A multi‐cable suspension manipulator moves a heavy payload through a large workspace by several spatially arranged cables with winches mounted on a crane system. All actuators are independently controllable to achieve a desired position and orientation of the payload with six degrees of freedom in space. As the payload is not fully constrained by the length of the cables, undesired load sway occurs. To transition the payload between predefined stationary equilibrium points, a feedforward control concept based on the input shaping method is presented. For the determination of the natural frequencies and damping ratios required for the design of an input shaper, a nonlinear dynamic model is linearised around a varying equilibrium point by an efficient procedure. The occurring kinematic actuator redundancy is resolved by optimisation. Experimental results from the prototype three‐cable suspension manipulator Cablev confirm the effectiveness of the approach.