Digital map and probability compensation model for repeated positioning error of feed axis

The randomness of the repeated positioning error of feed shaft is the main reason for the difficulty to address this error. The previous repeated loading and unloading suppression methods will easily do damages to parts. This paper sets out to investigate the probability distribution of all possible error values while feed shaft is at different positions, and determine the maximum value of probability error from the random errors. With feed axis at a certain position, first of all, we count the probability of each error based on a large number of random errors. We also draw the digital map of the repeated positioning error of this error with the error of positive and negative stroke measurement as the x-axis and y-axis coordinates and the probability of each error value as the z-axis coordinates. Secondly, based on the digital map of each position on the feed axis and combined with the dynamic optimization algorithm, we find the highest point on the map and take the coordinate point of this point as the starting point of each position compensation. Then, we set the error probability threshold to control the possibility of compensation errors and output the size and direction of the final compensation value. Finally, we start the compensation command and detect the compensated error. The error data that fail to meet the requirements will enter the probability statistics again, redraw the digital map and update the map. Through real-time detection and feedback, the digital map is dynamically improved to adapt to the changing environment. This probability compensation model of repeated positioning error can end the suppressing repeated positioning errors brought by repeatedly disassembling parts.


Introduction
Precision machine tools, which can process a variety of precision parts such as boxes and disks are widely used in military and civil industries. The feed shaft is the core transmission part of the machine tool [1], and the repeated positioning error of the feed shaft is one of the important indexes for the evaluation of machine tool. The requirements for precision machine tools are high speed, high precision, and high repeatability [2,3]. However, the randomness of the repeated positioning error of the linear axis makes it unable to build a model similar to deterministic compensation model of the positioning error. Thus, there is an urgent need for precision to reduce the repeated position error of feed shaft.
At present, there is little theoretical analysis on the repeated positioning error of the feed shaft, most of which focus on the experimental study of the influencing factors to the repeated positioning error, or reducing the repeated positioning error through the modeling and suppression of assembly error. Sun et al. [4] conducted experimental research on the mechanism of the repeated positioning error of the feed axis of the NC machine tool, determining the factors affecting the repeated positioning error of the feed axis through orthogonal experiments and proposing an assembly method to reduce the repeated positioning error of the feed axis. Szipka et al. [5] proposed a method for determining the repeatability of non-uniform parameters in the workspace under static and no-load conditions. They also made a detailed description of the multiaxial repeatability performance, which contributes to the understanding the root causes of performance changes in the manufacturing process. Mori et al. [6] also proposed a method of design and formation of workforce skills for machine tool assembly using simulations. They developed a simulation model based on the investigation of the actual machine tool workshop and applied several forms of labor skills into simulation. Considering the influence of joint surface deformation, Lu et al. [7] established a hybrid genetic algorithm and backpropagation neural network model to predict the assembly changes caused by joint surface deformation under different assembly conditions and parameters. Wang et al. [8] put forth an assembly performance method called pre-deformation. Since this method is technically based on machine tool assembly and collaborative computer aided engineering (CAE) analysis, it helps increase the assembly performance. He et al. [9] studied the propagation of machine tool changes caused by geometric errors during assembly and used the state space model to describe the change propagation in the process of machine tool assembly. This method has strong feasibility and practicality. Sun et al. [10] studied the influence of geometric errors on the repeatability of linear axis positioning, and proposed a mathematical model between the geometric errors of moving parts and the motion posture. Based on genetic algorithm, they determined the influence of geometric error on the repeatability of linear axis positioning. Due to the complexity of the repeated positioning error, most scholars attempt to make a breakthrough on the positioning error of the feed shaft such as thermal error. For example, models, such as neural network modeling, dynamic modeling, and finite element modeling [11][12][13][14][15], can accurately predict the positioning error of feed shaft, but these modeling methods are not well applied to reduce the repeated positioning error of the feed shaft. Based on the previous research, it is found that the lack of model of repeated positioning error model leads to a single means of restraining repeated positioning error. Usually, it exploits the experimental method to determine the possible influencing factors of repeated positioning error (such as preload and assembly accuracy) and the influence size of each factor, and makes adjustments by dissembling parts according to the experimental results.
