An Empirical Model and Feedforward Control of Laser Powder Bed Fusion

Quality control of the laser powder bed fusion (LPBF) parts remains an open research problem, as LPBF parts are prone to defects such as cracks, porosity [1], and poor surface finish [2,3]. Since LPBF part quality is strongly related to melt pool behavior [4], the regulation of melt pool geometry through laser power control is of particular interest. The laser power control problem can be cast as a melt pool regulation problem: the goal is to design the laser power profile to compensate for deviations in the melt pool temperature or geometry, ultimately reducing such effects as dross formation or overheating, e.g., in acute corners.

Prior research has shown that the melt pool can be regulated with a feedback controller based on measurements from a photodiode [5,6] or a camera [7–9] signal. However, feedback control in LPBF is challenging due to the high demands on the controller response time and the need for additional sensor-in-the-loop synchronization. Thus, the majority of existing literature focuses on the investigation of feedforward control strategies. Some of these strategies are model-free, such as the layer-to-layer data-driven control developed in Ref. [10] and the adjustment of the laser power based on geometry- and residual heat-based heuristics [11,12]. Others employ purely data-driven models: Yeung et al. [13] developed a regression model for melt pool size, while Ren and Wang [14] modeled the dynamics as a Gaussian process. The disadvantage of purely data-driven models is that they do not readily support control design strategies and are difficult to interpret. On the other hand, finite element process models are computationally expensive [4] and are poorly suited for process control. Recently, several reduced-order models have been reported in the literature [15,16], though no experimental validation was provided therein. A good example of an interpretable, control-oriented model of the melt pool behavior with experimental validation is presented in Ref. [17], where a model-based feedforward control application was demonstrated on a single-track scale. Thus, there is a lack of (a) interpretable, control-oriented models for melt pool in LPBF and (b) experimental application thereof on a part scale. From a physical understanding of LPBF, it is reasonable to expect that geometric features of the scan layer, i.e., sharp corners or narrow areas, can affect the melt pool behavior: if the laser quickly returns in proximity to the scanned point, it is likely that residual heat will not have dissipated completely. As a result, the melt pool increases in size at these locations, ultimately causing variability in the melt pool geometry. Currently, such melt pool variation issues are primarily addressed through a priori manual process parameter optimization. To address these gaps in current control design efforts in LPBF, this work focuses on (a) the development of a geometry-aware control-oriented model of the melt pool behavior and (b) the experimental implementation of feedforward control for LPBF based on this model.

The primary contributions of this work are the following:

1. Repeatable geometry-determined deviations of melt pool signatures (related to the decreasing scan line lengths) are reported. A control-oriented process model of these geometry-related deviations is identified from empirical data and experimentally validated. The model is shown to be applicable to different scanning geometries and scan patterns (Sec. 4).

2. A model-based feedforward controller is designed to suppress these deviations. The control problem is formulated as an optimization problem for laser power that varies line by line. The process output is predicted by the identified empirical model (Sec. 5.1).

3. The controller output is validated experimentally through builds of test parts. The experimental application of the designed controller on a part scale decreased geometric deviations in melt pool signal by 50% for different part geometries (Sec. 5.2).

As will be shown in Sec. 4, the geometry-dependent behavior of the melt pool can be modeled, given the laser power and the geometry of the scan pattern, i.e., the length of the nearby scan lines. Thus, the observed geometry-dependent measurement of the melt pool can be modeled, line by line, from empirical data, as function M(p_n, p_n−1, l_n, l_n−1), where p_n is the laser power, commanded while scanning line n, and l_n is the length of that line, while p_n−1 and l_n−1 correspond to the previous scan line (Fig. 1).

This research was performed on an open-architecture LPBF machine described in Ref. [10]. The machine is equipped with a SCANLAB intelliSCAN_de20 galvoscanner and a 400 W NdYAG laser and can build parts up to 50 × 50 mm² in cross section from commercially available metal powders, e.g., stainless steel. The supervisory control of the machine is achieved via america makes software [18], augmented with in-house developed c++ code. The low-level control of the scanning process, i.e., laser positioning and firing, is handled by the scanner control board. The scanning instructions for a layer are formatted as a text file containing a list of straight lines (scan file). Each line in the scan file is defined by start and end points, a laser power level, and a scanning speed value. Thus, the machine allows adjustment of the laser power within a layer on a line-by-line basis simply by modifying the input scan file.

To monitor the melt pool during the LPBF process, a coaxial camera-based setup, similar to that described in the literature [5,19,20], is integrated with the LPBF testbed. A Basler acA2000-165umNIR camera acquires 8-bit intensity images in the near-infrared band (800–950 nm) by looking at the melt pool through laser scanning optics. Each image is 64 × 64 pixels in size, with an instantaneous field of view of 22 µm per pixel. The camera acquires 2000 frames per second. A typical melt pool image (post-processed in matlab is shown in Fig. 2.

The melt pool location on the build plate at a given point in time was determined based on the nominal scan pattern, assuming constant scanning velocity and perfect trajectory tracking by the galvoscanner. Given the camera-assigned time stamp and the positional commands that define the layer scanning sequence, the 1D melt pool signature, such as C₁, can be transformed from a time series C₁(t) to a spatial map C₁(x, y) and plotted as a function of spatial coordinates in 2D, as illustrated in Fig. 3.

Remark. At length scales below 500 µm, the mapping of the nominal scan pattern to a measurement is imprecise due to the lack of positional feedback. Thus, in this work, images belonging to a line shorter than 0.5 mm were excluded from the model identification.

