Adaptive Digital Twin of Sheet Metal Forming via Proper Orthogonal Decomposition-Based Koopman Operator with Model Predictive Control

TL;DR

This paper addresses real-time digital twin construction for deformation-based sheet metal forming by coupling POD-based spatial reduction with a deep Koopman operator to linearize nonlinear dynamics in a lifted space, enabling MPC-based control. An online Recursive Least Squares update to the Koopman control matrix adapts the model to evolving material behavior, maintaining accuracy during operation. Experimental validation on a robotic English wheel demonstrates improved precision in achieving target geometries and shows how online updates mitigate model mismatch that arises from material hardening and process drift. The approach offers a generalizable, interpretable framework for adaptive, computationally efficient digital twins in nonlinear manufacturing systems, with potential extensions to more complex geometries and multi-modal sensing.

Abstract

Digital Twin (DT) technologies are transforming manufacturing by enabling real-time prediction, monitoring, and control of complex processes. Yet, applying DT to deformation-based metal forming remains challenging because of the strongly coupled spatial-temporal behavior and the nonlinear relationship between toolpath and material response. For instance, sheet-metal forming by the English wheel, a highly flexible but artisan-dependent process, still lacks digital counterparts that can autonomously plan and adapt forming strategies. This study presents an adaptive DT framework that integrates Proper Orthogonal Decomposition (POD) for physics-aware dimensionality reduction with a Koopman operator for representing nonlinear system in a linear lifted space for the real-time decision-making via model predictive control (MPC). To accommodate evolving process conditions or material states, an online Recursive Least Squares (RLS) algorithm is introduced to update the operator coefficients in real time, enabling continuous adaptation of the DT model as new deformation data become available. The proposed framework is experimentally demonstrated on a robotic English Wheel sheet metal forming system, where deformation fields are measured and modeled under varying toolpaths. Results show that the adaptive DT is capable of controlling the forming process to achieve the given target shape by effectively capturing non-stationary process behaviors. Beyond this case study, the proposed framework establishes a generalizable approach for interpretable, adaptive, and computationally-efficient DT of nonlinear manufacturing systems, bridging reduced-order physics representations with data-driven adaptability to support autonomous process control and optimization.

Figures (12)

• Figure 1: (a) The proposed framework for the dynamic process DT. (b) System setup for the robotic English wheel.
• Figure 2: (a) Vicon tracker locations on the blank sheet, the red circle indicates the contact point with the wheel at the beginning and the end for each cycle. (b) An example of the final workpiece. (c) Illustration of a cycle of the forming process when one toolpath is applied from the side view. The red dotted curve is the toolpath represented by the trajectory for the end effector to track.
• Figure 3: Illustration of Koopman operator theory. It shows that the evolution in the state space using nonlinear model $f$ is equivalent to the evolution of the observables in the lift space using linear observable $\mathcal{K}$. $g$ is a continuous, differentiable, and reversible function (diffeomorphism) that maps states to observables.
• Figure 4: First six Chebyshev polynomials of the first kind, $T_0(x)$ – $T_5(x)$, over the interval $[-1, 1]$.
• Figure 5: Dimensional reduction of sheet deformation using PCA. (a) The eigenvalue associated with each principal mode quantifies its contribution to the total variance, thereby indicating its importance. (b) Retained principal modes for this work. (c) Comparison between the reconstructed and the true deformation. (d) Reconstruction error on each Vicon location across all samples.