Analytical Calculation of the Position Loop Gain for Linear Motor CNC Machine Tool

Abstract

One of the most important factors which influence on the dynamical behavior of the linear motor servo drives for CNC machine tools is position loop gain or Kv factor. From the magnitude of the Kv-factor depends tracking or following error. In multi-axis contouring the following errors along the different axes may cause form deviations of the machined contours. Generally position loop gain Kv should be high for faster system response and higher accuracy, but the maximum gains allowable are limited due to undesirable oscillatory responses at high gains and low damping factor. Usually Kv factor is experimentally tuned on the already assembled machine tool. This paper presents a simple method for analytically calculation of the position loop gain Kv. A combined digital-analog model of the 4-th order of the position loop is presented. In order to ease the calculation, the 4-th order system is simplified with a second order model. With this approach it is very easy to calculate the Kv factor for necessary position loop damping. The difference of the replacement of the 4-th order system with second order system is presented with the simulation program MATLAB. Analytically calculated Kv factor is function of the nominal angular frequency  and damping D of the linear motor servo drive electrical parts (motor and regulator), as well as sampling period T. :The influence of nonlinearities was taken with the correction factor. Our investigations have proven that experimentally tuned Kv factor differs from analytically calculated Kv factor less than 10%, which is completely acceptable

Linear Electric Motors in Machining Processes

“…In general, a high static and dynamic stiffness can be achieved by raising the controller gains [22,30].…”

“…The most important variable that describes the behavior of a position control loop for computer numerically controlled (CNC) machine tool driven by a linear motor servo drive is position loop gain K v [30], which is the ratio of the command velocity (feed rate) v to the position control deviation (following error, tracking error, lag) ∆x, i.e.,…”

“…In general, the position loop gain K v should be high for faster system response and higher accuracy, but the maximum gains allowable are limited due to undesirable oscillatory responses at high gains and low damping factor [30]. Usually, the K v factor is experimentally tuned on the already assembled machine tool or can be calculated with the aid of the following equation [30]:…”

“…where ω is the angular frequency, D is the dimensionless damping constant (different than D v in eqns (1) and (16)) , ζ is the position control loop damping (0 < ζ < 1), and T s is the sampling period. In practice, the K v must be reduced up to 60% of that given by (19) due to the presence of nonlinearities [30].…”