A pre-generated matrix-based method for real-time robotic drilling chatter monitoring

Jianfeng TAO, Chengjin QIN, Dengyu XIAO, Haotian SHI, Chengliang LIU

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Abstract Currently,due to the detrimental effects on surface finish and machining system,chatter has been one crucial factor restricting robotic drilling operations, which improve both quality and efficiency of aviation manufacturing. Based on the matrix notch filter and fast wavelet packet decomposition,this paper presents a novel pre-generated matrix-based real-time chatter monitoring method for robotic drilling. Taking vibration characteristics of robotic drilling into account, the matrix notch filter is designed to eliminate the interference of spindle-related components on the measured vibration signal.Then, the fast wavelet packet decomposition is presented to decompose the filtered signal into several equidistant frequency bands, and the energy of each sub-band is obtained. Finally, the energy entropy which characterizes inhomogeneity of energy distribution is utilized as the feature to recognize chatter on-line, and the effectiveness of the presented algorithm is validated by extensive experimental data.The results show that the proposed algorithm can effectively detect chatter before it is fully developed.Moreover,since both filtering and decomposition of signal are implemented by the pre-generated matrices, calculation for an energy entropy of vibration signal with 512 samples takes only about 0.690 ms. Consequently, the proposed method achieves real-time chatter monitoring for robotic drilling, which is essential for subsequent chatter suppression.

KEYWORDS Aviation manufacturing;Fast wavelet packet decomposition;Matrix notch filter;Real-time chatter monitoring;Robotic drilling operations

1. Introduction

The enormous application potential of robotic drilling technology has attracted widespread attention from both industry and academia,especially from the field of aviation manufacturing.1However,industrial robots generally adopt a series joint structure. The stiffness of the robotic drilling system is lower than that of conventional computerized numerical control (CNC)machines tools and is closely related to the robot pose.2Consequently, it is more prone to unstable vibrations including chatter during the robotic drilling process.3The occurrence of chatter seriously affects the quality,accuracy and efficiency,accelerates the wear of the tool,and even result in the failure of cutting tool.4Chatter has become an important bottleneck restricting the practicality of robotic drilling technology. For these reasons, scholars have been devoted to modelling, analyzing, monitoring and suppressing such unstable vibration to ensure the quality and efficiency of robotic drilling.

Theoretically speaking, stable machining process can be achieved through the selection of optimal chatter-free machining parameters.5For the drilling dynamics and stability,Altintas is one of the pioneers. Based on the mechanistic modeling of drilling force, Roukema and Altintas6established a two degree-of-freedom delay differential equation for torsionalaxial chatter,and investigated drilling stability by both numerical methods and experiments. Later, a four-dimensional drilling chatter model was further proposed by Roukema and Altintas,7,8in which the kinematics and stability were analyzed in depth using time-domain method and frequency-domain method. Taking the process damping effect into account,Ahmadi and Altintas9developed a generalized drilling stability model. Nevertheless, errors between theoretical model and actual system are always unavoidable.For CNC machine tools and robotic drilling processes, the chatter mechanism are also different. Chatter may still occur even under the condition of selecting parameters according to the stability lobes diagrams.Therefore, it is necessary to monitor and recognize chatter as early as possible at the transition stage, so as to adopt appropriate suppression method to eliminate the negative effects of chatter.

