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.

翻译：细长结构（如杆件）即使在中等外部力（如重力）作用下也常表现出显著的非线性几何变形。这一特性导致其形态变化丰富多样，使其在软体机器人、海底电缆、装饰结等工程设计与应用中极具吸引力。先前研究表明，杆件的自然形状对其变形后的几何形态具有重要影响。因此，在制造和设计杆状结构时，必须考虑杆件的自然形状。本文聚焦于一个逆向问题：能否确定悬挂二维平面杆件的自然形状，使其变形后达到期望的目标形状？我们首先基于平面杆件平衡静力学建立理论框架，该框架可根据目标形状计算平面杆件的自然形状。此外，我们分析了不确定性（如数据噪声）对该理论框架精度的影响。结果表明，该理论框架在处理逆向问题中的不确定性方面存在缺陷，这一点在以往研究中常被忽视。为降低不确定性的影响，我们将平面杆件静力学与伴随方法相结合进行参数敏感性分析，构建了一个能够高效探索设计杆件自然形状并具有更强鲁棒性的学习框架。通过数值模拟验证了该框架的准确性和鲁棒性，为软体机器人及变形结构动画等领域的软结构逆向设计提供了重要见解。