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A model predictive path control algorithm of single-point incremental forming for non-convex shapes

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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.

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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

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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.

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Authors and Affiliations

  1. School of Mechanical & Mining Engineering, The University of Queensland, Brisbane, Queensland, 4072, Australia

    An He, Michael P. Kearney, Kristian J. Weegink, Chenhao Wang, Sheng Liu & Paul A. Meehan

Authors
  1. An He
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  2. Michael P. Kearney
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  3. Kristian J. Weegink
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  4. Chenhao Wang
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  5. Sheng Liu
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  6. Paul A. Meehan
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Correspondence to An He.

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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

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