Jacobian analysis of a long-span cable-driven manipulator and its application to forward solution

This paper addresses the Jacobian analysis of a long-span cable-driven manipulator. The Jacobian matrix that maps the infinitesimal change of the cable length coordinate to that of the end-effector coordinate is deduced and then employed to numerically achieve the forward solution of cable-driven manipulators in the static state or moving slowly enough to neglect their dynamics. The catenary curve is utilized to account for the cable sag effects. Therefore, determining the Jacobian matrix involves not only the geometrical constraints but also the equilibrium equations of the manipulator. The incremental relationship between the displacement of the end-effector and the cable length variation is derived according to the static equilibrium equations. Numerical examples of the feed supporting system of a large radio telescope show that the method proposed can successfully find the forward solution with a very high precision.