Remaining Useful Life Prediction of Ball Screw Using Precision Indicator
Hanbo Yang , Gedong Jiang , Zheng Sun , Zhuoni Zhang , Fei Zhao , Member, IEEE, Tao Tao , and Xuesong Mei
Abstract— Remaining useful life (RUL) prediction has been receiving increasing attention in industry and academia due to its advantages in ensuring safety, reducing costs, and improving efficiency. For industrial manufacturing, the ball screw is an indispensable high-precision servo component. There is an urgent need to predict its RUL to guarantee machining precision.
However, establishing a reliable and accurate physical model is a thorny issue as sophisticated mechanical characteristics are difficult to obtain. Besides, the limited degradation data reduces the prediction accuracy of data-driven methods. This article presents a hybrid RUL prediction architecture of ball screws.
First, to track the precision degradation process, the backlash is proposed as a novel nonlinear precision indicator. Then, physical- and data-driven methods are strategically leveraged to reveal the precision degradation process of the ball screw, where the stochastic degradation model is constructed and its initial para- meters are estimated by data-driven methods. The effectiveness of the method is experimentally demonstrated through the designed platform and compared with commonly used methods.
Index Terms— Backlash, ball screw, hybrid, precision, remaining useful life (RUL).
I. INTRODUCTION
A. Motivation
E
MERGING edge computing (EC), cyber–physical sys- tems (CPSs), and Industrial Internet-of-Things (IIoT) technologies have been vigorously promoting the automation level of enterprises, which puts forward higher requirements for the reliability of the machinery [1]–[4]. As a key step in predictive maintenance (PdM), remaining useful life (RUL) prediction not only provides effective decision support for maintenance strategies but also avoids catastrophic failures and casualties [5]. Therefore, RUL is a hot issue that has been attracting more and more attention in recent years [6].At the same time, the ball screw, as a high-precision transmission component that converts rotary motion into linear motion, acts a pivotal role in maintaining the normal operation
Manuscript received March 14, 2021; revised May 6, 2021; accepted May 24, 2021. Date of publication June 9, 2021; date of current version June 23, 2021. This work was supported in part by the National Key Research and Development Program of China under Grant 2018YFB1701200 and in part by the Science and Technology Major Project of Shaanxi Province China under Grant 2018zdzx01-01-01. The Associate Editor coordinating the review process was Qiang Miao.(Corresponding author: Zheng Sun.)
The authors are with the State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China, and also with the Shaanxi Key Laboratory of Intelligent Robots, Xi’an Jiaotong University, Xi’an 710049, China (e-mail: [email protected]; gdjiang@mail.
xjtu.edu.cn; [email protected]; [email protected]; ztzhao@
xjtu.edu.cn; [email protected]; [email protected]).
Digital Object Identifier 10.1109/TIM.2021.3087803
of machine tools [7]. However, ball screws are subject to severe working conditions during long-term operation. The precision and reliability are constantly threatened by degra- dation problems, such as grinding, pitting, and cracking [8].
Therefore, the RUL of the ball screw, i.e., the operating time to reach the precision threshold, should be accurately predicted.
The precision decreasing, as well as the stiffness decreasing of ball screws, will cause the instability of the servo controller and damage the quality of machine parts [9]. Similar to the concerns for other motion components, such as guide rails or bearings, the existing studies on the prognosis of ball screws mainly focus on their abrasion [6], [7]. The process of precision degradation is regretfully neglected. There is an urgent need for a method that can accurately and conveniently obtain the precision indicator (PI) of the ball screw.
Another issue is how to predict the RUL of the ball screw from the limited data, especially when there is no set of lifetime data. The ball screw is a critical and valuable component in machine tools; multiple groups of the run-to- failure test are always expensive. In addition, the physical properties of ball screws, such as mass, stiffness, and damping, are also difficult to obtain in industrial scenarios. Therefore, most of the physical-driven methods and data-driven methods are not sufficient to capture the nonlinearity and stochasticity of the degradation process from the limited data in less than one life cycle [10].
