# Gaussian Process (GP)-based Learning Control of Selective Laser Melting
Process ††thanks: The work was supported in part by the US National Science
Foundation under award CNS-1932370.

Farshid Asadi F. Asadi, A. Olleak, J. Yi, and Y. Guo are with the the
Department of Mechanical and Aerospace Engineering, Rutgers University,
Piscataway, NJ 08854 USA (e-mail: fa416@scarletmail.rutgers.edu).  Alaa Olleak
Jingang Yi  and Yuebin Guo

###### Abstract

Selective laser melting (SLM) is one of emerging processes for effective metal
additive manufacturing. Due to complex heat exchange and material phase
changes, it is challenging to accurately model the SLM dynamics and design
robust control of SLM process. In this paper, we first present a data-driven
Gaussian process based dynamic model for SLM process and then design a model
predictive control to regulate the melt pool size. Physical and process
constraints are considered in the controller design. The learning model and
control design are tested and validated with high-fidelity finite element
simulation. The comparison results with other control design demonstrate the
efficacy of the control design.

##  I Introduction

Metal additive manufacturing (AM) builds three-dimensional parts by melting
metal powders layer-by-layer using laser or other heating sources. Selective
laser melting (SLM) is one of the popular metal AM processes. Real-time
feedback control of metal AM is an enabling tool for reliable, robust, high-
quality process [1]. Although repetitive feature is used for control of metal
AM [1], few work exist for real-time control of SLM due to its complex process
features and lack of enabling, effective and reliable sensing techniques. The
goal of this paper is to develop a data-driven SLM process modeling and
control design.

Control of metal AM has been reported in past decade. For example, built on
repetitive nature of the AM process, iterative learning control is an
effective way to improve the fabrication of the metal AM processes (e.g., [2,
3]). The dynamic models for directed energy deposition (DED) processes are
mainly built on mass conversation and energy balance of the melt pool (e.g.,
[4, 5]). These physics-based model become complex for modeling SLM process
because of dynamic behavior of the powder melt pool, non-uniform heat
dissipation from the melt pool to the surrounding materials, and various
process parameters and scanning patterns. The recent work in [6] built a
simplistic physics-based model for real-time track-by-track control of SLM.
The effect of previous scanning track is considered in the next track control.
Several important assumptions are made for the melt pool geometry to derive
the model using the energy balance principle. Finite element (FE) method
(e.g.,[7]) and multi-scale multi-physics models (e.g., [8, 9]) are used to
obtain high-fidelity simulation for SLM process. It is however, impossible to
use these models to design real-time process control due to their high
computational costs. The recent work in [10], used data obtained from FE
analysis to derive a linear model of the process. In their work, the authors
used a repetitive control algorithm to control melt pool width for a multi-
track case in simulated FE software.

Surrogate models are developed to optimize the SLM process parameters for
stable melt pool and process windows of the laser power and velocity maps [11,
12, 13, 14]. SLM powder material characterization and process properties
(e.g., melt pool geometry, porosity, etc.) are also reported by using the
Gaussian process (GP) models [11, 12] and it is thus attractive to integrate
these development into the real-time control design. However, these data-
driven models do not incorporate temporal information and therefore cannot
directly be applied to capture dynamics of the SLM process. In this paper, we
extend the GP models to capture the dynamics feature of the SLM process and
propose a learning-based real-time process control.

Figure 1: (a) SLM process experimental setup. (b) Rectangular SLM scanning
strategy. (c) Layer-by-layer configuration for SLM process.

GP dynamics model (GPDM) and model predictive control (MPC) was proposed and
used in various applications [15, 16, 17]. In this paper, we take advantages
of the GP learning model of the SLM dynamics and develop an optimization-based
process control. The data-driven GP model is built on the calibrated FE
simulation with limited experimental data. The main contribution of the work
lies in the data-driven GP model and the design of real-time MPC control of
the stable melt pool in SLM.

