Abstract: With the rapid development of industrial automation and smart manufacturing, robotic manipulators have found widespread applications across multiple industries. As production processes continue to evolve, the tasks assigned to robot manipulators are becoming increasingly complex, placing higher demands on trajectory tracking performance. A robot manipulator is a highly coupled nonlinear system. Its operation is influenced by numerous uncertain factors, such as unmodeled dynamics, measurement noise, parameter perturbations, sampling delays, external disturbances, nonlinear friction, and input saturation. As an emerging advanced control strategy, prescribed performance control offers new insights into enhancing the trajectory tracking performance of robot manipulators. Prescribed performance control defines the desired performance metrics, such as convergence time, overshoot, and tracking accuracy, through a prescribed performance function for controller design and ensures that all performance metrics meet the predefined requirements. However, in robot manipulator systems with input saturation, the improvement in system performance is limited by the nonlinear saturation of the controller. Therefore, studying effective control strategies for simultaneously handling performance constraints and input saturation is of theoretical and practical significance. Therefore, a time-delay control strategy with flexible prescribed performance is proposed for robot manipulators, considering input saturation, to address the problems of multiple uncertainties and improper selection of prescribed performance function affecting the trajectory tracking performance. First, a monotonic tube-prescribed performance function with predefined finite-time convergence was designed, and an auxiliary system was developed to generate a nonnegative modification signal related to the input saturation error. Based on this design, the flexible prescribed performance function imposes flexibility constraints on the tracking error. When the input is saturated, the auxiliary system generates a non-negative correction signal, causing the performance function boundary to adaptively increase and reduce the saturation time of the actuator. When input saturation does not occur, the correction signal generated by the auxiliary system becomes zero, allowing the flexible performance function to revert to its original boundary. This provides a more effective and flexible constraint on the tracking error. Second, a time-delay control method and an adaptive radial basis function (RBF) neural network were employed to estimate and compensate for the uncertainties in the robot manipulator model and external disturbances, thereby enhancing trajectory tracking accuracy and stability. An error transformation function converts the constrained tracking error into an equivalent unconstrained vector, called the transformed error, based on which a flexibly prescribed performance time delay controller that accounts for input saturation is proposed by incorporating dynamic surface techniques. Using the Lyapunov stability theory, we demonstrated that the robot manipulator output can track the reference signal while satisfying the constraints of the flexibly prescribed performance function. Finally, a two-degree-of-freedom robot manipulator was selected for simulation verification to validate the effectiveness and robustness of the control strategy. Initial conditions close to the constraints were chosen to verify whether the auxiliary system can generate nonnegative modification signals. A mid-simulation application of stronger impulse disturbances verified the disturbance rejection capability of the control system. Compared with the three alternative control methods, the proposed flexibly prescribed time-delay control strategy demonstrated superior trajectory tracking accuracy, faster response speed, enhanced robustness, and stronger input saturation tolerance. Torque control exhibited smoothness without chatter, showcasing the greater engineering practicality of the method and actuator friendliness.