Continuum soft robots are nonlinear mechanical systems with theoretically infinite degrees of freedom (DoFs) that exhibit complex behaviors. Achieving motor intelligence under dynamic conditions necessitates the development of control-oriented reduced-order models (ROMs), which employ as few DoFs as possible while still accurately capturing the core characteristics of the theoretically infinite-dimensional dynamics. However, there is no quantitative way to measure if the ROM of a soft robot has succeeded in this task. In other fields, like structural dynamics or flexible link robotics, linear normal modes are routinely used to this end. Yet, this theory is not applicable to soft robots due to their nonlinearities. In this work, we propose to use the recent nonlinear extension in modal theory -called eigenmanifolds- as a means to evaluate control-oriented models for soft robots and compare them. To achieve this, we propose three similarity metrics relying on the projection of the nonlinear modes of the system into a task space of interest. We use this approach to compare quantitatively, for the first time, ROMs of increasing order generated under the piecewise constant curvature (PCC) hypothesis with a high-dimensional finite element (FE)-like model of a soft arm. Results show that by increasing the order of the discretization, the eigenmanifolds of the PCC model converge to those of the FE model.

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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.

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Model-based strategies are a promising solution to the grand challenge of equipping continuum soft robots with motor intelligence. However, finite-dimensional models of these systems are inherently inaccurate, thus posing pressing robustness concerns. Moreover, the actuation space of soft robots is usually limited. This article aims at solving both these challenges by proposing a robust model-based strategy for the shape control of soft robots with system uncertainty and input saturation. The proposed architecture is composed of two key components. First, we propose an observer that estimates deviations between the theoretical model and the soft robot, ensuring that the estimation error converges to zero within finite time. Second, we introduce a sliding mode controller to regulate the soft robot shape while fulfilling saturation constraints. This controller uses the observer's output to compensate for the deviations between the real system and the established model. We prove the convergence of the closed-loop with theoretical analysis and the method's effectiveness with simulations and experiments.@en

Soft robots are intrinsically underactuated mechanical systems that operate under uncertainties and disturbances. In these conditions, this letter proposes two versions of PID-like control laws with a saturated integral action for the particularly challenging shape regulation task. The closed-loop system is asymptotically stabilized and matched constant disturbances are rejected using a very reduced amount of system information for control implementation. Stability is assessed on the underactuated dynamic model through the Invariant Set Theorem for two relevant classes of soft robots, i.e., elastically decoupled and elastically dominated soft robots. Extensive simulation results validate the proposed controllers.

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Performing precise, repetitive motions is essential in many robotic and automation systems. Iterative learning control (ILC) allows determining the necessary control command by using a very rough system model to speed up the process. Functional iterative learning control is a novel technique that promises to solve several limitations of classic ILC. It operates by merging the input space into a large functional space, resulting in an over-determined control task in the iteration domain. In this way, it can deal with systems having more outputs than inputs and accelerate the learning process without resorting to model discretizations. However, the framework lacks so far a validation in experiments. This paper aims to provide such experimental validation in the context of robotics. To this end, we designed and built a one-link flexible arm that is actuated by a stepper motor, which makes the development of an accurate model more challenging and the validation closer to the industrial practice. We provide multiple experimental results across several conditions, proving the feasibility of the method in practice.@en

The intrinsically underactuated and nonlinear nature of continuum soft robots makes the derivation of provably stable feedback control laws a challenging task. Most of the works so far circumvented the issue either by looking at coarse fully-actuated approximations of the dynamics or by imposing quasi-static assumptions. In this letter, we move a step in the direction of controlling generic soft robots taking explicitly into account their underactuation. A class of soft robots that have no direct elastic couplings between the dynamics of actuated and unactuated coordinates is identified. Considering the actuated variables as output, we prove that the system is minimum phase. We then propose regulators that implement different levels of model compensation. The stability of the associated closed-loop systems is formally proven by Lyapunov/LaSalle techniques, taking into account the nonlinear and underactuated dynamics. Simulation results are reported for two models of 2D and 3D soft robots.

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