Time-optimal CNC tool paths : a mathematical model of machining

Abstract

Free-form surface machining is a fundamental but time-consuming process in modern manufacturing. The central question we ask in this thesis is how to reduce the time that it takes for a 5-axis CNC (Computer Numerical Control) milling machine to sweep an entire free-form surface in its finishing stage. We formulate a non-classical variational time-optimization problem defined on a 2-dimensional manifold subject to both equality and inequality constraints. The machining time is the cost functional in this optimization problem. We seek for a preferable vector field on a surface to obtain skeletal information on the toolpaths. This framework is more amenable to the techniques of continuum mechanics and differential geometry rather than to path generation and conventional CAD/CAM (Computer Aided Design and Manufacturing) theory. After the formulation, this thesis derives the necessary conditions for optimality. We decompose the problem into a series of optimization problems defined on 1-dimensional streamlines of the vector field and, as a result, simplify the problem significantly. The anisotropy in kinematic performance has a practical importance in high-speed machining. The greedy scheme, which this thesis implements for a parallel hexapod machine tool, uses the anisotropy for finding a preferable vector field.

(cont.) Numerical integration places tool paths along its integral curves. The gaps between two neighboring toolpaths are controlled so that the surface can be machined within a specified tolerance. A conservation law together with the characteristic theory for partial differential equations comes into play in finding appropriately-spaced toolpaths, avoiding unnecessarily-overlapping areas. Since the greedy scheme is based on a local approximation and does not search for the global optimum, it is necessary to judge how well the greedy paths perform. We develop an approximation theory and use it to economically evaluate the performance advantage of the greedy paths over other standard schemes. In this thesis, we achieved the following two objectives: laying down the theoretical basis for surface machining and finding a practical solution for the machining problem. Future work will address solving the optimization problem in a stricter sense.