Inverse Design of Planar Clamped-Free Elastic Rods from Noisy Data

Abstract

Slender structures, such as rods, often exhibit large nonlinear geometrical deformations even under moderate external forces (e.g., gravity). This characteristic results in a rich variety of morphological changes, making them appealing for engineering design and applications, such as soft robots, submarine cables, decorative knots, and more. Prior studies have demonstrated that the natural shape of a rod significantly influences its deformed geometry. Consequently, the natural shape of the rod should be considered when manufacturing and designing rod-like structures. Here, we focus on an inverse problem: can we determine the natural shape of a suspended 2D planar rod so that it deforms into a desired target shape? We begin by formulating a theoretical framework based on the statics of planar rod equilibrium that can compute the natural shape of a planar rod given its target shape. Furthermore, we analyze the impact of uncertainties (e.g., noise in the data) on the accuracy of the theoretical framework. The results reveal the shortcomings of the theoretical framework in handling uncertainties in the inverse problem, a fact often overlooked in previous works. To mitigate the influence of the uncertainties, we combine the statics of the planar rod with the adjoint method for parameter sensitivity analysis, constructing a learning framework that can efficiently explore the natural shape of the designed rod with enhanced robustness. This framework is validated numerically for its accuracy and robustness, offering valuable insights into the inverse design of soft structures for various applications, including soft robotics and animation of morphing structures.

Figures (9)

• Figure 1: Diagram illustrating the deformation process of an elastic rod with a clamped-free configuration under the influence of gravitational acceleration $g$. The detection and segmentation of the structure are likely to be performed using a finite number of pixel points, as yellow dotted points, with white noise present in the measurement process.
• Figure 2: The schematic of a planar rod under gravity and the forces acting on its element.
• Figure 3: Influence of the chosen surrogate model for expression $\theta(s)$. (a) The designed target shape -- letter "A". (b) The comparison of the deformed shapes computed from the natural shape solved by different fitted models. (c) The comparison between raw data and fitted data from the polynomial regression model with degrees 6, 11, and 16 for $\theta(s)$. (d) The comparison between raw data and fitted data for $\theta'(s)$. Note that the normalized arc length $\Tilde{s} = s/S$.
• Figure 4: Influence of measurement noise on inverse design solutions. (a) Schematic of measuring the target shape, the letter "A," using a sensor (e.g., camera). (b) Impact of measurement error $\sigma$ on the accuracy of different approaches. The theory baseline represents the result obtained from the theoretical framework without added noise; the noise baseline represents the result obtained from the theoretical framework with added noise; the proposed scheme is our proposed optimal method (the learning framework detailed in Sec. \ref{['subsec:learning_framework']}) designed to be robust against uncertainties.
• Figure 5: A randomly generated curve used to validate the effect of varying noise levels, characterized by the standard deviation $\sigma = 0.002$ (unit: m). The deformed shapes (green) are computed from the natural shape (blue) obtained by the fitted model. (a) Ground truth. (b) Predicted shape through optimal inverse design from noisy data. (c) Predicted shapes through inverse solver from noisy data.