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Mixed logical dynamical model for back bead width prediction of pulsed GTAW process with misalignment
Journal of Materials Processing Technology 210 (2010) 2036–2044
Contents lists available at ScienceDirect
Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Mixed logical dynamical model for back bead width prediction of pulsed GTAW process with misalignment Hongbo Ma ∗ , Shanchun Wei, Tao Lin, Shanben Chen Intelligentized Robotic Welding Technology Laboratory, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e
i n f o
Article history: Received 1 April 2010 Received in revised form 18 July 2010 Accepted 22 July 2010
Keywords: GTAW Prediction Back bead width Misalignment MLD
a b s t r a c t The purpose of this paper is to propose a kind of mixed logical dynamical (MLD) model to predict the back bead width of pulsed GTAW process with misalignment. Misalignment is considered as discrete input and the nonlinear welding process is approximated using piecewise linear models. A MLD model is then established and gives a good prediction quality of the back bead width of pulsed GTAW process with misalignment. The stability and reliability of the MLD model are tested by a closed loop control experiment. This study shows that the MLD framework is a good modeling method for pulsed GTAW process. Thus, a solid foundation for penetration control of welding process is established based on the MLD model. © 2010 Elsevier B.V. All rights reserved.
1. Introduction As is well known, welding quality control is a complicated problem in arc welding processes. Back bead width and penetration depth are major factors in the final weld quality for full penetration. Extensive researches have been done to sense and control of penetration depth both directly and indirectly. However, it is very difficult to directly measure the extent of penetration without sectioning the work piece. Hence, some estimation or prediction models are proposed to measure the penetration depth on-line in different welding applications. Hirai et al. (2003) gives a neural network model to estimate the penetration depth of pulsed MIG (metal inert-gas arc) welding. Tu et al. (1997) propose a model to relate the temperature, weld bead width, laser beam power and welding speed to penetration depth in laser welding. Zhang et al. (2008) recently gives a new multi-sensor data fusion model for online predication of underwater flux-cored arc welding penetration depth. On the other hand, direct penetration depth measurement is also studied by special sensor system. Kita and Ume (2007) use ultrasonic sensors to measure weld penetration depth by using a direct reflection of either a longitudinal or shear wave from the bottom of a weld bead. Moreover, penetration depth can be monitored by acoustic emission sensing as demonstrated (Duley and
∗ Corresponding author at: School of Materials Science and Engineering, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Minhang, Shanghai 200240, China. Tel.: +86 21 34202740 803; fax: +86 21 34202740 808. E-mail address: [email protected] (H. Ma). 0924-0136/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2010.07.022
Mao, 1994). Comparing to penetration depth, back bead width is more convenient to be measured both on-line and off-line. The on-line sensing and control of back bead width are crucial and challenging issues in automated welding. Direct back bead width measurement can be also realized by visual sensor monitoring from the backside. Baskoro et al. (2009) use an omnidirectional camera to monitor backside image of molten pool in pipe welding. Real-time root bead image feedback is also studied to measure backside bead width as demonstrated (Tsai et al., 2006). Fan et al. (2009) proposed a kind of multi-pass visual sensor, which can monitor backside weld pool for butt welding without backing plate. However, in most applications, it is very difficult to measure directly back bead width because of the existence of welding backing plate. Thus, many estimation models are built off-line and then used on-line to predict the back bead width. Linear MA (movingaverage) models were selected to predict weld depression and weld width as demonstrated (Zhang et al., 1996), where the back bead width can be determined with sufficient accuracy by average sag depression depth. Arc welding is characterized as inherently variable, nonlinear, time varying and having a strong coupling among welding parameters. So, it is very difficult to find a reliable mathematical model for arc welding by conventional modeling methods as said (Chen et al., 2000). Artificial intelligence methodology was developed for modeling and controlling the welding process because it could derive the control performance relying not on the mathematical process model but on human experience, knowledge and logic. Hence, researchers applied many artificial intelligence methods to estimate the back bead width using measurements acquired from the topside. Andersen et al. (1990) propose artifi-
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cial neural network model to evaluate the penetration of the weld pool, which is also studied by Chen et al. (2000) for bead-on-plate welding. Nagesh and Datta (2002) use artificial neural networks for the prediction of weld bead geometry and penetration in shielded metal-arc welding. Li et al. (2007) propose a generalized rough set method to modeling welding process. Huang and Chen (2006) propose a SVM-based fuzzy modeling method for the arc welding process, which is more easily to understand because of knowledge expressions by fuzzy rulers. In this study, a novel MLD modeling framework is established to estimate back bead width of pulsed GTAW process. The MLD modeling method stems from hybrid systems described by interacting physical laws, logical rules, and operating constraints. There are mainly three reasons for this study. (1) MLD modeling method is based on linear models. Comparing to nonlinear models, linear models are easier to acquire without losing prediction precision and their control schemes can be computationally tractable. Moreover, nonlinear welding process can be expressed by the combination of linear models and logical rules. (2) In practical production, welding process encounters many uncertainties, such as the errors of premachining, fitting of work piece, and in-process thermal distortions, which will change the seam state, such as seam gap and misalignment. However, most of the uncertainties are not included in traditional models. In the framework of MLD model, some of the uncertainties can be considered as discrete variables which can be expressed in model. (3) Logical rulers and operating constrains in welding process can be expressed in the MLD framework, which will benefit the solution of control scheme for welding process. This paper is organized as follows. In Section 2 a concise introduction to pulsed GTAW process is given, showing the welding device, welding parameters and sensing technology. In Section 3 a brief introduction of MLD model is given. Section 4 gives the modeling of pulsed GTAW process in detail. In Section 5, the prediction results of random experiments are reported to illustrate the effectiveness of the MLD model and a closed loop control experiment of butt welding is done to test the stability and reliability of the MLD model. Finally, some concluding remarks are made in Section 6.
