Equidistant Tool Path and Cartesian Trajectory Planning for Robotic Machining of Curved Freeform Surfaces

This article proposes an equidistant tool path planning strategy on curved freeform surfaces with the focus on robotic machining tasks. The strategy is based on the arc-length parameterization of the surface and can be considered as a variant of the iso-parametric path planning. This method allows us to directly consider a given Cartesian boundary in the parameter domain. Using an inverse interpolation scheme of the parameterization with cubic splines, the path planned in arc-length coordinates is mapped back in Cartesian coordinates. The equidistance property is also satisfied for larger path intervals and does not depend on the local curvature of the surface. The proposed trajectory planner ensures a constant velocity along the planned path on the freeform surface while limiting the maximum centripetal acceleration. However, general velocity profiles can be included depending on the actual machining application. Exploiting the differential geometric properties of the path and surface allows us to directly determine the orientation and angular velocity of the end-effector along the planned trajectory. The proposed methods are demonstrated with a robotic manipulator on a real curved freeform surface. Note to Practitioners—Robotic manipulators and five-axis computerized numerical control (CNC) machines provide high flexibility for machining processes on curved freeform surfaces, for example, polishing or grinding. For these applications, it is often desired that the distance between two subsequent machining paths is independent of the local geometry and surface curvature. This article proposes a novel method to plan equidistant coverage paths on a curved freeform surface. Therefore, we develop a transformation between the 3-D Cartesian coordinates and the 2-D parameters based on the arc-lengths along the surface. The geometry is given by CAD data or a 3-D scan of the part and is directly processed, making our approach particularly feasible for both large to small batch sizes and even individualized products. To cover an arbitrary area with parallel paths, its boundary—given in Cartesian coordinates—is transformed into arc-length parameters. The coverage path is planned in this parameter space, consisting of parallel, equidistant lines with interconnection paths. The path is transformed back in Cartesian coordinates using an inverse transformation. We plan the motion of the tool along this path, satisfying a desired feed rate depending on the actual machining application. The tool is slowed down when the maximum centripetal acceleration is reached. Considering the geometry of the path and the surface, we can also plan the tool’s orientation and its angular velocity during the manufacturing task. We demonstrate the path and motion planning approach on an experimental setup with a robotic manipulator.