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Dynamic nonlinear modeling of 3D weld pool surface in GTAW
Robotics and Computer-Integrated Manufacturing 39 (2016) 1–8
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Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim
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Dynamic nonlinear modeling of 3D weld pool surface in GTAW JIN Zeshi, LI Haichaon, JIA Guoqing, GAO Hongming State Key Laboratory of Advanced Welding and Joining, Harbin Institute of Technology, Harbin 150001, China
art ic l e i nf o
a b s t r a c t
Article history: Received 15 March 2015 Received in revised form 15 October 2015 Accepted 6 November 2015 Available online 10 December 2015
As the weld pool contains abundant information about the welding process, the skilled welder always utilizes the observed information to control the weld pool to get a good weld quality. In this paper, a realtime vision sensing technology is used as the eyes of the welder to observe the gas tungsten arc welding (GTAW) pool surface and a 3D reconstruction technology is used to reconstruct the observed weld pool surface in three-dimensions. The geometric parameters are obtained from the reconstructed pool surface. Step responses are conducted to acquire the relationship between the weld pool geometric parameters and the random variation of current, and the time-domain analysis of the system is done. Various random experiments are also conducted and the dynamic relationship of the weld current and the weld pool geometric parameters is established by means of a non-linear Hammerstein model. & 2015 Elsevier Ltd. All rights reserved.
Keywords: GTAW Weld pool surface 3D Hammerstein model Dynamic nonlinear modeling
1. Introduction The gas tungsten arc welding (GTAW) [1] is the primary welding process for precision joining of metals because of its process stability and excellent weld appearance. In manual GTAW process, the quality of the produced welds is affected by human factors [2], such as fatigue, stress and emotional issues, which will not exist in the automated welding machines. Furthermore, as the health issues of the human welder become a hot topic, the automated and robotic welding is more and more widely used in manufacturing industry. The improvement of intelligent capabilities in welding production will greatly reduce the dependence on human welders, which will liberate a great deal of workforce and reduce the demand of workers' welding skill level, also the production efficiency will be improved significantly. Welding quality control and seam tracking technology are the most fundamental topics in automated robotic welding [3], the perfect solution of the seam tracking and the welding quality control is the precondition of realizing the welding automation, and the welding quality control is the major difficulty. In the process of actual welding, the weld pool always contains a wealth of useful information, which tends to reflect the weld penetration and other factors related to welding quality [4]. The skilled welder always utilizes the observed information from the weld pool to control the welding quality. Thus it can be seen that an intelligent welding process control system can be established by imitating the control characteristics of skilled welder. So n
Corresponding author. E-mail address: [email protected] (H. LI).
http://dx.doi.org/10.1016/j.rcim.2015.11.004 0736-5845/& 2015 Elsevier Ltd. All rights reserved.
some sensing methods should be used as the eyes of the welder to observe the welding pool and obtain the useful information. Up to data the main sensing methods experimented include the ultrasonic sensing [5], the infrared sensing [6], X-ray sensing [7] and the vision sensing [8–14]. The vision sensing has better application prospects because its function is much more similar to the eyes of the human welder and it also can get more information of welding pool than others. According to the dimensions of the acquired welding pool geometric parameters, the vision sensing can be divided into 2D vision sensing and 3D vision sensing. In this paper, a vision sensing technology for 3D weld pool surface invented and developed at the University of Kentucky is used. This technology considers the weld pool surface as a mirror. A laser pattern is projected on the specular surface of the weld pool and its reflection from the specular surface is intercepted [15–17]. The reflection law is used to compute the weld pool surface that determines the reflection of the projected laser pattern. At the University of Kentucky, the technology has been developed to be able to measure the weld pool surface in real-time in 20 ms [18]. The 3D vision-based sensing system for GTAW process will be mentioned in the following section, which consists of an illumination laser with a 19-by-19 dot matrix structured light pattern, a camera, and an imaging plane. GTAW process is a multiple inputs multiple outputs, nonlinear, time-varying, and strong coupled process with many uncertainties [19,20]. The relationships of the inputs and the outputs are also nonlinear. In order to get the control system of the GTAW, the relationship of the input welding parameters and the output welding pool geometric parameters must be established at first. At the University of Kentucky, researches have been conducted to characterize the 3D weld pool surface using the proposed 3D weld
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pool sensing system [21], and the characterized weld pool parameters (weld pool width, length, and convexity) are utilized to estimate and control the back-side weld penetration [22,23]. Such 3D sensing system also found its application in learning human welder behaviors [24–27]. Specifically, in [28], a nonlinear model for 3D weld pool characteristic parameters is proposed for GTAW process. The cross couplings between the weld pool characteristic parameters are further modeled using a neuro-fuzzy model. It is found that this nonlinear model can provide detailed knowledge about the complex correlation between the weld pool surface and welding process inputs. However, implementing such complex nonlinear neuro-fuzzy model based control algorithm might impose additional challenge on controller design. Thus, a more practical approach to model the 3D weld pool parameters is of interest in current manufacturing industry. As the welding current is the most important welding parameter, this paper establishes a dynamic nonlinear model based on simple Hammerstein model to describe the correlation between the welding current and the weld pool geometry. The remainder of the paper is organized as follows. The vision-based sensing system, the welding process and the controlled variables are described in Section 2. In Section 3, the step responses are conducted and the time-domain analysis of the system is done. The dynamic relationship is identified by means of a non-linear Hammerstein model in Section 4. And the model analysis is also detailed in this section. Conclusion is finally drawn in Section 5.
