The thesis focused on the development and analysis of a distributed parameter model that apply to a class of an autonomous hyper-redundant slender robotic systems interacting with the environment. The class of robotic devices that will be implemented based on the modelling in this thesis, is intended to be autonomously deployed in unknown, unstructured environments, in which it has to accomplish different missions by being able to robustly negotiate unknown obstacles and unpredictable and unmodelled irregularities. Therefore the mechanical models presented here are inspired by some features of a class of organisms - millipedes and centipedes - that possess many of these capabilities. Specifically, these organisms posses flexible slender bodies whose shape morphs according to the curvature of the terrain on which they operate, and possess a highly redundant system of legs that couple the body with the terrain providing propulsion for forward or backward motion, with the high number of legs ensuring a robust distributed contact even on very irregular substrates. The mechanical model that naturally captures the structure of millipede bodies is the Timoshenko beam, which is therefore adopted here. Moreover, the coupling with the environment is modeled by a system of compliant elements, that provides a distributed support analogous to the one exerted by millipedes' legs; such support provides a distributed force that in a control framework is treated as the actuation for shape morphing, so that the body of the system deforms according to the curvature of the substrate. By using a Lagrangian mechanics approach, the evolution of the system is described in a suitable product Hilbert space, in which rigid body degrees of freedom and deformations are coupled. This formulation allows to pose a distributed parameter control problem in which shape morphing and locomotion are dictated by the interaction with the substrate, which in this case is approximated as rigid (that is, the profile of the substrate is not affected by the interaction with the system). Additionally, by modeling the material response of the substrate with a simple linear viscoelastic model, we pose an estimation problem in which, by measuring deformations and/or stresses on the body represented by the beam, we can infer the material properties of the substrate. In this case, the overall coupled system is modelled as a beam on a multi-layer viscoelastic foundation. Predictions of this sensor model are in good agreement with published results, suggesting that the system can be used in a versatile way as an autonomous agent operating in a generic environment, and simultaneously as a sensor that could inform the action of the system itself, or that could be used to monitor the environment. The modeling work done in this study opens the possibility for the implementation in engineering systems applied to environmental monitoring and health applications, in which we envision the system to be used to estimate material properties of living tissues, that can be correlated to the diagnosis of classes of diseases.

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