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In this paper, we present a semiglobal asymptotic stability analysis via Lyapunov theory for a new proportional-integral-derivative (PID) controller control scheme, proposed in this work, which is based on a fuzzy system for tuning the PID gains for robot manipulators. PID controller is a well-known set point control strategy for industrial manipulators which ensures semiglobal asymptotic stability for fixed symmetric positive definite (proportional, integral, and derivative) gain matrices. We show that semiglobal asymptotic stability attribute also holds for a class of gain matrices depending on the manipulator states. This feature increases the potential of the PID control scheme to improve the performance of the transient response and handle practical constraints in actual robots such as presence of actuators with limited torque capabilities. We illustrate this potential by means of a fuzzy self-tuning algorithm to select the proportional, integral, and derivative gains according to the actual state of a robotic manipulator. To the best of the authors' knowledge, our proposal of a fuzzy self-tuning PID regulator for robot manipulators is the first one with a semiglobal asymptotic stability proof. Real-time experimental results on a two-degree-of-freedom robot arm show the usefulness of the proposed approach.