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An accurate estimation of the dynamics efforts acting on a robot manipulator represents an important issue for both the analysis of its behavior and the synthesis of appropriate controllers. This paper proposes an iterative algorithm, which is based on the Newton-Euler approach, for the efficient evaluation of the manipulators' high-order kinematics and dynamics. In particular, the algorithm computes velocities, accelerations, and jerks of each link, while new dynamic equations are devised in order to evaluate the first derivative of generalized forces. Due to its moderate computational burden, the algorithm is suited to be used in online applications.