  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.