Chatter monitoring methodology measures physical signals sensitive to chatter online, and analyses the collected signals by the time-domain, frequency-domain, time-frequency analysis, and statistics methods. On this basis, timely detection of the chatter development process can be implemented with the selected appropriate chatter indicator. Researchers have utilized a variety of sensor signals to monitor and detect chatter, including motor current signals,10cutting force signals,11sound signals,12vibration signals,13and angular speed signals.14Among them, vibration signals have the advantages of low cost and easy measurement, and can fully reflect the chatter transition process. Consequently, vibration signals are more applicable under industrial conditions, and thus commonly used in machining chatter monitoring.15On the other hand, utilizing effective signal processing methods and extracting features that best describe the chatter development are also of critical importance for chatter detection. The development of chatter is a process in which the machining state transfers from smooth to unstable, and usually accompanied by variations in frequency components and energy distribution. During the stable machining process, the system energy is mainly affected by the spindle frequency and the corresponding harmonics. When chatter occurs, new frequency components, i.e., the chatter components, will appear near the system natural frequency, and the energy is absorbed to the chatter frequency gradually. Taking this into consideration, Tangjitsitcharoen16attempted to monitor chatter in the frequency domain according to the change of the frequency components. In addition, a review of the literature shows that the measured signals during machining processes are usually non-stationary, thus time-frequency processing methods have been frequently utilized for chatter monitoring. For instance,Thaler et al.17introduced a sound-based online method for chatter detection in band sawing, in which the measured sound signal was pre-processed with short-time Fourier transform (STFT). Based on the wavelet transform (WT) of the dynamic cutting forces, Tangjitsitcharoen et al.18presented an in-process chatter detection approach for ball end milling process. Cao et al.12developed a chatter detection method for milling operations, in which sound signals were processed by synchrosqueezing transform (SST) and the first-order singular value is selected as the chatter indicator. With the aid of high-resolution time-frequency representation of the measured vibration signal, Tao et al.15recommended a chroextracting transform (SET) based chatter identification approach for robotic drilling process. Besides, the variational mode decomposition (VMD)19and the empirical mode decomposition (EMD)20were also employed for chatter monitoring for milling process.

However, in-depth analysis indicates that the above works are still difficult to achieve real-time chatter monitoring due to relatively long computational time,which is critical for chatter suppression. In addition, the system dynamics, stiffness characteristics and chatter mechanism are quite different for CNC machine tools and robotic machining processes,resulting in differences in vibration characteristics.3,11For this reason,the existing monitoring methods that mainly based on the CNC machine tools cannot be applied directly to robotic drilling operations. During the early transition stage of chatter,the spindle frequency component and its corresponding harmonics still plays a major role. However, chatter components have appeared and are distributed in a wide frequency band.Therefore, it is necessary to accurately remove the spindlerelated components from the measured signals. To realize real-time and accurate monitoring of robotic drilling chatter,this paper proposes a novel pre-generated matrix-based method using the matrix notch filter and fast wavelet packet decomposition. The remainder of this paper is organized as follows. In Section 2, the self-designed robotic drilling system is presented, and the characteristics of the vibration signal for robotic drilling are investigated in-depth. In Section 3,the pre-generated matrix-based monitoring method is proposed.In Section 4,a description of chatter monitoring system is provided, and the experimental results and verification of proposed method are then presented. Finally, the conclusion is drawn in Section 5.

2. Robotic drilling system and vibration characteristics analysis

2.1. Robotic drilling system

As illustrated in Fig.1,the self-designed robotic drilling system consists of a KUKA KR 270 industrial robot, a sliding guide,a dedicated drilling end effector, a measurement control system, workpiece and corresponding holding fixture. The dedicated drilling end effector is mainly composed of a robot flange interface,a spindle unit,a feed unit,a normal detection unit,a visual measurement unit,and a pressure foot unit.Similar to traditional CNC machine tools, the drilling motion is achieved by the spindle and the feed unit. During the robotic drilling process, the presser foot is pressed tightly against the workpiece,thus increasing the stiffness of the system.The normal detection unit is utilized to measure the normal line of the aircraft envelope, which ensures the perpendicularity of the drilling hole. During the robotic drilling, two uncoated hard alloy steel drilling-countersinking tools with different geometric parameters were used.Both tool I and tool II have the 100°countersink angle. Besides, the tool I has 5 mm diameter and 130° point angle, while tool II has 6 mm diameter, and 120°point angle.

Fig.1 Designed robotic drilling system.

