Coupling numerical deformable models in global and reduced coordinates for the simulation of the direct and the inverse kinematics of soft robots

In this letter, we propose a method to combine the Finite Element Method (FEM) with Discrete Cosserat Modeling (DCM) to capture the mechanics and the actuation of soft robots. The FEM is used to simulate the non-linear behavior of the volume of the soft structure while the cable/rod used for the actuation is modeled using the DCM. The two models are linked using kinematic constraints without imposing meshing rules. We demonstrate that both direct and inverse kinematic models can be obtained by quadratic optimization. The originality of this coupling is that the FEM model uses global coordinates (the position of the nodes of its mesh in space) where the Cosserat model uses local coordinates (successive strain values). The coupling of these mechanical models allows to combine the best of each parametrization. On the one hand, FEM allows to capture the behavior of the volume structure of the robot while accounting for its geometry with a complex mesh. On the other hand, the DCM allows efficient modeling of 1D structures such as rods, (concentric) tubes, cables, etc. that are used to deform the volume structure of the soft robots. DCM handles large deformation, torsion and (in)-extensibility and is efficient to compute. Moreover, the approach is compatible with complementarity constraints introduced when modeling contact and friction of the robot with its environment as well as the self-collision.