**2.4.2.2 Coupled SMC**

156 Autonomous Underwater Vehicles

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

As mentioned previously, sliding mode control is a scheme that makes use of a discontinuous switching term to counteract the effect of dynamics that were not taken into

To examine how to apply sliding mode control to AUVs, firstly (24) is compacted to the

(

linear and nonlinear damping forces, gravitational and buoyancy forces and moments, and

*s c* =η + η

where *c* is positive, it can be seen that by setting *s* to zero and solving for *η* results in *η*

( ) ( ) <sup>0</sup> 0

regardless of initial conditions. Therefore, the control problem simplifies to finding a control

lim 0 ( ) *<sup>t</sup> s t* →∞

If *η* is now replaced by the difference between the current and desired states of the vehicle, it can be observed that application of a control law of this form will now allow for a reference

Within the kinetic equation of an AUV, (16), simplifications can be applied that will reduce the number of coefficients contained within the various matrices. These simplifications can be applied due to, for example, symmetries present in the body of the vehicle, placement of centres of gravity and buoyancy, and assumptions based on the level of effect a particular coefficient will have on the overall dynamics of the vehicle. Thus, the assumption of body

 η

 ηη

 η

+ = ( , , ) ( ) (28)

*t tee*<sup>−</sup> = (30)

= − > *T sT* ( , si ) () gn, , 0 ( ) (32)

(29)

= (31)

 τ η

, ,*t* contains the nonlinear dynamics, including Coriolis and centripetal forces,

*ct ct*

 ηη

being sufficiently large. Thus, it can be seen that the application of (32) will

*M ft*

η

η )η

η

 ηη

Two such variants of SMC are the *uncoupled* SMC and the *coupled* SMC.

This can be achieved by applying a control law in the form of (32),

τ

applied to underwater vehicles is given in Jalving (1994).

account at the design phase of the controller.

**2.4.2 Sliding mode control** 

form of (28),

where *f* ( ) η η

external disturbances.

law such that (31) holds.

result in *η* converging to zero.

trajectory to be tracked.

**2.4.2.1 Uncoupled SMC** 

with *T* ( ) η,η

If a sliding surface is defined as (29),

converging to zero according to (30)

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 ignored at the design phase of the control law.

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 frame according to (33),

$$
\vec{\eta} = \hat{\eta} - \eta\_d \tag{33}
$$

where ηˆ represents an estimate of the current position and orientation provided by the 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 the body frame.

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 implementing a controller in the body frame.

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 computationally demanding.
