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Automated machining with 5-axis robots require the generation of tool paths in form of positions of the tool tip and orientations of the tool at each position. Such a tool path can be described in form of two curves, one for the positional information and one for the orientational information, where the orientation is given by the vector that points from a point on the orientation curve to the respective point on the position curve. As the robots need to slow down for sharp turns, i.e., high curvatures in the tool path lead to slow processing, our goal is to generate tool paths with minimized curvatures and a guaranteed error bound. Starting from an initial tool path, which is given in form of polygonal representations of the position and orientation curves, we generate optimized versions of the curves in form of B-spline curves that lie within some error bounds of the input path. Our approach first computes an optimized version of the position curve within a tolerance band of the input curve. Based on this first step, the orientation curve needs to be updated to again fit the position curve. Then, the orientation curve is optimized using a similar approach as for the position curve, but the error bounds are given in form of tolerance frustums that define the tolerance in lead and tilt. For an efficient optimization procedure, our approach analyzes the input path and splits it into small (partially overlapping) groups before optimizing the position curve. The groups are categorized according to their geometric complexity and handled accordingly using two different optimization procedures. The simpler, but faster algorithm uses a local spline approximation, while the slower, but better algorithm uses a local sleeve approach. These algorithms are adapted to both the position and orientation curve optimization. Subsequently, the groups are combined into a 5-axis tool path in form of two G2-continuous B-spline curves over the same knot vector.

Efficient Curvature-optimized G2-continuous Path Generation with Guaranteed Error Bound for 5-axis Machining