Geometrically exact three-dimensional modeling of cable-driven parallel manipulators for end-effector positioning

A geometrically exact analytical model of cable-driven parallel manipulators operating in a three-dimensional space is formulated and discussed in this work. The model aims at in- vestigating the exact positioning of a rigid body in a Cartesian space by a m -cables-driven parallel manipulator taking into account the cables mass, their elasticity and, consequently, the sag effect. The direct kinematic problem is formulated by means of 3(m + 1) nonlinear equations expressing compatibility and equilibrium in the same number of unknowns, namely, 3 m components of the constraint reactions and 3 finite rotations. A novel approach to the solution of the inverse kinematic problem is formulated by adding a set of m target equations to obtain 4 m + 3 nonlinear algebraic equations solved in terms of an equal num- ber of unknowns. The latter are defined by the 3(m + 1) increments of the variables of the direct problem and m additional unknowns, i.e., the increments of the cable unstretched lengths. The elastostatic problem of point mass end-effector is then derived. Consequently, the direct and the inverse kinematic problems are formulated by means of 3 m and 4 m nonlinear equations, respectively, in the equal number of unknowns. Examples with an in- creasing number of cables are considered, showing that, although solving the inverse kinematic problem may imply very slack cables configurations, the proposed approach allows to find a solution.Funding acknowledgements: EU Research Fund for Coal and Steel 2017, Grant number 80068