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To compact the dynamic analysis and improve the computation efficiency, the advantages of Kanes equations and the product of exponential (POE) formulas were combined based on screw theory to propose a new effective dynamic modeling of robot manipulator with explicit geometric significance, and based on this dynamic model, a non-singular terminal sliding mode control was presented to achieve better performance. The velocity Jacobian matrix was addressed in the product of exponential form by introducing the screw theory, and the partial velocity of Kane’s equations was selected specifically from the suitable velocity Jacobian matrix. Then a dynamic modeling example based on Kane’s equations and screw theory was established, which led to be less complicated compared with other typically dynamic methods. With the proposed dynamic equations of serial robot manipulator, a fast non-singular terminal sliding mode (FNTSM) control was presented by introducing an improved fast non-singular terminal sliding mode surface, which was designed to ensure the fast convergence in global system state whether it was near to equilibrium or far away from the equilibrium. Then the stability analysis of the proposed method was performed by using Lyapunov stability theory. Finally, comparative experiments were implemented to demonstrate the effectiveness and robustness of the proposed approach.