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\title{Quality Monitoring in Robotised Welding \\using Sequential Probability Ratio Test}
\author{Stefan Adolfsson\thanks{Department of Signal Processing, University of
Karlskrona$\backslash$Ronneby and Department of Production and Materials Engineering, Lund
University}, Ali Bahrami\thanks{Department of Production and Materials Engineering, Lund University
and Technology Center of Kronoberg, Vaxjo}, Ingvar Claesson\thanks{Department of Signal Processing,
University of Karlskrona$\backslash$Ronneby }}
%\date{February, 1994}
\begin{document} \maketitle
\begin{abstract}
This paper addresses the problem of automatic monitoring the weld quality produced by robotised
short arc welding. A simple statistical change detection algorithm for the weld quality, recursive
Sequential Probability Ratio Test (SPRT), is used. The algorithm may equivalently be viewed as a
cumulative sum (CUSUM) - type test. The test statistics is based
upon the variance of the amplitude of the weld voltage. The performance of the algorithm is
evaluated using experimental data. The results obtained from the algorithm indicate that it is
possible to detect changes in the weld quality automatically and on-line.
\end{abstract}
\chapter{Introduction}
\section*{Background}
An ongoing process of automatization of the production lines is implemented in industry in order
to reduce production costs. Automatization of quality control should be seen as part of the cost
reduction, as also should be quality control of welding.
Monitoring systems of weld parameters, such as ADM III, Arc guard, and Weldcheck are commercially
available.~\cite{Agren(1995),Blakeley(1992)}. They all work in a similar way; voltage, current and
other process signals are measured, presented and compared with preset nominal values. An alarm is
triggered when the difference from preset values exceeds a given threshold.
In the field of short arc welding of steel, both physical analysis of the welding process
\cite{Amson(1965),Allum(1985)} and statistical analysis of measured welding signals have been made
\cite{Leino(1984),Gupta(1988),Adolfsson(1996)}.
The objective of the present paper is to detect changes in weld quality automatically in short
arc welding using signal processing methods.
In order to achieve an uniform weld quality the welding process must be stable. The process
stability, i.e the characteristics of the welding should not change in an uncontrolled manner.
Experiments have shown that optimal stability occurs when the number of short circuits per seconds
are at their maximum~\cite{Pinchuk(1980),Smith(1966)}.
Thus, a suitable parameter for detection of changes in the weld quality, is the variance of the
amplitude of the weld voltage. This parameter is used to form a test statistics and this in its
turn is fed into a recursive Sequential Probability Ratio Test (SPRT)
algorithm~\cite{Basseville(1993)}. The algorithm may equivalently be viewed as a cumulative sum
(CUSUM) - type test. The SPRT is optimal in the sense that it minimizes the worst mean delay for
detection given a specified probability of false alarm~\cite{Lorden(1971)}. In addition, storage
and computational requirements for the recursive SPRT are less as compared to fixed sample-size
tests.
The paper is organized as follows. Section 2 describes some experiments. Changes in the weld
quality is provoked in weldings in a controlled way while the weld voltage and current from this
process are monitored. Some changes of the the variance of the amplitude of the weld voltage during
a weldingpass are observed. Section 3 deals with the design of the recursive SPRT algorithm. The
section concludes by showing how the algorithms are used to detect defects in the weld joint.
Section 4 deals with tuning and estimation of parameters used in the algorithm. Robustness of the
proposed algorithm is also considered. The recursive SPRT is then evaluated using experimental
data. The paper concludes with a discussion of the performance of the method in section 5.
\chapter{Welding technology}
\section{Short arc GMA welding}
%\subsection{Short circuiting welding}
\noindent The GMA welding process employs a consumable wire electrode passing through a
copper contact tube. See figure~\ref{Pow_circ_EPSF}. Electric current supports an arc
flowing from the end of the electrode to the work piece. The electrode is melted by resistive
heating, and heat from the arc. The region surrounding the weld puddle is purged with shield gas
to prevent oxidation and contamination of the
weld joint~\cite{Allum(1985),Amin(1981),Amin(1983),Goldman(1962)}.
\begin{figure}[htb]
%\centerline{\includegraphics[width=10cm]{Powny.epsf}}
\caption{A schematic illustration of equipment for short arc GMA welding. The electric current of the weld process is
denoted $I$. The internal resistance and indictance of the welding source is denoted $R_{i}$ and $L_{i}$ respectively. The resistance
of the wire electrode stick-out, i.e the part of the electrode between the contact tube and the arc, is denoted $R_{e}$. The length of the
electrode stick-out and the arc length are denoted $\ell_{e}$ and
$\ell_{a}$ respectively. The voltage over the wire electrode stick-out is denoted $U_{e}$. The
voltage between electrode tip and work piece is called the arc voltage and is denoted
$U_{a}$}
\label{Pow_circ_EPSF}
\end{figure}
%\begin{figure}[htb]
%\centerline{\includegraphics[width=7cm]{'filnamn'}}
%\caption{'Bildtext'}
%\label{fig:gdelay}
%\end{figure}
The advantage of the short cicuiting welding is that the mean current, and thus the average heat input
to the work piece, is lower than in direct current (DC) GMA welding. Due to the smaller heat transfer, it is
possible to weld thinner plates with short arc GMA than with DC GMA welding.
\begin{figure}[htb]
%\centerline{\includegraphics[width=10cm]{stefan1.EPSF}}
\caption{A schematic illustration of the weld voltage and current in short-cicuiting welding. $T_{a}$
and $T_{s}$ denote the peak pulse time and background pulse time respectively; and $I_{p}$
and $I_{b}$ denote the peak current and background current respectively.}
\label{idpuls}
\end{figure}
To limit the heat input to the work piece, the open circuit voltage is set at a low value
compared to (DC) GMA welding. The electrode is molten and a small droplet is developed at the electode
tip. This part of the cycle is denoted `arc time' and represented by
$T_{a}$. \cite{Amin(1981),Amin(1983),Allum(1983)}.
During short cicuiting time, $T_{s}$, the voltage will decrease to almost zero volt and the current will
increase to its maximum value. At this stage the arc will extinguish and a droplet is detached and transferred to the work piece. The
main force for detaching a droplet and transferring it, is the electromagnetic force induced by the current~\cite{Lancaster(1986)}.
After the transferring of the droplet the arc is re-ignited and the cycle starts over again.
The weld voltage $U_{w}$, arc voltage, $U_{a}$, and the voltage over the wire electrode stick-out, $U_{e}$ are related
by
\begin{equation}
U_{w} = U_{a}+U_{e} ,
\end {equation}
\section{Optimal welding conditions}
In order to produce weld joints with uniform weld quality it is desirable that the welding process is stable. Inter alia should the
metal transfer from the electrode wire to the work piece occur under stable and regular conditions as possible. Experiments have shown
that optimum stability occurs when the number of short circuits per second are at their maximum given that the short circuit time exceeds
$1$ ms, see figure~\ref{transf}. Thus, the welding process is said, in this report, to operating under optimal welding conditions
when the number of short circuits per second are at their maximum. Deviation from the optimal condition leads to a greater probability of
spatter, uneven weld bed and other fusion defects. In this case the welding process is said to operate under non-optimal condition.\\
\begin{figure}[htb]
\centerline{{\includegraphics[width=6cm]{stefan2.EPSF}} }
\caption{Short arc transfer frequency}
\label{transf}
\end{figure}
\noindent The number of short circuits per second is controlled by the open circuit voltage $V_{oc}$. When the open circuit voltage is set
at greater value than under optimal condition, the metal transfere will either be globular or spray. Globular metal transfer is
deemed to be unstable, while spray transfer is considered as a naturally stable process but not suitable for welding thin
plates.\\
\noindent For the three main main metal transfer modes, short-circuiting, globular and spray, there is a correlation between the
waveform and mode of metal transfer. It can be seen, moving from short- circuiting to globular and spray transfer, that the variation
range on the welding current and voltage wavwform reduces. The minimum and maximum current approaches the mean welding current, See
figur~\ref{metaltransfer} part a-c.\\
\begin{figure}[htb]
%\centerline{\includegraphics[width=4cm]{stefan2.EPSF}}
\caption{Waveforms for metal transfer modes}
\label{metaltransfer}
\end{figure}
\noindent When the open circuit voltage is set at lower value lower than under optimal condition, the heat to melt down the electrode
during arc time is not sufficient. The electrode has, then, to melt down during short-circuiting time. Since the short-circuiting time
increases so will the peak current and the variation range on the weld
current increases compared to optimal welding condition. In this
case the variation range on the weld current increases but the variation range on weld voltage waveform decreases.\\
\noindent Thus, the developed algorithm in this report is based on the hypotheses that the variance or the AC power decreases when the
welding process not work under optimal condition.
