## 1 Introduction  
Robotic manipulators have become indispensable in the manufacturing industry, revolutionizing productivity and efficiency across a wide range of applications. These versatile machines play crucial roles in diverse manufacturing processes, including assembly, packaging, painting, material handling, and quality inspection. They find widespread use in industries such as automotive, electronics, pharmaceuticals, food processing, and logistics, where they excel at performing tasks like precision welding, cutting, drilling, palletizing, and order fulfillment [1; 2; 3; 4]. As the field of robotics continues to expand its applications, it faces challenges in operating within unstructured environments characterized by nonlinear dynamics and influenced by environmental factors such as gravity, friction, and external disturbances. Overcoming these challenges is essential to ensure the reliable and optimal performance of manipulators in such complex scenarios. This has prompted researchers in academia and industry to develop enhanced control methods that improve performance and enhance the stability of robot systems under negative influences. These advancements aim to ensure the robust and secure operation of robotic manipulators, safeguarding their efficiency and effectiveness in manufacturing settings. The robustness and reliability of robots can be enhanced through various control methodologies, such as PID control [5], computed torque control (CTC) [6; 7], fault-tolerant control (FTC) with PID [8; 9], neural network (NN)-based control [10; 11], disturbance observer-based control [12; 13; 14], sliding mode control (SMC) [15; 16; 17], and different forms of the SMC, like higher-order SMC [18; 19] and integral sliding mode control (ISMC) [20; 21; 22]. Among them, SMC and its various versions exhibit exceptional characteristics, including robustness against disturbances and uncertainties, finite-time convergence, and simplicity in design. These control techniques have found extensive applications across diverse systems, ranging from robot manipulators and underwater vehicles to spacecraft and electrical motors. However, ensuring fixed-time convergence for the system using conventional SMC methods is generally challenging. In addition, one primary hurdle is the occurrence of chattering, characterized by unwanted high-frequency oscillations or switching in the control signal due to the control law's discontinuity. Chattering can adversely impact system performance and result in mechanical wear. To address these limitations and ensure that the controlled error variables converge to equilibrium in a predefined time period, fixed-time control approaches have been introduced for robots [23; 24; 25; 26]. These controllers offer the advantage of achieving convergence within a fixed duration, thereby enhancing performance and reliability in robotic systems. By employing fixed-time control strategies, tracking errors can be effectively mitigated, enabling the system to meet stringent timing requirements in diverse applications. Fixed-time algorithms are specifically designed to ensure a bounded convergence time for the system, regardless of the initial state values. They have been developed based on various control frameworks, including SMC and backstepping control. For instance, fixed-time SMC algorithms have been proposed in specific studies [24; 26; 27], while fixed-time backstepping controls have been developed in other research papers [28; 29; 30]. These approaches provide valuable tools for achieving robust and time-constrained control in robotic systems. To further improve tracking performance and overcome the limitations of fixed-time controllers, the development of fixed-time adaptive controllers has gained attention [31; 32; 33]. These innovative approaches offer a reliable and practical solution for controlling robot manipulators in real-world applications. Notably, fixed-time adaptive SMC was successfully applied to robot manipulators, as demonstrated in Ref. [34].