Soft robots' ability to safely navigate complex environments motivates the development of algorithms for accurate environmental interaction assessment, enabling greater autonomy. Specifically, strain-based shape and force estimation of continuum robots with embedded soft sensors poses an open challenge mainly owing to continuous softness, anisotropic deformation, and non-linear properties. Mathematical description of deformable soft bodies and accurate estimation of external forces are crucial for achieving controllable and intelligent behaviors of these robots. In this paper, a kinetostatic strain-based modeling for rod-driven soft robots (RDSR) with embedded stretch sensors is proposed, which incorporates local strains, actuation variables, and external interactions. The strain model enables full shape estimation of the robot and prediction of strain variations in soft bodies. Building on this, we develop a force estimator based on predicted and measured sensor and actuator lengths to evaluate 3D external forces, accounting for both orthogonal and tangential components relative to the backbone. Moreover, we introduce a methodology using a novel ellipsoid representation to handle tangential forces that may become insensitive in certain singular configurations. This estimator allows us to either disregard such forces when they do not influence deformation or estimate them when they become observable. Our simulations and experiments demonstrate how this approach can be used to analyze the robot's configuration and successfully estimate external forces. Finally, it is demonstrated that when the continuum arm follows trajectories with higher strain sensitivity, tangential force estimation is significantly improved.

Soft robots, with their compliant and underactuated nature, pose significant challenges for real-time shape regulation. Practical implementations of these methods often rely on fully-actuated approximations, over-looking the underactuated nature of these continuum structures. This study experimentally validates model-based controllers through collocated control that explicitly address underactuation, incorporating gravity cancellation and elasticity compensation to outperform conventional PD/PID approaches. A new multi-segment soft robot with a passively actuated segment has been designed, enabling experimental validation and providing strong evidence of the controllers’ effectiveness. The work bridges theory and practice, offering a practical framework for real-time shape regulation applicable to diverse soft robotic systems.

Slender soft robots offer significant advantages for real-life applications, particularly in areas that require delicate and adaptable interaction with complex environments. However, their effectiveness and safety can be greatly limited in the absence of sensing capabilities. Hall effect sensors, known for their excellent sensitivity and compact design, offer an innovative solution for equipping soft manipulators with perceptive abilities. In this letter, we propose an optimized sensor-magnet arrangement that can estimate all 3 angular strains of a slender rod, including torsion and bending along orthogonal axes, using a single sensor-magnet pair. With optimized design and experimental data, we trained a neural network to accurately predict angular strains from the measured magnetic fields. Using the predicted strains at different points along the body, we reconstruct the 3D shape of the sensorized manipulator using a Piece-wise Constant Angular Strain (PCAS) model. Two manipulator designs were considered in this work: single-segment and three-segment. Experimental results indicate tip position errors of less than 2% of the total manipulator length for the single-segment soft robot and less than 5% for the three-segment soft robot. The inherent simplicity of our design enables easy scaling and replication while ensuring reliable strain measurements critical for accurate robot shape reconstruction.

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.

Model-based manipulation of deformable objects has traditionally dealt with objects while neglecting their dynamics, thus mostly focusing on very lightweight objects at steady state. At the same time, soft robotic research has made considerable strides toward general modeling and control, despite soft robots and deformable objects being very similar from a mechanical standpoint. In this work, we leverage these recent results to develop a control-oriented, fully dynamic framework of slender deformable objects grasped at one end by a robotic manipulator. We introduce a dynamic model of this system using functional strain parameterizations and describe the manipulation challenge as a regulation control problem. This enables us to define a fully model-based control architecture, for which we can prove analytically closed-loop stability and provide sufficient conditions for steady state convergence to the desired state. The nature of this work is intended to be markedly experimental. We provide an extensive experimental validation of the proposed ideas, tasking a robot arm with controlling the distal end of six different cables, in a given planar position and orientation in space.

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.

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.