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.

Suitable representations of dynamical systems can simplify their analysis and control. On this line of thought, this article aims to answer the following question: Can a transformation of the generalized coordinates under which the actuators directly perform work on a subset of the configuration variables be found? We not only show that the answer to this question is yes but also provide necessary and sufficient conditions. More specifically, we look for a representation of the configuration space such that the right-hand side of the dynamics in Euler-Lagrange form becomes [\boldsymbol{I}\; \boldsymbol{O}]{T}\boldsymbol{u}, being \boldsymbol{u} the system input. We identify a class of systems, called collocated, for which this problem is solvable. Under mild conditions on the input matrix, a simple test is presented to verify whether a system is collocated or not. By exploiting power invariance, we provide necessary and sufficient conditions that a change of coordinates decouples the input channels if and only if the dynamics is collocated. In addition, we use the collocated form to derive novel controllers for damped underactuated mechanical systems. To demonstrate the theoretical findings, we consider several Lagrangian systems with a focus on continuum soft robots.