2.2. Vibration characteristics analysis

To effectively monitor chatter for robotic drilling process,it is of vital importance to analyze the characteristics of the vibration signal during the chatter development. Therefore, plenty of experiments were carried out and the results were analyzed in depth. Fig. 2 presents the representative vibration signal of robotic drilling operations in both time domain and frequency domain. The diameter of the drilling hole was 6 mm and the workpiece was aluminum alloy 6061. In addition, the spindle speed was 2400 r/min and the feed rate was 2.4 mm/s. It can be seen from Fig.2 that the system transitions from stable cutting to unstable state, i.e., chatter. The amplitude of vibration increases rapidly, and the chatter components are dominant after chatter is fully developed. In order to further study the vibration characteristics of robotic drilling process, we select three typical time intervals, as shown in Fig. 2(c)-(h). In first interval S1,the spindle rotates and the pressure foot is pressed against the workpiece. It is shown that the amplitude of the vibration is relatively small in this interval, and the forced vibration is dominant. The dominant frequency of the vibration equals to four times of the spindle frequency. However,chatter components appear and are distributed in a wide frequency band. When the tool begins to drill the workpiece and the chatter occurs, the transition interval S2 appears. At this time, the amplitude of the chatter grow at a relatively fast rate, and exceeded that of the forced vibration. With the further development of chatter, the energy is almost absorbed to the chatter components in the interval S3, as shown in Fig. 2(h).

3. Proposed chatter monitoring method

3.1. Design of matrix notch filter

According to the analysis in the previous section,to detect the robotic drilling chatter at an early stage, accurately removing the spindle frequency component and its corresponding harmonics from the measured signals is needed. Notch filters are an effective means of eliminating narrowband or sinusoidal interference, and belong to the infinite impulse response(IIR) filter. The order of IIR filter is much lower than its corresponding finite impulse response(FIR) filter under the same frequency requirement.However,IIR filter suffers a long transition stage, making it difficult to obtain good filtering performance under conditions of short data. Consequently,researchers were trying to suppress the transition stage to optimize the conventional notch filter.21

The key to designing the notch filter is to obtain the transfer function with frequency response as close as possible to the ideal notch filter.22Define a signal of length N to be filtered as sin=[sin(1),sin(2),...,sin(N)],the output signal of the notch filter as sou=[sou(1),sou(2),...,sou(N)],then the notch matrix Fωnobtained based on certain optimal criterion satisfies:

where ωndenotes the notch frequency.

To solve the notch matrix,define a complex column vector as q(ω)=[1,exp(jω),exp(j2ω),...,exp(j(N-1)ω)]T.Obviously,the designed notch matrix should approximate the frequency response as close as possible to the ideal notch filter, namely

Suppose ρ as a small enough positive number, the bandpass of the designed notch filter will be Bp=[0, ωn-ρ]∪[0,ωn+ρ]. The first step is discretizing Bpinto M frequency points of equal distance, which can be expressed as

Fig.2 Representative vibration signal of robotic drilling process in both time domain and frequency domain.

By substituting the above discrete frequency points and the notch frequency ωninto the vector q(ω),the following matrices can be obtained:

where subscripts R and I denote real and imaginary parts of the complex number, respectively.

According to the characteristics of ideal notch filter, i.e.,Eq. (2), the matrices Q with size N×2 M, P and Fωnshould satisfy:

Let qiand fibe the ith column vector of QTand,respectively,i=1,2,...,N.By employing the least squares method,one can obtain:

Solving Eq.(6)by using Lagrangian multiplier method,the vector fican be obtained:

where G is defined by QQT.

On this basis, the notch matrix Fωncan be finally obtained as:

It is seen that the notch matrix Fωnmainly depends on the notch frequency frequency ωn.Therefore,the notch matrix Fωncan be pre-generated and directly utilized for efficient and accurate filtering of the measured signal.

3.2. Fast wavelet packet decomposition

Different from wavelet transform, the wavelet packet transform (WPT) is capable of further decomposing highfrequency components of the signal, thus it is a more precise decomposition method. Generally, both wavelet transform and wavelet packet transform are mainly realized by the well-known Mallat algorithm. However, the implementation of Mallat algorithm requires convolution operation, making it difficult to meet the real-time requirements for actual chatter monitoring.

Mathematically, wavelet transform belongs to linear transformation, which is essentially equivalent to matrix operation. Therefore, some scholars have proposed to utilize matrix operations instead of convolution operations to implement the wavelet transform.23Let X(t) be the signal to be decomposed and W be the wavelet transform matrix, then the wavelet decomposition of the signal X(t)can be calculated as:

where Wcdenotes the wavelet coefficient matrix.

Denote Wmas the wavelet transform matrix obtained by using the Mallat algorithm,i.e.,the multi-layer decomposition function wavedec in MATLAB, then the wavelet linear transformation W can be obtained as:

where Γkdenotes the kth column of the unit matrix INof the signal length N.

