This study focuses on enhancing the control precision and efficiency of a two-degree-of-freedom (2-DOF) space manipulator used for active space debris removal. The unpredictable space environment introduces large uncertainties, which introduces unique challenges beyond the capabilities of a standalone computed torque controller and degrades control performance. To address this problem, a robust controller is developed, integrating traditional techniques such as sliding mode and computed torque control with a Neural Network framework. This synergy leverages both methods' strengths—conventional controls' accuracy and Neural Network's adaptability. The integration of Neural Network-based sliding mode control complements the robustness of computed torque control by actively mitigating uncertainties and disturbances inherent in the space environment. The 2-DOF manipulator's state variables model the system dynamics, necessitating accurate relative motion estimation between the manipulator and debris. The global asymptotic stability of the developed algorithm is demonstrated through the Lyapunov theorem, guaranteeing error convergence to zero. The convergence, stability, precision, tracking errors, and responsiveness of the controller have been analysed and validated by the MATLAB Simulink simulations. The novel approach's performance effectiveness is substantiated by numerical simulations and a comparative analysis with conventional computed torque control. Outcomes highlight the superior precision and efficiency in manipulator tracking the trajectory, validating the integrated controller's potential for successful active space debris removal.