1 Introduction

Incremental sheet forming (ISF) is a process for the forming of sheet metal parts. As the part is shaped by a hemispherical tool and the driven toolpath, adjustments to the formed geometry can be made by simply altering the toolpath which makes it an excellent process for the production of sheet metal parts in small batch sizes as for example prototyping. However, the low geometrical accuracy of the process prevents a widespread industrial application. These errors can be compensated by toolpath adjustments based on the simulated part deviations [1]. Typically, finite element analysis (FEA) models are used to simulate the process focusing on a wide range of aspects as, e.g., forming forces [2], sheet thickness distribution [3], residual stress state [4], and springback [5]. However, finite element analysis suffers from a time-consuming modeling and calculation phase that contradicts the unique advantage of ISF, a fast production without the need for additional tooling [6]. Furthermore, FEA simulations of ISF are still not sufficiently accurate. Even recent and well-received publications exhibit simulation errors of over 1 mm [7]. The main reason for this is the complex nonlinear relationships of the individual process parameters [8]. This led to a rising amount of machine learning applications in ISF [9]. These have the advantage that after the time-consuming training phase, the calculation time is only a fraction of the time needed for FEA. However, all developed models are based on limited datasets as research institutes, in contrary to industrial companies, cannot produce a sufficient amount of parts and therefore data [10]. To simplify the gathering of transferable process data, the authors present a cluster analysis approach to detect missing part geometries in an existing process database. For achieving this, the local geometry around every toolpath point in a pre-existing database is abstracted by a surface representation consisting of fitted planes. Their parameter values are used as input for the cluster analysis. The Pearson correlation coefficient is used for identifying missing surface representations and therefore geometry features. Based on these geometry features, five new parts are derived and formed in 35 forming experiments. The established process database is published to enable other researchers the application of machine learning techniques without the need for time-consuming experimental series [11].

Fig. 1
figure 1

Forming setup at the LPS consisting of two KUKA KR360 robots driven by KUKA KRC2 controllers and two KUKA KR600 robots driven by KUKA KRC4 controllers which were used in the experiments [12]

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Fig. 2
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Forming principle of double-sided incremental sheet forming [12]

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Fig. 3
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Principle of the supporting angle \(\gamma \) ranging from \(\gamma _{0}\) to \(\gamma _{max}\) depending on the local wall angle \(\xi \) [12]

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Table 1 Exemplary machine learning approaches in ISF and the used databases and models
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1.1 Double-sided incremental forming

Over time, a wide variety of ISF process variants were invented. These can be distinguished by the type and amount of tools, machinery, toolpath strategies, and supporting elements such as heating devices and special clamping setups. The experiments were carried out by a roboforming setup (see Fig. 1). Roboforming belongs to the process variant double-sided incremental forming (DSIF) which is also known as duplex incremental forming (DPIF) [13]. In DSIF, a typically hemispherical tool actively forms the sheet, while a second tool supports the forming process on the backside of the sheet (see Fig. 2). These tools are mounted to industrial robots in the roboforming process. This way, the process benefits of the big workspace, wide distribution, high kinematic flexibility, and relatively low investment costs of industrial robots. On the contrary, the open kinematic chain of industrial robots leads to a low stiffness of the system and therefore deflections due to the forming forces what makes a stiffness compensation during the process mandatory [12]. The usage of two individual tools is accompanied by process parameters which are limited to DSIF. While the position of the forming tool during the forming process is fixed depending on the calculated toolpath, the tool center point of the supporting tool on the backside can be rotated around the forming tool to a certain degree called the supporting angle \(\gamma \) (see Fig. 3). When both tools are directly opposed to each other and their tool axes are aligned, the supporting angle is defined as \(\gamma _{0}\). Rotating the supporting tool outwards around the forming tool increases the supporting angle. At the maximum supporting angle \(\gamma _{max}\), the supporting tool cannot be further rotated without altering the part’s geometry. The range of this rotation is limited by the wall angle \(\xi \). Therefore, the supporting angle is defined as a percentage of the local wall angle. The supporting angle exhibits its advantage together with the second unique roboforming process parameter which is the supporting force \(F_{S}\). The supporting tool is force-controlled to apply a defined force in a certain direction on the backside of the sheet. This leads to superimposed stress which influences the material flow and improves the accuracy of the forming process and the maximum formable wall angle [14].

