Evaluation Method of Magnetic Field Stability for Robotic Arc Welding Based on Sample Entropy and Probability Distribution

Abstract

In this study, we analyzed the arc magnetic field to assess the stability of the arc welding process, particularly in robotic welding where direct measurement of welding current is challenging, such as under water. The characteristics of the magnetic field were evaluated based on low-frequency fluctuations and the symmetry of the signals. We used double-wire pulsed MIG welding for our experiments, employing Q235 steel with an 8.0 mm thickness as the material. Key parameters included an average voltage of 19.8 V, current of 120 A, and a wire feeding speed of 3.3 m/min. Our spectral analysis revealed significant correlations between welding stability and factors such as the direct current (DC) component and the peak power spectral density (PSD) frequency. To quantify this relationship, we introduced a novel approach using sample entropy and mix sample entropy (MSE) as new evaluation metrics. This method achieved a notable accuracy of 88%, demonstrating its effectiveness in assessing the stability of the robotic welding process.

1. Introduction

Arc welding is a versatile welding process used in various industries to join metals using an electric arc [1]. There are several types of arc welding, including gas metal arc welding (GMAW) [2], gas tungsten arc welding (GTAW) [3], and shielded metal arc welding (SMAW) [4]. Each type has its own advantages and limitations. For instance, GMAW is efficient for thick materials but may produce excessive spatter [5], while GTAW offers high precision for thin materials but is slower and more complex [6]. SMAW is portable and versatile but can be less efficient for long welds and produces more slag [7].

Robotic arc welding, a subset of arc welding, leverages robotic arms to perform welding tasks, offering enhanced precision, consistency, and automation [8]. This technique is increasingly being adopted in industries such as automotive, aerospace, and manufacturing. Despite its advantages, robotic arc welding faces challenges, particularly in environments where direct measurement of welding current is difficult, such as under water [9]. The stability of the welding process is crucial for ensuring high-quality welds and efficient operations.

The stability of the welding process can often be linked to the symmetry of arc magnetic field signals [10]. Symmetry, as a fundamental characteristic, provides insights into the regularity and predictability of the welding arc, which are essential for identifying and mitigating instabilities caused by factors such as bubbles or short circuits. The use of magnetic arc analysis aims to assess these instabilities by evaluating signal characteristics like the direct current (DC) component and the max power spectral density (PSD) frequency.

Entropy is used to assess the stability and complexity of the system by quantifying the disorder or uncertainty within the arc magnetic field signals. There are other techniques available for similar assessments, including oscillograms, probability density distribution, acoustic signals, current and voltage cyclograms, box plots with frequency histograms, ellipse parameters plotted on current, voltage cyclograms, and standard variation and coefficients of variation for welding current and arc voltage [11,12,13,14]. However, these traditional methods may not fully capture the intricacies of the welding process in environments like underwater welding. Our research introduces a novel approach using sample entropy and mix sample entropy (MSE) to provide a more precise gauge of the welding process’s stability.

Yao et al. studied process and parameter optimization of the double-pulsed gas metal arc welding (DP-GMAW) process [15]. A DP-GMAW process based on robot operation using twin-pulse XT DP control technology was employed to join stainless steel base plates. Panda et al. measured two bead dimensions, bead height and bead width, based on three input variables, peak current, wire feed speed, and travel speed, using the gas tungsten arc welding machine [16]. Zhang et al. described the laser-metal active gas (laser-MAG) hybrid welding of high-strength steel (HSS) formulated with 1.2 mm diameter austenitic stainless steel (ASS) filler wire [17]. Rossini et al. found the best combination of parameters considering the three tested welding conditions where the lead wire was parameterized in constant voltage mode and the trail wire was used for pulsed spray metal transfer with a torch angle of 90 degrees from the workpiece [18]. Wang et al. employed the four-index evaluation methods to determine arc stability [19]. They used oscillograms, cyclograms, probability distribution, variation coefficients of the welding current, and arc voltage to study the effect of ultrasonic waves on arc stability. Xia et al. implemented vision-based feedback control for the layer width during the wire arc additive manufacturing (WAAM) process [20]. Vasilev et al. presented the development and deployment of a novel multi-robot system for automated welding and in-process non-destructive examination (NDE) [21].