2. Pulsed GTAW process Pulsed GTAW is different from normal GTAW because it melts work piece using pulsed current and base current alternately. The weld pool is formed during pulsed current period and it is solidified during base current period. Finally, the weld seam is constituted by a lot of overlapping solider joints, shown in Fig. 1. Pulsed GTAW is commonly applied to thin plates of materials like aluminum, magnesium, aluminum alloy, magnesium alloy and so on, because these materials are easy to form high-melting-point oxide film on their surface. However, thermal deformation happens frequently for thin plates, such as misalignment. The heat input can be controlled by adjusting pulsed current, base current, pulsed current duration, base current duration, etc. Therefore, the size and quality of weld seam or the heat affected zone can be controlled. The experimental system of pulsed GTAW is shown in Fig. 1. The topside weld width Wf , topside weld length Lf and back bead width Wb are defined in Fig. 2. The weld pool image is captured by a CCD camera fixed behind the torch. Wf and Lf can be computed from the image and they can be adjusted by changing the welding parameters, such as pulsed current Ip . Normally, base current Ib stays constantly for a special welding process. Both Ib and Ip are the outputs of power supply and they can be controlled by the computer through D/A card. For pulsed GTAW process, Ib could be lower than Ip in order to maintain the arc. At the duration of base current, the arc light is
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Fig. 1. Experimental system diagram.
Fig. 2. Weld pool parameters.
relative low so that clear weld pool image can be acquired, shown in Fig. 3. However, Wb cannot be observed directly, as a result, in this study a kind of MLD model is proposed to predict Wb based on topside signals. 3. Mixed logic dynamic model Hybrid systems are dynamic systems that involve the interaction of continuous dynamics (modeled as differential or difference equations) and discrete dynamics (modeled by Petri Net or
Fig. 3. Clear image of weld pool.
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Automata). Hybrid systems have been a topic of intense research activity in recent years, primarily because of their potential importance in applications. Hybrid models are important to system analysis and control of hybrid systems. Bemporad and Morari (1999) presented a framework for modeling and control of hybrid systems described by interdependent physical laws, logic rules, and operation constraints, denoted as mixed logical dynamical (MLD) systems. MLD systems generalize a wide set of models, among which there are linear hybrid systems, finite state machines, some classes of discrete event systems, constrained linear systems, and nonlinear systems whose nonlinearities can be expressed (or, at least, suitably approximated) by piecewise linear functions. In MLD modeling method, a binary variable is assigned to each statement. If and only if the statement is true, the value of the binary variable is true. For example, statement xi means “topside weld pool length is too short” for welding process. Thus, a binary variable ıi is added to replace statement xi , expressed as (1). xi ≡ True ↔ ıi = 1
(1)
Combination of statements may be described with combination of binary variables, which can be suitably translated into linear inequalities involving binary variables ıi . Therefore, the system is modeled as an MLD through the following linear relations (2). x(t + 1) = At x(t) + B1t u(t) + B2t ı(t) + B3t z(t) y(t) = Ct x(t) + D1t u(t) + D2t ı(t) + D3t z(t)
4. Modeling of pulsed GTAW In this study, the model input and output of pulsed GTAW process are the pulsed current and back bead width respectively. Topside parameters are considered as the state variables of the model. The purpose of modeling pulsed GTAW process is tried to find a model to express the relation between the input, state and output of this process so that the back bead width can be evaluated by using this model. It is well known that the current topside and backside parameters are relative to those parameters of history times, shown in Fig. 4, where Wf , Lf , Wb at the time k + 1 not only have relationship with Ip (k + 1), but Wf , Lf , Wb , Ip at the times k and k − 1 also influence the values of Wf , Lf , Wb at time k + 1 because of the heat conduction. As a result, the model structure can be expressed as (3).
⎧ Wf (k) = f (Wf (k − 1)· · ·Wf (k − nWf ), Ip (k)· · ·Ip (k − nIp )) ⎪ ⎪ ⎪ ⎪ ⎨ Lf (k) = f (Lf (k − 1)· · ·Lf (k − nL ), Ip (k)· · ·Ip (k − nIp )) f
⎪ Wb (k) = f (Wb (k − 1)· · ·Wb (k − nWb ), Wf (k)· · ·Wf (k − nWf ), ⎪ ⎪ ⎪ ⎩ L (k)· · ·L (k − n ), I (k)· · ·I (k − n )) Lf
p
p
Material: 2 mm LF6; size: 400 × 200 mm2 Material: thorium–tungsten; diameter: 3.2 mm Pure Ar; flow rate: 15 L/min 50 A 17 mm/s 24 cm/min 2 Hz
Work piece Tungsten anode Shielded gas Base current Feed speed Weld speed Pulse frequence
Table 2 Chemical composition of the LF6 aluminum alloy.
where x is a state vector of the system and contains continuous and binary variables. y and u are respectively output and input vectors of the system, which r consist of continuous and discrete parts. The vectors ı ∈ 0, 1 l and z ∈ Rrc are auxiliary binary and continuous variables, respectively.
f
Table 1 Welding conditions of pulsed GTAW.
(2)
E2t ı(t) + E3t z(t) ≤ E1t u(t) + E4t x(t) + E5t
f
Fig. 4. The corresponding relations of welding parameters.
(3)
Ip
where Ip , Wf , If , Wb are pulsed current, topside width, topside length and back bead width respectively. k is the sampling time.
Chemical composition (mass fraction, %) Si
Fe
Mn
Mg
Zn
0.4
0.4
0.5–0.8
5.8–6.8
0.2
nIp , nWf , nLf , nWb are the effective intervals of Ip , Wf , Lf , Wb . In this paper, only one previous time is chosen, which means that nIp , nWf , nLf , nWb are all equal to 1. In model (3), Wf (k − 1) and Lf (k − 1) can be replaced by the measured values from images, however Wb (k − 1) only can be evaluated. The prediction error of Wb (k − 1) will be added to Wb (k). Thus, in our study the history values of back bead width are not considered in model (3), and then model (3) is changed into (4).