\chapter{Experiments}
\section{Aim of the experiments - provoke non optimal welding conditions}
The aim of the experiments is to provoke non optimal welding conditions in controlled manner while monitoring the weld voltage and
current from the process. Non optimal welding condition was provoked using a T-joint where gaps have been cut out in the standing
plate, see figure~\ref{arbetsstycke} part c. This specimen is
denoted a `T-joint with step disturbance'. During the step disturbance the welding process is operating under non optimal condition.
A second specimen shown in parts a and b is a T-joint with the standing plate in perfect contact with the laying plate. This specimen
was used to produce normal or reference weldings and is thus denoted a `reference T-joint'. During normal
welding the welding process is assumed to operate under optimal welding condition.
The specimens were each comprised of two rectangular $200 \times 10 \times 3$ mm plates of mild
steel SS $1312$. For the T-joint with step disturbance, the dimension of the gap was $ 2 \times 50$
mm. See figure~\ref{arbetsstycke} part c.
\begin{figure}[htb] \vskip 5mm
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 162mm by 160mm (arbets.epsf scaled 400)}}
\begin{picture}(0,0)(1.2,0.6)
%\begin{picture}(14.8,0)(1.2,0.6)
\put (17,30) {{\sf c)}}
\put (5,70) {{\sf a)}}
\put (54,70) {{\sf b)}}
\end{picture} \vskip -5mm
\caption{Steel T-joints provoking defects in weld joints. a) Reference T-joint, front view b)
Reference T-joint, side view c) T-joint with step disturbance, front view }
\label{arbetsstycke} \end{figure}
\section{Instrumentation}
The experimental setup is made up of a welding power source, a Motoman
robot carrying a welding torch, a positioner, a welding table and instrumentation for recording
weld voltage and current, see figure~\ref{setup1}. The welding torch is fixed in at angle of $45$
degrees to the welding table. The distance between the contact tube tip and the plate is $11$ mm.
\begin{figure*}[htb]
\vskip 0mm
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 158mm by 45.8mm (expset1.epsf scaled 750)}}
\begin{picture}(0,0)(0,0)
{
\put (19,18) {{\sf Power}}
\put (19,13.5) {{\sf source}}
\put (98,34) {{\sf measuring}}
\put (98,30) {{\sf buffer}}
\put (118,34) {{\sf data}}
\put (118,30) {{\sf acquisition}}
\put (68,14) {{\sf current}}
\put (68,10) {{\sf sensor}}
\put (34,20) {{\sf robot}}
\put (68,41) {{\sf voltage}}
\put (68,37) {{\sf measurement}}}
\end{picture} \vskip -5mm
\caption{The experimental setup is made up of welding power source a Motoman robot carrying a
welding torch, a positioner and instrumentation for recording welding voltage and current.}
\label{setup1} \end{figure*}
The weld voltage is measured between an electrode applied to the contact tube and a reference
electrode screwed into an aluminum plate which serves as an insulated welding
table~\cite{Adolfsson(1994)}. The current is measured with a current sensor, LEM Module LT 500-S,
equipped with a transformer. The sensor is mounted around the return conductor. The sampling
frequency is $8.192$ kHz, and the resulting lowpass filter has a cut frequency of $1.0$ kHz. The
data are then transferred for permanent storage to a personal computer.
Two different commercial welding equipment, Migatronic BDH S50 and Kemppi P500, were used for the
experiment. The wire feed rate was measured to be approximately $113-120$ mm/s and the nominal
welding speed was set at 10 mm/s. The filler wire material used in the experiment was ESAB OK 12.51
with a diameter of $1.0$ mm. The shielding gas used was Atal: $80$\%Ar/$20$\%CO$_{2}$. The flow rate of shielding gas
was set at 15 l/min.
\section{Measurments}
\subsection{Experimental procedure}
\noindent Before starting to measure, the specimen is positioned on the aluminum
plate and fixed by a fixture. Since the torch is attached to the Motoman
robot, the welding speed is
determined by the speed of the robot. The weld speed is set at $8$, $10$ and $12$mm/s, respectively. The robot
is started manually by the operator, who then starts the weld source. This in turn
sends a trigger signal to the acquisition system, which starts to record data. For the T-joint with step disturbance, the
operator starts the weld sex centimeter before the cut. See figure~\ref{arbetsstycke}. For the reference T-joint, the weld is started 2 cm
from the edge side. T length of the weld joint is approximately 13 cm.\\
\subsection{Recorded data}
\noindent For all measurements, 2 channels were used: one for the weld voltage and one for the
weld current. $90$ experiments in total were conducted during four days. Fifty seven
experiments were conducted for the T-joint with step disturbance and $33$ reference T-joint. The recording time
of the measured signals was $15$ s for step disturbance and
reference. As previously mentioned in the "Data acquisition system" section, the
sampling frequency was $8.192$ kHz, and the resulting lowpass filter had an upper frequency
limit of $1$ kHz.\\
\begin{table}[htbp]
\caption{Total number of experiments }
\begin{center}
\begin{tabular}{lcccc}
\hline
Welding Speed &Reference &Step \\ [0.15 cm]
8 m/s &2 &15 \\
10 m/s & $29$ &$28$ \\
12 m/s & $2$ &$14$ \\
\hline
SUM: & $33$ &$57$ \\
Total: & 90 & \\
\end{tabular}
\end{center}
\vskip -10 mm
\label{tests1}
\end{table}
\begin{table}[htbp]
\caption{Number of experiments conducted with Kemppi P500 - 960201}
\begin{center}
\begin{tabular}{lcccc}
\hline
Welding Speed &Reference &Step \\ [0.15 cm]
8 m/s &2 &5\\
10 m/s & $5$ &$5$ \\
12 m/s & $2$ &$5$ \\
\hline
\end{tabular}
\end{center}
\vskip -10 mm
\label{tests1}
\end{table}
\begin{table}[htbp]
\caption{Type and Number of experiments conducted with Kemppi P500 - 960304}
\begin{center}
\begin{tabular}{lcccc}
\hline
Welding Speed &Reference &Step \\ [0.15 cm]
8 m/s &0 &10 \\
10 m/s & $11$ &$10$ \\
12 m/s & $0$ &$9$ \\
\hline
\end{tabular}
\end{center}
\vskip -10 mm
\label{tests1}
\end{table}
\begin{table}[htbp]
\caption{Type and Number of experiments conducted with Migatronic BDH S50 - 960325}
\begin{center}
\begin{tabular}{lcccc}
\hline
Welding Speed &Reference &Step \\ [0.15 cm]
8 m/s &0 &0 \\
10 m/s & $7$ &$7$ \\
12 m/s & $0$ &$0$ \\
\hline
\end{tabular}
\end{center}
\vskip -10 mm
\label{tests1}
\end{table}
\begin{table}[htbp]
\caption{Type and Number of experiments conducted with Migatronic BDH S50- 960424}
\begin{center}
\begin{tabular}{lcccc}
\hline
Welding Speed &Reference &Step \\ [0.15 cm]
8 m/s &0 &0 \\
10 m/s & $6$ &$6$ \\
12 m/s & $0$ &$0$ \\
\hline
\end{tabular}
\end{center}
\vskip -10 mm
\label{tests1}
\end{table}
\chapter{Experimental Data Analysis}
The purpose of this chapter is partly to confirm, by examination of the waveform of weld voltage and current,
the assumption that the variance of the amplitude of the weld voltage and current decreases when the welding process deviviates
from the optimal welding condition. Partly to investigate some
parameters for monitoring short arc GMA welding such as, arc time, short arc time and
short circuit peak current, which are suggested in
published works in this field. Based on the performed investigation parmeters for monitoring are suggested.
\section{Time domain analysis of measurment data}
Figures~\ref{Fotoref} and \ref{Fotostepa} show
photos of samples of a typical reference T-joints and a typical T-joints with step disturbance.
Registrations of corresponding weld voltage and current are shown in figures~\ref{weldref1},
and \ref{weldstep1}. The position of disturbance in the weld joint is indicated at the
bottom of respectively voltage diagrams. Corresponding measured parameters
are shown in figures~\ref{weldrefa} -~\ref{weldrefc} and
figures~\ref{weldstep1a} to \ref{weldstep1c} respectively.
