Experimental investigation of the accuracy of the measuring method
The optical-flow-based analysis of the AFP process outputs translation vectors for each IR image pair. To ensure that the generated measured values can be compared with a reference value and evaluated for accuracy, a time difference \(\Delta t\) is assigned to each translation vector, which results from the recording frequency \({f}_{IR}\) of the IR camera.
Consequently, the material flow distance in each corresponding time interval of the lay-up can be determined. Since it can be assumed that the material flow equals the distance travelled by the end effector, these measurement pairs can be tested for correspondence.
Experimental investigation of the accuracy of the measuring method
The accuracy of the calculated material flow distance is estimated by comparing the derived machine parameters from the optical-flow-based process monitoring system to the recorded machine parameters of the AFP system. The measurement is performed during the lay-up of the third layer of a test laminate. In processes with slit-tape as substrate, significantly more thermal features can be found, since inhomogeneity in the temperature distribution occurs between individual tows due to very small air pockets, which result in warmer spots. These can be used as optical features by the algorithm for determining translation. An increase in the number of layers can have a reinforcing effect on the number and extent of the features, since thermal inhomogeneity in lower layers can also be seen to a lesser extent in the current layer. Consequently, in order to maximize the difficulty of finding characteristics and thus validate and evaluate the system under industry-relevant conditions, tape is used in the bottom layer.
Experimental measurements
In the experiment, courses of four ¼”-tows (HexPly® 8552 UD Carbon Prepreg, AS4 (12 K)) are applied over a length of \(l=500 \mathrm{mm}\). The relative humidity is \(24\%\) and the ambient temperature is \(21^\circ \mathrm{C}\) during lay-up. The target temperature of the substrate in the nip point is \(40^\circ \mathrm{C}\).The lay-up velocity of the individual courses varies, whereas the velocity during cutting the tow remains constant. The IR camera operates with a nominal frequency of \({f}_{IR}=75 \mathrm{Hz}\) and runs independently of the AFP system. This requires the time axes to be synchronized after conducting the experiments in order to evaluate the accuracy of the optical measurement procedure.
It is assumed that the distance covered by the end effector corresponds to the calculated material flow in the thermographic images. The measurements of both the machine control and the optical-flow-based measuring system are synchronized over time for two courses of different lay-up velocities, \({v}_{1}=0.3 \frac{m}{s}\) and \({v}_{2}=0.25 \frac{m}{s}\). The velocity at the end of every course during cutting the tows is \({v}_{c}=0.1 \frac{m}{s}\). The sampling frequency of the optical-flow-based measurement is by a factor of \(0.9\) lower than the sampling frequency \({f}_{R}=83.33 \mathrm{Hz}\) of the machine controller, due to the IR camera frequency of \({f}_{IR}=75\mathrm{ Hz}\).
In Fig. 5 the cumulative course length as well as the calculated velocities of the optical-flow based measurement and the AFP-head position and velocities from the machine control are synchronized over time. The illustrated course has a lay-up velocity of \({v}_{1}=0.3 \frac{m}{s}\) with a deceleration to the velocity of \({v}_{c}=0.1 \frac{m}{s}\) in the range of \(0.6 \mathrm{s}<t<1.5 \mathrm{s}\). Especially the calculation of the course length shows a high accuracy. The significantly higher scatter of the velocities is due to the slightly varying frequency of the camera.
Evaluation of the results
The evaluation of the optical measurement procedure—i.e. the quantified accuracy—is carried out by evaluating its precision and trueness, [13]. Precision herein describes a random scattering of measured values. Trueness is a measure of the deviation due to a systematic measurement error. Quantifying the accuracy of the measuring method is performed with a non-parametric method according to Passing and Bablok, which evaluates the power of a method in relation to a reference method, [14].
Each distance between two images can be assigned to a corresponding target distance of the machine control. The paths of the AFP-head from the machine control are interpolated cubically so that the required distance increments used as reference can be derived. All pairs of measurements—the material flow distance between two images and the associated distance acquired from the machine control—serve as data basis for the evaluation. The deceleration to the velocity required for the cut is not considered separately. Pairs of measured values with duplicates in the reference measurement are filtered out of the database, with the remaining pair of measured values being randomly selected.
Passing and Bablok recommend a preliminary correlation check. Here, the non-parametric rank correlation coefficient according to Spearman, [15], is used, whose correlation coefficient \(\rho =0.99937\) has been determined to a significance level of \(\alpha =0.01\). Therefore, as \(\rho \preccurlyeq 1\), the condition indicating a high correlation of the measurement pairs is met.