2. Experimental system and controlled parameter selection 2.1. Experimental system The experimental system is illustrated in Fig. 1, which consists of three parts, the vision-based sensing system, the moving system and the welding system. The moving system is controlled by a personal computer through the DAQ card and the motion controller. The adjusting range of the moving speed is 0–5 mm/s, and the adjustment of step length is 0.1 mm/s. The welding system is also controlled by the personal computer through the DAQ card. The analog output of the DAQ card realizes the control of the welding current. So the welding source can output different kinds of the current waveform through the program code that we write. The vision-based sensing system consists of laser generator, imaging plane, CCD camera, image acquisition card and the personal
computer. The laser power is 200 mW and the wavelength of the laser is 650 nm. It is used to project a 19-by-19 dot matrix structured light pattern on the weld pool region through the dot-array diffraction grating. As the surface of the weld pool just like a mirror, which can reflect the laser falls on it. So the dots project inside the weld pool will be reflected by the specular weld pool surface, and the other will not. The reflected light beam of the dots will be affected by the shape of the 3D weld pool surface, and on the other hand the shape of weld pool can also be calculated through the reflected light beam that obtained. An imaging plane is located at a 150 mm distance from the torch to get the spatial position of the reflected dots in a confirmed space plane. And a camera is needed behind the imaging plane to acquire the image of the reflected dots on the plane. The geometrical information of the weld pool will be calculated through the processed image by the personal computer. Fig. 2 shows the image of the reflected dots on the imaging plane and reconstructed 3D weld pool surface. 2.2. GTAW process In this paper, the weld pool is controlled mainly by the welding current with other welding parameters keeping invariant. When the welding current is respectively using 40 A, 50 A, 55 A, 60 A, 65 A and 70 A, the images of the reflected dots are shown in Fig. 3. As shown in the Fig. 3, the number of the reflected dots increases with the welding current, that means the size of the weld pool also increases with the welding current. When the current reaches 70 A, the dots gather to the middle position (see Fig. 3(f)). The phenomenon is caused by the concave of the weld pool, which is caused by the increasing of the penetration. So in the case of no wire filling welding and the current is too small, the weld pool is hardly to form, so it’s difficult to get the image of the reflected dots. But on the contrary, when the current is too large, it will be difficult to distinguish one point from the other, and the reconstruction of the 3D weld pool will also be impossible, just like Fig. 3(f) shown. Considering the above situation, the welding current should be suitable to ensure the size of the weld pool is big enough and the weld penetration is suitable. So the welding condition of the experiment in this paper is shown in Table 1. 2.3. Controlled parameters selection The controlled parameters are the physical quantities that are asked to keep the set value (close to a constant value or change
Fig. 1. 3D vision-based sensing experimental system.
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Fig. 2. Original image and the result of reconstructed 3D weld pool surface. (a) Original image and (b) 3D weld pool surface.
following some variable) in the experimental process. Actually, the welding quality largely depends on surplus height of the seam and the weld penetration, so the welding quality can be controlled through the control of the surplus height of the seam and the weld penetration. Surplus height of the seam and weld penetration relate to the volume of the weld pool higher than the workpiece, so the most direct way to control the surplus height of the seam and the weld penetration becomes to control the volume of the weld pool. As the laser dots arrange densely, the volume of the weld pool can be calculated through small volume that is divided according to the position of the point. Because every dot is close enough to its neighbors, the small volume can be approximately calculated by Eq. (1), and every area ( ΔS ) can be approximately equal in size. The estimation method of the weld pool volume is illustrated in Fig. 4.
(1)
dV = hi ⋅ΔS
Where hi is the height of the laser dot from the weld pool surface to the workpiece. So the volume of the weld pool can be calculated by:
V=
∑ hi⋅ΔS
(2)
The Eq. (2) can also be shown by the following formula:
Table 1 Welding process of no wire filling GTAW. Workpiece material Stainless steel Argon flow Workpiece dimensions 300 mm 40 mm 2 mm Welding current Tungsten electrode 2.4 mm Welding diameter speed
V = ΔS
∑ hi
5–6 L/min 40–60 A 1.2 mm/s
(3)
Where ∑ hi is the sum of the height from the reflected dots in the weld pool surface to the workpiece. As the area ( ΔS ) of every small volume is approximately equal in size, the ΔS can be approximately treated as constant. So the volume of the weld pool can also be expressed by ∑ hi . And the sum of the height ∑ hi is selected to be the main controlled parameter in this paper. However, the ∑ hi first increases and then decreases with the increase of welding current. The reason of that appearance is that the molten metal increase with the increase of welding current during the lack of penetration, which makes the volume of weld pool increased. As the welding current increases further, the weld pool will drip after the penetration occurred, which makes the volume of weld pool decreased. It means not only a value of the
Fig. 3. Images of reflected dots on imaging plane under different current. (a) 40 A, (b) 50 A, (c) 55 A, (d) 60 A, (e) 65 A, and (f) 70 A.
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Fig. 4. The estimation of the weld pool volume.
∑ hi corresponds to a state of the weld pool, so another controlled parameter should be introduced to distinguish the two states of the weld pool. The width of the weld pool increases with the increase of the welding current, which can be inferred from the Fig. 3. So the width of the weld pool is selected to be the auxiliary controlled parameter in this paper.
