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Motion control of liquid container considering an inclined transfer path
Control Engineering Practice 10 (2002) 465–472
Motion control of liquid container considering an inclined transfer path Ken’ichi Yano*, Shimpei Higashikawa, Kazuhiko Terashima Department of Production Systems Engineering, Toyohashi University of Technology, Hibarigaoka 1-1, Tempaku-cho, Toyohashi, 441-8580, Japan Received 30 March 2001; accepted 8 June 2001
Abstract The present paper is concerned with advanced control of liquid container transfer along an inclined transfer path, paying special attention to the suppression of sloshing (liquid vibration). To suppress sloshing in the container during acceleration and deceleration along the inclined transfer path, we present a method to actively control the container’s rotational motion. This system is useful for saving space in factories and optimizing foundry processes. The effectiveness of the proposed system is demonstrated through simulations and experiments. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Manufacturing processes; Process control; Factory automation; Control applications; HN control; Modelling
1. Introduction In this paper, control topics are focused on the liquid container transfer systems in the casting and steel industries. In practice, molten metal is generally moved from furnace to casting area by means of a self-transfertype cart such as a crane, a forklift or an automotive cart, after it has been poured into a ladle (Lindsay, 1983; Lerner & Laukhin, 2000). In automatic pouring processes, the transfer of molten metal includes the transfer of the ladle along molds as well as the transfer of the molds themselves. In these cases, it is important to shorten the total operational time in order to improve productivity. However, high-speed transfer causes the molten metal to slosh in both the ladle and the molds. This sloshing phenomenon is not only dangerous because of the overflow, but it also deteriorates the quality due to contamination and excessive cooling of the molten metal. Further, in automatic pouring, sloshing disturbs the positioning of the dropping point of the fluid (Lindsay, 1983; Burditt & Bralower, 1989; Terashima & Yano, 1999). Therefore, it is necessary to
*Corresponding author. Tel.: +81-532-44-6699; fax: +81-532-446690. E-mail addresses: [email protected] (K. Yano), [email protected] (K. Terashima).
construct a liquid container transfer system that will suppress sloshing as well as facilitate high-speed transfer. Many studies have been published with respect to the suppression of sloshing (Housner, 1963; Abramson, Chu, & Kana, 1966; Barron & Chng, 1989). However, few studies have focused on both sloshing suppression and high-speed transfer. In recent studies, modelling and optimal control for several engineering specifications using LQI control for the transfer of rectangular containers, taking into consideration the suppression of sloshing, are presented (Hamaguchi, Terashima, & Nomura, 1994). To construct a high-speed transfer system for liquid containers that reduces endpoint residual vibration and has the robustness to withstand change in the static liquid level, an HN feedback control system was applied to this process (Yano, Yoshida, Hamaguchi, & Terashima, 1996; Yano & Terashima, 2001). Also presented is an active control method that takes into account the rotational motion of the container in addition to the linear transfer motion; this method can completely suppress sloshing throughout the entire transfer process (Yano & Terashima, 2001). Feddema et al. (1997) also presented the transfer system by using a robot arm, and Grundelius and Rernhardsson (1999) and Grundelius (2000) applied the system to an industrial packaging machine. Recently, the development of space-saving factories and the optimization of foundry processes have been in high demand from the casting and steel industries. To
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answer this demand, research on analysis and control of the various transfer paths of liquid containers has become necessary. With respect to the shapes and transfer paths of the containers, historically, either narrow or three-dimensional rectangular containers that follow a straight path have been studied. By contrast, in recent years, the transfer of cylindrical containers and transfers via curved paths have been of increasing interest. The optimum motion control system for an automotive, cart-based, cylindrical container was studied, with the main focus being the suppression of sloshing on a curved transfer track (Terashima, Hamaguchi, & Kaneshige, 1995; Hamaguchi, Yamamoto, & Terashima, 1997b). Furthermore, the optimum shape for such a container was studied by minimizing the performance index and considering both the resonant frequency of first-mode sloshing and the volume of liquid (Hamaguchi, Ronda, & Terashima, 1997a). However, no studies have reported on sloshing suppression along the inclined transfer paths, in spite of the importance of such research to factories that rely on large numbers of inclined paths. Therefore, in this paper, our purpose is to establish a liquid container transfer system that effectively reduces sloshing along an inclined transfer path. To avoid overflow and sloshing during such a transfer, the sloshing model comprises two degrees of freedom (both linear and rotational container motion). Furthermore, active control using additional rotational motion is achieved by HN control. The proposed new system will also be useful for eliminating sloshing during acceleration and deceleration intervals on the inclined transfer path. The effectiveness of the proposed system is demonstrated through simulations and experiments.
2. Experimental apparatus The elimination of sloshing during acceleration and deceleration intervals is not possible in mechanisms for linearly moving containers with only a single degree of freedom (Hamaguchi et al., 1994; Yano et al., 1996; Yano & Terashima, 2001). Hence, a transferring machine with two degrees of freedom was proposed in our previous paper by adding the function of rotational motion to the container (Yano & Terashima, 2001). In the present paper, a transferring machine with two degrees of freedom is applied to an inclined transfer path. A schematic diagram of the new experimental apparatus is shown in Fig. 1. The size of the three-dimensional rectangular container is 0.14 m0.14 m0.3 m (lengthwidthheight). A straight transfer path with a constant slope is used in this study, and the angle of the inclined transfer path is set to 5.01: The container is moved using a DC
Fig. 1. Schematic diagram of liquid container transfer system with inclined transfer path.
servomotor with a timing belt, and the velocity and position of the container are controlled by means of the voltage applied to the motor. With respect to the rotational motion, the container is rotated by another DC servomotor using a timing belt. The position of the container is detected by a potentiometer fitted to a pulley, and the angle of the container is detected by an encoder fitted to a rotational axis. Displacement of the liquid level is detected through changes in resistance between two stainless steel electrodes. The electrodes are installed 5 103 m away from the side wall of the container. This position was used because the displacement of the liquid level in the first-order mode sloshing can be most distinctly observed near the side walls of the container, and because the sensor had to be slightly detached from the side wall in order to avoid the effects of the wall. Considering the similarity law in fluid dynamics, water was adopted as the object liquid in the experiment, because the Reynolds number of water at normal temperature is almost the same as that of molten iron metal or molten aluminum metal at high temperature. For example, the kinematic viscosities of iron molten metal at 1350 and 1400 K are 1.365106 and 1.237106 mPa s, respectively, while that of water is 1.0 106 mPa s at 293 K. Therefore, the fluid behavior of the molten metal can be fairly accurately predicted by the behavior of water. It enhances the design safety if the control design is conducted using a fluid with lower damping characteristics, and for this reason water was used as the target liquid in this study.
