Control analysis and design for robot manipulators require the knowledge of their dynamic model. In the first part of this paper, the Lagrange-Euler (L-E) formulaion has been developed to separate the generalized forces/torques due to rotational motion from that due to the translational motion. Therefore a new form for the dynamic equations of the robot manipulators has been obtained. In the second part, the generalized d’Alembert (GD) method has been used to develop the equations of motion for robot manipulators with revolute and/or prismatic joints. Also, the rotational effect and the translational effect in the new form of the GD method are separated. The develpoed equations, by both methods, when applied to robot manipulators result in an efficient and explicit set of closed form second order nonlinear differential equations. They give fairly well structured equations of motion suitable for control analysis and manipulator design. Using the proposed models, the rotational effects and translational effects have been studied separately and a simplified model has been obtained, a computational algorithm has been established and a computer software has been developed to perform the necessary calculations to obtain the generalized force required for each joint to follow any pre-specified trajectory. As an application of the proposed methods, a simplified model for a Stanford manipulator has been obtained and it is found that the results of the normal and the simplified models are very close and the simplified model can be accepted even with end-effector load variation and with different trajectories. The simplified model has the advantage of less computational time which is more appropriate for control purposes. Also, the results obtained by the developed L-E and GD methods are the same.