Communications on Applied Nonlinear Analysis 

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Nonlinear Improved Adaptive Optimized Grey Model for Real-Time 

Monitoring and Control of Weld Bead Geometry in Robot-Assisted 

Wire Arc Additive Manufacturing 

 

Satish Kumar Parimi1*, L Suvarna Raju 1,2, G. Mallaiah3 

1Department of Mechanical Engineering, Vignan's Foundation for Science, Technology and Research, Andhra Pradesh, 

India 

2Department of Mechanical Engineering Education, National Institute of Technical Teachers Training & Research 

(NITTTR), Bhopal, India 

3Department of Mechanical Engineering, KITS, Singapur, Huzurabad. 

Corresponding author*: satishniyogi1972@gmail.com  

 

Article History: 

Received: 19-09-2024 

Revised: 27-10-2024 

Accepted: 07-11-2024 

Abstract: 

WAAM with directed energy deposition process and live monitoring of weld bead shape gives 

researchers and manufacturers real-time information, allowing them to regulate the metal 

deposition process layer by layer effectively. This work created an improved adaptive 

optimized grey model (IAOGM), for online monitoring of metal deposits' height and depth in 

each layer. The IAOGM(1,N) model takes fewer training samples and has limited information. 

To improve prediction accuracy, the training data was constantly updated by eliminating old 

data and introducing newer data. The framework's parameters comprised welding time, current, 

the absolute difference in current at 5-second intervals, and arc force. The best accurate weld 

bead height and depth estimates were obtained by calculating the root mean square error 

(RMSE) for various parameter combinations. The interplay of time, current, and arc force was 

discovered to have a considerable impact on weld bead diameters. Using these parameters, the 

model predicted weld bead height and depth with MAPE, RMSE, and MAE, values of 5.48, 

3.32, and 7.56 respectively, when compared to experimental data. This technique enables users 

to properly balance process parameters and achieve desired weld bead heights without the 

requirement for substantial training. 

Keywords: WAAM, Non-linear data model, Improved adaptive optimized grey model, Weld 

bead characteristics, Online monitoring 

 

1. Introduction: 

The advancement of Industry 5.0 transforms the conventional additive manufacturing(AM) methods 

with the use of robots and AI-based smart machines with or without the intervention of human beings 

to enhance sustainability. Technologies such as IoT-enabled smart machines and big data analytics are 

in industry 4.0 transformed into Robot-assisted AI-based smart machines with human-computer 

interaction (HCI) in concern with sustainability aspects. Additive manufacturing is attracting industry 

and researchers' attention globally due to its progressive and potential benefits[1]. As per the survey 

of the global additive market CAGR (Compound Annual Growth Rate) will be 22% in the next 5 years. 

The current USD 14.5 billion valued Additive manufacturing market will be projected to be USD 69 

billion by 2030. Material versatility is one of the concerns of AM technologies to be addressed and 

studied further, especially with metal additive manufacturing. Due to excellent material utilization, 



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ease of flexibility, and its potential with high deposition it to be used in a vast range of applications 

such as automotive, textile, medical, jewelry, customized appliances, and even the most expensive 

aerospace industry because of its less buy-to-fly(BTF) ratio.[2] AM more often referred as 3D printing 

with metal as filament can be treated as metal additive manufacturing(MAM). The production across 

the industries are revolutionizing with MAM technologies. Among the MAM technologies, Direct 

energy deposition(DED) is renowned for its efficient material deposition. In the recent times, the ease 

of AM is gaining attention of researchers along with the industries. The complex intricate shapes are 

developed by process of material deposition as layer by layer[3]. Wire plus arc additive 

manufacturing(WAAM) adopts DED technology because of the advent features such as, ease of 

operation, and increased productivity with efficient material deposition which grasp the attention of 

researchers[4]. Using robots in manufacturing will enhance productivity with precision in performing 

works such as the robotic gas metal arc welding process(GMAW) which is based on DED technology 

with high potential beneficial aspects along with the automation. The dimensional accuracy of the 

deposited weld bead profiles is also an important concern, resulting in poor build structure of the weld 

bead. Surface quality is another barrier to the wide adoption of WAAM in all manufacturing sectors, 

affecting the building structure with voids and defects [5]. Further proper selection of process 

parameters is essential to obtain the homogeneous structure of the thin wall geometry produced by the 

WAAM process[6]. Before the selection of optimal process parameters researchers have to take a call 

on the modeling techniques available such as physics-driven models and data-driven models to control 

and monitor weld bead profiles. Most physics-driven models are based on the finite element 

method(FEM) based and data-driven models rely on the previous experimental history of the weld 

bead deposition in the formation of structures. As the FEM is derived from physics-driven modeling 

it requires a lot of computational effort and time along with the complex analytical equations. Hence 

data-driven models are preferable for convenient usage along with the control and monitoring of the 

weld bead profiles with enhanced surface quality and build structure integrity. 