Fig. 4
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Base geometry that was used in the experimental series. It exhibits a wide range of convex and concave curvatures [28]

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2 Machine learning in incremental sheet forming

The recently simplified accessibility due to numerous frameworks accompanied by rising computational processing capabilities led to a rising amount of machine learning applications in a wide amount of research fields. The same trend can be observed in ISF [9]. Machine learning has been used to predict a wide amount of process values such as springback [29], forming accuracy [30], formability [31], forming forces [32], surface roughness [33], and thickness distribution [34]. Despite the high amount of machine learning approaches, the prediction quality and especially the transferability are still lacking. While reviewing the state of the art, Harfoush et al. [9] and Nagargoje et al. [10] independently of each other observed that only limited datasets were available and therefore used for training at the research institutes. Both concluded that it is mandatory to build up bigger and wider distributed databases in the future. An overview of the data used in selected publications is given in Table 1. A more detailed summary can be found in [10]. It can be observed that in most publications, the forming parameters such as the step depth were varied during the experiments. However, typically only one to three different geometries were formed. Furthermore, these were mostly truncated cones or rectangular frustums which are very simple geometries. While these are sufficient for demonstrating the general applicability of the chosen approach, the transferability is limited as the models are only valid for this geometry or slightly different ones.

For ensuring the transferability of the machine learning models, it is mandatory to establish a broadly diversified process database although research institutes lack the capacities for forming a huge amount of parts. One possibility for solving this contradiction is variations in the parts itself. While a rectangular frustum mainly consists of four inclined planes and therefore only data for these, altering the wall angle or rounding the surface yields new formed features without increasing the initial experimental expenditure. Low et al. used this technique by constructing plateaus in their parts and varying the radii and wall angles on each side and plateau. This way, they could cover a wide amount of process states with only three different geometries [15]. Carette and Duflou used a more advanced approach to generate a diversified database [18]. They systematically created different spheres and ellipses by specifying a range for different diameters, height and coordinate and angle offsets and choosing parameter combinations according to fractional factorial design of experiments. Störkle et al. pursued a similar approach for establishing a process database [28]. As this is the initial database used for the systematic extension carried out for this publication, it is described in detail hereafter.

Fig. 5
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Extrusion of the base geometry in five degree steps with angles ranging from 30 to 60 degree [28]

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Table 2 Experimental design used for forming the initial process database [28]
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2.1 Initial database

For covering a wide range of geometry features, Störkle et al. designed a specific base geometry (see Fig. 4) [28]. This geometry exhibits different convex and concave curved segments with equal lengths as well as a straight side and multiple corners which are rounded after extrusion. This base geometry was afterwards extruded in five degree steps with angles ranging from 30 to 60 degree resulting in seven different parts with varying wall angles (see Fig. 5). Lower angles would have caused an intersection of the part with itself, while higher might have exceeded the forming limit. Afterwards, a design of experiments was conducted. Additionally to the wall angle and therefore the geometry, the supporting force, supporting angle, and step depth were varied. For the supporting force, a range between 100 and 500 N was chosen as a lower force has a neglectable effect and a higher force would worsen the forming accuracy [35]. The supporting angle was varied between 0 and 1. For achieving a feasible forming force and forming accuracy, the maximum step depth was 0.6 mm. As the industrial robots have a positioning accuracy of 0.1 mm, this was chosen for the minimum step depth as a lower value would only increase the forming time. Via latin hypercube sampling, an experimental design with 35 individual forming experiments was conducted (see Table 2). During all experiments, both tools had a 4 mm radius, and DC04 steel of 0.8 mm thickness was formed. Afterwards, the still clamped sheets were digitized with a Comet 5 2 M white light stripe projector of Steinbichler Optotechnik GmbH.

3 Database extension

Despite the variation of the curvature in the base geometry, the wall angle and various process parameters and the vast amount of 35 individual forming experiments, is still limited in its geometry features. Plateaus and small concave radii are missing for example. For identifying missing geometry features, the authors present a cluster analysis approach. In the cluster analysis, similar geometry shapes are clustered. Afterwards, gaps between the clusters are found via distance metrics. These gaps represent missing geometry shapes which are used for constructing new parts to be formed.

