The trajectory tracking control problem of underwater robot is addressed in this paper. In general, an accurate thrust modeling is very difficult to establish for underwater robot in practice. Hence, the control voltage of thruster is designed directly as the input of system by the controller in this article. First, Taylor's polynomial is used to transform the form of trajectory tracking error system of underwater robot to the form of affine nonlinear systems, whose input is the control voltage of thruster. Then, according to the principle of sliding mode control, and using the local recurrent neural network to estimate the unknown item of affine system online, an adaptive sliding mode control is proposed. Aiming at the chattering problem which is caused by sliding mode control item, we propose a switch gain adjust method based on exponential function. It was proved that the trajectory tracking error of the underwater robot control system is uniformly ultimately bounded through Lyapunov theory. The feasibility and effectiveness of the proposed approach is demonstrated with trajectory tracking experiments of the experimental prototype of underwater robot.