Abstract
A broadly applicable formulation for identifying the swept profiles (SWP) generated by subsets of a toroidal surface is presented. While the problem of locating the entire SWP of a torus has been extensively addressed in the literature, this rarely addressed problem is of significance to NC machining with non-standard shape of milling tools. A torus, generated by revolving a circle about an axis coplanar with the circle, is made up of inner and outer parts of a tube. The common use of the torus is in a fillet-end mill which contains only the fourth quadrant of a cross section of the tube. However, in the industrial applications the different regions of the torus geometry appear. Especially we can see this on the profile cutters, such as the corner-rounding and concave-radius end mills. Also to the best of our knowledge, the interior of the torus-tube is either neglected or represented by B-spline curves in literature. In case of common milling tool surfaces such as sphere, cylinder and frustum there exists only one SWP in any instance of a tool movement. But, in case of the toroidal surface there exist two sophisticated SWPs and we need to consider only one of them in tool swept envelope generation. Therefore, considering the complexity of five-axis tool motions there is a need not only to distinguish the front from the rear of the cutter but also the exterior from the interior of a tube. This paper presents a methodology and algorithms for analytically formulating the SWP of any sub-set of the torus in five-axis tool motions. By introducing the rigid body motion theory, two moving frames along with a fixed frame are defined. Arbitrary poses of a tool between tool path locations are interpolated by a spherical linear interpolation (slerp) whose effect is a rotation with uniform angular velocity around a fixed rotation axis. For the problem of NC simulation, by using the envelope theory the closed-form solutions of swept profiles are formulated as two-unit vector functions.