Fig. 6
figure 6

The ROI of toolpath point transformed into a local coordinate system (a) and the corresponding surface representation consisting of \(5*5\) fitted planes (b)

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3.1 Surface representation

In order to identify similar geometry shapes in the cluster analysis, the mathematical description of the part is used as the input variables. The typical description used in CAD is non-uniform rational B-splines. These require extensive computation time and implementation expenditure when used in machine learning due to their complexity, especially if the whole part is taken into account [36]. Therefore, the mathematical description of the part needs to be transformed in order to decrease its complexity while still maintaining all relevant information about its shape. In ISF, the forming accuracy of an area is highly dependent on the local geometry as, for example, radii stiffen the geometry and reduce subsequent deformation. This led to approaches where the geometry was divided into individual features such as radii or planar surfaces. Abstract descriptions of the features (several length and angle metrics) and their spatially relationship to each other were used as the input for machine learning approaches [37, 38]. This abstraction made the prediction highly transferable. However, the division in different feature categories exhibits its limits when free-form geometries need to be classified. El Salhi et al. solved this problem by introducing geometry representations which are applicable to all geometry shapes [39]. The general idea involves the transformation of the part into a two-dimensional grid where every grid field contains the corresponding height. Every grid field represents one individual data set. The influence of the surrounding local geometry can be incorporated by surface representations of the adjacent grid fields. El Salhi et al. developed three individual surface representations. The local geometry matrix divides the adjacent grid fields into different categories based on their distance on the grid and the height difference towards the center grid field. In the local distance measure representation, the distances to critical points for the part accuracy, such as edges, form the representation. Finally, in the point series representation, the height values of the adjacent grid fields are taken into account in the order of a spiral. We take up on these ideas while also neutralizing some of the existing flaws. Using a fixed grid has the disadvantage that a machine learning prediction is only valid for the center of the individual grid field. If these are offset according to the prediction, it is technically challenging to calculate a new toolpath based on the grid. Instead of using a fixed grid and calculating a surface representation for every grid field, we use the toolpath points as a base for the representation. Therefore, the toolpath points can be adjusted according to the prediction. Around every toolpath point, we define a region of interest (ROI) of \(100*100~mm\) (see Fig. 6). The ROI is divided in a \(5*5\) grid where every grid field is described by a fitted plane. Therefore, the heights and normal vectors of the individual grid resemble the surface representation and are the input of the cluster analysis. Beforehand, the surface representation is transformed into a local coordinate system where the Y-axis points towards the next toolpath point while the Z-axis maintains its original direction. The X-axis is set accordingly. This way, the transferability of the surface representation is ensured. Otherwise, rotating the part in the clamping frame would yield completely different local surface representations, although the forming result would be the same. The initial database consists of 526,734 toolpath points. For every toolpath point, the surface representation consisting of \((5*5)*(1+2)=75\) ((grid size) * (fitted plane height + fitted plane normal vector components) parameters was calculated.

3.2 Cluster analysis

After calculating the surface representations for every toolpath point, the data was used to conduct a cluster analysis. Based on distance metrics, similar data is identified and clustered. This is used afterwards to fill the gaps between the clusters with new part geometries and forming experiments. As it is the goal to cluster similar part geometries, the surface representation coefficients are chosen as the cluster variables j with a total amount of J. The input of the cluster analysis are the objects k with their total amount of K. These resemble the 526,734 individual toolpath points. Their parameter values are denoted as p. As the cluster analysis functions based on distance metrics, the parameter values of the objects need to be of the same dimension. Otherwise, the results are distorted. We have chosen the surface representation coefficients and therefore height and angles as cluster variables. This results in a huge variance due to their different dimensions which can be seen in Fig. 7. To fix this, we have standardized the parameter values s into x which have a variance of 1 and a mean value of 0 according to

$$\begin{aligned} x_{kj}=\dfrac{(s_{kj}-\bar{s}_{j})}{\sigma _{j}} \end{aligned}$$
(1)
Fig. 7
figure 7

Mean value and variance of the cluster variables and therefore parameters of the surface representation before (top, mid) and after (bottom) the standardization

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Fig. 8
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Correlation of the individual cluster variables before and after the principal component analysis

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Furthermore, it needs to be ensured that the cluster variables are independent of each other as this would otherwise yield to an unequal weighting during the cluster analysis. This is especially important due to the fact that adjacent grid fields have similar surface representation values. Otherwise, there would be sharp edges in the sheet which can not be formed by ISF. The correlation between the cluster variables was removed via principal component analysis (PCA) where new independent cluster variables are calculated without losing any information [40]. In the first step, the covariance of two standardized cluster variables is calculated:

$$\begin{aligned} cov(x_{1},x_{2})=\dfrac{1}{K}\sum _{k=1}^{K}(x_{1k}-\bar{x}_{1})*(x_{2k}-\bar{x}_{2}) \end{aligned}$$
(2)

This is done for all possible combinations. Afterwards, their eigenvectors are calculated in the eigenvector matrix V:

$$\begin{aligned} V=ev \begin{pmatrix} cov(x_1,x_1) &{} ... &{} cov(x_1,x_K)\\ ... &{} {} &{} ...\\ cov(x_K,x_1) &{} ... &{} cov(x_K,x_K) \end{pmatrix} \end{aligned}$$
(3)

