The Calculation of Tool Marks Taking into Account Errors of CNC Machines and Tools

Algorithms for analytic calculations of the surfaces machined by various machine-tools are described in this article. Using the proposed tool mark modeling methods, it is possible to take into account geometrical accuracy of the machine and the tools.  The authors consider specific features of modeling tool marks that occur in cutting and abrasive machining, and propose the equations that can constitute the basis for static and dynamic calculations of the machine-attachment-tool-workpiece system. The expressions for calculating radius vectors of cutting edges of various tools, an algorithm for isolating self-intersections when milling, and algorithms for identifying surfaces that show cutting and abrasive tool marks are presented in the article.  The expressions obtained are applicable to machines with serial kinematics, and are not applicable to machine with parallel kinematics. They can be used to determine the balance of the machine accuracy, identify the factors affecting the accuracy, introduce correction in CNC machines, and perform structural analysis and synthesis of metal-cutting machines. The calculation is based on mathematical tools borrowed from the calculation of kinematics of multi-axis manipulators, where the position and movement of bodies are described by homogeneous coordinates.

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