## 3 Mathematical Models of Pouring Motion
### The Pouring Process Model  
The pouring process model \(P_{f}\) in Figure 2 represents the volume balance of the topmost liquid volume in the ladle, which can be shown as the input-output relation from angular velocity \(\omega\) to flow rate \(q\) m\({}^{3}\)/s of the outflow liquid. The cross section of the  
Figure 2: Block diagram of motorized pouring motion.  
pouring process at tilting angle \(\theta\) is presented in Figure 3, where the volume balance of the topmost liquid volume in the ladle can be given as follows:  
\[\frac{dV_{r}(t)}{dt} = -q(t)-\frac{\partial V_{s}(\theta(t))}{\partial\theta}\omega(t),\ \ (V_{r}\geq 0,\ \theta\geq\theta_{s}), \tag{3}\]  
where \(V_{r}\) m\({}^{3}\) is the liquid volume over the pouring mouth, \(V_{s}\) m\({}^{3}\) is the liquid volume under the pouring mouth, and \(\theta_{s}\) deg is the angle of the ladle at the start of the liquid outflow. Accordingly, volume \(V_{r}\) m\({}^{3}\) can be approximated as follows:  
\[V_{r}(t) \approx A(\theta(t))h(t),\ \ (h(t)\geq 0), \tag{4}\]  
where \(A\) m\({}^{2}\) is the upper surface of the liquid in the ladle, and \(h\) m is the height of the liquid over the pouring mouth. As presented in Figure 3, surface \(A\) is changed by tilting angle \(\theta\) deg of the ladle.  
Using [PERSON]'s principle, flow rate \(q\) at liquid height \(h\) is given as follows:  
\[q(t)=c\int_{0}^{h(t)}L_{f}(h_{a})\sqrt{2 gh_{b}}dh_{b},\ \ (0<c\leq 1,\ h_{a}=h(t)-h_{b}), \tag{5}\]  
where \(L_{f}\) m is the width of the pouring mouth at height \(h_{a}\) m from the bottom edge of the pouring mouth (see Figure 4a), \(h_{b}\) m is the depth at the pouring mouth from the surface of the liquid in the ladle, \(g\) m/s\({}^{2}\) is the acceleration of gravity, and \(c\) is the flow-rate coefficient. Figure 4a and Equation (5) show the relation between the flow rate and the liquid height in the pouring mouth with the generalization of cross-sectional shape. In more detail, since the geometry of the pouring mouth is represented on the basis of the bottom edge of the pouring mouth, the width \(L_{f}\) is defined by the function of the height \(h_{a}\). On the one hand, the flow velocity \(\sqrt{2 gh_{b}}\) depends on the depth \(h_{b}\) from the surface of the liquid. In design of the flow-rate control, the relation in Equation (5) can be implemented by the interpolation method [22] or the look-up table [39]. Accordingly, the flow rate \(q(t)\) can be calculated by giving the liquid height \(h(t)\) to the implementation of Equation (5) each sampling time. In the case that the cross-sectional shape of the pouring mouth is rectangle as shown in Figure 4b, the flow rate \(q\) shown in Equation (5) can be simplified as follows:  
\[q(t)=\frac{2}{3}cL_{f}\sqrt{2 gh(t)^{3}},\ \ (0<c\leq 1). \tag{6}\]  
In this study, since the ladle with the rectangular pouring mouth was used in the experiment, Equation (6) was applied as the relation between the flow rate and the liquid height on the pouring mouth. The flow-rate coefficient \(c\) can be identified by fitting the simulation result to the experimental result of the weight of the outflow liquid measured by the load cell. In this study, we obtained \(c=0.75\).  
From Equations (3), (4) and (6), the dynamics of liquid height over the pouring mouth in the pouring process were derived as follows:  
\[\frac{dh(t)}{dt}=-\frac{q(h(t))}{A(\theta(t))}-\frac{1}{A(\theta(t))}\bigg{(} \frac{\partial V_{s}(\theta(t))}{\partial\theta(t)}+\frac{\partial A(\theta(t ))}{\partial\theta(t)}h(t)\bigg{)}\omega(t),\ (h\geq 0,\ \theta\geq\theta_{s}). \tag{7}\]  
The actual weight \(W\) kg of the outflow liquid can be represented as:  
\[\frac{dW(t)}{dt}=\rho q(t), \tag{8}\]  
where \(\rho\) kg/m\({}^{3}\) is the density of the liquid. In this study, since we used water as a target liquid, \(\rho=1.0\times 10^{3}\) kg/m\({}^{3}\) was applied.