Abstract: Based on the analysis of the causes of positioning errors in economical CNC punching machines, a method for positioning error compensation by software is proposed. A mathematical model for positioning error compensation is established and has been successfully applied in practice. This method provides a reference for positioning error compensation in various economical point-and-click CNC machine tools. Keywords: CNC punching machine; positioning error compensation; point-and-click CNC 1 Introduction To meet the growing demand for food, beverage, and chemical product packaging, we developed the JK-20 economical CNC punching machine, solving the problems of low productivity, high material consumption, and easy workplace accidents associated with using ordinary punching machines for sheet metal stamping in the light industry. However, when stamping printed sheet metal on this CNC punching machine, the printing is required to be located at the center of the stamped part, which places high demands on the positioning accuracy of the CNC punching machine. To solve the problem of low positioning accuracy of this economical open-loop CNC punching machine, we analyzed the causes of its positioning errors. Based on the characteristics of the stamping process, we dynamically measured the positioning error of its worktable in advance and used software to compensate for the positioning error, thereby improving its processing accuracy. 2. Causes of Positioning Errors The positioning errors generated by the worktable driven by the power stepper motor through the lead screw mainly include: (1) Errors caused by the nonlinear distribution of the x and y coordinates of the worktable throughout the entire range of displacement due to the lead screw drive. This error has relative stability within a certain period of time, but it will change as the system is put into use for longer periods of time and as the transmission pairs wear. (2) Backlash error generated by the lead screw; (3) Errors caused by the non-perpendicularity of the x and y axis guide rails; (4) Errors caused by step loss and overshoot of the power stepper motor. The causes of step loss and overshoot of the stepper motor are: overload, unsuitable acceleration and deceleration during start-up and stop, and small static locking current value. When selecting a stepper motor, the load torque of the system should not exceed the torque-frequency characteristic line of the stepper motor, and a certain margin should be left. At the same time, appropriate acceleration and deceleration laws and curves and static locking current should be selected. In this way, step loss and overshoot of the stepper motor are rare. Therefore, we only analyze and compensate for the first three positioning errors. [color=#000000][b]3 Positioning Error Compensation[/b] 3.1 Nonlinear Displacement Error Compensation The nonlinear displacement error of the worktable, which is distributed throughout the x and y coordinate directions caused by the lead screw drive, is a constant error over a certain period of time. Therefore, after the worktable is installed and adjusted, the positioning error of the worktable should be dynamically measured and processed in advance. Measurement can be performed using an inductive synchronizer, a digital display device, or a laser measuring device. During processing, error compensation can be performed based on the pre-measured positioning error. Since its nonlinear displacement error curve is basically invariant, we can take a series of discrete points on the x-axis with a certain precision based on the pre-measured positioning error, and establish a table of the compensation values ​​(in pulse equivalents) of the discrete points according to the error, and store it in the computer memory. For example, if the error of a certain point is 2.1 pulse equivalents, we can take the compensation value as 2 pulse equivalents (the compensation pulse equivalent is taken as an integer multiple of the error, that is, the error value (in pulse equivalents) is rounded to the integer). That is, when the worktable moves to this point, the control pulse is reduced by 2 pulses. Theoretically, this can control the table accuracy within 0.5 pulse equivalents. Based on the above compensation principle, the table displacement can be compensated point-by-point. However, point-by-point compensation requires a large amount of storage space in the microcomputer and sufficient processing speed, which poses certain difficulties for systems developed using microcontrollers in practical applications. Considering the point-to-point control characteristic of stamping, only the workpiece positioning points (stamping points) need to be compensated. Based on this stamping characteristic, it greatly facilitates the compensation of positioning errors by microcomputer software. Let the table linear error be shown in Figure 1a, and Figure 1b show the stamping points x1, x2, ..., xk, xk+1... during processing. A nonlinear displacement error compensation table can be established from Figure 1a. [ color=#000000]Figure 1 Forward Feeding Positioning Nonlinear Displacement Error Compensation [color=#000000]When the worktable feeds in the x direction, assuming the starting point is from the absolute origin, and the stamping process is carried out at x1, x2, ..., xk, xk+1, ..., when stamping point x1, the theoretical number of pulses that the microcomputer should issue is: Lx1＝X1／δ where δ——pulse equivalent. Due to the existence of nonlinear displacement error, when the microcomputer issues Lx1 pulses, the feeding in the x direction is not at point x1. From the table, the nonlinear displacement error of point x1 is Ex1 (in pulse equivalent), and the actual number of pulses that the microcomputer should issue is: LB（X1）＝Lx1－Ex1 （2） Due to the error compensation of point x1, the nonlinear error curve from point x1 to point x2 is translated into the dashed line in Figure 1a. Therefore, when stamping point x2, the actual number of pulses that the microcomputer should send is: LB(X2) = (Lx2 - Lx1) - (Ex2 - Ex1) = Lx2 - LB(x1) - Ex2. Similarly, after the error compensation of point X2, the nonlinear error curve from point x2 to point x3 becomes the dashed line in Figure 1a. Therefore, when stamping point x3, the actual number of pulses that the microcomputer should send is: LB(x3) = (Lx3 - Lx2) - (Ex3 - Ex2) = Lx3 - [LB(x2) + LB(x1)] - Ex3 (4) By induction, we can obtain the actual