\subsubsection*{Short arc transfer frequency}
Figure~\ref{weldrefa} and figure~\ref{weldstep1a} part a, shows
measured short circuit transfer frequency curve of a reference
T-joints and a T-joints with step disturbance, respectively.
During short-circuit time melted electrod materials are transfered to the work piece, see figur~\ref{idpuls}.
When the metal transfer from
the electrode wire to the work piece occur at regular intervals the probability of stable weld process is increased. The metal transfers
are reflected in the weld voltage as almost zero voltage events of $2$
ms in figure~\ref{cweldref1} and figure~\ref{cweldstep1}. According to
previous described hypothesis the number of short-circuit per second is expected to decrease when the welding process is disturbed.
Figure~\ref{weldstep1a} part a verify this assumption. In figure~\ref{idpuls} it can be read of that the
number of short-circuit per second shall be 100 when the welding process working under optimal conditions. In figure~\ref{weldstep1a} is the
number a bit greater but in broad outline verify the assumed hypothesis. During the step disturbance the number of short-cicuting
decreases to almost half compared to optimal welding conditions. In figur~\ref{idpuls} it can
be read off that the process is working in stubb-in or globular mode, i.e. in the A area or in area B.
\subsubsection*{Weld voltage during arc time}
To decide if the process is operating in stubb-in, dip, globular or
spray transfer mode area, the mean
value for arc time voltage is calculated a reference
T-joints and a T-joints with step disturbance, see figure~\ref{weldrefa}
and figure~\ref{weldstep1a} part a, respectively. If the arc time
voltage is greater than arc time voltage under optimal welding
condition then then process is operating in area C i.e. globular
mode. If the reverse is true the process is operating in area A, i.e
stubb-in mode. Figure~\ref{weldstep1a} shows that, if not apparently, that the process are in area A, stubbing in mode, during step disturbance which indicate that the electrode tip is not heated sufficient enough.
\subsubsection*{Arc time}
Figure~\ref{weldrefa} and figure~\ref{weldstep1a} part d shows arc
time and part e shows the overall trend of the arc time. To obtain the
overall trend of the arc time a medianfilter of length 100 was applied to the estimated arc time sequence in part d.
In figur~\ref{weldstep1a} part e the arc time has increased from 7 ms to 14 ms during step disturbance. This observation is in line
with previous observation above. Compare figure~\ref{weldstep1a} part
a with figure~\ref{weldstep1a} part e. Assume that the short-arc time is
constant under optimal as well as non optimal condition and the arc time is redouble under nonoptimal condition then will the number of
short-circuiting per second decrease to half as much.
\subsubsection*{Short-circuiting time}
Figure~\ref{weldstep1b} part b shows that the short-circuiting time inreases in mean from 2.2 ms under optimal condition to 2.8 ms during step disturbance.
If the weld voltage is to low the energy in the arc is not sufficient to melt the electrode and form a droplet that is suited for short
circuit that will occur next. When this happens, an excessive amount of time must be spent in the short circuit phase in order to generate
the heat necessary to melt the electrode and releas the droplet.
\subsubsection*{Short circuit peak current }
Figure~\ref{weldstep1b} part d confirm the results in Figure~\ref{weldstep1b} part b. When the short-circuiting time increases so do the short circuit current peak.
Both of these figures suggest that larger droplet is detached from the
electrode for each short-circuiting cycle. An physical account of this
phenomenon is given in chapter 8.
\subsubsection*{Mean and variance}
As mention in section 2.8 the variance of the amplitude of the weld voltage might be a suitable parameter for detection of
changes in the weld quality. When the variance of the weld voltage is larger than the variance
during optimal conditions spatter has occurred. When the variance of the weld voltage is less than
the variance during optimal conditions the number of short circuits per second are not at their
maximum, indicating that the welding process is disturbed. \\
\noindent The weld voltage is divided into $k$ sections, with $N=1024$ samples in each section. The AC power
is calculated for each section. The estimated mean and variance is shown in figure~\ref{weldstep1c} parta and b. Note the decrease in
the variance during step disturbance, indicating no optimal stability.
\clearpage
%referens
\begin{figure}[htb]
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 270mm by 180mm (XP230206F.epsf scaled 550)}}
\vskip 0 mm
% \centerline{{\scaledpicturee 270mm by 180mm (XP230206R.epsf scaled 550)}}
\begin{picture}(160,0)(0,0)
\put (0,210) {\sf a)}
\put (0,110) {\sf b)}
\end{picture}
\vskip 0 mm
\caption{{\sl Reference T-joint: Photo of the a) front and b) rear side of a welded joint.}}
\label{Fotoref} \end{figure}
\clearpage
\begin{figure}[htb] \vskip -10mm
\setlength{\unitlength}{1mm}
%\centerline{\includegraphics[width=6cm]{curref1.eps }}
\vskip 4mm
%\centerline{\includegraphics[width=6cm]{curref1.eps }}
\begin{picture}(160,0)(0,0) {\footnotesize
\put(75,2){\sf Position (mm)}
\put(40,70) {\sf \shortstack{W\\ e\\ l\\d\\ \\ v \\o \\ l \\ t \\ a \\ g \\ e \\ \\
(V)}}
\put (40,14) {\sf \shortstack{W\\ e\\ l\\d\\ \\ c \\u \\ r \\ r \\ e \\ n \\ t \\
\\ (A)}} \put (30,105) { a)}
\put (30,54) { b)}}
\end{picture} \vskip -2 mm
\caption{{\sl Reference T-joint: a) Measured current and b) measured voltage.}}
\label{weldref1} \end{figure}
\begin{figure}[htb]
\vskip 4mm
\setlength{\unitlength}{1mm}
%\centerline{\includegraphics[width=6cm]{volcur.eps }}
\begin{picture}(0,0)(0,4) {\footnotesize
%\begin{picture}(0,0)(0,0) {\footnotesize
\put(77,6){\sf Time (s)}
\put (40,37) {\sf \shortstack{V\\o \\ l \\ t\\ a\\ g \\ e \\ \\ (V)}}
\put (40,13) {\sf \shortstack{C\\u \\ r \\ r\\ e\\ n \\ t \\ \\ (A)}}
\put (32,57) {\sf a)}
\put (32,31) {\sf b)}}
\end{picture}
\vskip -2 mm
\caption{Reference T-joint: Close up view of a) measured weld voltage and b) weld current.}
\label{cweldref1}
\end{figure}
\vskip -10 mm
\clearpage
\begin{figure}[htb] \vskip -3mm
\setlength{\unitlength}{1mm}
%\centerline{\includegraphics[width=6.5cm]{Cshort2.eps }}
\vskip 4mm
% \centerline{\includegraphics[width=6.5cm]{kkort.eps }}
\vskip 4mm
%\centerline{\includegraphics[width=6.5cm]{arctime.eps }}
\begin{picture}(160,0)(0,0)
\put (32,130) {\sf \shortstack{ S\\h\\o\\r\\t\\ \\c\\i\\r\\c\\u\\i\\t }}
\put (38,133) {\sf \shortstack{ t\\r\\a\\n\\s\\f\\e\\r\\s\\ \\(N o/s) }}
\put (32,172) {{\sf a)}}
\put (32,112) {\sf b)}
\put (32,85) {\sf c)}
\put (32,54) {\sf d)}
\put (32,27) {\sf e)}
\put (40,100) {\shortstack{ \sf V \\ o \\l.\\(V) }}
\put (40,70) {\sf \shortstack{ \sf V \\ o \\l.\\(V)}}
\put (40,41) {\shortstack{ \sf T \\i\\m\\e\\(s) }}
\put (40,12) {\sf \shortstack{ \sf T \\ i \\m\\e \\(s) }}
\end{picture}
\vskip 0mm
\caption{Reference T-joint: a) Measured short circuit transfer frequency curve and b) estimated mean voltage during arc time . c)
Medianfilter of length 100 applied to the estimated mean voltage during arc time. d) arc time and e) medianfilter of length 100
applied to arc time sequence}
\label{weldrefa} \end{figure}
