Abstract
The geometrical error of the formed part is one of the most significant limitations that restricts the widespread application of incremental sheet forming (ISF) in aerospace industry. The geometry of ISF parts is dependent upon the tool path, so its correction can improve the part precision. Previous research has utilized model predictive control approach to achieve this, but the method was restricted to simple convex shapes. In this study, the tool path and the formed shape were parameterized and the analytical models of geometry responses relative to tool perturbations were proposed. Then, a model predictive control algorithm was developed, aiming at reducing the geometrical errors of the parts with complex non-convex shapes in the ISF process. Experimental validation of the developed control algorithm was carried out by forming a complex shape by single-point incremental forming. The results show that the developed control algorithm greatly reduced the geometrical error in the closed-loop process.
Similar content being viewed by others
Abbreviations
- h :
-
Notation is related to horizontal module of the control algorithm
- v :
-
Notation is related to vertical module of the control algorithm
- k :
-
Notation is related to kth step of the ISF process
- ρ :
-
Notation is related to ρth step in the prediction horizon
- i :
-
Notation is related to ith tool path point in a tool path contour
- τ :
-
Notation is related to τth horizontal geometry representation point
- η :
-
Notation is related to ηth vertical geometry representation point
- ∗ :
-
Optimization of the related notation
- \( \hat{\mkern6mu} \) :
-
Predicted value of the related variable
- ¯:
-
Nominal value of the related variable
- w:
-
Reference state of a geometry representation point
- y :
-
Measured state of a geometry representation point
- u v :
-
Tool step-depth between neighboring steps
- u h :
-
Tool step-over between neighboring steps
- c :
-
Tool path contour
- z :
-
Tool path depth
- m :
-
Total number of sampling points on a tool path contour
- r :
-
Radius of the round end of the tool
- α :
-
Wall angle of target shape
- p :
-
Total number of prediction horizon
- n :
-
Total number of steps in an ISF process
- G :
-
Total number of vertical geometry representation points
- S :
-
Total number of horizontal geometry representation points
- n h :
-
Unit normal vector of a tool path contour
- J :
-
Cost of an optimization problem
- λ :
-
Weighting factor in the cost function
- Ω:
-
Boundary of inequality constraint of the cost function
- I :
-
Identity matrix
- ≔ :
-
Definition
- ∀:
-
For all
- ‖·‖2 :
-
ℓ2-norm
- Bold :
-
Vectors or matrices
- Regular:
-
Scalars
References
Lu H, Kearney M, Li Y, Liu S, Daniel WJT, Meehan PA (2015) Model predictive control of incremental sheet forming for geometric accuracy improvement. Int J Adv Manuf Technol 82(9–12):1781–1794
Liu Z, Li Y, Meehan PA (2014) Tool path strategies and deformation analysis in multi-pass incremental sheet forming process. Int J Adv Manuf Technol 75(1–4):395–409
Li Y, Chen X, Liu Z, Sun J, Li F, Li J, Zhao G (2017) A review on the recent development of incremental sheet-forming process. Int J Adv Manuf Technol 92(5–8):2439–2462
Zhai W, Li Y, Cheng Z, Sun L, Li F, Li J (2020) Investigation on the forming force and surface quality during ultrasonic-assisted incremental sheet forming process. Int J Adv Manuf Technol 106:2703–2719. https://doi.org/10.1007/s00170-019-04870-0
Lu H, Liu H, Wang C (2019) Review on strategies for geometric accuracy improvement in incremental sheet forming. Int J Adv Manuf Technol 102(9–12):3381–3417
Wei H, Zhou L, Heidarshenas B, Ashraf IK, Han C (2019) Investigation on the influence of springback on precision of symmetric-cone-like parts in sheet metal incremental forming process. Int J Light Mater Manuf 2(2):140–145
Lu H, Kearney M, Liu S, Daniel WJT, Meehan PA (2016) Two-directional toolpath correction in single-point incremental forming using model predictive control. Int J Adv Manuf Technol 91(1–4):91–106
Lu H, Kearney M, Wang C, Liu S, Meehan PA (2017) Part accuracy improvement in two point incremental forming with a partial die using a model predictive control algorithm. Precis Eng 49:179–188
Fu Z, Mo J, Han F, Gong P (2012) Tool path correction algorithm for single-point incremental forming of sheet metal. Int J Adv Manuf Technol 64(9–12):1239–1248
Micari F, Ambrogio G, Filice L (2007) Shape and dimensional accuracy in single point incremental forming: state of the art and future trends. J Mater Process Technol 191(1–3):390–395
