三坐标测量机测量误差分析 Analysis of Measurement Errors for Coordinate Measuring Machines

​作为高精度测量设备，三坐标测量机的测量误差问题一直存在，，为了进一步提高该设备的应用价值，相关学者针对三坐标测量机的各类误差，提出了相应的补偿方法，尽可能的消除各类误差，得到准确的结果。

As high-precision measuring equipment, coordinate measuring machines (CMMs) are inherently subject to measurement errors. To further enhance the application value of such equipment, relevant scholars have proposed corresponding compensation methods targeting various types of errors inherent to CMMs, so as to eliminate these errors to the maximum extent and obtain accurate measurement results.

01-三坐标测量机常见误差类型

01 Common Types of Errors in Coordinate Measuring Machines

在相关理论基础上，三坐标测量机的误差类型可以分为两类，即静态误差、动态误差，其中静态误差的特点在于其误差值会始终保持在稳定水平，而动态误差则会随着存在时间的增长而增加，所以在误差补偿角度上，应当先对两种误差进行了解，再选择相应的方法。下文将介绍三坐标测量机静态误差、动态误差的产生原因以及事例表现。

Based on relevant theoretical foundations, measurement errors of Coordinate Measuring Machines (CMMs) fall into two main categories: static errors and dynamic errors. Static errors feature constant, stable error magnitudes over time, whereas dynamic errors grow as the operating duration elapses. From the perspective of error compensation, it is therefore necessary to first clarify the characteristics of these two error types before selecting corresponding correction strategies. The following sections elaborate on the generation mechanisms and typical manifestations of static and dynamic errors in CMMs.

（1）静态误差

（1）Static Errors

三坐标测量机的静态误差产生原因一般在于：外部因素对设备结构造成了瞬时性影响，此类影响带来的误差因为影响转瞬即逝，所以不容易发生变化，但这一表现不代表静态误差的影响力不大，因为在通常情况下静态误差的误差值要大于动态误差的初始值，乃至动态误差经过一段时间发展后也无法超过静态误差值，所以应当对静态误差保持重视。例如，在三坐标测量机测量当中，其测头测针存在磨损现象，此时就会形成静态误差）

Static errors of coordinate measuring machines (CMMs) generally stem from transient external influences exerted on the machine structure. Since such disturbances are fleeting, the resultant error magnitude remains stable and barely varies over time. Nevertheless, this stable characteristic does not imply static errors exert negligible impacts. In most cases, the magnitude of static errors exceeds the initial value of dynamic errors; even after prolonged operation, dynamic errors cannot grow larger than static errors. Accordingly, sufficient attention shall be paid to static errors. For instance, stylus wear on the CMM probe during measurement will introduce static errors.

（2）动态误差

（2）Dynamic Errors

三坐标测量机动态误差的产生原因有很多，例如温度、灰尘、人工等外在因素，此类误差在大部分情况下都会随着时间的延长而增加误差值，但在特殊情况下会表现出不稳定的动态化表现。例如：在三坐标测量机测量当中，周边的温度、灰尘会随着时间累积而增长，相应引起的测量误差值也会随之增长，这即为动态误差的常规表现；在人工因素下，介于人工不稳定性的特征，其来点测量速度会不平衡，但具体表现却无法预测，由此就形成了不稳定的动态误差表现。

Dynamic errors of coordinate measuring machines (CMMs) originate from numerous sources, including external environmental factors such as ambient temperature and dust, as well as human factors. In most cases, the magnitudes of such errors rise with extended measuring time, yet they may exhibit unstable dynamic characteristics under special working conditions. Two typical cases are illustrated as follows:

During CMM measurement, ambient temperature fluctuation and dust accumulation aggravate over time, which correspondingly increases the resultant measurement error magnitudes — this represents the regular manifestation of dynamic errors. In terms of human factors, operators display inconsistent probing speeds due to inherent human instability, and the resultant error variation is unpredictable, thereby generating dynamic errors with unstable performance.

此外，在静态误差与动态误差的综合角度上，静态误差本身虽然不会因为时间增长而发生变化，但在其他因素条件下，其会出现动态性表现，例如测头测针磨损现象就会在长期应用当中愈发严重，这即为一种动态性表现，针对这一现象，在误差补偿中影响以当前静态误差值为基准来进行计算，以保障计算结果准确性。

Furthermore, from the perspective of coupling static and dynamic errors, static errors themselves remain invariant over time. Nevertheless, they may demonstrate dynamic behaviors under the influence of other factors. A typical example is progressive stylus wear of the probe during long-term equipment operation, which constitutes one such dynamic characteristic. To address this phenomenon, error compensation calculations shall be performed based on the real-time magnitude of static errors, so as to guarantee the accuracy of calculated results.

02三坐标测量机常见误差补偿方法

02 Common Error Compensation Methods for Coordinate Measuring Machines

结合相关理论得知，三坐标测量机误差补偿方法主要有三种，即温度补偿法、软件修正补偿法、测量力误差补偿法，对此下文将进行相关分析。

Combined with relevant theoretical research, it can be concluded that there are three primary error compensation methods for coordinate measuring machines (CMMs), namely temperature compensation, software correction compensation and measuring force error compensation. Relevant analyses of the above methods will be presented in the following sections.

2.1 温度补偿法

2.1 Temperature Compensation Method

温度是引起三坐标测量机误差的重要因素，且在此因素条件下，静态误差与动态误差可能同时存在，所以在测量当中采用温度补偿法是消除测量误差的重要举措。在相关理论当中，温度补偿法分为三个部分，即标温下结构参数标定、温度实时采集、误差系数补偿，下文将对三个步骤的具体内容进行分析。

Temperature constitutes a critical factor inducing measurement errors in coordinate measuring machines (CMMs). Under thermal influences, static errors and dynamic errors may coexist simultaneously. Accordingly, adopting the temperature compensation method during measurement serves as an essential measure to eliminate measurement deviations. Theoretically, the temperature compensation approach consists of three core procedures: calibration of structural parameters under standard temperature, real-time temperature acquisition, and error coefficient compensation. Detailed explanations of the three procedures will be elaborated in the subsequent text.