\clearpage
\begin{figure}[h] \vskip -3mm
\setlength{\unitlength}{1mm}
%\centerline{\includegraphics[width=6.5cm]{Imax.eps }}
\vskip 4mm
%\centerline{\includegraphics[width=6.5cm]{kkort.eps }}
\begin{picture}(160,0)(0,0)
\put(75,2){{\sf Position (mm)}}
\put (32,176) {{\sf a)}}
\put (32,149) {{\sf b)}}
\put (32,117) {\sf c)}
\put (32,88) {\sf d)}
\put (32,58) {\sf e)}
\put (32,29) {\sf f)}
\put (40,165) {\shortstack{ \sf C \\ u \\r\\(A) }}
\put (40,136) {\sf \shortstack{ \sf C \\ u \\r\\(A)}}
\put (40,106) {\shortstack{ \sf V \\ o \\l\\(V) }}
\put (40,76) {\sf \shortstack{ \sf V \\ o \\l\\(V)}}
\put (40,44) {\shortstack{ \sf C\\u\\r\\ \\(A) }}
\put (40,15) {\sf \shortstack{ \sf C \\ u \\r\\ \\(A) }}
\end{picture}
\vskip 0mm
\caption{Reference T-joint: a) Measured short circuit current peak and b) medianfilter of length 100 applied to the short circuit
current peak sequence in part a. c) Measured short-circuit time and b) medianfilter of length 100 applied to the short-circuit
time current peak sequence in part a}
\label{weldrefb} \end{figure}
\clearpage
\begin{figure}[h] \vskip -3mm
\setlength{\unitlength}{1mm}
%\centerline{\includegraphics[width=6.5cm]{Imax.eps }}
\vskip 4mm
%\centerline{\includegraphics[width=6.5cm]{kkort.eps }}
\begin{picture}(160,0)(0,0)
\put(75,2){{\sf Position (mm)}}
\put (32,176) {{\sf a)}}
\put (32,149) {{\sf b)}}
\put (32,117) {\sf c)}
\put (32,88) {\sf d)}
\put (32,58) {\sf e)}
\put (32,29) {\sf f)}
\put (40,165) {\shortstack{ \sf C \\ u \\r\\(A) }}
\put (40,136) {\sf \shortstack{ \sf C \\ u \\r\\(A)}}
\put (40,106) {\shortstack{ \sf V \\ o \\l\\(V) }}
\put (40,76) {\sf \shortstack{ \sf V \\ o \\l\\(V)}}
\put (40,44) {\shortstack{ \sf C\\u\\r\\ \\(A) }}
\put (40,15) {\sf \shortstack{ \sf C \\ u \\r\\ \\(A) }}
\end{picture}
\vskip 0mm
\caption{Reference T-joint : a) Mean of the weld voltage and b) estimated variance of weld voltage. c) Mean of the weld current and d) estimated variance of weld current.}
\label{weldrefc} \end{figure}
\clearpage
% stegstūrning 1
\begin{figure}[htb]
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 270mm by 180mm (XP230206F.epsf scaled 550)}}
\vskip 0 mm
% \centerline{{\scaledpicturee 270mm by 180mm (XP230206R.epsf scaled 550)}}
\begin{picture}(160,0)(0,0)
\put (0,210) {\sf a)}
\put (0,110) {\sf b)}
\end{picture}
\vskip -20 mm
\caption{{\sl T-joint with step disturbance No. 1 : Photo of the a) front and b) rear side of a welded joint.}}
\label{Fotostepa} \end{figure}
\clearpage
\begin{figure}[htb] \vskip -10mm
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 170.1mm by 150mm (curref1.eps scaled 350)}}
% \centerline{{\scaledpicturee 170.1mm by 150mm (voltref1.eps scaled 350)}}
\begin{picture}(160,0)(0,0) {\footnotesize
\put(65,-2){\sf Position (mm)}
\put (30,13) {\sf \shortstack{W\\ e\\ l\\d\\ \\ v \\o \\ l \\ t \\ a \\ g \\ e \\ \\ (V)}}
\put (30,72) {\sf \shortstack{W\\ e\\ l\\d\\ \\ c \\u \\ r \\ r \\ e \\ n \\ t \\ \\ (A)}} \put
(30,119) { a)}
\put (30,59) { b)}}
\end{picture} \vskip -2 mm
\caption{{\sl T-joint with step disturbance No. 1: a) Measured current and b) measured voltage.}}
\label{weldstep1} \end{figure}
\begin{figure}[htb]
\vskip 0mm
\setlength{\unitlength}{1mm}
%\centerline{{\scaledpicturee 164.1mm by 150mm (volcur.eps scaled 350)}}
\begin{picture}(0,0)(0,4) {\footnotesize
%\begin{picture}(0,0)(0,0) {\footnotesize
\put(30,3){\sf Time (s)}
\put (0,33) {\sf \shortstack{V\\o \\ l \\ t\\ a\\ g \\ e \\ \\ (V)}}
\put (0,9) {\sf \shortstack{C\\u \\ r \\ r\\ e\\ n \\ t \\ \\ (A)}}
\put (-4,52) {\sf a)}
\put (-4,27) {\sf b)}}
\end{picture}
\vskip -2 mm
\caption{During step disturbance: Close up view of a) measured weld voltage and b) weld current.}
\label{cweldstep1}
\end{figure}
\clearpage
\begin{figure}[htb] \vskip -3mm
\setlength{\unitlength}{1mm}
%\centerline{\includegraphics[width=6.5cm]{Cshort2.eps }}
\vskip 4mm
\centerline{\includegraphics[width=6.5cm]{kkort.eps }}
\vskip 4mm
%\centerline{\includegraphics[width=6.5cm]{arctime.eps }}
\begin{picture}(160,0)(0,0)
\put (32,130) {\sf \shortstack{ S\\h\\o\\r\\t\\ \\c\\i\\r\\c\\u\\i\\t }}
\put (38,133) {\sf \shortstack{ t\\r\\a\\n\\s\\f\\e\\r\\s\\ \\(N o/s) }}
\put (32,172) {{\sf a)}}
\put (32,112) {\sf b)}
\put (32,85) {\sf c)}
\put (32,54) {\sf d)}
\put (32,27) {\sf e)}
\put (40,100) {\shortstack{ \sf V \\ o \\l.\\(V) }}
\put (40,70) {\sf \shortstack{ \sf V \\ o \\l.\\(V)}}
\put (40,41) {\shortstack{ \sf T \\i\\m\\e\\(s) }}
\put (40,12) {\sf \shortstack{ \sf T \\ i \\m\\e \\(s) }}
\end{picture}
\vskip 0mm
\caption{T-joint with step disturbance No 1: a) Measured short circuit
transfer frequency curve and b) estimated mean voltage during arc time . c)
Medianfilter of length 100 applied to the estimated mean voltage during arc
time. d) arc time and e) medianfilter of length 100 applied to arc time
sequency}
\label{weldstep1a} \end{figure}
\clearpage
\begin{figure}[h] \vskip -3mm
\setlength{\unitlength}{1mm}
%\centerline{\includegraphics[width=6.5cm]{Imax.eps }}
\vskip 4mm
%\centerline{\includegraphics[width=6.5cm]{kkort.eps }}
\begin{picture}(160,0)(0,0)
\put(75,2){{\sf Position (mm)}}
\put (32,176) {{\sf a)}}
\put (32,149) {{\sf b)}}
\put (32,117) {\sf c)}
\put (32,88) {\sf d)}
\put (32,58) {\sf e)}
\put (32,29) {\sf f)}
\put (40,165) {\shortstack{ \sf C \\ u \\r\\(A) }}
\put (40,136) {\sf \shor
tstack{ \sf C \\ u \\r\\(A)}}
\put (40,106) {\shortstack{ \sf V \\ o \\l\\(V) }}
\put (40,76) {\sf \shortstack{ \sf V \\ o \\l\\(V)}}
\put (40,44) {\shortstack{ \sf C\\u\\r\\ \\(A) }}
\put (40,15) {\sf \shortstack{ \sf C \\ u \\r\\ \\(A) }}
\end{picture}
\vskip 0mm
\caption{T-joint with step disturbance No 1: a) Measured short circuit current peak and b) medianfilter of length 100 applied to the short circuit
current peak sequence in part a. c) Measured short-circuit time and b) medianfilter of length 100 applied to the short-circuit
time current peak sequence in part a}
\label{weldstep1b}
\end{figure}
\clearpage
\begin{figure}[h] \vskip -3mm
\setlength{\unitlength}{1mm}
%\centerline{\includegraphics[width=6.5cm]{Imax.eps }}
\vskip 4mm
%\centerline{\includegraphics[width=6.5cm]{kkort.eps }}
\begin{picture}(160,0)(0,0)
\put(75,2){{\sf Position (mm)}}
\put (32,176) {{\sf a)}}
\put (32,149) {{\sf b)}}
\put (32,117) {\sf c)}
\put (32,88) {\sf d)}
\put (32,58) {\sf e)}
\put (32,29) {\sf f)}
\put (40,165) {\shortstack{ \sf C \\ u \\r\\(A) }}
\put (40,136) {\sf \shortstack{ \sf C \\ u \\r\\(A)}}
\put (40,106) {\shortstack{ \sf V \\ o \\l\\(V) }}
\put (40,76) {\sf \shortstack{ \sf V \\ o \\l\\(V)}}
\put (40,44) {\shortstack{ \sf C\\u\\r\\ \\(A) }}
\put (40,15) {\sf \shortstack{ \sf C \\ u \\r\\ \\(A) }}
\end{picture}
\vskip 0mm
\caption{T-joint with step disturbance No 1: a) Mean of the weld voltage and b) estimated variance of weld voltage. c) Mean of the weld current and d) estimated variance of weld current.}
\label{weldstep1c} \end{figure}
\clearpage
\section{Spectral domain analysis of measurment data}
The spectra of recordings from normal welding condition can be compared with spectra of recordings during step disturbance in
the search for relevant characteristics.\\
\noindent The weld voltage and current has been decimated to the samplings frequency 4092 kHz. The lowpassfilter
used in the experiments has a cut-off frequency at 1 kHz, so the figures shows the spectra in the range 0-1 kHz.