##  II Data-Driven SLM Process Models

###  II-A SLM Process Dynamics

SLM is a powder bed fusion (PBF) process where the powder particles absorb the
energy delivered by the moving laser beam, start to melt, and form a melt
pool. Fig. 1 illustrates the experimental setup of a typical SLM process. The
laser beam movement is controlled by mirrors that direct the beam in the $x$
and $y$ directions at high speeds and the laser beam is shot on powder
particles layers. During a SLM process, a thin layer of materials powder is
distributed over the solid substrate by the recoater. Under the laser beam
scanning, a desired pattern is fused to the substrate. For perfect bonding,
the melt pool must be deeper than the powder layer thickness and wider than
hatch spacing (i.e., the distance between center lines of two adjacent
tracks). After the entire layer is scanned according to the scanning strategy,
the next powder layer is added and the process is repeated until the entire
part is built. Fig. 1 shows one layer scanning SLM process and Fig. 1
illustrates the multiple layer building structure.

It was reported that over one hundred parameters affect the quality of parts
manufactured by SLM, among which the most important parameters are laser power
and scanning speed. Furthermore, residual heat effect due to the subsequent
scanning tracks might result in high melt pool size variations over the entire
layer. Thus, real-time process monitoring and feedback control are needed to
achieve a stable and consistent melt pool sizes. Process monitoring can be
conducted through fast thermal imaging by using CCD camera, pyrometer, and
light sensitive diodes, e.g., [18]. These sensing suites can be used to
provide information about the temperature distribution, and melt pool size,
and therefore to develop a feedback control enhancing the process behavior. In
this work, a single layer with multi bidirectional tracks are considered for
SLM modeling and control.

###  II-B Gaussian Processes Modeling

GP modeling is used to obtain data-driven model of the SLM process dynamics.
We here present a short summary of the GP models and readers can refer to [19]
for further details. GP is a powerful technique to derive analytical models
from relatively small data set. We consider a noisy observation of the process
as

$y=f(\bm{x})+\eta,$ (1)

where $y\in\mathbb{R}$ is a scalar output observation,
$\bm{x}\in\mathbb{R}^{n}$ is an $n$-dimensional independent variable,
$f(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}$ is the underlying unknown
model, and $\eta\sim\mathcal{N}(0,\sigma^{2}_{n})$ is zero mean Gaussian
observation noise with variance $\sigma^{2}_{n}$. Furthermore, we assume that
$f(\cdot)$ has a Gaussian probability distribution prior. Without loss of
generality, $f(\cdot)$ is considered as a Gaussian process with the (zero)
mean and covariance function

$f(\bm{x})\sim\mathcal{GP}(0,k(x,x^{\prime})),$ (2)

where
$k(\bm{x},\bm{x}^{\prime}):\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$
is a positive definite covariance function. In this work, a squared
exponential function is used with the following form

$k(\bm{x},\bm{x}^{\prime})=\sigma^{2}_{f}\exp\begin{pmatrix}-\frac{(\bm{x}-\bm{x}^{\prime})^{T}\bm{L}^{-1}(\bm{x}-\bm{x}^{\prime})}{2}\end{pmatrix},$
(3)

where $\sigma_{f}$ and $\bm{L}=\mathop{\rm
diag}\nolimits\left(l_{1},\dots,l_{n}\right)$, $l_{i}>0$, $i=1,\ldots,n$, are
hyper parameters to be determined.

Having $m$ observation pairs $(y_{i},\bm{x}_{i})$, $i=1,\ldots,m$, and the
assumption of Gaussian prior over $f(\cdot)$, we construct the following joint
distribution of training data set and prediction pair as

$\begin{bmatrix}\bm{Y}\\\
f^{*}\end{bmatrix}\sim\mathcal{N}\begin{pmatrix}\mathbf{0},\begin{bmatrix}\bm{K}(\bm{X},\bm{X})+\sigma^{2}_{n}\bm{I}_{m}&\bm{K}(\bm{X},\bm{x}^{*})\\\
\bm{K}(\bm{x}^{*},\bm{X})&K(\bm{x}^{*},\bm{x}^{*})\end{bmatrix}\end{pmatrix},$
(4)

where $\bm{Y}=\begin{bmatrix}y_{1},\dots,y_{m}\end{bmatrix}^{T}$ and
$\bm{X}=\begin{bmatrix}\bm{x}_{1},\dots,\bm{x}_{m}\end{bmatrix}^{T}$ are
training data sets, $\bm{x}^{*}$ and $f^{*}$ are prediction pair, and
$\bm{K}(\cdot,\cdot)$ are corresponding covariance given as follows.