3. Modelling 3.1. Sloshing model In the case of a straight transfer path, sloshing in a three-dimensional rectangular container can be
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approximated as a two-dimensional phenomenon as long as there are no sudden changes in the acceleration (Hamaguchi et al., 1994). Therefore, in the present study, the sloshing phenomenon is described by a pendulum-type sloshing model (Hamaguchi et al., 1994; Yano & Terashima, 2001). The rotational motion is added to the pendulum-type sloshing model, and a new model is proposed as a sloshing model that considers both the inclined transfer path and the rotational motion. The principles of this model are shown in Fig. 2. By considering the moment balance around the fulcrum of a pendulum, the following equation is derived: J
d2 y cdðy ZÞ c cos2 y mgc sin y ¼ c dt2 dt þ mxc . cos f cos y mxc . sin fsin y
d2 Z cos y ð1Þ dt2 where J is the moment of inertia (J ¼ mc2 ), O the fulcrum of a pendulum, Or the center of revolution, D the distance between the center of the revolution and the center of gravity, y the angle of the pendulum model from horizontal, Z the angle of the container, Z. the angular acceleration of the container rotation, f the angle of the transfer path, and x. is the acceleration in linear transfer applied to the container. This model contains an equivalent coefficient c of viscosity, considering both the viscosity of the liquid and the friction between the liquid and the wall. It also contains the mass m of the liquid, the gravitational acceleration g; and the equivalent length c of the mcD
pendulum. The parameters c and c for the above model were determined by experiments, where c was also theoretically obtained based on the natural frequency determined by the perfect fluid theory (Hamaguchi et al., 1994). Linearization of Eq. (1) around yC0 gives Eq. (2). c 1 1 D y. ¼ ðy’ Z’ Þ ðg þ x. sin fÞy þ x. cos f Z. : ð2Þ m c c c In Eq. (2), the forces Dc Z. ; added by the rotational motion of the container, and xy . sin f; added by the transfer motion of the container, are neglected, because these values are assumed to be very small. Hence, Eq. (3) is given as a final approximate sloshing model. c g 1 y. ¼ ðy’ Z’ Þ y þ x. cos f: ð3Þ m c c 3.2. System model In the DC servomotor, a transfer function Gm ðsÞ from the input voltage e1 ðtÞ to the velocity v1 ðtÞ of the container transfer is given as a first-order lag model, shown in Eq. (4). V1 ðsÞ Km ¼ ; ð4Þ Gm ðsÞ ¼ E1 ðsÞ Tm s þ 1 where Tm is the time constant and Km is the gain. These parameters were obtained by adding a step-wise input to the apparatus and then changing the output data from the position of the container to the velocity, as shown in Table 1. A transfer function Gr ðsÞ from the input voltage e2 ðtÞ to the angular velocity v2 ðtÞ of the container rotation is given as a second-order lag model, as shown in Eq. (5). The model parameters were obtained by adding a couple of sinusoidal-wave inputs to the apparatus, is also shown in Table 1. Gr ðsÞ ¼
V2 ðsÞ Kr o2n ¼ 2 : E2 ðsÞ s þ 2zon s þ o2n
ð5Þ
Then, a linear vector state space equation in Eq. (6) is obtained from Eqs. (3) – (5) where u1 ðtÞ is the control input of a linear transfer, u2 ðtÞ the input of rotational motion and h the displacement of water level at the wall
Table 1 Motor parameters Parameter
Fig. 2. The sloshing model considering an inclined transfer path and rotational motion.
Motor gain, Km Time constant, Tm Motor gain, Kr Damping factor, z Natural angular frequency, on
Value 0.0912 0.0227 0.5807 0.3778 41.446
Unit m/s V s rad/s V F rad/s
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of the container, approximately represented by h ¼ Ly: This model is controllable and observable. x’ ¼ Ax þ Bu;
y ¼ Cx
where 2
0
0
0
0
0
c m
0
0
0 0
1 0
0 1
0 0
0 0
o2n 0
2zon 0
0 0
0
0
0
0
0
1
6 g 6 c mc 6 6 0 6 0 6 A¼6 0 0 6 6 6 0 0 6 6 0 0 4 0 0 " B¼
0
Km cTm
0
0
0
Km Tm
0
0
0
Kr o2n
0
0
C¼
x¼ y
y¼ h
L 0
0 0
0 0
0 1
y’
Z Z’
Z.
x
x’
x
T
;
u ¼ u1
u2
0 0
3
7 cT1m cosf 7 7 7 0 7 7 7 0 7 7 7 0 7 7 1 5 T1m
0
0 0
L 0
0
cos f
0 "
ð6Þ
#T
#
Fig. 3. Validity of the sloshing model and motor model.
T
T
These sloshing parameters were also identified by adding a couple of sinusoidal-wave inputs to the apparatus. The results of parameter identification are shown in Table 2. The validity of the sloshing model is demonstrated in Fig. 3. As input voltages, sinusoidal waves are applied to the case in which liquid level is hs ¼ 0:14 m, where these inputs are amplitude 2 V, frequency 0.3 Hz for linear transportation and amplitude 0.5 V, and frequency 0.5 Hz for rotation. The figures show the following: (a) the control input of the transfer u1 and the rotation u2 ; (b) sloshing in the simulation result by the nonlinear model expressed in Eq. (1), the simulation result by the linear model expressed in Eq. (3) and the experimental result; (c) the position of the container; and (d) the angle of the container. The experimental results are shown by solid lines and the simulation results are shown by dashed lines. The results show that the responses of the model and the real plant agree strongly. Therefore, the validity of the proposed sloshing model is demonstrated. At the same time, since the results of the nonlinear model expressed in Eq. (1) and of the linear model expressed in Eq. (3) are very similar, as shown in Fig. 3(b), the validity of the linearization is also demonstrated. Although some random vibration is found in the experimental result in Fig. 3(b), this is a hardware problem caused by a dead zone due to the backlash of the rotational motion.