W (k) = f (W (k − 1)· · ·W (k − nW ), Ip (k)· · ·Ip (k − nI f
f
f
f
p ))
Lf (k) = f (Lf (k − 1)· · ·Lf (k − nLf ), Ip (k)· · ·Ip (k − nIp ))
(4)
Wb (k) = f (Wf (k)· · ·Wf (k − nWf ), Lf (k)· · ·Lf (k − nLf ), Ip (k)· · ·Ip (k − nIp ))
Pulsed GTAW process is commonly characterized as nonlinear system with uncertainties. However, exact nonlinear mathematical model is difficult to obtain. For practical application, the control problem of nonlinear model is more complex than linear model. In our study, linear models based on MLD framework are proposed. The nonlinear model of pulsed GTAW process can be replaced by piecewise linear models. In order to analyze the nonlinear characteristics of pulsed GTAW process, five groups of random pulsed currents are considered to excite the dynamics of the process. The welding conditions of random experiments are listed in Table 1. Tables 2 and 3 give the chemical composition and physical properties of the LF6 aluminum alloy respectively.
Table 3 Physical properties of the LF6 aluminum alloy. Density (g/cm3 )
Specific heat capacity (J/kg ◦ C) 100 ◦ C
Thermal conductivity (W/m ◦ C) 25 ◦ C
Linear expansion efficient (×10−6 /◦ C) 20–100 ◦ C
Resistivity (×10−6 ◦ C) 20 ◦ C
2.64
921
117.2
23.7
6.73
H. Ma et al. / Journal of Materials Processing Technology 210 (2010) 2036–2044
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Fig. 5. Experimental data. (a) Pulsed current; (b) topside width; (c) topside length; (d) back bead width.
The experimental input, state and output variables are shown in Fig. 5, where each group is separated by dotted lines. The state variables in Fig. 5 are calculated from images at every cycle. The output variable in Fig. 5 is manually measured off-line after experiments. The mean values and variances of pulsed currents are given in Table 4, where ‘Mean( )’ and ‘Var( )’ are commands in Matlab. Linear ARX (auto-regressive with exogenous input) models are used to identify the relations between Ip and Wf , Lf . The measured and predicted values of Wf , Lf together with their prediction errors are shown in Figs. 6 and 7 respectively. The absolute average prediction error and root mean square error of Wf are 0.1358 mm and 0.1794 mm. The absolute average prediction error and root mean
square error of Lf are 0.1743 mm and 0.2232 mm. The prediction errors of Wf and Lf are very small so that the relation between Ip and Wf , Lf in (4) can be expressed as linear models, shown in (5).
W (k) = 0.8631 × W (k − 1) + 0.02045 × I f
f
p (k)
− 0.01247 × Ip (k − 1)
Lf (k) = 0.8913 × Lf (k − 1) + 0.0125 × Ip (k) − 0.005993 × Ip (k − 1)
A linear model for Wb is identified the same as Wf and Lf , shown in (6), and the measured and predicted values of Wb together with its prediction error are shown in Fig. 8. Wb (k) = −0.01397 × Ip (k) + 0.007821 × Ip (k − 1) − 0.4727 × Wf (k) + 0.7375 × Wf (k − 1) − 0.3702 × Lf (k) + 0.9517 × Lf (k − 1)
Table 4 Mean value and variance of pulsed current for each group. Group number
1
2
3
4
5
Mean( ) Var( )
124.6 82.1
126.1 120.8
116.1 86.8
119.3 88.5
119.9 96.4
(5)
(6)
The absolute average prediction error and root mean square error of Wb are 0.6588 mm and 0.9966 mm. The prediction errors are very large so that linear model (6) is not appropriate for
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Fig. 6. Measured, predicted and error values of topside width.
Fig. 7. Measured, predicted and error values of topside length.
Fig. 8. Measured, predicted and error values of back bead width.
the prediction of Wb . In this study, model (6) is considered as a general model and modifications based on model (6) are given according to the nonlinear dynamics of Wb . There are mainly three kinds of nonlinearities according to our analysis.
(1) As the decreasing of pulsed current, Wb decreases to zero suddenly sometimes instead of decreasing linearly and gradually, shown in Fig. 5(d). The topside lengths at those points where Wb is equal to zero are shown in Fig. 9. Wb = 0 can be estimated according to topside length. Wb is equal to zero when Lf is less than 5.6 mm,
Fig. 9. Topside length (when Wb = 0).
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otherwise Wb is not equal to zero when Lf is larger than 5.6 mm in these experiments. Here, it is convenient to introduce a logical variable ı1 defined as (7). [ı1 (k) = 1] ↔ [Lf (k) < 5.6]
(7)
(2) In random experiment, there are operating point and permitted range of pulsed current. For example, the operating point and permitted range of pulsed current are 125 A and [110 A, 140 A] for group 1. Because of the heat accumulation in welding process, the measured back bead width will be bigger than the predicted value estimated by linear model when the pulsed current stays higher than the operating point for several sampling times. Otherwise, the measured back bead width will be smaller than the predicted value estimated by linear model when the pulsed current stays lower than the operating point for several sampling times. Here, a real variable z1 is defined as (8), where Ipop is the operating point of pulsed current and n is equal to 3 in this paper. z1 =
n
(Ip (k − i) − Ipop )
(8)
Fig. 10. Image of misalignment.
i=0
As a result, parameters 1 and 2 are introduced into the model of Wb in order to correct the predicted value of linear model, shown in (9). Here, two logic variables are introduced shown in (10), where zset1 and zset1 are two thresholds.
⎧ Wb (k) = 1 f (Wf (k)· · ·Wf (k − nWf ), Lf (k)· · ·Lf (k − nLf ), ⎪ ⎪ ⎪ ⎨ Ip (k)· · ·Ip (k − nIp )) ← z1 > zset1
⎪ Wb (k) = 2 f (Wf (k)· · ·Wf (k − nWf ), Lf (k)· · ·Lf (k − nLf ), ⎪ ⎪ ⎩
(9)
Ip (k)· · ·Ip (k − nIp )) ← z1 < zset2
[ı2 = 1] ↔ [z1 > zset1 ] [ı3 = 1] ↔ [z1 < zset2 ]
(10)
(3) Except for the natural nonlinear of pulsed GTAW process, uncertainties in process also cause the nonlinearity of Wb , such as the existence of seam gap and misalignment. In this paper, the seam gap is small enough that it can be neglected. However, misalignment is very common in our experiments and it can be calculated from images, shown in Fig. 10, where d is an index of misalignment in pixel level. In welding process, misalignment is equal to the increase of plate thickness and it will cause the decrease of back bead width. Hence, logic inputs ub1 and ub2 are introduced to describe the size of misalignment in (11).
