Feedforward Vector-Level Laser Power Scheduling

• The paper presents a feedforward control method that assigns laser power per scan vector to maintain consistent melt pool metrics in LPBF.
• It integrates empirical and physics-based models to predict geometry-induced thermal variations and optimize processing parameters.
• Experimental validation shows reductions in melt pool footprint variability and porosity, illustrating improved process quality over constant-power scanning.

Searching arXiv for the cited LPBF and related papers to ground the article in published work. Feedforward vector-level laser power scheduling is a control and toolpath-planning approach in which laser power is assigned at the level of individual scan vectors, or thermally defined vector segments, before execution so that predictable geometry-dependent disturbances are compensated proactively rather than reactively. In laser powder bed fusion (LPBF), the method is used to regulate melt pool behavior across a layer or build by exploiting the known scan order, vector lengths, local support conditions, and predicted thermal history. Published formulations range from empirical geometry-dependent scheduling laws based on melt pool footprint measurements to physics-based frameworks that couple part-scale thermal prediction with reduced-order melt pool models (Shkoruta et al., 2022, Kirschbaum et al., 16 Jul 2025).

1. Conceptual definition and control objective

In LPBF, vector-level scheduling addresses the fact that constant-power scanning does not produce constant melt pool conditions. Under a continuous meander hatch, the melt pool “footprint” increases in regions where hatch vectors become short, such as sharp corners, narrow necks, and tapering sections. The reported interpretation is that shorter vectors reduce revisit time to nearby previously scanned material, increase residual heat coupling, and enlarge the melt pool footprint. With scan line length $l$ and scan speed $V$, the average revisit time is approximated as

$\Delta t = \frac{l}{V}.$

Shorter $l$ therefore implies smaller $\Delta t$, less cooling time, and a larger footprint signal (Shkoruta et al., 2022).

The control objective is to assign one laser power value to each vector so that a chosen melt pool metric remains approximately constant despite geometry-induced thermal variation. In the empirical formulation, the regulated quantity is the melt pool footprint $C_1$, extracted from coaxial near-infrared images. In the later physics-based formulation, the regulated quantity is melt pool area $A_c$, predicted from local subsurface temperature, scan speed, and laser power. In both cases, feedforward is appropriate because the disturbance is toolpath-determined, available before scanning, and sufficiently repeatable to justify offline compensation (Shkoruta et al., 2022, Kirschbaum et al., 16 Jul 2025).

“Vector-level” has a specific meaning in this literature. In the 2022 empirical work, the unit of control is the scan line, and the laser power level during a scan of a line is constant. In the 2025 physics-based work, a vector can itself be a subdivision of a line according to expected local thermal conditions, so that scheduling remains path-wise and machine-aligned while resolving finer thermal structure than layerwise control. This places vector-level scheduling between coarse layerwise parameter selection and fully continuous intra-vector modulation (Shkoruta et al., 2022, Kirschbaum et al., 16 Jul 2025).

2. Empirical geometry-dependent scheduling in LPBF

The empirical line of work defines the melt pool footprint from thresholded coaxial image pixels as

$C_\alpha = \sum_{r,c} \mathds{1}[I(r,c)\ge \alpha], \qquad \alpha=\{1,100\},$

with the control-oriented model built on the low-threshold footprint metric

$C_1 = \sum_{r,c} \mathds{1}[I(r,c)\ge 1].$

The reported geometric trend is exponential in scan line length:

$C_1(l)=C_\infty+\Delta C \exp(-l/r),$

where $V$0 is the asymptotic footprint for long lines, $V$1 is the short-line excess footprint, and $V$2 is the decay length scale. For one scan example, fitting this form gave RMSE $V$3 and $V$4, and the paper states that the model performs similarly on other parts, geometries, and directions (Shkoruta et al., 2022).

To make the model usable for control, the coefficients are parameterized by laser power $V$5:

$V$6

with

$V$7

$V$8

$V$9

The reported regressions yielded $\Delta t = \frac{l}{V}.$0 for $\Delta t = \frac{l}{V}.$1, $\Delta t = \frac{l}{V}.$2 for $\Delta t = \frac{l}{V}.$3, and $\Delta t = \frac{l}{V}.$4 for $\Delta t = \frac{l}{V}.$5. This produces a static geometry-power map that is partly separable into a baseline term and a geometry-dependent exponential term, but not fully separable into $\Delta t = \frac{l}{V}.$6 because the geometry term itself depends on power (Shkoruta et al., 2022).