Select Daubechies 8(db8)as the wavelet basis function,the wavelet transform matrix W can be further deduced as follows:where λ denotes the number of layers of signal decomposition.

Considering that wavelet packet decomposition also belongs to linear transformation,the wavelet packet decomposition matrix Wpcan be similarly constructed. Let Wmpbe the wavelet packet decomposition matrix obtained by using the Mallat algorithm.Following a similar way,the wavelet packet linear transformation Wpcan be obtained as:

On the basis of the wavelet packet decomposition function wpdec in MATLAB,the wavelet packet decomposition matrix Wpcan be further deduced as:

It should be noted that the function wpdec is employed to construct the wavelet packet tree. Accordingly, to obtain the wavelet packet coefficients, it is necessary to use the function wpcoef to extract corresponding wavelet packet coefficients,that is:

Finally,the fast wavelet packet decomposition of the signal X(t), i.e., the wavelet packet decomposition coefficient matrix Wpc, can be obtained as:

Obviously, the wavelet packet decomposition matrix Wpcan be pre-computed and stored. In terms of chatter monitoring application, it can be directly imported for rapid signal decomposition,thus greatly improving the computational efficiency.Compared with conventional wavelet packet decomposition based on Mallat algorithm, the main advantage of the proposed fast wavelet packet decomposition algorithm based on pre-generated decomposition matrix is that it greatly reduces computational time. To verify the computational efficiency of the proposed method,comparisons of computational time with the conventional methods are conducted, as shown in Table 1. It is implemented on a desktop computer (Intel Core i3-6500 3.3 GHz, 4.0 GB of DDR3L RAM, Windows10 OS)when processing signal with a length of 512.It demonstrates that the proposed can significantly reduce computational time.

Table 1 Computational time of different methods when processing signal with a length of 512.

3.3. The proposed monitoring method

Chatter in machining processes arises from a self-excitation mechanism. Chatter will cause changes in frequency components and energy distribution. During stable machining, the energy of the dynamic system is mainly affected by its spindle frequency and the corresponding harmonics. When chatter appears,new dominant frequency component will appear near the natural frequency of the cutting system, and the energy is gradually absorbed by the chatter frequency. For robotic drilling process, chatter components emerge and distribute in a wide frequency range at the beginning, and the amplitude is smaller than the spindle-related frequency. However, with the development of chatter, the amplitude of the chatter increases at a relatively fast rate.After a short time,the amplitude and energy of the chatter will quickly equal and exceed the forced vibration. Consequently, in order to better recognize the chatter, it is of vital importance to accurately remove the spindle frequency and its corresponding harmonic components from the measured vibration signals.

Taking the above analysis into account, a novel robotic drilling chatter monitoring algorithm based on the matrix notch filter and fast wavelet packet decomposition is presented.The matrix notch filter is designed to eliminate the disturbance of spindle frequency and corresponding harmonic components to the vibration signal.Then,the filtered vibration signal is decomposed into several equal-distance bands by the fast wavelet packet decomposition algorithm, and the energy of each sub-band is calculated. Finally, the energy entropy which characterizes the inhomogeneity of energy distribution is utilized as the indicator to recognize chatter on-line.

The following is a detailed description of the specific steps of the proposed chatter monitoring algorithm.In the first step,the signal length N is determined according to the signal sampling period T and the time interval h of the chatter monitoring:

Suppose uin=[uin(1), uin(2),..., uin(N)] is the measured vibration signal,and ωn1,ωn2,...,ωnkare the spindle frequency and corresponding harmonics.The corresponding notch matrices Fωn1，Fωn2，···，Fωnkare designed by the presented method in Section 2.1. By removing the spindle frequency and its corresponding harmonics, the filtered vibration signal uou=[uou(1),uou(2),...,uou(N)]can be obtained as follows:

Then, the filtered vibration signal uois decomposed into several equal-distance bands by the fast wavelet packet decomposition algorithm. When the signal is decomposed by the λlayer wavelet packet, the signal with equal frequency bandwidth of 2λis obtained. Let Wpckbe the kth wavelet packet coefficient obtained by the fast wavelet packet decomposition presented in Section 2.2, one can obtain:

