• This is a Time-varying Continuous Approach.

The dynamics of an underactuated ship is

The coordinate transformations

are first used to transform the ship system to

where $c = m_{11}/m_{22}$ and $d = d_{22}/m_{22}$. Then the following nontrivial coordinate transformations

are applied to the transformed ship system to obtain the new system:

Let $k_2, k_3, k_\mu$ and $k_r$ be strictly positive constants such that $1 \geq k_2 \geq k_3$. The control $\Omega_1$ and $\Omega_{2\mu}$ are designed in the paper as

where

It is proven that the closed loop system is GAS at the origin in the paper.

Remark: The stabilizer design mentioned above is nontrivial, it is difficult to extend the control design scheme to solve a trajectory-tracking problem, and the convergence of the stabilizing errors to zero is slow. In addition, the physical meaning of feedback is not clear.