Description of Machine Tool Errors Using Screw Coordinates

Abstract

The kinematic behavior of a mechanism may be modeled using a number of different formalisms. In this paper, the description of machine tool errors is examined using screw coordinates. The six components of error associated with each axis motion, three rotational errors about the three coordinate axes and three translational errors of the origin along the coordinate axes, are shown to be identically the six components of an “error twist,” or screw displacement. This “error twist” is a motor, and the six components of the motor are, in general, functions of the axis position. Thus, the “error twist” is a motor valued function of a scalar variable which describes axis errors in an extremely compact form. Furthermore, this approach allows clearer insights into errors which arise due to the Abbé effect. Using screw-coordinates to describe the machine errors, it becomes clear that the so-called Abbé errors cannot be eliminated in the general case, even if the tool point is collinear with the machine scale. However, the relative importance of these errors can be easily determined since the location of the instantaneous axis of rotation is easily found. Expressions are developed which allow the translational components of the errors to be measured at any convenient points on the body, including differing points for each component, and properly transformed into error motions of any desired point. Finally, a method for determining the total error at the tool point due to the individual error components of each of the members is presented.