Estimation of the power spectral density of weld voltage and current is based on the periodograms method. This is described in
Appendix A. The method is implemented with the command `spectrum' in the MATLAB Signal Processing
Toolbox ~\cite{Krauss(1993)}. With this command, the $8192$ data samples are divided into
$16$ sections, with $1024$ points in each section. The sections are detrended. In order to
reduce the effect of spectral leakage, a Hanning data window is applied to the sections of the
signal prior to computing the periodogram. To lower the variance of the estimate, the modified
periodograms of the sections of the signal are averaged $95$ \% confidence interval is
also calculated and plotted by the routine.
\noindent The result of the power estimation of the weld voltage and current
for normal weld of a reference T-joint, and for a welding a T-joint with step,
disturbance is shown in figure~\ref{Fig: Powerspec} part a - d. A visual comparison between the four power spectral densities shows that the main
difference is above 70 Hz.
\begin{figure}[htb] \vskip 0mm
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 200.1mm by 123.8mm (specref.eps scaled 350)}
% {\scaledpicturee 160.1mm by 123.8mm (specstep2.eps scaled 350)}}
\begin{picture}(160,0)(0,0)
{\footnotesize
{\footnotesize \put (-1.5,10) { \sf \shortstack{ P\\ o\\ w\\e\\r \\ \\S \\ p \\
e \\ c \\ t \\ r
\\ u\\m\\ \\($ {\sf {V^{2} \over Hz}}$)}}
\put (76,10) {\sf \shortstack{P\\ o\\ w\\e\\r \\ \\S \\ p \\ e \\ c \\ t \\ r \\
u\\m\\ \\($ {\sf {V^{2} \over Hz}}$)}}
\put (28,1) {\sf frequency (Hz)}
\put (106,1) {\sf frequency (Hz)}}
\put (2,57) {\sf a)}
\put (78,57) {\sf b)}}
\end{picture} \vskip 0 mm
\caption{{\sl Power spectral densities of a) the weld voltage from a reference T-joint during
normal welding, b) the weld voltage from a T-joint with step
disturbance during step disturbance, c) the weld current from a
reference T-joint during normal welding and d) weld current from a
T-joint with step disturbance during step disturbance.
The dotted curve represents the $95$ $\%$ confidence limits. Note
the main spectral diffrence above 70 Hz}}
\label{Fig: Powerspec} \end{figure}
\section{Variance of filtered data}
\subsubsection*{Filtering the data}
The main difference in the power spectra of the weld voltage and
current for normal welds, and for the
welds during step-disturbance, occurs for frequencies over $70$ Hz. To
enhance the difference between normal weld and welds during
step disturbance the weld
voltage and current is highpass-filtered with a discrete-time filter with the following
specification~\cite{HPman}: The maximum pass-band ripple of the magnitude of the
discrete-time filter, $ \leq 0.1$ dB. The minimum stop-band attenuation of the discrete-time
filter is 60 dB. The stop band edge of the discrete-time filter is $f_{1}=50$ kHz. The
pass-band edge of the discrete-time filter is $f_{2}=80$ Hz. This filter was designed with an elliptic filter of an order $8$ in
MATLAB, Signal Processing Toolbox~\cite{Krauss(1993)}. The magnitude response of the elliptic filter is
shown in figure~\ref{passfilt}.\\
\noindent In order not to distort the phase of the output relative to the input, the phase-shift of
the filter should be zero. One technique for achieving this is to process the data forwards and
then backwards through the same filter~\cite{Oppenheim(1989)}. A more thorough description of
zero-phase filter operation is presented in Appendix
B. Figure~\ref{filteredsignals1} part a - d shows the result of applying the highpass filter to the weld voltage and weld current.\\
\begin{figure}[htb]
\vskip 5mm
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 180.1mm by 123.8mm (phasmagn.eps scaled 350)}}
\begin{picture}(160,0)(0,0)
\end{picture}
\vskip -5 mm
\caption{{\sl Magnitude for the designed elliptic bandpass filter used to filter the weld
voltage. }}
\label{passfilt} \end{figure}
\begin{figure}[htb] \vskip 5 mm
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 180.1mm by 123.8mm (voltstep12.eps scaled 350)}
% {\scaledpicturee 180mm by 123.8mm ( Banpassvolt3.eps scaled 350)}}
\begin{picture}(160,0)(0,0)
{\footnotesize
\put (28,0) {\sf Position (mm)}
\put (99,0) {\sf Position (mm)} {\footnotesize
\put (1,54) {\sf a)}
\put (72,54) {\sf b)}
\put (1,10) {\sf \shortstack{W\\ e\\ l\\d\\ \\ v \\o \\ l \\ t \\ a \\ g \\ e \\ \\ (V)}}
\put (72,11) {\sf \shortstack{W \\ e \\ l \\ d \\ \\c \\ u \\ r \\ r \\ e \\ n \\ t \\ \\ (A)}}}}
\end{picture}
\caption{{ \sl a) Weld voltage and b) bandpass-filtered ( 3.0 - 12.8 kHz ) weld voltage from
welding a T-joint with step disturbance. }}
\label{filteredsignals1} \end{figure}
\subsubsection*{Variance of filtered data}
The variance of the amplitude of the filtered weld voltage and current might be a suitable parameter for detection of
changes in the weld quality. When the variance of the weld voltage is larger than the variance
during optimal conditions spatter has occurred. When the variance of the weld voltage is less than
the variance during optimal conditions the number of short circuits per second are not at their
maximum, indicating that the welding process is disturbed. \\
\noindent The weld voltage is divided into sections, with $N=1024$
samples in each section. The variance for the filtered weld voltageand current
is calculated for each section. The estimated mean and variance is
shown in figure~\ref{weldstep1c} part a and b. Note the decrease in
the variance for both weld voltage and curret during step disturbance, indicating no optimal stability.
\noindent The estimated AC power is shown in figure~\ref{weldstep2} part b. Note the decrease in
mean of the AC power estimate $y_{i}$ during step disturbance, indicating no optimal stability. Same algorithm is also applied to the filtered weld current to obtain an estimate of AC power for filtered weld current.The
estimated AC power is shown in figure~\ref{weldstep2} part b. Note the decrease in mean of the AC power estimate during step disturbance, indicating no optimal stability.
\begin{figure}[htb] \vskip 5 mm
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 180.1mm by 123.8mm (voltstep12.eps scaled 350)}
% {\scaledpicturee 180mm by 123.8mm ( Banpassvolt3.eps scaled 350)}}
\begin{picture}(160,0)(0,0)
{\footnotesize
\put (28,0) {\sf Position (mm)}
\put (99,0) {\sf Position (mm)} {\footnotesize
\put (1,54) {\sf a)}
\put (72,54) {\sf b)}
\put (1,10) {\sf \shortstack{W\\ e\\ l\\d\\ \\ v \\o \\ l \\ t \\ a \\ g \\ e \\ \\ (V)}}
\put (72,11) {\sf \shortstack{W \\ e \\ l \\ d \\ \\c \\ u \\ r \\ r \\ e \\ n \\ t \\ \\ (A)}}}}
\end{picture}
\caption{{ \sl a) ac power}}
\label{filteredsignals1} \end{figure}
\section{Selected parameters for monitoring short-arc GMA welding}
The observations described above are typical of the weldings, though
considerable deviations from the normal behavior can
occur. The normal pattern for reference T-joint and T-joint with step
disturbance are shown in figure~\ref{weldrefa}-~\ref{weldrefc} and
figure~\ref{weldstep1a} -~\ref{weldstep1c}, respectively. Deviation from the normal pattern
for a T-joint with step disturbance are shown in
figure~\ref{weldstep2a}-~\ref{weldstep2c}. These last figures shows
no decreas in numbers of short circuitings per second and no
increase in arc and short-circuiting time as well as maximum
current. Figure shows, however, a decreas in the variance during step
disturbance, despite the fact that there is no change or in numbers of short circuitings per second,in maximum current, arc and short-circuiting time as well as maximum
current. Further discussion of these phenomenon se chapter 8. \\
\noindent A detection algorithm based on the variance of the amplitude of
the filtered weld voltage and current seems possible.