Table 2 Model parameters for the sloshing model Parameter
Value
Unit
Liquid level, hs Angle of inclined path, f Length of the pendulum, c Coefficient of viscosity, c Mass of liquid, m Distance between Or and G; D
0.140 5.0 0.0442 1.88 2.744 0.02
m deg m N s/kg kg m
4. Robust controller design When a liquid container is transferred on an inclined transfer path, a transferring machine with one degree of freedom may cause overflow and contamination of the molten metal in terms of only acceleration control for linear container transfer. Therefore, a HN controller with two control inputs was applied to the design of the container transfer so that the free surface of the liquid could be kept parallel to the bottom plate of the container, thus keeping the difference between the angle of the pendulum and the angle of the container at zero throughout the transfer interval. This system was also designed to provide the robustness necessary to withstand the measurement noise and the higher-mode sloshing. A HN loop shaping method (Zames & Francis, 1983; Doyle, Glover, Khargonekar, & Francis 1989) was applied to the present liquid container transfer system in order to guarantee such robustness. The liquid level was
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hs ¼ 0:14 m, and an augmented closed loop system was constructed as shown in Fig. 4. These specifications are represented in the following equation: WR R ð7Þ WN T og: WS S
469
where WR11 ¼
9:6 102 ð5s þ 1Þð10s þ 1Þ ; ðs þ 40Þðs þ 80Þ
WR12 ¼
2:7 103 ðs þ 1Þð10s þ 1Þ : ðs þ 60Þðs þ 15Þ
N
First of all, in order to enhance the robustness for the higher-mode sloshing, the uncertainty weight WR is given to be proper and monotonically increased, as shown in Fig. 5. Its value is shown in Eq. (8). " # WR11 WR12 WR ¼ ; ð8Þ 0 0
Fig. 4. Augmented closed loop system for HN control design.
Next, WN is added to negate the effect of measurement noise, as shown in Eq. (9), based on the actual noise characteristics of measurement of both h and x: " # WN11 0 WN ¼ ; ð9Þ 0 WN22 where WN11 ¼
5500ð2s þ 1Þ2 ; ðs þ 100Þ2
WN22 ¼
1:1 103 ð10s þ 125Þ : s þ 125
Lastly, the sensitivity weight WS is considered. Each diagonal element of WS is given as a different value, as shown in Eq. (10), because this system is a regulator system for the suppression of sloshing and also a servosystem for the position of the container. In particular, the weighting function for the position of the container is set to be a high-gain, at around o ¼ 0; in order that it can operate as a servo-system. " # WS11 0 WS ¼ ; ð10Þ 0 WS22 where
Fig. 5. Frequency responses of the sloshing model and the uncertainty weight WR :
WS11 ¼
ar ; sþ1
WS22 ¼
as : s þ 104
There is, however, a trade-off between these two features, because the transfer part features one input, but two outputs. So, in order to make a fair comparison with the conventional results, which are the case of f ¼ 0 and the one-degree-of-freedom system (Yano & Terashima, 2001), a servo-gain as of WS was provided to yield the same transfer time as the conventional results. A regulator gain ar will be given hereafter. As a result, ar and as were set to 15,000 and 50, respectively. A MATLAB Robust Control Toolbox was used in the present control design. As a result of synthesis, a 16th-order controller was obtained, as shown in Eq. (11), where the optimum value of g was 1.105104 : " # Ku1 eh ðsÞ Ku2 eh ðsÞ KðsÞ ¼ ; ð11Þ Ku1 ex ðsÞ Ku2 ex ðsÞ
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Fig. 7. Simulation results on the inclined transfer path. Fig. 6. Frequency response of the controller obtained by using the HN control theory.
where " u¼
u1 u2
#
" ¼
Ku1 eh ðsÞ
Ku2 eh ðsÞ
Ku1 ex ðsÞ Ku2 ex ðsÞ
#"
eh ex
# :
ð12Þ
In the equations, u1 is the input for the container transfer, u2 the input for the container rotation, eh the displacement error of water levels, and ex the position error of the container. The frequency response of the controller is shown in Fig. 6. From these results, the gains from both the position error ex of the container and the displacement error eh of water levels to the control input u1 are locally low at approximately the resonance frequency of sloshing. Therefore, the controller for u1 is a notchfilter type, which does not excite sloshing during the transfer. On the other hand, the gain from the position error ex of the container to the control input u2 for rotation is locally high, at approximately the resonance frequency of sloshing. Therefore, in this system the container is actively rotated to suppress the sloshing generated by acceleration.
