Engineering Innovations, ISSN: 2813-1002, Vol. 19, pp 101-106

doi: 10.4028/p-JOBaY6

© 2026 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2025-09-08

Accepted:
2025-09-28

Online:
2026-06-11

Bayesian Strategy for the Volume Accuracy of Ink Droplets Continuous Estimation during Printing Based on Prior Knowledge with Earth Mover's Distance

The original version of this paper is available on https://www.scientific.net/EI.19.101.pdf

Deyi Kong 1,a, Jiankui Chen 1, b*, Jiacong Xiong 1,c

1 The State Key Laboratory of Intelligent Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

a  d202280271@hust.edu.cn, b  chenjk@hust.edu.cn, cjiacongx@hust.edu.cn

Keywords: IJP OLED, volume accuracy, bayes inference, continuous evaluation.

Abstract

Monitoring the volume accuracy of ink droplets during printing is important for OLED panel production. This article uses data from ink droplet observation volumes before printing to build a membership distribution histogram and intuitively describes the volume status of multiple nozzles. A histogram in conjunction with the Earth Mover's Distance (EMD) indicator is used to design prior knowledge, which describes a subjective estimate of ink droplet volume distribution based on experimental data. And using the likelihood function, combining the sampled data of the ink droplets observation volume, obtains the volume accuracy estimate of the ink droplets during printing. This article proposes a Bayesian strategy for the volume accuracy of ink droplet continuous estimation during printing based on prior knowledge with EMD. This strategy can randomly inspect any number of samples of nozzles in the production process and get a reasonable estimate of volume after each sampling observation, which can realize continuous estimation and adjustment for the volume accuracy of ink droplets during printing.

Introduction

The original version of this paper is available on https://www.scientific.net/EI.19.101.pdf

Inkjet printing technology is widely used in the manufacturing of various flexible electronic products, especially in the manufacturing process of OLED display panels. Pixel volume filling defect is the main obstacle for the mass production of IJP OLED. With the gradual increase of display resolution and panel size, it is necessary to adopt multi-printhead splicing to increase the number of printing nozzles to improve IJP OLED's production efficiency [1]. Due to solute deposition, parameter change, and other interference factors, the volume of ink droplets may change, resulting in a decrease in volume accuracy of ink droplets during printing and panel defects [2]. Detecting the ink droplet volume of part nozzles to estimate the volume accuracy of all during the printing process is a reasonable solution with advantages.

The key step of IJP OLED production is observing precisely the volume of ink droplets before printing. For different application scenarios and process requirements, there are many ways to measure the volume of an ink droplet. At present, ink droplet volume measurement methods are mainly classified as offline measurement and online measurement [3]. At offline measurement, white light interferometry has the best accuracy. For droplets with a diameter of 50 microns, white light interferometry can achieve measurement accuracy that is suitable for obtaining a certain amount of average volume of deposition droplets [4]. Online measurement method including visual, laser phase doppler analysis (PDA), and the ultrasonic attenuation method. For the 10 pL of spherical droplets, the PDA method equivalent measurement precision can reach [5]. Traditional visual methods are difficult to achieve the equivalent measurement accuracy of. Compared with the visual method, the ultrasonic attenuation method measures the volume of an ink droplet within an error [6]. To achieve higher measurement accuracy, the time or cost of measurement is also increased. To continuously estimate the volume accuracy of ink droplets during printing, measures need to be taken to balance efficiency and precision.

This article proposes a Bayesian strategy for the volume accuracy of ink droplet continuous estimation during printing based on prior knowledge with EMD, which can realize the combination of subjective information and objective information during the printing process. This strategy can get more accurate estimates, without relying on a large amount of data training. This strategy can be used even if the process parameters and equipment change, which is efficient in realizing the stable control of the volume accuracy of ink droplets during continuous printing.

Description method of the ink droplet volume accuracy of multiple nozzles.

The original version of this paper is available on https://www.scientific.net/EI.19.101.pdf

Due to parameter fluctuation, installation accuracy, measurement accuracy, and other reasons, continuous measurement of the ink droplets' volume at a selected nozzle will result in different measurement results. In order to estimate the observation accuracy, we can use the median error to estimate the accuracy of the measurement result. At this scene, the median error of measuring result is defined as the times of continuous ink droplet volume measurement on a certain nozzle in a short period of time under the same conditions, Δ1,Δ2,…,Δn represents the square of the difference between the measured value and the measured mean value. The median error m is:

m=±[Δ1+Δ2+…+Δn]n(1)

m can reflect the measurement accuracy of the observed method and also represent the change of ink droplet volume in a short time scale. For a measuring result Va of a single nozzle, we can use [Va−m,Va+m] to represent its true value's possible range. In order to improve observation efficiency, we expect fewer observed frequencies. And we want to estimate the measuring result based on m.