3. Step responses The step response experiment is done with the welding current as the input signal, and the step span of the welding current is 20 A. The specific welding parameters are shown in Table 1. The welding current of the positive step experiment changes from 40 A to 60 A, and it changes from 60 A to 40 A in the negative step experiment. The vision-based sensing system is used to reconstruct the 3D GTAW pool surface, and the ∑ hi and the width are obtained through the 3D GTAW pool surface. The interval of the weld pool image acquisition is 1 s. The change of the width and the change of the ∑ hi during the positive step experiment and the negative step experiment are plotted, respectively, in Fig. 6 and Fig. 8. The change of the reflected dots and the corresponding reconstructed 3D weld pool surface are shown in Fig. 5 and 7. It can be seen from Figs. 6 and 8 that there is no oscillation of the response output curves, so the system can be described by a first-order system, and the corresponding transfer function can be represented by:
G (s ) =
K e−τs 1 + Ts
(4)
Where T is the time constant of step responses, K is the gain coefficient, and τ is the time-delay constant. The key to system
identification is to determine the parameters T , K and τ . Due to the recognition method is relatively mature, so the MATLAB system identification toolbox is directly used to identify the parameters by the results of the step responses. The result of the parameter identification is shown in Table 2, where the parameters ( T , K and τ ) of the corresponding transfer functions with the positive and negative steps are presented. Through the analysis of the identification result, some conclusions are drawn as follows: Welding process is a multivariable and strongly coupled process, and the geometric parameters of 3D weld pool are affected by many factors that coupled together. While keeping other welding parameters constant, the whole variant trend of weld pool width and the height-sum is increasing with the increase of welding current. The time-delay of the weld pool response to the welding current is small. The system is a nonlinear system, and it is difficult to describe the system through a simple linear model, so a nonlinear dynamic model must be established.
4. Dynamic nonlinear modeling The dynamic nonlinear model between welding current and geometric parameters of weld pool is established through model identification by using a simple Hammerstein model. The established model is also validated through other experimental data in this section. 4.1. Design of identification signal The random experiment is done to identify the model by using white noise of welding current as input signal, whose average
Fig. 5. The change of the reflected dots (a) and the corresponding reconstructed 3D weld pool surface and (b) in positive step experiment.
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Fig. 6. The change of the weld pool width (a) and the change of the weld pool height sum and (b) in positive step experiment.
Fig. 7. The change of the reflected dots (a) and the corresponding reconstructed 3D weld pool surface and (b) in negative step experiment.
value is time-independent. According to the results of the previous experiment, the weld pool can be large enough without dripping by using the welding current near 55 A (see Fig. 3). So the variation range of welding current is limited between 50 A and 60 A, the output frequency of welding current is 1/3 Hz, and the other welding parameters remain unchanged as shown in Table 1. The MATLAB is used to generate the white noise as input signal, which is shown in Fig. 9. The previous mentioned vision-based system is used to reconstruct the 3D weld pool in the random experiment, and the pool width and the sum of pool height are obtained through the reconstructed pool in every moment. Figs. 10 and 11 have shown the change of the width and the height sum by using the random
current as the input. 4.2. Dynamic system identification and modeling It can be found that there are some nonlinear factors in the controlled parameters through step responses, which means that the controlled system is the nonlinear system. Specifically, the main steps of the system identification are as follows: The model structure is determined through prior experiments and purpose; identifying structure and parameters of the system model by collecting and processing input and output signal; finally, testifying the model. And the most important part is to identify the structure and parameters of the system model. The single input
Fig. 8. The change of the weld pool width (a) and the change of the weld pool height sum and (b) in negative step experiment.
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Table 2 The parameters of the corresponding transfer functions. Weld pool parameters Positive step
Negative step
K T τ K T τ
width
Height-sum
0.4121 5.754 0.187 0.449 8.774 0.0722
7.3 27.81 0.336 0.2797 1.431 0.044
Fig. 11. The sum of weld pool height.
Fig. 9. The input signal of welding current.
Fig. 12. Structure diagram of modular model.
Fig. 13. Structure diagram of simple Hammerstein model.
G (z ) =
( ) ( )
z − dB z − 1 A z −1
(6) B (z −1) = 1 + b1z −1 + ⋅⋅⋅ + bnb z −nb .
Where and The expression of the dynamic system model can be expressed as formula (7) by means of least square method. A (z −1) = 1 + a1z −1 + ⋅⋅⋅ + ana z −na
Fig. 10. Weld pool width.
single output nonlinear system has several commonly used structures, such as the series model of Volterra, the modular model and so on. And the modular model is mainly composed of dynamic linear module and static nonlinear module, which is shown in Fig. 12. Hammerstein model is a typical modular model, which is mainly composed of dynamic linear module and static nonlinear module in series. It includes simple Hammerstein model and generalized Hammerstein model. The generalized Hammerstein model can conduct a comprehensive and detailed description about the nonlinearity of the system, but at the same time, it also has a complex form and a lot of parameters. Although, the accuracy of the simple Hammerstein model is a bit lower than the generalized Hammerstein model, but it can still describe some nonlinear process well. Furthermore, it also has a simpler form and less parameter than the generalized Hammerstein model, which make it widely used in industrial production. The structure of the simple Hammerstein model is depicted in Fig. 13. The memoryless nonlinear function can be expressed by the following formula based on Taylor series expansion:
x (k ) = C0 + C1u (k ) + C2 u2 (k )
(5)
And the transfer function of dynamic linear system can be represented as shown in formula (6) [29].
y (k ) = − a1y (k − 1) − ⋯ − a na y ( k − na ) + b + c0 u (k − d) + ⋯ 2
+ cnb u ( k − d − nb ) + d0 u (k − d)2 + ⋯ + dnb u ( k − d − nb ) = φT (k ) θ
(7)
T
Where φ (k ) is data vector, θ is parameter vector that needs to be estimated. ε (k ) is used to stand for the difference between the actual output and the estimated output, which is called residual. It can be calculated by Eq. (8).
