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