（1）标温下结构参数标定。在温度引起的静态误差当中，需要先确认正常温度条件下，设备最低误差参数，即三坐标测量机未受温度影响之前，其所有结构的参数数值，确认之后即可得到一个标准的对比参数集，主要用于之后的误差值对比当中。

（1）Calibration of Structural Parameters at Standard Temperature

For static errors induced by temperature variations, it is necessary to first determine the minimum error parameters of the machine under nominal temperature conditions. These parameters refer to the characteristic values of all machine structures before the CMM is subjected to thermal disturbances. After measurement and confirmation, a standard reference parameter set can be established, which is primarily adopted for subsequent comparison of error magnitudes.

（2）温度实时采集。在完成上述参数标定工作之后，需要对当前存在温度误差的三坐标测量机。依照标定参数集中的每个参数项进行实际测量、采集，以得到设备当前的参数集合。之后将参数标定结果与实际采集结果相互对比，即可得到当前三坐标测量机误差与标准参数之间的差距，以供后续补偿调整。

（2）Real-Time Temperature Acquisition

After completing the aforementioned parameter calibration, the coordinate measuring machine (CMM) with current thermal errors shall be measured and sampled item by item in accordance with each parameter contained in the calibrated parameter set to obtain the real-time parameter set of the equipment. By comparing the calibrated reference parameters with the practically sampled parameters, the deviation between the current CMM measurement errors and the standard parameters can be quantified, which provides a basis for subsequent compensation and adjustment.

（3）误差系数补偿。根据上述温度实时采集与标定参数的对比结果，可以采用温度热变形误差公式来进行补偿，在补偿过程当中，要将对比得出的三坐标测量机实际误差与参数误差的比值代入公式当中，通过数值补偿将设备参数调整到与标定参数一致的水平下即可。

（3） Error Coefficient Compensation

Based on the comparison results between real-time temperature sampling data and calibrated structural parameters mentioned above, the thermal deformation error formula of the CMM can be applied for compensation. During the compensation process, the ratio of the actual machine error to the parametric error derived from the comparison shall be substituted into the formula. The equipment parameters can then be adjusted to match the calibrated reference values via numerical compensation.

此外，介于上述分析可见，温度补偿法的应用相对复杂，且需要进行多项对比，那么为了保障该方法应用顺利，建议采用仿真软件来模拟标定参数设备与误差设备，以此得到两个模型，借助模型的直观性可以简化其中计算过程。

In addition, it can be seen from the above analysis that the implementation of the temperature compensation method is relatively complicated and requires numerous comparative calculations. To ensure the smooth implementation of this method, it is recommended to adopt simulation software to model the machine under calibrated parameters and the error-affected machine separately, so as to generate two numerical models. The intuitiveness of these models can greatly simplify the relevant calculation procedures.

2.2 软件修正补偿法

2.2 Software Correction Compensation Method

软件修正补偿法是一种针对三坐标测量机动态误差来进行补偿的方法，在实际应用当中十分常见。具体应用上，介于上述动态误差的温度、环境、人工分析结果可见，动态误差可以被分为两个类型，这两个类型被称为实时性动态误差、非实时性动态误差，那么针对实时性动态误差，在软件修正补偿法当中，直接对现场误差数据进行调整，使其与标准参数一致即可；针对非实时性误差，则利用采集系统来获得设备当前误差系数，再结合标准系数进行矫正即可。

The software correction compensation method is a widely adopted technique dedicated to compensating dynamic errors of coordinate measuring machines (CMMs). As demonstrated by the foregoing analysis on thermal, environmental and human-induced dynamic errors, dynamic errors can be categorized into two types: real-time dynamic errors and non-real-time dynamic errors.

For real-time dynamic errors, this software-based compensation approach directly adjusts on-site error data to align with standard parameters. For non-real-time dynamic errors, the data acquisition system is utilized to acquire the current error coefficients of the equipment, followed by correction in combination with standard coefficients.

2.3测量力误差补偿法

2.3 Measuring Force Error Compensation Method

在三坐标测量机应用当中，如果其测量力存在异常容易导致侧杆弯曲，相应引起误差现象，此类误差属于非实时性动态误差，那么针对此类误差，除了上述提到的软件修正补偿法以外，还可以采用测量力误差补偿法来进行调整。具体应用当中，首先利用仿真软件（或其他建模软件）来建立三坐标测量机的测头、测杆结构模型，其次将当前测量力输入模型当中，在看测杆是否弯曲、弯曲程度，如果存在弯曲，则减弱模型中的测量力，直至测杆不弯曲，且满足测量需求位置，最终将模型测量力参数输入三坐标测量机当中即可起到应有效果。

In the application of coordinate measuring machines (CMMs), abnormal measuring force may cause stylus deflection and the resultant measurement errors. Such errors are categorized as non-real-time dynamic errors. Apart from the aforementioned software correction and compensation method, the measuring force error compensation method can also be adopted for adjustment.

In practical implementation, the procedures are as follows. First, establish structural models for the probe and stylus of the CMM using simulation software or other modeling tools. Second, input the actual measuring force into the model and analyze whether stylus deflection occurs as well as its magnitude. If deflection is detected, reduce the measuring force parameter in the model iteratively until the stylus remains straight while reaching the required measuring position. Finally, import the optimized measuring force parameters from the model into the CMM to realize the error compensation effect.