Geometric Computing for 5-Axis Sculptured Surface Machining

Geometric Methods for 5-axis milling are currently a topic of international research efforts. Only rudimentarily are present algorithms capable of utilizing the full movability of the milling tool in order to achieve an optimal manufacturing process, and simultaneously of avoiding undercuttings - which means unwanted collisions of the tool with the workpiece. The problems which naturally arise in this context fall into two groups: local and global ones. The former include the correct characterization of all positions of the milling tool relative to the workpiece, such that locally undercutting is impossible. The criteria which are already implemented and used in practice, often fail to achieve this goal, whereas a more careful geometric investigation shows an elegant and simple solution. A second local problem is to find, among all admissible positions of our tool, an optimal one, which roughly means that the part of the workpiece finished after performing one milling step is as large as possible. From the geometer`s viewpoint global problems are much more interesting: While finishing one part of the workpiece, the tool must not cut into another, thereby destroying its (the tool`s) previous or future work; and also no part of the workpiece should be an obstruction to shaping another more remote one. The collision test in its most general form is very time consuming and our aim is to replace it completely by more sophisticated methods which have their origin in elementary geometric topology. Here the results of the research project "The Geometry of NC Milling" which was funded by the Austrian Science Fund (FWF), are very helpful and it seems promising to pursue further the ways of development shown during work in that project. Our research project "Computational line geometry" which is to be carried out simultaneously, will provide tools and algorithms for efficiently treating problems of line geometry which will naturally occur in connection with 5-axis milling. The final aim of this research project will be the design of a motion planning algorithm for 5-axis milling. Its development and testing in practice will be carried out in cooperation with several institutions which have the necessary equipment and have also contributed to the theoretical foundations of the field. These include Brigham Young University/Utah, Technion/Haifa, and Seoul National University/South Korea.

The research project deals with the development of geometric algorithms for the optimization of CNC machining of sculptured surfaces. The research goal is to optimize 5-axis milling motion planning, both with respect to machining time and surface quality. The results include a differential geometric study of the local contact situation between the milling tool surface and the surface to be machined. This leads to optimal positions of the milling tool and optimal motion directions for the milling motion. Moreover we investigated estimates of the machined strip width and their application to the optimization of the cutting tool motion. For the description of the tool motion we developed a new approach based on subdivision algorithms and active contours. This resulted in a very fruitful interplay between geometry, kinematics, Computer Vision and Computer Aided Geometric Design. The methods developed here have more application areas than just CNC machining of sculptured surfaces. Extensions to the solution of other geometric optimization problems are investigated within the research project P16002-N05.