method of Fletcher and Reeves (1964), combined with a clipping-off technique (Quintana & Davison, 1974). In the calculation, the angle of the inclined transfer path is assumed to be f ¼ 0 and the position of the container is assumed to be controlled by only the control input u1 : An optimal trajectory was calculated in the following way. Firstly, a terminal condition, as shown in Eq. (13), is imposed to achieve both the target velocity (0.5 m/s) and the suppression of sloshing at the end-time of the acceleration. An optimal control problem that considers the restriction of the input magnitude of the actuator in the experimental transfer system can be formulated as Eq. (14), where the cost function J is formulated as a quadratic form in order to minimize the square norm of state error at the end-time of the acceleration interval, and where tA½to ; tf is the interval of acceleration. ’ T ¼ ð0:5; 0; 0ÞT ; xt ¼ ðx; ð13Þ ’ y; yÞ f
mina s:t:
J ¼ ðxtf xðtf ÞÞT Wðxtf xðtf ÞÞ 2 3 2 3 a x’ d6 7 6 7 ð14Þ y’ 4y5 ¼ 4 5; dt g c ’ 1 ’y y cosy sin y þ a cos y m
5. Control results The simulation results are shown in Fig. 7, where the transportation distance is 1 m, the liquid level is hs ¼ 0:14 m, and the sampling time is 0.01 s. In this study, in order to achieve a high-speed transfer system as well as the reduction of endpoint residual vibration, a reasonable reference trajectory for the position control of the container was determined using the optimization
c
c
’ T ¼ ð0; 0; 0ÞT ; xðt0 Þ ¼ ðx; ’ y; yÞ jajp2:0 m=s2 : Secondly, a nonlinear optimal theory based on the Fletcher and Reeves method, combined with a clippingoff technique, is applied to solve the optimal control problem with constrained inputs. The interval of deceleration is known to be able to turn over the input of acceleration. Finally, an optimal reference trajectory
K. Yano et al. / Control Engineering Practice 10 (2002) 465–472
is obtained by integrating the velocity curve for the total interval. This method gives a minimum time of 0.34 s for the acceleration interval, where W ¼ diag½106 ; 106 ; 106 : The effectiveness of this optimal reference trajectory was shown in the literature (Yano & Terashima, 2001). In the simulations, the 1 m transfer time of this active transfer is intended to be the same as that of the conventional transfer (Yano & Terashima, 2001) in the case of f ¼ 0 using a one-degree-of-freedom system without the rotation of the container. In this paper, the straight path transfer of f ¼ 0 with one degree of freedom is called the normal transfer. The container is actively rotated in order to avoid overflow and sloshing, including during acceleration and deceleration. As a result, sloshing is almost completely suppressed throughout the entire transfer. To show an idea of this control system, the illustration of the liquid container transfer control is shown in Fig. 8, where (a) shows the start point; (b) the acceleration interval; (c) the constant velocity interval; (d) the deceleration interval and (e) the end of the transfer. As seen from this figure, the container is actively rotated to suppress the sloshing during acceleration (b) and deceleration (d). At other intervals, the container is always tilted to keep it at the same angle with the slope for the inclined ground in order to avoid overflow and sloshing. The experiment was conducted under the same conditions as the simulation. As seen in Fig. 9, this control system almost satisfies requirements such that the water level does not oscillate throughout the entire transfer interval on the inclined transfer path, i.e., the acceleration, constant-velocity and deceleration intervals, as well as after arrival. Furthermore, it is also clarified that the proposed method can apply to a downslope transfer path, as seen in Fig. 10. In the design for a down-slope transfer path, the parameters of the weighting functions were set to different values, because the time constant of the motor model was varied.
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Fig. 9. Experimental result on up-slope transfer path.
Fig. 10. Experimental result on down-slope transfer path.
In this paper, the angle of the inclined transfer path was fixed at 5.01: In fact, this proposed system can be applied to other angles.
6. Conclusion
Fig. 8. The active liquid container transfer control on the inclined transfer path.
In this paper, rotational motion was added to the pendulum-type sloshing model, resulting in the proposed sloshing model, which accounts for an inclined transfer path and rotational motion. By adding active
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rotational motion to the container, sloshing was eliminated during acceleration and deceleration along the inclined path, and the endpoint residual vibration was reduced. The effectiveness of the proposed system was demonstrated through simulations and experiments.
References Abramson, H. N., Chu, W. H., & Kana, D. D. (1966). Some studies of nonlinear lateral sloshing in rigid containers. Journal of Applied Mechanics, Transactions of the ASME, 777–784. Barron, R., & Chng, S. W. R. (1989). Dynamic analysis and measurement of sloshing of fluid in containers. Journal of Dynamic Systems, Measurement, and Control, Transactions. of the ASME, 111, 83–90. Burditt, M. F., & Bralower, P. M. (1989). Good pouring practice contributes to quality casting. Modern Casting, 59–63. Doyle, J. C., Glover, K., Khargoneker, P. P., & Francis, B. A. (1989). State-space solutions to standard H2 and HN control problems. IEEE Transactions on Automatic Control, 34, 831–847. Feddema, J. T., Dohrmann, C. R., Parker, G. G., Robinett, R. D., Romero, V. J., & Schmidt, D. J. (1997). Control for slosh-free motion of an open container. IEEE Control Systems Magazine, 17, 29–36. Fletcher, R., & Reeves, C. M. (1964). Function minimization by conjugate gradients. Computer Journal, 7, 149–154. Grundelius, M. (2000). Iterative optimal control of liquid slosh in an industrial packaging machine, Proceedings of the 39th IEEE CDC, CD-ROM. Grundelius, M., & Rernhardsson, B. (1999). Motion control of open containers with slosh constraints. Proceedings of the 14th IFAC World Congress (pp. 487–492). Hamaguchi, M., Ronda, W., & Terashima, K. (1997a). Optimum shape design of container for liquid transfer considering the
damping of sloshing. Proceedings of FLUCOME’97 (pp. 639–644). Hamaguchi, M., Terashima, K., & Nomura, H. (1994). Optimal control of liquid container transfer for several performance specifications. Journal of Advanced Automation Technology, 6, 353–360. Hamaguchi, M., Yamamoto, M., & Terashima, K. (1997b). Modeling and control of sloshing with swirling in a cylindrical container during a curved path transfer, Proceedings of ASCC’97, I (pp. 233–236). Housner, G. W. (1963). The dynamic behavior of water tanks. Bulletin of Seismological Society of America, 53, 381–387. Lerner, Y., & Laukhin, N. (2000). Development trends in pouring technology. Foundry Trade Journal, 11, 16–21. Lindsay, W. (1983). Automatic pouring and metal distribution system. Foundry Trade Journal, pp. 151–165. Quintana, V. H., & Davison, E. J. (1974). Clipping-off gradient algorithms to compute optimal controls with constrained magnitude. International Journal of Control, 20, 243–255. Terashima, K., & Hamaguchi, M., & Kaneshige, A. (1995). Motion control of an automotive cart based cylindrical container considering suppression of liquid oscillation on a curve track transfer. Proceedings of IFAC Workshop on Motion Control (pp. 845–852). Terashima, K., & Yano, K. (1999). Sloshing-suppression control of tilting-type automatic pouring machine. Proceedings of the 14th IFAC World Congress, O (pp. 25–30). Yano, K., & Terashima, K. (2001). Robust liquid container transfer control for complete sloshing suppression. IEEE Transactions on Control Systems Technology, 9 (pp. 483–493). Yano, K., Yoshida, T., Hamaguchi, M., & Terashima, K. (1996). Liquid container transfer considering the suppression for the change of liquid level. Proceedings of the 13th IFAC World Congress B, (pp. 85–601). Zames, G., & Francis, B.A. (1983). Feedback, minimax sensitivity and optimal robustness. IEEE Transactions on Automatic Control, 28 (pp. 585–601).