The primary motivation of this work is to accurately predict the RUL of ball screws by combing the merits of the data- and physical-driven methods. A suitable indicator concerning precision is required to reveal the degradation process. Then, the physical model coupled with the PI can be employed to characterize the nonlinearity and randomness of the degrada- tion curve. The information mining capabilities of data-driven methods will be leveraged to estimate the unknown parameters from the limited data. The particle filter (PF) is suitable to deal with this issue since it combines the acquired data with the expert knowledge in empirical models [11], which will be applied to reveal the precision degradation process of the ball screw.
B. Related Works
Generally, RUL prediction methods can be categorized into data-driven, physical-driven, and hybrid methods.
Data-driven methods, e.g., artificial neural networks (ANNs) [12], [13], the Gaussian process regression (GPR) [14], [15], and support vector machine (SVM) [16], [17], leverage data
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mining to establish the correlation between the acquired data and RUL. To cope with the complexity and uncertainties in prognostics, Cheng et al. [18] proposed a novel ensemble long short-term memory neural network (ELSTMNN) model.
Verl et al.[19] proposed the sensor-less automated condition monitoring algorithm based on the signals from the control drive (e.g., position, speed, current, and so on) to monitor the wear status of the ball screw. Li et al. [20] conducted in-depth research and experimental analysis on the sensor-less and sensor-rich strategies and used GPR to predict the RUL of ball screws. Mao et al. [21] utilized Pearson’s correlation coefficient to divide the state of bearings and then applied the least-squares SVM (LSSVM) to establish the RUL prediction model for the fast-degradation state. Data-driven methods rely on historical data throughout the life cycle, while the new machinery lacks historical data to train the model and infer its degradation curve [22].
Physical-driven methods aim to establish a mathemati- cal or physical model to capture the changes in physical prop- erties induced by the deterioration of the machinery [23]. For instance, through monitoring the change of ball pass frequency, Tsai et al. [24] detected the preload loss of the ball screw caused by long-term operating. Feng and Pan [25] established a mathematical model of the ball screw and diagnosed the preload variation by the slide of peak frequency and the magnitude of the peak frequency. Xi et al. [26] developed a multi-mass model to simulate the frequency shift of the machine axis dependent on wear. However, a reliable physical model is inseparable from the accurate characterization of the mechanical properties of the ball screw, which is difficult to acquire in industrial scenarios.
Hybrid methods leverage the advantages of both data-driven and physical-driven methods to perform RUL prediction [27].
Wanget al.[28] obtained different representative data through the data-driven method and established the exponential model to extrapolate the degradation curve. Ahmad et al. [22]
selected the appropriate regression model through the growth rate of the health indicator. To achieve better performance in accuracy and convergency, Baptista et al. [29] integrated data-driven methods with Kalman-based models for RUL prediction. Skordilis and Moghaddass [30] proposed a hybrid state-space model to describe the evolution of the system’s operating status and its degradation over time.
C. Contributions of This Article
In this article, a hybrid prognostic method for ball screws based on the PI is proposed. Investigation reveals that the backlash generated by wear is considered to be a significant indicator reflecting the precision degradation status of the ball screw [9], [26]. Therefore, the backlash is proposed as a novel nonlinear PI.
Initially, signals acquired from the control system are processed to construct the PI. Subsequently, the dynamic model considering the nonlinearity and stochasticity of the degradation process is established. Then, the initial para- meters in the dynamic model are estimated using limited PIs, i.e., backlashes from the initial time to the predic- tion time. In PF, the parameters of the dynamic model are
Fig. 1. Backlash phenomenon.
updated to track the degradation process. Ultimately, the effectiveness of the method is experimentally demonstrated through the designed platform. The proposed method makes four contributions.
1) The backlash is proposed as the PI, which successfully exhibits a novel nonlinear indicator to characterize the underlying precision degradation trend of the ball screw.
In addition, the proposed PI is not affected by artificial labels, such as the linear indicator, but is generated from the nature precision degradation perspective.
2) A hybrid ball screw RUL prediction method is presented.
