## 5 Flow Rate Estimation by Decentralization of Kalman Filters
### Simulations  
Flow-rate estimation by DKFs was performed in the simulation for the tilting-ladle-type automatic pouring machine. The ladle used in this study is presented in Figure 8.  
The model parameters of the ladle were obtained from the three-dimensional data of the ladle. In Figure 9, the horizontal area \(A(\theta)\) m\({}^{2}\) on the pouring mouth and volume \(V_{s}(\theta)\) m\({}^{3}\) under the pouring mouth (presented in Figure 3) are indicated in (a) and (b), respectively. Figure 9c,d are the derivatives of (a) and (b) for the tilting angle, respectively. Figure 8: Geometry of ladle (inside dimension). The derivatives \(\partial A(\theta)/\partial\theta\) and \(\partial V_{s}(\theta)/\partial\theta\) can be derived as follows:  
\[\frac{\partial A(\theta)}{\partial\theta}=\frac{A(\theta(k+1))-A(\theta(k))}{ \Delta\theta},\ \ \frac{\partial V_{s}(\theta)}{\partial\theta}=\frac{V_{s}(\theta(k+1))-V_{s}( \theta(k))}{\Delta\theta}, \tag{32}\]  
where \(\Delta\theta\) is the sampling interval of the tilting angle, and \(k\) is the sampling number, which has the relation of \(\theta(k)=k\Delta\theta\). In this study, we used \(\Delta\theta=1.0\) deg. The volume \(V_{s}(\theta)\) is decreased with increasing the tilting angle \(\theta\). The horizontal area \(A(\theta)\) is increased with increasing the tilting angle until reaching the bottom of the ladle, \(\theta\leq 40\) deg. In the tilting angle over 40 deg, it is decreased with increasing the tilting angle. In particular, it is decreased as the quadratic curve by increasing the area of the channel to the pouring mouth. To estimate the angular velocity of the motor for tilting the ladle, the covariance of process noise \(Q_{t}\) and the covariance of observation noise \(R_{t}\) were assumed as follows:  
\[Q_{t} = \mathrm{diag}(1.0\times 10^{-3}\ \mathrm{deg}^{2}/\mathrm{s}^{2},1.0\times 10^{-7}\ \mathrm{deg}^{2}), \tag{33}\] \[R_{t} = 1.84\times 10^{-7}\ \mathrm{deg}^{2}. \tag{34}\]  
To estimate the acceleration of the movement of the ladle on the \(z\)-axis, the covariance of process noise \(Q_{z}\) and covariance of the observation noise \(R_{z}\) were assumed as follows:  
\[Q_{z} = \mathrm{diag}(1.0\times 10^{-8}\ \mathrm{m}^{2},1.0\times 10^{-4} \ \mathrm{m}^{2}/\mathrm{s}^{2}), \tag{35}\] \[R_{z} = 5.0\times 10^{-6}\ \mathrm{m}^{2}. \tag{36}\]  
Similarly, to estimate the state of the pouring process, the covariance of the process noise \(Q_{f}\) and the covariance of the observation noise \(R_{f}\) were assumed as follows:  
\[Q_{f} = \mathrm{diag}(25\ \mathrm{m}^{2},1.0\ \mathrm{kg}^{2},1.0\ \mathrm{kg}^{2})\times 10^{-11}, \tag{37}\] \[R_{f} = 5.0\times 10^{-4}\ \mathrm{kg}^{2}. \tag{38}\]  
In the simulations, the feedforward flow-rate control [22] was constructed to realize the desired flow rate, The reference flow rate applied to the flow rate control is presented in Figure 10.  
Figure 8:0 Model parameters of ladle. (**a**) Horizontal area on pouring mouth; (**b**) Volume under pouring mouth; (**c**) Derivative of horizontal area \(A\) with respect to angle \(\theta\); (**d**) Derivative of volume \(V_{s}\) with respect to angle \(\theta\). The simulation results obtained without any disturbances are presented in Figure 11, where (a) indicates the input command applied to the motor for tilting the ladle, and (b) and (c) indicate the angular velocity and the tilting angle of the ladle, respectively. The black solid lines are the results simulated using the motor model, and the chained lines are the results estimated using the SSKF. Furthermore, (d-f) indicate the liquid height, the pouring flow rate, and the outflow weight, respectively. The dotted magenta lines are the reference values, and the black solid lines are the results simulated using the pouring process model. The chained green lines are the results estimated using the EKF. In Figure 11b, the amplitude of angular velocity of the ladle is within 12 deg/s. Therefore, the splash of the liquid in the ladle could be suppressed into the level that has no influence on the pouring motion.