# Implementation and Experimental Verification of Flow Rate Control Based on Differential Flatness in a Tilting-Ladle-Type Automatic Pouring Machine

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## Abstract

**:**

## 1. Introduction

## 2. Tilting-Ladle-Type Automatic Pouring Machine

## 3. Modeling of the Tilting-Ladle-Type Automatic Pouring Machine

#### 3.1. Motor Model on the $\Theta $-Axis

#### 3.2. Motor Models on the X- and Z-Axes

#### 3.3. Pouring Process Model

^{3}denotes the liquid volume under the pouring mouth, and $A\left(\theta \right)$ m

^{2}denotes the area of the liquid plane, which is horizontal to the pouring mouth. The volume ${V}_{s}\left(\theta \right)$ and area $A\left(\theta \right)$ are dependent on the tilt angle $\theta $. ${V}_{r}$ m

^{3}denotes the volume over the area $A\left(\theta \right)$, and h m denotes the liquid height at the pouring mouth. When the liquid level is above the lip of the pouring mouth, it flows out of the ladle at the flow rate q m

^{3}/s. The pouring process model, ${P}_{F}$, can be described by the mass balance of the liquid in the ladle and Bernoulli’s theorem, as

^{2}is the acceleration of gravity.

^{3}denotes the density of the liquid in the ladle.

#### 3.4. Load Cell Model

^{2}denote the weight and acceleration of the part moving along the Z-axis, respectively. The acceleration, ${a}_{z}$, can be derived as ${a}_{z}={\dot{v}}_{z}$ from Equation (3), and ${w}_{z}^{\prime}$ kg denotes the weight which appears in the load cell data during the vertical motion of the ladle. Moreover, we assume that the vibration of the load cell is small because the natural frequency of the load cell is much higher than the dynamics of the pouring motion. Therefore, the dynamics, ${P}_{L}$, of the load cell are modeled simply by a linear first-order system:

## 4. Design of the Pouring Flow Rate Control Based on Differential Flatness

- ${K}_{t}\ne 0,\phantom{\rule{3.33333pt}{0ex}}\frac{\partial {V}_{s}\left(\theta \right)}{\partial \theta}+\frac{\partial A\left(\theta \right)}{\partial \theta}F\ne 0$;
- ${\beta}_{0,1}>0$;
- the reference trajectory is set as ${F}^{\ast}=0$, within finite time;
- the control input is switched to ${u}_{t}=0$ at ${F}^{\ast}=0$;
- the control input is within the limitations ${u}_{tlower}\le {u}_{t}\le {u}_{tupper}$, where ${u}_{tlower}$ and ${u}_{tupper}$ are the lower and the upper bounds in the control input, respectively.

^{3}/s denotes the initial flow rate and ${q}_{f}^{\ast}$ m

^{3}/s denotes the terminal flow rate of transition. We assume that the relation between the flow rate q and liquid height h at pouring mouth is uniquely derived.

## 5. Implementation of the Flow Rate Control System

#### 5.1. Synchronous Control

#### 5.2. State Estimation in an Automatic Pouring Machine

#### 5.3. Model Parameter Identification

^{3}, as water is considered to be the target liquid in this study.

#### 5.4. Control Parameters

#### 5.5. Design of the Reference Trajectory

^{3}/s. The reference trajectory of the liquid height at the pouring mouth, by which the target flow rate can be achieved, is depicted in Figure 9.

^{3}/s, and the flow rate from 12–15 s stays at ${q}_{f2}^{\ast}=1.50\times {10}^{-4}$ m

^{3}/s.

## 6. Experimental Verification

## 7. Conclusions and Future Works

- From the viewpoint of practical application, a SISO-nonlinear-type controller design based on differential flatness was applied, to ensure the flow rate control of the automatic pouring machine;
- while implementing the flow rate control using the feedback control scheme, it was necessary to measure the states of the automatic pouring machine—however, it is difficult to directly measure the states of high-temperature molten metal. Therefore, in this study, the Kalman filter approach was applied for estimating the state variables of the automatic pouring machine;
- the steady-state and extended Kalman filters were decomposed for simple construction of the state estimate of the automatic pouring machine;
- in the experiments related to the pouring process without disturbance, both the feed-forward flow rate control based on the inverse pouring model, and the flow rate control based on differential flatness, achieved a flow rate by which the outflow liquid precisely tracked with the reference trajectory: We conclude that the pouring model described in this study could precisely represent the actual pouring process in the automatic pouring machine;
- in the experiments on a pouring process with disturbance, the tracking performance of the flow rate in the pouring machine could be improved by implementing flow rate control based on differential flatness; and
- a majority of the tracking errors of the flow rate control based on differential flatness were caused by the estimation errors of the extended Kalman filter.