Motion control of liquid container considering an inclined transfer path Ken’ichi Yano*, Shimpei Higashikawa, Kazuhiko Terashima Department of Production Systems Engineering, Toyohashi University of Technology, Hibarigaoka 1-1, Tempaku-cho, Toyohashi, 441-8580, Japan Received 30 March 2001; accepted 8 June 2001
Abstract The present paper is concerned with advanced control of liquid container transfer along an inclined transfer path, paying special attention to the suppression of sloshing (liquid vibration). To suppress sloshing in the container during acceleration and deceleration along the inclined transfer path, we present a method to actively control the container’s rotational motion. This system is useful for saving space in factories and optimizing foundry processes. The effectiveness of the proposed system is demonstrated through simulations and experiments. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Manufacturing processes; Process control; Factory automation; Control applications; HN control; Modelling
1. Introduction In this paper, control topics are focused on the liquid container transfer systems in the casting and steel industries. In practice, molten metal is generally moved from furnace to casting area by means of a self-transfertype cart such as a crane, a forklift or an automotive cart, after it has been poured into a ladle (Lindsay, 1983; Lerner & Laukhin, 2000). In automatic pouring processes, the transfer of molten metal includes the transfer of the ladle along molds as well as the transfer of the molds themselves. In these cases, it is important to shorten the total operational time in order to improve productivity. However, high-speed transfer causes the molten metal to slosh in both the ladle and the molds. This sloshing phenomenon is not only dangerous because of the overflow, but it also deteriorates the quality due to contamination and excessive cooling of the molten metal. Further, in automatic pouring, sloshing disturbs the positioning of the dropping point of the fluid (Lindsay, 1983; Burditt & Bralower, 1989; Terashima & Yano, 1999). Therefore, it is necessary to
*Corresponding author. Tel.: +81-532-44-6699; fax: +81-532-446690. E-mail addresses: [email protected] (K. Yano), [email protected] (K. Terashima).
construct a liquid container transfer system that will suppress sloshing as well as facilitate high-speed transfer. Many studies have been published with respect to the suppression of sloshing (Housner, 1963; Abramson, Chu, & Kana, 1966; Barron & Chng, 1989). However, few studies have focused on both sloshing suppression and high-speed transfer. In recent studies, modelling and optimal control for several engineering specifications using LQI control for the transfer of rectangular containers, taking into consideration the suppression of sloshing, are presented (Hamaguchi, Terashima, & Nomura, 1994). To construct a high-speed transfer system for liquid containers that reduces endpoint residual vibration and has the robustness to withstand change in the static liquid level, an HN feedback control system was applied to this process (Yano, Yoshida, Hamaguchi, & Terashima, 1996; Yano & Terashima, 2001). Also presented is an active control method that takes into account the rotational motion of the container in addition to the linear transfer motion; this method can completely suppress sloshing throughout the entire transfer process (Yano & Terashima, 2001). Feddema et al. (1997) also presented the transfer system by using a robot arm, and Grundelius and Rernhardsson (1999) and Grundelius (2000) applied the system to an industrial packaging machine. Recently, the development of space-saving factories and the optimization of foundry processes have been in high demand from the casting and steel industries. To
0967-0661/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 6 1 ( 0 1 ) 0 0 1 0 7 - 1
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answer this demand, research on analysis and control of the various transfer paths of liquid containers has become necessary. With respect to the shapes and transfer paths of the containers, historically, either narrow or three-dimensional rectangular containers that follow a straight path have been studied. By contrast, in recent years, the transfer of cylindrical containers and transfers via curved paths have been of increasing interest. The optimum motion control system for an automotive, cart-based, cylindrical container was studied, with the main focus being the suppression of sloshing on a curved transfer track (Terashima, Hamaguchi, & Kaneshige, 1995; Hamaguchi, Yamamoto, & Terashima, 1997b). Furthermore, the optimum shape for such a container was studied by minimizing the performance index and considering both the resonant frequency of first-mode sloshing and the volume of liquid (Hamaguchi, Ronda, & Terashima, 1997a). However, no studies have reported on sloshing suppression along the inclined transfer paths, in spite of the importance of such research to factories that rely on large numbers of inclined paths. Therefore, in this paper, our purpose is to establish a liquid container transfer system that effectively reduces sloshing along an inclined transfer path. To avoid overflow and sloshing during such a transfer, the sloshing model comprises two degrees of freedom (both linear and rotational container motion). Furthermore, active control using additional rotational motion is achieved by HN control. The proposed new system will also be useful for eliminating sloshing during acceleration and deceleration intervals on the inclined transfer path. The effectiveness of the proposed system is demonstrated through simulations and experiments.
