1 Overview

The primary objective of machining is to achieve the optimal balance between precision, cost, and efficiency. To reach this goal, one of the most critical technologies that require urgent research and development is online precision measurement, especially in environments characterized by a variety of products and small batch production. Online measurement technology holds significant importance as it is an essential component of integrated process and measurement systems, playing a crucial role in ensuring part quality and boosting productivity. Internationally, the significance of online measurement has long been recognized, with extensive research and widespread application in industrial settings.

Online measurement of part machining accuracy can be categorized into two scenarios. The first involves directly measuring the machined surface during the process, allowing for the acquisition of the required accuracy index upon completion. This is the ideal situation for online measurement. The second scenario involves measuring the workpiece while it remains installed on the machine tool after the machining process. In ultra-precision machining, thermal deformation significantly impacts accuracy, necessitating constant temperature oil circulation or cooling of the cutting fluid during the process. However, due to the limitations of current 0.01μm sensors, many ultra-precision parts are still measured using traditional offline methods, which often result in costs equal to or exceeding the processing cost itself.

Given these challenges, this paper focuses on the second scenario to develop online measurement capabilities for parts, essentially utilizing the lathe as a coordinate measuring machine. Due to the high precision of submicron ultra-precision lathes, when combined with appropriate measuring instruments, online accuracy measurement becomes feasible. This not only expands the application scope of the machine tool but also addresses part measurement challenges. Today, the trend in machining quality assurance is to integrate online measurement and statistical quality control, aiming to ensure product quality closer to the machining process, thereby reducing post-processing rework. Achieving this requires reliable online measurement with high efficiency and accuracy to enable timely decisions and necessary compensations. Therefore, studying online measurement technology for machining accuracy holds great practical significance.

2 Error Source Analysis Affecting Online Measurement Accuracy

The purpose of online measurement is to verify whether the machined part meets the required precision standards. If it does, the part is removed; otherwise, compensation is applied until the desired accuracy is achieved. Accurate measurement of machining accuracy requires equipment with at least one order of magnitude higher precision than the part being measured. In ultra-precision machining, the environment for online measurement is similar to the machining environment. Ensuring measurement accuracy relies on error compensation. While uncorrected online measurements may not meet the 10 times rule, compensated measurements can achieve sufficient accuracy to be meaningful. The lathe, used as a coordinate measuring machine, is affected by sensor accuracy, measurement strategy, and data processing techniques. Advanced design features such as air bearings, granite structures, and precise temperature control help reduce thermal deformation errors. The main error sources affecting measurement accuracy are machine tool geometric errors, totaling 21 items—six per moving part and three between axes. Identifying these errors is crucial for achieving high-precision online measurements. Some error sources have minimal impact, such as δ(x), γ(x), δY(z), γ(z), δY(Ф), and α(Ф). Other errors like βXZ do not affect measurement accuracy and are included in other parameters.

List of 21 Lathe Error Sources

No. Symbol Geometry Meaning No. Symbol Geometry

1 δx(x) X-axis positioning error 12 γ(z) Z-axis pitch error

2 δy(x) X-axis Y-axis straightness error 13 δx(Ф) Spindle X-bounce

3 δz (x) Z axis straightness error 14 δy (Ф) Spindle Y traverse

4 α(x) X axis rollover error 15 δz (Ф) Spindle Z direction gyration

5 β(x) X axis yaw error 16 α ( Ф) Spindle rotation angle X error

6 (x) X-axis tilt error 17 β (Ф) Spindle rotation Y rotation error

7 δz(z) Z-axis positioning error 18 γ(Ф) Spindle rotation around Z rotation error

8 δx(z) Z X axis straightness error 19 αz Main spindle and Z axis parallelism error

9 δy(z) Z axis Y linearity error 20 βx Main spindle and X axis squareness error

10 α(z) Z axis rollover error 21 βxz X axis sum Z axis perpendicularity error

11 β(z) Z axis yaw error

3 Error Source Identification and Modeling

Lathes primarily process cylindrical surfaces, end faces, tapers, and spherical parts. Without considering spindle rotation errors, only X-direction error compensation is needed for cylindrical surface measurements. End face measurements require Z-direction compensation, while taper and spherical surfaces need both X and Z-direction compensation. Error compensation models must be two-dimensional. Capacitive sensors can also compensate for spindle rotation errors. There are typically two methods for identifying error compensation: offline identification using homogeneous coordinate transformation rules, or direct measurement of the machined surface. The first method is time-consuming and affected by modeling assumptions, while the second is limited to specific parts. This paper combines both approaches to improve accuracy and save time.

3.1 Error Source Identification

1 X-direction error compensation amount identification:

Six error sources affect X-direction measurement accuracy: δX(z), αZФ, β(z), α(z), δX(x), and β(x). The X-direction error compensation amount is expressed as:

δX=δX(x)+δX(z)+αZФ.z+β(z).WZ+α(z).Wh+β(x).TZ (1)

Where z is the distance from the origin of the Z-slide; WZ is the distance of the measuring point from the suction cup in the Z direction; TZ is the distance from the Z-sensor to the center of gravity of the X-slide; Wh is the vertical distance from the center of gravity of the measuring point on the workpiece to the Z-slide center. By turning a cylindrical surface in the Z direction and measuring the busbar error, four error sources can be identified: δX(z), αZФ, β(z), and α(z). The error compensation amount is given by:

Δ-column X=δX(z)+αZФ.z+β(z).WZ+α(z).Wh (2)

Figure 1 Cylindrical busbar error measurement

From equations (1) and (2), δX(x) and β(x) need to be identified. δX(x) is ensured by the lathe’s full closed-loop system, while β(x) is identified offline using a laser measuring device. Thus, the X-direction error compensation amount can be calculated as:

Δx=f(x,z)=δ Column X+β(x).TZ (3)

2 Z-direction error compensation identification:

The error components affecting Z-direction measurement accuracy include δz(x), βxФ, α(x), β(x), β(z), and δz(z). The error compensation amount is expressed as:

Δz=δz(x)+βxФ.x+α(x).Th+β(x).Tx+β(z).WX+δz(z) (4)

When measuring the end face, four error sources affect the accuracy: βxФ, δz(x), α(x), and β(x). The formula is:

Δ plane X=δ(x)+βxФ.x+α(x).Th+β(x).Tx (5)

To identify these, a 200mm diameter end face is turned, and a prism is mounted on the tool post. The tool moves along the Z-direction, and the sensor measures the straightness. Using the least squares method, the angle between the laser and the X slide motion is fitted. The error compensation amount from the workpiece center to the measurement facet is obtained as:

Δ surface Z=δ line (x)+x .βxФ

Multiple measurements are taken to overcome random errors. From equations (4) and (5), β(z) and δz(z) are identified offline, while δz(z) is ensured by the lathe’s full-closed-loop system. The Z-direction error compensation amount is:

Δz=f(x,z)=δ face Z+β(z).TX (6)

3.2 Error Source Modeling

Considering that a BP neural network with enough hidden nodes can approximate any nonlinear function with high accuracy, this paper uses a neural network to build an error compensation model. John C. Ziegert first applied this method but failed to solve the training sample problem. This paper uses MATLAB5.1’s neural network toolbox, making the process convenient. All networks use the BP algorithm with Levenberg-Marquardt optimization. The hidden layer uses the “tansig” function, and the output layer uses the “purelin” function.

1 X-direction error compensation modeling:

The Z position of the slide plate is the input, and δ column X is the output. A neural network A fits the function δ1X=δ column X=f(z). The X position is the input, and β(z).TZ is the output. Neural network B fits δ2X=β(z).TZ=f(x). δ1X is the error compensation model for cylindrical surface and busbar straightness measurement.

Fig. 2 Topological diagram of single input and single output neural network

2 Z-direction error compensation model:

X position x is the input, and δ surface Z is the output. Neural network C fits δ1Z=δ plane Z=f(x). Z position z is the input, and β(z).TX is the output. Neural network D fits δ2Z=β(z).TX=f(z). δ1Z is the error compensation model for planar measurement.

In actual processing, the frequently used area is 100mm×100mm. With a screw pitch of 5mm, the straightness error shows a cycle. To accurately model, the interval is divided into 51×51 points, requiring 2601 training samples. The training sample calculation method is as follows:

X-axis error compensation amount training sample calculation:

When measuring δ-column X, sampling points (δ-column X(zj), zj) with zj=0, 2, 4,..., 100 mm are obtained. Similarly, for β(x).Tz, 51 points are sampled. The training sample for X-direction error compensation is:

δ(xi,zj)=δ2X(xi)+δ1X(zj) xi,zj=0,2,4,...,100(mm)

i,j=0,1,2,...,50;

The same applies for Z-direction error compensation:

δZ(xi,zj)=δ1Z(xi)+δ1Z(zj) xi,zj=0,2,4,...,100(mm)

i,j=0,1,2,...,50;

Thus, the neural network training sample is [(δX(xi,zj), δZ(xi,zj)), (xi,zj)]. A double-input, double-output neural network can fit the online measurement error compensation amount in the lathe machining space. The model excludes the influence of δX(x), δZ(z), and its topological structure is shown in Fig. 3. For spherical surfaces, the error compensation amounts in the X and Z directions at (xi, zj) are:

δX=δX(xi,zj)−δX(xi-1,zj-1) (9)

δZ = δZ (xi, zj) - δZ (xi-1, zj-1) (10)

Fig. 3 Topology diagram of double-input, double-output neural network

4 Online Measurement Experiments and Results Analysis

To validate the model, online measurements were performed on two machined parts using an inductor with 0.01 μm accuracy:

1. Turning a cylindrical surface (busbar 100mm): The online straightness error was 0.26μm (0.10μm after processing). With the spindle mounted on a codewheel, the cylinder error was 0.38μm (0.21μm after compensation), matching 0.40μm on a TAYLOR cylindrimeter.

2. Turning an end face (diameter 100mm): The non-compensated flatness error was 0.80μm, and the offline measurement was 0.78μm. After compensation, the flatness error was 0.12μm. The measurement principle involved adjusting the CNC lathe spindle speed to 60r/min, replacing the tool with a capacitance sensor, and sampling via an optical isolator. The sensor trajectory followed an Archimedean spiral, with the polar equation ρ=-(θ+π)/2π, θ ∈ [-π, -97π]. Converting to rectangular coordinates allows calculating the flatness error. Sampling 481 points with a 0.1s interval ensures accurate error compensation. The sampling interval can be adjusted based on actual needs.

From the results, the online measurement accuracy after error compensation is satisfactory. If spindle rotation errors are also compensated, the accuracy improves further. However, without high-precision probes, online measurement of spherical surfaces is not possible. With high-precision probes and software, the lathe can function as a high-precision coordinate measuring machine.

5 Conclusion

The measurement results demonstrate that the neural network-based error compensation model proposed in this paper effectively improves online measurement accuracy, is simple to implement, and is practical. A well-designed training sample covers the entire machining space, enabling automated online measurement. Sensors can be stored on the tool holder, automatically switching to measurement mode during machining. The goal of manufacturing technology is to ensure machining accuracy entirely through the machine tool, integrating machining and measurement. Online measurement plays a vital role in quality assurance and productivity. Although challenges remain, such as developing high-precision sensors and optimizing data processing strategies, online measurement technologies will continue to evolve and offer promising applications in the future.

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