- To obtain high tracking performance, the state estimation approach of the pouring machine should be improved in future studies;
- in the developed flow rate control, the flow rate of the outflow liquid was indirectly controlled, based on the liquid height at the pouring mouth. If any disturbances were observed in the relation between the flow rate and the liquid height at the pouring mouth, the tracking performance of the flow rate control may have degraded. Further, direct flow rate control must be achieved to construct a high-precision automatic pouring machine; and
- in practical pouring processes, the characteristics of the pouring material are variable with temperature, the added substance, and so on. Therefore, a control approach with computational intelligence, such that the controller can be adapted autonomously to the pouring environment, should be developed in the future work.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

SISO | Single input and single output. |

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**Figure 1.**Photos of tilting-ladle-type automatic pouring machines. (

**a**) An automatic pouring machine that is actually applied in a casting plant, and (

**b**) the automatic pouring machine that has been developed in the present laboratory of the authors.

**Figure 3.**Signal diagram for the tilting-ladle-type automatic pouring machine. (

**a**) Drive system of the $\Theta $-axis, with pouring process and load cell; (

**b**) drive system of the Z-axis; and (

**c**) drive system of the X-axis.

**Figure 4.**Geometric parameters of pouring process. (

**a**) Parameters related to the ladle shape, and (

**b**) parameters related to the pouring mouth.

**Figure 6.**Signal diagram of the entire control system for applying flow rate control to the tilting-ladle-type automatic pouring machine.

**Figure 7.**Illustrations of ladle motions in the X- and Z-directions, controlled synchronously to tilt the ladle in the $\Theta $-direction: (

**a**) Ladle motion without synchronous control, and (

**b**) ladle motion with synchronous control.

**Figure 8.**Model parameters related to the ladle shape. (

**a**) Area of liquid plane which is horizontally located on the pouring mouth, $A\left(\theta \right)$. (

**b**) Partial derivative of the area $A\left(\theta \right)$, with respect to the angle $\theta $. (

**c**) Liquid volume under the tip of the pouring mouth, ${V}_{s}\left(\theta \right)$. (

**d**) Partial derivative of area ${V}_{s}\left(\theta \right)$, with respect to the angle $\theta $. (

**e**) Ideal flow rate ${q}_{id}$ of the outflow liquid, with respect to the liquid height h at the pouring mouth.

**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).

**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.

**Figure 11.**Illustration of the disturbance applied in the experiments, which was caused by a difference between the tilt angles at the beginning of pouring the liquid in the experiments and the controller design.

**Figure 12.**Experimental results of flow rate control to pouring process with 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.

**Figure 13.**Comparative results of liquid height at pouring mouth. (

**a**) Liquid height in developed flow rate control. (

**b**) Liquid height in conventional feed-forward flow rate control.

**Table 1.**Specifications of drive systems in the tilting-ladle-type automatic pouring machine developed by the authors.

X-Direction | Z-Direction | $\mathbf{\Theta}$-Direction | |
---|---|---|---|

Type of motor | Brushless DC motor | Brushless DC motor | Brushless DC motor |

Control mode in servo amplifier | Velocity mode | Velocity mode | Velocity mode |

Motion range | 0–0.3 m | 0–0.4 m | 0–70 deg |

Maximum torque of motor | 1.22 Nm | 1.22 Nm | 11.5 Nm |

Pitch of ball screw | 0.02 m | 0.02 m | - |

Maximum velocity of motor | 0.77 m/s | 0.38 m/s | 371 deg/s |

X-Axis | Z-Axis | $\mathbf{\Theta}$-Axis | |
---|---|---|---|

Gain, K | 1.00 | 1.00 | 1.00 |

Time constant, T | 0.050 s | 0.050 s | 0.022 s |

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## Share and Cite

**MDPI and ACS Style**

Noda, Y.; Sueki, Y.
Implementation and Experimental Verification of Flow Rate Control Based on Differential Flatness in a Tilting-Ladle-Type Automatic Pouring Machine. *Appl. Sci.* **2019**, *9*, 1978.
https://doi.org/10.3390/app9101978

**AMA Style**

Noda Y, Sueki Y.
Implementation and Experimental Verification of Flow Rate Control Based on Differential Flatness in a Tilting-Ladle-Type Automatic Pouring Machine. *Applied Sciences*. 2019; 9(10):1978.
https://doi.org/10.3390/app9101978

**Chicago/Turabian Style**

Noda, Yoshiyuki, and Yuta Sueki.
2019. "Implementation and Experimental Verification of Flow Rate Control Based on Differential Flatness in a Tilting-Ladle-Type Automatic Pouring Machine" *Applied Sciences* 9, no. 10: 1978.
https://doi.org/10.3390/app9101978