The stability of the arc welding process is a critical factor for ensuring the quality of welded products. Over the years, numerous studies have been conducted to monitor the stability of welding processes. Cayo et al. studied a non-intrusive gas metal arc welding (GMAW) process quality monitoring system using acoustic sensing [22]. Luksa et al. collected process signals online through an experimental platform and analyzed welding quality based on signal variations [23]. Hermans et al. collected electrical signals during the welding process and analyzed the frequency of short circuits, concluding that the welding process was stable when the short circuit frequency matched the frequency of the weld pool oscillations [24]. Adolfsson et al. utilized a repeated sequential probability ratio test (SPRT) algorithm to detect minor changes in weld voltage, demonstrating the potential for automatic online detection of changes in weld quality [25].

While these studies have addressed certain aspects of process stability, they primarily focused on detecting abnormal phenomena during welding rather than quantitatively analyzing the overall process stability. Moreover, arc welding is a highly nonlinear and multi-field coupled process. Relying solely on limited electrical signals may not fully capture its complex characteristics. Hence, it is essential to incorporate additional signals that contain sufficient process information to effectively evaluate and achieve high-quality welding products.

Arc magnetic field signals generated during arc welding processes contain a wealth of valuable information. Skilled welders can assess the stability of the welding process solely based on observations of the arc magnetic field. Wang et al. (2020) conducted a study on GMAW and utilized arc magnetic field signals to investigate arc and droplet behaviors, identifying irregular changes in arc magnetic fields as indicators of process instability [26]. Martim et al. also explored the relationship between other operational variables of the welding process and the generation of magnetic fields in various welding methods [27,28]. This relationship indicates a close association between arc magnetic field signals and the stability of the arc welding process, making them a valuable tool for evaluating arc welding process stability.

In the past, arc magnetic field signals have been characterized using time-domain representations, frequency spectrograms, and power spectrograms. However, these conventional representations do not fully capture the simultaneous relationship between time, frequency, and power spectral density. Guan et al. conducted online monitoring of weld quality using different indicators in both the time and frequency domains of the arc magnetic field [29]. They suggested that combining the advantages of both domains is crucial, as each domain analysis has its strengths and weaknesses, and neither approach alone comprehensively captures all the signal characteristics.

To overcome this limitation, magnetic field spectral analysis technology was developed, introducing the use of spectrograms or sonograms. Spectrograms provide the visual characterization of variations in magnetic field signals or other signals as a function of time or other selected variables. In this study, the spectrogram, which also represents the magnetic field spectrum, is a colorful plane figure generated through the spectral decomposition of magnetic field signals. Time is represented on the X-axis and frequency on the Y-axis, and each pixel corresponds to the power spectral density (PSD) of the signal at a specific time and frequency. Information about signal stability was obtained by magnetic field spectral analysis, allowing for observation of variations in power spectral density and frequency waveforms. This approach enables a comprehensive assessment of the stability of the magnetic field signals during arc welding processes.

The spectrogram is a widely recognized time–frequency distribution method extensively used for analyzing signals exhibiting varying power in both time and frequency domains. It has found wide applications in various fields, including diagnostics of electrical and mechanical systems (Oravec et al., 2021, Moon et al., 2018) [30,31], evaluation of ferromagnetic materials (Tian et al., 2022, Maciusowicz et al., 2019) [32,33], and measurement of energetic-electron-driven emissions (Wang et al., 2023) [34], among others. The versatility and effectiveness of the spectrogram make it a valuable tool for signal analysis in diverse scientific and engineering disciplines.

Given the aforementioned research and applications, utilizing magnetic field spectral analysis technology to analyze arc magnetic field signals and evaluate the welding process’s stability is indeed feasible. However, there are certain limitations associated with this approach. The pixelated nature of the information presented in the magnetic field spectrum and the dependence on the analysts’ experience and interpretation standards may lead to considerable errors in the analysis. Additionally, the information obtained from the spectrogram may lack repeatability, making it challenging to conduct consistent and reliable analysis and post-processing. As a result, these disadvantages impose significant constraints on the practical application of arc magnetic field spectral analysis technology. Efforts are required to address these challenges and improve the robustness and reliability of the analytical process.

In recent years, approximate entropy has been used extensively to investigate the stability of welding processes. Tolle et al. observed an increase in the approximate entropy of the arc voltage with wire feed speed in GMAW [35]. Similarly, Cao et al. proposed the use of approximate entropy to quantify the stability of arc and welding processes in short-circuiting GMAW [36,37]. Additionally, Nie et al. employed neural networks to predict the approximate entropy of pulsed metal inert gas (MIG) welding of aluminum alloy, and achieved promising results [38]. Zhang et al. evaluated the effect of adaptive control using approximate entropy, and found that low approximate entropy values corresponded to better adaptive control performance [39].