Physical- and data-driven methods are strategically leveraged and effectively integrated. The degradation model considering the nonlinearity and stochasticity is constructed, and its initial parameters are estimated by data-driven methods.
3) An effective PI calculation method is introduced based only on signals available in the controller. In the semi- closed loop control mode, the grating and encoder signals are applied to track the precision degradation process. The proposed PI has more practical significance for industrial scenarios because it does not require the installation of external sensors (in addition to the built- in sensors, i.e., grating and encoder), as well as the acquisition and analysis of high-frequency data.
4) The proposed method provides the theoretical basis for the ball screw RUL prediction with limited data, as, in actual industrial applications, a large number of failure tests are often not allowed due to time and security reasons.
The remainder of this article is organized as follows.
Section II introduces the backlash phenomenon and the basic theory of PF. The proposed RUL prediction architecture is given in Section III. Then, in Section IV, the proposed method is validated through the experimental investigations. Finally, Section V draws the conclusion and discusses future work.
II. THEORETICALBACKGROUND
A. PI
As depicted in Fig. 1, the principle of precision loss caused by backlash is given as follows.
1) The nut is transferred from initial state 1 to state 2 through the rotation of the screw.
2) At state 2, the screw starts to reverse to state ˆ2, while the position of the nut remains unchanged.
3) Then, the nut starts to move in the reverse direction from state ˆ2 and finally reaches state 3.
During the transition from state 2 to state ˆ2, due to the existence of the backlash, the screw rotates within a certain angle, while the nut remains stationary. This angle can be converted into the corresponding linear length, which is the backlash of the ball screw. In low-power mechanical systems, the backlash phenomenon will interfere with the accuracy and stability of the position loop in the control system, whereas, in high-power systems, they are likely to excite impulse torque, which will further aggravate the growth of cracks and the increase in wear area [31]. Consequently, the backlash is the suitable PI to characterize the degradation process of the ball screw.
B. Basic Theory of PF
In PF, a dynamic system can be characterized through the state-space model comprising the following state transition and measurement function [32]:
xk = fk(xk−1, vk−1)
yk =hk(xk,nk) (1) where xk is the system state at time k, vk−1 stands for the process noise at time k −1, fk(·) is a nonlinear function describing the correlation betweenxk andxk−1,nkis the mea- surement noise, yk is the observed value, andhk(·)represents the measurement function.
The procedure to estimate the posterior probability density function (PDF) of the current state xk has the following prediction and update function:
p(xk|y1:k−1)=
p(xk,xk−1|y1:k−1)d xk−1
=
p(xk|xk−1,y1:k−1)p(xk−1|y1:k−1)d xk−1
=
p(xk|xk−1)p(xk−1|y1:k−1)d xk−1 (2) p(xk|y1:k)= p(yk|xk,y1:k−1)p(xk|y1:k−1)
p(yk|y1:k−1)
= p(yk|xk)p(xk|y1:k−1) p(yk|y1:k−1) with p(yk|y1:k−1)=
p(yk|xk)p(xk|y1:k−1)d xk (3) where p(xk|y1:k−1) denotes the prior PDF of xk, p(xk|xk−1) is the state transition PDF, p(xk−1|y1:k−1)is the state PDF at timek−1, p(xk|y1:k)is the posterior PDF ofxk, and p(yk|xk) represents the likelihood function.
Given the nature of the Bayesian inference, the first line can be transformed to the second line in (2), whereas the transformation from the second line to the third line in (2) and the first line to the second line in (3) are the properties of the Markov process. However, the analytical solution for (2) and (3) is intractable due to the nonlinear and non- Gaussian systems [32]. Therefore, PF draws a set of particles {xik,i =1,2, . . . ,n}sampled fromq(xk|xk−1y1:k)to estimate the posterior PDF at time k
p(xk|y1:k)≈ n
i=1
wikδ
xk−xki
(4)
Fig. 2. Flowchart of the proposed hybrid RUL prediction method.
whereδ denotes the Dirac delta function andwik is the weight ofxki and is updated and normalized as follows:
wik=wik−1p yk|xki
p xki|xki−1
q
xki|xki−1,y1:k
, wik= wik
N i=1wik
. (5) For simplicity, q(xk|xk−1y1:k) is set to be equal to p(xk|xk−1); then, the weight is updated as follows:
wik=wik−1p yk|xki
. (6)
To deal with the particle degradation problem in the PF, the inverse cumulative distribution function (CDF) is applied for resampling [23].