The adjustment of assembly sequence can reduce the repeated positioning error to a certain extent. However, with the use times of the machine tool growing, each part of the machine tool will change. Then, parts must be disassembled and assembled again to adapt to the new changes. This method of repeated assembly is of blindness, which does allow researchers to fully grasp the distribution law of errors. In addition, repeated disassembly and assembly is very easy to damage parts [16]. In view of the above problems, this paper makes analyses from the direction of experimental measurement, data modeling, and error compensation. It also uses the measured data to build a digital map of repeated positioning error, and combines the dynamic optimization algorithm to build a probability compensation model of repeated positioning error of the feed shaft and test the error compensation. The method of error probability compensation can efficiently eliminate the disadvantages brought by repeatedly disassembling parts. The model construction process is shown in the Fig. 1.

Digital map drawing of repeated positioning error
The key to constructing probability compensation model is to draw the digital map of repeated positioning error. Based on a large number of repeated positioning error data, we found the frequency of some error at a certain position, and further calculated the probability of this error. The digital map of repeated positioning error is drawn using the statistical probability distribution of the positive and negative direction error of the feed axis, and the drawing method is shown in Fig. 2.
The construction process of repeated positioning error digital map has the following steps: Step 1, arrange the laser interferometer according to the measurement position; Step 2, move the feed shaft to drive the reflector, where the reflector is fixed at the origin of the feed system coordinate system; Step 3, use the position of the mirror to measure the error of the feed axis; Step 4, repeat step 2 and step 3 for N times, and record the measurement error value of each time; Step 5, count the error distribution frequency of each position of positive stroke and negative stroke according to the recorded measurement error value; Step 6, calculate the error distribution probability of each position according to the error distribution frequency of each position; Step 7, build the coordinate axis of the specific location digital map. Among them, x-axis coordinate represents the positive stroke error value of the position, y-axis represents the negative stroke error value of the position, and z-axis represents the probability of taking the error value; Step 8, draw a digital map of the repeated positioning error of each position.
In order to obtain a complete digital map, it is necessary to draw three-dimensional points outside the coordinate axis. Nonetheless, this point is of no practical significance. To ensure that the data points outside the coordinate axis do not affect the search results, as shown in Fig. 2, the following conditions should be met: where p 1-3 represents the probability of error points outside the coordinate axis, and p 1 and p 3 represent the probability of error points on the coordinate axis.
The x-axis and y-axis represented by the positive stroke and negative stroke intersect at the origin, and the corresponding z-coordinate value takes the maximum value of the x-axis and y-axis at the intersection. It will be mentioned in the later analysis that the origin corresponds to no compensation, and taking the maximum probability of positive and negative stroke 0 error at the origin can reduce the probability of error compensation.
In order to provide data support for the construction of digital map, the displacement error test experiment of feed drive system is carried out. The x-axis feed drive system of a horizontal processing center is used as the experimental object, and the laser interferometer is used as the experimental instrument. The laser head was placed and fixed on the outer side of the machine, the interference mirror was placed on the inner side of the machine, and the reflection mirror was fixed on the x-axis and moved with the nut. Adjust the laser head, the interference mirror and the reflection mirror in a straight line. The specific arrangement of the devices is as shown in Fig. 3. The actual moving distance of the feeding shaft was determined by the laser's shooting and reflection distance, and the deviation of the moving distance could

Fig. 2
Construction process of digital map of repeated positioning error be determined by comparing with the theoretical moving distance of the feeding shaft.
