Manufacturing of free-form geometries using 5-axis Computer Numerically Controlled (CNC) machining brings challenges in path- and motion-planning as one typically wants to minimize the manufacturing time of the object under consideration, while keeping the machining error within fine machining tolerances that ranges in tens of microns. We propose an optimization-based pipeline that, for a given toroidal and/or cylindrical flat-end cutter, simultaneously optimizes its milling paths together with its local positioning represented by the rotation and tilt functions. The proposed strategy is validated on a variety of benchmark surfaces, with different hyperparameters for the objective function and initial conditions, showing that our results provide high-quality approximations of free-form geometries using by-construction non-colliding motions of the given tool. • Optimize toolpaths, rotation, and tilt together for 5-axis toroidal/flat tools. • Evolution-based framework (CMA-ES) for the multiobjective optimization problem. • Optimizes path distribution to improve accuracy and reduce overall length. • Validated on various benchmarks, including industrial free-form geometries.

We propose a computational framework for motion planning for 5-axis CNC machining of free-form surfaces. Given a reference surface, a set of contact paths on it, and a shape of a toroidal cutting tool as input, the proposed algorithm designs the tool motions that are by construction locally and globally collision-free, and offers a trade-off between approximation quality and tool wear using an optimization-based framework. The proposed algorithm first quickly constructs 2D time-tilt configuration spaces along each contact path, detecting regions that are collision-free. The configuration spaces are then merged into a single time-tilt configuration space to find a global tilt function to control the overall motion of the tool. An initial collision-free tilt function in B-spline form is first estimated and then optimized to minimize the machining error while distributing the tool wear as uniformly as possible along the entire cutting edge of the tool while staying in the collision-free region. Our algorithm is validated on both synthetic free-form surfaces and industrial benchmarks, showing that one can considerably reduce the tool wear without degrading the machining accuracy.

Existing flank milling path-planning methods typically lead to tiny gaps or overlaps between neighboring paths, which causes artifacts and imperfections in the workpiece. We propose a new multi-strip path-planning method for 5-axis flank milling of free-form surfaces which targets G 1 (tangent-plane) continuity of the neighboring strips along shared boundaries. While for some geometries one cannot achieve G 1 continuity and high approximation quality at the same time, our optimization framework offers a good trade-off between machining accuracy in terms of distance error and the G 1 connection of neighboring strips. We demonstrate our algorithm on synthetic free-form surfaces as well as on industrial benchmark datasets, showing that we are able to meet fine industrial tolerances and simultaneously significantly reduce the kink angle of adjacent strips, and consequently to improve the surface finish in terms of smoothness.

CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow one to adapt the tool motion optimally to the surface to be produced. We aim at a high-quality surface finish, thereby reducing the need for hard-to-control post-machining processes such as grinding and polishing. Our work is based on a careful geometric analysis of curvature-adapted machining via so-called second order line contact between tool and target surface. On the geometric side, this leads to a new continuous transition between "dual" classical results in surface theory concerning osculating circles of surface curves and osculating cones of tangentially circumscribed developable surfaces. Practically, it serves as an effective basis for tool motion planning. Unlike previous approaches to curvature-adapted machining, we solve locally optimal tool positioning and motion planning within a single optimization framework and achieve curvature adaptation even for convex surfaces. This is possible with a toroidal cutter that contains a negatively curved cutting area. The effectiveness of our approach is verified at hand of digital models, simulations and machined parts, including a comparison to results generated with commercial software.

We continue the study of rational envelope (RE) surfaces. Although these surfaces are parametrized with the help of square roots, when considering an RE patch as the medial surface transform in 4D of a spatial domain it yields a rational parametrization of the domain’s boundary, i.e., the envelope of the corresponding 2 -parameter family of spheres. We formulate efficient algorithms for G 1 data interpolation using RE surfaces and apply the developed methods to rational skinning and blending of sets of spheres and cones/cylinders, respectively. Our results are demonstrated on several computed examples of skins and blends with rational parametrizations.

Minkowski Pythagorean hodograph (MPH) curves provide a means for representing domains with rational boundaries via the medial axis transform. Based on the observation that MPH curves are not the only curves that yield rational envelopes, we define and study rational envelope (RE) curves that generalise MPH curves while maintaining the rationality of their associated envelopes. To demonstrate the utility of RE curves, we design a simple interpolation algorithm using RE curves, which is in turn used to produce rational surface blends between canal surfaces. Additionally, we initiate the study of rational envelope surfaces as a surface analogy to RE curves.