A systematic modeling approach based on skew ray tracing for optical measurement systems using geometrical optics and its applications in a six-DOF geometrical error measurement system for linear stages

Precision manufacturing machines with submicron accuracy are essential for driving innovation in industries and technologies. To meet this requirement, the measurement systems used to calibrate these machines must achieve exceptionally high accuracy. It has been observed that angular errors contribute to additional parallel errors, commonly referred to as Abbe and Bryan errors. To mitigate these issues, researchers have proposed various multi-degree-of-freedom (DOF) geometric errors (GEs) measurement systems based on optical principles. Among these, six-DOF GEs measurement systems typically comprise an interferometer and multiple measurement modules that utilize a collimated laser. The interferometer measures positioning errors using wave optics, while the measurement modules address the remaining five-DOF GEs using geometrical optics. However, despite the maturity of these systems, the mathematical models employed in the latter are often characterized by complex analytical derivations, making it difficult for equipment manufacturers to adapt these systems to their specific equipment. To address this challenge, this paper proposes a systematic modeling approach for multi-DOF GEs measurement systems using geometrical optics. The proposed approach facilitates the derivation of a six-DOF GEs analysis model through numerical methods, thereby eliminating the need for complex symbolic expressions that can impede understanding. The models are first validated through a series of simulations and subsequently applied to a six-DOF GEs measurement system proposed in this paper.