Estimation and Control of the Geometric Error in a Linear Interpolator With Parabola Blending

Given a sequence of G01 codes, a linear interpolator outputs a refined sequence of G01 codes obeying the inequality constraints imposed upon the velocity, acceleration, etc., of the machining tool, and the tracking error, geometric error, etc., between the two sequences. While the output G01 sequence is usually obtained from a continuous motion by sampling along the trajectory by a constant interpolation period, a simple strategy of generating the blending curve between two concatenated line segments under the velocity and axis-wise acceleration constraints of the machining tool, is to use parabolas — trajectories of constant-acceleration motions. This paper considers the estimation and control of the geometric error in such a linear interpolator. Classical model of chord error by approximating parabolas with their contact circles leads to incorrect result on the geometric error, if the latter is taken as the superposition of (i) the error of approximation of the input G01 trajectory by parabolas, and (ii) the chord error caused by sampling along the blended C1-smooth trajectory. By computing the geometric error directly without accumulating the approximation error and the chord error, we realize correct geometric error control by establishing inequality constraints on the accelerations of the motion. This work is supported partially by 2011CB302404, NSFC 10925105, 60821002/F02.