A prismatic part in Fig. 4 was designed to investigate if a quick return of the laser in the proximity of the previously scanned track would influence the melt pool behavior. The part consists of four stacked prisms G1–G4, triangular in cross section. There are four orientations of the scanning raster pattern throughout the part (i.e., four different hatch angles S1–S4). The whole build included four parts (P1–P4) of the same geometry. This way, a large variety of corner scan patterns, with repetition, could be studied: 174 layers were built for each part P1–P4, with each scan pattern S1–S4 repeating at least 10 times. Process parameters were set as follows: laser power 225 W, scanning velocity 800 mm/s, hatch spacing 90 µm, and continuous meander (snake-like) linear scan pattern. This test will be referred to as the TP (triangle prism) test.

The coaxial images acquired from the TP repeatedly showed geometry-dependent behavior: C₁ would raise in a corner, if the laser was scanning “into” it. The C₁ was found to exponentially depend on the length of the scan line (Fig. 5), i.e., $C1=a+bexp(−l)$⁠, where l is the scan line length. The effect was independent of the orientation of the scan pattern, the orientation of the part within the process chamber, the corner angle, or the areal size of the triangle.

To investigate the dependence of model parameters on input laser power, four different laser powers (150 W, 175 W, 200 W, and 225 W) were used to build eight cubes, with each power replicating twice. Each cube was scanned at three different angles: $30deg,135deg$⁠, and 60 deg (Fig. 6). The hatch spacing, scanning velocity, and meander scan were repeated from the TP test. This test is referred to as Cubes in the following text.

Figure 8 illustrates the model fit to one of the validation layers from Cubes. The coaxial signal C₁ exhibits high variation; thus, a filtered signal had to be used to quantify the geometry-related trend and the model's fit to it. It appears that the identified model captured the in-layer signal trend well.

For the in situ demonstration of the feedforward control, two additional geometries were used. The Star and Wave geometries are shown in Fig. 9. They were designed with curved edges to distinguish them from the simple shapes of TP and Cubes to further support the efficacy of the proposed controller. Star was scanned at $0deg$ and $90deg$ (horizontally and vertically), while Wave was scanned at three different angles (cycling through $0deg$⁠, $45deg$⁠, $0deg$⁠, and $135deg$⁠). The nominal laser power was set to 200 W, while the scanning speed and hatch spacing remained the same as in other tests.

The control problem, as discussed in Sec. 2, is to find the optimal vector of line-by-line powers $P~≜[p0,p1,…,pL]$⁠, such that the difference between all values of C₁(n) and the reference C_ref is minimized over the whole layer, on a line-by-line basis, subject to lower and upper power limits.

Given the model (5) and scan pattern geometries, the optimal line-by-line power profiles were found for the Star and Wave parts, via matlab with the fmincon function, for all unique scan patterns of these two geometries. Power limits were defined as 150 W and 225 W, and the desired level for C₁ was set to C_ref = 1500 (the average observed value of C₁ in open-loop tests).

The optimized laser power profiles are shown in Figs. 10 and 11. These power profiles, once applied in situ, should have reduced or completely eliminated the geometry-dependent behavior of the coaxial signature C₁.

The optimized power profiles were subsequently tested experimentally. Each of the layer scan files for the Star and Wave was updated offline, such that each scan line received an appropriate power command, as defined by the solution of the optimization problem (1). Figures 12 and 13 compare C₁ signals from the controlled and the open-loop layers of the parts for different scan directions. The Star (Fig. 12) test showed a reduction in 2-norm errors for each scan path: 28.0% and 24.6%, respectively. Similarly, the results from the Wave (Fig. 13) test showed a reduction of 25.1%, 25.5%, and 14.0% for each scan direction, demonstrating the efficacy of optimized power profiles. The geometry-dependent behavior of C₁ in corners and thin sections of the Star and Wave was visibly reduced in the controlled layers.

To assess the reduction of the geometry-related variation, the C₁ signal was filtered with the filter F (as in Fig. 8) due to high levels of variance. Once the filtered signal F(C₁) was obtained, the standard deviation of F(C₁), σ(F(C₁)) (commonly defined), was used to quantify the range of signal change. Figure 14 shows the σ(F(C₁)) for 20 layers of the controlled Star and Wave and compares it with the corresponding open-loop data. It is evident that, for both geometries, variation in the C₁ signal was reduced, with consistently lower values of σ(F(C₁)). The reduction of in-layer geometry-related variation in C₁ was approximately 50%. Thus, the model-based feedforward control was successful in regulating the undesirable melt pool behavior, and the control approach was experimentally validated.

In this letter, the behavior of the melt pool characteristics (such as the area of the melt pool footprint) was shown to depend on the geometry of the layer scan pattern. It was demonstrated that the footprint exponentially increases in size in the areas of a part where scan lengths shorten, e.g., in corners or narrow sections. The existence of such exponential behavior was observed on different parts and scan geometries, including the ones with non-straight edges. To capture this behavior, the empirical model, which incorporates variable laser power, was developed and experimentally validated. This model further enabled the application of a line-by-line model-based feedforward controller to regulate the footprint area. To reduce the signal deviations, optimal laser power profiles were calculated given the empirical model. These power profiles were then evaluated experimentally, for different geometries. The experimental evaluation of the proposed model-based feedforward control scheme demonstrated that such an approach reduces the geometry-induced changes in footprint area two-fold as compared to the open-loop operation.

This work was supported by NSF CMMI Award #2222250.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.