High Speed Cornering based on Real-Time P-H Curve CNC Interpolators

Abstract: In the present research, a trajectory planning strategy is developed for maintaining the tool positioning accuracy in high speed contouring applications. Real-Time Pythagorean-hodograph (P-H) curve CNC interpolators are used for high speed corner machining. There are large contouring errors around sharp corners when low-bandwidth servo controllers are used. Also backlashes can impose additional errors during tracking a corner. It is possible to construct a P-H curve in the region of sharp corners in order to decrease the amount of cornering error. The over- and under-corner P-H approaches are revised when low-bandwidth servo controllers (such as P-Pl servo control) and high bandwidth servo controllers (such as sliding mode control) are used, respectively. The geometric parameters of the over- and under-corner P-H curve, such as end-points, end-derivatives and correction angles along with the feed rate along the modified tool path are computed by Pattern Search Algorithm in order to reduce the maximum cornering error.
The over-corner P-H curve, constructed for the region of sharp corner, implemented by constant velocity interpolation. Besides, the first deceleration phase of motion is devised by both jerk-limited and quadratic velocity interpolation algorithms. In the under-corner P-H curve approach, the constructed P-H curve is devised by constant velocity interpolation algorithm and the deceleration phases of motion are implemented by jerk-limited algorithm.
The proposed approach introduces one arc-length parametric P-H curve for the whole corner without feed rate fluctuations along the actual tool path. The real-time interpolations of P-H curve are computed accurately and efficiently by the Newton-Raphson method.
The developed algorithm is implemented for symmetrical and unsymmetrical corners with different angles. With respect to sharp tool path, total machining time is increased by a small amount, but the cornering error is reduced to the allowable tolerance limit. The results of simulation, such as the cornering error and the total cornering time, are compared with previously published methods. It has been observed that the developed over corner PH approach, even with the backlash of the feed drive mechanism, can substantially reduce the amount of cornering error for P-PI servo controller with backlash. In addition, it has been observed that the developed under-corner P-H curve approach; even with the backlash of the feed drive mechanism, and guide way friction consideration can substantially preserve the amount of cornering error within the allowable tight tolerance limit.