[20 pixel > d > 10 pixel] → ub1 [d > 20 pixel] → ub2
(11)
Table 5 Identification results of parameters 1 , 2 , 3 , 4 . 1
2
3
4
1.1
0.9
0.95
0.8
Because of the logic inputs ub1 and ub2 , parameters 3 and 4 are introduced into the model of Wb in order to correct the predicted value of linear model when misalignment happens, shown in (12).
⎧ Wb (k) = 3 f (Wf (k)· · ·Wf (k − nWf ), Lf (k)· · ·Lf (k − nLf ), ⎪ ⎪ ⎪ ⎨ Ip (k)· · ·Ip (k − nIp )) ← ub1 = 1
⎪ Wb (k) = 4 f (Wf (k)· · ·Wf (k − nWf ), Lf (k)· · ·Lf (k − nLf ), ⎪ ⎪ ⎩
(12)
Ip (k)· · ·Ip (k − nIp )) ← ub2 = 1
Based on the analysis above, the experiment data is divided into 9 sects according to the discrete dynamics, which is involved with discrete events ı1 , ı2 , ı3 and discrete inputs ub1 , ub2 . Parameters 1 , 2 , 3 , 4 are identified by using linear least squares method based on each sect of experiment data and the results are shown in Table 5. Finally, the MLD model for pulsed GTAW process can be easily obtained as the format of Eq. (2) benefiting from the software HYSDEL (hybrid system description language) as demonstrated (Torrisi and Bemporad, 2004). Due to the limited space, this paper does not give the numerical values of the matrices in (2).
Fig. 11. The prediction result of back bead width.
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Fig. 12. Dynamics of model selector, discrete inputs and discrete events. (a) Model selector; (b) discrete input ub1 ; (c) discrete input ub2 ; (d) discrete event ı1 ; (e) discrete event ı2 ; (f) discrete event ı3 .
H. Ma et al. / Journal of Materials Processing Technology 210 (2010) 2036–2044
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Fig. 13. Schematic diagram of PID controller.
5. Experiment results According to MLD model, the predicted values of topside width and length have been given in Figs. 6 and 7. The predicted value of back bead width based on MLD model is shown in Fig. 11. The dynamics of model selector, discrete inputs and discrete events are shown in Fig. 12. The absolute average prediction error and root mean square error of Wb are 0.3322 mm and 0.4091 mm. The result illustrates the effectiveness of the MLD model, which can conveniently be used for penetration control of pulsed GTAW process. In order to evaluate the stability and reliability of the MLD model, a PID (proportion integral differential) controller is designed for butt welding. The experiment condition is the same as Table 1. The schematic diagram of PID control experiment is shown in Fig. 13, where Wbest is the reference value of Wb , Wbpre is the predicted value of Wb , eW is the offset of Wb . W f , L¯ f , d¯ are the measured b
topside width, topside length and misalignment by visual sensor. The topside and backside pictures of weld seam after welding are given in Fig. 14. The back bead widths were measured manually by using Vernier Caliper in each cycle and were shown in Fig. 15, where Wbset is set by 5.5 mm in this experiment. The experiment result shows this MLD model gives a reliable ability to predict the back bead width of pulsed GTAW process. 6. Conclusions
Fig. 14. Topside and backside pictures of plate after welding. (a) Topside picture; (b) backside picture.
The modeling and control of welding process are crucial issues in the quality control of welding. Although extensive researches have been done to find feasible approaches for modeling of welding process, more practical solutions are still strongly needed. In this paper, a novel MLD modeling method is studied to express the dynamics of pulsed GTAW process. This study will make the modeling of welding process more easily and give a novel understanding of welding process from hybrid system. In the future, more dynamics in welding process will be included by using MLD mod-
Fig. 15. The measured back bead width of the seam.
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eling method and the predictive control of welding process will be studied. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 60874026, the Natural Science Foundation of Shanghai under Grant No. 08ZR1409500 and the Shanghai Sciences & Technology Committee under Grant No. 09JC1407100. References Andersen, K., Cook, G.E., Karsai, G., Ramaswamy, K., 1990. Artificial neural networks applied to arc welding process modeling and control. IEEE Transactions on Industry Applications 26, 824–830. Baskoro, A.S., Masuda, R., Kabutomori, M., Suga, Y., 2009. An application of genetic algorithm for edge detection of molten pool in fixed pipe welding. International Journal of Advanced Manufacturing Technology 45, 1104–1112. Bemporad, A., Morari, M., 1999. Control of systems integrating logic, dynamics, and constraints. Automatica 35, 407–427. Chen, S.B., Lou, Y.J., Wu, L., Zhao, D.B., 2000. Intelligent methodology for sensing, modeling and control of pulsed GTAW. Part 1. Bead-on-plate welding. Welding Journal 79, 151–163. Duley, W.W., Mao, Y.L., 1994. The effect of surface condition on acoustic emission during welding of aluminium with CO2 laser radiation. Journal of Physics D: Applied Physics 27, 1379–1383.