Data-driven models are considered as prominent alternatives to model the weld bead profile using the 

general-additive-model method to predict the weld bead dimensional characteristics[5]. The weld 

bead's dimensional characteristics along with the structural integrity and quality of the weld bead are 

significantly influenced by the input process parameters of the equipment most of the GMAW 

processes are combined with a Degree of freedom robot to automate the process and the parameters 

are weld speed, feed rate, input voltage for the process, with the equal significance of time for cooling 

in between weld beads or layers[7]. The effect of inter-pass temperature will affect the grain size in 

the microstructure of the weld bead which affects the mechanical performance parameters of the 

structure[8]. In few studies, it was identified that the multilayer weld bead deposition enhances the 

energy deposition which implies that deposition happens with a certain quantum of arc force which 

further leads to an increase in depth penetration of the weld bead[9]. The homogeneous weld bead 

formation will depend on the fine-tuning of process parameter selection, few researchers have 

identified that weld bead irregular formation at the starting and at the end of the bead is a generic 

problem with the WAAM and attention is required to control this phenomenon. Even though this 

critical issue is complex, it has been addressed by a geometrical leveling strategy by depositing an 

additional run of weld bead to compensate for the irregularities which consumes additional time to 

complete[10].  The uniformity of the weld bead is monitored with a passive vision system which is 



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integrated with PID controller to monitor and control the formation based on the feedback obtained 

from the most recently deposited weld bead [11]. In most situations, the innovative model has the 

highest forecasting precision, indicating that optimizing the beginning condition and background value 

may significantly improve the gray model's flexibility and prediction accuracy[12].From the literature, 

it is summarized that few researchers have reported limited number of studies on modeling methods 

for the estimation of wall structure geometry in WAAM process. But, the quality of the deposited 

structure and dimensional accuracy depends on the qualitative deposition of molten metal with 

appropriate arc force, hence the monitoring of weld bead deposition requires much attention. Hence, 

in the present study, in-process monitoring of weld bead deposition using novel robust in-process 

online grey forecasting model(OGM) frame work is adopted to monitor the weld bead geometry, such 

as width of the weld bead (WWB), height of the weld bead(HWB)& depth of the weld bead(DWB) in 

formation of thin wall structure in wire arc additive manufacturing. The input process parameters such 

as deposition time, arc force and current are considered as independent variables. The output process 

parameters such as width, height, and depth of the weld bead are forecasted by OGM. 

 

Fig.1 Wire arc additively manufactured Thin wall 

1.1 Objective of the work: 

In forming a thin wall structure formed by multiple weld beads layer by layer, the dimensional 

characteristics such as HWB, WWB, and DWB are dependent on process parameters used for the 

deposition of weld beads. The formation of weld bead throughout the deposition is not uniform, and 

the same kind of structure can be observed in the literature also, as the extreme positions of the weld 

bead differ from the middle segments of it, and the same is reported in the literature as well[10]. In the 

formation of intricate shapes via the WAAM process the varying cross-sectional weld bead deposition 

is considered as typical and deposition of like-wise weld beads to develop thin walls with varied 

dimensions requires adaptive weld bead formation with optimal process parameters as mentioned[13]. 

Accordingly, the most influencing process parameters in Robotic WAAM are wire feed, weld speed, 

current, and torch angle are selected for the present work. In the present work, the torch angle is taken 

as 79 degrees, which is optimal as per the existing literature. The objective of the present work to 

develop a novel and robust online monitoring and control system for weld bead dimensional 

characteristics with reduced MAPE value between forecasted and experimental results. So, the 

accuracy of the proposed model should be improved. In addition, a comparative study is to be made 

between the normal thin wall structure and the application of the proposed framework model.  

 

 



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1.2. Factors affecting the weld bead: 

The parameters such as width and depth of weld beads are influenced by the mode of metal transfer, 

and its associated arc force[14]. The arc force in the metal transfer is a composition of forces from an 

electrode to the substrate is explained in this section. 