%stegstūrning 2
\begin{figure}[htb]
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 270mm by 180mm (XP230206F.epsf scaled 550)}}
\vskip 0 mm
% \centerline{{\scaledpicturee 270mm by 180mm (XP230206R.epsf scaled 550)}}
\begin{picture}(160,0)(0,0)
\put (0,210) {\sf a)}
\put (0,110) {\sf b)}
\end{picture}
\vskip -20 mm
\caption{{\sl T-joint with step disturbance No. 2 : Photo of the a) front and b) rear side of a welded joint.}}
\label{Fotostepb} \end{figure}
\clearpage
\begin{figure}[htb] \vskip -10mm
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 170.1mm by 150mm (curref1.eps scaled 350)}}
% \centerline{{\scaledpicturee 170.1mm by 150mm (voltref1.eps scaled 350)}}
\begin{picture}(160,0)(0,0) {\footnotesize
\put(65,-2){\sf Position (mm)}
\put (30,13) {\sf \shortstack{W\\ e\\ l\\d\\ \\ v \\o \\ l \\ t \\ a \\ g \\ e \\ \\ (V)}}
\put (30,72) {\sf \shortstack{W\\ e\\ l\\d\\ \\ c \\u \\ r \\ r \\ e \\ n \\ t \\ \\ (A)}} \put
(30,119) { a)}
\put (30,59) { b)}}
\end{picture} \vskip -2 mm
\caption{{\sl T-joint with step disturbance No. 2 : a) Measured current and b) measured voltage.}}
\label{weldstep2} \end{figure}
\begin{figure}[htb]
\vskip 0mm
\setlength{\unitlength}{1mm}
%\centerline{{\scaledpicturee 164.1mm by 150mm (volcur.eps scaled 350)}}
\begin{picture}(0,0)(0,4) {\footnotesize
%\begin{picture}(0,0)(0,0) {\footnotesize
\put(30,3){\sf Time (s)}
\put (0,33) {\sf \shortstack{V\\o \\ l \\ t\\ a\\ g \\ e \\ \\ (V)}}
\put (0,9) {\sf \shortstack{C\\u \\ r \\ r\\ e\\ n \\ t \\ \\ (A)}}
\put (-4,52) {\sf c)}
\put (-4,27) {\sf d)}}
\end{picture}
\vskip -2 mm
\caption{During step disturbance: c) Measured voltage and d) measured current.}
\label{cweldstep2}
\end{figure}
\clearpage
\begin{figure}[htb] \vskip -3mm
\setlength{\unitlength}{1mm}
%\centerline{\includegraphics[width=6.5cm]{Cshort2.eps }}
\vskip 4mm
% \centerline{\includegraphics[width=6.5cm]{kkort.eps }}
\vskip 4mm
\centerline{\includegraphics[width=6.5cm]{arctime.eps }}
\begin{picture}(160,0)(0,0)
\put (32,130) {\sf \shortstack{ S\\h\\o\\r\\t\\ \\c\\i\\r\\c\\u\\i\\t }}
\put (38,133) {\sf \shortstack{ t\\r\\a\\n\\s\\f\\e\\r\\s\\ \\(N o/s) }}
\put (32,172) {{\sf a)}}
\put (32,112) {\sf b)}
\put (32,85) {\sf c)}
\put (32,54) {\sf d)}
\put (32,27) {\sf e)}
\put (40,100) {\shortstack{ \sf V \\ o \\l.\\(V) }}
\put (40,70) {\sf \shortstack{ \sf V \\ o \\l.\\(V)}}
\put (40,41) {\shortstack{ \sf T \\i\\m\\e\\(s) }}
\put (40,12) {\sf \shortstack{ \sf T \\ i \\m\\e \\(s) }}
\end{picture}
\vskip 0mm
\caption{T-joint with step disturbance No 2: a) Measured short circuit
transfer frequency curve and b) estimated mean voltage during arc time . c)
Medianfilter of length 100 applied to the estimated mean voltage during arc
time. d) arc time and e) medianfilter of length 100 applied to arc time
sequency}
\label{weldstep2a} \end{figure}
\clearpage
\begin{figure}[h] \vskip -3mm
\setlength{\unitlength}{1mm}
%\centerline{\includegraphics[width=6.5cm]{Imax.eps }}
\vskip 4mm
%\centerline{\includegraphics[width=6.5cm]{kkort.eps }}
\begin{picture}(160,0)(0,0)
\put(75,2){{\sf Position (mm)}}
\put (32,176) {{\sf a)}}
\put (32,149) {{\sf b)}}
\put (32,117) {\sf c)}
\put (32,88) {\sf d)}
\put (32,58) {\sf e)}
\put (32,29) {\sf f)}
\put (40,165) {\shortstack{ \sf C \\ u \\r\\(A) }}
\put (40,136) {\sf \shortstack{ \sf C \\ u \\r\\(A)}}
\put (40,106) {\shortstack{ \sf V \\ o \\l\\(V) }}
\put (40,76) {\sf \shortstack{ \sf V \\ o \\l\\(V)}}
\put (40,44) {\shortstack{ \sf C\\u\\r\\ \\(A) }}
\put (40,15) {\sf \shortstack{ \sf C \\ u \\r\\ \\(A) }}
\end{picture}
\vskip 0mm
\caption{T-joint with step disturbance No 2: a) Measured short circuit current peak and b) medianfilter of length 100 applied to the short circuit
current peak sequence in part a. c) Measured short-circuit time and b) medianfilter of length 100 applied to the short-circuit
time sequence in part a}
\label{weldstep2b} \end{figure}
\clearpage
\begin{figure}[h] \vskip -3mm
\setlength{\unitlength}{1mm}
%\centerline{\includegraphics[width=6.5cm]{Imax.eps }}
\vskip 4mm
%\centerline{\includegraphics[width=6.5cm]{kkort.eps }}
\begin{picture}(160,0)(0,0)
\put(75,2){{\sf Position (mm)}}
\put (32,176) {{\sf a)}}
\put (32,149) {{\sf b)}}
\put (32,117) {\sf c)}
\put (32,88) {\sf d)}
\put (32,58) {\sf e)}
\put (32,29) {\sf f)}
\put (40,165) {\shortstack{ \sf C \\ u \\r\\(A) }}
\put (40,136) {\sf \shortstack{ \sf C \\ u \\r\\(A)}}
\put (40,106) {\shortstack{ \sf V \\ o \\l\\(V) }}
\put (40,76) {\sf \shortstack{ \sf V \\ o \\l\\(V)}}
\put (40,44) {\shortstack{ \sf C\\u\\r\\ \\(A) }}
\put (40,15) {\sf \shortstack{ \sf C \\ u \\r\\ \\(A) }}
\end{picture}
\vskip 0mm
\caption{T-joint with step disturbance No 2: a) Mean of the weld voltage and b) estimated variance of weld voltage. c) Mean of the weld current and d) estimated variance of weld current.}
\label{weldstep2c} \end{figure}
\chapter{Fault detection algorithm}
The objective of the fault detection algorithm is to test wether the
welding process is working in optimal or in non opimal condition. The
non optimal welding condition is provoked by a T-joint with step
disturbance. The
detection algorithm is based on the assumption that any change in the
weld process will cause a decreas in the variance in the filtered data. To avoid confusion of ideas the variance
of the amplitude of the filtered weld voltage and current is
henceforth denoted AC power for weld voltage and AC power for weld
current. \\
\section{Test parameter}
\noindent The high pass filtered weld voltage is divided into $k$ sections, with $N=1024$ samples in each section. The AC power
is calculated for each section and is given an index, $i$, defined by the position in the
sequence. The AC power is estimated as follows:
\begin{equation} y_{i}={1 \over N-1 } \sum_{p=1}^{N} (v_{p} - \overline{v})^{2}
\end{equation}
\noindent where $v_{p }$ is the filtered weld voltage, $N$ is the number of data points and
$\overline{v}$ is the mean of the filtered weld voltage calculated as
\begin{equation}
\overline{v}={1 \over N } \sum_{l=1}^{N} v_{l}
\end{equation}
\noindent The estimated AC power is shown in figure~\ref{weldstep2} part b. Note the decrease in
mean of the AC power estimate $y_{i}$ during step disturbance, indicating no optimal stability. The
sequence ${\bf y} =(y_{0}, y_{1}, \ldots y_{k})$ is assumed to be identical, independent and
Gaussian distributed with mean value $\mu$ and constant variance $\sigma^{2}$.\\
\noindent Same algorithm is also applied to the filtered weld current to obtain an estimate of AC power for filtered weld current.The
estimated AC power is shown in figure~\ref{weldstep2} part b. Note the decrease in mean of the AC power estimate $y_{i}$ during step
disturbance, indicating no optimal stability. The obtained sequence sequence ${\bf y} =(y_{0}, y_{1}, \ldots y_{k})$ is also assumed to be
identical, independent and Gaussian distributed with mean value $\mu$
and constant variance $\sigma^{2}$.
