Multi-Point Positioning Control of Small Plate-Like Flexible Objects Using FEM-Based Constrained QP

Precise multi-point positioning during robotic assembly of small, thin, plate-like flexible components is essential for applications like electronics manufacturing, yet challenging due to large elastic deformations, strong coupling effects, and control-induced oscillations. To address these challenges, this paper presents a control method integrating an efficient finite-element method (FEM) kinematic model with constrained quadratic programming (QP). The FEM kinematics, derived using a corotational formulation and static condensation, yields a state-dependent deformation Jacobian matrix (DJM). This DJM is then utilized by the QP controller to systematically minimize multi-point positioning errors while respecting actuator limits. To further enhance robustness against oscillations arising from model uncertainties or unmodeled dynamics, the method incorporates an online module for detecting and attenuating vibrations using adaptive damping and smoothing. Simulations with a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$20 \times 5\times 0.25\ \mathbf{mm}^{3}$</tex> polyimide plate demonstrate reliable convergence, achieving a final root-mean-square error below 0.01 mm for three-, five-, and seven-point tasks and outperforming a conventional Jacobian pseudo-inverse baseline. The method therefore offers an efficient, high-precision approach to automated assembly involving underactuated flexible objects.