The mode of metal transition from the filler wire to the substrate consists of a composition of forces 

which are described in different theories explained[15]. The static force theory focuses on the 

detaching phenomenon of metal drop-in metal transfer mode, when the metal drop retaining force is 

less than the static detaching force, the metal drop detaches from the electrode and sticks on the 

substrate[16]. This theory also states that metal drop detaching is under the association of gravity force, 

the current-produced electromagnetic force and plasma arc drag force while retaining force under the 

surface tension of the molten metal drop[17]. 

The gravity force of the metal drop is due to self-weight, which inhibits the ability to detach from feed 

wire and it is calibrated by the equation (1) and calculated as 3.55 × 10-3 N. As the molten filler metal 

drop transition is dependent on the arc current and the flow of metal drops is less, the globular mode 

of metal transfer occurs, and its volume (𝑣) is under gravitational force. The diameter of a droplet is 

twice of the mild steel filler wire supplied, with the density (𝜌𝐷) and force of gravity (g) taken as 

0.00785 gmm-3 and 7.83 gmm-3 respectively[18]. In the present work the experiments are performed 

with conventional industry power ratings and process parameters are selected [19]. 

𝑃𝑔 =  𝑣 𝜌𝐷 g    (1) 

The electromagnetic force associated is determined[18] by equation (2) 

Pemf = 
CI2

4π
μ

0
       (2) 

Magnetic permeability (μ
0
) in the calibration of the electromagnetic force is 

considered as 12.57 × 10-7 NA-2. The coefficient of globular mode 

arc is treated as 1 Jones et al as for the molten metal droplet[18]. The plasma arc drag force on the 

metal droplet is described by equation (3)  

Pp = 
π

2
(R2 − r2) Cdv

p

2
ρ

p
    (3) 

The plasma drag constant (Cd) is 0.44 and it is calculated as 5.11 × 10-3  N, where vp is the speed rate 

and ρ
p
 is the density of the droplet of radius R[20]. 

The net arc force (PA) required to exceed molten metal drop retaining force and to detach from the tip 

of the nozzle to the substrate is given by equation   (4) 

PA = Pg + Pemf  + Pp   (4) 

1.3 Discrete Grey Forecasting Model: 

The grey system theory was proposed by the renowned professor Deng, to predict the future data 

effectively from the small portions of existing data. Classical grey modeling techniques are widely 

used in parameter estimation and structural optimization[12][21]. Initially, the GM (1,1) model is only 



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applicable to exponential series data. Later, researchers increased the model's flexibility to deal with 

nonlinear data[22]. In addition, an adaptive and optimized grey Bernoulli model was presented to cope 

with restricted time series data containing stochastic disturbance. To address the time delay effect, a 

time-delayed and power-driven (GMTDPD) grey model is presented, as well as an improved adaptive 

optimized grey model (IAOGM) for prediction models containing nonlinear, changing data sequences. 

The randomness of the raw time data series will vanish after performing accumulated generation 

operations repeatedly. 

Usually, Grey forecasting models utilize the limited data sets in 0th order as indicated as DGM(0,N), 

where 0 represents the order of accumulation and N represents the number of variables (both dependent 

and independent). The first order accumulated generation operation (I-AGO) is indicated as 

DGM(1,N)[23]. In general the Discreete grey forecasting model is shown as follows[24]: 

{𝑋𝑖
(0)

} = {𝑋𝑖
(0)(1), 𝑋𝑖

(0)(2), 𝑋𝑖
(0)(3), … . . , 𝑋𝑖

(0)(𝑛)}       (5) 

Where n represents the number of data samples in the sequence, and 𝑋𝑖
(0)

 is the number of the variable 

sequences.  

As per the definition of grey prediction model the I-AGO series of 𝑋𝑖
(0)

 is given by (): 

{𝑋𝑖
(1)} = {𝑋𝑖

(1)(1), 𝑋𝑖
(1)(2), 𝑋𝑖

(1)(3), … . . , 𝑋𝑖
(1)(𝑛)}       (6) 

In DGM(1,N), if N=4, the grey forecasting model consists of single dependent variable and three 

independent variables. The sequence of the data variables are given as: 

Sequence of the data variable -1: {𝑋1

(1)} = {𝑋1

(1)(1), 𝑋1

(1)(2), 𝑋1

(1)(3),… . . , 𝑋1

(1)(𝑛)} (7) 

Sequence of the data variable -2: {𝑋2

(1)} = {𝑋2

(1)(1), 𝑋2

(1)(2), 𝑋2

(1)(3),… . . , 𝑋2

(1)(𝑛)} (8) 