However, these studies also highlighted certain limitations related to data length, embedding dimension, and the relationship between approximate entropy and welding signal stability. To address these limitations, Richman et al. introduced a new method for measuring time-series complexity based on the comparison between approximate entropy and sample entropy [40]. They concluded that sample entropy offered better consistency and accuracy with faster computing speed compared with approximate entropy. This improvement is attributed to sample entropy excluding self-match values and reducing errors in the analysis of welding signal stability.

At present, arc magnetic field detection and related research are still at the analytical stage and not ready for practical applications. Most studies have focused on exploring the characteristics of arc magnetic fields and their correlation with welding quality. Limited efforts have been made to quantitatively analyze the stability of the welding process using arc magnetic fields. Therefore, the main objective of this study is to develop a novel approach that utilizes our proposed experimental platform to collect original arc magnetic field signals. These signals are then processed by vector summation and analyzed through the sample entropy algorithm. We can quantitatively assess the stability of arc magnetic fields during the welding process by employing this method. The ultimate goal is to design new quantitative indicators and spectrograms that effectively evaluate the stability of arc magnetic fields in welding processes, such as the direct current component, and the frequency of the max power spectral density, enabling accurate quantification of arc welding process stability. In a word, this paper primarily aims to evaluate welding quality using the magnetic field generated during robotic arc welding, especially in situations where visual observation is not convenient, and to apply entropy algorithms to the maximum PSD time-domain sequences to assess welding stability.

2. Experimental Process and Method

2.1. Arc Magnetic Field Signal Acquisition and Characteristics Analysis

In this study, we employed a proprietary synchronous multi-information acquisition device to gather arc magnetic field signals. The data collection was carried out using the Narda ELT-400 (Pfullingen, Germany) exposure level tester, an industrial magnetic field sensor equipped with a B-field probe. The sensor can achieve a resolution of 1 nT and a frequency response spanning a range from 0 Hz to 400 kHz. The sampling frequency of ADC chosen for this research was 20 kHz. Our experimental setup included a custom-made welding inverter designed specifically for double-wire pulsed MIG welding. Figure 1 illustrates the comprehensive experimental platform used in our study.

The double-wire pulsed MIG welding platform consisted of a welding power supply (LORCH S5-RobotMIG, Lorch Schweißtechnik GmbH, Auenwald, Germany), a wire feeder, a six-axis robot (FANUC M-10iA, FANUC Corporation, Oshino, Japan), shielding gas systems (98% Ar + 2% CO₂), a magnetic field sensor (Narda ELT-400, Narda Safety Test Solutions GmbH, Pfullingen, Germany), an ADC (NI USB-6363, National Instruments Corporation (NI), Austin, TX, USA) and a control computer. The welding process utilized Q235 steel with an 8.0 mm thickness for bead-on-plate welds. We used a welding wire with a 1.2 mm diameter, specifically ER316. Also, 98% Ar + 2% CO₂ served as the shielding gas, at a flow rate of 15 L/min. The average voltage, current, and wire feeding speed were recorded as 19.8 V, 120 A, and 3.3 m/min, respectively.

To explore the efficacy of the stability assessment of arc magnetic field signals through the DC component and max PSD frequency in the arc magnetic field spectrograms, we selected five arc magnetic field signals with identical operational parameters but varying welding speeds for in-depth analysis. These samples were obtained using the experimental platform depicted in Figure 1. We focused on the signals within the 0 to 5 s time frame after the arc initiation to ensure consistency in the results and facilitate analysis. Subsequently, the data were processed using LabVIEW 2018 [41].

Table 1. The parameter values of the five samples used in this study.

Parameter	Sample 1	Sample 2	Sample 3	Sample 4	Sample 5
Ip (A)	310	310	310	310	310
Ib (A)	90	90	90	90	90
Tp (s)	0.001	0.001	0.001	0.001	0.001
Tb (s)	0.00989	0.00989	0.00989	0.00989	0.00989
v (cm/min)	15	20	30	40	50

provides the specific values of these five key parameters for each of the five samples. The welding parameters in Table 1 were selected based on preliminary experiments. By keeping all other parameters constant and varying only the welding speed, we were able to intentionally create different degrees of arc magnetic stability for focused analysis.

Throughout the experimental process, several crucial current parameters were specified, including peak current (I_p), base current (I_b), peak time (T_p), base time (T_b), and welding speed (v). A comprehensive explanation of these parameters is shown in Figure 2.