According to the wavelet packet coefficient Wpck, the energy EN,kcorresponding to the kth band can be calculated,namely:

On this basis, working as the indicator of chatter monitoring, the energy entropy MNF-FWPE of the filtered signal can be obtained as:

The highlight of the proposed algorithm is that the notch matrices and the wavelet packet decomposition matrix can be pre-generated and directly utilized for the signal filtering and decomposition. Compared with conventional notch filter and the wavelet packet decomposition based on Mallat algorithm, the proposed pre-generated matrix-based algorithm can greatly reduce the computation time of signal filtering and decomposition, thus effectively guaranteeing the realtime performance of chatter monitoring. In addition, taking vibration characteristics of robotic drilling into account, the spindle frequency and its corresponding harmonics are accurately removed by the designed matrix notch filter from the measured signals for high detection accuracy. During the robotic drilling process, the energy entropy of the measured vibration signal is calculated by using the proposed method,and compared with the chatter threshold. When the energy entropy exceeds the threshold, the robotic drilling process is regarded as unstable and the subsequent chatter suppression should be conducted as soon as possible. It should be noted that the chatter threshold is determined based on plenty of robotic drilling experiments,which ensures the selected threshold are applicable for different robotic drilling condition.

4. Experimental verification and analysis

4.1. Chatter monitoring system

To validate the proposed method, robotic drilling tests with different cutting conditions were conducted on the selfdesigned robotic drilling system.The workpiece was aluminum alloy AL7075 and AL6061. The chatter detection for robotic drilling process was implemented by the chatter monitoring system, as shown in Fig. 3. A PCB 356A24 accelerometer was utilized to measure the vibration signals. In order to obtain the signal closest to the tool vibration, the PCB accelerometer was mounted on the fixed front end face of the spindle closest to the drilling-countersinking. In addition,working as the data acquisition system, the Crystal Instruments CoCo-80 was adopted to acquire the measured acceleration signals.Considering the high frequency characteristics of the robotic drilling chatter,the sampling frequency were set as 10240 Hz.

4.2. Experimental results and verification

The effectiveness of the proposed chatter monitoring algorithm is verified by a large number of tests with different drilling parameters. During the robotic drilling tests, the spindle speed varied from 1800 r/min to 4500 r/min and the feed rate from 0.9 mm/s to 9.6 mm/s.Meanwhile,the drilling depth was set as 6 mm and the pressing force was set as 0.12 MPa.Besides, the robotic drilling tests were conducted without lubrication and cooling.

To realize real-time and accurate detection of chatter, a novel monitoring algorithm based on the matrix notch filter and fast wavelet packet decomposition is proposed. First, the matrix notch filter is designed to accurately remove the spindle frequency and its corresponding harmonic components from the measured vibration signals.Then,the filtered vibration signal is decomposed into 16 equal-distance frequency bands by the fast wavelet packet decomposition algorithm, and the energy of each sub-band is calculated. Finally, the energy entropy is utilized to capture the inhomogeneity of energy distribution during the transition process of chatter. During the chatter monitoring, the measured vibration signal within 50 ms is processed every 25 ms. Since the sampling frequency was set as 10240 Hz, the length of the sliding window is selected as 512 sample data with overlapping. Besides, the Daubechies 8 (db8) is chosen as the wavelet basis function.

Fig.3 Chatter monitoring system.

It should be pointed out that the determination of threshold has an important influence on the chatter monitoring results.The threshold is affected by many factors, such as machining parameters, workpiece material and tool wear. Over-high threshold will lead to a lag of detection time,making it difficult to identify the chatter at an early stage. In contrast, if the threshold is too low,the robustness will be worse and the false alarm will be easily caused. Therefore, the selection of threshold should guarantee high detection accuracy and make the detection time of chatter as early as possible. It should also be applicable for different robotic drilling conditions. To determine a reasonable chatter threshold, 480 robotic drilling experiments involving different cutting tools, workpiece materials and cutting parameters were carried out. The experimental data was divided into two parts:one was used to determine the threshold, and the other was used to verify the rationality of the threshold. When determining the threshold value, the time of chatter occurrence was first determined by combining the amplitude variation and time-frequency spectrum of vibration signals,and then the energy entropy of the corresponding time period was calculated. After processing extensive experimental data, a reasonable threshold was obtained. To ensure the rationality and applicability of the threshold,the other part of the data was adopted for verification.Through analysis of a large number of experimental data, it was found that the energy entropy of the system keeps near a relatively stable value when chatter occurs and changes slightly over a wide range of cutting parameters. Finally, the threshold was determined to be 2×105m2/s4.

