## Case study allocation problem formulation for an AUV with control surfaces

Some underwater vehicles perform all their missions at forward speed. In these applications, the vehicle hull design is streamlined so as to reduce hull drag, and the preferred type of control surface is the hydrofoil or fin. Hydrofoils produce lift, which is the useful force for controlling the motion of the vehicle. The side effect of lift generation, however, is drag — in other words, drag is the price we pay to obtain lift. Hence, for vehicles with several mounted control surfaces, the control allocation seeks the implementation of the demanded generalised forces while minimising the foil-induced drag. In this section, we formulate the control allocation problem for an AUV with two fixed thrusters and hydrofoil control surfaces.

Figure 4 shows INFANTE — an AUV built and operated by the Insituto Supetior Tecnico de Lisboa, Portugal. This AUV has two fixed thrusters at the stern, and six control surfaces: two horizontal fins mounted on the bow quarter, two horizontal fins mounted on the stern quarter, and two rudders mounted vertically behind the propellers.

Standard hydrofoil theory, see for example Marchaj (2000), establishes that the lift force produced by the hydrofoils is directed perpendicular to the incoming flow while the drag force is directed along the incoming flow direction. The magnitude of the lift and drag forces can be modelled as, where pw is the water density, A is the area of the hydrofoil, Uf is the fluid velocity relative to the hydrofoil, CL and CD are the lift and drag coefficients respectively (measured experimentally), and S is the angle of attack between the hydrofoil and the incoming flow. Table 2 shows the different variables associated with the different control actuators considered in this case study. Notice that for the positive angle deflection of the control surfaces we use the right-hand rule along the direction of the rotation axis towards the tip.

Variable |
Description |
Positive convention |

Spb |
Port bow fin angle |
Forward edge down |

Ssb |
Starboard bow fin angle |
Forward edge up |

Sps |
Port stern fin angle |
Forward edge down |

Sss |
Starboard stern fin angle |
Forward edge up |

Spr |
Port rudder angle |
Forward edge to port |

S sr |
Starboard rudder angle |
Forward edge to port |

Tp |
Port thuster thust |
Forward |

Ts |
Starboard thuster thust |
Forward |

Table 2. Manipulated variables associated with the different actuators of the AUV shown in Figure 4.

Table 2. Manipulated variables associated with the different actuators of the AUV shown in Figure 4.

For the control allocation problem, we will assume that the velocity Uf is either measured or estimated. We will also assume that the vehicle manoeuvres slowly from its equilibrium operational condition at forward speed. Hence, we can neglect the small drift angles; and thus, the lift and drag forces of the different hydrofoils can be considered to act along the x-and y-direction of the body-fixed coordinate system attached to the vessel. Furthermore, under the slow manoeuvring assumption and small drift angle, the angle of attack S of the hydrofoils can be approximated by the mechanical angle of rotation of the hydrofoils. For the particular vehicle under study, we can consider motion control objectives in 5DOF (surge, heave, pitch, roll, and yaw). With these objectives, the fins can be used to control heave, pitch and roll, the rudders to control yaw, and the thrusters to control surge. Then, we can simplify the allocation problem by taking a three-step approach:

1. Solve the allocation of the fins to obtain the deflection angles that implement the desired heave force and pitch and roll moments while minimising the induced drag.

2. Compute rudder angles based on the demanded yaw moment.

3. Compute thrust demand for the thrusters based on the demanded surge force while compensating for the fin and rudder induced drag forces.

The separation into these three steps simplifies the optimisation problem associated with the allocation. The first step results in a quadratic programme with linear constraints since only the lift forces are used. Then the rudders are used only for controlling the heading or yaw.

2 fL

Finally, after computing the fin and rudder deflection angles, the thrust can be computed to implement the desired surge force and to compensate for the drag forces of the fins and rudders.

The above allocation scheme could be interpreted as a feed-forward compensation for the side effects of the fin and rudder drag induced forces. Step 1: fin Allocation

Based on the above assumptions and the adopted positive convention for the variables shown in Table 1, we obtain the following vector of fin commands and force configuration matrix for heave, pitch and roll allocation

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