## 1 Introduction
To enhance the robustness of a control scheme, several observer-based control methods have been shown to effectively tackle external disturbances by estimating and compensating for them in the control scheme [16; 17]. [PERSON] et al. [18] proposed a fixed-time SMC, which is on the basis of a disturbance observer that considers the underwater vehicle trajectory tracking and external disturbance compensations. In the fixed-time disturbance observer proposed, it estimates not only the trajectory tracking deviation, but it also compensates for the time-varying environmental disturbance. In addition, a smoother switch was designed for the fixed-time sliding mode surface, which is used to guarantee the system state continuity of the sliding mode surface. [PERSON] and [PERSON] [19] proposed a disturbance observer based on a nonsingular fast terminal sliding mode controller (NFTSMC), which further improves the approach speed in existing NFTSMC and has a smooth external disturbance with good robustness. [PERSON] et al. [20] proposed a control method that combines a disturbance-observer-based controller (DOBC) with a fuzzy-adaptive S-surface controller to estimate and reject disturbances and unmodeled dynamics. [PERSON] et al. [21] proposed a nonlinear controller based on a disturbance observer for a hybrid aerial underwater vehicle (HAUV) prototype to control its takeoff, landing, and maneuvering when faced with environmental disturbances. The control scheme was shown to achieve an improved accuracy of tracking and robustness to uncertain, varying external disturbances compared to the PID controller. The disturbance observer has a good anti-interference ability against external disturbances and can effectively compensate for them, but it cannot improve the system's reach time to achieve a fast response. Introducing a sliding mode control can effectively compensate for this shortcoming, but SMC exhibits a chattering problem, and the system's response time and the chatter amplitude differences between systems are difficult to reconcile. In this paper, we propose a scheme to tackle the trajectory tracking problem and the environmental disturbance problem of ROVs by proposing a combined fast sliding mode control and a fixed-time disturbance observer control scheme. The main contributions of this work can be summarized as follows:  
* An innovative fast reaching law (FRL) is presented to tackle the chatter problem of conventional SMC. The presented FRL can both achieve fast response as well as reduce the system chatter, which solves the shortcomings of the ERL;
* A novel fixed-time disturbance observer is presented in order to compensate for ROVs model parameter errors and time-varying external disturbances. A Lyapunov analysis is used to demonstrate the stability of the system;* To demonstrate the superior performance of the FSMC-FDO, a comparative study between the FSMC-FDO and the CSMC trajectory tracking is performed by simulation. The remainder of the paper is organized as follows: the mathematical model of the underwater robot is described in Section 2. The control scheme is designed and stability analysis is presented in Section 3. Simulation results and comparative analysis are given in Section 4. Finally, conclusions are summarized in Section 5.