Repeated positioning error is related to the statistics and possibility distribution of positioning error. Before digital map drawing, the relationship between repeated positioning error and positioning error is explored. Repeatedly move the feed drive system to measure the positioning accuracy at different locations and repeat positioning accuracy as shown in Fig. 4. By comparing the positioning errors and repeated positioning errors at different positions of the drive system, it can be seen that the repeat positioning errors are positively correlated with the positioning errors. It can be seen from the relative difference curves of repetitive positioning error and positioning error that the positioning error is 2-3 times of the repeated positioning error, and the relative difference generally tends to be stable.
As shown in Fig. 3(a) and (b), the object of this digital map drawing is the feed axis of the horizontal CNC machining center, and a total of 11 digital maps are drawn. Among them, the digital map of 240 mm position and 640 mm position is shown in Fig. 3(c). As for the x-axis and y-axis in Fig. 3(c), a negative value represents that the actual motion  fails to reach the ideal point, a positive value represents that the actual motion exceeds the ideal point, and 0 represents that the actual motion just reaches the ideal point. The Z-axis in Fig. 3 (c) represents the probability of taking the error value. After comparing the two digital maps, it can be seen that the morphology of maps at different locations varies greatly, which may be single peak morphology or multi peak morphology. These conditions also reflect that the probability distribution of repeated positioning error at each position differs.
Repeated positioning error of feed drive system is affected by many factors, such as assembly accuracy, preload, thermal load, feedback control system accuracy, etc. Digital map topography is the result of a combination of factors. Among the many influencing factors, the assembly accuracy of the feed system has the greatest influence on the repeat positioning error. For example, the smaller the standard deviation of the pitch and yaw angles of the guide rail, the smaller the repeat positioning error. Thermal load will increase the assembly error and further affect the repeat positioning error by causing thermal deformation of structural parts. The accuracy of feedback control system determines the upper limit of error, and a high-precision control system is necessary to reduce the repetitive positioning error. In this paper, the object machine tool is equipped with high-precision grating and cooling system, and also has high assembly accuracy, which ensures that the error probability distribution of each experiment tends to be consistent.
In addition to the above factors, the type of feed drive and the parameters of the main corresponding components will also affect the repeat positioning error. The feed drive system studied in this paper is equipped with grating ruler. When the datum of feed system is unchanged and the workshop is at constant ambient temperature, the error of grating ruler will not change due to the change of ambient temperature. At this time, the repeated positioning error is caused by the relative motion error of nut and screw. When the temperature of the processing environment changes constantly, the error of grating ruler changes due to thermal induction error, which changes the measuring datum of the screw, resulting in the change of the measuring result and a new repeated positioning error.
The digital map at 240 mm was extracted for further analysis. As shown in Fig. 5 and Fig. 6, there are also significant differences in the probability distribution of positive travel error and negative travel error at the same position, and the positions where the maximum probability occurs are also different. It shows that it is of necessity to place the positive and negative travel repeated positioning errors on different coordinate axes since it ensures that the probability distributions of the two direction errors will not mutually influence.
The map shape is determined by the error probability of each position. Draw a digital map of the repeated positioning error of each position of the feed axis to form a map database for subsequent calls.

Construction of probability compensation model for repeated positioning error
In actual processing, the position of the feed shaft is constantly changing, and the digital map of different positions is also different, which demands that the search environment should change constantly. Also, the target value of search is also changing continuously. It can be seen that this is a dynamic optimization problem. This paper addresses this problem through the dynamic particle swarm optimization algorithm based on sensitive particles. The flow of the dynamic particle swarm optimization algorithm based on sensitive particles is shown in Fig. 7. The fitness function of dynamic particle swarm optimization algorithm is: Combined with the digital map obtained, the search process of the dynamic optimization algorithm is described in detail in Fig. 8. The change curve in Fig. 8 illuminates the sum of the fitness of position sensitive particles in the positive stroke of feed shaft. In each iteration step, the x-axis coordinates and y-axis coordinates of sensitive particles remain unchanged, and whether the map changes are judged according to the sum of z-axis coordinates of all sensitive particles, namely fitness sum. If the map is not changed, the search process will be skipped, and the optimal value is the results of the last search. If the map changes, the ordinary particles will be reinitialized, and the optimal value will be obtained according to the standard particle swarm search process. All particles will approach the particles with high fitness until all reach the optimal position.
