According to the complexity of motion modeling and low control accuracy of the continuum robot due to its flexibility and redundancy of freedom, a dynamic modeling method of the cable-driven continuum robot with nonlinear friction was proposed. Firstly, the kinematics model of the continuum robot was derived to analyze the relationship between the end pose and joint parameters, and the velocity kinematics model was presented for the convenience of mechanical analysis. Then, a dynamic model with considering of the inertial force, elastic force, gravity force and driving force, was established based on the Kane’s method to analyze the mapping relationship between its motion and the force applied to the robot. Moreover, the force transmission characteristics of the cable-pulley system were analyzed by the improved Capstan equation, and a feedforward control strategy based on the established motion model and the nonlinear friction was presented to compensate the loss of the transmission force. Finally, the simulations and motion experiments were carried out to verify the established model and the proposed control method of the continuum robot. The results showed that the established dynamic model with nonlinear friction can well reflect the deformation characteristics of the continuum robot, which can be used as a reference for the shape reconstruction of the robot in future research.