Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions

AbstractAlgorithms are developed, based on topological principles, to evaluate the boundary and “internal structure” of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the “internal” envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in R3, and other forms of geometrical convolution, are briefly discussed

A Novel Geometric Theory of On-Machine Tool Measurement and Practical, Optimal Approaches to Highly Accurate and Efficient On-Machine Measurement

Modern industry trends to smart machining that improves productivity at a low cost. The kernel technology of intelligent manufacturing is the automatic on-machine measurement (OMM). When applying OMM technology to computer numerical control (CNC) machines, in-situ measurement takes place in the machining environment without the need of unloading the tool and the part. However, adverse measurement environment, limitations on the efficiency of data capturing and processing, and diversified measured objects render efficient and accurate OMM very difficult. Holistic solutions are needed to advance OMM technology and therefore many scientific topics are involved. This work primarily focuses on geometric modeling of the on-machine cutting tool measurement and kinematic modeling for the calibration process of both the probe and the machine. On-machine cutting tool measurement often takes place on a laser tool setter. However, the geometry principles of the gauging mechanisms of laser tool setters are complicated and had not been studied before. This dissertation modeled such a gauging mechanism and presented virtual simulations of the measurement processes on laser tool setters based on geometry principles. The virtual simulations can predict and compensate the measurement errors, allowing for accurate tool setter calibration processes in practical situations. For cutting tool measurement, the tool length characteristic curve for measurement of round-insert mills is discovered. The derivation of the tool length characteristic curve was carried out by modeling the geometries of tool length measurement processes on a laser tool setter. Based on this characteristic curve, an accurate and efficient approach to measuring lengths of mills with round inserts and bottom cutting edge wear is proposed. Current techniques for probe calibration and machine calibration assume the impractical situations where either the machine is accurate or the location of the probe is accurately known. To address these drawbacks, the actual kinematic model of a six-axis belt grinding CNC machine with a customized add-on probe is built in this dissertation. Using this model along with a specially designed artifact can facilitate the simultaneous calibration of the probe position and the machine geometry error