Using the digital map of the repeated positioning error of the 11 positions of the feed axis, combined with the dynamic particle swarm optimization algorithm, the compensation points (i.e., the maximum probability point or the optimal value point) of the positive and negative strokes of each position of the feed axis are obtained, as shown in Fig. 10(a).
The size of the compensation value needs to be set before compensation, and the size of the compensation value depends on the compensation target. The compensation target of this study is that the repeated positioning error after compensation is between ± 1 µm. Since the probability compensation model provided in this paper has probability compensation errors and if the compensation direction is opposite or the error, the compensation will exceed the compensation target, there is a need to control the error Position 160 mm: Positive compensation point is − 1 µm, and negative compensation point is 1 µm. The positive direction compensation is 1 µm, and the error rate is 4%. The negative direction is compensated by 1 µm, and the error rate is 48%, so the negative direction is not compensated.
The difference probability distribution is shown in Table 1 and Table 2.
Position 240 mm: Positive compensation point is − 1 µm, positive compensation is 1 µm, and error rate is 0. The negative compensation point is − 1 µm, the negative compensation is 1 µm, and the error rate is 18%.
Position 320 mm: Positive compensation point is − 3 µm. If the direct positive compensation is 3 µm, the error rate is 40%, and if the compensation is 2 µm, the error rate is 22%, so the compensation is 1 µm, and the error rate is 4%. The forward compensation 1 µm command is executed according to the threshold of out of tolerance probability. The negative compensation point is 0 µm. Thus, the negative direction is not compensated. The positive error distribution is shown in Table 3.
Position 400 mm: Positive compensation point is − 2 µm, positive compensation is 2 µm, and error rate is 20%.   Negative compensation point is − 1 µm, negative compensation is 1 µm, error rate is 4%. Position 480 mm: Positive compensation point is − 1 µm, positive compensation is 1 µm, and error rate is 16%. Negative compensation point is − 1 µm, and the error rate of negative compensation of 1 µm is 44%. Hence, the negative direction is not compensated.
Position 640 mm: Positive compensation point is − 3 µm, positive compensation error rate of 3 µm is 40%, and compensation error rate of 2 µm is 20%. Therefore, compensation is 2 µm. The negative compensation point is − 1 µm, the negative compensation is 1 µm, and the error rate is 14%.
Position 800 mm: Positive compensation point is − 2 µm, and if positive compensation is 2 µm, error rate is 24%. If the compensation is 1 µm and error rate is 5%, positive compensation will be 1 µm. If the negative compensation point is − 3 µm and the negative compensation is 3 µm, error rate is 48%; if the compensation is 2 µm, error rate is 28%; if the compensation is 1 µm, error rate is 12%. Thus, the negative compensation is 1 µm.
The compensation direction and compensation value of each position are shown in Fig. 9 and the probability of compensation out of tolerance at each position is counted, as shown in Fig. 10(b).

Application of probability compensation model
We have obtained a probability compensation model under specific working conditions through the above analysis, which includes the size and direction of compensation value. However, in the process of practical application, the working conditions are constantly changing, that is, the error probability distribution of each position is constantly changing, which requires a digital map model base covering all working conditions. Moreover, different processing requirements make different requirements for the range of repeated positioning error, which requires that the model can adapt to the adjustment of error probability threshold and error range. Combined with  Fig. 11. The so-called "self-learning" refers to that the new repeated positioning error can participate in digital map rendering and update the digital map model library in real time. First, the processing digital map library is constructed, which corresponds to the working condition and the digital map, matches the working condition and calls the corresponding working condition digital map when starting the processing task. Then, the digital map search algorithm is embedded in the control system of the processing center. After the feed axis movement, the control system implements the search of the corresponding digital map, transfers the optimal results to the compensation system, and executes the compensation command with the search value as the compensation value. Finally, after each period of operation, the machine tool needs to test the positioning error of the feed axis, input the new error data into the digital map library, and update the map library so that the compensation system can adapt to the changes of environment and working conditions in real time.