Fig. 4 shows the time-domain diagram, the monitoring result and the time-frequency spectrogram for the vibration signal with spindle speed 4500 r/min and feed rate 3.2 mm/s.The workpiece was aluminum alloy AL7075. It is seen from Fig. 4(a) that the amplitude of the vibration signal increases rapidly after t=0.176 s,and chatter occurs during the robotic drilling process under this machining parameter combination.The corresponding surface quality of the hole being machined is presented in Fig. 5. Obvious chatter vibration patterns can be seen from the Fig. 5, and the number of vibration waves is about 20. According to the corresponding high-resolution spectrum of synchroextracting transform (SET),23,24i.e.,Fig. 4(c), obvious chatter frequency appeared around t=0.183 s. At the beginning, energy entropy is relatively small and grows slowly. However, it can be seen from Fig. 4(b) that the energy entropy of the vibration signal increases rapidly after t=0.126 s. Therefore, the selected statistical energy entropy is very sensitive to the change of energy distribution. As shown in Fig. 4(b), the chatter is recognized at around t=0.140 s. Comparison between the time when chatter is detected and the time when obvious chatter frequency occurs shows that the proposed monitoring algorithm can effectively recognize the chatter before it is fully developed,which leaves valuable time for the subsequent chatter suppression measures. Moreover, since both the filtering and decomposition of the signal are implemented by the pre-generated notch matrix and wavelet packet decomposition matrix, the calculation of an energy entropy takes only about 0.685 ms.For the method using conventional notch filter and the wavelet packet decomposition based on Mallat algorithm, it takes almost 390.267 ms to compute an energy entropy. Consequently,the proposed pre-generated matrix-based robotic drilling chatter monitoring method can meet the real-time requirements for machining chatter detection.

Fig.4 Monitoring result and SET spectrogram for the vibration signal with spindle speed 4500 r/min and feed rate 3.2 mm/s.

Fig.5 Surface quality of the drilling hole with spindle speed 4500 r/min and feed rate 3.2 mm/s.

Then, different spindle speed and feed rate combinations were utilized. The monitoring result and the SET spectrogram for the vibration signal with spindle speed 4200 r/min and feed rate 1.4 mm/s is illustrated in Fig. 6. The workpiece was also aluminum alloy AL7075.According to Fig.6(a),the amplitude of the measured vibration signal is relatively small in the initial stage. After t=0.722 s, the amplitude of the measured vibration signal increases faster, and the chatter occurs. It can be seen from Fig. 6(a) that after a sharp increase the chatter has been fully developed at t=0.842 s. According to the corresponding SET spectrum Fig. 6(c), the obvious chatter frequency appears at t=0.722 s; and the chatter frequency dominates after t=0.842 s. Fig. 7 shows the corresponding surface quality of the drilling hole.It is seen that chatter leaves obvious vibration marks on the surface of the drilling hole,and the number of chatter waves is about 25. According to the detection result Fig. 6(b), the proposed monitoring algorithm recognizes the occurrence of chatter vibration near t=0.372 s. The result demonstrates that the vibration state of the robotic drilling system can be well recognized by the proposed chatter monitoring method. In addition, calculating an energy entropy value takes only about 0.690 ms, which is basically the same as the previous case. By contrast, the method using conventional notch filter and the wavelet packet decomposition based on Mallat algorithm takes approximately 393.362 ms. Therefore, it is concluded that real-time chatter monitoring can be achieved by the proposed method.

Fig.6 Monitoring result and SET spectrogram for vibration signal with spindle speed 4200 r/min and feed rate 1.4 mm/s.