Sequence of the data variable -3: {𝑋3

(1)
} = {𝑋3

(1)(1), 𝑋3

(1)(2), 𝑋3

(1)(3),… . . , 𝑋3

(1)(𝑛)} (9) 

Sequence of the data variable -4: {𝑋4

(1)
} = {𝑋4

(1)(1), 𝑋4

(1)(2), 𝑋4

(1)(3),… . . , 𝑋4

(1)(𝑛)}        (10) 

Where  {𝑋𝑚
1 (𝑝)} = ∑ 𝑋𝑚

(0)
 (𝑛), 𝑝 = 1,2,3, … . , 𝑞𝑝

𝑛−1  

The mean of the sequence, generated from 𝑋𝑚
1 (𝑝), now 

{𝑍𝑚
(1)

} = {𝑍𝑚
(1)(2), 𝑍𝑚

(1)(3), 𝑍𝑚
(1)(4),… . . , 𝑍𝑚

(1)(𝑛)} if N= 4, then m=1,2,3 &4  

Where  

𝑍𝑚
(1)

 (𝑝) =  
𝑋𝑚
1 (𝑝−1)+𝑋𝑚

1 (𝑝)

2
, 𝑝 = 2,3,4, … 𝑞, then the model DGM (1, N) is given as: 

𝑋𝑚
1 (𝑝) + 𝑎𝑍𝑚

(1)
 (𝑝) = ∑ 𝑋𝑚

(1)𝑁
𝑚  (𝑝)  

Further, the least square method is used to estimate the sequence of the parameters of DGM (1, N) as 

follows: 

�⃑� = [𝑎, 𝑏1,𝑏2, …… . , 𝑏𝑁]𝑇 



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Where a is the system coefficient and b terms are the driving coefficients of the model. It can be written 

as  

�⃑� = (𝐵𝑇𝐵)−1𝐵𝑇𝑌 

Where the B matrix and Y matrix are estimated as 

B=

[
 
 
 
 −𝑧1

(1)(2) −𝑧2
(1)(2) ⋯ −𝑧𝑁

(1)(2)

−𝑧1
(1)(3)
⋯

−𝑧1
(1)(𝑚)

−𝑧2
(1)(3)
⋯

−𝑧2
(1)(𝑚)

⋯
⋯
⋱

−𝑧𝑁
(1)(3)
⋯

−𝑧𝑁
(1)(𝑚)]

 
 
 
 

   𝑌 =  

[
 
 
 
 𝑋1

(0)(2)

𝑋1

(0)(3)

⋮

𝑋1

(0)(𝑚)]
 
 
 
 

  

The robust grey forecasting models are the potential in forecasting the dependent parameters which is 

the response of the DGM (1, N) with the reduced MAPE(). The Model is estimated as: 

𝑋1

(0)(𝑘) + 𝑎𝑧1
(1)(𝑚) =  ∑ 𝑏𝑖𝑋𝑖

(1)𝑁
𝑖=2 (𝑘) + ℎ1(𝑘 − 1) + ℎ2  

Where ℎ1, ℎ2 𝑎𝑟𝑒 𝑙𝑖𝑛𝑒𝑎𝑟 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑟𝑒𝑦 𝑚𝑜𝑑𝑒𝑙. The parameter sequence of the Predicting 

model with the coefficient  𝑟 ⃗⃗   is described as: 

𝑟 ⃗⃗ = [𝑎, 𝑏1, 𝑏2, 𝑏3, … . , 𝑏𝑁,    ℎ1, ℎ2 ]
𝑇 

𝑟 ⃗⃗ = (𝐵𝑇𝐵)−1𝐵𝑇𝑌 

Where 

 B = 

[
 
 
 
 𝑋1

(1)(2) 𝑋2

(1)(2) ⋯  𝑋𝑁
(1)(2) −𝑧1

(1)(2)         1    1

𝑋1

(1)(3) 𝑋2

(1)(3) ⋯  𝑋𝑁
(1)(3) −𝑧1

(1)(3)          2   1

⋮

𝑋1

(1)(𝑚)

⋮

𝑋2

(1)(𝑚)

⋮           ⋮              ⋮                    ⋮    1

⋱  𝑋𝑁
(1)(𝑚) −𝑧1

(1)(𝑚)  𝑚 − 1   1]
 
 
 
 

   𝑌 =  

[
 
 
 
 𝑋1

(0)(2)

𝑋1

(0)(3)

⋮

𝑋1

(0)(𝑚)]
 