2. Experimental apparatus The elimination of sloshing during acceleration and deceleration intervals is not possible in mechanisms for linearly moving containers with only a single degree of freedom (Hamaguchi et al., 1994; Yano et al., 1996; Yano & Terashima, 2001). Hence, a transferring machine with two degrees of freedom was proposed in our previous paper by adding the function of rotational motion to the container (Yano & Terashima, 2001). In the present paper, a transferring machine with two degrees of freedom is applied to an inclined transfer path. A schematic diagram of the new experimental apparatus is shown in Fig. 1. The size of the three-dimensional rectangular container is 0.14 m0.14 m0.3 m (lengthwidthheight). A straight transfer path with a constant slope is used in this study, and the angle of the inclined transfer path is set to 5.01: The container is moved using a DC
Fig. 1. Schematic diagram of liquid container transfer system with inclined transfer path.
servomotor with a timing belt, and the velocity and position of the container are controlled by means of the voltage applied to the motor. With respect to the rotational motion, the container is rotated by another DC servomotor using a timing belt. The position of the container is detected by a potentiometer fitted to a pulley, and the angle of the container is detected by an encoder fitted to a rotational axis. Displacement of the liquid level is detected through changes in resistance between two stainless steel electrodes. The electrodes are installed 5 103 m away from the side wall of the container. This position was used because the displacement of the liquid level in the first-order mode sloshing can be most distinctly observed near the side walls of the container, and because the sensor had to be slightly detached from the side wall in order to avoid the effects of the wall. Considering the similarity law in fluid dynamics, water was adopted as the object liquid in the experiment, because the Reynolds number of water at normal temperature is almost the same as that of molten iron metal or molten aluminum metal at high temperature. For example, the kinematic viscosities of iron molten metal at 1350 and 1400 K are 1.365106 and 1.237106 mPa s, respectively, while that of water is 1.0 106 mPa s at 293 K. Therefore, the fluid behavior of the molten metal can be fairly accurately predicted by the behavior of water. It enhances the design safety if the control design is conducted using a fluid with lower damping characteristics, and for this reason water was used as the target liquid in this study.
3. Modelling 3.1. Sloshing model In the case of a straight transfer path, sloshing in a three-dimensional rectangular container can be
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approximated as a two-dimensional phenomenon as long as there are no sudden changes in the acceleration (Hamaguchi et al., 1994). Therefore, in the present study, the sloshing phenomenon is described by a pendulum-type sloshing model (Hamaguchi et al., 1994; Yano & Terashima, 2001). The rotational motion is added to the pendulum-type sloshing model, and a new model is proposed as a sloshing model that considers both the inclined transfer path and the rotational motion. The principles of this model are shown in Fig. 2. By considering the moment balance around the fulcrum of a pendulum, the following equation is derived: J
d2 y cdðy ZÞ c cos2 y mgc sin y ¼ c dt2 dt þ mxc . cos f cos y mxc . sin fsin y
d2 Z cos y ð1Þ dt2 where J is the moment of inertia (J ¼ mc2 ), O the fulcrum of a pendulum, Or the center of revolution, D the distance between the center of the revolution and the center of gravity, y the angle of the pendulum model from horizontal, Z the angle of the container, Z. the angular acceleration of the container rotation, f the angle of the transfer path, and x. is the acceleration in linear transfer applied to the container. This model contains an equivalent coefficient c of viscosity, considering both the viscosity of the liquid and the friction between the liquid and the wall. It also contains the mass m of the liquid, the gravitational acceleration g; and the equivalent length c of the mcD
pendulum. The parameters c and c for the above model were determined by experiments, where c was also theoretically obtained based on the natural frequency determined by the perfect fluid theory (Hamaguchi et al., 1994). Linearization of Eq. (1) around yC0 gives Eq. (2). c 1 1 D y. ¼ ðy’ Z’ Þ ðg þ x. sin fÞy þ x. cos f Z. : ð2Þ m c c c In Eq. (2), the forces Dc Z. ; added by the rotational motion of the container, and xy . sin f; added by the transfer motion of the container, are neglected, because these values are assumed to be very small. Hence, Eq. (3) is given as a final approximate sloshing model. c g 1 y. ¼ ðy’ Z’ Þ y þ x. cos f: ð3Þ m c c 3.2. System model In the DC servomotor, a transfer function Gm ðsÞ from the input voltage e1 ðtÞ to the velocity v1 ðtÞ of the container transfer is given as a first-order lag model, shown in Eq. (4). V1 ðsÞ Km ¼ ; ð4Þ Gm ðsÞ ¼ E1 ðsÞ Tm s þ 1 where Tm is the time constant and Km is the gain. These parameters were obtained by adding a step-wise input to the apparatus and then changing the output data from the position of the container to the velocity, as shown in Table 1. A transfer function Gr ðsÞ from the input voltage e2 ðtÞ to the angular velocity v2 ðtÞ of the container rotation is given as a second-order lag model, as shown in Eq. (5). The model parameters were obtained by adding a couple of sinusoidal-wave inputs to the apparatus, is also shown in Table 1. Gr ðsÞ ¼
V2 ðsÞ Kr o2n ¼ 2 : E2 ðsÞ s þ 2zon s þ o2n
ð5Þ
Then, a linear vector state space equation in Eq. (6) is obtained from Eqs. (3) – (5) where u1 ðtÞ is the control input of a linear transfer, u2 ðtÞ the input of rotational motion and h the displacement of water level at the wall
Table 1 Motor parameters Parameter
Fig. 2. The sloshing model considering an inclined transfer path and rotational motion.
Motor gain, Km Time constant, Tm Motor gain, Kr Damping factor, z Natural angular frequency, on
Value 0.0912 0.0227 0.5807 0.3778 41.446
Unit m/s V s rad/s V F rad/s
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of the container, approximately represented by h ¼ Ly: This model is controllable and observable. x’ ¼ Ax þ Bu;
y ¼ Cx
where 2
0
0
0
0
0
c m
0
0
0 0
1 0
0 1
0 0
0 0
o2n 0
2zon 0
0 0
0
0
0
0
0
1
6 g 6 c mc 6 6 0 6 0 6 A¼6 0 0 6 6 6 0 0 6 6 0 0 4 0 0 " B¼
0
Km cTm
0
0
0
Km Tm
0
0
0
Kr o2n
0
0
C¼
x¼ y
y¼ h
L 0
0 0
0 0
0 1
y’
Z Z’
Z.
x
x’
x
T
;
u ¼ u1
u2
0 0
3
7 cT1m cosf 7 7 7 0 7 7 7 0 7 7 7 0 7 7 1 5 T1m
0
0 0
L 0
0
cos f
0 "
ð6Þ
#T
#
Fig. 3. Validity of the sloshing model and motor model.