III. PROPOSEDARCHITECTURE
As illustrated in Fig. 2, the hybrid prognostic method for ball screw based on the PI encompasses four major modules, i.e., backlash acquisition, physical model construction, para- meters estimation, and RUL prediction, the details of which are presented in Sections III-A–III-D.
A. Backlash Acquisition
The essential point for RUL prediction is to obtain the displacement difference between the nut and the ball screw in reverse motion. However, directly capturing this displace- ment difference in micron-level requires simultaneous high- frequency and high-resolution sampling of the encoder and grating signals, which is intricate to achieve in actual oper- ation. Our investigation reveals that, if the encoder signal is used as the feedback signal of the ball screw and the grating signal is used as the measurement value of the nut position, the backlash can be calculated straightforwardly by comparing the displacement difference at the target point. The specific procedures are given as follows.
Initially, in the semiclosed-loop servo control mode, the screw drives the nut in the direction to the target point 1, and the corresponding measurement valuex(t1)of the grating is read. The servo controller then sends an instruction with the positive displacement of L, and the ball screw keeps moving in the same direction until it reaches target point 2.
Then, the measurement signal of the grating x(t2) is read.
Ultimately, the servo controller sends the command with reverse displacementL, and the ball screw travels against the direction until it reaches the destination, recording the grating signalx(t3). Due to the existence of backlash and pitch error, the relationship between x(t1)andx(t3)can be expressed as
x(t3)=x(t1)+ t3
t1
fvdt+ t3
t1
rvdt with
t3
t1
fvdt+
t3
t1
rvdt =b|x(t2)+ t3
t1
∂pe
∂t dt (7)
where fv is the forward speed, rv is the reverse speed, b represents the backlash, and pe denotes the pitch error.
Furthermore, the total pitch error is the cumulative pitch error within the equivalent length of the backlash as follows:
t3 t1
∂pe
∂t dt= t2
t1
pe dt+ t3
t2
pedt= b|x(t2)
0
∂pe
∂x dx. (8) In (8), the pitch error generated in the micron-level motion can be ignored, and then, (7) can be simplified as
x(t3)=x(t1)+b|x(t2). (9) The distance traveled from the grating position x(t2) to target point 2 is short, and it can be determined that the backlash of the two places is approximately equal; then, the backlash at target point 2 can be derived as
b|x(t2)≈b|target point2=x(t3)−x(t1). (10) Given the inconsistent contact surface characteristics at different positions, the backlash varies slightly at different measurement locations [9]. The backlash of the ball screw can be calculated by selecting n target points {backlashj,
j =1,2, . . . ,n}
backlash= 1 n
n j=1
backlashj. (11)
B. Physical Model Construction
The inherent randomness in manufacturing and operation leads to the stochastic degradation process [33], which can be represented with the following state-space model:
xk = x0+θtkη+σB(tk)
yk = xk+n (12)
where xo denotes the initial state of the dynamic model, θ is the time-varying parameter following N
μθσθ2
,ηis degrada- tion index that represents the nonlinearity of the degradation process, σB(tk) is the Brownian motion (BM) following the normal distribution N
0, σ2tk
, and n is the measurement noise subject to N(0, τ2). In this case, the stochasticity is characterized by BM and measurement noise.
