid height, presented in (**a**), into Equation (6). At first, we conducted experiments on the flow rate control without any disturbance; because there was no disturbance during the experiments, pouring the liquid from the ladle began according to the controller design, with \(\theta=20\) deg. The experimental results are depicted in Figure 10.  
Figure 10a-c show the input signals to the servomotors of the \(\Theta\)-, \(X\)-, and \(Z\)-axes, respectively. Figure 10d-f show the tilt angle of the ladle on the \(\Theta\)-axis and the positions of the ladle on the \(X\)- and \(Z\)-axes, respectively. Figure 10g,h show the angular velocity of the tilting ladle and the acceleration of the ladle along the \(Z\)-axis, respectively. These are estimated by the steady-state Kalman filters. Figure 10i-k denote the liquid height at the pouring mouth, the flow rate of the outflow liquid, and the weight of the outflow liquid, respectively, which were simulated by the pouring model with the input signals shown in Figure 10a-c. In Figure 10a-k, the blue and orange lines represent the results obtained using the developed flow rate control based on the differential flatness and the conventional feed-forward flow rate control, respectively. In Figure 10i-k, the broken yellow lines represent the reference trajectories, which are also presented in Figure 9. Figure 10l denotes the weight of the outflow liquid, which was measured using the load cell. In this graph, the blue and orange lines represent the experimental results obtained using the developed and conventional flow rate controls, respectively. The yellow and purple lines represent the simulation results obtained by the developed  
Figure 10: Experimental results of flow rate control to pouring process without disturbance. (**a**) Input signal added to motor of the \(\Theta\)-axis. (**b**) Input signal added to motor of the \(X\)-axis. (**c**) Input signal added to motor of the \(Z\)-axis. (**d**) Angle of tilting ladle on \(\Theta\)-axis. (**e**) Position of ladle on \(X\)-axis. (**f**) Position of ladle on \(Z\)-axis. (**g**) Angular velocity on \(\Theta\)-axis, estimated by steady-state Kalman filter. (**h**) Acceleration on \(Z\)-axis, estimated by steady-state Kalman filter. (**i**) Liquid height at pouring mouth in simulation, using pouring model. (**j**) Flow rate of outflow liquid in simulation, using pouring model. (**k**) Weight of outflow liquid in simulation, using pouring model. (**l**) Weight of outflow liquid, measured by load cell. and conventional flow rate controls, respectively. As can be observed from Figure 101, because the simulation results were fitted with the experimental results, the pouring process in the tilting-ladle-type automatic pouring machine can be precisely represented by using the pouring model derived in this study.