In order to further verify the effectiveness of the proposed MNF-FWP-based chatter monitoring algorithm,experimental data of different workpiece materials and cutting parameters were analyzed. Fig. 8 illustrates the time-domain diagram,the monitoring result and the SET spectrogram for the vibration signal with spindle speed 3000 r/min and feed rate 3.0 mm/s. Now the workpiece was aluminum alloy AL6061. It can be seen from Fig. 8(a) that the amplitude of the vibration signal increases substantially after t=0.534 s,and the chatter occurs during the robotic drilling process. The corresponding surface quality of the drilling hole is presented in Fig.9.It is seen that chatter leaves obvious chatter marks on the surface of the drilling hole,and the number of vibration waves is approximately 32. As illustrated in Fig. 8(b), the energy entropy is relatively small and fluctuates slightly at the beginning stage. After t=0.225 s, the energy entropy increases rapidly, but then starts to decrease at t=0.375 s.It decreases for a short period of time and then increases rapidly again.Finally,the chatter is recognized at t=0.482 s. According to corresponding SET spectrum Fig. 8(c), the apparent chatter frequencies appear near t=0.647 s. Therefore, the proposed MNF-FWP-based monitoring method can effectively recognize the chatter at its early transition stage. Moreover, the calculation of an energy entropy value takes only about 0.690 ms, thus the proposed algorithm can well meet the real-time requirements of chatter monitoring.

The time-domain diagram, the monitoring result and the SET spectrogram for the vibration signals with spindle speed 3000 r/min and feed rate 3.5 mm/s is shown in Fig. 10. The workpiece was aluminum alloy AL6061 as well.It can be seen from Fig. 10(a) that the vibration signal increases faster after t=0.440 s, chatter occurs under this spindle speed and feed rate combination.Fig.11 shows the surface quality of the hole being machined. It is seen that chatter leaves about 35 vibration waves on the surface of the drilling hole. According to the corresponding SET spectrum Fig.10(c),the apparent chatter frequencies appear around t=0.508 s.As shown in Fig.10(b),the chatter is recognized at t=0.380 s.Therefore,the current vibration state of the robotic drilling system can be well recognized by the proposed chatter monitoring algorithm. In addition, since the filtering and decomposition of the signal are implemented by the pre-generated matrices, it takes only about 0.690 ms to calculate an energy entropy of the vibration signal with 512 samples.Consequently,the results validate that the proposed pre-generated matrix-based chatter monitoring method can greatly reduce the computation time of signal filtering and decomposition, and achieve real-time robotic drilling chatter monitoring.

Fig.7 Surface quality of the drilling hole with spindle speed 4200 r/min and feed rate 1.4 mm/s.

Fig.8 Monitoring result and SET spectrogram for the vibration signal with spindle speed 3000 r/min and feed rate 3.0 mm/s.

Fig.9 Surface quality of the drilling hole with spindle speed 3000 r/min and feed rate 3.0 mm/s.

Fig.10 Monitoring result and SET spectrogram for the vibration signal with spindle speed 3000 r/min and feed rate 3.5 mm/s.

Fig.11 Surface quality of the drilling hole with spindle speed 3000 r/min and feed rate 3.5 mm/s.

5. Conclusions

In this work,we present a real-time chatter monitoring method for robotic drilling based on the matrix notch filter and the fast wavelet packet decomposition. In the first step, the matrix notch filter is designed to remove the spindle frequency component and its harmonics from the measured vibration signal.Then, the filtered vibration signal is decomposed into several equidistant frequency bands by the proposed fast wavelet packet decomposition algorithm. On this basis, the energy of each sub-band can be obtained, and the energy entropy which characterizes the energy uniformity is utilized as the indicator to detect the chatter on-line.The effectiveness of the proposed method have been verified by robotic drilling experiments with different workpiece materials and cutting parameters, and the results show that the proposed monitoring algorithm can effectively recognize the chatter at its early transition stage. Moreover, calculation for an energy entropy of the vibration signal with 512 samples takes only about 0.690 ms,while the method that uses conventional notch filter and the wavelet packet decomposition based on Mallat algorithm needs almost 390.300 ms. Therefore, the proposed pre-generated matrixbased method can greatly reduce the computation time of signal filtering and decomposition, and could meet the real-time requirements of chatter monitoring for robotic drilling operations.It can be concluded that the proposed method gives consideration to both real-time and accurate detection for robotic drilling chatter; thus, it achieves high industrial application value.

Acknowledgement

This study was supported by the National Key R&D Program of China (No. 2017YFB1302601 and 2018YFB1702503).