## Rov

Fig. 1. Block diagram illustrating the control allocation problem.

The paper presents a survey of control allocation methods with focus on mathematical representation and solvability of thruster allocation problems. The paper is useful for university students and engineers who want to get an overview of state-of-the art control allocation methods as well as advance methods to solve more complex problems.

1.1 Problem formulation

Consider an underwater vehicle (Fossen, 2002):

that is controlled by designing a feedback control law of generalized control forces:

where a e Rp is a vector azimuth angles and u e Rr are actuator commands. For marine vehicles, some control forces can be rotated an angle about the z-axis and produce force components in the x- and y-directions, or about the y-axis and produce force components in the x- and z-directions. This gives additional control inputs a which must be computed by the control allocation algorithm. The control law uses feedback from position/ attitude n = [x,y,zand velocity v = [u,v,w,p,q,r]T as shown in Figure 1.

For marine vessels with controlled motion in n DOF it is necessary to distribute the generalized control forces t to the actuators in terms of control inputs a and u. Consider (1.2) where B(a) e Rnxr is the input matrix. If B has full rank (equal to n) and r > n, you have control forces in all relevant directions, this is an over-actuated control problem. Similarly, the case r < n is referred to as an under-actuated control problem.

Computation of a and u from t is a model-based optimization problem which in its simplest form is unconstrained while physical limitations like input amplitude and rate saturations imply that a constrained optimization problem must be solved. Another complication is actuators that can be rotated at the same time as they produce control forces. This increases the number of available controls from r to r+p.

2. Actuator models

The control force due to a propeller, a rudder, or a fin can be written

where k is the force coefficient and u is the control input depending on the actuator considered; see Table 1. The linear model F=ku can also be used to describe nonlinear monotonic control forces. For instance, if the rudder force F is quadratic in rudder angle 8, that is

F = hS | S |, the choice u = S | S |, which has a unique inverse S = sign(u)J\u\, satisfies (1.3).

Actuator

u

a

fT

Main propeller/longitudinal thrusters

pitch/rpm 