Fan, C., Lv, F., Chen, S., 2009. Visual sensing and penetration control in aluminum alloy pulsed GTA welding. The International Journal of Advanced Manufacturing Technology 42, 126–137. Hirai, A., Kaneko, Y., Nagasawa, K., Yamane, S., Oshima, K., 2003. Neural network model of weld pool in pulsed MIG welding. Quarterly Journal of the Japan Welding Society 21, 227–233. Huang, X., Chen, S., 2006. SVM-based fuzzy modeling for the arc welding process. Materials Science & Engineering A 427, 181–187. Kita, A., Ume, I.C., 2007. Measuring on-line and off-line noncontact ultrasound time of flight weld penetration depth. Welding Journal 86, 9s–17s. Li, W.H., Chen, S.B., Lin, T., Du, Q.Y., 2007. A generalized rough set modeling method for welding process. Journal of Shanghai Jiaotong University (Science) 12, 319–322. Nagesh, D.S., Datta, G.L., 2002. Prediction of weld bead geometry and penetration in shielded metal-arc welding using artificial neural networks. Journal of Materials Processing Technology 123, 303–312. Torrisi, F.D., Bemporad, A., 2004. HYSDEL—a tool for generating computational hybrid models for analysis and synthesis problems. IEEE Transactions on Control Systems Technology 12, 235–249. Tsai, C.H., Hou, K.H., Chuang, H.T., 2006. Fuzzy control of pulsed GTA welds by using real-time root bead image feedback. Journal of Materials Processing Technology 176, 158–167. Tu, J.F., Lankalapalli, K.N., Gartner, M., Leong, K.H., 1997. On-line estimation of laser weld penetration. Journal of Dynamic Systems, Measurement, and Control 119, 791. Zhang, W., Wang, G., Shi, Y., Zhong, B., 2008. On-line predication of underwater welding penetration depth based on multi-sensor data fusion. In: 2008 7th World Congress on Intelligent Control and Automation, pp. 1108–1113. Zhang, Y.M., Kovacevic, R., Li, L., 1996. Adaptive control of full penetration gas tungsten arc welding. IEEE Transactions on Control Systems Technology 4, 394–403.
Contents lists available at ScienceDirect
Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Mixed logical dynamical model for back bead width prediction of pulsed GTAW process with misalignment Hongbo Ma ∗ , Shanchun Wei, Tao Lin, Shanben Chen Intelligentized Robotic Welding Technology Laboratory, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e
i n f o
Article history: Received 1 April 2010 Received in revised form 18 July 2010 Accepted 22 July 2010
Keywords: GTAW Prediction Back bead width Misalignment MLD
a b s t r a c t The purpose of this paper is to propose a kind of mixed logical dynamical (MLD) model to predict the back bead width of pulsed GTAW process with misalignment. Misalignment is considered as discrete input and the nonlinear welding process is approximated using piecewise linear models. A MLD model is then established and gives a good prediction quality of the back bead width of pulsed GTAW process with misalignment. The stability and reliability of the MLD model are tested by a closed loop control experiment. This study shows that the MLD framework is a good modeling method for pulsed GTAW process. Thus, a solid foundation for penetration control of welding process is established based on the MLD model. © 2010 Elsevier B.V. All rights reserved.
1. Introduction As is well known, welding quality control is a complicated problem in arc welding processes. Back bead width and penetration depth are major factors in the final weld quality for full penetration. Extensive researches have been done to sense and control of penetration depth both directly and indirectly. However, it is very difficult to directly measure the extent of penetration without sectioning the work piece. Hence, some estimation or prediction models are proposed to measure the penetration depth on-line in different welding applications. Hirai et al. (2003) gives a neural network model to estimate the penetration depth of pulsed MIG (metal inert-gas arc) welding. Tu et al. (1997) propose a model to relate the temperature, weld bead width, laser beam power and welding speed to penetration depth in laser welding. Zhang et al. (2008) recently gives a new multi-sensor data fusion model for online predication of underwater flux-cored arc welding penetration depth. On the other hand, direct penetration depth measurement is also studied by special sensor system. Kita and Ume (2007) use ultrasonic sensors to measure weld penetration depth by using a direct reflection of either a longitudinal or shear wave from the bottom of a weld bead. Moreover, penetration depth can be monitored by acoustic emission sensing as demonstrated (Duley and
∗ Corresponding author at: School of Materials Science and Engineering, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Minhang, Shanghai 200240, China. Tel.: +86 21 34202740 803; fax: +86 21 34202740 808. E-mail address: [email protected] (H. Ma). 0924-0136/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2010.07.022
Mao, 1994). Comparing to penetration depth, back bead width is more convenient to be measured both on-line and off-line. The on-line sensing and control of back bead width are crucial and challenging issues in automated welding. Direct back bead width measurement can be also realized by visual sensor monitoring from the backside. Baskoro et al. (2009) use an omnidirectional camera to monitor backside image of molten pool in pipe welding. Real-time root bead image feedback is also studied to measure backside bead width as demonstrated (Tsai et al., 2006). Fan et al. (2009) proposed a kind of multi-pass visual sensor, which can monitor backside weld pool for butt welding without backing plate. However, in most applications, it is very difficult to measure directly back bead width because of the existence of welding backing plate. Thus, many estimation models are built off-line and then used on-line to predict the back bead width. Linear MA (movingaverage) models were selected to predict weld depression and weld width as demonstrated (Zhang et al., 1996), where the back bead width can be determined with sufficient accuracy by average sag depression depth. Arc welding is characterized as inherently variable, nonlinear, time varying and having a strong coupling among welding parameters. So, it is very difficult to find a reliable mathematical model for arc welding by conventional modeling methods as said (Chen et al., 2000). Artificial intelligence methodology was developed for modeling and controlling the welding process because it could derive the control performance relying not on the mathematical process model but on human experience, knowledge and logic. Hence, researchers applied many artificial intelligence methods to estimate the back bead width using measurements acquired from the topside. Andersen et al. (1990) propose artifi-
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cial neural network model to evaluate the penetration of the weld pool, which is also studied by Chen et al. (2000) for bead-on-plate welding. Nagesh and Datta (2002) use artificial neural networks for the prediction of weld bead geometry and penetration in shielded metal-arc welding. Li et al. (2007) propose a generalized rough set method to modeling welding process. Huang and Chen (2006) propose a SVM-based fuzzy modeling method for the arc welding process, which is more easily to understand because of knowledge expressions by fuzzy rulers. In this study, a novel MLD modeling framework is established to estimate back bead width of pulsed GTAW process. The MLD modeling method stems from hybrid systems described by interacting physical laws, logical rules, and operating constraints. There are mainly three reasons for this study. (1) MLD modeling method is based on linear models. Comparing to nonlinear models, linear models are easier to acquire without losing prediction precision and their control schemes can be computationally tractable. Moreover, nonlinear welding process can be expressed by the combination of linear models and logical rules. (2) In practical production, welding process encounters many uncertainties, such as the errors of premachining, fitting of work piece, and in-process thermal distortions, which will change the seam state, such as seam gap and misalignment. However, most of the uncertainties are not included in traditional models. In the framework of MLD model, some of the uncertainties can be considered as discrete variables which can be expressed in model. (3) Logical rulers and operating constrains in welding process can be expressed in the MLD framework, which will benefit the solution of control scheme for welding process. This paper is organized as follows. In Section 2 a concise introduction to pulsed GTAW process is given, showing the welding device, welding parameters and sensing technology. In Section 3 a brief introduction of MLD model is given. Section 4 gives the modeling of pulsed GTAW process in detail. In Section 5, the prediction results of random experiments are reported to illustrate the effectiveness of the MLD model and a closed loop control experiment of butt welding is done to test the stability and reliability of the MLD model. Finally, some concluding remarks are made in Section 6.