\section{Algorithm}
Let $ {\bf y} =(y_{0}, y_{1}, \ldots y_{k})$ denote a random sample of scalar random variables
of AC power, each of which is Gaussian distributed:
\begin{equation} p_{\theta}(y_{i})= \frac{\displaystyle 1}{\displaystyle \sigma \sqrt{2 \pi}} e^{-
\frac{\displaystyle (y_{i} -
\mu)^2}{\displaystyle 2 \sigma^2}}
\end{equation}
\noindent The welding process is known to operate under either normal $(\theta =
\mu_{0})$ or fault $(\theta = \mu_{1})$ conditions where $ \mu_{0} > \mu_{1}$. Furthermore, we
assume that prior to $t=0$,
$\theta = \mu_{0}$ and it may only change to $\theta = \mu_{1}$ at one of the $n$ sampling
instants. Consider the problem of testing $ k+1$ hypotheses $H_{0}, H_{1} \ldots H_{k}$
\begin{equation}
\begin{array}{llll} H_{0}:&\theta= \mu_{0} & for & 1 \leq i \leq k
\\ & & & \\ H_{j}:&\theta= \mu_{0} & for & 1 \leq i \leq j-1
\\ &\theta= \mu_{1} & for & j \leq i \leq k
\end{array}
\end{equation}
\noindent If the instant of change $j$ is fixed, then the Sequential Probability Ratio Test (SPRT)
between $H_{0}$ and $H_{j}$ is based on a comparison of the likelihood
ratio~\cite{Basseville(1993)}:
\begin{equation} S_{j}^{k}= \sum _{i=j}^{k} s_{i}
\end{equation}
\noindent where
\begin{equation} s_{i}=\ln
\frac{\displaystyle p_{\mu_{1}(y_{i})}}{\displaystyle p_{\mu_{0}(y_{i})}}
\end{equation}
\noindent to a threshold $h$. At the sampling instant $k$, $S_{j}^{k}$ is computed. If
$S_{j}^{k} \geq h$ a defect in the weld joint is detected. In the scalar independent case
$S_{j}^{k}$ is recursively updated as:
\begin{equation} S_{j}^{k+1}= S_{j}^{k}+s_{i}
\end{equation}
\noindent In the case of a change in the mean value $\mu$ of an independent Gaussian random
sequence
$y_{k}$ with known variance $\sigma^{2}$, the sufficient statistics $s_{i}$ is calculated as
\begin{equation} s_{i}= \frac{\displaystyle \mu_{1}-\mu_{0}}{\displaystyle \sigma^2}(y_{i}-
\frac{\displaystyle \mu_{1}+\mu_{0}}{\displaystyle 2})
\end{equation}
\noindent which we write as
\begin{equation} s_{i}=\frac{\displaystyle (\mu_{1}-\mu_{0})^2}{\displaystyle \sigma^2}(y_{i}-
\mu_{0}- \frac{\displaystyle \nu}{\displaystyle 2})
\end{equation}
\noindent where
\begin{equation}
\nu = \mu_{1}- \mu_{0}
\end{equation}
\noindent is the change in magnitude. The SPRT is optimal with respect to the worst mean delay,
when error probability for false alarms goes to zero. The instant of change $j$ is in fact unknown,
but may be estimated using the maximum likelihood principle~\cite{Rao(1973)}, leading to the
decision function and alarm instant:
\begin{equation} g_{k}=\max_{0 \leq j \leq k} S_{j}^{k} \label{eq:gk}
\end{equation}
\begin{equation} t_{a}= \min \{k:g_{k} \geq h \}
\end{equation}
\noindent The algorithm has been formulated as a set of parallel SPRT's, but may equivalently be
viewed as repeated SPRT or a CUSUM - type test. The connection between these alternative points of
view has been investigated by~\cite{Basseville(1993)}. The decision function $g_{k}$ introduced
in~\ref{eq:gk} becomes in repeated SPRT formulation
\begin{equation} g_{k}=[g_{k-1}+s_{i}]^{+}
\end{equation}
\noindent and in the Gaussian case
\begin{equation} g_{k}=[g_{k-1}+\frac{\displaystyle
\mu_{1}-\mu_{0}}{\displaystyle \sigma^{2}}(y_{k}-\frac{\displaystyle
\mu_{1}+\mu_{0}}{\displaystyle 2})]^{+}
\end{equation}
\noindent where $(x)^{+}=\sup(0,x)$. The alarm threshold $h$ is chosen by a tradeoff between worst
mean delay for detection, $\tau$ and false alarm probability $\alpha$. The CUSUM algorithm is
optimal when $\alpha$ goes to zero~\cite{Lorden(1971)}:
\begin{equation}
\label{eq:tau}
\begin{array}{llll}
\tau \sim \frac{\displaystyle \ln \alpha^{-1}}{\displaystyle K(\mu_{1},\mu_{0})} & when & \alpha
\rightarrow &0\\
\end{array}
\end{equation}
\noindent where
\begin{equation} K(\mu_{1},\mu_{0})=E_{\mu_{1}}[\ln
\frac{\displaystyle p_{\mu_{1}}(\displaystyle y_{i})}{\displaystyle p_{\mu_{0}}(y_{i})}]
\end{equation}
\noindent is the Kullback information. In Gaussian case the Kullback information is
\begin{equation} K(\mu_{1},\mu_{0})=\frac{(\mu_{1}-\mu_{0})^2}{\sigma^2}
\end{equation}
\noindent Due to Wald's inequality the alarm threshold satisfy
\begin{equation}
\alpha=e^{-h} \label{eq:eh}
\end{equation}
\noindent and thus the alarm threshold $h$ is easy to obtain~\cite{Wald(1947)}. The complete fault
detection algorithm may be summarized as follows:\\
\noindent {\sl Algorithm:} For each section $k$ of $1024$ data samples:
\begin{itemize}
\begin{enumerate}
\item calculate AC power $y_{i}$
\item calculate $g_{k}=[g_{k-1}+s_{i}]$
\item {\bf if} $g_{k} \leq 0$ {\bf then} $g_{k}=0$
\item {\bf if} $g_{k} \geq h$ {\bf then}\\ $Alarm$\\
$g_{k}=h$\\
\end{enumerate}
\end{itemize}
\chapter{Evaluation and Tuning}
\section{Evaluation}
\noindent In order to evaluate the proposed detector, two batches, each of 180 samples of the
parameter $y_{i}$ originating from weld voltage from normal welds and welds during step disturbance
respectively, were used, see figure~\ref{Fig:detectP}. A sample length of $184$ and welding speed
at 10 mm/s corresponds approximately to a
$20$ cm weld joint.