number of pulses that the microcomputer should send when stamping any point xk, that is, the x-direction error compensation formula is: [color=#000000][/color] [color=#000000][font=Arial]where LB(xk) is the actual displacement (in pulses) that needs to be traveled in the x-direction from point k-1 to point k after compensation. Lxk is the theoretical coordinate value of the current point (point k) in the x-direction (distance from the origin). LB(xi) — The actual displacement required from point i-1 to point i in the x-direction before the current point k, after compensation. Exk — The error value corresponding to point xk on the nonlinear error curve in the x-direction. From equation (5), it can be seen that LB(xk) is only the sum of the theoretical coordinate value of the current point, the error value of the current point, and the actual displacement from the previous stamping point to the current point after compensation of all stamping points before the current point. The error value of the current point can be found from the established table. If the point does not exist in the table, it can be obtained by interpolation using the previous and next points in the table. The sum of the actual displacement from the previous stamping point to the current point after compensation of all stamping points before the current point can be stored in an accumulator. Therefore, this formula is very convenient for error compensation. Based on the above principle, the general formula for error compensation in the y-direction can also be derived: Error compensation formulas (5) and (6) are derived under the condition that the worktable is fed and positioned along the positive direction from the absolute origin. However, in actual processing, the worktable is not only fed and positioned along the positive direction from the absolute origin, but also fed and positioned along the negative direction. The general feeding and positioning process is shown in Figure 2. Whether error compensation formulas (5) and (6) are valid under normal circumstances will be discussed below. [color=#000000][/color][color=#000000][align=center]□ Absolute Origin ○ Relative Machining Point ↓ Y Feeding Positioning Direction → X Feeding Positioning Direction Figure 2 Schematic Diagram of Stamping Feeding Positioning Machining Process[/color][/align] As shown in Figure 3a, the worktable starts from the absolute origin and feeds and positions itself along the positive direction. The positioning machining points are x1, x2, ..., xn. After stamping at point n, the material is fed in the y direction, and then from point n, it is fed and positioned along the negative direction. The positioning machining points are xn+1, xn+2, ..., xn+k... When machining point n: [color=#000000][/color] Figure 3 shows the nonlinear error compensation for feeding and positioning. Due to the error compensation at point n, the error curve from point n to point n+1 is shifted to the dashed line in Figure 3b. When feeding and positioning to process point n+1, the actual number of pulses that the microcomputer should send is: [color=#000000][/color] [font=Arial] Due to the error compensation at point n+1, the error curve from point n+1 to point n+2 is shifted to the dashed line in Figure 3b. When feeding and positioning to process point n+2, the actual number of pulses that the microcomputer should send is: [/font] [color=#000000][/color] [font=Arial] By induction, when feeding and positioning to process any point n+k, the actual number of pulses that the microcomputer should send, i.e., the error compensation formula in the x-direction, is: [/font] [color=#000000][/color] [font=Arial][color=#000000] It can be seen that formulas (5) and (6) have universal significance. 3.2 Backlash Error Compensation Due to the gap between the lead screw and nut pair, a backlash error will inevitably occur when the worktable reverses direction, affecting the worktable's feeding and positioning accuracy. The gap between the lead screw and nut pair has two characteristics: (1) It has relative stability, that is, the gap is a constant within a certain range; (2) It increases accordingly with the wear of the mechanical transmission. Therefore, the gap is measured in advance, and the positioning error caused by the backlash is compensated by software using the statistical average value of the backlash. In the software design, only one direction register needs to be designed to determine whether the worktable changes direction. The backlash error of the lead screw and nut is eliminated by not compensating when the direction does not change and compensating once every time the direction changes. 3.3 Positioning Error Compensation Caused by Non-Perpendicularity of the x and y Axes The previous analysis discussed the compensation of the nonlinear displacement error distributed throughout the x and y coordinate directions of the worktable caused by the lead screw drive. The positioning error compensation caused by the non-perpendicularity of the x and y axes is not included. Considering that the geometric error caused by the two axes not being perpendicular is a linear function, its value is: E′xn＝Lyn.α／2 （11） E′yn＝Lxn.α／2 （12） By superimposing E′xn and E′yn into Exn and Eyn in equations (5) and (6), the positioning error caused by the x and y axes not being perpendicular can be eliminated at the same time. The compensation formula for this positioning error is as follows: 4. Conclusion This positioning error compensation method was applied to two CNC punch presses we developed. The stepper motor pulse equivalent of these CNC punch presses is 1/15mm. When punching 75mm printing round boxes and 60mm printing round box lids, the punching speed is 180 times per minute, and the positioning accuracy reaches ±0.10mm, meeting the manufacturer's technical specifications. This positioning error compensation method is simple, reliable, and easy to implement using software programming. Without adding any hardware, it can improve the positioning accuracy of CNC punch presses to a certain extent. After the system has been in use for a certain period, the positioning error of the worktable can be dynamically measured again, the table can be corrected, and the changes in positioning error caused by wear of transmission components can be eliminated. This article uses CNC punching machines as an example, but its application can be extended to error compensation for all CNC machine tools.