2. Pulsed GTAW process Pulsed GTAW is different from normal GTAW because it melts work piece using pulsed current and base current alternately. The weld pool is formed during pulsed current period and it is solidified during base current period. Finally, the weld seam is constituted by a lot of overlapping solider joints, shown in Fig. 1. Pulsed GTAW is commonly applied to thin plates of materials like aluminum, magnesium, aluminum alloy, magnesium alloy and so on, because these materials are easy to form high-melting-point oxide film on their surface. However, thermal deformation happens frequently for thin plates, such as misalignment. The heat input can be controlled by adjusting pulsed current, base current, pulsed current duration, base current duration, etc. Therefore, the size and quality of weld seam or the heat affected zone can be controlled. The experimental system of pulsed GTAW is shown in Fig. 1. The topside weld width Wf , topside weld length Lf and back bead width Wb are defined in Fig. 2. The weld pool image is captured by a CCD camera fixed behind the torch. Wf and Lf can be computed from the image and they can be adjusted by changing the welding parameters, such as pulsed current Ip . Normally, base current Ib stays constantly for a special welding process. Both Ib and Ip are the outputs of power supply and they can be controlled by the computer through D/A card. For pulsed GTAW process, Ib could be lower than Ip in order to maintain the arc. At the duration of base current, the arc light is
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Fig. 1. Experimental system diagram.
Fig. 2. Weld pool parameters.
relative low so that clear weld pool image can be acquired, shown in Fig. 3. However, Wb cannot be observed directly, as a result, in this study a kind of MLD model is proposed to predict Wb based on topside signals. 3. Mixed logic dynamic model Hybrid systems are dynamic systems that involve the interaction of continuous dynamics (modeled as differential or difference equations) and discrete dynamics (modeled by Petri Net or
Fig. 3. Clear image of weld pool.
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Automata). Hybrid systems have been a topic of intense research activity in recent years, primarily because of their potential importance in applications. Hybrid models are important to system analysis and control of hybrid systems. Bemporad and Morari (1999) presented a framework for modeling and control of hybrid systems described by interdependent physical laws, logic rules, and operation constraints, denoted as mixed logical dynamical (MLD) systems. MLD systems generalize a wide set of models, among which there are linear hybrid systems, finite state machines, some classes of discrete event systems, constrained linear systems, and nonlinear systems whose nonlinearities can be expressed (or, at least, suitably approximated) by piecewise linear functions. In MLD modeling method, a binary variable is assigned to each statement. If and only if the statement is true, the value of the binary variable is true. For example, statement xi means “topside weld pool length is too short” for welding process. Thus, a binary variable ıi is added to replace statement xi , expressed as (1). xi ≡ True ↔ ıi = 1
(1)
Combination of statements may be described with combination of binary variables, which can be suitably translated into linear inequalities involving binary variables ıi . Therefore, the system is modeled as an MLD through the following linear relations (2). x(t + 1) = At x(t) + B1t u(t) + B2t ı(t) + B3t z(t) y(t) = Ct x(t) + D1t u(t) + D2t ı(t) + D3t z(t)
4. Modeling of pulsed GTAW In this study, the model input and output of pulsed GTAW process are the pulsed current and back bead width respectively. Topside parameters are considered as the state variables of the model. The purpose of modeling pulsed GTAW process is tried to find a model to express the relation between the input, state and output of this process so that the back bead width can be evaluated by using this model. It is well known that the current topside and backside parameters are relative to those parameters of history times, shown in Fig. 4, where Wf , Lf , Wb at the time k + 1 not only have relationship with Ip (k + 1), but Wf , Lf , Wb , Ip at the times k and k − 1 also influence the values of Wf , Lf , Wb at time k + 1 because of the heat conduction. As a result, the model structure can be expressed as (3).
⎧ Wf (k) = f (Wf (k − 1)· · ·Wf (k − nWf ), Ip (k)· · ·Ip (k − nIp )) ⎪ ⎪ ⎪ ⎪ ⎨ Lf (k) = f (Lf (k − 1)· · ·Lf (k − nL ), Ip (k)· · ·Ip (k − nIp )) f
⎪ Wb (k) = f (Wb (k − 1)· · ·Wb (k − nWb ), Wf (k)· · ·Wf (k − nWf ), ⎪ ⎪ ⎪ ⎩ L (k)· · ·L (k − n ), I (k)· · ·I (k − n )) Lf
p
p
Material: 2 mm LF6; size: 400 × 200 mm2 Material: thorium–tungsten; diameter: 3.2 mm Pure Ar; flow rate: 15 L/min 50 A 17 mm/s 24 cm/min 2 Hz
Work piece Tungsten anode Shielded gas Base current Feed speed Weld speed Pulse frequence
Table 2 Chemical composition of the LF6 aluminum alloy.
where x is a state vector of the system and contains continuous and binary variables. y and u are respectively output and input vectors of the system, which r consist of continuous and discrete parts. The vectors ı ∈ 0, 1 l and z ∈ Rrc are auxiliary binary and continuous variables, respectively.
f
Table 1 Welding conditions of pulsed GTAW.
(2)
E2t ı(t) + E3t z(t) ≤ E1t u(t) + E4t x(t) + E5t
f
Fig. 4. The corresponding relations of welding parameters.
