Abstract
In terms of CNC machining task, high feedrate and high accuracy are of great significance. In this paper, an integrated jerk-limited method of minimal time trajectory planning with confined contour error (IJLMTPC) is proposed. Firstly, an optimal control method is utilized to obtain the minimal time feedrate profile. To compensate the contour error, it is added into the optimal control framework as a constraint. Based on the high-order transfer functions of every axis servo model, the contour errors constraint is formulated as the function of tracking errors. However, the optimal control problem (OCP) is difficult to be solved. Hence, the OCP is transformed into two convex subproblems. Since the axial jerk elements are included in the contour error constraint, they are no longer considered as dependent constraints in the two subproblems specially. Afterwards, two subproblems are discretized with control vector parameterization. Consequently, two subproblems are more efficient to be solved with nonlinear programming, and the minimal time feedrate is obtained. Secondly, considering that the OCP is solved on the domain of curve parameter, it is inconvenient for real-time interpolation. The resulting feedrate from solving the OCP is transformed into the corresponding profile on time domain, while the limited tangent jerk is imposed on. Hence, the feedrate is smoothed. Finally, to alleviate the fluctuation of feedrate, the real-time interpolation is performed based on the parameter correction b-spline. Two benchmark tool paths are adopted to test the proposed scheme, and the effectiveness is verified.
Similar content being viewed by others
References
Song D, Zhong Y, Ma JW (2020) Look-ahead-window-based interval adaptive feedrate scheduling for long five-axis spline toolpaths under axial drive constraints. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 234(13):1656–1670
Tajima S, Sencer B, Shamoto E (2018) Accurate interpolation of machining tool-paths based on FIR filtering. Precis Eng 52:332–344
Lin M, Lin W, Lin Y, Lee C, Lee J (2019) Smooth Feedrate Planning and Trajectory Generation for Five-axis Tool Path. In: 2019 IEEE 15th International Conference on Control and Automation (ICCA), Edinburgh, UK, vol 2019, pp 405–410
Yang J, Yuen A (2017) An analytical local corner smoothing algorithm for five-axis CNC machining. Int J Mach Tools Manuf 123:22–35
Dong J, Stori JA (2006) A generalized time-optimal bidirectional scan algorithm for constrained feed-rate optimization. J Dynam Syst Measure Cont 128(2):379–390
Bharathi A, Dong J (2016) Feedrate optimization for smooth minimum-time trajectory generation with higher order constraints. Int J Adv Manufact Technol 82(5):1029–1040
Sencer B, Dumanli A, Yamada Y (2018) Spline interpolation with optimal frequency spectrum for vibration avoidance. CIRP Ann 67(1):377–380
Farouki RT, Pelosi F, Sampoli ML (2019) Optimization of corner blending curves. Comput Aided Des 117:102739
Yang Y, Liao L, Yang H, Li S (2020) An optimal control strategy for multi-UAVs target tracking and cooperative competition. IEEE/CAA J Automatica Sinica 99:1–15
Elbanhawi M, Simic M (2014) Sampling-Based Robot motion planning: a review. IEEE Access 2(1):56–77
Tang X, Chen J (2016) Direct trajectory optimization and costate estimation of infinite-horizon optimal control problems using collocation at the flipped legendre-gauss-radau points. IEEE/CAA J Automatica Sinica 3(2):174–183
Ji C, Kong M, Li R (2019) Time-Energy Optimal trajectory planning for variable stiffness actuated robot. IEEE Access 7:14366–14377
Ye P, Zhang Y, Xiao J, Zhao M, Zhang H (2019) A novel feedrate planning and interpolating method for parametric toolpath in Frenet-Serret frame. Int J Adv Manufact Technol 101(5-8):1915–1925
Erkorkmaz K, Chen QG, Zhao MY, Beudaert X, Gao XS (2017) Linear programming and windowing based feedrate optimization for spline toolpaths. CIRP Ann 66(1):393–396
Zhang Q, Li SR, Guo JX, Gao XS (2016) Time-optimal path tracking for robots under dynamics constraints based on convex optimization. Robotica 34(9):2116–2139
Zhang Q, Li SR, Guo JX (2014) Minimum time trajectory optimization of CNC machining with tracking error constraints. Abstract Appl Anal 2014(5):835098