ε (k ) = y (k ) − y^ (k ) = y (k ) − φT (k ) θ^
(8)
So the sum of the squared residuals function J is used as the performance indicator, which is shown in formula (9). A batch algorithm based on least square method is used to determine the model structure parameters θ^ , which make the value of function J to reach the minimum. Fig. 14(a) and (b) shows the minimum of the function J correspond to different values of N 、 d to the width and the height-sum.
J=
2
∑k = 1 ε2 (k) = ∑k = 1 ⎡⎣ y (k) − φT (k) θ^⎤⎦ L
L
(9)
To the dynamic model of the width, it can be found in Fig. 14 (a) that the value of the function J reaches the minimum, when d = 0, N = 5; And to the dynamic model of the height-sum, it can be found in Fig. 14(b) that the value of the function J also reaches the minimum, when d = 0, N = 5. So the parameter vector θ can be
Z. JIN et al. / Robotics and Computer-Integrated Manufacturing 39 (2016) 1–8
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Fig. 14. Change curve of minimal J to width model (a) and change curve of minimal J to height-sum model (b).
y1 [k ] = 0.5927y1 [k − 1] + 0.371 × y1 [k − 2] − 0.0433 × y1 [k − 3] − 0.0379 × y1 [k − 4] − 0.1496 × y1 [k − 5] + 5.6261 − 0.1995 × u [k ] − 0.0056 × u [k − 1] − 0.1536 × u [k − 2] + 0.055 × u [k − 3] + 0.0253 × u [k − 4] + 0.0671 × u [k − 5] + 0.0025 × u [k ] × u [k ] − 0.0003 × u [k − 1] × u [k − 1] Fig. 15. Fitting situation of width model.
+ 0.0013 × u [k − 2] × u [k − 2] − 0.0005 × u [k − 3] × u [k − 3] − 0.0003 × u [k − 4] × u [k − 4] − 0.0005 × u [k − 5] × u [k − 5]
(11)
y2 [k ] = 0.3339y2 [k − 1] + 0.3275 × y2 [k − 2] + 0.0284 × y2 [k − 3] + 0.191 × y2 [k − 4] − 0.2 × y2 [k − 5] + 11.3284 − 0.0536 × u [k ] − 0.0778 × u [k − 1] − 0.1153 × u [k − 2] + 0.148 × u [k − 3] − 0.1997 × u [k − 4] − 0.1983 × u [k − 5]
Fig. 16. Fitting situation of height-sum model.
+ 0.0014 × u [k ] × u [k ] + 0.0005 × u [k − 1] × u [k − 1] calculated by Eq. (10).
θ^ = ϕT ϕ
−1 T
( )
ϕ Y
+ 0.0009 × u [k − 2] × u [k − 2] − 0.0015 × u [k − 3] × u [k − 3] (10)
+ 0.0019 × u [k − 4] × u [k − 4] + 0.0024 × u [k − 5] × u [k − 5]
Where ϕ = [φT (1) , φT (2)⋅⋅⋅φT (L )]T , Y = [y (1) , y (2)⋅⋅⋅y (L )]T and L is the number of the sample points. The θ^ can be calculated by the Eq. (10). So the dynamic relationship between the welding current and width can be obtained as the θ^ is known. And the dynamic relationship between the welding current and height-sum can also be obtained in a similar way. The relationships are shown in formulae (11) and (12), and besides that, the fitting situations of the two models are shown in Figs. 15 and 16.
(12)
4.3. Model validation In this section, other experimental data are used to validate the established models. The results of the validations are shown in Figs. 17 and 18. Form the figures, it can be seen that the established models are relatively reliable, and the changing trend of the width and the height-sum that estimated fits the measured changing trend well.
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response to the welding current is small. Finally, a dynamic nonlinear model based on simple Hammerstein is established to relate the welding current with the weld pool width and the weld pool height-sum.
References
Fig. 17. Validation result of width model.
Fig. 18. Validation result of height-sum model.