(3)
Ip
where Ip , Wf , If , Wb are pulsed current, topside width, topside length and back bead width respectively. k is the sampling time.
Chemical composition (mass fraction, %) Si
Fe
Mn
Mg
Zn
0.4
0.4
0.5–0.8
5.8–6.8
0.2
nIp , nWf , nLf , nWb are the effective intervals of Ip , Wf , Lf , Wb . In this paper, only one previous time is chosen, which means that nIp , nWf , nLf , nWb are all equal to 1. In model (3), Wf (k − 1) and Lf (k − 1) can be replaced by the measured values from images, however Wb (k − 1) only can be evaluated. The prediction error of Wb (k − 1) will be added to Wb (k). Thus, in our study the history values of back bead width are not considered in model (3), and then model (3) is changed into (4).
W (k) = f (W (k − 1)· · ·W (k − nW ), Ip (k)· · ·Ip (k − nI f
f
f
f
p ))
Lf (k) = f (Lf (k − 1)· · ·Lf (k − nLf ), Ip (k)· · ·Ip (k − nIp ))
(4)
Wb (k) = f (Wf (k)· · ·Wf (k − nWf ), Lf (k)· · ·Lf (k − nLf ), Ip (k)· · ·Ip (k − nIp ))
Pulsed GTAW process is commonly characterized as nonlinear system with uncertainties. However, exact nonlinear mathematical model is difficult to obtain. For practical application, the control problem of nonlinear model is more complex than linear model. In our study, linear models based on MLD framework are proposed. The nonlinear model of pulsed GTAW process can be replaced by piecewise linear models. In order to analyze the nonlinear characteristics of pulsed GTAW process, five groups of random pulsed currents are considered to excite the dynamics of the process. The welding conditions of random experiments are listed in Table 1. Tables 2 and 3 give the chemical composition and physical properties of the LF6 aluminum alloy respectively.
Table 3 Physical properties of the LF6 aluminum alloy. Density (g/cm3 )
Specific heat capacity (J/kg ◦ C) 100 ◦ C
Thermal conductivity (W/m ◦ C) 25 ◦ C
Linear expansion efficient (×10−6 /◦ C) 20–100 ◦ C
Resistivity (×10−6 ◦ C) 20 ◦ C
2.64
921
117.2
23.7
6.73
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Fig. 5. Experimental data. (a) Pulsed current; (b) topside width; (c) topside length; (d) back bead width.
The experimental input, state and output variables are shown in Fig. 5, where each group is separated by dotted lines. The state variables in Fig. 5 are calculated from images at every cycle. The output variable in Fig. 5 is manually measured off-line after experiments. The mean values and variances of pulsed currents are given in Table 4, where ‘Mean( )’ and ‘Var( )’ are commands in Matlab. Linear ARX (auto-regressive with exogenous input) models are used to identify the relations between Ip and Wf , Lf . The measured and predicted values of Wf , Lf together with their prediction errors are shown in Figs. 6 and 7 respectively. The absolute average prediction error and root mean square error of Wf are 0.1358 mm and 0.1794 mm. The absolute average prediction error and root mean
square error of Lf are 0.1743 mm and 0.2232 mm. The prediction errors of Wf and Lf are very small so that the relation between Ip and Wf , Lf in (4) can be expressed as linear models, shown in (5).
W (k) = 0.8631 × W (k − 1) + 0.02045 × I f
f
p (k)
− 0.01247 × Ip (k − 1)
Lf (k) = 0.8913 × Lf (k − 1) + 0.0125 × Ip (k) − 0.005993 × Ip (k − 1)
A linear model for Wb is identified the same as Wf and Lf , shown in (6), and the measured and predicted values of Wb together with its prediction error are shown in Fig. 8. Wb (k) = −0.01397 × Ip (k) + 0.007821 × Ip (k − 1) − 0.4727 × Wf (k) + 0.7375 × Wf (k − 1) − 0.3702 × Lf (k) + 0.9517 × Lf (k − 1)
Table 4 Mean value and variance of pulsed current for each group. Group number
1
2
3
4
5
Mean( ) Var( )
124.6 82.1
126.1 120.8
116.1 86.8
119.3 88.5
119.9 96.4
(5)
(6)
The absolute average prediction error and root mean square error of Wb are 0.6588 mm and 0.9966 mm. The prediction errors are very large so that linear model (6) is not appropriate for
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Fig. 6. Measured, predicted and error values of topside width.
Fig. 7. Measured, predicted and error values of topside length.
Fig. 8. Measured, predicted and error values of back bead width.
the prediction of Wb . In this study, model (6) is considered as a general model and modifications based on model (6) are given according to the nonlinear dynamics of Wb . There are mainly three kinds of nonlinearities according to our analysis.
(1) As the decreasing of pulsed current, Wb decreases to zero suddenly sometimes instead of decreasing linearly and gradually, shown in Fig. 5(d). The topside lengths at those points where Wb is equal to zero are shown in Fig. 9. Wb = 0 can be estimated according to topside length. Wb is equal to zero when Lf is less than 5.6 mm,
Fig. 9. Topside length (when Wb = 0).
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otherwise Wb is not equal to zero when Lf is larger than 5.6 mm in these experiments. Here, it is convenient to introduce a logical variable ı1 defined as (7). [ı1 (k) = 1] ↔ [Lf (k) < 5.6]
(7)
(2) In random experiment, there are operating point and permitted range of pulsed current. For example, the operating point and permitted range of pulsed current are 125 A and [110 A, 140 A] for group 1. Because of the heat accumulation in welding process, the measured back bead width will be bigger than the predicted value estimated by linear model when the pulsed current stays higher than the operating point for several sampling times. Otherwise, the measured back bead width will be smaller than the predicted value estimated by linear model when the pulsed current stays lower than the operating point for several sampling times. Here, a real variable z1 is defined as (8), where Ipop is the operating point of pulsed current and n is equal to 3 in this paper. z1 =
n
(Ip (k − i) − Ipop )
(8)
Fig. 10. Image of misalignment.
i=0
As a result, parameters 1 and 2 are introduced into the model of Wb in order to correct the predicted value of linear model, shown in (9). Here, two logic variables are introduced shown in (10), where zset1 and zset1 are two thresholds.
