Spline Algorithm for Modeling Cutting Errors on Turning Centers

摘要

Turned parts on turning centers are made up of features with profiles defined by arcs and lines. An error model for turned parts must take into account not only individual feature errors but also how errors carry over from one feature to another. In the case where there is a requirement of tangency between two features, such as a line tangent to an arc or two tangent arcs, any error model on one of the features must also satisfy a condition of tangency at a boundary point between the two features. Splines, or piecewise polynomials with differentiability conditions at intermediate or knot points, adequately model errors on features and provide the necessary degrees of freedom to match constraint conditions at boundary points. The problem of modeling errors on features becomes one of least squares fitting of splines to the measured feature errors subject to certain linear constraints at the boundaries. The solution of this problem can be formulated uniquely using the generalized or pseudo inverse of a matrix. This is defined and the algorithm for modeling errors on turned parts is formulated in terms of splines with specified boundary constraints.