Figure 3a presents the arc magnetic field signal, where the background magnetic flux density measured is expected to be less than 0.4 μT, which is considered negligible and does not require noise reduction processing of the raw signal. The raw signal consists of three channels corresponding to the magnetic flux intensity in the X-Y-Z directions. To explore the frequency domain characteristics of magnetic induction intensity, it is essential to conduct a fast Fourier transform (FFT) on the synthesized magnetic flux density obtained from the three axes. The averaged vector sum of the magnetic flux density, B, is calculated as follows:

$$B=\sqrt{B_X^2+B_Y^2+B_Z^2},$$

(1)

The pretreated signals obtained through vector summation are shown in Figure 3b. The signals from the three axes in Figure 3a have been combined into a single signal in Figure 3b. The signal characteristics are preserved after vector summation. At 1.5 s, the magnetic flux density reaches its maximum value in Figure 3a,b, indicating that the signal characteristics are well retained after the vector summation process.

After applying FFT to the arc magnetic field signals [42], the PSD is obtained and represented on a logarithmic scale, as shown in Figure 3c. It is evident that the frequencies of the arc magnetic field signals are primarily found below 500 Hz, with a significant power concentration within the frequency range of 0 to 300 Hz, particularly at 0 Hz (DC component), which can reflect the characteristics of DC signals. Above 1000 Hz, the power per hertz ratio gradually decreases.

To overcome the limitations of traditional analysis methods that only focus on time-domain or frequency-domain scenarios, we applied windowing techniques to the signals, which enabled us to study the overall spectral variation with time. This was achieved using short-time Fourier transform (STFT). Given that the stability analysis of the welding process primarily focuses on factors such as bubbles and short circuits, a high temporal resolution becomes essential. Therefore, for this study, we specifically selected a Hamming window with a length, n, of 4700 to ensure accurate and detailed analysis [43].

After applying STFT to the arc magnetic field signals, as shown in Figure 3b, the resulting spectrogram is presented in Figure 4. The time variable is represented along the X-axis, while the frequency is denoted along the Y-axis. Each pixel on the spectrogram indicates the PSD of the signal at a specific time and frequency. The corresponding color of the pixel changes as the PSD varies. Deeper colors on the spectrogram correspond to relatively high PSD values at the given point. The graph clearly illustrates that frequencies with high PSD values are mainly concentrated within the range of 300 Hz.

Around 1.5 s, the high PSD value distributions are irregular and scattered, indicating the presence of magnetic fields of unknown origin during that period. Between 1.6 s and 5 s, a uniform and regular PSD distribution is observed, creating a band around 90 Hz and the harmonic frequencies, resulting in similar colors in that frequency range. The results obtained from the spectrogram analysis consistently align with the actual scenarios, indicating that the stability of the arc magnetic field signals can be monitored by analyzing the PSD variations within the spectrogram.

Figure 5a shows the corresponding 3D representation of the arc magnetic field signals. The unstable intervals observed in the spectrogram indicate instances of magnetic flux density signal variations, suggesting the presence of bubbles or short circuits at specific frequencies and times. Particularly, the significant occurrence of a bubble or short circuit is observed around 1.5 s, with a corresponding frequency centered at 87 Hz. The peak amplitude of the signal is measured at 2.08 dBμT. Between 1.6 s and 5 s, the arc magnetic field demonstrates relative stability, with no single peaks present in this time interval. The DC component of the arc magnetic field is displayed in Figure 5a. It reveals that a less stable magnetic field is associated with a more irregular DC component. Since the DC component of the arc magnetic field can be analyzed separately, the focus is on the PSD between 20 Hz and 300 Hz, as shown in Figure 5b, which also applies to the remaining 3D figures in this paper.

In stable welding processes, the arc magnetic field exhibits a steady, low-noise pattern, and the corresponding waveform appears regular. Simultaneously, the max PSD exhibits minimal changes and demonstrates high regularity on the time–frequency interface. On the other hand, in unstable welding processes, the arc magnetic field experiences slight instability due to the presence of bubbles and short circuits. Consequently, the max PSD shows larger variations in the spectrogram. The more unstable the arc magnetic field signals are, the more changes and irregularities can be observed in the max PSD.

2.2. DC Component of Magnetic Field and Frequencies of Max PSD Sample Entropy Evaluation Method

The analysis of the spectrogram of the arc magnetic field confirmed that the stability of the arc welding process is directly associated with the regularity of the DC component of the magnetic field and the frequencies of max PSD in the time–frequency domain, which can reflect the characteristics of AC signals. Higher levels of instability in the welding process result in a broader distribution of the DC component of the magnetic field and the frequencies of the max PSD values. Entropy has been employed as the primary variable to quantify this disordered behavior. In this study, a novel methodology is introduced that leverages the concept of sample entropy to evaluate the stability of the arc welding process based on the distribution of the DC component and max PSD frequencies in the arc magnetic field.

