A Novel Method to Reduce the Laser Drilling Time for Hole Cluster considering the Influence of Residual Vibration of the Sample Stage

In large areas of laser drilling, the residual vibration occurs when the sample stage moves in PTP (Point to Point) movement. It affects the surface quality and processing efficiency of the holes. +e common solution for this problem is to set the laser irradiation delay time by the controller to wait for the residual vibration attenuation, but the whole drilling circle will increase. In this paper, a new method is introduced to reduce the laser drilling circle. By setting the allowable threshold of the residual vibration for the subsequent process, the sum of the time in deceleration segment of the trapezoidal moving profile, and the time when the residual vibration attenuates below the amplitude threshold (ST) can be minimized as the optimization goal. +e results show that for a given operating speed, there is always an optimum acceleration value for the deceleration segment of the trapezoidal moving profile, which minimizes the ST value. Further, the delay time for laser irradiation can also be estimated according to the optimal acceleration during laser drilling.


Introduction
Laser processing holds the characteristics of noncontact, low pollution, and high efficiency. In recent years, with the development of laser technology, laser processing has become ever more in-depth in the fields of welding [1][2][3], hard or brittle material cutting [4][5][6][7], drilling [8][9][10][11][12], and surface structure texturing [13][14][15]. e performance requirements of laser processing products are higher and higher, such as ultracomplex structure, super surface of hydrophobic and oleophobic and high aspect ratio microholes, etc., which induces higher performance requirements for laser processing equipment. Laser processing equipment is usually integrated by laser systems, motion control systems, optical path systems, and galvanometer systems. ese subsystems are usually produced by different manufacturers and are relatively independent, which is necessary to be effectively integrated. Laser processing is achieved through cooperative control between these subsystems. e delay problem in the cooperative control process determines the performance of the equipment and affects the processing efficiency. It is of profound significance to reduce the delay problem in the cooperative control process. For a better understanding, Table 1 provides an explanation of the abbreviations in this paper. Figure 1(a) shows a common laser processing equipment that is equipped with a laser source, a galvanometer, and a motion control system. Figure 1(b) shows the common PTP path for laser drilling. Figure 1(c) is the schematic diagram of the vibration of the laser beam relative to the workpiece caused by residual vibration. Usually, the sample stage achieves motion positioning and then the galvanometer controls the laser drilling. When the galvanometer is positioned, the controller triggers the laser shutter to emit several laser pulses for drilling. However, the completion of positioning action is always accompanied by the generation of residual vibration, as shown in Figure 1(d). e effect of vibration on holes' quality is not negligible if the laser is turned on immediately, especially when the design size of the hole is small. e common solution to this problem is to set the laser irradiation delay so that the laser pulses are irradiated after the attenuation of the residual vibration [16]. When the motion actuator is a robot arm, the residual vibration tends to be larger and the decay time is longer, which seriously affects the entire drilling efficiency [17,18].
Nowadays, in the field of laser drilling, many motion controllers still use the simple trapezoidal moving profile to control the motion of the sample stage due to the complexity  of motion control algorithms and product cost. Although this type of move profile is simple and easy to be operated, residual vibration will inevitably occur when the sample stage stops, which affects the processing quality and processing efficiency [19][20][21]. e residual vibration is related to the deceleration section of the trapezoidal moving profile. So how to effectively control the acceleration value of the deceleration section to make the motion positioning time and the delay time for laser irradiation shortest is of great significance for shortening the whole drilling cycle. At present, most scholars' researches focus on how to eliminate the residual vibration and they have done a lot of meaningful researches. Some scholars studied the high-order trajectory smoothing algorithm, such as the fifth-order polynomial trajectory to make the acceleration and jerk (the derivative of acceleration with time) continuous, so as to reduce the acceleration fluctuation during the motion [22]. Some scholars study the optimization algorithm to minimize the energy of the jerk for the whole running trajectory under certain speed and acceleration constraints.
en the obtained jerk trajectory is used to generate smooth acceleration and velocity trajectory, which make the sample stage run smoother [23][24][25]. Some scholars use filtering methods such as low-pass filtering, band-pass filtering, input shaping, etc. to make the spectrum of the acceleration trajectory have lower energy near the natural frequency of the sample stage, thereby reducing the vibration during the motion of the table [26][27][28][29]. ere are also some scholars using feedforward plus feedback control to improve servo performance. e residual vibration is regarded as the disturbing term to the command position. e filter model needed to eliminate the residual vibration is obtained by solving the inverse model of the system dynamics model [30][31][32].
All of the above researches are considered to reduce residual vibration completely by modulating the acceleration during motion. ere is no guarantee that these methods will make the whole processing time smaller, but it may make the time larger. Moreover, the methods above researches need to be written to the motion controller before used, which is time-consuming and laborious and very difficult for the laser application. Most motion controllers in the laser drilling field are closed to the user at the control algorithm level and do not support user-defined motion control algorithms. Otherwise, it may cause security risks [33].
In actual laser drilling, the trapezoidal moving profile is the most common velocity profile and almost all controllers support this velocity profile. In fact, when the residual vibration is attenuated to a certain amplitude which does not affect the quality of the sample, the laser can be irradiated. e decay time of the residual vibration is related to the magnitude of the excitation force when the motion stage is stopped, that is to say, it is related to the magnitude of the acceleration value of the deceleration section. erefore, there may be an acceleration value to let the ST get the minimum, so as to shorten the whole drilling circle. In this paper, the problem is deeply studied from the perspective of shortening the whole drilling cycle rather than from the perspective of eliminating the residual vibration.
e ST value is minimized as the optimization goal, and the optimal acceleration is obtained. e results demonstrate that the optimum acceleration of the deceleration section is only related to the dynamic parameters of the sample stage at a given operating speed. When the dynamic parameters are obtained, the optimal acceleration of the deceleration section is just set according to the speed. e optimal delay time before laser irradiation can also be estimated according to the optimal acceleration. e content of this paper is organized as follows. e second part gives the model of the impulse response for the second-order underdamped system and the system parameters identification method based on the ARMA (autoregressive moving average) method. e third part gives the method for reducing the ST value proposed in this paper. e fourth part introduces the experimental design and experimental equipment of this paper and explains the system parameters identification results based on ARMA.
en the experimental results of the method for reducing the whole laser drilling circle in this paper are given. e last part gives the conclusions of this paper. Figure 2 is a schematic diagram of the mechanical dynamics such as the sample stage, which can be described by a typical second-order system. e transfer function of a typical second-order system can be described as follows:

Second-Order Underdamped System Dynamics Model.
where ω n is the undamped natural frequency and ξ is the damping ratio. For a second-order underdamped system, 0 < ξ < 1, and ξ determines the degree of vibration attenuation of the system, and ω n determines the period of vibration attenuation [33]. When the sample stage moves in a trapezoidal moving profile, the residual vibration occurs at the moment the sample stage stops. e form of the residual vibration usually appears as the attenuation vibration of a second-order underdamped system for the impulse input. By identifying the parameters of the system model, the residual vibration can be mathematically described, which provide conditions for the planning algorithms of optimal move profile in further time.
e response of the second-order system to the input can be expressed as follows: When U(s) � K, that is, the input is a pulse signal, the output Y(s) � KG(s), then the time response signal of the second-order system for the pulse input is obtained by inverse Laplace transform: Shock and Vibration 3 Using system identification algorithms such as ARMA, STD (sparse time domain), etc., the model parameters can be obtained through the measurement and analysis of the residual vibration signal.

ARMA Time Series Analysis Method for Solving System
Parameters. In the process of obtaining the dynamic parameters of the sample stage, the self-excited force generated by the high-speed impact of the sample stage is used to generate the residual vibration, and the residual vibration signal is collected. e ARMA time series analysis method is used to obtain the inherent spectrum and damping coefficient of the sample stage. Compared with the traditional hammer method, the parameter identification method based on residual vibration signal analysis can better describe the system state in the actual working environment [34,35]. e ARMA time series analysis method is a method for describing the ordered random vibration data by using model parameters, thereby obtaining system parameters; the details of this method being available in the literature [36,37]. e parametric model includes an autoregressive (AR) model and a moving average (MA) model. e AR model describes the dynamic characteristics of the system and the MA moving is related to external disturbances and inputs. Usually, a system model composed by a p-order AR model and a q-order MA model can be expressed as ARMA(p, q), and its expression is as follows: In the above expression, x t is the time domain signal, ϕ i is the autoregressive coefficient, θ l is the moving average coefficient, and ε t is the white noise.
Since the ARMA process {x t } has a unique stationary solution expressed by the following: en In equation (7), G j is an impulse response function. From equations (6) and (8), the following expressions can be derived: In the above expression, the item E(x t x t−k ) is the k-order autocorrelation function of the time domain signal x t and can be expressed by r k . e item E(ε t−l ε t−j−k ) is the k-order autocorrelation function of the white noise and can be expressed by the following: e σ 2 is the variance of the white noise. It can be further derived from equations (10) and (11) In the above expression, when j < 0, G j � 0. When l � k + j ∉ {1, 2, . . ., q}, θ l � 0. e r k is the k-order autocorrelation function of the time domain signal. Usually, the difference order k of the ARMA model satisfies k ≫ max(p, q + 1), which means k + j ≫ q, then the right term in equation (12) is equal to 0, which gives the following: Let the length of the autocorrelation function be L. Usually, L is much larger than M. en, the following expressions can be derived:

Shock and Vibration
which is as follows: So the autoregressive coefficient ϕ can be expressed as follows: When the autoregressive coefficient ϕ k is obtained, the dynamic parameters of the system can be calculated by the transfer function expression of the ARMA model. e transfer function form of the ARMA model is expressed as follows: e characteristic equation of the above formula can be expressed as follows: e relationship between the transfer function poles and the system dynamics parameters can be expressed as in the following equation, in which z k is the conjugate of z k * : e expression of the modal frequency w k and damping ratio ξ k can be obtained from equation (19), as shown in the following equation, where s k is the k-order modal frequency and Δt is the sampling interval:

Time Optimal Trapezoidal Moving Profile under Residual Vibration Threshold
For the PTP motion using the trapezoidal moving profile, it is not true that the larger the acceleration value of the deceleration section is, the shorter the time for the PTP motion is, if the attenuation time of the residual vibration should be considered. For example, in the field of precision laser drilling, the residual vibration often has an influence on the shape of the hole [38]. erefore, it is necessary to set a settling time to reduce the influence of residual vibration. Generally, the greater the acceleration of the deceleration section is, the shorter the time taken by the deceleration section will be, but the larger the amplitude of the residual vibration will be and the longer the settling time is required. It is of great significance to set the optimal acceleration value of the deceleration section to obtain the minimum ST value. Figure 3 is a schematic diagram of the residual vibration of the trapezoidal move profile. e motion profile indicated by the red and blue curves is different only in the acceleration value of the deceleration section. e distance S des of the PTP motion and the maximum speed V max remain constant. Assuming that the positioning time ST is T − , the following expression can be achieved when there is a constant velocity segment: In equation (22), T a − represents the deceleration period time, and T v represents the time required for the residual vibration to decay below the set threshold. Obviously, when the acceleration section adopts the same acceleration and the set operation speed is constant, the positioning cycle T takes the minimum value T min when the T − takes its minimum value. ereby, for a complete trapezoidal move profile, it is only to require the minimum value T − to minimize the total positioning cycle T.
As can be seen from Figure 3, at the moment of the sample stage stops, the acceleration will be abrupt, resulting in an instantaneous inertial force that acts on the sample stage and causing the residual vibration of the table.
e following expression can be obtained by Newton's second law: In the case of a certain V max , the larger the a − is, the smaller the T a − will be. As can be seen from equation (23), the larger the F will be, which will cause the larger impact, then the longer the time T v when the residual vibration reaches the set amplitude threshold will be. erefore, there should be an optimal acceleration a * − to make T − get the minimum value T _min : For the impulse response of the second-order underdamped system, the envelope exhibits the characteristic of the exponential decay curve, as shown in Figure 4. e Shock and Vibration envelope of the oscillation can be expressed by the following equation: Assuming that the amplitude threshold allowed by the process is S Vlimit and the relationship between the proportional coefficient K for the impulse response and the acceleration value a − of the deceleration section is as the following equation, λ is the proportional coefficient between K and acceleration a − :

Shock and Vibration
If en, the following expression can be obtained from equations (26) and (27): Let zT − /zT a− � 0, the following equations can be obtained: It can be seen from equation (30) that when T a − � 1\ξw n , T − takes the minimum value. e required acceleration a − is represented by the following equation: It can be seen that for the different speeds V max , there is always an optimal acceleration a − to guarantee the shortest execution time of PTP movement in the case of meeting the residual vibration threshold requirement. It also shows that the optimal value is dependent on the system dynamics parameters ω n and ξ.