Saleem O, Rizwan M (2020) ILC-Adapted parameter optimization of cross-coupled single-input fuzzy tracking controllers for an X-Y positioning table. J Chin Inst Eng 43(6):519–531
El Khalick MA, Uchiyama N (2011) Discrete-time model predictive contouring control for biaxial feed drive systems and experimental verification. Mechatronics 21(6):918–926
Kornmaneesang W, Chen SL (2020) MPC-Based robust contouring control for a robotic machining system. Asian J Control 2020:1–13
El Khalick MA, Uchiyama N, Sano S (2013) Sliding mode contouring control design using nonlinear sliding surface for three-dimensional machining. Int J Mach Tools Manuf 65:8–14
Kim H, Okwudire CE (2020) Simultaneous servo error pre-compensation and feedrate optimization with tolerance constraints using linear programming. Int J Adv Manufact Technol 109(2):809–821
Du D, Liu Y, Guo X, Yamazaki K, Fujishima M (2010) An accurate adaptive NURBS curve interpolator with Real-Time flexible Acceleration/Deceleration control. Robot Comput Integr Manuf 26(4):273–281
Tsai MC, Cheng CW (2003) A Real-Time predictor corrector interpolator for CNC machining. J Manufact Sci Eng-Trans ASME 125(3):449–460
Jahanpour J, Alizadeh MR (2015) A novel acc-jerk-limited nurbs interpolation enhanced with an optimized s-shaped quintic feedrate scheduling scheme. Int J Adv Manufact Technol 77(9-12):1889–1905
Liu H, Liu Q, Sun P, Liu Q, Yuan S (2017) A polynomial equation-based interpolation method of NURBS tool path with minimal feed fluctuation for high-quality machining. Int J Adv Manufact Technol 90(9):2751–2759
Lu L, Zhang L, Gu Y, Zhao J (2017) A parametric interpolation method with minimal feedrate fluctuation by nonuniform rational basis spline. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231(18):3301–3317
Heng M, Erkorkmaz K (2010) Design of a NURBS interpolator with minimal feed fluctuation and continuous feed modulation capability. Int J Mach Tools Manuf 50(3):281–293
Lei WT, Sung MP, Lin LY, Huang JJ (2007) Fast real-time NURBS path interpolation for CNC machine tools. Int J Mach Tools Manuf 47(10):1530–1541
Zhang Q, Li SR, Guo JX (2012) Smooth time-optimal tool trajectory generation for CNC manufacturing systems. J Manuf Syst 31(3):280–287
Zhang Q, Li SR (2013) Efficient computation of smooth minimum time trajectory for CNC machining. Int J Adv Manufact Technol 68(1-4):683–692
Zhang Q, Li S (2011) Minimum time feed-rate optimization along predefined toolpath with acceleration constraints on each axis. Proceedings of the 30th Chinese Control Conference, (2011):5462–5467
Zhao K, Kang ZJ, Guo XB (2020) Smooth trajectory generation for predefined path with pseudo spectral method. IEEE Access 8:158735–158744
Zhang Q, Gao XS (2016) Practical feedrate optimization for planar high precision contouring. In: 2016 IEEE International Conference on Information and Automation (ICIA), Ningbo, China, vol 2016, pp 1872–1877
Koren Y, Lo Ch-Ch (1991) Variable-Gain Cross-Coupling Controller for Contouring. CIRP Annals 40(1):371–374
Zhao K, Li SR, Kang ZJ (2019) Smooth minimum time trajectory planning with minimal feed fluctuation. Int J Adv Manufact Technol 105(9-12):1099–1111
Wu JC, Zhou HC, Tang XQ, Chen JH (2012) A NURBS interpolation algorithm with continuous feedrate. Int J Adv Manufact Technol 59(5-8):623–632
Dong J, Wang T, Li B, Ding Y (2014) Smooth feedrate planning for continuous short line tool path with contour error constraint. Int J Mach Tools Manuf 76:1–12
Tsai MS, Nien HW, Yau HT (2008) Development of an integrated look-ahead dynamics-based nurbs interpolator for high precision machinery. Comput Aided Des 40(5):554–566
Yang M, Yang JX, Ding H (2018) A high accuracy on-line estimation algorithm of five-axis contouring errors based on three-point arc approximation. Int J Mach Tools Manuf 130-131:73–84
Funding
This work is supported by the National Natural Science Foundation of China under Grant 61573378.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Availability of data and material
The manuscript has no associated data and materials.
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhao, K., Li, S., Kang, Z. et al. Smooth trajectory generation based on contour error constraint and parameter correction b-spline. Int J Adv Manuf Technol 119, 4359–4373 (2022). https://doi.org/10.1007/s00170-021-08367-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00170-021-08367-7