The control-oriented reformulation introduces line-to-line thermal carryover:

$\Delta t = \frac{l}{V}.$7

Here, $\Delta t = \frac{l}{V}.$8 is the current line’s steady-state contribution, while the exponential term is interpreted as the residual effect of the previous line’s power and length. Geometry enters through vector length, and transitions are represented through explicit dependence on the previous vector. The model was developed for continuous meander hatch scanning and does not explicitly parameterize corner angle, curvature, or jump time. Curved and discontinuous geometries were nevertheless used for validation, suggesting practical robustness within the limits of the line-length abstraction (Shkoruta et al., 2022).

3. Physics-based thermal and melt-pool formulations

A later formulation replaces the purely empirical geometric law with a modular, physics-based framework that decouples part-scale thermal behavior from local melt pool physics. The architecture couples a conduction-only finite-difference thermal model to a reduced-order analytical melt pool model. The thermal model predicts vector-level temperature fields and, in particular, the scalar subsurface temperature one layer below the upcoming vector; the melt pool model maps that scalar thermal state and the process settings to melt pool area. This decoupling is presented as the reason the method remains computationally tractable while still accounting for thermal history, short vectors, thin walls, holes, and overhangs (Kirschbaum et al., 16 Jul 2025).

The governing thermal equation is

$\Delta t = \frac{l}{V}.$9

with a hemispherical Goldak heat source, convection at the top surface, insulation at the side surfaces, and constant temperature at the bottom. The discretized model is rearranged into state-space form,

$l$0

and then lumped over a vector,

$l$1

so that control acts at vector boundaries rather than at every thermal timestep. The scalar quantity passed to the melt pool model is the average temperature one layer below the vector footprint,

$l$2

Overhangs are treated separately by a 1D powder-conduction approximation, and interlayer dwell is approximated by 1D heat transfer in $l$3 followed by a 2D Gaussian blur in $l$4 (Kirschbaum et al., 16 Jul 2025).

The reduced-order melt pool model is Rosenthal-based. The reported approximations are

$l$5

with melt pool area approximated as a triangle plus a half circle,

$l$6

This makes the optimization one-dimensional in power for each vector. Reported calibration results for the analytical model are $l$7, $l$8 for Inconel 718 and $l$9, $\Delta t$0 for 316L stainless steel, with width-fit $\Delta t$1 values of $\Delta t$2 and $\Delta t$3, and length-fit $\Delta t$4 values of $\Delta t$5 and $\Delta t$6, respectively. The authors state that the framework relies more on capturing relative trends with $\Delta t$7, $\Delta t$8, and $\Delta t$9 than on exact absolute area prediction (Kirschbaum et al., 16 Jul 2025).

4. Scheduling laws and optimization structure

The empirical LPBF controller computes a vector of line-by-line powers

$C_1$0

by minimizing predicted deviation from a constant melt pool footprint reference:

$C_1$1

The implementation reported in the paper uses MATLAB fmincon, with bounds

$C_1$2

and $C_1$3, chosen as the average value of $C_1$4 for the nominal scan. Computation time is given as approximately $C_1$5 s on a ThinkPad T495 laptop. The power is constant during each scanned line, and no additional rate constraints on power changes are described (Shkoruta et al., 2022).

The physics-based scheduler uses a different optimization structure. For each vector $C_1$6, it solves the scalar bounded problem

$C_1$7

subject to machine power bounds $C_1$8. In the reported experiments, $C_1$9 is kept constant throughout the print and is tuned so that optimized power remains near the nominal process point in a bulk region. The resulting controller is sequential rather than globally optimized over all vectors at once: predict $A_c$0, solve for current-vector power, propagate the thermal state across the vector, and continue (Kirschbaum et al., 16 Jul 2025).

The two formulations therefore differ in both state representation and optimization granularity. The empirical controller uses a compact previous-line residual model and a multi-line constrained optimization over a layer or pattern. The physics-based controller uses forward simulation of a thermal state and a one-dimensional optimization per vector. A plausible implication is that the latter trades global optimality for explicit thermal interpretability and extensibility to features such as overhangs and layer-to-layer accumulation (Shkoruta et al., 2022, Kirschbaum et al., 16 Jul 2025).