This method involves several steps. First, the DC component is separated from the arc magnetic field in the spectrogram, and the parts affected by windowing at the beginning and end are removed. Subsequently, the frequency line of the max PSD is determined. The max PSD frequency line is formed by connecting the frequencies corresponding to the max PSD points within each time interval in the time–frequency domain. Then, the sample entropy method is employed to analyze the time coordinates and values of each point in both the DC component and the max PSD frequency sequence, to quantify their levels of disorder. The sample entropy method is further employed to multiply the DC component with the base-processed max PSD frequency after normalization. Finally, the proposed evaluation criterion is applied to assess the stability of the arc welding process.

The comprehensive calculation procedures are shown in Figure 6.

Procedure 1: acquisition of arc magnetic field signals throughout the welding process using the experimental platform.

Procedure 2: preprocessing the collected arc magnetic field data by applying vector summation to combine the signals from the three channels.

Procedure 3: construction of the spectrogram of the arc magnetic field signals, providing insights into the time, frequency, and PSD relationships.

Procedure 4: separation of the DC component from the spectrogram.

Procedure 5: determination of the max PSD at each time interval through mathematical comparison of the PSD values within 20 Hz to 300 Hz. This step involves the identification of the corresponding time and frequency, and the definition of the max PSD frequency line in the time–frequency domain. As a result, a sequence of max PSD frequency values can be obtained.

Procedure 6: mitigation of the effects of amplitude variations on the quantitative results by preliminary computing of the average of the DC component and the max PSD frequency by using the following equations:

$${\overline{B}}_{DC}={\displaystyle \sum _{i=1}^N{B}_{DC}\left(i\right)/N},$$

(2)

$${\overline{f}}_{PSD\_\max }={\displaystyle \sum _{i=1}^N{f}_{PSD\_\max }\left(i\right)/N},$$

(3)

where $B_{DC}\left(i\right)$ represents the ith element in the DC component, N signifies the total number of elements within the time sequence, $f_{PSD\_\max }\left(i\right)$ is the ith element in the max PSD frequency, ${\overline{B}}_{DC}$ denotes the average of the DC component, and ${\overline{f}}_{PSD\_\max }$ stands for the average of the max PSD frequency.

Procedure 7: determination of the sample entropy of the DC component and max PSD frequency, as follows [44]:

$$B_{DC\_SaEn}=SampEn\left(B_{DC},m,r,N\right),$$

(4)

$$f_{SaEn}=SampEn\left(f_{PSD\_\max },m,r,N\right),$$

(5)

where $B_{DC\_SaEn}$ and $f_{SaEn}$ are the sample entropy of the DC component and the corresponding max PSD frequency, respectively, for the data sequence $B_{DC}$ and $f_{PSD\_\max }$ used in Equations (2) and (3), m and r are the vector dimensions and the threshold for calculating the sample entropy, respectively, and $SampEn\left(B_{DC},m,r,N\right)$ and $SampEn\left(f_{PSD\_\max },m,r,N\right)$ are functions that we proposed and used in Matlab, and are described in detail in our previous publications [45].

Procedure 8: calculation of the standard deviation of the DC component and max PSD frequency, respectively, as follows:

$$B_{DC\_std}=\sqrt{{\displaystyle {\sum }_1^N{\left[B_{DC}\left(i\right)-{\overline{B}}_{DC}\right]}^2}/\left(N-1\right)},$$

(6)

$$f_{std}=\sqrt{{\displaystyle {\sum }_1^N{\left(f_{PSD\_\max }\left(i\right)-{\overline{f}}_{PSD\_\max }\right)}^2}/\left(N-1\right)},$$

(7)

where $B_{DC\_std}$ and $f_{std}$ denote the standard deviation of the DC component and the max PSD frequency sample entropy, respectively.

Procedure 9: calculation of the DC component sample entropy (DCSE) and the frequency sample entropy (FSE), as follows:

$$DCSE=B_{DC\_SaEn}\times B_{DC\_std},$$

(8)

$$FSE=f_{SaEn}\times f_{std},$$

(9)

Procedure 10: calculation of the base-processed max PSD frequency, as follows:

$$f_{PSD\_\max \_b}\left(i\right)=\left|f_{PSD\_\max }\left(i\right)-f_{base}\right|,$$

(10)

where $f_{PSD\_\max \_b}\left(i\right)$ is the base-processed max PSD frequency value at each point, and $f_{base}$ represents the max PSD frequency value with the most repetitions as the base frequency.

