Design on Odd-even Steps Third Order Approach Interpolation Algorithm for Logarithmic Curve

Logarithmic curve interpolation is essential for precise machining conical cutters with constant rake and spiral angle in CNC machine. In this paper, a novel odd-even steps third order approach algorithm is presented to realize the logarithmic interpolation. In each interpolating cycle there are odd and even steps divided. At the odd step, the interpolating point moves in tangent direction for an instruction interpolation length. The even step is in succession to interpolate same length through the coordinate increments computed by using the Taylor Mean-value Theorem in third order approach algorithm for the interpolation point approaching the ideal curve closely. The third order approach interpolating formulae for calculating coordinate increments are deduced in the results with simplified arithmetic operations that can be quickly computed for real time application. Test results indicate the third order approach algorithm is featured with high accuracy, suitable to be executed by the 32-bit micro-processor in ARM9 embedded CNC system for multi-axis servo control to accurate grind the conical teeth.