5. Conclusion In this paper a real-time vision sensing technology is used as the eyes of the welder to observe the gas tungsten arc welding (GTAW) pool surface and a 3D reconstruction technology is used to reconstruct the observed weld pool surface in three-dimensions. Different welding currents have been used, and the results of the experiment have been analyzed. The optimal welding current range is about 40–60 A during the experiment. The number of the reflect dots increases with the welding current, which indicates the increase of the welding pool size. The large welding current leads to the gathering of the reflected dots on the imaging plane, which indicates the concave of the weld pool. The controlled parameters width and height-sum are selected through the analysis. The height-sum is selected as the main controlled parameter, and the width is selected as the auxiliary controlled parameter. The step responses have been done as the priori experiment, and the relationships between the welding current and the geometrical information of 3D weld pool surface are analyzed. While keeping other welding parameters constant, the whole variant trend of weld pool width and the height-sum is increasing with the increase of welding current. The time-delay of the weld pool
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Full length Article
Dynamic nonlinear modeling of 3D weld pool surface in GTAW JIN Zeshi, LI Haichaon, JIA Guoqing, GAO Hongming State Key Laboratory of Advanced Welding and Joining, Harbin Institute of Technology, Harbin 150001, China
art ic l e i nf o
a b s t r a c t
Article history: Received 15 March 2015 Received in revised form 15 October 2015 Accepted 6 November 2015 Available online 10 December 2015
As the weld pool contains abundant information about the welding process, the skilled welder always utilizes the observed information to control the weld pool to get a good weld quality. In this paper, a realtime vision sensing technology is used as the eyes of the welder to observe the gas tungsten arc welding (GTAW) pool surface and a 3D reconstruction technology is used to reconstruct the observed weld pool surface in three-dimensions. The geometric parameters are obtained from the reconstructed pool surface. Step responses are conducted to acquire the relationship between the weld pool geometric parameters and the random variation of current, and the time-domain analysis of the system is done. Various random experiments are also conducted and the dynamic relationship of the weld current and the weld pool geometric parameters is established by means of a non-linear Hammerstein model. & 2015 Elsevier Ltd. All rights reserved.
Keywords: GTAW Weld pool surface 3D Hammerstein model Dynamic nonlinear modeling
1. Introduction The gas tungsten arc welding (GTAW) [1] is the primary welding process for precision joining of metals because of its process stability and excellent weld appearance. In manual GTAW process, the quality of the produced welds is affected by human factors [2], such as fatigue, stress and emotional issues, which will not exist in the automated welding machines. Furthermore, as the health issues of the human welder become a hot topic, the automated and robotic welding is more and more widely used in manufacturing industry. The improvement of intelligent capabilities in welding production will greatly reduce the dependence on human welders, which will liberate a great deal of workforce and reduce the demand of workers' welding skill level, also the production efficiency will be improved significantly. Welding quality control and seam tracking technology are the most fundamental topics in automated robotic welding [3], the perfect solution of the seam tracking and the welding quality control is the precondition of realizing the welding automation, and the welding quality control is the major difficulty. In the process of actual welding, the weld pool always contains a wealth of useful information, which tends to reflect the weld penetration and other factors related to welding quality [4]. The skilled welder always utilizes the observed information from the weld pool to control the welding quality. Thus it can be seen that an intelligent welding process control system can be established by imitating the control characteristics of skilled welder. So n
Corresponding author. E-mail address: [email protected] (H. LI).
http://dx.doi.org/10.1016/j.rcim.2015.11.004 0736-5845/& 2015 Elsevier Ltd. All rights reserved.
some sensing methods should be used as the eyes of the welder to observe the welding pool and obtain the useful information. Up to data the main sensing methods experimented include the ultrasonic sensing [5], the infrared sensing [6], X-ray sensing [7] and the vision sensing [8–14]. The vision sensing has better application prospects because its function is much more similar to the eyes of the human welder and it also can get more information of welding pool than others. According to the dimensions of the acquired welding pool geometric parameters, the vision sensing can be divided into 2D vision sensing and 3D vision sensing. In this paper, a vision sensing technology for 3D weld pool surface invented and developed at the University of Kentucky is used. This technology considers the weld pool surface as a mirror. A laser pattern is projected on the specular surface of the weld pool and its reflection from the specular surface is intercepted [15–17]. The reflection law is used to compute the weld pool surface that determines the reflection of the projected laser pattern. At the University of Kentucky, the technology has been developed to be able to measure the weld pool surface in real-time in 20 ms [18]. The 3D vision-based sensing system for GTAW process will be mentioned in the following section, which consists of an illumination laser with a 19-by-19 dot matrix structured light pattern, a camera, and an imaging plane. GTAW process is a multiple inputs multiple outputs, nonlinear, time-varying, and strong coupled process with many uncertainties [19,20]. The relationships of the inputs and the outputs are also nonlinear. In order to get the control system of the GTAW, the relationship of the input welding parameters and the output welding pool geometric parameters must be established at first. At the University of Kentucky, researches have been conducted to characterize the 3D weld pool surface using the proposed 3D weld
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pool sensing system [21], and the characterized weld pool parameters (weld pool width, length, and convexity) are utilized to estimate and control the back-side weld penetration [22,23]. Such 3D sensing system also found its application in learning human welder behaviors [24–27]. Specifically, in [28], a nonlinear model for 3D weld pool characteristic parameters is proposed for GTAW process. The cross couplings between the weld pool characteristic parameters are further modeled using a neuro-fuzzy model. It is found that this nonlinear model can provide detailed knowledge about the complex correlation between the weld pool surface and welding process inputs. However, implementing such complex nonlinear neuro-fuzzy model based control algorithm might impose additional challenge on controller design. Thus, a more practical approach to model the 3D weld pool parameters is of interest in current manufacturing industry. As the welding current is the most important welding parameter, this paper establishes a dynamic nonlinear model based on simple Hammerstein model to describe the correlation between the welding current and the weld pool geometry. The remainder of the paper is organized as follows. The vision-based sensing system, the welding process and the controlled variables are described in Section 2. In Section 3, the step responses are conducted and the time-domain analysis of the system is done. The dynamic relationship is identified by means of a non-linear Hammerstein model in Section 4. And the model analysis is also detailed in this section. Conclusion is finally drawn in Section 5.
