**2.4.1 PID control**

The fundamentals of PID control is that an error signal is generated that relates the desired state of the plant to the actual state (26),

$$e(t) = \mathbf{x}\_d(t) - \mathbf{x}(t) \tag{26}$$

Where *e*(*t*) is the error signal, *xd*(*t*) is the desired state of the plant, and *x*(*t*) is the current state of the plant, and this error signal is manipulated to introduce a corrective action, denoted *τ*(*t*), to the plant.

PID control is named due to the fact that the three elements that make up the corrective control signal are: *proportional* to the error signal by a factor of *KP*, a scaled factor, *KI*, of the *integral* of the error signal, and a scaled factor, *KD*, of the *derivative* of the error signal, respectively (27).

$$
\pi \left( t \right) = K\_P e \left( t \right) + K\_I \int\_0^t e \left( \mathcal{A} \right) d\mathcal{A} + K\_D \frac{d}{dt} e \left( t \right) \tag{27}
$$

PID control is best suited to linear plants, yet has also been adopted for use on nonlinear plants even though it lacks the same level of performance that other control systems possess.

Fully Coupled 6 Degree-of-Freedom Control of an Over-Actuated Autonomous Underwater Vehicle 157

symmetries allows reduced level of coupling between the various DoFs. An uncoupled SMC therefore assumes that no coupling exists between the various DoFs, and that simple manoeuvring is employed such that it does not excite these coupling dynamics (Fossen, 1994). The effect this has on (16) is to remove all off-diagonal elements within the various matrices which significantly simplifies the structure of the mathematical model of the vehicle (Fossen, 1994), and therefore makes implementation of a controller substantially

Although the removal of the off-diagonal elements reduces the computational complexity of the uncoupled SMC, it also causes some limitation to the control performance of AUVs, particularly those operating in highly dynamic environments and required to execute complex manoeuvres. Taking these two factors into account, these off-diagonal coupling terms will have an influence on the overall dynamics of the vehicle, and therefore cannot be

Coupled SMC is a new, novel control law that retains more of the coupling coefficients present in (16) compared to the uncoupled SMC (Kokegei et al., 2008, 2009). Furthermore, even though it is unconventional to design a controller in this way, the body frame is selected as the reference frame for this controller. This selection avoids the transformations employed in (24) and (25) used to rotate the vehicle model from the body frame to the NED frame although it does require that guidance and navigation data be transformed from the NED frame to the body frame. By defining the position and orientation error in the NED

> ˆ η =η η

navigation system, and *ηd* represents the desired position and orientation provided by the guidance system, a single rotation is required to transform this error from the NED frame to

In general, desired and current velocity and acceleration data are already represented in the body frame, and as such, no further rotations are required here for the purposes of

By comparing the number of rotations required to transform the vehicle model into the NED frame, as seen in (25), for the uncoupled control scheme with the single rotation required by the coupled control scheme to transform the guidance and navigation data into the body frame, it can be seen that the latter has less rotations involved, and is therefore less

The role of the control law is to generate a generalised force to apply to the vehicle such that a desired state is approached. This force, *τ*, for underwater vehicles consists of six components, one for each DoF, as seen in (23). The control allocation system is responsible for distributing this desired force amongst all available actuators onboard the vehicle such that this generalised 6 DoF force is realised. This means that the control allocation module must have apriori knowledge of the types, specifications, and locations, of all actuators on the vehicle.

ˆ represents an estimate of the current position and orientation provided by the

− *<sup>d</sup>* (33)

easier.

**2.4.2.2 Coupled SMC** 

frame according to (33),

where η

the body frame.

ignored at the design phase of the control law.

implementing a controller in the body frame.

computationally demanding.

**3. Control allocation** 

However, due to the wide use and acceptance of PID control for use in controlling a wide variety of both linear and nonlinear plants, it is very much employed as the "gold standard" that control systems are measured against. An example of a PID-based control strategy applied to underwater vehicles is given in Jalving (1994).
