## 7 Experimental Verifications  
The flow-rate estimation method by DKFs was applied to the laboratory-type automatic pouring machine (see Figure 16), and the appropriateness of this flow-rate estimation approach was verified via the pouring experiments described in the current section. The conditions of the experiments were the same as for the simulations noted in Section 5.2. The flow-rate feedforward controller was also applied to control the flow rate based on the reference pattern (see Figure 10). In the first experiment, the flow-rate estimation was performed using an ideal condition, with no error between the ideal and the actual tilting angle at the start of the liquid outflow. The experimental results are presented in Figure 17, where (a) is the input command applied to the motor for tilting the ladle; (b) and (c) indicate the ladle's angular velocity and angle, respectively; and (d) and (e) indicate the liquid height in the pouring mouth of the ladle and the pouring flow rate, respectively. In Figure 17d,e, the dotted magenta lines indicate the reference patterns, and the dashed cyan lines indicate the simulation results obtained from the pouring process model by applying the results of (b) and (c). The chained green lines show the estimated results gained by using the EKF in the flow-rate estimation via DKFs. Figure 17f presents the weight of the outflow liquid. The black solid line is the weight measured by the load cell, and the remaining lines are shown in the same manner as (d) and (e). The state variables in the pouring motion were estimated in real time, as presented in Figure 17. To verify the appropriateness of the flow-rate estimation by DKFs, we applied the two other types of flow-rate estimation methods described in Section 6. Moreover, to verify the validity of the system parameters in DKFs, we also applied DKFs with the different covariance of process noise \(Q_{f}\) to Equation (37). The covariance of the process noise was given as follows:  
\[Q_{f} = \text{diag}(250\ \text{m}^{2},1.0\ \text{kg}^{2},1.0\ \text{kg}^{2}) \times 10^{-11}. \tag{46}\]  
The process noise in the liquid height on the pouring mouth was increased as compared with Equation (37). In Figure 18a-d, the black solid lines are the simulation result obtained from the pouring process model, and these results were the same as the dashed line in Figure 17e. The solid green lines in Figure 18a-d indicate the results of flow-rate estimations using the EKF with the covariance of the process noise in Equation (37) that in Equation (46), the derivative of the load cell data, and the visible camera, respectively. As presented in Figure 18a,b, the estimated results were similar to the simulated result. The result in Figure 18c shows that the flow rate exhibited a higher level of noise compared with  
Figure 16: Laboratory-type automatic pouring machine. the result estimated by the EKF. Additionally, the response was delayed by the low-pass filter applied for noise reduction. The flow rate of the outflow liquid from the ladle does not have a negative value (i.e., \(q\geq 0\) m\({}^{3}\)/s). However, it was confirmed that the flow rate during the back-tilting motion of the ladle from 16 s to 17 s indicates the negative value as shown in Figure 18c. This response was caused by the fact that the load cell data was influenced by the movement of the ladle on the \(z\)-axis as shown in Figure 6.  
Figure 17: Experimental results using automatic pouring machine (tilting angle error at the start of liquid outflow: \(+0\) deg). (**a**) Input command to motor for tilting ladle; (**b**) Angular velocity of tilting ladle; (**c**) Angle of tilting ladle; (**d**) Liquid height on pouring mouth; (**e**) Flow rate of outflow liquid from ladle; (**f**) Weight of outflow liquid from ladle. Figure 18: Comparison of flow-rate estimations in experiments with tilting angle error at the start of liquid outflow: \(+0\) deg.