Al-Ghamdi KA, Hussain G (2014) The pillowing tendency of materials in single-point incremental forming: experimental and finite element analyses. Proc Inst Mech Eng B J Eng Manuf 229(5):744–753
Ambrogio G, Costantino I, De Napoli L, Filice L, Fratini L, Muzzupappa M (2004) Influence of some relevant process parameters on the dimensional accuracy in incremental forming: a numerical and experimental investigation. J Mater Process Technol 153-154:501–507
Attanasio A, Ceretti E, Giardini C, Mazzoni L (2008) Asymmetric two points incremental forming: improving surface quality and geometric accuracy by tool path optimization. J Mater Process Technol 197(1–3):59–67
Attanasio A, Ceretti E, Giardini C (2006) Optimization of tool path in two points incremental forming. J Mater Process Technol 177(1–3):409–412
Taherkhani A, Basti A, Nariman-Zadeh N, Jamali A (2018) Achieving maximum dimensional accuracy and surface quality at the shortest possible time in single-point incremental forming via multi-objective optimization. Proc Inst Mech Eng B J Eng Manuf 233(3):900–913
Chang Z, Li M, Li M, Chen J (2019) Investigations on a novel quadratic spiral tool path and its effect on incremental sheet forming process. Int J Adv Manuf Technol 103(5–8):2953–2964
Hirt G, Ames J, Bambach M, Kopp R, Kopp R (2004) Forming strategies and process Modelling for CNC incremental sheet forming. CIRP Ann 53(1):203–206
Fiorentino A, Giardini C, Ceretti E (2015) Application of artificial cognitive system to incremental sheet forming machine tools for part precision improvement. Precis Eng 39:167–172
Fiorentino A, Feriti GC, Giardini C, Ceretti E (2015) Part precision improvement in incremental sheet forming of not axisymmetric parts using an artificial cognitive system. J Manuf Syst 35:215–222
Ambrogio G, Filice L, De Napoli L, Muzzupappa M (2005) A simple approach for reducing profile diverting in a single point incremental forming process. Proc Inst Mech Eng B J Eng Manuf 219(11):823–830
Allwood JM, Music O, Raithathna A, Duncan SR (2009) Closed-loop feedback control of product properties in flexible metal forming processes with mobile tools. CIRP Ann 58(1):287–290
Hao W, Duncan S (2011) Constrained model predictive control of an incremental sheet forming process,” in 2011 IEEE International Conference on Control Applications (CCA), pp 1288–1293
Karelovic P, Putz E, Cipriano A (2015) A framework for hybrid model predictive control in mineral processing. Control Eng Pract 40:1–12
Cychowski M, Szabat K, Orlowska-Kowalska T (2009) Constrained model predictive control of the drive system with mechanical elasticity. IEEE Trans Ind Electron 56(6):1963–1973
Mayne DQ (2014) Model predictive control: recent developments and future promise. Automatica 50(12):2967–2986
Jones E, Oliphant E, Peterson P. SciPy: open source scientific tools for Python, www.scipy.org
Dierckx P (1982) Algorithms for smoothing data with periodic and parametric splines. Comput Graphics Image Process 20(2):171–184
Hao W, Duncan S. Optimization of tool trajectory for incremental sheet forming using closed loop control, pp 779–784
Bambach M (2010) A geometrical model of the kinematics of incremental sheet forming for the prediction of membrane strains and sheet thickness. J Mater Process Technol 210(12):1562–1573
Andersen MS, Dahl J, Vandenberghe L. CVXOPT: a Python package for convex optimization. Available at cvxopt.org. 16 Nov 2019
Creaform. VXelements: 3D software platform and application suite. www.creaform3d.com/en/metrology-solutions/3d-applications-software-platforms
Arun KS, Huang TS, Blostein SD (1987) Least-squares fitting of two 3-d point sets. IEEE Trans Pattern Anal Mach Intell 9(5):698–700
Besl PJ, McKay ND (1992) A method for registration of 3-D shapes. IEEE Trans Pattern Anal Mach Intell 14(2):239–256
Acknowledgments
The authors acknowledge Queensland Government, Boeing Research & Technology–Australia, The University of Queensland, and QMI Solutions for the support and collaboration through the Advanced Queensland Innovation Partnerships Project 2016000418. The first author acknowledges The University of Queensland for financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
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
He, A., Kearney, M.P., Weegink, K.J. et al. A model predictive path control algorithm of single-point incremental forming for non-convex shapes. Int J Adv Manuf Technol 107, 123–143 (2020). https://doi.org/10.1007/s00170-020-04989-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00170-020-04989-5