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Abstract and Applied Analysis
Volume 2014, Article ID 835098, 15 pages
Research Article

Minimum Time Trajectory Optimization of CNC Machining with Tracking Error Constraints

1College of Information and Control Engineering, China University of Petroleum (East China), Qingdao 266580, China
2National Center for Mathematics and Interdisciplinary Sciences (NCMIS), Chinese Academy of Sciences, Beijing 100190, China

Received 22 April 2014; Accepted 30 June 2014; Published 20 July 2014

Academic Editor: Chong Li

Copyright © 2014 Qiang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


An off-line optimization approach of high precision minimum time feedrate for CNC machining is proposed. Besides the ordinary considered velocity, acceleration, and jerk constraints, dynamic performance constraint of each servo drive is also considered in this optimization problem to improve the tracking precision along the optimized feedrate trajectory. Tracking error is applied to indicate the servo dynamic performance of each axis. By using variable substitution, the tracking error constrained minimum time trajectory planning problem is formulated as a nonlinear path constrained optimal control problem. Bang-bang constraints structure of the optimal trajectory is proved in this paper; then a novel constraint handling method is proposed to realize a convex optimization based solution of the nonlinear constrained optimal control problem. A simple ellipse feedrate planning test is presented to demonstrate the effectiveness of the approach. Then the practicability and robustness of the trajectory generated by the proposed approach are demonstrated by a butterfly contour machining example.