Histogram {h} can represent a mapping of a multidimensional vector to a group of non-negative real numbers. These vectors represent a fixed interval of the histogram. We can divide the ink droplets observation volume data into M intervals. The midpoint of intervals is the average volume Vavg  of all ink droplets, and the distance between each interval is 2m. The center Vhi of each interval {hi} respectively Vavg −(M−1)m,…,Vavg −2m,Vavg ,Vavg +2m,…,Vavg +(M−1)m,hi represents the nozzles number of each interval, and i(1≤i≤M) is the index of the interval.

The membership degree of the ink droplet observation volume for each interval is wij.N is the number of nozzles, and the membership degree wij of the NO. j nozzle ink droplet observation volume Vj for interval hi is as follows:

ij={0i>11i=1wij={0i<M1i=Mwij={Vj−Va−12mi=a−1Va−Vj2mi=a0i≠a−1,aVha−1≤Vh1≤VhM. . . . . yi=Vha(1<a<M)(2)

The amount of each interval that can be transferred to a normalized result is:

hi={whi=∑j=1Nwij/N}(3)

Vhi can be a multidimensional vector describing the status of an ink droplet. Drop point deviation, Angle, velocity, and other characteristics of the ink droplet can also be used as indicators to describe the print status. In this paper, only the volume of the ink droplet is studied. Fig. 1(a) shows a set of
experimental test results. Fig. 1(b) shows five times of continuous ink droplet volume measurement on a certain nozzle in a short period under the same conditions. Fig. 1(c) shows converting the observation ink droplet volume to a membership histogram, which makes it easy to get the status of the ink droplet volume.

[picture]

Fig. 1. Observation volume data converted to a histogram. (a) a set of experimental test results. (b) multi-measurements from a certain nozzle. (c) Convert to histogram representation.

Fig. 1. Observation volume data converted to a histogram. (a) a set of experimental test results. (b) multi-measurements from a certain nozzle. (c) Convert to histogram representation.

[ dij ] represents the conversion cost between different intervals, and is represented here by the Euclidean distance between the mean volume represented by different intervals:

dij=|Vhi−Vhj|(4)

WORK(P,Q,F)=∑i=1a∑j=1bdijfij(5)

WORK(P,Q,F) denotes the cost of transformation between distributions P and Q. The above formula is subject to the following constraints:

fij≥01≤i≤a,1≤j≤b∑i=1afij≤whpi1≤i≤a∑j=1bfij≤whqj1≤j≤b∑i=1a∑j=1bfij=min(∑i=1awhpi,∑j=1bwhqj).

(6)(7)(8)(9)

Equation (5) restricts the transformation of the distribution from P to Q and vice versa. Equation (7) shows the maximum weight that P can transfer. Equation (8) limits the upper limit of transformation targets in Q. Equation (9) forces to movement of as much supply as possible. Equations (6)-(9) can be solved as a linear programming problem.

Solve the distribution transformation problem in the background of the corresponding transportation problem, and find the optimal flow F . The distance of EMD is defined as the result of the normalization of the total flow:

EMD(P,Q)=∑i=1a∑j=1bdijfij∑i=1a∑j=1bfij.(10)

Bayesian continuous estimation strategy.

The original version of this paper is available on https://www.scientific.net/EI.19.101.pdf

We can obtain all possible distribution results through enumeration, and obtain the prior probability P(ΩQ) of any distribution Q as follows:

p(ΩQ)=EMDQ−EMDmaxEMDmax−EMDmin(11)

Here, EMDQ is the distance between distribution Q and the ink droplet volume observation distribution P before printing, EMDmin  is the minimum distance among all distributions, and EMDmax  is the maximum among.

Likelihood function can introduce objective information of observation results into the Bayesian process. L(Ω)→P(V∣Ω), the likelihood function is proportional to the probability of observing the experimental result V in the printing process under a given distribution. V is an ink droplet volume observation result in the printing process. For the observed volume of ink droplet Va, combined with the median error and the actual volume of ink droplet Va∈(Va−m,Va+m), if Va≥Vavg+(M−1)m or Va≤Vavg −(M−1)m, we classify them into two extreme distributions {h1} and {hM}, and the corresponding likelihood function L(Ω) is:

L(Ω)={whMVa≥VhMwhi−1+whi+whi−1( Vhi−1−Va)+whi(Va−Vhi)2 mVhi−1<Va<Vhi(1≤i≤M)wh1Va≤Vh1(12)

The Bayesian rule can be used to combine the prior distribution with the observation information. An estimation approaching the truth can be obtained gradually from the sampling results. To ensure the reliability of estimates, we use the mean of the posterior estimates as the optimal estimation. H is the number of samples of the posterior distribution, M is the number of intervals, the result obtained is the ink droplets volume accuracy estimation of all nozzles w1,w2,…,wM. Estimation of the posterior distribution P(w1,w2,…,wM|[1,…,k]) is:

p(w1…wM∣V[1…k])=∑i=1Hw1ip(Ωi∣V[1..k]),∑i=1Hw2ip(Ωi∣V[1…k]),…∑i=1HwMip(Ωi∣V[1…k])(13)

Experimental verification.