In order to examine the practical applicational effect of the model, the previously obtained probability compensation model is detected by another five groups of repeated positioning error data. The first group of data used for error detection is shown in Table 4.
The compensation data shown in Fig. 9 was applied to the error data in the Table 4. The compensation process and results are as follows.

Positive stroke error compensation process:
Position 0 mm: actual error is 0 µm, and compensation is 0 µm; Position 80 mm: actual error is 0 µm, and compensation is 0 µm; Position 160 mm: the actual error is − 2 µm, the positive direction compensation is 1 µm, the reduced error is 1 µm, and the error after compensation is − 1 µm; Fig. 11 Modeling and application of probability compensation model Table 4 Test data of the first group of models Measuring point (mm)  0  80  160  240  320  400  480  560  640  720  800 Positive stroke error (µm) Position 240 mm: the actual error is − 1 µm, the positive direction compensation is 1 µm, the reduced error is 1 µm, and the error after compensation is 0 µm; Position 320 mm: the actual error is − 3 µm, the positive direction compensation is 1 µm, the reduced error is 1 µm, and the error after compensation is − 2 µm; Position 400 mm: the actual error is − 3 µm, the positive direction compensation is 2 µm, the reduced error is 2 µm, and the error after compensation is − 1 µm; Position 480 mm: the actual error is − 1 µm, the positive direction compensation is 1 µm, the reduced error is 1 µm, and the error after compensation is 0 µm; Position 560 mm: the actual error is − 2 µm, the positive direction compensation is 1 µm, the reduced error is 1 µm, and the error after compensation is − 1 µm; Position 640 mm: the actual error is − 3 µm, the positive direction compensation is 2 µm, the reduced error is 2 µm, and the error after compensation is − 1 µm; Position 720 mm: the actual error is − 3 µm, the positive direction compensation is 1 µm, the reduced error is 1 µm, and the error after compensation is − 2 µm, which exceeded the maximum preset error; Position 800 mm: the actual error is − 2 µm, the positive direction compensation is 1 µm, the reduced error is 1 µm, and the error after compensation is − 1 µm.

Negative stroke error compensation process:
Position 0 mm: actual error is 0 µm, and compensation is 0 µm; Position 80 mm: actual error is 0 µm, and compensation is 0 µm; Position 160 mm: actual error is 1 µm, and compensation is 0 µm; Position 240 mm: actual error is 1 µm, the negative direction compensation is 1 µm, the increased error is 1 µm, and the compensated error is 2 µm, exceeding the maximum preset error; Position 320 mm: actual error is 0 µm, and compensation is 0 µm; Position 400 mm: actual error is − 1 µm, the negative direction is compensated by 1 µm, the error is reduced by 1 µm, and the error after compensation is 0 µm; Position 480 mm: actual error is 0 µm, and compensation is 0 µm; Position 560 mm: actual error is 0 µm, the negative direction is compensated by 1 µm, the error is increased by 1 µm, and the error after compensation is 1 µm; Position 640 mm: actual error is − 1 µm, the negative direction is compensated by 1 µm, the error is reduced by 1 µm, and the error after compensation is 0 µm; Position 720 mm: actual error is − 2 µm, the negative direction is compensated by 2 µm, the error is reduced by 2 µm, and the error after compensation is 0 µm; Position 800 mm: actual error is − 2 µm, the negative direction is compensated by 1 µm, the error is reduced by 1 µm, and the error after compensation is − 1 µm.
