The equation of Motion by Gibbs - Appell formulation has been used the least for deriving the dynamic equations of manipulator robots. So, in this paper a new systematic method for deriving the equation of motion of n - rigid robotic manipulators with revolute - prismatic joints (R - P - J) is considered. The equation of motion for this robotic system is obtained based on (G - A) formulation. All the mathematical operations are done by only 3×3 and 3×1 matrices. Also, all dynamic characteristics of a link are expressed in the same link local coordinate system. Based on the developed formulation, an algorithm is proposed that recursively and systematically derives the equation of motion. Finally, a computational simulation for a manipulator with three (R - P - J) is presented to show the ability of this algorithm in deriving and solving high degree of freedom of robotic system.