A new approach to find gouge free tool positions for a toroidal cutter for Bezier surfaces in Five axis machining

Abstract

We implement and test a multi-point machining tool positioning technique that positions the tool using only a variation on gouge checking. The result is a method that is roughly twice as fast as an earlier method that performed a numerical search to find a tool position with multiple points of contact with the design surface. A GPU implementation provides an additional factor of ten speedup. Verification of the method was done via simulation and machining and measuring physical parts.

Tool positioning in five-axis machining on a tensor product surface using circular rays

“…Figure 3 shows the stage wise process of finding the two points of contact for our vertical and circular ray firing method, as well as illustrating an earlier method for comparison. The toroidal tool is placed over the Bézier surface at T 0 C as the center of the tool and both the points of contact P and Q over the Bézier surface where the tool is supposed to touch the surface without any gouging are shown in the Figure 3 To find the first point of contact P with this method, vertical rays are fired from the surface towards the tool; compare this to the method given by Singh et al [9], in which, the rays are fired from the toroidal surface towards the Bézier surface. After computing the drop distance d, the tool is dropped vertically by the drop distance and touches the Bézier surface at point P giving the location and position of the pseudo-insert.…”

“…To compute the second point of contact, previous approached tilted the tool about the axis of pseudoinsert until the tool touches the surface at Q, as shown in Figure 3(b). This rotation approach either requires solving non-linear transcendental equations [8], or iteratively rotating the Bézier surface about the axis of the pseudo-insert until the surface touches the toroidal cutter at Q [9].…”

“…Over the shadow of the tool a parametric grid with the limits (u min ,u max ) and (v min ,v max ) is created which is then discretized with δu and δv, as shown in Figure 4. Additional implementation details can be found in [15].…”

“…Singh et al [9] omitted the step of using Newton's method and found the first point of contact by just doing a form of gouge checking by firing rays from the tool to the design surface. Singh et al then did a similar form of gouge checking/ray casting, using binary search to find the second point of contact.…”

“…In this work a method is developed that is purely based on gouge checking (ray casting) without using Newton's method to position the tool over the Bézier surface, and without using binary search to find the second point of contact. Our method finds the first point of intersection by firing rays from the design surface to the torus (i.e., in the opposite direction of Singh et al [9]), and then finds the second point of contact by firing "circular rays" from points on the design surface to the torus. While in general, such line/circle-intersect-torus calculations require solving fourth degree equations, in both cases, we choose special rays (both linear and circular) that intersect the torus in at most two locations and result in quadratic equations rather than quartic equations.…”