C. Data-Driven Methods for Initial Parameter Estimation
The unknown parameter =
μθ, σθ2, η, σ2, τ2 in the state-space model is crucial to accurately describe the degra- dation process of the ball screw. Existing studies assume that the initial value is 0 to characterize a perfect state without any degradation [6], [33]. However, for the ball screw, despite the preload condition specified by the manufacturer, the backlash cannot be eliminated due to the combined effect of static fric- tion and elastic deformation. Consequently, the initial state is denoted as x0 in this article. Assuming that thek-dimensional measurement data y = [y1,y2, . . .yk]T have been obtained,
according to the characteristics of BM, the multivariate normal distribution ofy is given as follows:
y∼
⎛
⎜⎜
⎝[x0,x0, ...x0]T+μθ
t1η,t2η, ...tkηT
σθ2
⎡
⎢⎢
⎣
t1ηt1η t1ηt2η · · · t1ηtkη t2ηt1η t2ηt2η · · · t2ηtkη
· · · · tkηt1η tkηt2η · · · tkηtkη
⎤
⎥⎥
⎦+
⎡
⎢⎢
⎣
τ2 0 · · · 0 0 τ2 · · · 0
· · · · 0 0 · · · τ2
⎤
⎥⎥
⎦
+σ2
⎡
⎢⎢
⎣
min(t1,t1) min(t1,t2) · · · min(t1,tk) min(t2,t1) min(t2,t2) · · · min(t2,tk)
· · · · min(tk,t1) min(tk,t2) · · · min(tk,tk)
⎤
⎥⎥
⎦
⎞
⎟⎟
⎠. (13) The mean matrix and the covariance matrix of y can be represented as follows:
µy =[x0,x0, ...x0]T+μθ
t1η,t2η, ...tkη
T
y
=σθ2
⎡
⎢⎢
⎣
t1ηt1η t1ηt2η · · · t1ηtkη t2ηt1η t2ηt2η · · · t2ηtkη
· · · · tkηt1η tkηt2η · · · tkηtkη
⎤
⎥⎥
⎦
+
⎡
⎢⎢
⎣
τ2 0 · · · 0 0 τ2 · · · 0
· · · · 0 0 · · · τ2
⎤
⎥⎥
⎦
+σ2
⎡
⎢⎢
⎣
min(t1,t1) min(t1,t2) · · · min(t1,tk) min(t2,t1) min(t2,t2) · · · min(t2,tk)
· · · · min(tk,t1) min(tk,t2) · · · min(tk,tk)
⎤
⎥⎥
⎦. (14) Then, the maximum likelihood estimation (MLE) of the multivariate normal distribution can be expressed as follows:
lnL
|µy,
y
= −K
2 ln(2π)−1 2ln|
y
|
−1 2
y−µy
T
y
−1 y−µy
. (15)
However, (15) involves the complex matrix operation, so the establishment and solution of the partial differential equations of (15) are complicated. If (15) is appropriately deformed, the optimal value of can be easily acquired, i.e., invert (15) and then apply the quasi-Newton algorithm to acquire the optimal initial parameters [34]
ˆ =
μˆθ,σˆθ,η,ˆ σˆ2,τˆ2
=arg min
−lnL
|µy,
y
.
(16) D. RUL Prediction Process
In the online prediction process, observations are used to estimate and update unknown parameters as a form of the PDF, which is expressed by the following form [23]:
p(|y)∝L(y|)p() (17)
Fig. 3. Designed experimental system.
where = [,x] denote the unknown parameters, y is the observed data, L(y|)is the likelihood function conditional on the given, and p() is a prior function for.
Based on (17), (6) can be extended as wki =wik−1p
yk|xki, ik
=wik−1L yk|ik
. (18) The RUL Rk at timek can be expressed as follows:
Rk=inf{Rk :x(Rk+k)≥ε|x1:k} (19) where represents the preset failure threshold based on the maximum tolerable backlash, x1:k is the estimated state value from the initial time to timek, andx(Rk+k)is the predicted value for the state at time Rk+k. Then, the following state transition function can be applied to predict the subsequential state value for theith particle:
xi(j+1)t+k−xijt+k
=θki
((j+1)t+k)ηik−(jt+k)ηik
+σkiB((j+1)t+k)−σkiB(jt+k) (20) wheret is the sampling interval,x(ij+1)t+k is the estimated state value of the ith particle at time (j +1)t +k, and j ∈ N. Once the predicted value exceeds the threshold, the transition process is stopped. A set of RULs {Rki,i = 1,2, . . . ,n}can be obtained from (19). Ultimately, the PDF of the RUL can be acquired as follows:
p(Rk|x1:k)≈ N
i=1
wkiδ
Rk−Rki
. (21)
IV. EXPERIMENT ANDDISCUSSION
A. Introduction to the Experimental System
As illustrated in Fig. 3, this experimental system is designed to demonstrate the RUL prediction method of the ball screw based on the PI. To make the operating condition during the experiment more suitable for the real industrial scenarios and accelerate the wear process, the experimental environment is treated as follows.
