To address the high-precision trajectory tracking control problem of manipulators subject to time-varying disturbances and model uncertainties in engine oil leak detection scenarios, this paper proposes an adaptive nonlinear extended state observer-based sliding mode control (ANESO-SMC) scheme. First, the kinematic model of the engine inspection system and the dynamic model of the manipulator with uncertainties are established. Next, a nonlinear extended state observer with adaptive bandwidth parameters is designed to estimate and compensate for the total system disturbance. Meanwhile, a sliding mode controller employing the arctangent function in place of the signum function is developed to ensure system stability during observer convergence. Then, based on Lyapunov stability theory, it is proved that both the observer estimation error and the bandwidth parameter error are uniformly ultimately bounded, and the tracking error of the closed-loop system converges asymptotically under the considered task conditions . Finally, the proposed scheme is comprehensively compared with nonsingular fast terminal sliding mode control (NFTSMC), extended state observer-based disturbance rejection control (ESO-DRC), and its adaptive variant (AESO-DRC) through simulation experiments. The results demonstrate that the proposed method exhibits favorable performance in convergence speed, tracking accuracy, and robustness against uncertainties under the considered operating conditions.