\begin{figure}[htbp]
\vskip 8mm
\setlength{\unitlength}{1mm}
\centerline{{\scaledpicturee 180.1mm by 123.8mm (acpowny.eps scaled 350)}}
\begin{picture}(0,0)(4,4)
\put (3,54) {\sf a)}
\put (3,27) {\sf b)}
\put (36,4) {\sf Index (i) }
\put (5,45) {\sf \shortstack{ y$_{i}$}}
\put (5,20) {\sf \shortstack{ y$_{i}$}}
\end{picture}
\vskip -2 mm
\caption{The AC power $y_{i}$ during normal weld. The AC power $y_{i}$ during step disturbance. The
AC power $y_{i}$ is based on 1024 samples of the weld voltage.}
\label{Fig:detectP}
\end{figure}
The estimated AC power of the weld voltage $y_{i}$ is assumed to be identically, independent,
Gaussian distributed with mean value
$\mu_{0}$ and $\mu_{1}$ under normal and fault condition respectively. The variance
$\sigma$ is assumed to be constant under both conditions. For each batch of data, mean value and
variance are estimated, see table~\ref{estpar}.
\begin{table}[htb] {\small
\caption{List of estimated parameters.}
\centering{ \begin{tabular}{lcc} \hline
\multicolumn{3}{c}{\sc Estimated Parameters} \\ \hline & mean value $\hat{\mu}$ & variance
$\hat{\sigma}^{2}$ \\ Normal conditions & $56.4$ & $6.76$ \\
Fault conditions & $47.2$ & $9.10$ \\\hline \end{tabular}}
\label{estpar}} \end{table}
$\chi^{2}$ tests shows that the AC power $y_{i}$ under fault condition, in contrast to normal
conditions, is neither independent nor Gaussian. Furthermore, the variance is not equal under the
both condition, see table 1. The algorithm is still chosen, because the algorithm is robust
with respect to independent and Gaussian assumption as well as demand for equal
variance~\cite{Bagshaw(1975a), Bagshaw(1975b)}. In addition, storage and computational requirements
for the recursive SPRT are moderate.
\section{Tuning}
In the proposed algorithm the only tuning parameter is the threshold $h$. Using
formula~\ref{eq:tau} we can compute worst mean delay for detection, $\tau$ and false alarm
probability $\alpha$ and use them for choosing a relevant
alarm threshold, $h$. If the false alarm probability
$\alpha$ is set at $10^{-9}$, the alarm threshold
$h$ is calculated to be $h=21$. But, since $y_{i}$ under fault conditions is correlated and can
not be assumed Gaussian and we assume
$\sigma^{2}=6.76$ , the alarm threshold $h$ is set conservatively. Thus, in order to maintain the
false alarm probability, $\alpha \geq 10^{-9}$ the alarm threshold, $h$ is set at $25$.
\chapter{Results}
\subsubsection*{Test of the SPRT algorithm}
The recursive SPRT algorithm was tested on $31$ specimens. A total of 15 experiments were
conducted for reference T-joint and sixteen experiments were conducted for the T-joint with step
disturbance. The recording time of the measured signals was $15$ s.
The test was designed as follows: When the alarm turns on and there is a step disturbance, the test
results in a detection; and when the alarm does not turn on, there is a nondetection. If the alarm
turns on and there is no step disturbance, the result is a false alarm.
\subsubsection*{Results}
The results of the test are shown in table 2. Typical behavior for a T-joint with a step
disturbance, is shown in figure~\ref{weldstep2}. The top diagram of the figure, part a, shows the
weld voltage, and part b shows the weld current. Part c shows the corresponding AC power $y_{i}$ and
the actual position of the step disturbance along the weld joint. Part d of the figure shows
decision function $g_{i}$ and the $Alarm$.
\begin{table}[htbp] {\small
\caption{{The results of the test of the SPRT algorithm. Welding speed $=$ 10 m/s }}
\centering{ \begin{tabular}{lcccc} \hline Type of T-joint &Reference &Step \\ [0.15 cm]
Number of specimens &15 &16 \\Detection &
$15$ &$16$ \\ Nondetection & $0$ &$0$ \\ False Alarm & $0$ &$0$ \\ \hline
\end{tabular}}}
\vskip -2 mm
\label{perf2} \end{table}
\clearpage
\begin{center}
{\LARGE \sc Reference T-joint }
\end{center}
\begin{figure}[h] \vskip 10mm
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 170.1mm by 150mm (vlcrrf.eps scaled 600)}}
% \centerline{{\scaledpicturee 170.1mm by 150mm (acpowref2.eps scaled 600)}}
\begin{picture}(160,0)(0,0)
{\large
\put(65,-2){{\sf Position (mm)}}
\put (15,140) {\sf \small \shortstack{ V \\o\\ l \\ t \\ a \\ g \\ e\\ \\
(V) }}
\put (15,100) {\sf \small \shortstack{ C \\u \\ r \\ r \\ e \\ n\\ t\\ \\
(A) }}
\put (10,176) {{\sf a)}}
\put (10,134) {\sf b)}
\put (10,84) {\sf c)}
\put (10,40) {\sf d)}
\put (15,67) {\shortstack{ \sf $P_{k}$}}
\put (15,25) {\sf \shortstack{ \sf $G_{K}$}}}
\end{picture}
\vskip -28mm
% \caption{}
\label{welds1} \end{figure}
\clearpage
\begin{figure}[htb]
\vskip 0mm
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 164.1mm by 150mm (vlcrst.eps scaled 350)}}
%\centerline{{\scaledpicturee 164.1mm by 150mm (acpowstep2.eps scaled 350)}}
\begin{picture}(0,0)(4,2) {\footnotesize
\put(35,3){\sf Position (mm)}
\put (7,85) {\sf \shortstack{ V \\o \\ l \\ t \\ a \\
g \\ e \\ \\ (V)}}
\put (7,60) {\sf \shortstack{C \\u\\ r \\ r \\ e\\ n\\ t \\
\\ (A)}}
\put (3,106) {\sf a)}
\put (3,80) {\sf b)}
\put (3,52.0) {\sf c)}
\put (3,26.0) {\sf d)}
\put (7,43) {\sf \shortstack{y$_{i}$}}
\put (7,18) {\sf \shortstack{g$_{i}$}}}
\end{picture}\vskip -2 mm
\caption{Illustration of the detection of step disturbance: Measured weld voltage and current
are shown in part a and b respectively. The corresponding AC power $y_{i}$ and the actual
position of the step disturbance are shown in part c. The decision function $g_{i}$ and
$Alarm$ is shown in part d. }
\label{weldst}
\end{figure}
\clearpage
\centerline{\LARGE \sc T-joint with step disturbance }
\begin{figure}[h] \vskip 10mm
\setlength{\unitlength}{1mm}
% \centerline{{\scaledpicturee 170.1mm by 150mm (vlcrst.eps scaled 600)}}
% \centerline{{\scaledpicturee 170.1mm by 150mm (acpowstep3.eps scaled 600)}}
\begin{picture}(160,0)(0,0)
{\large
\put(65,-2){{\sf Position (mm)}}
\put (15,140) {\sf \small \shortstack{ V \\o\\ l \\ t \\ a \\ g \\ e\\ \\
(V) }}
\put (15,100) {\sf \small \shortstack{ C \\u \\ r \\ r \\ e \\ n\\ t\\ \\
(A) }}
\put (10,176) {{\sf a)}}
\put (10,134) {\sf b)}
\put (10,84) {\sf c)}
\put (10,40) {\sf d)}
\put (15,67) {\shortstack{ \sf $P_{k}$}}
\put (15,25) {\sf \shortstack{ \sf $G_{K}$}}}
\end{picture}
\vskip -28mm
% \caption{}
\label{welds1} \end{figure}
\clearpage
\chapter{Discussion}
\subsubsection*{monitoring indices}
\subsubsection*{arbetspunkt}
The proposed recursive SPRT algorithm is designed to detect step disturbance in welds.
The algorithm could, however, be used to detect other disturbances in the weld process as well.
Furthermore, to enhance the performance of the algorithm, other parameters such as short-circuit time
and short arc time can easily be incorporated into the algorithm.
\subsubsection*{kombinera parametrar. strom o. spaanning}
\subsubsection*{physical account- droplet}
\subsubsection*{tuning parameters for other experiment}
\subsubsection*{nackdel med algorithmen - var olika gor olika
experiment situaner. algoritmen estimera inte foandringen}
\subsubsection*{tuning parameters - minimum jump}
\subsubsection*{samma algorithm fr andra parametrar }
\bibliographystyle{unsrt}
\bibliography{WT,Book}
\end{document}