In this paper, we construct two kinds of nonlinear mapping functions (MF) to deal with the problem of time-varying yet asymmetric output constraint for parametric non-linear systems. Unlike the Barrier Lyapunov Function (BLF) methods, the developed adaptive backstepping control has the following properties: 1) the issue of constraint can be coped with without requiring the asymmetric BLF (ABLF), making the design analysis less complicated; and 2) the situation without output constraint can also be handled. In addition, it is shown that the error converges to zero asymptotically by utilizing the Lyapunov stability theorem.