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The main goal of this paper is to present a mathematical model for inverse dynamic equation of elastic robotic manipulator with revolute-prismatic joints. Due to the fact that there is no limitation on the number of mechanical arms, the proposed model must be extracted based on a systematic and automotive algorithm. Also according to the high computational complexity, the equations should be formed by a recursive formulation. Hence, a recursive and systematic methodology for deriving the equation of motion of elastic robotic arm with revolute-prismatic joints is presented. The inverse dynamic equations for this robotic system are obtained based on Gibbs-Applle formulation. All dynamic expressions of a link are expressed in the same link local coordinate system. Finally, in order to show the ability of this formulation in deriving and solving the equation of motion of such systems, a computational simulation for a flexible single robotic arm with revolute-prismatic joint is presented.