Updating a turning center error model by singular value decomposition

Abstract

The precision of manufacturing using machine tools depends on the accuracy of the relative position of the cutting tool with respect to the workpiece. Kinematic modeling of machine tools is used to describe this relative position. This motion can be modeled by homogeneous coordinate transformation matrices composed of both rotational elements as well as positional offset elements of the associated coordinates. The rotation and translation components of the homogeneous transformations are considered to be functions of nominal tool position and machine temperatures. In general these functions are low order polynomials in terms of position and temperatures. The coefficients are usually calculated with least squares curve fitting techniques. The data for these fits are obtained by measuring actual coordinate positions and temperatures on the machine tool based upon desired programmed nominal coordinates. The process of measuring and modeling the errors of machine axis positions as functions of nominal positions and temperatures is referred to as machine tool characterization. The geometric-thermal models developed through machine tool characterization may not fully predict the errors encountered by a machine tool during machining. The data from machine characterization usually provides the structure to derive the basic form of the equations used to model the various error components used in the homogeneous matrices. This process of model updating involves determining the residual systematic errors of the machine tool and applying an algorithm to update the geometric-thermal model coefficients. The updating algorithm described in this report begins with adding perturbation terms to the characterization coefficients of the geometric-thermal model. These coefficients are estimated by an “inverse” process, using residual systematic errors, determined from part measurements on a coordinate measuring machine. The main tool used in identifying the perturbation terms is called a generalized or pseudo inverse matrix. This matrix is applied to the residual error vector to obtain a “best” approximate solution to the least squares problem.