T
T
These sloshing parameters were also identified by adding a couple of sinusoidal-wave inputs to the apparatus. The results of parameter identification are shown in Table 2. The validity of the sloshing model is demonstrated in Fig. 3. As input voltages, sinusoidal waves are applied to the case in which liquid level is hs ¼ 0:14 m, where these inputs are amplitude 2 V, frequency 0.3 Hz for linear transportation and amplitude 0.5 V, and frequency 0.5 Hz for rotation. The figures show the following: (a) the control input of the transfer u1 and the rotation u2 ; (b) sloshing in the simulation result by the nonlinear model expressed in Eq. (1), the simulation result by the linear model expressed in Eq. (3) and the experimental result; (c) the position of the container; and (d) the angle of the container. The experimental results are shown by solid lines and the simulation results are shown by dashed lines. The results show that the responses of the model and the real plant agree strongly. Therefore, the validity of the proposed sloshing model is demonstrated. At the same time, since the results of the nonlinear model expressed in Eq. (1) and of the linear model expressed in Eq. (3) are very similar, as shown in Fig. 3(b), the validity of the linearization is also demonstrated. Although some random vibration is found in the experimental result in Fig. 3(b), this is a hardware problem caused by a dead zone due to the backlash of the rotational motion.
Table 2 Model parameters for the sloshing model Parameter
Value
Unit
Liquid level, hs Angle of inclined path, f Length of the pendulum, c Coefficient of viscosity, c Mass of liquid, m Distance between Or and G; D
0.140 5.0 0.0442 1.88 2.744 0.02
m deg m N s/kg kg m
4. Robust controller design When a liquid container is transferred on an inclined transfer path, a transferring machine with one degree of freedom may cause overflow and contamination of the molten metal in terms of only acceleration control for linear container transfer. Therefore, a HN controller with two control inputs was applied to the design of the container transfer so that the free surface of the liquid could be kept parallel to the bottom plate of the container, thus keeping the difference between the angle of the pendulum and the angle of the container at zero throughout the transfer interval. This system was also designed to provide the robustness necessary to withstand the measurement noise and the higher-mode sloshing. A HN loop shaping method (Zames & Francis, 1983; Doyle, Glover, Khargonekar, & Francis 1989) was applied to the present liquid container transfer system in order to guarantee such robustness. The liquid level was
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hs ¼ 0:14 m, and an augmented closed loop system was constructed as shown in Fig. 4. These specifications are represented in the following equation: WR R ð7Þ WN T og: WS S
469
where WR11 ¼
9:6 102 ð5s þ 1Þð10s þ 1Þ ; ðs þ 40Þðs þ 80Þ
WR12 ¼
2:7 103 ðs þ 1Þð10s þ 1Þ : ðs þ 60Þðs þ 15Þ
N
First of all, in order to enhance the robustness for the higher-mode sloshing, the uncertainty weight WR is given to be proper and monotonically increased, as shown in Fig. 5. Its value is shown in Eq. (8). " # WR11 WR12 WR ¼ ; ð8Þ 0 0
Fig. 4. Augmented closed loop system for HN control design.
Next, WN is added to negate the effect of measurement noise, as shown in Eq. (9), based on the actual noise characteristics of measurement of both h and x: " # WN11 0 WN ¼ ; ð9Þ 0 WN22 where WN11 ¼
5500ð2s þ 1Þ2 ; ðs þ 100Þ2
WN22 ¼
1:1 103 ð10s þ 125Þ : s þ 125
Lastly, the sensitivity weight WS is considered. Each diagonal element of WS is given as a different value, as shown in Eq. (10), because this system is a regulator system for the suppression of sloshing and also a servosystem for the position of the container. In particular, the weighting function for the position of the container is set to be a high-gain, at around o ¼ 0; in order that it can operate as a servo-system. " # WS11 0 WS ¼ ; ð10Þ 0 WS22 where
Fig. 5. Frequency responses of the sloshing model and the uncertainty weight WR :
WS11 ¼
ar ; sþ1
WS22 ¼
as : s þ 104
There is, however, a trade-off between these two features, because the transfer part features one input, but two outputs. So, in order to make a fair comparison with the conventional results, which are the case of f ¼ 0 and the one-degree-of-freedom system (Yano & Terashima, 2001), a servo-gain as of WS was provided to yield the same transfer time as the conventional results. A regulator gain ar will be given hereafter. As a result, ar and as were set to 15,000 and 50, respectively. A MATLAB Robust Control Toolbox was used in the present control design. As a result of synthesis, a 16th-order controller was obtained, as shown in Eq. (11), where the optimum value of g was 1.105104 : " # Ku1 eh ðsÞ Ku2 eh ðsÞ KðsÞ ¼ ; ð11Þ Ku1 ex ðsÞ Ku2 ex ðsÞ
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Fig. 7. Simulation results on the inclined transfer path. Fig. 6. Frequency response of the controller obtained by using the HN control theory.
where " u¼
u1 u2
#
" ¼
Ku1 eh ðsÞ
Ku2 eh ðsÞ
Ku1 ex ðsÞ Ku2 ex ðsÞ
#"
eh ex
# :
ð12Þ
In the equations, u1 is the input for the container transfer, u2 the input for the container rotation, eh the displacement error of water levels, and ex the position error of the container. The frequency response of the controller is shown in Fig. 6. From these results, the gains from both the position error ex of the container and the displacement error eh of water levels to the control input u1 are locally low at approximately the resonance frequency of sloshing. Therefore, the controller for u1 is a notchfilter type, which does not excite sloshing during the transfer. On the other hand, the gain from the position error ex of the container to the control input u2 for rotation is locally high, at approximately the resonance frequency of sloshing. Therefore, in this system the container is actively rotated to suppress the sloshing generated by acceleration.