$\displaystyle\bm{K}(\bm{X},\bm{X})=\begin{bmatrix}k(\bm{x}_{1},\bm{x}_{1})&\dots&k(\bm{x}_{1},\bm{x}_{m})\\\
\vdots&\ddots&\vdots\\\
k(\bm{x}_{m},\bm{x}_{1})&\dots&k(\bm{x}_{m},\bm{x}_{m})\end{bmatrix}\in\mathbb{R}^{m\times
m},$

$\bm{K}(\bm{x}^{*},\bm{X})=[k(\bm{x}^{*},\bm{x}_{1})\,\cdots\,k(\bm{x}^{*},\bm{x}_{m})]=\bm{K}^{T}(\bm{X},\bm{x}^{*})\in\mathbb{R}^{1\times
m}$, and $K(\bm{x}^{*},\bm{x}^{*})=k(\bm{x}^{*},\bm{x}^{*})$.

Having the joint distribution (4), we use Bayesian rule to calculate posterior
distribution of $f^{*}$ as a function of $\bm{x}^{*}$, conditional on training
data set $\bm{Y}$ and $\bm{X}$ as

$f^{*}|\bm{X},\bm{Y},\bm{x}^{*}\sim\mathcal{N}\begin{pmatrix}\bar{f}^{*},\mathop{\rm
cov}\nolimits(f^{*})\end{pmatrix},$ (5)

where mean
$\bar{f}^{*}=\bm{K}(\bm{x}^{*},\bm{X})\left(\bm{K}(\bm{X},\bm{X})+\sigma^{2}_{n}\bm{I}_{m}\right)^{-1}\bm{Y}$
and covariance function $\mathop{\rm
cov}\nolimits(f^{*})=K(x^{*},x^{*})-\bm{K}(\bm{x}^{*},\bm{X})\left(\bm{K}(\bm{X},\bm{X})+\sigma^{2}_{n}\bm{I}_{m}\right)^{-1}\bm{K}(\bm{X},\bm{x}^{*})$.
We then write the marginal likelihood probability for the training data set as

$\log
p(\bm{Y}|\bm{X},\bm{\Theta})=-\frac{1}{2}\bm{Y}^{T}\bm{K}_{Y}^{-1}\bm{Y}-\frac{1}{2}\log\left|\bm{K}_{Y}\right|-\frac{n}{2}\log(2\pi),$

where $\bm{K}_{Y}=\bm{K}(\bm{X},\bm{X})+\sigma^{2}_{n}\bm{I}_{m}$,
$\bm{\Theta}=\\{\sigma_{f},\sigma_{n},\bm{L}\\}$ is the hyper-parameter set,
and $|\bm{K}_{Y}|$ denotes the determinant of $\bm{K}_{Y}$. Maximizing
marginal likelihood with respect to $\bm{\Theta}$ would generate the
appropriate hyper parameters.

Figure 2: (a) SLM process FEA overview. (b) Melt pool area extraction from
thermal solution.

###  II-C GP-based SLM Process Dynamics

We consider the GP model for the SLM dynamics. For the SLM process, one of the
main performance metrics is the consistent melt pool cross sectional area,
denoted as $x(t)$ at time $t$. Among many process parameters, laser power and
scanning speed are the most influential process control parameters [7].
Therefore, we consider laser power and scanning speed, denoted as $p(t)$ and
$v(t)$ respectively, as control inputs for the SLM dynamics. Additionally, it
is observed that substrate initial temperature affects the process states such
as cross sectional area [6]. As a result, surface temperature, denoted as
$T(t)$ at time $t$, of the selected scanning point before laser beam hits, is
considered as a measurable disturbance to the process dynamics 111At each
sampling time, temperature of next laser spot (plus laser beam nominal radius)
is used for this purpose.. This treatment is similar to that in [6].
Therefore, SLM process dynamics is formulated in discrete time as follows

$x_{k+1}=f(x_{k},T_{k},p_{k},v_{k}),$ (6)

where $x_{k}$, $T_{k}$, $p_{k}$, and $v_{k}$ are the melt pool cross sectional
area, surface temperature of melt pool surrounding, laser power, and laser
scanning speed at the $k$th step, respectively, $k\in\mathbb{N}$.

