Efficient Error-bounded Curvature Optimization for Smooth Machining Paths

JVRB, 15(2018), no. 2.: Automated machining with 3-axis robots requires the
generation of tool paths in form of positions of the tool
tip. For 5-axis robots, the orientations of the tool at
each position needs to be provided, as well. Such a
tool path can be described in form of two curves, one
for the positional information (as for 3-axis machining)
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 the form of polygonal representations of the
position and orientation curves, we generate optimized
versions of the curves in the 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. The outcome of this first step can directly be
applied to 3-axis machining. Based on this first step,
for 5-axis machining 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 the 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 complete tool path in the form of G2-
continuous B-spline curves, where we have one such
curve for 3-axis machining and two such curves defined
over the same knot vector for 5-axis machining.
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