Analysis and control of the underactuation in continuum soft robots: a kinematic independent approach

This thesis addresses the challenges involved in formulating the dynamics of continuum soft robots and their nonlinear control in the underactuated regime. The objectives are twofold: to advance control-oriented modeling approaches for soft robots and to elucidate how the intrinsic underactuation of these systems can be explicitly managed within a model-based framework to ensure guaranteed system performance. Although this work primarily focuses on continuum soft robots, many theoretical findings apply to a broader class of underactuated Lagrangian systems. First, we propose a dynamic modeling approach for holonomically constrained serial soft robots that is inherently recursive and independent of the kinematics used to describe system motion. This formulation enables the immediate derivation of an inverse dynamics procedure with computational complexity linear in the number of degrees of freedom. Since no specific kinematic model is required, the proposed framework represents a first step towards a unified modeling approach for continuum soft robots, analogous to the Newton-Euler method for rigid manipulators. Additionally, it makes the kinematics an additional design parameter for control design. We demonstrate that it is possible to simulate models not handled by popular existing methods. This modeling framework serves—though not strictly necessary—as a foundational layer for subsequent derivations. Furthermore, we distribute an expandable open-source library implementing the inverse dynamics procedure. We then introduce a transformation of the generalized coordinates to bring the dynamics into a collocated form, where each input directly affects one, and only one, equation of motion. This change of coordinates is motivated by the distributed nature of the actuation forces in soft robotics, which generally influence all degrees of freedom. The coordinates that yield the collocated form, referred to as actuation coordinates, allow the system properties to be studied as if the actuation were restricted to only some degrees of freedom. This simplifies control design and analysis and provides direct access to control techniques for other underactuated robots. Furthermore, these coordinates have a physical interpretation, corresponding to the coordinates on which the actuators directly perform work, often making them measurable through proprioceptive sensors. As the kinematics becomes a design parameter, it is crucial to assess which dynamic model is best suited for a specific control objective. We address this challenge by introducing a systematic approach based on nonlinear modal theory, which is independent of both the number of degrees of freedom and the adopted discretization. The motivation for using modes lies in their ability to encapsulate intrinsic properties of the dynamics, much as they do in linear systems. By exploiting the collocated form, we solve the regulation problem, as it has remained an open challenge. Moreover, shape regulation allows to demonstrate the usefulness of the theoretical tools developed in previous chapters. We propose several families of feedback control laws and provide conditions under which they are provably stable, ensuring convergence of the actuation coordinates to the desired set-point. These regulators, along with their performance when derived from different discretization techniques, are experimentally validated on a continuum soft robot platform designed in-house. The experiments rely on findings from previous chapters, providing empirical validation.