An analytical quadratic model for the geometric error of a machine tool

In the analysis of geometric error of a machine, errors (especially angular errors) of a joint (axis) are assumed to remain constant with movement along the joint. The assumption, as actual measurements indicate, is not justifiable. Substantial variations can be observed. In this paper an analytical model for the prediction of geometric error of a machine tool is presented. Unlike previously proposed models, this model allows for the variation of errors along the machine's joints (axes). The model, developed using rigid body kinematics, relates the error vector at a point in the machine tool work space to the coordinates of that point by the dimensional and form errors of the individual links and joints of the machine's kinematic scheme. Shape and joint transforms are developed for inaccurate machine elements. An expression is developed for the case where the individual joint errors vary linearly with movement along the joint (or axis). The expression is quadratic. A comparison, made between the errors predicted by the model which allows for variation of individual errors and one that does not, indicates that the higher order terms of the expression make significant contributions to the predicted error. Finally, a method for estimating the coefficients of the model from the error obtained on a workpiece is proposed.