GP is used to estimate $f$ in (6) using training data set. To train the GP
models, process data sets are necessary and in many cases, experiments are
needed to collect these data sets. However, running SLM experiments is
expensive and in many cases, high-fidelity numerical simulation such as finite
element analysis has been extensively used as an alternative means to study
SLM processes.

###  II-D Finite Element Analysis

Finite Element (FE) analysis is widely used to simulate and predict the
thermal history during SLM process. In this work, FE models are developed to
provide information about the melt pool size and temperature history to build
the GP model and validate the control design. At the $i$th node in the FE
model, the addition of heat by the laser beam is represented using the
volumetric heat source considering the Gaussian profile for the beam shape,
that is, the heat flux is calculated as

$Q_{i}=\frac{6\sqrt{3}ap_{k}}{r_{s}^{2}c\pi\sqrt{\pi}}e^{-3\left[\bigl{(}\frac{\Delta
x_{i}}{r_{s}}\bigr{)}^{2}+\bigl{(}\frac{\Delta
y_{i}}{r_{s}}\bigr{)}^{2}+\bigl{(}\frac{\Delta z_{i}}{c}\bigr{)}^{2}\right]},$
(7)

where $\Delta x_{i}$, $\Delta y_{i}$, and $\Delta z_{i}$ are the distances
between the $i$th node and laser beam center at the $k$th step, $a$ is the
absorptivity, $p_{k}$ is the laser power at the $k$th step, $r_{s}$ is the
laser beam radius, and $c$ is the penetration depth. For computational
efficiency and accuracy, the adaptive remeshing framework in [20] along with
tetrahedral mesh are utilized, where a fine mesh of average size $25\,\mu m$
is used at the high temperature gradient regions (e.g., melt pool). Fig. 2
illustrates the mesh configuration during scanning the first track. The
implicit Ansys® MAPDL® solver is used to solve the thermal problem. The
movement of the laser beam is simulated by multiple time steps, where the
single step represents laser movement distance. Fig. 2 illustrates the melt
pool area and temperature distribution in the FE model.

##  III SLM Process Control

Linear time varying MPC is used in the present work for its constraint
handling capability and relative low computational cost. Considering the SLM
dynamics (6), we denote the GP estimate of nonlinear function $f$ as
$\bar{f}_{GP}$ and the control input vector
$\bm{u}_{k}=[p_{k}\;v_{k}]^{T}\in\mathbb{R}^{2}$. The discrete-time linearized
system is then given in the following form,

$x_{k+1}=A_{d}x_{k}+\bm{B}_{d}\bm{u}_{k},$ (8)

where $A_{d}$ and $\bm{B}_{d}$ are calculate as

$A_{d}=\frac{\partial\bar{f}_{GP}}{\partial
x}\Big{|}_{(x_{k},\bm{u}_{k},T_{k})},\;\bm{B}_{d}=\frac{\partial\bar{f}_{GP}}{\partial\bm{u}}\Big{|}_{(x_{k},\bm{u}_{k},T_{k})},$

and $T_{k}$ is the surface temperature of melt pool surrounding at the $k$th
step.

The MPC problem is formulated as a constrained optimization and at the $k$th
step, the formulation is given as

$\displaystyle\min_{\bm{U}}\quad$
$\displaystyle\sum_{i=0}^{H-1}\left(Qx_{k+i}^{2}+\bm{u}_{k+i}^{T}\bm{R}\bm{u}_{k+i}\right)+Q_{f}x_{k+H}^{2},$
(9a) $\displaystyle s.t.\quad$ $\displaystyle
x_{i+1}=A_{d}x_{i}+\bm{B}_{d}\bm{u}_{i},$ (9b) $\displaystyle x_{l}\leq
x_{i}\leq x_{u},\,p_{l}\leq p_{i}\leq p_{u},\,v_{l}\leq v_{i}\leq v_{u},$ (9c)
$\displaystyle x_{l}\leq x_{k+H}\leq x_{u},\;i=k,\ldots,k+H-1,$ (9d)

where $\bm{U}=\\{\bm{u}_{k},\ldots,\bm{u}_{k+H}\\}$ is the control input set
and $H$ is the state prediction horizon. In (9), $x_{l}$ ($x_{u}$), $p_{l}$
($p_{u}$), and $v_{l}$ ($v_{u}$) are the lower (upper) bounds for the melt
pool cross sectional area, laser power, scanning speed, respectively. MPC
control bounds should be chosen based on the optimal parameters of the process
to ensure adequate adhesion between melted powder and the substrate and also
to avoid keyhole formation [21].

