Theoretical justification for calculating the absolute coordinates of a tool for dimensional setting of CNC machines to enable untended technologies

Introduction. The key obstacle to the development of “smart” mechanical engineering enterprises and the implementation of untended technologies lies in the “digital divide” (i.e., the gap between fully digitized process planning and the empirical setting of tool coordinates) during the dimensional setting of CNC machine tools. The design contour of pre-production engineering (PPE) has been completely digitized, but tool setup coordinates are still calculated empirically using trial runs because there is no theory describing the behavior of the non-stationary tolerance zone for a batch of workpieces in the absolute coordinate system of the machine. Objective. To theoretically substantiate the existence and uniqueness of the absolute coordinate of a cutting tool, guaranteeing that every workpiece in the batch, starting from the very first one, is machined within tolerance under the influence of multicomponent errors. Methods. The study is based on the systems analysis of technological dimensional chains and elasticity theory. The key methodological approach is axiomatic; four basic axioms are formulated (on the batch of parts, stationarity of the machine coordinate system, non-stationarity of the tolerance zone, and the generation of geometry). These axioms serve as the foundation for rigorous proofs of subsequent theorems. Results. Based on the adopted axioms, a theorem on the existence of a stationary “0–0 boundary” in the CNC coordinate system was formulated and proven for the first time. This boundary serves as a stationary reference datum linking the position of the non-stationary tolerance zone with the rigid machine coordinate system. Its uniqueness as a condition for minimum guaranteed stock removal is proven. An equation for the balance of tolerance and multicomponent errors is derived, which is a fundamental condition for process feasibility. An “accuracy life” criterion is introduced and justified to quantitatively evaluate the safety margin of the technological system and to limit cutting parameters. Scientific novelty. For the first time, the existence of an absolute setting coordinate has been analytically proven based on a system of axioms formalizing the non-stationarity of the tolerance zone and the rigidity of a CNC system. It is shown that the “0–0 boundary” is the only point that guarantees process determinism under parametric uncertainty. Practical significance. The developed theory provides a methodological basis for creating digital twins of operations and intelligent CAM modules that automatically calculate setting parameters. This eliminates trial machining cycles, reduces the time required for the production process, and paves the way for the implementation of untended technologies in small- and medium-scale production.