In this paper an ℒ1 adaptive back-stepping controller, using a barrier Lyapunov function (BLF), and considering the position-velocity constraints is investigated. The purpose of this development is to remove the filter C(s) in the design process and reduce the complexity of ℒ1 adaptive method to achieve a fast adaptation and robustness for practical applications. The BLF is also used to constrain the system output. The performance of the proposed controller is evaluated numerically on two robotics scenarios, namely a single-link manipulator, and a 6-DOF remotely operated vehicle (ROV). The results in both scenarios confirm that the tracking and estimation error for both position and velocity output of the ROV outperforms compared to the standard ℒ1 adaptive control method. The results also show the efficacy of the proposed method in removing time-varying disturbances and uncertainties. It also demonstrates a satisfying tracking performance in front of uncertainty on the input gain and unknown nonlinear dynamics.