The modelling of volumetric errors of machine tools has been widely used by the method of Homogeneous Transformation Matrix (HTM) based on rigid body kinematics analysis for a long time. Without spindle induced errors and thermal errors, to establish a closed-loop HTM for a three-axis machine tool, the total of 21 geometric errors are the primary elements to be known. It is well known that the measured points of translational errors are directly related to the volumetric error at the tool cutting point through rigid body kinematics. In the generalized HTM method, however, this relationship is missing. This report, therefore, proposes a new comprehensive approach to formulate the volumetric errors based on the famous Abbe principle in order to derive the error term in the motion direction, and Bryan principle to derive the error terms in orthogonal to the motion direction. The proposed methodology is simple in concept, rational in physical meaning and easy in implementation. Experimental results, including faster multi-degree-of-freedom error measurements, volumetric error analysis and compensation on a testbed of small machine tool, validate the correctness and effectiveness of this method.

长期以来，基于刚体运动学分析的均质转换矩阵（HTM）方法已广泛使用机床的体积误差建模。在没有主轴引起的误差和热误差的情况下，要为三轴机床建立闭环HTM，总共需要知道21种几何误差。众所周知，平移误差的测量点通过刚体运动学与刀具切削点处的体积误差直接相关。但是，在广义HTM方法中，缺少这种关系。因此，本报告提出了一种新的综合方法，以著名的阿贝原理为基础来计算体积误差，以便得出运动方向上的误差项，并根据Bryan原理得出与运动方向正交的误差项。所提出的方法概念简单，物理意义合理，易于实施。实验结果，包括更快的多自由度误差测量，体积误差分析以及在小型机床测试台上的补偿，验证了该方法的正确性和有效性。