 
 
 

 

The output response of the IOGM (1, N) model is estimated by the equation mention below: 

𝑋 ̌1
(0)(𝑘) =  𝜇1 ∑𝑏𝑖𝑋𝑖

(1)

𝑁

𝑖=2

(𝑘) + 𝜇2 𝑋 ̌1
(1)(𝑘 − 1) + 𝜇3 𝑘 + 𝜇4, k =  2,3,4, … n 

Where 𝜇1 = (1 + 0.5𝑎)−1  

            𝜇2 = (1+ 0.5𝑎)−1 

𝜇3 = ℎ1(1+ 0.5𝑎)(1+ 0.5𝑎)−1 

𝜇4 = (ℎ
2
− ℎ1)(1 + 0.5𝑎)−1 

Accordingly, HWB, DWB, WWB responses are estimated using the above equation. 

The GM (1,1) model is frequently employed for analysis of time series due to its ability to adapt to 

uncertain systems with little data. information. This section will discuss the concepts and faults of the 

original GM (1,1) model. The IAOGM (1,1) model is proposed to address the primary error causes of 

GM (1,1), which are based on the starting and background values. This IAOGM (1,1) model's precise 

techniques for coordinating two optimum approaches will be investigated and analysed. 



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The IAOGM (1,1) model improves two elements based on the intrinsic error sources of the GM (1,1) 

model, as discussed in Section 3.1. On the one hand, with the intention to gain a Preliminary Condition 

After considering all observations and prioritizing the most recent information, a new weighted series 

is used to create a modified starting condition that aligns with the evolving system. Simpson's formula 

is used to construct the background value, which is crucial for enhancing the grey model's applicability 

to high-growth exponential sequences. It addresses integral issues. Simpson's formula is used in 

sequences to provide background values that can manage integral difficulties. The IAOGM (1,1) 

model's processes are outlined in the stages below, leveraging its superiority. Figure 2 illustrates the 

summary flowchart for the IAOGM (1,1) model. 

 

The improved adaptive optimized grey model IAOGM: 

 

 

 



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2. Experimentation: 

In the present work, experimentation was carried out using an industrial-grade IRB 1520 MIG robot 

as shown in the Fig.2. Robot-assisted wire arc additive manufacturing used as an alternative to metal 

additive manufacturing due to the ease of operation and most resourceful, instead of expensive metal 

3D printers in the market. As the equipment of industry-standard, industrial power rating was used in 

the work to study the deposition of weld beads as layers. The controller is equipped with a teach 

pendant, an instrument to work with the robot. The geometrical motions involved in the deposition of 

the weld beads to form the structure are customized and pre-programmed using rapid code. As shown 

in the figure a metal substrate of customized size as per the structure to be formed, is rigidly clamped 

to the fixed platform of the robot. The operating requirements of the robot such as voltage, weld feed 

speed, weld speed, and shielding gas supply are taken as 18V, 4 m/min, 0.3 m/min and 20 l/min. The 

filler wire of 1.2 mm diameter of AISI 1018 steel material is used. AISI 1018 steel is widely used in 

industries due to its ability to uniform weld beads and also offers superior machining characteristics 

with excellent weldability properties.  

 

Fig.2 IRB 1520 Robot cell 

The experiment was performed in two phases. In the first phase, the substrate was rigidly fixed as 

shown in the figure with the help of clamps as also simulated in the simulation. The weld beads were 

on the substrate with the help of a robot as shown in Fig.1. At every 4-second instance during the weld 

bead formation the current was measured with the help of a kempi arc controller. The average feed 

rate is also measured along with the current. After the formation of the initial layer, the weld bead was 

sectioned at 5 different locations with the wire EDM, and the corresponding weld bead characteristics 

were measured using stereo zoom microscopy. From the first phase, it was observed different weld 

bead characteristics such as height, width, and depth. In the later phase, weld beads were deposited on 

the substrate using a multi-bead formation strategy with a time halt of 5 seconds among the layup of 

12 fully overlapped multi-weld beads. Similarly, the dimensions were measured at 5 locations as 

measured in the 1st phase. The measured features at 5 locations are tabulated. 

 

 



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Table 1. Weld bead dimensional features at 5 different locations 

S. 

No. 

Weld 

Bead 

No. 

