Trajectory Tracking Performance Transition Analysis from Polar to Cartesian Coordinates

Trajectory tracking of underactuated marine vessels has become an intense research area in the nonlinear control community over the past two decades. In the authors’ previous work, an asymptotic tracking scheme was proposed by introducing two polar coordinate transformations, which can transform two-input-three-output underactuated system model into a certain two-input-two-output one. Therefore, it becomes possible to solve the trajectory tracking problem of these kinds of underactuated system using general backstepping method. However, since the polar coordinate transformation does not satisfy to be a diffeomorphism, the proposed tracking scheme in the polar frame cannot guarantee the same tracking performance in the Cartesian frame. With this consideration, in this paper the authors aim to analyze and establish a generalized condition under which the proposed tracking scheme in the previous work can guarantee the same tracking property in the Cartesian frame. The analysis results are verified through numerical studies as well.