Cable-driven hyper-redundant robots have been adopted in many fields for accessing harsh and confined environments that may be inaccessible or dangerous for humans. The cable actuation strategy makes the robot hardware safer and increases the robot payload reducing its weight. In this paper, a novel design of a fully actuated cable-driven hyper-redundant robot has been proposed. This solution is a pulleyless design that decreases the mechanical complexity, allowing to have a compact arm diameter and avoid tension losses on the cables during the motion. Three different joint designs have been taken into account and experiments have been carried to study their performances.The kinematics for the n-joint robot has been formulated, and two cable routing optimization methods, based on a genetic algorithm, have been proposed and applied to a five-joint robot.