(n) 

T 

(min) 
I (A) 

ΔI 

(A) 

AFX 

10^-

3 (N) 

DWB 

(mm) 

WWB 

(mm) 

HWBi 

(mm) 
HWBi 

Total 

HWB 

1 1 5 190.5 0 12.35 1.28 3.81 2.7 0 2.7 

2 1 10 174.2 16.3 11.71 1.03 3.06 3.43 0 3.43 

3 1 15 180.8 6.6 11.95 1.34 3.99 3.05 0 3.05 

4 1 20 186.4 5.6 12.15 1.52 4.53 2.53 0 2.53 

5 1 25 191.9 5.5 12.36 1.75 5.22 2.72 0 2.72 

6 1 30 197.3 5.4 12.57 1.98 5.91 2.35 0 2.35 

7 2 35 191.5 5.8 12.35 1.3 3.87 3.02 2.7 5.72 

8 2 40 175.4 16.1 11.76 1.06 3.15 2.75 3.43 6.18 

9 2 45 179.9 4.5 11.92 1.32 3.93 2.66 3.05 5.71 

10 2 50 186 6.1 12.14 1.55 4.62 2.7 2.53 5.23 

11 2 55 193.2 7.2 12.41 1.76 5.25 1.98 2.72 4.7 

12 2 60 198.3 5.1 12.61 2.01 6 2.02 2.35 4.37 

13 3 65 191.2 7.1 12.34 1.33 3.96 2.31 5.4 7.71 

14 3 70 173 18.2 11.67 1.1 3.27 3.29 6.86 10.15 

15 3 75 183.2 10.2 12.04 1.31 3.9 3.22 6.1 9.32 

16 3 80 185.8 2.6 12.13 1.54 4.59 2.02 5.06 7.08 

17 3 85 194.1 8.3 12.15 1.8 5.37 2.83 5.44 8.27 

18 3 90 198.6 4.5 12.14 2.03 6.06 2.06 4.7 6.76 

19 4 95 192 6.6 12.17 1.32 3.93 3.88 9.28 13.16 

20 4 100 173.7 18.3 12.18 1.13 3.36 3.03 9.89 12.92 

21 4 105 184.1 10.4 12.18 1.34 3.99 2.97 9.07 12.04 

22 4 110 185.3 1.2 12.17 1.62 4.83 3.13 8.19 11.32 

23 4 115 192.4 7.1 12.15 1.82 5.43 2.19 7.63 9.82 

24 4 120 196.5 4.1 12.14 2.14 6.39 2.14 6.84 8.98 

25 5 125 188.8 7.7 12.16 1.4 4.17 1.78 5.28 7.06 

26 5 130 172.7 16.1 12.17 1.14 3.39 3.2 4.17 7.37 

27 5 135 182.3 9.6 12.18 1.4 4.17 2.78 4.37 7.15 

28 5 140 188 5.7 12.16 1.61 4.8 3.35 4.74 8.09 

29 5 145 191.7 3.7 12.13 1.82 5.43 2.92 4.01 6.93 

30 5 150 197.5 5.8 12.12 2.14 6.39 3.15 4.28 7.43 

31 6 155 191.2 6.3 12.15 1.38 4.11 2.91 3.16 6.07 

32 6 160 175.4 15.8 12.16 1.14 3.39 3.03 8.31 11.34 

33 6 165 181.6 6.2 12.16 1.4 4.17 2.42 6.59 9.01 

34 6 170 187 5.4 12.16 1.65 4.92 1.39 5.76 7.15 

35 6 175 192.8 5.8 12.16 1.83 5.46 1.63 6.37 8 

36 6 180 195.6 2.8 12.16 2.15 6.42 2.16 6.17 8.33 



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37 7 185 193.1 2.5 12.16 1.44 4.29 2.52 5.68 8.2 

