Tool positioning algorithms for improved machining of triangulated surfaces

Abstract

Multipoint 5-axis machining (MPM) with a radiused end mill is ideal for efficient machining of sculptured surfaces. Multipoint tool positioning using radiused end mill cutter is challenging especially when the input part geometry is taken in parametric form. Known methods use optimization techniques to determine the orientation of the tool axis relative to an orthogonal coordinate system. These optimization methods result in higher order transcendental equations that have many solutions, are ill behaved and lack robustness.

This thesis presents a method for multipoint machining of sculptured surfaces that is robust, reliable and can be implemented in an environment that is usable in industry. The method is based on a new concept in which the tool positioning is divided into two steps. In the first step the tool is dropped along a fixed axis onto the part surface and in the second step the tool is rotated onto the part surface in a manner that maintains the first point of contact. This sub-division of the tool positioning method into two steps reduces the complexity of the resulting equations. To further simplify the mathematical model, the surface of the part is modeled as a series of connected triangles. The triangulation allows the piecewise representation of the surface to any desired accuracy, thereby simplifying the toolpath generation equations. The impact of the simplification of the part surface definition is that at each tool position numerous checks involving faces, edges and vertices have to be done. Although the number of computations increases, their complexity reduces.

The proposed method is called, Drop and Tilt Method (DTM). The detailed concept of DTM approach is presented in chapter 3 and the solutions of the mathematical model are obtained using a symbolic algebra package. This implementation is used to develop and test the toolpaths for a number of test parts. These test parts are validated in simulation and by machining of sample parts.

In the next step a new method for numerical implementation of DTM, that does not require a symbolic environment, is developed. The numerical DTM method uses a Bisection algorithm in conjunction with the generalized tool drop algorithm and tilt about pseudo-insert axis algorithm to determine the 5-axis tool position. The numerical DTM method is computationally faster compared to the original analytical DTM method. The complete mathematical model used for numerical implementation of Drop and Tilt method is presented in chapters 4 and 5 along with sample toolpaths developed for test models that are used to validate the proposed model in simulation and with machined samples.

The edge-torus tangency found in the numerical implementation is computationally expensive and thus, an efficient method for edge-torus tangency is developed. The developed edge-torus tangency model is presented in chapter 6 along with the testing and validation of this new edge-torus tangency model.

Overall, the DTM method is computational simple and efficient method for generating the toolpaths for 5-axis machining of the triangulated models. The accuracy of the developed 5-axis tool positioning method is tested in simulation and by actual machining of various test parts using DMG 80-P Hi-Dyn 5-axis tilt-rotary table CNC machine. The simulations and machining results confirm that the method is suitable for industrial use.