Introduction

Thruster modeling and control is the core of underwater vehicle control and simulation, because it is the lowest control loop of the system; hence, the system would benefit from accurate and practical modeling of the thrusters. In unmanned underwater vehicles, thrusters are generally propellers driven by electrical motors. Therefore, thrust force is simultaneously affected by motor model, propeller map, and hydrodynamic effects, and besides, there are many other facts to consider (Manen & Ossanen, 1988), which make the modeling procedure difficult. To resolve the difficulties, many thruster models have been proposed.

In the classical analysis of thrust force under steady-state bollard pull conditions, a propeller's steady-state axial thrust (T) is modeled proportionally to the signed square of propeller shaft velocity (Q),T=c1Q | Q | (Newman, 1977). Yoerger et al. (Yoerger et al., 1990) presented a one-state model which also contains motor dynamics. To represent the four-quadrant dynamic response of thrusters, Healey et al. (Healey et al., 1995) developed a two-state model with thin-foil propeller hydrodynamics using sinusoidal lift and drag functions. This model also contains the ambient flow velocity effect, but it was not dealt with thoroughly. In Whitcomb and Yoerger's works (Whitcomb & Yoerger, 1999a; Whitcomb & Yoerger, 1999b), the authors executed an experimental verification and comparison study with previous models, and proposed a model based thrust controller. In the two-state model, lift and drag were considered as sinusoidal functions, however, to increase model match with experimental results, Bachmayer et al. (Bachmayer et al., 2000) changed it to look-up table based non-sinusoidal functions, and presented a lift and drag parameter adaptation algorithm (Bachmayer & Whitcomb, 2003). Blanke et al. (Blanke et al., 2000) proposed a three-state model which also contains vehicle dynamics. Vehicle velocity effect was analyzed using non-dimensional propeller parameters, thrust coefficient and advance ratio. However, in the whole range of the advance ratio, the model does not match experimental results well.

In the former studies, there are three major restrictions. First, thruster dynamics are mostly modeled under the bollard pull condition, which means the effects of vehicle velocity or ambient flow velocity are not considered. However, while the thruster is operating, naturally, the underwater vehicle system is continuously moving or hovering against the current. In addition, the thrust force would be degraded by up to 30% of bollard output due to ambient flow velocity. Therefore, the bollard pull test results are only valid at the beginning of the operation, and the ambient flow velocity induced by vehicle movement or current must be taken into consideration. Moreover, non-parallel ambient flow effects have received less attention in previous works (Saunders & Nahon, 2002). These are dominant when an underwater vehicle changes its direction, or when an omni-directional underwater vehicle with non-parallel thrusters like ODIN (Choi et al., 1995) is used. Non-parallel ambient flow effects could be modeled simply by multiplying the ambient flow by the cosine function, but experimental results have been inconsistent. Second, in the models including the ambient flow effect, the thrust equations are derived from approximations of empirical results without concern for physical and hydrodynamic analysis. This leads to a lack of consistency in the whole thrust force map, especially, when the directions of thrust force and ambient flow velocity are opposite. Third, most of the previous models contain axial flow velocity of the thruster, because the models are usually based on Bernoulli's equation and momentum conservation. However, measuring axial flow velocity is not feasible in real systems, so we cannot apply those equations directly to the controller. Hence, in Fossen and Blanke's work (Fossen & Blanke, 2000), the authors used an observer and estimator for the axial flow velocity. And, Whitcomb and Yoerger (Whitcomb & Yoerger, 1999b) used the desired axial velocity as an actual axial flow velocity for the thrust controller. Those approaches, however, increase the complexity of controller. To resolve the above restrictions, in this article, we mainly focus on steady-state response of thrust force considering the effects of ambient flow and its incoming angle, and propose a new thruster model which has three outstanding features that distinguish it from other thruster models. First, we define the axial flow velocity as the linear combination of ambient flow velocity and propeller shaft velocity, which enables us to precisely fit the experimental results with theoretical ones. The definition of axial flow gives a physical relationship between the momentum equation and the non-dimensional representation, which has been widely used to express the relation between ambient flow velocity, propeller shaft velocity, and thrust force. Also, the modeling requires only measurable states, so it is practically feasible. Second, we divide the whole thrust force map into three states according to the advance ratio. The three states, equi-, anti-, and vague directional states, explain the discontinuities of the thrust coefficient in the non-dimensional plot. While the former approaches failed to consider anti- and vague directional states, the proposed model includes all of the flow states. Here, we define the value of border status between anti- and vague directional states as Critical Advance Ratio (CAR) where the patterns of streamline change sharply. The details will be given in Section 3. Third, based on the two above features, we develop the incoming angle effects to thrust force. Incoming angle means the angle between ambient flow and thruster, which is easily calculated from vehicle velocity. If the incoming angle is 0 degree, the thrust force coincides with the equi-directional state, or if the angle is 180 degree, the thrust force coincides with the vague or anti-directional state according to the advance ratio. It should be pointed out that the mid-range of incoming angle cannot be described by a simple trigonometric function of advance ratio. So we analyze the characteristics of incoming angle, and divide the whole angle region into the three states above. Also, for the border status among the states, Critical Incoming Angle (CIA) is defined.

This chapter is organized as follow: In Section 2, the thruster modeling procedure will be explained and a new model for the thruster is derived. Section 3 addresses three fluid states with CAR and CIA, and explains the physical meanings. Then Section 4 describes the matching results of experiments with the proposed model simulation, and compares these results with conventional thrust models. Section 5 describes the thruster controller based on the proposed model. Finally, concluding remarks will summarize the results.

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