Procedure 11: calculation of the mix sample entropy (MSE) by multiplication, as follows:

$$f_{PSD\_\max \_b}\left(i\right)=\left|f_{PSD\_\max }\left(i\right)-f_{base}\right|,$$

(11)

$$M_{SaEn}=SampEn\left(M_{\mathrm{nor}},m,r_{\mathrm{mul}},N\right),$$

(12)

$${\overline{M}}_{\mathrm{nor}}={\displaystyle \sum _{i=1}^N{M}_{\mathrm{nor}}\left(i\right)/\left(N\right)},$$

(13)

$$M_{std}=\sqrt{{\displaystyle {\sum }_1^N{\left[M_{\mathrm{nor}}\left(i\right)-{\overline{M}}_{\mathrm{nor}}\right]}^2}/\left(N-1\right)},$$

(14)

$$MSE=M_{SaEn}\times M_{std}.$$

(15)

where $M_{\mathrm{nor}}\left(i\right)$ is the time series multiplied by $B_{DC}\left(i\right)$ and $f_{PSD\_\max \_b}\left(i\right)$, normalized separately, $B_{DC\_\min }$ are the minima of $B_{DC}\left(i\right)$, $B_{DC\_\mathrm{span}}$ is the interval suitable for the normalization of $B_{DC}\left(i\right)$, $f_{PSD\_\max \_\mathrm{span}}$ is the interval suitable for the normalization of $f_{PSD\_\max \_b}\left(i\right)$, $M_{SaEn}$ is the sample entropy of $M_{\mathrm{nor}}\left(i\right)$, $r_{\mathrm{mul}}$ is the threshold for calculating $M_{SaEn}$, ${\overline{M}}_{\mathrm{nor}}$ is the average of $M_{\mathrm{nor}}\left(i\right)$, and $M_{std}$ stands for the standard deviation of $M_{\mathrm{nor}}\left(i\right)$.

The evaluation criteria proposed in this study are DCSE, FSE, and MSE. DCSE reflects the instability level of the DC component of the magnetic flux intensity, while FSE reflects the instability level of the AC component of the magnetic flux intensity. The rationale for incorporating $B_{DC\_SaEn}$, $B_{DC\_std}$, $f_{SaEn}$, $f_{std}$, $M_{SaEn}$, and $M_{std}$ in the evaluation is that they exhibit similar variation tendencies, and their product has a more pronounced effect on the evaluation. By calculating the value of DCSE, FSE, and MSE, the stabilities of different arc magnetic field signals are determined by comparing their respective MSE values.

3. Results and Discussion

Figure 7 shows the arc magnetic field waveform spectrogram of Sample 1 after vector summation. Notable fluctuations in the magnetic field throughout the process with distinct stages from 0.1 s to 2.9 s and from 3.2 s to 4.7 s were observed. The most prominent variation was evident at 0.6 s. Figure 8 illustrates the welding seam of Sample 1. Spatters at around 0.7 s and 3.2 s were observed, primarily caused by short circuits or bubbles. These events lead to a sudden increase in current, which in turn causes a spike in the arc magnetic field, matching the fluctuations shown in Figure 7.

The 3D representation of the processed signal after STFT is displayed in Figure 9. The peaks observed in the image indicate abrupt changes in the arc magnetic field signal. These peaks sporadically emerge along with substantial magnetic field variations, signifying an unstable signal with significant fluctuations. The amplitude of the largest variation reached approximately 1.3 μT, while the entire signal exhibited intense variations with numerous fluctuations exceeding 0.5 V. Peaks were observed around 0.13 s, 0.74 s, 1.34 s, and 4.72 s, which agree with the observations made in Figure 7.

Figure 10 illustrates the arc magnetic field waveform spectrogram of Sample 2 after vector summation. The magnetic field signal suddenly increased to an amplitude of about 45 μT at 0.04 s to 1.7 s, and at 2.5 s and 4.7 s. In summary, this spectrogram demonstrates unsteady magnetic field variation with minor fluctuations. Figure 11 shows the welding seam of Sample 2. Spatters at around 0.7 s and 1.7 s were observed, consistent with the fluctuations shown in Figure 10.

Figure 12 displays the corresponding 3D image of Sample 2 after STFT processing. This signal is relatively more stable than Sample 1. Notably, undulating ridges are observed near 90 Hz and 180 Hz, indicating that the PSD of the arc magnetic field is not uniform.

Figure 13 shows the arc magnetic field waveform spectrogram of Sample 3 after vector summation. The PSD changes are relatively low. Only a slight magnetic field variation is evident at 1 s. Figure 14 illustrates the welding seam of Sample 3. Spatters at around 0.8 s and 2.1 s were observed, consistent with the fluctuations shown in Figure 13.

