Current Lagrangian models have been developed under rigid body considerations, therefore, body deformation properties have not been fully incorporated into these models. Thus, key deformation properties such as density variation, variable CoM, and time-varying inertia tensor are missing. In addition, the highly nonlinear dynamical couplings arising from deformation are also missing in deformable body-based formulations. In this paper, Lagrangian and quasi-Lagrangian models of a non-inertial soft robot are developed under the essential modeling assumption that distance among particles varies. Such non-rigid body assumption introduces deformation properties that lead to a clear description of free-motion deformation dynamics subject to reaction forces. Altogether, our proposal gives rise to sound qualitative mathematical models that incorporate deformation phenomena, with well-posed structural properties. To exemplify quantitatively the usefulness of the proposed models, two simulation scenarios are presented and discussed. In the first one, the soft robot is fixed to the ground, while the second scenario, a soft robot is attached to a 3D free-flying rigid frame.

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