Unlike traditional robots, soft robots can navigate narrow and complex environments while interacting safely and compliantly with their surroundings. The potential of these abilities motivates the development of algorithms that can accurately assess environmental interactions, making soft robots more autonomous. There is growing interest in estimating external loads in soft robots by matching precise shape data with mechanics models. This approach aims to enhance the robots’ ability to adapt to unpredictable forces and environments, improving real-time decision-making and responses. In this article, we propose a novel approach for simultaneous shape and force estimation of tendon-driven slender soft robot-based solely on tendon displacements and tensions. Our approach introduces a kinetostatic model for slender soft robots based on the geometric variable-strain method, builds upon the Cosserat rod theory. The length of the threadlike actuators (tendons) is defined as a function of the deformation of the soft robot. By solving the inverse problem using this kinetostatic model and the length and tension of the tendons, we can estimate both the external force and the shape of the soft robot simultaneously. We validate our approach with an experimental prototype, achieving accurate force and shape estimation results. This method efficiently integrates actuation and sensing in a small and integrated form, which can be highly beneficial for applications requiring compact designs.

Soft robotic systems pose a significant challenge for traditional modeling, estimation, and control approaches, primarily owing to their inherent complexity and virtually infinite degrees of freedom (DoFs). This work introduces an innovative method for dynamically estimating the states of tendon-actuated soft manipulators. Our technique merges the Geometric Variable-Strain (GVS) approach with a kinematic formula that links the length variation of tendons to the deformations of the manipulator and a nonlinear observer design based on state-dependent Riccati equation (SDRE). In our methodology, the soft links are represented by Cosserat rods, and the robot's geometry is parameterized by the strain field along its length. Consequently, its infinite dimensions can be described by utilizing multiple degrees of freedom, depending on the required precision. This enables us to estimate the states (pose and velocity) of tendon-actuated soft manipulators solely based on tendon displacements and actuator forces. Through simulation, we demonstrate the convergence of our estimation method across various DoFs and actuator numbers, revealing a trade-off between the number of DoFs and required actuators for observing system states. Furthermore, we validate our approach with an experimental prototype of 25 cm in length, achieving an average tip position error during dynamic motion of 1.79 cm-less than 7% of the overall body length.

While soft robots offer advantages in adaptability and safe interaction, their modeling remains challenging. This paper presents a novel, data-driven approach for model order reduction of slender soft robots using autoencoder-parameterized strain within the Geometric Variable Strain (GVS) framework. We employ autoencoders (AEs) to learn low-dimensional strain parameterizations from data to construct reduced-order models (ROMs), preserving the Lagrangian structure of the system while significantly reducing the degrees of freedom. Our comparative analysis demonstrates that AE-based ROMs consistently outperform proper orthogonal decomposition (POD) approaches, achieving lower errors for equivalent degrees of freedom across multiple test cases. Additionally, we demonstrate that our proposed approach achieves computational speed-ups over the high-order models (HOMs) in all cases, and outperforms the POD-based ROM in scenarios where accuracy is matched. We highlight the intrinsic dimensionality discovery capabilities of autoencoders, revealing that HOM often operate in lower-dimensional nonlinear manifolds. Through both simulation and experimental validation on a cable-actuated soft manipulator, we demonstrate the effectiveness of our approach, achieving near-identical behavior with just a single degree of freedom. This structure-preserving method offers significant reductions in the system degrees of freedom and computational effort while maintaining physical model interpretability, offering a promising direction for soft robot modeling and control.