The measuring accuracy of coordinate measuring machines (CMMs) will be affected by the different geometrical and dynamic errors, including the deviations associated to the axis displacement, the working table and the part to be measured. This work is focused on the analysis of the influence of the position errors, repeatability errors and reversibility errors in 3-axis FXYZ coordinate measuring machines, and it will be developed by a numerical model that is known as EE-based stochastic model. This model implements a new error index that is named equivalent error (EE), which will integrate the totality of machine errors of the CMMs and will allow a global description of all these error sources by means of a unique error parameter. The results obtained by this numerical model have been compared with the application of a traditional method, and it was probed that the EE-based model makes possible an increase of a 13.29% in the linear modelling of the performance of CMMs from the machine errors considered in this work, which implies a relevant improvement for the analysis and description of the effect of the distinct error sources on the achievable measuring accuracy of CMMs. For this reason, the EE-based model will be of special interest for industrial applications such as the quality control to be applied inside the production systems dedicated to manufacture mechanical components of high dimensional accuracy.

This research presents a robust multivariate statistical framework for the non-destructive prediction of geomechanical state parameters in quartz-rich coastal sands through acoustic emission (AE) monitoring. Granular media under symmetrical compressive stress function as complex natural systems, where microscopic energy dissipation—arising from particle rearrangement and grain microcracking—radiates as transient elastic waves. To decode these stochastic processes, 24 confined uniaxial compression tests were conducted across diverse soil typologies and moisture contents (0–12%). A high-dimensional data matrix was constructed, integrating 13 geotechnical variables with 48 acoustic descriptors formulated through three distinct temporal aggregations: stage-specific, history average and weighted history average. The statistical results identify the logarithmic effective vertical stress (${\log }_{10}\left({{\mathsf{\sigma }}^{\prime }}_{\mathrm{v}}\right)$) and the cumulative axial strain ($\mathsf{\epsilon }$) as the most significant geomechanical drivers, exhibiting Pearson correlation coefficients $\left|p\right|$ ≥ 0.85 with acoustic activity. In the acoustic domain, the analysis reveals that Signal Strength (${\mathrm{S}}_{\mathrm{s}}$) and cumulative energy ($\mathrm{E}$) flux are the most reliable predictors for volumetric deformation, while the amplitude ($\mathrm{A}$), b-value ($\mathrm{b}$), and average frequency ($\mathrm{F}$) emerge as critical indicators for identifying the transition between spatial rearrangement and the onset of grain fragmentation. Furthermore, the inclusion of dimensionless parameters, particularly earliness ($\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}$), enhances model stability by standardising waveform symmetry across varying stress regimes. High-order polynomial regression models (up to the third degree) were derived, demonstrating that the statistical complexity of acoustic signatures allows for the high-fidelity inference of the soil matrix’s initial and state parameters. This methodology establishes a unified mathematical architecture for the in situ characterisation of granular skeletons, balancing computational efficiency with predictive power in intricate geological domains.