Auto-triangulation and auto-trilateration. Part 1. Fundamentals

Accuracy of a machine tool limits the dimensional accuracy of parts machined on it; thus, the errors in machine tools need to be monitored and controlled. Conventional methods of observing machine tool errors use high precision artifacts, accurate displacement transducers, or a combination of both. Such systems often require high capital investment, trained technicians, and long set-up times. Further, the maintenance of these systems raises even more difficult problems. Therefore, it is desirable to have an error observation technique whose measurement accuracy has reduced dependency on the accuracy of the instruments used. In this work, a machine tool error identification scheme using an inaccurate linear displacement transducer with well-known and much-used trilateration (or triangulation) technique is provided. Because of the inaccuracy of the transducer, the trilateration technique alone would not be enough to accurately estimate the errors. However, accurate estimation of such errors is achieved by exploiting redundancy and transitivity relationships that enable successive information exchange between machine tool and transducer across machine workspace with the use of a strategically constructed error calibration map. In this part of the work, the proposed scheme is investigated in two-dimensional space, and the mathematical basis for auto-triangulation and auto-trilateration is developed.