Method of measuring precession details on a coordinate measuring machine

The article focuses on the development and implementation of an effective methodology for measuring high-precision parts on coordinate measuring machines (CMM). The proposed approach addresses the challenges associated with complex geometry measurements under variable environmental conditions by combining advanced mathematical modeling techniques with adaptive error compensation algorithms. The mathematical foundation is based on the application of tensor formalism in Riemannian space, which allows for more precise model-ing of geometric errors using Christoffel symbols and covariant derivatives of the error po-tential. This approach significantly improves the accuracy of spatial positioning. A refined stochastic error model, incorporating Stratonovich integrals and fractional Brownian motion, provides a more accurate description of random processes occurring in the measurement sys-tem. To enhance overall measurement accuracy, an adaptive correction algorithm is pro-posed, based on Itô stochastic differential equations with Fourier-Bessel series expansion, which ensures efficient compensation of systematic errors. In addition, the measurement tra-jectory optimization is formulated as a variational problem, taking into account holonomic and non-holonomic constraints, enabling the optimal positioning strategy. A thermoelastic deformation model based on sixth-rank tensors and Green’s functions is developed to account for temperature-induced deformations, ensuring the reliability of measurements under ther-mal instability. Experimental verification confirmed the effectiveness of the proposed method-ology, demonstrating a 15-20% reduction in systematic errors, a 10-15% decrease in random errors, and an approximately 20% improvement in the accuracy of measurement uncertainty estimation. The proposed methodology combines theoretical advancements with practical so-lutions, providing a robust tool for improving the accuracy and reliability of coordinate measurements in industrial metrology applications.