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In this study, a fuzzy logic controller is designed in order to use in trajectory control of a robot manipulator. The considered robot manipulator consists of a rotating-prismatic joint housing a sliding flexible arm that carries a concentrated mass at the tip end. The tip end of the flexible arm traces a multi-straight-line path. This study is aimed to use a fuzzy logic controller in controlling the trajectory traced by the tip end of the flexible arm so as to reduce the vibrations induced in the flexible arm. The designed fuzzy controller is aimed to control both the position of the tip end of the flexible robot arm while the tip end traces a multi-straight-line path and the vibrations induced in the flexible arm. Numerical simulations obtained by using a developed computer program are presented and physical trend of obtained numerical results are discussed. The performance of the fuzzy logic control system is evaluated on the basis of the simulation results.