A cable-driven redundant manipulator (CDRM) composed of quaternion joints has important applications in confined space, including minimal invasive surgery, aircraft parts assembly, environment exploring, and so on. Benefitting from the unique joint characteristic and cable routing, it can achieve a larger workspace with fewer driving modules than traditional universal joint CDRM. However, the positioning accuracy of the end-effector suffers from the lack of joint feedback information and the delay effect of cable driving mechanisms. In this paper, we propose an equivalent sensing method and design corresponding sensors for each quaternion joint, and develop a high precision controller for the whole manipulator. The motion sensing of the quaternion joint is achieved by establishing the kinematics between its bending pose and middle limb joints. To realize real-time estimation, the fitting technique is adopted. To improve the efficiency of path planning, a geometric iterative inverse kinematics approach for quaternion joint CDRM is proposed based on the isosceles trapezoid simplified model. Furthermore, an accurate controller is designed by combining the feedforward gain and modified PID feedback control. Finally, an 8DOF CDRM prototype with four quaternion joints is developed, and the experiment verifies the effectiveness of the proposed method.