Although the MPC control design is built on the GP model $\bar{f}_{GP}$, the
validation of the control performance is through the FE simulation. The use of
controller with FE models was previously implemented using COMSOL®
Multiphysics in [10]. While COMSOL® and MATLAB® can be integrated through
LiveLink®, this is not an option with Ansys®. In this work, the temperature
solution is written by Ansys® after each step is solved in batch mode. The
Python framework, which is integrated and implemented between simulations
steps, is responsible for melt pool area calculation and passing the
temperature values at location ahead of the melt pool to the MPC controller.
The controller updates the laser power and the FE solver proceeds with these
updates for the following simulation step. The process is repeated until
scanning the entire powder layer is complete. Detailed results will be
demonstrated in the next section.

##  IV Simulation Results

###  IV-A GP Model Performance

We take the IN625 metal SLM process in [6] as an example to demonstrate the GP
model training and validation. The IN625 thermal properties are taken from [6]
and the value of the process parameters are shown in Table I. The implicit
Ansys® APDL® solver is used to solve the thermal problem. The simulation step
time is $\Delta T=50$ $\mu$s.

TABLE I: Process parameters used in IN625 metal SLM simulation. Parameter |  $p$ (W) |  $v$ (mm/s) | Hatch space (mm) |  $r_{s}$ ($\mu$m) | $a$ |  $c$ ($\mu$m)
---|---|---|---|---|---|---
Value | $250$ | $800$ | $0.1$ | $50$ | $0.4$ | $3$

Figure 3: FE simulation profiles and results for Case #9 for GP construction.
(a) Laser power $p(t)$ and scanning speed $v(t)$ profiles. (b) Melt pool cross
sectional area $x(t)$. (c) Surface temperature $T(t)$ of the melt pool
surrounding.

Figure 4: (a) Comparison results between the trained GP model prediction and
the SE simulation using validation data set. (b) Comparison results of the GP-
based model prediction over time and the FE simulation of the melt pool cross
sectional area for Case #2 profile.

In order to collect representative data for training the GP model, four single
track and five double track simulations are conducted with various
combinations of scanning speed and laser power profiles and different hatch
spaces (for double track simulations). Table II lists these simulation
experiments and their configurations. In all of the test cases, each track
length is $5$ mm and the power and scanning speed profiles are constrained
within optimal process parameters for SLM process (e.g., [21]). The scanning
speed and laser power, along with melt pool cross sectional area and scanning
point initial temperature, are recorded at sampling time $\Delta T$. As an
example, Fig. 3 shows the laser power $p(t)$ and scanning speed $v(t)$, cross
sectional area $x(t)$ and melt pool temperature profiles for Case #9.

TABLE II: FE simulation summary for GP model training Case No. | Power | Scanning speed | Hatch space (mm)
---|---|---|---
Single track |  |  |
# $1$ | Constant | Constant | N/A
# $2$ | Sinusoidal | Constant | N/A
# $3$ | Constant | Sinusoidal | N/A
# $4$ | Sinusoidal | Sinusoidal | N/A
Double track |  |  |
# $5$ | Constant | Constant | $0.1$
# $6$ | Constant | Constant | $0.15$
# $7$ | Constant | Constant | $0.05$
# $8$ | Sinusoidal | Profile | $0.1$
# $9$ | Profile | Profile | $0.1$

From all nine cases FE simulation, a total of 1649 data sets were collected,
among which 100 data sets were used for the GP training and the rest of 1549
data points are used for model validation. Fig. 4 shows the comparison of the
GP model prediction and the FE simulation results. The average of the
prediction error of the GP model is around $2.06$ % for the melt pool cross
sectional area. Furthermore, coefficient of determination $R^{2}=99.5\%$ is
achieved on the test set. To demonstrate the prediction capability of the GP
model, we conduct model-based forward simulation using the GP model. In this
test, the GP model is used to predict the melt pool area over time for Case #2
in Table II. Fig. 4 shows the comparison results between the GP model
prediction and the FE simulation. It is clear that the GP model predicts the
FE simulation results closely.