38 7 190 173.3 19.8 12.15 1.18 3.51 2.63 10.94 13.57 

39 7 195 183.7 10.4 12.15 1.43 4.26 2.63 9.22 11.85 

40 7 200 185 1.3 12.15 1.67 4.98 2.91 8.67 11.58 

41 7 205 193.8 8.8 12.15 1.83 5.46 3.51 9.88 13.39 

42 7 210 196.4 2.6 12.15 2.15 6.42 2.02 8.19 10.21 

43 8 215 189.5 6.9 12.15 1.5 4.47 2.92 8.6 11.52 

44 8 220 176 13.5 12.15 1.18 3.51 4.37 15.31 19.68 

45 8 225 179.7 3.7 12.15 1.41 4.2 2.65 11.87 14.52 

46 8 230 187.3 7.6 12.15 1.67 4.98 3.23 11.9 15.13 

47 8 235 194.2 6.9 12.15 1.84 5.49 1.75 11.63 13.38 

48 8 240 199 4.8 12.15 2.16 6.45 1.11 9.3 10.41 

49 9 245 192.1 6.9 12.15 1.55 4.62 3.47 12.07 15.54 

50 9 250 171.4 20.7 12.15 1.22 3.63 3.1 18.41 21.51 

51 9 255 181.9 10.5 12.15 1.47 4.38 2.94 14.81 17.75 

52 9 260 184.5 2.6 12.15 1.72 5.13 1.91 13.81 15.72 

53 9 265 194.7 10.2 12.15 1.86 5.55 2.85 14.48 17.33 

54 9 270 198.4 3.7 12.15 2.16 6.45 3.41 12.71 16.12 

55 10 275 190.5 7.9 12.15 1.58 4.71 2.33 14.4 16.73 

56 10 280 174.2 16.3 12.15 1.22 3.63 2.64 21.05 23.69 

57 10 285 180.8 6.6 12.15 1.51 4.5 3.22 18.03 21.25 

58 10 290 186.4 5.6 12.15 1.73 5.16 3.16 16.97 20.13 

59 10 295 192.1 5.7 12.15 1.88 5.61 2.43 16.91 19.34 

60 10 300 197.3 5.2 12.15 2.16 6.45 2.07 14.78 16.85 

61 11 305 184.5 12.8 12.15 1.78 5.31 3.47 17.87 21.34 

62 11 310 194.7 10.2 12.15 1.24 3.69 3.1 24.15 27.25 

63 11 315 198.4 3.7 12.15 1.53 4.56 2.94 20.97 23.91 

64 11 320 190.5 7.9 12.15 1.74 5.19 1.91 18.88 20.79 

65 11 325 174.2 16.3 12.15 1.89 5.64 2.85 19.76 22.61 

66 11 330 180.8 6.6 12.15 2.34 6.99 3.41 18.19 21.6 

67 12 335 186.4 5.6 12.15 1.8 5.37 2.33 20.2 22.53 

68 12 340 192.1 5.7 12.15 1.26 3.75 2.64 26.79 29.43 

69 12 345 197.3 5.2 12.15 1.55 4.62 3.22 24.19 27.41 

70 12 350 189.5 7.8 12.15 1.76 5.25 3.16 22.04 25.2 

71 12 355 176 13.5 12.15 1.91 5.7 2.43 22.19 24.62 

72 12 360 179.7 3.7 12.15 2.36 6.82 2.07 20.26 22.33 

 

2.1. Validation of the proposed model: 

To accurately predict future trends, it's important to use proper testing criteria after estimating 

outcomes using competing models. As a result, the testing criteria should be able to detect the 



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discrepancy between real observations and estimated values. This article uses four statistical 

assessment indicators: Mean absolute percentage error (MAPE), and root mean squared error (RMSE), 

and mean absolute error (MAE), which are described below. 

MAPE = 
1

𝑛
∑

|�̂�0(𝑘)−𝑥0(𝑘)|

𝑥0(𝑘)
𝑋 100%𝑛

𝑘=1  

RMSE = √
1

𝑛
∑ (�̂�0(𝑘) − 𝑥0(𝑘))2𝑛

𝑖=0  

MAE = 
1

𝑛
∑ (�̂�0(𝑘) − 𝑥0(𝑘))2𝑛

𝑘=0  

where the 𝑥(0)(𝑘) is experimental  value, and the ̂ 𝑥(0)(𝑘) is predicted value. 

Low values for indicators indicate better predicting ability with greater precision. Table 2 shows the 

accuracy levels calculated using MAPE values. 

3. Results and Discussions 

3.1. Prediction of HWB, WWB, DWB 

Fig.3 describes the temporal evolution of the measured parameter "HWB (mm)" for 12 distinct bead 

categories (BEAD-1 to BEAD-12) over 60 seconds. The measurements were recorded at 5-econd 

intervals. All bead categories exhibit a progressive increase in HWB values with time.  

 

Fig.3. HWB for 12 distinct bead categories (BEAD-1 to BEAD-12) over a period of 60 Seconds. 

This indicates a dynamic process influencing the bead characteristic, such as swelling, growth, or 

reaction to an external stimulus. BEAD-12 consistently shows the highest HWB values across all time 

points, suggesting either a unique composition or interaction compared to other bead types. 

Conversely, categories like BEAD-1 and BEAD-2 display the lowest HWB values, highlighting a 

significant disparity in behaviour or properties.  

