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

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.