For receiving the PCA matrix P containing the independent cluster variables p, V is multiplied with the matrix X which contains all standardized parameter values:

$$\begin{aligned} P=V^{T}\times X^{T} \end{aligned}$$
(4)

When comparing the covariance and therefore correlation of the cluster variables before and after the PCA, it can be seen that the cluster variables are now independent of each other (see Fig. 8). Thus, the data preparation is complete, and the cluster analysis can begin. The Pearson correlation coefficient r was chosen for the distance metric as it is independent of variable units [41]. It can be calculated as the ratio of the covariance to the standard deviation:

$$\begin{aligned} r_{k1,k2}=\dfrac{\sum \limits _{j=1}^{J}(p_{jk1}-\bar{p}_{k1})*(p_{jk2}-\bar{p}_{k2})}{\sqrt{\sum \limits _{j=1}^{J}(p_{jk1}-\bar{p}_{k1})^2*\sum \limits _{j=1}^{J}(p_{jk2}-\bar{p}_{k2})^2}} \end{aligned}$$
(5)
Fig. 9
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Scree plot of the original database with 150 clusters chosen for the analysis

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A Pearson correlation coefficient of 0 means that both objects are dissimilar, while 1 or \(-1\) indicate similarity. For preforming the cluster analysis itself, the k-means clustering algorithm was chosen [42]. The algorithm needs the desired amount of clusters to be found as an input. In the scree plot (see Fig. 9), no clear kink can be seen what would indicate the amount of clusters. Due to the huge amount of toolpath points and the similarity of their surface representations as the toolpath points are directly to each other, one can not clearly distinguish individual geometry shapes and therefore clusters. As a compromise, a cluster amount of 150 was chosen for the k-means clustering.

Fig. 10
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Principle of the surface generation

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Fig. 11
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Order of the generation of new patches

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Fig. 12
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New test geometries derived from the generated surface representations with the lowest maximum Pearson correlation coefficients

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3.3 Surface generation

After performing the cluster analysis, new geometry shapes respectively gaps between the found clusters can be identified. Although, simply searching for data points with the lowest Pearson correlation coefficient would yield geometry shapes that are impossible to form with ISF as these are obviously not present in the database. These could contain offsets between the patches or too steep wall angles, for example. Instead, we have pursued a different approach by generating formable geometry surfaces and identifying the ones with the lowest Pearson correlation coefficient. Starting point for the surface generation is the patch in the middle of the grid. As the toolpath point defines the zero point, the fitted plane should go through the zero point. To achieve this, we define a gradient vector \(g_1\) that intersects with the zero point (see Fig. 10). The direction of the gradient vector is determined by the rotation angle \(\beta _1\). Afterwards, the final gradient vector is specified by the inclination angle \(\alpha _1\). Therefore, the middle surface patch can be described by the gradient vector \(g_1\). Afterwards, the starting point of the second surface patch is determined by projecting the y-axis onto the first fitted plane. The gradient vector \(g_2\) is specified in a similar manner. For achieving formable surfaces, the values of the inclination angle and of the rotation angle were limited to

$$\begin{aligned} 0^{\circ }< & {} \alpha _{1,2} < 55^{\circ } \end{aligned}$$
(6)
$$\begin{aligned} 225^{\circ }< & {} \beta _{1} < 315^{\circ } \end{aligned}$$
(7)
$$\begin{aligned} \beta _{2}= & {} \beta _{1} - 270^{\circ } \end{aligned}$$
(8)
Table 3 Parameter limits for the experimental design
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Table 4 Experimental design for expanding the process database
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Afterwards, the remaining surface patches were generated according to the same principle. The order of their generation followed the arrows in Fig. 11. In the end, the gaps in the remaining 8 grid fields were closed by creating a gradient vector between the projected axes of the adjacent grid fields. In total, 1,000,000 different surface representations were calculated. For each of these, the Pearson correlation coefficient to each of the 526,734 original toolpath point surface representations was calculated and the maximum value was saved. As a high value indicates a high similarity, this also implies that the correlation is worse for all other surfaces. Therefore, the 1,000,000 generated surfaces were sorted to find the ones with the lowest maximum Pearson correlation coefficients which are therefore the ones most dissimilar to the initial process database. These were used afterwards to derive 5 new test geometries based on 15 individual surface representations (see Fig. 12).

