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.

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.