TABLE I
SPECIFICATIONS FOR THEPLATFORM
1) The ball screw is degreased before leaving the factory.
Except for the very limited antirust oil in the beginning stage, the ball screw is no longer lubricated during the experiment.
2) In the real industrial environment, the wear speed of the guide is much lower than that of the ball screw [20].
In order to alleviate the impact of the guide wear on the measurement results, the guides are lubricated regularly.
3) A load of 100 kg has been added to the table directly above the nut.
The whole feed system is fixed to the vibration damping platform, the table moves horizontally along the guide rail, and the specifications for the designed platform are given in Table I. The ball screw reciprocates throughout the entire life cycle at a speed of 80 mm/s and a stroke of 250 mm.
The backlash at 10, 50, 90, 130, 170, 210, and 240 mm was measured with a time interval of two days.
Unlike the abrasion that is concerned with the failure of other mechanical equipment, the ball screw is applied as a high precision transmission component, and its failure threshold is based on the PI, i.e., backlash. In the precision transmission process, the backlash above 10 μm is generally considered that the ball screw is in the fault state [20], [35]. Therefore, the accelerated life test (ALT) of the ball screw was stopped when the backlash calculated according to (7)–(11) exceeded 10 μm, which was then preset as the failure threshold in the prediction process.
B. Backlash Acquisition
The acquisition of the initial state is the premise of MLE for the unknown parameters in the stochastic degradation model. Ignoring the measurement error, the initial state can be approximated by the measured value. The measurement process of the backlash at position 50 mm is shown in Fig. 4.
It should be pointed out that the first forward motion is to make the ball screw switch to state 1 in Fig. 1 so that the backlash is only on one side of the nut due to extrusion.
In addition, before the ALT, the backlash at different posi- tions was measured five times to obtain the mean and standard deviation. As illustrated in Fig. 5, the measured backlash fluctuated between 1.60 and 2.60 μm from positions 10 to 240 mm, while the standard deviation hits the peak value of 0.55 at position 90 and 240 mm, respectively. In the initial state, the ball screw exhibits consistency, and the fluctuation of the mean value at different positions is within 1μm, which is mainly the measurement error caused by the grating with
Fig. 4. Measurement process of the backlash at 50 mm.
Fig. 5. Backlash at different locations before the ALT.
TABLE II
MLE RESULTS FOR THEUNKNOWNPARAMETERS
1-μm resolution. Concerning the mean value of all the mea- sured backlashes, the initial state is set to 2.09μm.
C. RUL Prediction for the Ball Screw
The unknown parameters in the dynamic model are esti- mated with the PI from the initial time to the current time and are updated in the PF. The MLE results for the unknown parameters are given in Table II. The MLE values of degra- dation indexηfor the three models in the initial, middle, and final stages are 1.6158, 1.4367, and 1.3705, respectively. The degradation index η acts a pivotal role in determining the degradation speed of the ball screw. In the initial stage, the estimation result of η peaks at 1.6158 due to the limited measurement data; after that, the estimated value ofηbecomes smaller as the measurement data increases. By contrast,
Fig. 6. Prediction results at different stages. (a) Initial stage. (b) Middle stage. (c) Final stage.
μθ increases steadily with the degradation process of the ball screw. While the MLE results for the remaining three unknown parameters σθ2, σ2, and τ2 associated with the variance fluctuate over three different stages.
