Minimum zone evaluation of conicity error using minimum potential energy algorithms

Abstract

Current algorithm system of the Coordinate Measuring Machine (CMM) widely adopts the least-squares (LS) method for form error quality inspections. This LS algorithm provides an approximate solution for form error evaluation only, with possible actual error overestimation and further leading some of acceptable mechanical components to be rejected. Moreover, the existing method for the evaluation of conicity error is actually infrequent. Thus, the paper elaborates a heuristic approach using an umbrella-shaped mechanical model for precisely evaluating the minimum zone conicity error based on minimum potential energy principle. The umbrella-shaped mechanical model consists of two coaxial and equal-vertex-angle conical surfaces by a fictitious spring connection. All measured data points are enclosed within two conical surfaces which are mathematically determined by the coordinates of 7 active data points. To allow the error assessment being conducted precisely, the non-linear model provides seven degrees of freedom to handle various possible situations. The shrinking spring reduces simulated mechanical system potential energy, yielding two new coaxial cones with smaller normal separation and new contact points on conical surfaces. The system eventually reaches a stable state with a minimum elastic potential energy. The normal separation between such two conical surfaces is minimum zone of conical form error. A direct searching technique based on the derived minimum zone criterion is demonstrated, and a fast and flexible computational algorithm is also proposed.