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In this paper, considering a braided continuum soft-robot, whose radialdeformation is constrained but elongation is assumed, a quasi-Lagrangian modelis proposed that meets the Lagrangian models properties, including a well-posedinput matrix. Actuation is considered throughout three inner pressure cambers,and torsional effects are neglected. The closed-form analytical model isobtained using a scalar varying mass density field, previously neglected in theliterature, which produces on one hand a varying center of mass, whichgenerally does not lay in the backbone curve, and one the other hand acoordinate-dependent inertial tensor. The Lagrangian approach enforces thebasic skew symmetric property, thus exhibiting passivity. The advantage ofdealing with all these effects together display the following distinctfeatures: \textit{i)} the Lagrangian soft-robot dynamic model is similar to theLagrangian rigid-robot case; \textit{ii)} the non-linear system is affine inthe control input; \textit{iii)} the continuum deformable body stands for asegment of constant curvature, when interconnected with other segments ofdifferent constant curvature each, would leads to a quasi-continuum$n$-segments variable curvature soft-robot, yet preserving the aforementionedprevious features of one segment. \redc{Representative simulations and videosare shown, in open- and closed-loop.

On the Modelling of Soft-robots as Quasi-Continuum Lagrangian Dynamical Systems with Well-posed Input Matrix