⎧ Wb (k) = 1 f (Wf (k)· · ·Wf (k − nWf ), Lf (k)· · ·Lf (k − nLf ), ⎪ ⎪ ⎪ ⎨ Ip (k)· · ·Ip (k − nIp )) ← z1 > zset1
⎪ Wb (k) = 2 f (Wf (k)· · ·Wf (k − nWf ), Lf (k)· · ·Lf (k − nLf ), ⎪ ⎪ ⎩
(9)
Ip (k)· · ·Ip (k − nIp )) ← z1 < zset2
[ı2 = 1] ↔ [z1 > zset1 ] [ı3 = 1] ↔ [z1 < zset2 ]
(10)
(3) Except for the natural nonlinear of pulsed GTAW process, uncertainties in process also cause the nonlinearity of Wb , such as the existence of seam gap and misalignment. In this paper, the seam gap is small enough that it can be neglected. However, misalignment is very common in our experiments and it can be calculated from images, shown in Fig. 10, where d is an index of misalignment in pixel level. In welding process, misalignment is equal to the increase of plate thickness and it will cause the decrease of back bead width. Hence, logic inputs ub1 and ub2 are introduced to describe the size of misalignment in (11).
[20 pixel > d > 10 pixel] → ub1 [d > 20 pixel] → ub2
(11)
Table 5 Identification results of parameters 1 , 2 , 3 , 4 . 1
2
3
4
1.1
0.9
0.95
0.8
Because of the logic inputs ub1 and ub2 , parameters 3 and 4 are introduced into the model of Wb in order to correct the predicted value of linear model when misalignment happens, shown in (12).
⎧ Wb (k) = 3 f (Wf (k)· · ·Wf (k − nWf ), Lf (k)· · ·Lf (k − nLf ), ⎪ ⎪ ⎪ ⎨ Ip (k)· · ·Ip (k − nIp )) ← ub1 = 1
⎪ Wb (k) = 4 f (Wf (k)· · ·Wf (k − nWf ), Lf (k)· · ·Lf (k − nLf ), ⎪ ⎪ ⎩
(12)
Ip (k)· · ·Ip (k − nIp )) ← ub2 = 1
Based on the analysis above, the experiment data is divided into 9 sects according to the discrete dynamics, which is involved with discrete events ı1 , ı2 , ı3 and discrete inputs ub1 , ub2 . Parameters 1 , 2 , 3 , 4 are identified by using linear least squares method based on each sect of experiment data and the results are shown in Table 5. Finally, the MLD model for pulsed GTAW process can be easily obtained as the format of Eq. (2) benefiting from the software HYSDEL (hybrid system description language) as demonstrated (Torrisi and Bemporad, 2004). Due to the limited space, this paper does not give the numerical values of the matrices in (2).
Fig. 11. The prediction result of back bead width.
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Fig. 12. Dynamics of model selector, discrete inputs and discrete events. (a) Model selector; (b) discrete input ub1 ; (c) discrete input ub2 ; (d) discrete event ı1 ; (e) discrete event ı2 ; (f) discrete event ı3 .
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Fig. 13. Schematic diagram of PID controller.
5. Experiment results According to MLD model, the predicted values of topside width and length have been given in Figs. 6 and 7. The predicted value of back bead width based on MLD model is shown in Fig. 11. The dynamics of model selector, discrete inputs and discrete events are shown in Fig. 12. The absolute average prediction error and root mean square error of Wb are 0.3322 mm and 0.4091 mm. The result illustrates the effectiveness of the MLD model, which can conveniently be used for penetration control of pulsed GTAW process. In order to evaluate the stability and reliability of the MLD model, a PID (proportion integral differential) controller is designed for butt welding. The experiment condition is the same as Table 1. The schematic diagram of PID control experiment is shown in Fig. 13, where Wbest is the reference value of Wb , Wbpre is the predicted value of Wb , eW is the offset of Wb . W f , L¯ f , d¯ are the measured b
topside width, topside length and misalignment by visual sensor. The topside and backside pictures of weld seam after welding are given in Fig. 14. The back bead widths were measured manually by using Vernier Caliper in each cycle and were shown in Fig. 15, where Wbset is set by 5.5 mm in this experiment. The experiment result shows this MLD model gives a reliable ability to predict the back bead width of pulsed GTAW process. 6. Conclusions
Fig. 14. Topside and backside pictures of plate after welding. (a) Topside picture; (b) backside picture.
The modeling and control of welding process are crucial issues in the quality control of welding. Although extensive researches have been done to find feasible approaches for modeling of welding process, more practical solutions are still strongly needed. In this paper, a novel MLD modeling method is studied to express the dynamics of pulsed GTAW process. This study will make the modeling of welding process more easily and give a novel understanding of welding process from hybrid system. In the future, more dynamics in welding process will be included by using MLD mod-
Fig. 15. The measured back bead width of the seam.
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eling method and the predictive control of welding process will be studied. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 60874026, the Natural Science Foundation of Shanghai under Grant No. 08ZR1409500 and the Shanghai Sciences & Technology Committee under Grant No. 09JC1407100. References Andersen, K., Cook, G.E., Karsai, G., Ramaswamy, K., 1990. Artificial neural networks applied to arc welding process modeling and control. IEEE Transactions on Industry Applications 26, 824–830. Baskoro, A.S., Masuda, R., Kabutomori, M., Suga, Y., 2009. An application of genetic algorithm for edge detection of molten pool in fixed pipe welding. International Journal of Advanced Manufacturing Technology 45, 1104–1112. Bemporad, A., Morari, M., 1999. Control of systems integrating logic, dynamics, and constraints. Automatica 35, 407–427. Chen, S.B., Lou, Y.J., Wu, L., Zhao, D.B., 2000. Intelligent methodology for sensing, modeling and control of pulsed GTAW. Part 1. Bead-on-plate welding. Welding Journal 79, 151–163. Duley, W.W., Mao, Y.L., 1994. The effect of surface condition on acoustic emission during welding of aluminium with CO2 laser radiation. Journal of Physics D: Applied Physics 27, 1379–1383.
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