Figure 15 illustrates the 3D representation of Sample 3 obtained after STFT. The signal demonstrates a higher level of stability compared with the previous samples. The ridges near 90 Hz and 180 Hz are less undulating, indicating a more consistent PSD distribution in those frequency bands. However, some rounded-off ridges can be observed at around 1.2 s, 3.82 s, and 4.26 s, suggesting minor fluctuations during that timeframe.

In Figure 16, the arc magnetic field waveform of Sample 4 after vector summation is displayed, revealing relatively small amplitudes in the arc magnetic field and minimal changes in PSD. The magnetic field variation remained consistently low throughout the entire welding process. Figure 17 shows the welding seam of Sample 4. An undercut at around 4.2 s is noticeable, which matches the fluctuation shown in Figure 16.

Figure 18 depicts the 3D representation obtained after STFT processing. It is evident that the peaks near 90 Hz and 180 Hz exhibit a smoother profile, with no abrupt changes, except at around 4.37 s. The magnetic field demonstrates only slight variations, indicating a stable signal characterized by steady magnetic field behavior.

Figure 19 displays the waveform of Sample 5, which exhibits the lowest amplitude in the arc magnetic field and no significant changes in the PSD. Figure 20 shows the welding seam of Sample 5, with practically no welding defects except for some small spatters.

Figure 21 depicts the corresponding 3D image after STFT processing. The ridges near 90 Hz and 180 Hz appear to be the flattest, with no abrupt changes observed.

From the graphical representations, images, and analytical data presented earlier, it is clear that the stabilities of arc magnetic field signals showed gradual improvements going from Sample 1 to Sample 5.

The arc magnetic field spectrograms of the five samples are displayed in Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26. The white lines with blue hollow circles in the spectrograms represent the max PSD frequency lines.

In Figure 22, the max PSD frequency lines exhibit highly irregular patterns with significant amplitude changes. In contrast, Figure 23 shows irregular lines with comparatively lower amplitude variations compared with Figure 22. Figure 24 and Figure 25 display more regular max PSD frequency lines with smaller variation ranges. The variation range in Figure 24 is slightly larger than that in Figure 25. From 0 s to 0.8 s and from 3.8 s to 5 s, the max PSD frequency lines exhibit meandering behavior, corresponding to the ridges observed at 1.20 s, 3.82 s, and 4.26 s in Figure 15. Figure 26 demonstrates the least variation of the max PSD frequency lines. Based on these analytical results, it can be roughly inferred that the overall welding process is stable. However, a detailed qualitative description cannot be obtained at this stage.

For a more significant comparison of the five signals, the five max PSD frequency lines can be overlaid in one graph, as illustrated in Figure 27.

According to Figure 27, the max PSD frequency of Sample 1 is larger, exhibiting random peaks around 0.25 s, 1.77 s, 3 s, 3.5 s, and 4.2 s, with highly irregular variations. Sample 2 shows relatively regular max PSD frequency patterns, but still has some random fluctuations and mutations. Sample 3 demonstrates a regular max PSD frequency with less variation compared with the previous two samples, and no significant mutations are observed. Sample 4 exhibits more regular max PSD frequencies with only small variations around 1 s, 3.6 s, and 4.4 s. Sample 5 has the most regular max PSD frequency with the least variations. However, these variations are of short duration, and the phenomena are not clear.

Figure 28 shows the overlaid DC component curves of the five samples. It can be observed that the curves become increasingly flat going from Sample 1 to Sample 5.

Figure 27 shows that all frequencies are within 300 Hz. The base-processed frequencies in the range of 0 to 300 Hz are normalized to 1 by dividing by 300 Hz. From Figure 28, the difference between the maximum and minimum values of the DC component is within 10 dBμT. Each DC component line is normalized to the range of 0 to 1 by dividing by 10 dBμT after baseline correction. Finally, the normalized DC component and base-processed max PSD frequency for each sample is multiplied, with the results shown in Figure 29. It can be observed that the lines become increasingly flat and low going from Sample 1 to Sample 5.

In summary, based on the curves in this graph, the stability of Sample 1 is the worst and Sample 2 is only slightly better than Sample 1, whereas Sample 5 exhibits the best stability. The stability of different samples can be determined by analyzing their corresponding max PSD lines. For more reliable stability analysis results, the method proposed in Section 3 can be employed as described below.