###  IV-B MPC Performance

Using the GP model, the MPC is implemented by only considering the laser power
as the control input with constant scanning speed $v=800$ mm/s. The values of
the MPC design parameters are listed in Table III. To show the performance of
the MPC controller, a $4\times 10$ mm track test is designed. The MPC
controller is used to update the laser power in FE simulation to maintain
constant melt pool cross sectional area over all tracks. Also, a differential
plus direct feedback from the initial temperature is added to the MPC command.
The challenge in multi-track SLM process is the high initial temperature after
first track is completed; see Fig. 3. The increased initial temperature leads
to increased melt pool cross sectional area and therefore, feed forward
control of the process is inefficient and closed-loop control is needed. Fig.
5 shows the transition of the melt pool area and laser power profiles under
the MPC design. Open-loop performance (under constant laser power input) is
also included for comparison purpose.

TABLE III: Values of the MPC design parameters Parameter |  $p_{l}/p_{u}$ (W) |  $\Delta p_{l}/\Delta p_{u}$ (W) |  $x_{l}/x_{u}$ (mm2) | $Q$ | $R$ | $Q_{f}$ | $H$
---|---|---|---|---|---|---|---
Value | $0/350$ | $-350/350$ | $0/0.5$ | $1$ | $0.1$ | $20$ | $20$
TABLE IV: Control performance Set point | Max overshoot | Max undershoot | Steady state $\left|e\right|_{avg}$
---|---|---|---
$0.09$ | $30\%$ | $36.37\%$ | $\pm 8.16\%$

Despite the oscillation in the closed-loop response, it can be seen that the
MPC successfully attenuates melt pool overshoot in start of new tracks, that
is, at around $13$, $26$, $39$ ms in Fig. 5. Also, as shown in Fig. 5, the
power level decreases significantly at the beginning of each new track. This
is due to energy accumulation of multiple tracks. These results agree with
these reported in [6] that used the same material and feedback signal in their
work. Furthermore, after the sudden drop at the location of each new track,
the gradual increase of the laser power along each individual track is also in
accordance with the experiments in [6]. This gradual increase happens due to
the exponential drop of initial temperature along each new track; see Fig. 3.
It is obvious that under the MPC closed-loop control, the variation of the
melt pool cross sectional area profiles shows consistently stable results.
Table IV further summarizes the closed-loop performance. Comparing with the
control performance in [6, Table 5, five track case], the overshoot of the
response under the MPC is $10$ % higher, the undershoot is however reduced by
$40$ %. The max overshoot and undershoot are calculated as
$\frac{A_{max,min}}{A_{\textit{set point}}}$. The average steady state error
is also calculated as root mean square error (RMSE) between melt pool area and
the set point.

Figure 5: Control performance comparison under MPC and constant power inputs.
(a) Melt pool area $x(t)$ profiles. (b) Input laser power $p(t)$.

The computational burden of the control algorithm is really important in SLM
process, due to high speed nature of the SLM process. The proposed algorithm
takes on average $1\>ms$ to be computed. This result is achieved in python,
run on a core i5® laptop with 4 GB ram. This ensures that the algorithm can be
run on real time devices with required control frequency. Note that the
simulation results in the paper are done with higher sampling frequency due to
limitations in the FEA time length.

##  V Conclusion and Future Work

In this paper, a data driven GP model was proposed for the SLM process. The
modeling development was also used in MPC framework to regulate the melt pool
cross sectional areas in multi-track SLM scanning fabrication. The main
challenge in multi-track scanning scheme was the melt pool area due to the
high initial temperature of substrate generated by previous scans. We used
high-fidelity FE simulation that has been validated by experiments (see [20])
to generate the training data sets for the GP models. The GP model and the MPC
performance were demonstrated to successfully maintain a stable, consistent
melt pool cross sectional area during multi-track scanning process. Comparison
with the existing feed forward SLM control was also presented. We are
currently trying to implement the proposed GP model and real-time control on
an SLM machine experimentally. Inclusion of other real-time sensing and laser
scanning speed control is also among the ongoing research directions.

## Acknowledgments

The authors would like to thank Ansys® for providing the software, and
Professor Zhimin Xi of Rutgers University for his helpful discussions and
comments.

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