Fig. 13
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Comparison of the scree plots of the initial and the extended database

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Table 5 Clustering metrics of the initial and the extended database for various amounts of clusters
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3.4 Forming experiments

For extending the process database, 35 new forming experiments were conducted to double the amount of parts formed. The experimental design was determined via latin hypercube sampling. The limits of the forming experiments can be seen in Table 3. In comparison with the experimental design of the initial database (see Table 2), the maximum supporting force was reduced to 300 N as higher values greatly reduced the forming accuracy. All other parameter limits were identical. The same applies to the tool radii of 4 mm and the DC04 sheets with a thickness of 0.8 mm to ensure the comparability and therefore compatibility of the data. The wall angle as a parameter was replaced by the systematically derived test geometries. This led to the experimental design which can be seen in Table 4. During the forming experiments E49, E61, and E63, the parts cracked due to unknown reasons. The exact moment when the cracking occurred is unclear as the industrial robots prevent a clear view of the forming zone. Therefore, these forming experiments are deleted from the database. All remaining and successfully formed parts were measured in its still clamped state after forming, and the geometric accuracy of every toolpath point in normal direction was calculated. In total, the extended database consists of 982,806 toolpath points and their surface representations.

4 Validation

For validating the achieved diversification of the database, the cluster analysis was repeated for the extended database. When comparing the scree plot of the cluster analysis of the initial and the extended database (see Fig. 13), it can be observed that the mean squared error of the extended database is distinctly higher. Although this seems counterintuitively at first, this indicates a more diversified database as a higher amount of clusters is necessary for achieving a similar clustering performance. The same phenomenon can be noticed for different clustering metrics where the diversification of the database results in worse metrics (see Table 5). The Davies Bouldin score describes the average ratio of the distances within a cluster to the distances to the closest clusters [43]. A low value indicates a good clustering. For the same amount of 150 clusters, the extended database exhibits a worse score. When the amount of clusters during the cluster analysis is increased, the score improves until it is almost even for 1000 clusters. Therefore, it can be concluded that the extended database is more diversified. Higher cluster amounts than 1000 were not taken into account due to strongly increasing calculation times. The sum of squared errors is also decreasing with the amount of chosen clusters hinting that the extended database contains a higher amount of clusters. Although, its meaningfulness is limited as the amount of toolpath points of the extended database is nearly twice as high what automatically results in a higher error sum. The silhouette score is defined as the mean ratio of the average distances within a clusters to the average distances to the closest cluster where a high score value indicates a good clustering [44]. One can observe an increase of the score alongside with the amount of clusters. Although, from 500 to 1000 clusters, there can be seen a small decrease. Lastly, the Calinski Harabasz score describes the ratio of the sum of squared distances between clusters to the sum of squared distances within a cluster for all clusters [45]. Typically, a high value indicates a good amount of clusters, although a kink is necessary for clear information gain. As no kink is present in the data, the Calinski Harabasz score yields no further information. While the optimum amount of clusters can not be derived based on the given scores, the extended database seems to be harder to cluster even with a high amount of clusters. Furthermore, all clustering metrics worsened what indicates that the given data exhibits less clear gaps between the clusters. As this was the initial goal of the database extension, it can be concluded that the data has been successfully diversified.

5 Conclusion

Recently, numerous machine learning approaches were developed for predicting various process values of incremental sheet forming such as forming accuracy or surface roughness. In most cases, the prediction performance and generalizability are limited due to the fact that only a limited amount of forming experiments were carried out for gathering process data for the training process. Research institutes lack the capabilities for performing big experimental series. Furthermore, there are no incremental sheet forming datasets publicly available. Therefore, the authors presented a cluster analysis approach to systematically extend an existing process database while gaining the maximum amount of new information content in the process data. To do so, the part geometries in the existing database were transformed into mathematical surface representations of the toolpath points which resembled the data used in the k-mean cluster analysis. Afterwards, 1,000,000 artificial surface representations were computed by an algorithm, and the ones missing from the database were identified based on their Pearson correlation coefficient towards the beforehand detected clusters. From that, five parts were derived and formed in 35 forming experiments which extended the process database from 526,734 to 982,806 toolpath points. By repeating the cluster analysis, it could be validated that the database is distinctly more diversified. Even for a cluster amount of 1000, compared to the 150 clusters of the original cluster analysis, various clustering metrics such as the Davis Bouldin score indicate a worse clustering. Therefore, the process data in the database is harder to cluster and better distributed what increases the transferability of machine learning approaches. For enabling the application of machine learning approaches in incremental sheet forming for a wide range of users, the developed database consisting of 70 individual forming experiments is published. This first publicly available database forms therefore a crucial step for the application of machine learning techniques in incremental sheet forming, especially for transfer learning.