**2.4 Control laws**

154 Autonomous Underwater Vehicles

The 6 1 × vector of gravitational and buoyancy forces and moments are represented in (16)

( ) ( ) ( ) ( ) ()

*W B W B W B*

sin cos sin cos cos cos cos cos sin

θ φ

θ φ

θ

sin cos cos

θ φ

θ

(20)

θ φ

*g g gg r xyz* <sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> (21)

*b b bb r x* = *y z* (22)

= *XYZKMN* (23)

cos sin sin

( )( )

*zW zB xW xB*

θ

*gb gb*

θ

*g yW yB zW zB*

*g b g b*

() ()

θ φ

Here, *W* is the weight of the vehicle, determined using *W=mg* where *m* is the dry mass of the vehicle and *g* is the acceleration due to gravity. *B* is the buoyancy of the vehicle which is due to how much fluid the vehicle displaces while underwater. This will be determined by the size and shape of the vehicle. Vectors determining the locations of the centre of gravity and the centre of buoyancy, relative to the origin of the body frame, are given by (21) and (22)

*<sup>T</sup> <sup>b</sup>*

[ ] *<sup>b</sup> <sup>T</sup>*

[ ]*<sup>T</sup>*

Here, the translational forces affecting surge, sway and heave are *X*, *Y*, and *Z* respectively, and the rotational moments affecting roll, pitch and yaw are *K*, *M* and *N* respectively.

Overall, (16) provides a compact representation for the nonlinear dynamic equations of motion of an underwater vehicle, formulated in the body frame. By applying the rotations

> η

 νηη

Within (24), the equations in (25) contain the rotations of the various matrices from the body

1

*C J C MJ J J*

 ν

ηη

( ) () () ()() ()

<sup>=</sup> ⎡ ⎤ <sup>−</sup> ⎣ ⎦

1

 η

− −−

(

 ηη

 ηη

 η τ η ω

1 1

 η

(25)

+ ( , , ) + + =+ ( ) ( ) ( ) (24)

The 6 1 × vector of control input forces is denoted by *τ*, and is given by (23).

contained within (8), (16) can be formulated in the NED frame as given in (24).

*MC D g*

( ) ( ) ( )

− −

 η

 η ν

 ηη

 ητ

*T T T T T*

*M J MJ*

=

= = =

( ) () () ()

− −

() ()()

() ()

*D J DJ g Jg J*

− −

τ

 η  νηη

The 6 1 × vector of external disturbances is denoted by *ω*.

, ,

νη

η

η

η

ν η

η

η

τ η

η

η

η

frame to the NED frame.

η )η *xW xB yW yB*

φ

<sup>⎡</sup> <sup>−</sup> <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> − − <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> − − <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>=</sup> ⎢−− +− <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> − +− <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> −− − − <sup>⎣</sup> <sup>⎦</sup>

*g b g b*

by *g*(*η*), and determined using (20).

respectively.

( )

η Various control strategies, and therefore control laws, have been implemented for AUV systems.

The benchmark for control systems would be the classical proportional-integral-derivative (PID) control that has been used successfully to control many different plants, including autonomous vehicles. PID schemes are, however, not very effective in handling the nonlinear AUV dynamics with uncertain models operating in unknown environments with strong wave and current disturbances. PID schemes are therefore only generally used for very simple AUVs working in environments without any external disturbances.

An alternative control scheme known as sliding mode control (SMC), which is a form of variable structure control, has proven far more effective and robust at handling nonlinear dynamics with modelling uncertainties and nonlinear disturbances. SMC is a nonlinear control strategy which uses a nonlinear switching term to obtain a fast transient response while still maintaining a good steady-state response. Consequently, SMC has been successfully applied by many researchers in the AUV community. One of the earliest applications of using SMC to control underwater vehicles was conducted by Yoerger and Slotine wherein the authors demonstrated through simulations studies on an ROV model, the SMC controller's robustness properties to parametric uncertainties (Yoerger & Slotine, 1985). A multivariable sliding mode controller based on state feedback with decoupled design for independently controlling velocity, steering and diving of an AUV is presented in Healey and Lienard (1993). The controller design was successfully implemented on NPS ARIES AUV as reported in Marco and Healey (2001).
