Experimental low-speed positioning system with VecTwin rudder for automatic docking (berthing)

Abstract

A VecTwin rudder system comprises twin fishtail rudders with reaction fins to increase its performance. With a constant propeller revolution number, the vessel can execute special low-speed maneuvers like hover, crabbing, reverse, and rotation. Such low-speed maneuvers are termed dynamic positioning (DP), and a DP vessel should be fully/overly actuated with several thrusters. This article introduces a novel and experimental VecTwin positioning system (VTPS) without making the ship fully/overly actuated. Unlike the usual dynamic positioning system (DPS), the VTPS is developed for low-speed operations in a calm harbor area. It is designed upon the assumption that the forces due to the interaction between the rudders, the propeller, and the hull are linear with the rudder angles within a range around the hover rudder angle. The linear relationship is obtained through linear regression of the results from several CFD simulations. The VTPS implements a PID controller that regulates the actuator forces to achieve the given low-speed positioning objective. It was tested in combined automatic docking and position-keeping experiments where disturbances from the environment exist. It shows promising potential for a practical application but with further improvements.

Appendix

1.1 On the linearity assumption for a wider range of rudder angle: 60 degree to 105 degree

Fig. 19

Comparison between the interpolated surface from additional CFD simulations and the regression plane (first row of $\tilde{\textbf{V}}$) for rudder angle between 60° and 105°

Fig. 20

Comparison between the interpolated surface from additional CFD simulations and the regression plane (second row of $\tilde{\textbf{V}}$) for rudder angle between 60° and 105°

The command–force relationship for the rudders $\left( \tilde{\textbf{V}}\right)$ was obtained via multiple linear regression of the results from nine CFD simulations (Table 2): pairs of rudder angles $\textbf{u}_{\delta }$ (2) between 70 and 80°, i.e., linearization around the hover rudder angle $\textbf{u}_{\delta \textrm{h}}$ (8). This linearity is assumed to be true for a wider range angle: 60° to 105°. Let $\mathcal {R}$ be the set of all $\textbf{u}_{\delta }$ within this wider range.

Alternative to physical tests, this assumption is verified by extending the CFD simulations for additional $\textbf{u}_{\delta }$: between 40° and 105°. Let $\mathcal {A}$ be the finite set of this additional $\textbf{u}_{\delta }$. One can linearly interpolate the values between two adjacent simulation results (red dots) to form an interpolated surface. This is shown in Figs. 19 and 20 where the interpolated surface (black) is compared with the regression plane (multicolored): the image of transformation/linear mapping $\tilde{\textbf{V}}$ (5) given the set $\mathcal {R}$ as the domain.

From Fig. 19, the regression plane gives a good estimation of the $X_{\textrm{CT}}$ for $\textbf{u}_{\delta }$ around the hover angle (8): 60°–90°. For $\textbf{u}_{\delta }$ above 90°, the regression overestimates the $X_{\textrm{CT}}$. This is desirable in low-speed operations because it gives aggressive actions to maintain the low speed. For $\textbf{u}_{\delta }$ below 60°, as expected, the regression underestimates the $X_{\textrm{CT}}$, i.e., the slope should be steeper: first row of $\tilde{\textbf{V}}$ should be larger in magnitude. This implies that for operations at normal speed and/or smaller $\textbf{u}_{\delta }$, a different $\tilde{\textbf{V}}$ should be constructed. From Fig. 20, the regression plane also gives a good estimation of the $Y_{\textrm{CT}}$ for $\textbf{u}_{\delta }\in \mathcal {R}$. As one can see, the slopes with respect to $Y_{\textrm{CT}}$ change sign when $\textbf{u}_{\delta }$ is lower than 60°. Thus, it is safe to say that the linear relationship is acceptable for any $\textbf{u}_{\delta }\in \mathcal {R}$: between 60° and 105°, as verified by the CFD simulations and validated (tested in a closed-loop feedback control scenario) in the experiments.

As a final note, with these additional data points, one may be tempted to obtain a single command–force relationship that covers the whole range of $\textbf{u}_{\delta }$. This is generally not recommended as it will move the intercept of the regression away from the hover rudder angle $\textbf{u}_{\delta \textrm{h}}$; effectively contradicting the fact that at $\textbf{u}_{\delta \textrm{h}}$ the resultant forces are zero such that the ship hovers.

1.2 On the extension of the CFD results as an estimation for any arbitrary ship with VecTwin rudders

This appendix explains a rather elementary way to extend the CFD results for any arbitrary scale model (or a full-scale ship). Due to the facts that (a) the forces shown in Table 2 are due to the propeller–rudder–hull interaction, (b) the propeller revolution $n$ is constant, and (c) the ship’s speed is very low, one can nondimensionalize the forces with the rudder properties. It can be done via the following equations (the prime symbol denotes the nondimensionalized quantities),

$$\begin{aligned} X_{\textrm{CT}}' = \frac{X_{\textrm{CT}}}{\rho A_{\textrm{R}}u_{\textrm{R}}^2} \quad \quad \text {and} \quad \quad Y_{\textrm{CT}}' = \frac{Y_{\textrm{CT}}}{\rho A_{\textrm{R}}u_{\textrm{R}}^2}, \end{aligned}$$

(39)

where $A_{\textrm{R}}$ is the profile area of the movable part of the rudder and $u_{\textrm{R}}$ is the longitudinal fluid inflow velocity at the rudder.

The above nondimensionalization of $X_{\textrm{CT}}$ and $Y_{\textrm{CT}}$ (that include hull forces) with $A_{\textrm{R}}$ is determined based on the following reasons. At a bollard pull condition, the hull forces can be smaller than the forces of the rudders and the propeller. Moreover, they depend on the flow reflected by the rudders, thus the nondimensionalization with $A_{\textrm{R}}$. On the other hand, $u_{\textrm{R}}$ captures the effect of $n$ and can be approximated as [34],

$$\begin{aligned} u_{\textrm{R}}=k_x\sqrt{\frac{8C_1\mu }{\pi }}nD_\textrm{P}, \end{aligned}$$

(40)

where $k_x$ is the coefficient of fluid inflow acceleration at the rudder, $C_1$ is the intercept of the regression of the propeller thrust coefficient $K_{\textrm{T}}$, $D_\textrm{P}$ is the diameter of the propeller, and $\mu$ is the ratio between $D_\textrm{P}$ and the height of the rudder.

Given the geometric properties of the rudder and the propeller, one can extend/estimate the command–force relationship for any arbitrary ship by scaling the $X_{\textrm{CT}}'$ and $Y_{\textrm{CT}}'$ appropriately following (39). Since the relationship is linear, this is equivalent to scaling the $\tilde{\textbf{V}}$ (5) with the same scaling term. Equivalent dimensional analysis can also be done for the bow thruster, i.e., scaling the $C_{\textrm{B}}$ (11) according to similarity law. As a caveat, due to the limited studies on this matter, the applicability of this very straightforward dimensional analysis is yet to be validated and hence requires extensive investigations.