Fig.4, & Fig.5, reveal the temporal progression of "WWB (nm)" values for 12 bead categories (BEAD-

1 to BEAD-12) measured at intervals of 5 seconds, spanning a total duration of 60 seconds. The WWB 

values for all bead categories exhibit a stepwise increase over time, indicating a consistent growth or 

change in the measured parameter across the 60-second duration. This could reflect a time-dependent 

process, such as swelling, absorption, or another reaction occurring in the beads. Although all beads 



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follow a similar overall trend, slight variations are observed among different bead categories: BEAD-

12 and BEAD-6 appear to have the highest WWB values at each time interval. BEAD-1 and BEAD-2 

show relatively lower WWB values compared to other groups, suggesting differing material properties 

or responses to the experimental conditions. 

 

Fig.4. WWB for 12 distinct bead categories (BEAD-1 to BEAD-12) over a period of 60 Seconds. 

 

Fig.5. DWB for 12 distinct bead categories (BEAD-1 to BEAD-12) over a period of 60 Seconds. 

Table.2 Experimental vs Predicted errors 

Phase Method DGM(1,1) OGM(1,1) IAOGM (1,1) 

 MAPE 5.65 4.36 3.85 

Experimental RMSE 12.45 11.5 12.72 
 MAE 17.23 18.34 18.56 

 MAPE 8.34 7.47 5.48 

Predicted RMSE 3.54 4.52 3.32 

 MAE 6.45 5.74 7.56 



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Fig.6. Comparison of experimental and predicted methods 

DGM (1,1), OGM (1,1), and IAOGM (1,1)—using different error metrics across experimental and 

predicted phases. In the experimental phase, IAOGM (1,1) outperforms the other methods in terms of 

Mean Absolute Percentage Error (MAPE) with the lowest value of 3.85, indicating higher accuracy. 

However, OGM (1,1) shows the lowest Root Mean Square Error (RMSE) at 11.5, suggesting slightly 

better error consistency compared to the others, whereas DGM (1,1) achieves the lowest Mean 

Absolute Error (MAE) at 17.23. For the predicted phase, IAOGM (1,1) again demonstrates superior 

performance with the lowest MAPE (5.48) and RMSE (3.32), highlighting its robust forecasting 

ability. However, OGM (1,1) yields the lowest MAE (5.74) in this phase, slightly edging out the other 

methods in absolute error terms. Overall, IAOGM (1,1) shows a balance of high accuracy and error 

consistency across both phases, making it the most reliable method among the three as shown in Fig.6 

 

Fig. 7.  HWB prediction using IAOGM 

The proposed IAOGM (1, N) model successfully monitors the height of the weld beads as show shown 

in Fig.7 that deposits as layer by layer, utilizing real-time data to improve accuracy. 

4. Conclusions: 

This study demonstrates the effectiveness of an Improved Adaptive Optimized Grey Model (IAOGM) 

for real-time monitoring and prediction of weld bead shape in the WAAM (Wire Arc Additive 

Manufacturing) process using directed energy deposition. From the study, the following conclusions 

have been derived.  

 



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1. The developed IAOGM (1, N) model successfully monitored the height and depth of metal 

deposits layer by layer, leveraging real-time data updates to improve prediction accuracy. 

Experimental results validated the model's ability to provide reliable predictions, with the best 

performance achieved in the predicted phase, where MAPE, RMSE, and MAE values were 5.48, 3.32, 

and 7.56, respectively, demonstrating superior accuracy compared to other forecasting methods like 

DGM (1,1) and OGM (1,1). 

2. The analysis of the provided graphs further substantiates these findings. The WWB (nm) and 

HWB (nm) trends highlighted the temporal changes in weld bead parameters, illustrating how process 

conditions such as time, current, and arc force significantly influence bead dimensions.  

3. The IAOGM (1, N)'s ability to incorporate these factors into its predictive framework allowed 

for precise estimation of weld bead characteristics, as demonstrated by the low error margins observed 

across all metrics. 

4. The integration of this model with WAAM processes enables manufacturers to effectively 

regulate metal deposition in real time, reducing reliance on extensive training data and manual 

intervention. This approach not only ensures optimal weld bead quality but also provides a scalable 

solution for future applications in additive manufacturing, where precision and adaptability are critical.  

5. By continually updating training data and considering parameter interdependencies, the 

IAOGM (1, N) framework presents a significant advancement in achieving desired weld bead 

dimensions, paving the way for more efficient and reliable WAAM processes. 

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