Singularity Conditions for Continuum Parallel Robots

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.…”

“…Once a solution of Eq. ( 6) is identified, its stability, and its singularities can be evaluated by the method presented in [26]. The Jacobian matrix J of Eq.…”

“…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]).…”

“…Our workspace-evaluation algorithm is based on the iterative solution of the inverse geometrico-static problem (IGSP) [26]. In order to certify the existence of a unique solution of the IGSP at each step of the workspace computation algorithm, and thus certify that all the computed configurations are characterized by the same working mode, we propose to apply the Kantorovich theorem.…”

“…Finally, we propose an adaptive-grid workspace-exploration algorithm. At each step of the workspace algorithm, the grid is refined in case the solution of the IGSP is not certifiable, so that a refined grid is used only where necessary (in contrast to the fixed grid flooding algorithm of [26]): the overall computation time is therefore reduced. The algorithm is then employed to identify stable and unstable workspace regions, and to detect singularities that determine workspace boundaries as in [26].…”