Experiment
In order to obtain the dynamic parameters of the sample stage, two different move profiles are used to excite the sample stage, and the residual vibration signal after the sample stage stops is obtained by the signal acquisition device. e designed moving profiles are shown in Figure 5, which adopts a triangular motion profile and trapezoidal motion profile, respectively. e experimental setup is illustrated in Figure 6. e data acquisition equipment belongs to the NI company, and the data acquisition software is developed based on the LabVIEW environment. During the experiment, the sampling frequency is set as 1000 Hz. e acceleration sensor is ICP acceleration sensor. e measurement frequency range is 0.5-10000 Hz. e acceleration range is ±50 g. e machine tool is a three-axis ultraviolet nanosecond laser processing equipment, including a sample stage, a galvanometer, and so on. Figure 7 shows the results of the residual vibration in the triangular moving profile, and Figure 8 shows the results in the trapezoidal move profile. It can be seen that the sample stage stops accompanied by the residual vibration, which is as shown by the curve FVSF. e larger the acceleration of the deceleration section is, the larger the amplitude of the residual vibration will be.  Shock and Vibration It can be seen from the PSD diagram that the dominant mode frequency of the sample stage is about 7 Hz. In order to facilitate the identification of the system parameters, the band-pass filtering method is used to process the frequency band near the dominant mode. en the frequency domain integration method is used to quadraticize the acceleration signal to obtain the displacement vibration signal of the residual vibration, which is used for the identification of system parameters. e ARMA method is used to identify the system model parameters. e results are presented in     Figure 9 shows the ARMA residual vibration identification results under the triangular move profile and Figure 10 shows the ARMA residual vibration identification results under the trapezoidal move profile. e natural frequency w n and damping coefficient ξ of the sample stage identified in the two move profiles are shown in Tables 2 and 3. Finally, w n � 44.55, ξ � 0.04356, and λ � 0.004783 are taken as the dynamic parameters of the sample stage.
It can be obtained from equation (31), when w n � 44.55 and ξ � 0.04356, then a − � 1.940598 V max . Assuming V max � 0.75 m s −1 , then the optimal acceleration of the deceleration section is a − � 1.46 m s −2 and the deceleration section time is T a − � 0.515 s. e evaluated optimal delay time for laser irradiation is T v � 0.5850 s, according to equation (28), which means the amplitude of the residual vibration will decay below the threshold when the decay time is greater than T v . Figure 11 shows the experimental results when a − takes different values. e threshold of the residual vibration is set to 0.1 um. e results (Figures 11(c) and 11(d)) show that when T a − � 0.515 s, i.e., a − � 1.46 m s −2 , the experimental result of the T v is 0.5869 s, which is very close to the evaluated optimal delay time.
e time T − � T a − + T v � 1.1019 s takes the minimum value, which proves the correctness of the conclusions in this paper. Obviously, this is useful for reducing the settling time and improving processing efficiency.
Although the difference in the values for T − between the three experiments is small, this is because the laser drilling machine of this paper uses a marble bed, which results in a smaller initial amplitude and a faster decay for the residual vibration. If the robotic arm or nonmarble bed structure of the laser processing equipment is used for positioning, the experimental results will be more obvious.

Conclusion
PTP motion is common in the industry. When the trapezoidal velocity trajectory is used in laser drilling, the sample stage will generate residual vibration when positioning completes. In order to reduce the influence of residual vibration on laser processing quality, it is necessary to set delay time to make the residual vibration attenuate, which causes a drop in overall machining efficiency. In this paper, based on the theoretical analysis, we have found that the acceleration value of the deceleration section for the trapezoidal moving profile can minimize the ST value, and the decay time after the sample stage positioning for laser irradiation can also be evaluated so that the whole laser drilling circle can be shorter. e conclusions provided in this paper are simple and effective. Only the dynamic parameters of the sample stage are needed to obtain the optimal acceleration value of the deceleration section and the delay time for laser irradiation at a given operating speed, which effectively shortens the execution time of the whole laser drilling circle and improves the processing efficiency. In further research, the accurate acquisition methods of the dynamic parameters of the sample stage and the accurate time of the laser irradiation delay will be studied in more detail. At the same time, the conclusions of this paper will be applied to design the process instruction of large-area laser drilling and help to improve the overall drilling efficiency.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.