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This article presents a novel recursive smooth trajectory (RST) generation algorithm for application in robotics and in particular for unmanned aerial vehicles (UAVs). RST builds the trajectory recursively as a smooth polynomial path, thus a closed form trajectory satisfying any arbitrary dynamic limitation that translates into kinematic constraints (e.g., position, velocity, acceleration, etc.). Uncertainties and perturbations in the constraints are also discussed, modeled, and finally included in the calculation of the polynomial trajectory coefficients. Moreover, due to its recursive formulation, RST enables an immediate extension toward the solution of a trajectory optimization problem. In particular, it is shown that the minimum-snap piecewise polynomial trajectory can be interpreted as a special case of RST. The effectiveness of the proposed algorithm is demonstrated numerically via two illustrative scenarios. Its application to a UAV structure is also discussed to highlight the advantage of a smooth path over a piecewise one. Finally, the computational complexity and memory requirements are analyzed. [ABSTRACT FROM AUTHOR]