As a field-oriented portable three-dimensional coordinate-measuring equipment, laser tracers have a measuring range of up to 20 m, and their measurement accuracy can reach 0.2 μm+0.3 μm/m. Laser tracers are suitable for calibrating geometrical errors in coordinate-measuring machines (CMMs) and machine tools. According to ISO 9000 requirements, measurement uncertainty must be modeled based on the sources of uncertainty and their effects during laser tracer measurements. The most commonly used “Guide to the Uncertainty in Measurement (GUM)” can accurately assess uncertainty in straightforward linear models. Meanwhile, the Monte Carlo method offers significant advantages for complex and nonlinear models that include stochastic processes. To solve two major challenges, i.e., the inability to completely separate individual error sources from other error sources and the difficulty in uncertainty evaluation caused by inconsistent measurement strategies, the uncertainty of the calibrated CMM of the virtual laser tracer multi-station measurement system was evaluated using the adaptive Monte Carlo method by constructing a virtual laser tracer multi-station measurement system and a virtual CMM system.

Virtual prototyping technology was used to create a virtual laser tracer multi-station measurement system model, which can be simulated to model the virtual multi-station testing process under different environmental conditions. The mechanical structures of the laser tracer measurement system, electronic control system, and software system were tightly integrated for the optimization and virtual test simulation of the laser tracer multi-station measurement system. The application of the adaptive Monte Carlo method to evaluate uncertainty presupposes the determination of a measurement model corresponding to the input and output quantities. Using a redundant measurement method, a laser tracer was used to construct a laser tracer multi-station measurement model via a time-sharing transfer station. The CMM was calibrated using a laser tracer multi-station measurement technique. Because the ranging principle of the laser tracer is based on laser interferometric ranging, the effect of temperature change on the laser wavelength must be considered in actual measurements. Therefore, a measurement model was established by considering the maximum permissible error of indication (MPE_E) of the CMM and the error due to the relative interference length provided by the laser tracer. Finally, the uncertainty of the laser tracer multi-station measurement system was evaluated using an adaptive Monte Carlo method. The contribution of the uncertainty of individual error sources to the measurement system was analyzed and quantified.

As shown in Fig. 5, the output quantity approximately obeys the normal distribution, which is consistent with the default normal distribution of the GUM method. When the inclusion probability P=95%, the inclusion factor is 1.96, the measurement uncertainty in the x-direction is 4.2 μm and the inclusion interval is (299.9943 mm, 300.0055 mm), that in the y-direction is 1.7 μm and the inclusion interval is (399.9976 mm, 400.0023 mm), and that in the z-direction is 7.1 μm and the inclusion interval is (-325.0054 mm, -324.9951 mm). The results of the adaptive MCM evaluation show a normal distribution. Based on the results shown in Table 2, the MPE_E of the CMM contributed the most significantly to the measurement uncertainty, with an influence degree of 78.0%, and is the main source of measurement uncertainty in the virtual laser tracer multi-station measurement system. This indicates that the accuracy of the CMM directly affects the reliability of the measurement results. Additionally, the measurement accuracy of the laser tracer and the error introduced by the temperature resulted in a non-negligible uncertainty in the measurement system.

In this study, based on the mechatronics system of a virtual prototype, a virtual laser tracer multi-station measurement system was constructed. By analyzing the measurement uncertainty sources of this virtual measurement system, individual error sources were separated from other error sources and substituted into the established measurement model. The measurement uncertainty of this virtual system was evaluated using the adaptive Monte Carlo method. Experimental results show that, unlike the GUM method for evaluating measurement uncertainty, a more comprehensive and accurate evaluation result, including the actual sampling information, can be obtained using the adaptive MCM. Moreover, this approach does not require one to simplify the measurement model or calculate the sensitivity coefficients. The error introduced by the MPE_E, which is the accuracy index of the CMM, is the main source of uncertainty in this measurement system. The uncertainty evaluation method proposed herein can be similarly used for the error calibration of machine tools and industrial robot ends, which can significantly reduce the cost and provide support for the improvement of measurement efficiency and accuracy, the evaluation and optimization of measurement task solutions, and the application guidance of engineering.