The calculation of sample entropy involves determining three parameters: m, r, and N. Different sample entropies are obtained using different values of m and r. According to Lake et al. [44], the sample entropy distribution follows a normal pattern when m is small and r is large, and a small number of lost data points does not significantly affect the entropy calculation. After several trials, we selected $m=2$, $r=0.1$, $r_{\mathrm{mul}}=0.01$, $N=501$, $B_{DC\_\mathrm{span}}=10\mathrm{dB}\mathsf{\mu }\mathrm{T}$, and $f_{PSD\_\max \_\mathrm{span}}=300\mathrm{Hz}$ for the calculation, and the results are presented in

Table 2. Sample entropy evaluation results for the five samples studied.

Sample No.	${\mathit{B}}_{\mathit{D}\mathit{C}\_\mathit{s}\mathit{t}\mathit{d}}$	${\mathit{B}}_{\mathit{D}\mathit{C}\_\mathit{S}\mathit{a}\mathit{E}\mathit{n}}$	DCSE	${\mathit{f}}_{\mathit{s}\mathit{t}\mathit{d}}$	${\mathit{f}}_{\mathit{S}\mathit{a}\mathit{E}\mathit{n}}$	FSE	${\mathit{M}}_{\mathit{s}\mathit{t}\mathit{d}}$	${\mathit{M}}_{\mathit{S}\mathit{a}\mathit{E}\mathit{n}}$
1	1.177	0.5071	0.5966	51.31	0.5561	28.79	0.03083	1.662 × 10−1
2	0.7213	0.2932	0.2115	44.86	0.4431	19.88	0.02506	1.121 × 10−1
3	0.3561	0.2172	0.07734	41.71	0.4319	18.01	0.01100	6.334 × 10−2
4	0.3153	0.2013	0.06346	29.94	0.2861	8.565	0.005655	1.709 × 10−2
5	0.1720	0.1119	0.01925	17.92	0.2095	3.754	0.001831	6.961 × 10−3

.

Based on the calculations presented in Table 2, the values of each parameter decrease sequentially from Sample 1 to Sample 5. Furthermore, the corresponding values of DCSE, FSE, and MSE, which directly indicate the stability of the arc magnetic field, clearly demonstrate that the stability of the welding process improves from Sample 1 to Sample 5.

Based on the presented analysis and data, our proposed spectral method, including DC component and max PSD frequency sample entropy, proves to be highly effective for evaluating the stability of arc magnetic field signals. This method can be widely applied in assessing the stability of arc welding processes. The method offers reliable and convincing quantitative results. To further validate its effectiveness, additional studies were conducted, involving the collection and evaluation of 100 distinct arc magnetic field signals, which contained 20 specimens for each experimental set under the same conditions shown in Table 1. These samples comprised 29 unstable cases, 44 ordinary cases, and 27 stable cases, based on the knowledge of electrical signal stability, welding process observation, and welding seam quality. The method proposed in this paper was applied to these samples, and the spectrum containing the DC component and max PSD frequency sample entropy was determined for each sample.

After computing the final spectrum containing the DC component and the max PSD frequency sample entropy for each sample, the results were categorized into three zones based on the MSE criteria. MSE values below 0.0002 were classified as indicating stability, values lower than 0.0015 and equal to or above 0.0002 were categorized as indicating ordinary stability, and values equal to or above 0.0015 were classified as indicating instability. The scatter plot for the 100 samples is presented in Figure 30.

The distribution of MSE values is consistent with our expectations, with only 12 values falling outside their corresponding zones. Therefore, we can conclude that the accuracy rate reaches 88%. It is also noteworthy that the values of the remaining 12% deviate only minimally, indicating a high level of reliability of this method.

4. Conclusions

In this study, we analyzed the arc magnetic field to assess the stability of the arc welding process, particularly in robotic welding scenarios under water where direct measurement of welding current is challenging. We used double-wire pulsed MIG welding on Q235 steel with an 8.0 mm thickness by changing wire feeding speed.

Our results demonstrated that low-frequency fluctuations and the symmetry of the arc magnetic field signals are crucial indicators of welding process stability. By employing spectral analysis, we identified significant correlations between welding stability and factors like the direct current (DC) component and the peak power spectral density (PSD) frequency. We introduced sample entropy and mix sample entropy (MSE) as new metrics to quantify these relationships, achieving an accuracy of 88% in our assessments.

Furthermore, the differences in magnetic field stability, as shown in Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26, were intentionally caused by varying welding speeds. These differences highlight the potential of using magnetic field measurements to assess welding quality. Our findings suggest that analyzing the time–frequency signals of the magnetic field can effectively reflect the stability and quality of the welding process.

In conclusion, the use of entropy-based metrics and spectral analysis provides a reliable method for evaluating welding stability and quality. Future research will focus on refining these techniques and expanding their application to diverse welding environments, including underwater and automated robotic welding systems.