5. Identification, calibration, and experimental evidence

The empirical model was identified on an open-architecture LPBF machine with a SCANLAB intelliSCAN$A_c$1 20 galvoscanner, a 400 W NdYAG laser, and a build area up to $A_c$2 mm$A_c$3, using cobalt chrome alloy, $A_c$4 mm/s scan speed, $A_c$5 hatch spacing, and a Basler acA2000-165umNIR coaxial camera operating at $A_c$6 kHz in the $A_c$7–$A_c$8 nm band. Exploratory geometry studies used stacked triangular prisms; power-dependent calibration used eight cubes at $A_c$9, $C_\alpha = \sum_{r,c} \mathds{1}[I(r,c)\ge \alpha], \qquad \alpha=\{1,100\},$0, $C_\alpha = \sum_{r,c} \mathds{1}[I(r,c)\ge \alpha], \qquad \alpha=\{1,100\},$1, and $C_\alpha = \sum_{r,c} \mathds{1}[I(r,c)\ge \alpha], \qquad \alpha=\{1,100\},$2 W; and validation used star and wave geometries not used for model fitting. The parameterized model was validated on the last three layers of the cubes, corresponding to 24 separate scans not used in training, with $C_\alpha = \sum_{r,c} \mathds{1}[I(r,c)\ge \alpha], \qquad \alpha=\{1,100\},$3 across all 24 validation scans after median filtering. In feedforward validation on 20 controlled layers for each geometry, within-layer variation of the filtered footprint signal $C_\alpha = \sum_{r,c} \mathds{1}[I(r,c)\ge \alpha], \qquad \alpha=\{1,100\},$4 was reduced by approximately $C_\alpha = \sum_{r,c} \mathds{1}[I(r,c)\ge \alpha], \qquad \alpha=\{1,100\},$5, or “reduced by a factor of two,” relative to constant-power scanning (Shkoruta et al., 2022).

The physics-based framework uses a more modular calibration workflow. The melt pool model is identified from single-track scans on thin metal plates mounted on a heated baseplate, with sweeps over power $C_\alpha = \sum_{r,c} \mathds{1}[I(r,c)\ge \alpha], \qquad \alpha=\{1,100\},$6 W, speed $C_\alpha = \sum_{r,c} \mathds{1}[I(r,c)\ge \alpha], \qquad \alpha=\{1,100\},$7 mm/s, and baseplate temperature $C_\alpha = \sum_{r,c} \mathds{1}[I(r,c)\ge \alpha], \qquad \alpha=\{1,100\},$8, $C_\alpha = \sum_{r,c} \mathds{1}[I(r,c)\ge \alpha], \qquad \alpha=\{1,100\},$9, $C_1 = \sum_{r,c} \mathds{1}[I(r,c)\ge 1].$0, and $C_1 = \sum_{r,c} \mathds{1}[I(r,c)\ge 1].$1. All parameter combinations were tested twice, for $C_1 = \sum_{r,c} \mathds{1}[I(r,c)\ge 1].$2 single-track scans per material. The thermal model is then calibrated on a 2D “stepped pyramid” geometry with 333 vectors, using the normalized area-variation metric

$C_1 = \sum_{r,c} \mathds{1}[I(r,c)\ge 1].$3

to tune the heat-input factor $C_1 = \sum_{r,c} \mathds{1}[I(r,c)\ge 1].$4 and the target melt pool area $C_1 = \sum_{r,c} \mathds{1}[I(r,c)\ge 1].$5. Reported values are $C_1 = \sum_{r,c} \mathds{1}[I(r,c)\ge 1].$6 for Inconel 718, $C_1 = \sum_{r,c} \mathds{1}[I(r,c)\ge 1].$7 for 316L stainless steel, and $C_1 = \sum_{r,c} \mathds{1}[I(r,c)\ge 1].$8 for both materials. The authors emphasize that under 3000 vectors per material were scanned on 2D plates to calibrate both models (Kirschbaum et al., 16 Jul 2025).

