has become one of the common methods to solve problems of nonlinear systems. However, many adaptive control schemes cannot achieve satisfactory results due to two reasons: one is complex uncertainties, such as system parameter uncertainties and external disturbances. The other is the complexity explosion caused by the derivation of the virtual controller. In short, adaptive control is more suitable for trajectory tracking control of the nonlinear AUV system, with two issues being handled. A key technique of trajectory tracking control is to adopt a suitable method to resolve the effects of uncertainties. In practical engineering applications, uncertainties and disturbances are inevitable in AUV systems due to unmodeled dynamics, model uncertainties, and environmental impacts, such as ocean currents and waves [15; 16]. Most of these uncertainties and disturbances are hard to measure and adversely affect tracking performance, rendering great challenges in controller design [17]. Estimating these uncertainties and compensating for them in controller design has drawn great attention. A high-gain observer is employed for unknown internal and external disturbances, which is compensated for by output feedback control law [18]. An extended states observer is developed for system uncertainties in path-following control of under-actuated AUVs [19; 20]. However, it may cause differential peak problems due to high gain, which affects control performance. Neural network approximation capabilities are utilized to estimate uncertainties and environmental disturbances, which are compensated for by adaptive control methods [21; 22]. However, the performance of the neural network is heavily dependent on the number of neural network nodes, resulting in a computationally intensive system that is not conducive to practical engineering applications. The super twisting algorithm is adopted to solve the influence of nonlinear environmental disturbances in controller design [23]. However, controller parameter adjustment is complicated. Another key technique of trajectory tracking control is to design a controller to handle the input constraints, i.e., input saturation, which can occur due to the limitation of physical conditions in practical applications. The input constraints may make the controller unable to achieve the corresponding output, which may result in system instability and even control failure. Usually, two common approaches are used to tackle the input constraints: compensation with an auxiliary term [24] and approximate with a smooth function [25]. As to the input saturation caused by differentiating a composite reference trajectory (i.e., virtual input) during the application of the backstepping technique, using command/integral filters is an effective way to deal with input saturation and guarantee good performance [26; 27; 28]. In addition to the adaptive control method used for under-actuated AUV, controller design and system stability analysis may suffer from computational difficulty as the increasing of system order. The development of the backstepping algorithm makes the feedback adaptive control systematic and simplifies the complex controller design [29; 30; 31]. However, the required form of strict feedback is difficult for the under-actuated AUV system. Regarding this problem, a composite system, such as a singularly perturbed system and cascade system, in the form of lower order subsystems and their interconnecting structures, is adopted to simplify the control system [32; 33]. Thus, based on the above discussion, a backstepping-based adaptive control strategy employing disturbance observer and command filtered technique is proposed for trajectory tracking of AUVs with system uncertainties and environmental disturbances. They are summarized below. (1) A composite system, which consists of two subsystems with an interconnection function, is established instead of a kinematic tracking error dynamic. Then, it decouples the tracking error subsystem from the surge force and the yaw torque to the traditional solution.