Advanced path interpolation in high performance motion control systems

In the path following motion control system, the reference tool path of the machine tool is geometric curves predetermined by applications. The reference motion command for servo control system is generated based on the geometric tool path and the feedrate. The command generation module is called path interpolation. It is a classical problem in the motion control system. However, a clear formula is still missing in the literature. In this paper, path interpolation problem for the generic parametric tool path is formulated clearly in a mathematical way. By examining the parametrization of the tool path, the problem turns out to be an initial value problem for the ordinary differential equation system (ODE). From this formula, traditional interpolation algorithms, like digital differential analyzer (DDA) and digital increment algorithm, are easily derived. Numerical computation algorithms, such as the truncated Taylor expansion algorithm, the Runge-Kutta algorithm and the predictor-corrector algorithm, can be utilized. Considering reliability, accuracy and complexity, all those algorithms are analyzed and compared from three perspectives - theoretical comparison based on numerical analysis, simulation and experiments. Although the Taylor expansion algorithms and the predictor-corrector algorithms are widely used in the literature, the Runge-Kutta algorithms are highly recommended from our analysis and comparison. The Runge-Kutta algorithm of order 4 is recommended, if the computation resource is available.