**2.4.2 Sliding mode control**

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 account at the design phase of the controller.

To examine how to apply sliding mode control to AUVs, firstly (24) is compacted to the form of (28),

$$M\_{\eta} \left( \eta \right) \ddot{\eta} + f \left( \dot{\eta}, \eta, t \right) = \tau\_{\eta} \left( \eta \right) \tag{28}$$

where *f* ( ) η η, ,*t* contains the nonlinear dynamics, including Coriolis and centripetal forces, linear and nonlinear damping forces, gravitational and buoyancy forces and moments, and external disturbances.

If a sliding surface is defined as (29),

$$s = \dot{\eta} + c\eta \tag{29}$$

where *c* is positive, it can be seen that by setting *s* to zero and solving for *η* results in *η* converging to zero according to (30)

$$
\eta\left(t\right) = \eta\left(t\_0\right)e^{ct\_0}e^{-ct} \tag{30}
$$

regardless of initial conditions. Therefore, the control problem simplifies to finding a control law such that (31) holds.

$$\lim\_{t \to \!\!\to \!\!\!\times} s(t) = 0$$

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

$$\tau = -T\left(\dot{\eta}, \eta\right) \text{sign}\left(s\right) , T\left(\dot{\eta}, \eta\right) > 0\tag{32}$$

with *T* ( ) η,η being sufficiently large. Thus, it can be seen that the application of (32) will result in *η* converging to zero.

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 trajectory to be tracked.

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

#### **2.4.2.1 Uncoupled SMC**

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 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 easier.