method of Fletcher and Reeves (1964), combined with a clipping-off technique (Quintana & Davison, 1974). In the calculation, the angle of the inclined transfer path is assumed to be f ¼ 0 and the position of the container is assumed to be controlled by only the control input u1 : An optimal trajectory was calculated in the following way. Firstly, a terminal condition, as shown in Eq. (13), is imposed to achieve both the target velocity (0.5 m/s) and the suppression of sloshing at the end-time of the acceleration. An optimal control problem that considers the restriction of the input magnitude of the actuator in the experimental transfer system can be formulated as Eq. (14), where the cost function J is formulated as a quadratic form in order to minimize the square norm of state error at the end-time of the acceleration interval, and where tA½to ; tf is the interval of acceleration. ’ T ¼ ð0:5; 0; 0ÞT ; xt ¼ ðx; ð13Þ ’ y; yÞ f
mina s:t:
J ¼ ðxtf xðtf ÞÞT Wðxtf xðtf ÞÞ 2 3 2 3 a x’ d6 7 6 7 ð14Þ y’ 4y5 ¼ 4 5; dt g c ’ 1 ’y y cosy sin y þ a cos y m
5. Control results The simulation results are shown in Fig. 7, where the transportation distance is 1 m, the liquid level is hs ¼ 0:14 m, and the sampling time is 0.01 s. In this study, in order to achieve a high-speed transfer system as well as the reduction of endpoint residual vibration, a reasonable reference trajectory for the position control of the container was determined using the optimization
c
c
’ T ¼ ð0; 0; 0ÞT ; xðt0 Þ ¼ ðx; ’ y; yÞ jajp2:0 m=s2 : Secondly, a nonlinear optimal theory based on the Fletcher and Reeves method, combined with a clippingoff technique, is applied to solve the optimal control problem with constrained inputs. The interval of deceleration is known to be able to turn over the input of acceleration. Finally, an optimal reference trajectory
K. Yano et al. / Control Engineering Practice 10 (2002) 465–472
is obtained by integrating the velocity curve for the total interval. This method gives a minimum time of 0.34 s for the acceleration interval, where W ¼ diag½106 ; 106 ; 106 : The effectiveness of this optimal reference trajectory was shown in the literature (Yano & Terashima, 2001). In the simulations, the 1 m transfer time of this active transfer is intended to be the same as that of the conventional transfer (Yano & Terashima, 2001) in the case of f ¼ 0 using a one-degree-of-freedom system without the rotation of the container. In this paper, the straight path transfer of f ¼ 0 with one degree of freedom is called the normal transfer. The container is actively rotated in order to avoid overflow and sloshing, including during acceleration and deceleration. As a result, sloshing is almost completely suppressed throughout the entire transfer. To show an idea of this control system, the illustration of the liquid container transfer control is shown in Fig. 8, where (a) shows the start point; (b) the acceleration interval; (c) the constant velocity interval; (d) the deceleration interval and (e) the end of the transfer. As seen from this figure, the container is actively rotated to suppress the sloshing during acceleration (b) and deceleration (d). At other intervals, the container is always tilted to keep it at the same angle with the slope for the inclined ground in order to avoid overflow and sloshing. The experiment was conducted under the same conditions as the simulation. As seen in Fig. 9, this control system almost satisfies requirements such that the water level does not oscillate throughout the entire transfer interval on the inclined transfer path, i.e., the acceleration, constant-velocity and deceleration intervals, as well as after arrival. Furthermore, it is also clarified that the proposed method can apply to a downslope transfer path, as seen in Fig. 10. In the design for a down-slope transfer path, the parameters of the weighting functions were set to different values, because the time constant of the motor model was varied.
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Fig. 9. Experimental result on up-slope transfer path.
Fig. 10. Experimental result on down-slope transfer path.
In this paper, the angle of the inclined transfer path was fixed at 5.01: In fact, this proposed system can be applied to other angles.
6. Conclusion
Fig. 8. The active liquid container transfer control on the inclined transfer path.
In this paper, rotational motion was added to the pendulum-type sloshing model, resulting in the proposed sloshing model, which accounts for an inclined transfer path and rotational motion. By adding active
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rotational motion to the container, sloshing was eliminated during acceleration and deceleration along the inclined path, and the endpoint residual vibration was reduced. The effectiveness of the proposed system was demonstrated through simulations and experiments.
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damping of sloshing. Proceedings of FLUCOME’97 (pp. 639–644). Hamaguchi, M., Terashima, K., & Nomura, H. (1994). Optimal control of liquid container transfer for several performance specifications. Journal of Advanced Automation Technology, 6, 353–360. Hamaguchi, M., Yamamoto, M., & Terashima, K. (1997b). Modeling and control of sloshing with swirling in a cylindrical container during a curved path transfer, Proceedings of ASCC’97, I (pp. 233–236). Housner, G. W. (1963). The dynamic behavior of water tanks. Bulletin of Seismological Society of America, 53, 381–387. Lerner, Y., & Laukhin, N. (2000). Development trends in pouring technology. Foundry Trade Journal, 11, 16–21. Lindsay, W. (1983). Automatic pouring and metal distribution system. Foundry Trade Journal, pp. 151–165. Quintana, V. H., & Davison, E. J. (1974). Clipping-off gradient algorithms to compute optimal controls with constrained magnitude. International Journal of Control, 20, 243–255. Terashima, K., & Hamaguchi, M., & Kaneshige, A. (1995). Motion control of an automotive cart based cylindrical container considering suppression of liquid oscillation on a curve track transfer. Proceedings of IFAC Workshop on Motion Control (pp. 845–852). Terashima, K., & Yano, K. (1999). Sloshing-suppression control of tilting-type automatic pouring machine. Proceedings of the 14th IFAC World Congress, O (pp. 25–30). Yano, K., & Terashima, K. (2001). Robust liquid container transfer control for complete sloshing suppression. IEEE Transactions on Control Systems Technology, 9 (pp. 483–493). Yano, K., Yoshida, T., Hamaguchi, M., & Terashima, K. (1996). Liquid container transfer considering the suppression for the change of liquid level. Proceedings of the 13th IFAC World Congress B, (pp. 85–601). Zames, G., & Francis, B.A. (1983). Feedback, minimax sensitivity and optimal robustness. IEEE Transactions on Automatic Control, 28 (pp. 585–601).