The RULs of the ball screw are predicted based on the PDF of the particle set. Fig. 6 illustrates the prediction results at the beginning, middle, and final stages, respectively. From day 1 to day 14, the PI curve shows a slight upward trend although there are some fluctuations. Then, the observed values are employed for the MLE of the model in the initial stage.
Given the fact that only the measured data in the initial slow degradation stage are encompassed, the prediction result has a large error, with a predicted RUL of 24 days and a true value of 32 days. Furthermore, the PIs mount moderately to 6.4, which is effectively captured by the second prediction model. The predicted RUL is 16 days, while the actual RUL is 18 days. For the third model, the predicted curve is more consistent with the degradation trend of the ball screw since it contains backlashes from the initial moment to the predicted point of the final stage. The predicted RUL is ten days, which is in perfect agreement with the real RUL. Experimental inves- tigations demonstrate that the estimated unknown parameters are capable of characterizing the degradation process, and the RUL prediction is accurate especially in the middle and subsequent degradation stages.
In addition, the updating process of the degradation indexη is given in Fig. 7. The variation of this parameter reflects the nonlinear change of the ball screw in the degradation process, where a larger value represents a faster degradation speed.
In the prediction process of different groups, η is updated posteriorly based on the prior data, while the confidence interval keeps narrowing with the updating time and, finally,
TABLE III
PREDICTIONRESULTS FORBALLSCREW
Fig. 7. Updating process for the unknown parametersη. (a) Initial stage.
(b) Middle stage. (c) Final stage.
converges to a constant, which is 1.6119, 1.4365, and 1.3718, respectively. In the initial stage, the degradation index is overestimated and has a large confidence interval due to the limited data. In the following stages, as the volume of data increases, this value will decrease and have a narrower confi- dence interval to approximate the actual degradation process.
D. Method Comparison
In order to demonstrate the effectiveness of the proposed method, the prediction results are compared with the data- driven approach in [20], the widely used physical-driven approach in [23], where the parameters are selected manually, and the hybrid approach in [28].
Two score functions are used to comprehensively evaluate the prediction results. The first type of the evaluation function is expressed as follows [36]:
Score1= 1 3
3 i=1
Ai
Ai =
e−ln(0.5)Er5i Eri ≤0
eln(0.5)Er20i Eri >0 (22) where Er=100×(RULtrue−RULpredicted)/RULtrue. A higher score represents a better prediction result.
The second type of the evaluation function is the mean square error (MSE), which is defined as
Score2=1 3
3 i=1
h2i (23)
wherehi =RULtrue−RULpredicted. Conversely, a lower score of the second function represents better performance.
Fig. 8. Prediction results of different methods at different stages. (a) Initial stage. (b) Middle stage. (c) Final stage.
In [20], GPR is applied for ball screw RUL prediction.
In the physical-driven approach [23], the initial parameters for the stochastic model are selected manually. As depicted in Fig. 8, the predicted results of the two methods in three different stages all belong to late predictions (i.e., the predicted RUL is larger than the actual RUL). As shown in Table III, the prediction accuracy of [20] and [23] is relatively low, where the values of Score1 are the same, which is 0.0008, and the values of Score2 are 225.333 and 333.3333, respectively.
In [28], a hybrid prognostic method is proposed, in which a data-driven method is used to determine the fitting curve, and the exponential model is constructed to characterize the degra- dation process. On the contrary, the predicted results using this strategy in three stages all belong to early predictions (i.e., the predicted RUL value is smaller than the actual RUL value).
Compared with the prediction results in [20] and [23], the prediction results in [28] have been improved, where the values
of Score1 and Score2 are 0.3012 and 50.6667, respectively.
However, the exponential model will grow explosively over time, reaching the failure threshold prematurely in the predic- tion process.
The superiority of the proposed method is validated as it obtains the highest value of Score1 and the lowest value of Score2 among them. Therefore, the precision degradation process can be effectively tracked, which has more practical significance in industrial scenarios.
