An Analytical Formulation for the Geometrico-Static Problem of Continuum Planar Parallel Robots

Abstract

In this paper, we provide an analytical formulation for the geometricostatic problem of continuum planar parallel robots. This formulation provides to an analytical computation of a set of equations governing the equilibrium configurations. We also introduce a stability criterion of the computed configurations. This formulation is based on the use of Kirchhoff's rod deformation theory and finite-difference approximations. Their combination leads to a quadratic expression of the rod's deformation energy. Equilibrium configurations of a planar parallel robot composed of two hinged flexible rods are computed according to this new formulation and compared with the ones obtained with state-of-the-art approaches. By assessing equilibrium stability with the proposed technique, new unstable configurations are determined.

Singularity Conditions for Continuum Parallel Robots

“…More general assumptions, like the Cosserat ones [3], could We derive here the equations for the 3D problem. Equations for the 2D case can be found in [4].…”

Workspace Computation of Planar Continuum Parallel Robots

“…where V e and V l represent beam's deformation energy and the potential energy of concentrated and distributed loads acting on the beam, computed by approximating derivatives with finite differences (see [25], [26] for their expressions). We assume that the PCPR comprises n beams, as in Fig.…”

“…We want to include several aspects usually of interest for workspace evaluation that stateof-the-art approaches cannot include simultaneously, namely equilibrium stability, singularity identification, strain limits, and external load influence. For this purpose, we decided to adopt the modelling strategy proposed in [25], since it allows us to apply external loads to any robot component, to enforce generic geometric constraints, to assess equilibrium stability, and to identify singular configurations (by the procedure introduced in [26]).…”

“…With this theorem, it is possible to ensure that a computed solution lies in the neighborhood of a previously computed configuration, and to certify the convergence of the numerical method used to find the new IGSP solution. By employing an energy-based modelling strategy combined with finite differences [25], we can obtain an analytical expression of the constants required for the solution certification through the Kantorovich theorem that may be not trivial to obtain with other approaches (i.e. the continuous formulation in [2], which may require the solution of an optimization problem).…”

A Gazebo Simulator for Continuum Parallel Robots

“…Where W g 𝑑 𝑝 and W ext 𝑑 𝑝 are gravity and external wrenches acting on the distal plate. Finally, the section (r 𝐷𝐺 𝑀 |r 𝐼𝐺 𝑀 ) allows to handle both DGM and IGM with the same numerical framework [9]. It constrains either q or ⟨p 𝑑 𝑝 , ea 𝑑 𝑝 ⟩ depending on the problem type.…”