2. Experimental system and controlled parameter selection 2.1. Experimental system The experimental system is illustrated in Fig. 1, which consists of three parts, the vision-based sensing system, the moving system and the welding system. The moving system is controlled by a personal computer through the DAQ card and the motion controller. The adjusting range of the moving speed is 0–5 mm/s, and the adjustment of step length is 0.1 mm/s. The welding system is also controlled by the personal computer through the DAQ card. The analog output of the DAQ card realizes the control of the welding current. So the welding source can output different kinds of the current waveform through the program code that we write. The vision-based sensing system consists of laser generator, imaging plane, CCD camera, image acquisition card and the personal
computer. The laser power is 200 mW and the wavelength of the laser is 650 nm. It is used to project a 19-by-19 dot matrix structured light pattern on the weld pool region through the dot-array diffraction grating. As the surface of the weld pool just like a mirror, which can reflect the laser falls on it. So the dots project inside the weld pool will be reflected by the specular weld pool surface, and the other will not. The reflected light beam of the dots will be affected by the shape of the 3D weld pool surface, and on the other hand the shape of weld pool can also be calculated through the reflected light beam that obtained. An imaging plane is located at a 150 mm distance from the torch to get the spatial position of the reflected dots in a confirmed space plane. And a camera is needed behind the imaging plane to acquire the image of the reflected dots on the plane. The geometrical information of the weld pool will be calculated through the processed image by the personal computer. Fig. 2 shows the image of the reflected dots on the imaging plane and reconstructed 3D weld pool surface. 2.2. GTAW process In this paper, the weld pool is controlled mainly by the welding current with other welding parameters keeping invariant. When the welding current is respectively using 40 A, 50 A, 55 A, 60 A, 65 A and 70 A, the images of the reflected dots are shown in Fig. 3. As shown in the Fig. 3, the number of the reflected dots increases with the welding current, that means the size of the weld pool also increases with the welding current. When the current reaches 70 A, the dots gather to the middle position (see Fig. 3(f)). The phenomenon is caused by the concave of the weld pool, which is caused by the increasing of the penetration. So in the case of no wire filling welding and the current is too small, the weld pool is hardly to form, so it’s difficult to get the image of the reflected dots. But on the contrary, when the current is too large, it will be difficult to distinguish one point from the other, and the reconstruction of the 3D weld pool will also be impossible, just like Fig. 3(f) shown. Considering the above situation, the welding current should be suitable to ensure the size of the weld pool is big enough and the weld penetration is suitable. So the welding condition of the experiment in this paper is shown in Table 1. 2.3. Controlled parameters selection The controlled parameters are the physical quantities that are asked to keep the set value (close to a constant value or change
Fig. 1. 3D vision-based sensing experimental system.
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Fig. 2. Original image and the result of reconstructed 3D weld pool surface. (a) Original image and (b) 3D weld pool surface.
following some variable) in the experimental process. Actually, the welding quality largely depends on surplus height of the seam and the weld penetration, so the welding quality can be controlled through the control of the surplus height of the seam and the weld penetration. Surplus height of the seam and weld penetration relate to the volume of the weld pool higher than the workpiece, so the most direct way to control the surplus height of the seam and the weld penetration becomes to control the volume of the weld pool. As the laser dots arrange densely, the volume of the weld pool can be calculated through small volume that is divided according to the position of the point. Because every dot is close enough to its neighbors, the small volume can be approximately calculated by Eq. (1), and every area ( ΔS ) can be approximately equal in size. The estimation method of the weld pool volume is illustrated in Fig. 4.
(1)
dV = hi ⋅ΔS
Where hi is the height of the laser dot from the weld pool surface to the workpiece. So the volume of the weld pool can be calculated by:
V=
∑ hi⋅ΔS
(2)
The Eq. (2) can also be shown by the following formula:
Table 1 Welding process of no wire filling GTAW. Workpiece material Stainless steel Argon flow Workpiece dimensions 300 mm 40 mm 2 mm Welding current Tungsten electrode 2.4 mm Welding diameter speed
V = ΔS
∑ hi
5–6 L/min 40–60 A 1.2 mm/s
(3)
Where ∑ hi is the sum of the height from the reflected dots in the weld pool surface to the workpiece. As the area ( ΔS ) of every small volume is approximately equal in size, the ΔS can be approximately treated as constant. So the volume of the weld pool can also be expressed by ∑ hi . And the sum of the height ∑ hi is selected to be the main controlled parameter in this paper. However, the ∑ hi first increases and then decreases with the increase of welding current. The reason of that appearance is that the molten metal increase with the increase of welding current during the lack of penetration, which makes the volume of weld pool increased. As the welding current increases further, the weld pool will drip after the penetration occurred, which makes the volume of weld pool decreased. It means not only a value of the
Fig. 3. Images of reflected dots on imaging plane under different current. (a) 40 A, (b) 50 A, (c) 55 A, (d) 60 A, (e) 65 A, and (f) 70 A.
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Fig. 4. The estimation of the weld pool volume.
∑ hi corresponds to a state of the weld pool, so another controlled parameter should be introduced to distinguish the two states of the weld pool. The width of the weld pool increases with the increase of the welding current, which can be inferred from the Fig. 3. So the width of the weld pool is selected to be the auxiliary controlled parameter in this paper.