V. CONCLUSION
In this article, a hybrid prognostic method for ball screws is proposed. A novel nonlinear PI characterizing the underlying degradation trend of the ball screw is introduced. Physical- and data-driven methods are effectively integrated for RUL prediction. The stochastic mathematical model is constructed from the precision perspective, and its unknown parameters are estimated from the limited data using the data-driven method.
The effectiveness of the proposed method is validated through the designed test platform, which can precisely track the non- linearity and stochasticity of the degradation process from dif- ferent stages. Compared with the state of the art, the proposed method showed a competitive performance. In addition, the proposed method is suitable for industrial scenarios because it only relies on the built-in sensors, i.e., grating and encoder, and neither requires the installation of external sensors nor the acquisition and analysis of high-frequency data.
However, some limitations should be considered. For instance, to achieve PdM, not only the RULs of the ball screw should be predicted but also its status should be monitored in real time. Due to the harsh industrial environment, the invasion of foreign objects, such as iron chips and the impact induced by misoperation, will change the original degradation mecha- nism of the ball screw, resulting in accidental failures. Another challenge is how to build a precision degradation model under time-varying conditions.
In the future, seamless integration between prognostic algorithms and real-time monitoring systems deserves to be focused on, as well as the enabling EC and DT technologies for timely and in-depth analysis of time-varying data.
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Hanbo Yangreceived the B.Eng. degree in mechan- ical engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 2016. He is currently pursuing the Ph.D. degree with the School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, China.
He has several technical articles published in inter- national journals, such as the IEEE INTERNET OF THINGSJOURNALand MEASUREMENT. He is also an active reviewer for measurement. His research interests include smart manufacturing, prognostics,
and health assessment.
Gedong Jiangreceived the B.E. degree in mechan- ical engineering and the Ph.D. degree in mechanical engineering from Xi’an Jiaotong University, Xi’an, China, in 1992 and 1998, respectively.
She is currently a Professor with the School of Mechanical Engineering, Xi’an Jiaotong University.
She was a Visiting Scholar with the University of Manitoba, Winnipeg, MB, Canada, in 2010. Her research interests include intelligent manufacturing, condition monitoring, and robotics.
Zheng Sun received the Dr.-Ing. degree from the University of Stuttgart, Stuttgart, Germany, in 2017.
Since 2018, he has been an Assistant Professor at the Institute for Robotics and Intelligent Sys- tems, Xi’an Jiaotong University, Xi’an, China. His research interests focus on the high dynamic motion control for servo systems and the cyber–physical manufacturing systems.
Zhuoni Zhangreceived the B.Eng. degree from the School of Mechanical Electronic and Information Engineering, China University of Mining and Tech- nology, Beijing, China, in 2018. She is currently pursuing the M.Eng. degree with the Institute of Robotics and Intelligent Systems, Xi’an Jiaotong University, Xi’an, China.
Her research focuses on condition monitoring in the production line.
Fei Zhao(Member, IEEE) received the Ph.D. degree in mechanical engineering from Xi’an Jiaotong Uni- versity, Xi’an, China, in 2013.
He joined the School of Mechanical Engineering, Xi’an Jiaotong University, and the Shaanxi Key Lab- oratory of Intelligent Robots, in 2017. His research interests include robot kinematics and condition monitoring for intelligent manufacturing equipment.
Tao Tao received the B.S. degree in mechani- cal engineering, the M.S. degree in mechanical engineering, and the Ph.D. degree in mechanical engineering from Xi’an Jiaotong University, Xi’an, China, in 1986, 1999, and 2005, respectively.
He is currently a Professor with the School of Mechanical Engineering, Xi’an Jiaotong University.
His research interests include high dynamic motion control for servo systems and error compensation for machine tools.
Xuesong Meireceived the Ph.D. degree in mechan- ical engineering from Xi’an Jiaotong University, Xi’an, China, in 1991.
He joined the School of Mechanical Engineer- ing, Xi’an Jiaotong University. He is currently the Director of the Shaanxi Key Laboratory of Intelli- gent Robots, Xi’an Jiaotong University. His research interests include intelligent manufacturing, robotics, and methods for precision laser processing.