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UBC Theses and Dissertations

Spline interpolation and contour error pre-compensation for 5-axis machining Alexander, Yuen


This thesis presents experimentally verified smooth spline interpolation and contour error pre-compensation algorithms developed for 5-axis machine tools. In the smooth spline interpolation algorithm, the cutter location data from the computer-aided manufacturing system is first fitted independently to decouple the relative changes in position and orientation. The tool tip positions are fitted to a quintic B-spline, achieving geometric jerk continuity. Next, the tooltip orientations are fitted to another quintic B-spline in spherical coordinates, achieving geometric jerk continuity and feasible orientations at all points of the spline. The nonlinear relationship between spline parameters and displacement are approximated with 9th order and seventh order feed correction splines for position and orientation, respectively. The 9th order feed correction spline is fit adaptively to minimize fitting error while preserving C³ continuity. The seventh order feed correction spline is optimized to minimize jerk while preserving C³ continuity as well. In the contour error pre-compensation algorithm, the position commands generated in the smooth spline interpolation algorithm, are first fitted to piecewise quintic splines while respecting velocity, acceleration and jerk continuity at the spline joints. The transfer function of each servo drive is kept linear by compensating the disturbance effect of friction with a feed-forward block. Using the analytically represented 5-axis, splined tool path, splined tracking errors and kinematic model of the five-axis machine tool, contouring errors are predicted ahead of axis control loops. The contouring errors are decoupled into three linear and two rotary drives, and the position commands are modified before they are sent to servo drives for execution. The methods developed in this thesis have been evaluated on a 5-axis machining center with a tilting-table configuration, and are directly applicable to other 5-axis kinematic configurations such as spindle-tilting or hybrid configurations. The experiments show improvements in fitting accuracy, reduction in vibrations, reduction in tracking errors, and significant reductions in contour error for five-axis tool paths.

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