3. Step responses The step response experiment is done with the welding current as the input signal, and the step span of the welding current is 20 A. The specific welding parameters are shown in Table 1. The welding current of the positive step experiment changes from 40 A to 60 A, and it changes from 60 A to 40 A in the negative step experiment. The vision-based sensing system is used to reconstruct the 3D GTAW pool surface, and the ∑ hi and the width are obtained through the 3D GTAW pool surface. The interval of the weld pool image acquisition is 1 s. The change of the width and the change of the ∑ hi during the positive step experiment and the negative step experiment are plotted, respectively, in Fig. 6 and Fig. 8. The change of the reflected dots and the corresponding reconstructed 3D weld pool surface are shown in Fig. 5 and 7. It can be seen from Figs. 6 and 8 that there is no oscillation of the response output curves, so the system can be described by a first-order system, and the corresponding transfer function can be represented by:
G (s ) =
K e−τs 1 + Ts
(4)
Where T is the time constant of step responses, K is the gain coefficient, and τ is the time-delay constant. The key to system
identification is to determine the parameters T , K and τ . Due to the recognition method is relatively mature, so the MATLAB system identification toolbox is directly used to identify the parameters by the results of the step responses. The result of the parameter identification is shown in Table 2, where the parameters ( T , K and τ ) of the corresponding transfer functions with the positive and negative steps are presented. Through the analysis of the identification result, some conclusions are drawn as follows: Welding process is a multivariable and strongly coupled process, and the geometric parameters of 3D weld pool are affected by many factors that coupled together. While keeping other welding parameters constant, the whole variant trend of weld pool width and the height-sum is increasing with the increase of welding current. The time-delay of the weld pool response to the welding current is small. The system is a nonlinear system, and it is difficult to describe the system through a simple linear model, so a nonlinear dynamic model must be established.
4. Dynamic nonlinear modeling The dynamic nonlinear model between welding current and geometric parameters of weld pool is established through model identification by using a simple Hammerstein model. The established model is also validated through other experimental data in this section. 4.1. Design of identification signal The random experiment is done to identify the model by using white noise of welding current as input signal, whose average
Fig. 5. The change of the reflected dots (a) and the corresponding reconstructed 3D weld pool surface and (b) in positive step experiment.
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Fig. 6. The change of the weld pool width (a) and the change of the weld pool height sum and (b) in positive step experiment.
Fig. 7. The change of the reflected dots (a) and the corresponding reconstructed 3D weld pool surface and (b) in negative step experiment.
value is time-independent. According to the results of the previous experiment, the weld pool can be large enough without dripping by using the welding current near 55 A (see Fig. 3). So the variation range of welding current is limited between 50 A and 60 A, the output frequency of welding current is 1/3 Hz, and the other welding parameters remain unchanged as shown in Table 1. The MATLAB is used to generate the white noise as input signal, which is shown in Fig. 9. The previous mentioned vision-based system is used to reconstruct the 3D weld pool in the random experiment, and the pool width and the sum of pool height are obtained through the reconstructed pool in every moment. Figs. 10 and 11 have shown the change of the width and the height sum by using the random
current as the input. 4.2. Dynamic system identification and modeling It can be found that there are some nonlinear factors in the controlled parameters through step responses, which means that the controlled system is the nonlinear system. Specifically, the main steps of the system identification are as follows: The model structure is determined through prior experiments and purpose; identifying structure and parameters of the system model by collecting and processing input and output signal; finally, testifying the model. And the most important part is to identify the structure and parameters of the system model. The single input
Fig. 8. The change of the weld pool width (a) and the change of the weld pool height sum and (b) in negative step experiment.
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Table 2 The parameters of the corresponding transfer functions. Weld pool parameters Positive step
Negative step
K T τ K T τ
width
Height-sum
0.4121 5.754 0.187 0.449 8.774 0.0722
7.3 27.81 0.336 0.2797 1.431 0.044
Fig. 11. The sum of weld pool height.
Fig. 9. The input signal of welding current.
Fig. 12. Structure diagram of modular model.
Fig. 13. Structure diagram of simple Hammerstein model.
G (z ) =
( ) ( )
z − dB z − 1 A z −1
(6) B (z −1) = 1 + b1z −1 + ⋅⋅⋅ + bnb z −nb .
Where and The expression of the dynamic system model can be expressed as formula (7) by means of least square method. A (z −1) = 1 + a1z −1 + ⋅⋅⋅ + ana z −na
Fig. 10. Weld pool width.
single output nonlinear system has several commonly used structures, such as the series model of Volterra, the modular model and so on. And the modular model is mainly composed of dynamic linear module and static nonlinear module, which is shown in Fig. 12. Hammerstein model is a typical modular model, which is mainly composed of dynamic linear module and static nonlinear module in series. It includes simple Hammerstein model and generalized Hammerstein model. The generalized Hammerstein model can conduct a comprehensive and detailed description about the nonlinearity of the system, but at the same time, it also has a complex form and a lot of parameters. Although, the accuracy of the simple Hammerstein model is a bit lower than the generalized Hammerstein model, but it can still describe some nonlinear process well. Furthermore, it also has a simpler form and less parameter than the generalized Hammerstein model, which make it widely used in industrial production. The structure of the simple Hammerstein model is depicted in Fig. 13. The memoryless nonlinear function can be expressed by the following formula based on Taylor series expansion:
x (k ) = C0 + C1u (k ) + C2 u2 (k )
(5)
And the transfer function of dynamic linear system can be represented as shown in formula (6) [29].
y (k ) = − a1y (k − 1) − ⋯ − a na y ( k − na ) + b + c0 u (k − d) + ⋯ 2
+ cnb u ( k − d − nb ) + d0 u (k − d)2 + ⋯ + dnb u ( k − d − nb ) = φT (k ) θ
(7)
T
Where φ (k ) is data vector, θ is parameter vector that needs to be estimated. ε (k ) is used to stand for the difference between the actual output and the estimated output, which is called residual. It can be calculated by Eq. (8).
