Integrating linearized inverse eigenvalue method and grey relational analysis for optimizing a kirigami-inspired micro-CMM probe

Micro-coordinate measuring machines (micro-CMMs) are essential for high-precision dimensional metrology in micro-device manufacturing, but their tactile probes often suffer from dynamic instability due to small size and high compliance. This paper proposes a two-stage optimization framework for a kirigami-inspired micro-CMM probe that integrates the Linearized Inverse Eigenvalue (LINE) method with Grey Relational Analysis (GRA). In the first stage, LINE is used as an inverse eigenvalue tool to tune three key geometric parameters (sheet thickness, hinge width, hinge length) so that the fundamental natural frequency is shifted to a higher, target range without resorting to trial-and-error design sweeps. In the second stage, GRA aggregates three competing performance metrics – natural frequency, tip compliance, and maximum von Mises stress – into a single Grey Relational Grade to identify the best compromise design. Finite element analysis (FEA) shows that the optimized probe achieves a 41% increase in fundamental frequency (from 512 Hz to 725 Hz) and a 22% reduction in maximum stress (from 425 MPa to 330 MPa under a 5 mN load), while maintaining adequate compliance for low-force probing. Quasi-static strain-gauge experiments on the fabricated probe confirm the predicted deformation patterns and directional compliance, supporting the proposed design methodology. The two-stage LINE–GRA framework provides a systematic and generalizable approach for designing high-performance compliant mechanisms in precision micro-metrology.