## 7 Experimental Verifications
(**a**) Estimated flow rate by EKF with covariance as shown in Equation (37); (**b**) Estimated flow rate by EKF with covariance as shown in Equation (46); (**c**) Estimated flow rate by differentiating weight of outflow liquid measured by load cell; (**d**) Estimated flow rate by visible camera. The noise in the flow rate estimated by the visible camera (as presented in Figure 18d) was smaller than that estimated by differentiating the load cell data (see Figure 18c). Hence, we confirmed that the flow rate could be estimated roughly using the visible camera. The real-time flow-rate estimation is required to process within 0.02 s for realizing the high-precision real-time flow rate control [32]. The processing times in the flow-rate estimations using DKFs and the derivative of the load cell data were within 0.02 s. The processing time in the flow-rate estimation using the visible camera was 0.06 s by taking the time for the frame rate of the camera and the image processing. Figure 19 presents the experimental results for the pouring conditions alongside the tilting angle error at the start of the liquid outflow which was +3 deg. Figure 20 presents the estimated results of the flow rate for comparing the three types of flow-rate  
We compared quantitatively the accuracies of each flow-rate estimation. Since the simulation results of the flow rate were generated by the pouring process model with the experimental conditions, we consider that the simulation results of the flow rate represented the actual flow rate in the experiments faithfully. The accuracies of the flow rate estimations were evaluated by the integral absolute error (IAE) of the estimated and the simulated flow rates as follows:  
\[IAE = \sum_{i=0}^{N}|q_{esti}-q_{simi}|\Delta T, \tag{47}\]  
where \(i\) is the sampling number and \(N\) is the total number of the sampling in the estimation. \(\Delta T\) is the sampling interval of the flow rate estimations. \(q_{est}\) and \(q_{simi}\) are the estimated and the simulated flow rates shown in Figures 18 and 20. Table 1 presents the results of IAE. In Table 1, the IAEs in the ideal condition with no error between the ideal and the actual tilting angles at the start of the liquid outflow and the condition with +3 deg error between the ideal and the actual tilting angle at the start of the liquid outflow were shown. Furthermore, we also compared quantitatively the amounts of noises in the estimated flow rates. The amounts of the noises were evaluated by the total variation (TV) [40] of the estimated flow rates as follows:  
\[TV = \sum_{i=1}^{N}|q_{esti+1}-q_{esti}|, \tag{48}\]  
Table 2 presents the results of TV. This table is shown in the same manner as in Table 1. However, TV as shown in Equation (48) includes not only the amounts of the noises but also the variation of the set-point changes in the flow rate. The set points in the flow rate are changed monotonically as shown in Figure 10. Therefore, we used the modified TV (mTV) [41] for eliminating the variation of the set-point changes from TV as  
\[Q_{f} = \text{diag}(250\ \text{m}^{2},1.0\ \text{kg}^{2},1.0\ \text{kg}^{2}) \times 10^{-11}. \tag{46}\]0 Comparison of flow-rate estimations in experiments with tilting angle error at the start of liquid outflow: \(+3\) deg. (**a**) Estimated flow rate by EKF with covariance as shown in Equation (37); (**b**) Estimated flow rate by EKF with covariance as shown in Equation (46); (**c**) Estimated flow rate by differentiating weight of outflow liquid measured by load cell; (**d**) Estimated flow rate by visible camera. \[Q_{f} = \text{diag}(250\ \text{m}^{2},1.0\ \text{kg}^{2},1.0\ \text{kg}^{2}) \times 10^{-11}. \tag{46}\]1  
where \(q_{simn}\) is the extreme point of the set-point changes which has the shape of one pulse. \(q_{simN}\) and \(q_{sim0}\) are the final and initial values of the set-points in the flow rate, respectively. These parameters were obtained from the simulation result of the flow rate which does not have the noises.