Validation of the physics-based scheduler was performed on a 562-layer “cologne bottle” geometry containing thin walls, overhangs, holes, and changing cross-section, using an open-architecture PANDA 11 LPBF machine with a 500 W IPG Photonics 1070 nm fiber laser, $C_1 = \sum_{r,c} \mathds{1}[I(r,c)\ge 1].$9 spot size, and coaxial, photodiode, and IR monitoring. The main reported results are a 62% reduction in geometric inaccuracy in key dimensions, a 16.5% reduction in overall porosity, and a 6.8% average decrease in photodiode variation across geometrically complex layers 100–550. In a key overhang layer, layer 488, the photodiode signal variation was reported as $C_1(l)=C_\infty+\Delta C \exp(-l/r),$0 nominally and $C_1(l)=C_\infty+\Delta C \exp(-l/r),$1 under feedforward control, a 38% reduction. The whole-part porosity reduction was reported as 17% for Inconel 718 and 16% for 316L stainless steel, while overhang-region porosity reduction was 61% and 24%, respectively (Kirschbaum et al., 16 Jul 2025).

6. Assumptions, limitations, and terminological boundaries

Several limitations recur across the LPBF literature. The empirical model is machine-, material-, sensor-, and process-specific; it uses only the previous line’s length and power; it does not explicitly model jumps, corner angles, or broader thermal memory; and it was trained with vectors shorter than $C_1(l)=C_\infty+\Delta C \exp(-l/r),$2 mm excluded because image-to-position mapping was unreliable at that scale. Raw $C_1(l)=C_\infty+\Delta C \exp(-l/r),$3 measurements exhibit substantial high-frequency variation, and the reported validation trends rely on median filtering. In addition, the fit quality of $C_1(l)=C_\infty+\Delta C \exp(-l/r),$4 and $C_1(l)=C_\infty+\Delta C \exp(-l/r),$5 is limited, with $C_1(l)=C_\infty+\Delta C \exp(-l/r),$6 and $C_1(l)=C_\infty+\Delta C \exp(-l/r),$7, respectively (Shkoruta et al., 2022).

The physics-based framework addresses some of those deficiencies but introduces others. Its thermal model neglects radiation, Marangoni convection, latent heat, and detailed powder-bed effects; the analytical melt pool model inherits the assumptions of Rosenthal’s solution; powder conduction in overhangs is represented crudely; and the interlayer dwell approximation is explicitly approximate. The controller tracks a constant melt pool area target even though constant area is not claimed to be the true optimum for geometry, porosity, microstructure, or mechanical properties. Validation was performed on one open-architecture machine, two materials, and one benchmark 3D geometry (Kirschbaum et al., 16 Jul 2025).

A common misconception is to equate vector-level feedforward scheduling with continuous intra-vector modulation. The LPBF studies do not do that. In the empirical work, power is constant over each line. In the physics-based work, one optimized power is assigned per vector, where vectors may be subdivided according to thermal context, but the method remains discretized at vector granularity rather than continuously along the scan path (Shkoruta et al., 2022, Kirschbaum et al., 16 Jul 2025).

A second misconception is terminological. Outside LPBF, closely related phrases appear in optical wireless communication, but they refer to different objects of control. One study uses a DNN to generate a vector-valued power allocation over outer-common, inner-common, and private hierarchical rate-splitting messages in an infrared VCSEL network; its output layer size is $C_1(l)=C_\infty+\Delta C \exp(-l/r),$8, with a 10,000-sample dataset and 60%/20%/20% training-validation-test split (Alazwary et al., 2023). Another develops one-step-ahead mobility-aware power control for VCSEL-based indoor OWC, where the controlled quantity is a per-user next-slot power vector $C_1(l)=C_\infty+\Delta C \exp(-l/r),$9 computed from predicted mobility state and predicted channel matrix at a 100 ms control interval; the reported gains are approximately 30–35% average energy-efficiency improvement over CPC and ConsPC and approximately 45–65% over RPC (Ncube et al., 24 Apr 2026). These works are relevant as examples of feedforward vector-valued laser power allocation, but they are not direct per-emitter temporal scheduling in the strict LPBF sense.

A plausible implication of the published record is that feedforward vector-level scheduling is best viewed as a complement to feedback rather than a replacement for it. The LPBF papers emphasize proactive cancellation of predictable, geometry-driven disturbances, while leaving residual errors, drift, powder variation, and unmodeled events as natural targets for future closed-loop correction (Shkoruta et al., 2022, Kirschbaum et al., 16 Jul 2025).