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 investigating 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, 3m 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 4m+3 nonlinear algebraic equations solved in terms of an equal number 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 3m and 4m nonlinear equations, respectively, in the equal number of unknowns. Examples with an increasing 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.