The original version of this paper is available on https://www.scientific.net/EI.19.101.pdf

As shown in Fig. 2(a), the experiment is carried out in the independent domestic IJP OLED Equipment, which uses a QS256 inkjet printer head. And we use this equipment under the common process data. Fig. 2(b) shows the electron-hole transport layer (HTL) of the OLED panel printed. Fig. 2(c) shows the method we use for ink droplet observation, which takes an image by two cameras together. And this method takes more than 15 minutes to measure the ink droplet volume of the selected 256 nozzles. The basic parameter settings of the test process are shown in Table 1.

[picture]

Fig. 2. Experiment condition. (a) Independent IJP OLED equipment. (b) OLED panel after printing. (c) Observed method.

Fig. 2. Experiment condition. (a) Independent IJP OLED equipment. (b) OLED panel after printing. (c) Observed method.

Table 1. Experiment Parameters.

Parameter | Value | Parameter | Value
Nozzle number | 256 | Print height(mm) | 0.57
Nominal volume(pL) | 11.5 | Print frequency(kHz) | 20
Midian error(pL) | 0.15 | Print speed(mm/s) | 100
Nominal ink droplet velocity(m/s) | 3 |  |

According to the Bayesian theory, it is necessary to integrate the L(Ω)P(Ω) under the prior distribution to obtain the complete posterior distribution, which is very difficult to calculate the high-dimensional integral. In order to facilitate the calculation of posterior results, the prior distribution can be discretized with the minimum unit 0.02 , and the set of the prior distribution can be expressed as Q=0.02×{(q1,wq1),(q2,wq2),…,(qM,wqM)},(wq1+wq2+…+wqM=50),wq1,wq2,…,wqM is integer. Sampling enough nozzles would obviously yield a more accurate estimate. And we can clearly know that the ink droplet volume state does not change a lot in a short time. So we want to validate the Bayesian strategy under the condition of fewer nozzles and longer time. As shown in Fig. 3, the experiment is designed to conduct ink droplet observation of all nozzles at Day 1, Day 3, Day 10, Day 11, Day 15, and Day 18. And before the ink droplet observation of all nozzles, we sample 15% nozzles and use the Bayesian strategy to estimate the inkjet droplet volume accuracy based on the data of the last ink droplet observation of all nozzles.

[picture]

Fig. 3. Experimental test data.

Fig. 3. Experimental test data.

As shown in Fig. 4, the Bayesian strategy we propose achieves an overall good performance. Over the whole, when the number of sampling nozzles reaches 10%, the deviation estimated for each interval is less than 10%. If the number of samples is further increased to about 15%, the deviation estimated for each interval is less than 5%. And if the status of ink droplet volume changes little, which also means the interval between two status check times is short, a suitable estimation can be obtained from a small number of samples. And even after a long time, we can still get obvious results based on a sample of less than 15%. Compared with full observation of all nozzles, efficiency can be greatly improved by the Bayesian method we proposed.

[picture]

Fig. 4. (a) Estimation of day 3. (b) Estimation of day 10. (c) Estimation of day 15. (d) Estimation of day 18 .

Fig. 4. (a) Estimation of day 3. (b) Estimation of day 10. (c) Estimation of day 15. (d) Estimation of day 18 .

Conclusion

The original version of this paper is available on https://www.scientific.net/EI.19.101.pdf

In this paper, a Bayesian strategy for inkjet printing volume accuracy continuous evaluation based on an EMD prior is proposed, which can guarantee the stability of ink droplet volume accuracy and not affect the production efficiency. Compare the methods based on the model and artificial neural network, this method has obvious advantages. First, it has universality for different process parameters and equipment parameters, so this can be applied in any printing scenario and equipment. Second, this method doesn't need a lot of data for training, and only with limited data and computational processes can it achieve good results. In addition, this method can be combined with a control strategy and decision strategy to achieve the dynamic control of ink droplet volume accuracy. With the improvement of inkjet print accuracy, I think more and more sophisticated electronic components can be manufactured by inkjet printing.

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The original version of this paper is available on https://www.scientific.net/EI.19.101.pdf

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