Figure 6 shows the linear regression diagram of the optical-flow-based measurement of the distance \({s}_{0}\) against the reference path measurement from machine control to a significance level of \(\alpha =0.01\). Table 1 shows the descriptive statistics for the two methods and the calculated model coefficients. The value of the intercept is \(0.999 {\mu m}\) with a confidence interval including \(0\). From this, it is derived that there is no systematic difference between the two methods. The gradient coefficient is equal to \(0.997\) with a confidence interval including \(1\). This is interpreted as to that the proportional difference between the two methods is equal to \(1\). Consequently, to a significance level of \(\alpha =0.01\), no systematic or proportional differences between the two methods exist.
Table 1 The descriptive statistics of the linearity test for the material flow measurements For the test to be valid, the assumption of linearity must be checked as well. This is done by a cumulative sum test according to Passing and Bablok, [14]. In Fig. 7 the results of the linearity test are shown. Since the cumulative sum does not exceed the critical values \(c=\pm 1.63\) at the significance level of \(\alpha =0.01\) the hypothesis of a linear relationship between the two measurement methods holds.
Analysis of the results
A critical analysis is performed on the Bland Altman diagram, Fig. 8, in which the deviations of the optical-flow-based measurement from the reference measurement are plotted against the reference distance, [16]. The average deviation of the measuring differences to the reference distance is \(\mu =-0.162\, {\upmu {\rm m}}\) and spreads with a standard deviation of \(\sigma =16.8 \, {\upmu {\rm m}}\). The majority of the differences lie within the \(1.96\, {\sigma}\)-interval and all differences lie within the \(\frac{1}{2} pixel\)-interval. This confirms the subpixel accuracy of the optical measuring method.
It can further be seen that the scattering of measured values increases with increasing reference distance, which in turn is proportional to the end effector velocity. This indicates a direct dependency of the precision of the optical-flow measuring algorithm from the process velocity.
The described relative material-flow measuring method for locating IR-images along the end effector path shows high precision and trueness. Nevertheless, it should be checked whether the optical measuring method is inferior to the absolute measurements of the machine control, since the measurement uncertainty correlates with the absolute number of individual measurements taken. It is therefore desirable to minimize the overlap of successive IR images. In the test case, the maximum diameter of the error ellipse is approx. \(5 \mathrm{mm}\), while the image section is approx. \(50 \mathrm{mm}\) long. Thus, an overlapping of the IR images of \(20\%\) is deemed sufficient. At a constant material flow velocity \(v\) the number of measurements can thus be reduced by adjusting the recording frequency according to:
$${f}_{IR}=\frac{v}{0.05 m\bullet 0.8}$$
The number of material flow measurements \(n\) for a specific course length \(l\) can be determined by:
$$n= \frac{l}{0.05 m\bullet 0.8}$$
In turn, the measurement uncertainty of independent dispersing quantities can be approximated by:
$${u}_{O}=\sqrt{{\sum }_{i=1}^{n}{\left(\frac{dl}{d{\delta }_{y,i}}\bullet {u}_{i}\right)}^{2}}with \frac{dl}{d{\delta }_{y,i}}=1 , { u}_{i}=const$$
$${u}_{O}={ u}_{i}\bullet \sqrt{n}=5\bullet { u}_{i}\bullet \sqrt{l}$$
In Fig. 9 the uncertainty of the position assignments to the images depending on the course length are compared. The slash-dotted lines are the average expected uncertainty of the position assignment, where:
$${u}_{O}({ u}_{i}=\sigma )$$
$${u}_{R}:={\sigma }_{R}=\frac{v}{2\bullet \sqrt{3}\bullet {f}_{R}}=const.$$
The dotted lines are the maximal expected uncertainty of the position assignment, where maximal uncertainty of the optical-flow based measurement is approximated by the \(6\sigma\)-interval:
$${u}_{O}({ u}_{i}=6\bullet \sigma )$$
$${u}_{R}:=\frac{v}{2\bullet {f}_{R}}=const.$$
When considering the maximal expected uncertainty, the relative position of the last image of courses of \(12.755 \mathrm{m}\) length is equal for both methods with an uncertainty of \(1.8 \mathrm{mm}\) at a constant lay-up velocity of \(v=0.3 \frac{m}{s}\). Furthermore, the color indicates the distribution density of the uncertainties, where the uncertainty of machine control assignment is equally distributed compared to the optical-flow based normally distributed uncertainty.