Based on the above analysis, a group of positive and negative stroke includes 22 measurement data in total, and the compensation command is started 15 times. The original data has a total error of 28 µm. After compensation, the error is reduced by 16 µm, accounting for 57% of the total error. After compensation, the error is increased by 2 µm, accounting for 7% of the total. Among them, there is a rise of error in the negative stroke position 240 mm and the position 560 mm after compensation, and the positive stroke position 720 mm and the negative stroke position 240 mm exceed the maximum preset error after compensation. It can be seen from Fig. 10(b) that the negative stroke position 240 mm and the position 560 mm are both at approximate compensations out of tolerance positions.
In order to briefly describe the compensation process, only the first group of detection data and compensation process are exhibited, and the other four groups of detection data and compensation process are omitted. However, the methods involved are consistent. The remaining four groups of measurement data are analyzed according to the above method, and the compensation results are shown in Fig. 12.
It can be seen from Fig. 12 that the error fluctuation is significantly reduced after compensation. There are 88 sets of data in the rest of the four groups of experiments, and the compensation command is started for 60 times. A total of 121 µm errors were found in the original data. After compensation, the total error was reduced by 114 µm, accounting for 94% of the reduced error. The error increased by 4 µm, accounting for 3% of the increased error. The size of the error of 3 µm remained unchanged, but the direction changed. After compensation, a total of 6 groups of data exceeds the maximum preset error, and the error rate accounts for 7%.
According to the definition of calculating the repeated positioning error of the linear axis in the international standard ISO230-2, the repeated positioning error before and after compensation is calculated respectively based on the above five groups of measured data and compensated data. The calculation process is as follows.
The feed shaft moves back and forth for five times in a certain direction, and the error X i, j of the jth movement to P i is where P i , j are the actual arrival position of the feed axis, and P i is the target position of the feed axis.
Average position deviation X ave_i of feed axis at P i is The reverse clearance B i at position P i is Among them, the symbol ↑ represents positive motion and ↓ represents negative motion.
The unidirectional standard uncertainty S i of the position P i is further obtained as Fig. 12 Error comparison before and after compensation model application The unidirectional repeated positioning error R i at position P i is Finally, the bi-directional repeated positioning error R at position P i is obtained as The repeated positioning error of each point of the feed axis before and after compensation is shown in Fig. 13.
It can be seen from Fig. 13 that the repeated positioning error after compensation is generally lower than that before compensation. The compensation effect of different positions is also significantly different, among which 160 mm, 480 mm and 640 mm positions have the most apparent compensation effect and reduce the repeated positioning error by more than 1 µm. The main reason for this difference is that the error digital map obtained at different positions is different, combined with the control of the maximum preset error and out of tolerance probability, resulting in different amplitude and direction of different position compensation.

Conclusions
The difficulty to tackle this error is rooted in the randomness of repeated positioning error. In the past, the repeated positioning error suppression method of feed shaft is mainly to adjust the assembly sequence repeatedly. This method is characterized by blindness and tendency to damage part. In view of the above problems, the probability compensation research of repeated positioning error is carried out. Combined with the random distribution characteristics of error and dynamic optimization algorithm, the ideal error of different positions is determined in the form of probability, and further compensation is conducted. Based on the results and analysis, the following conclusions can be drawn: (1) A description method of repeated positioning error of feed shaft is proposed. Based on a large number of random errors, we draw a digital map of repeated positioning errors. With the error and probability as coordinates, the possibility of each error is displayed, which lays a data basis for error compensation. (2) The probability compensation model of repeated positioning error of feed shaft is constructed. We extract the digital map and use the dynamic optimization algorithm to search the peak value and its coordinates of the map. Furthermore, we determine the size and direction of the compensation value according to the actual compensation demand. (3) A dynamic modeling method of probability compensation model for repeated positioning error is described.
We build a digital map model base in which new errors automatically participate in digital map rendering and update the map base in real time, thus realizing the dynamic update of the model.