ε (k ) = y (k ) − y^ (k ) = y (k ) − φT (k ) θ^
(8)
So the sum of the squared residuals function J is used as the performance indicator, which is shown in formula (9). A batch algorithm based on least square method is used to determine the model structure parameters θ^ , which make the value of function J to reach the minimum. Fig. 14(a) and (b) shows the minimum of the function J correspond to different values of N 、 d to the width and the height-sum.
J=
2
∑k = 1 ε2 (k) = ∑k = 1 ⎡⎣ y (k) − φT (k) θ^⎤⎦ L
L
(9)
To the dynamic model of the width, it can be found in Fig. 14 (a) that the value of the function J reaches the minimum, when d = 0, N = 5; And to the dynamic model of the height-sum, it can be found in Fig. 14(b) that the value of the function J also reaches the minimum, when d = 0, N = 5. So the parameter vector θ can be
Z. JIN et al. / Robotics and Computer-Integrated Manufacturing 39 (2016) 1–8
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Fig. 14. Change curve of minimal J to width model (a) and change curve of minimal J to height-sum model (b).
y1 [k ] = 0.5927y1 [k − 1] + 0.371 × y1 [k − 2] − 0.0433 × y1 [k − 3] − 0.0379 × y1 [k − 4] − 0.1496 × y1 [k − 5] + 5.6261 − 0.1995 × u [k ] − 0.0056 × u [k − 1] − 0.1536 × u [k − 2] + 0.055 × u [k − 3] + 0.0253 × u [k − 4] + 0.0671 × u [k − 5] + 0.0025 × u [k ] × u [k ] − 0.0003 × u [k − 1] × u [k − 1] Fig. 15. Fitting situation of width model.
+ 0.0013 × u [k − 2] × u [k − 2] − 0.0005 × u [k − 3] × u [k − 3] − 0.0003 × u [k − 4] × u [k − 4] − 0.0005 × u [k − 5] × u [k − 5]
(11)
y2 [k ] = 0.3339y2 [k − 1] + 0.3275 × y2 [k − 2] + 0.0284 × y2 [k − 3] + 0.191 × y2 [k − 4] − 0.2 × y2 [k − 5] + 11.3284 − 0.0536 × u [k ] − 0.0778 × u [k − 1] − 0.1153 × u [k − 2] + 0.148 × u [k − 3] − 0.1997 × u [k − 4] − 0.1983 × u [k − 5]
Fig. 16. Fitting situation of height-sum model.
+ 0.0014 × u [k ] × u [k ] + 0.0005 × u [k − 1] × u [k − 1] calculated by Eq. (10).
θ^ = ϕT ϕ
−1 T
( )
ϕ Y
+ 0.0009 × u [k − 2] × u [k − 2] − 0.0015 × u [k − 3] × u [k − 3] (10)
+ 0.0019 × u [k − 4] × u [k − 4] + 0.0024 × u [k − 5] × u [k − 5]
Where ϕ = [φT (1) , φT (2)⋅⋅⋅φT (L )]T , Y = [y (1) , y (2)⋅⋅⋅y (L )]T and L is the number of the sample points. The θ^ can be calculated by the Eq. (10). So the dynamic relationship between the welding current and width can be obtained as the θ^ is known. And the dynamic relationship between the welding current and height-sum can also be obtained in a similar way. The relationships are shown in formulae (11) and (12), and besides that, the fitting situations of the two models are shown in Figs. 15 and 16.
(12)
4.3. Model validation In this section, other experimental data are used to validate the established models. The results of the validations are shown in Figs. 17 and 18. Form the figures, it can be seen that the established models are relatively reliable, and the changing trend of the width and the height-sum that estimated fits the measured changing trend well.
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response to the welding current is small. Finally, a dynamic nonlinear model based on simple Hammerstein is established to relate the welding current with the weld pool width and the weld pool height-sum.
References
Fig. 17. Validation result of width model.
Fig. 18. Validation result of height-sum model.
5. Conclusion In this paper a real-time vision sensing technology is used as the eyes of the welder to observe the gas tungsten arc welding (GTAW) pool surface and a 3D reconstruction technology is used to reconstruct the observed weld pool surface in three-dimensions. Different welding currents have been used, and the results of the experiment have been analyzed. The optimal welding current range is about 40–60 A during the experiment. The number of the reflect dots increases with the welding current, which indicates the increase of the welding pool size. The large welding current leads to the gathering of the reflected dots on the imaging plane, which indicates the concave of the weld pool. The controlled parameters width and height-sum are selected through the analysis. The height-sum is selected as the main controlled parameter, and the width is selected as the auxiliary controlled parameter. The step responses have been done as the priori experiment, and the relationships between the welding current and the geometrical information of 3D weld pool surface are analyzed. While keeping other welding parameters constant, the whole variant trend of weld pool width and the height-sum is increasing with the increase of welding current. The time-delay of the weld pool
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