**3.2 Actuators**

The force applied to a vehicle due to the various actuators of a vehicle can be formulated as (34),

$$
\pi = T\mathbb{K}u \tag{34}
$$

where, for an AUV operating with 6 DoF with *n* actuators, *T* is the actuator configuration matrix of size 6 × *n* , *K* is the diagonal force coefficient matrix of size *n n* × , and *u* is the control input of size 1 *n*× . Actuators are the physical components that apply the desired force to the vehicle, and the particular configuration of these actuators will determine the size and structure of *T*, *K* and *u*, with each column of *T*, denoted *ti*, in conjunction with the corresponding element on the main diagonal of *K*, representing a different actuator.

A vast array of actuators are available to underwater vehicle designers, the more typical of which include propellers, control fins and tunnel thrusters, and each has their own properties that make them desirable for implementation within AUVs. For all the following actuator descriptions *lx* defines the offset from the origin of the actuator along the x-axis, *ly* defines the offset along the y-axis, and *lz* defines the offset along the z-axis.

#### **3.2.1 Propellers**

Propellers are the most common actuators implemented to provide the main translational force that drives underwater vehicles. These are typically located at the stern of the vehicle and apply a force along the longitudinal axis of the vehicle. The structure of *ti* for a propeller is given in (35).

$$t\_i = \begin{bmatrix} 1 & 0 & 0 & 0 & l\_z & -l\_y \end{bmatrix}^T \tag{35}$$

As can be seen from (35), if the propeller is positioned such that there is no y-axis or z-axis offset, the force produced will be directed entirely along the x-axis of the vehicle, with no rotational moments produced.

#### **3.2.2 Control surfaces**

Control surfaces, or control fins, are actuators that utilise Newton's Third Law of motion to apply rotational moments to the vehicle. These surfaces apply a force to the water which causes a deflection in the water's motion. Hence, the water must also apply a force to the control surface. Due to this force being applied at a distance from the centre of gravity of the vehicle, a rotational moment is produced that acts on the vehicle. The typical configuration for control surfaces on an AUV is to have four independently controlled fins arranged in two pairs orientated horizontally and vertically at the stern of the vehicle. The structure of *ti* for the horizontal fins is given in (36),

$$t\_i = \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{1} & l\_y & -l\_x & \mathbf{0} \end{bmatrix}^T \tag{36}$$

and for vertical fins given in (37).

158 Autonomous Underwater Vehicles

The role of the control allocation module is to generate the appropriate signals to the actuators in order for the generalised force from the control law to be applied to the vehicle. Since the vehicle under consideration is over-actuated, which means multiple actuators can apply forces to a particular DoF, the control allocation is responsible for utilising all available actuators in the most efficient way to apply the desired force to the vehicle. Power consumption is of particular importance for all autonomous vehicles, as it is a key factor in determining the total mission duration. The control allocation is therefore responsible for

The force applied to a vehicle due to the various actuators of a vehicle can be formulated as

where, for an AUV operating with 6 DoF with *n* actuators, *T* is the actuator configuration matrix of size 6 × *n* , *K* is the diagonal force coefficient matrix of size *n n* × , and *u* is the control input of size 1 *n*× . Actuators are the physical components that apply the desired force to the vehicle, and the particular configuration of these actuators will determine the size and structure of *T*, *K* and *u*, with each column of *T*, denoted *ti*, in conjunction with the

A vast array of actuators are available to underwater vehicle designers, the more typical of which include propellers, control fins and tunnel thrusters, and each has their own properties that make them desirable for implementation within AUVs. For all the following actuator descriptions *lx* defines the offset from the origin of the actuator along the x-axis, *ly*

Propellers are the most common actuators implemented to provide the main translational force that drives underwater vehicles. These are typically located at the stern of the vehicle and apply a force along the longitudinal axis of the vehicle. The structure of *ti* for a propeller

As can be seen from (35), if the propeller is positioned such that there is no y-axis or z-axis offset, the force produced will be directed entirely along the x-axis of the vehicle, with no

Control surfaces, or control fins, are actuators that utilise Newton's Third Law of motion to apply rotational moments to the vehicle. These surfaces apply a force to the water which causes a deflection in the water's motion. Hence, the water must also apply a force to the control surface. Due to this force being applied at a distance from the centre of gravity of the vehicle, a rotational moment is produced that acts on the vehicle. The typical configuration

*T <sup>i</sup> z y <sup>t</sup>* <sup>=</sup> <sup>⎡</sup> *l l* <sup>−</sup> <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> (35)

= *TKu* (34)

applying the desired forces to the vehicle, while minimising the power consumed.

τ

corresponding element on the main diagonal of *K*, representing a different actuator.

defines the offset along the y-axis, and *lz* defines the offset along the z-axis.

1000

**3.1 Role** 

**3.2 Actuators** 

**3.2.1 Propellers** 

is given in (35).

rotational moments produced.

**3.2.2 Control surfaces** 

(34),

$$t\_i = \begin{bmatrix} 0 & 1 & 0 & -l\_z & 0 & l\_x \end{bmatrix}^T \tag{37}$$

These structures of *ti* show that horizontal surfaces produce a heave force as well as roll and pitch moments, while vertical surfaces produce a sway force as well as roll and yaw moments. What must be considered here is that the force being produced by control surfaces relies on the vehicle moving relative to the water around it. If the vehicle is stationary compared to the surrounding water, control surfaces are ineffective. However, if the vehicle is moving relative to the surrounding water, these actuators are capable of applying forces and moments to the vehicle while consuming very little power.

### **3.2.3 Tunnel thrusters**

The previously mentioned limitation of control surfaces can be overcome by the use of tunnel thrusters. These thrusters are usually implemented by being placed in tunnels transverse to the longitudinal axis of the vehicle. Similar to control surfaces, the typical arrangement is to position two horizontal tunnel thrusters equidistant fore and aft of the centre of gravity, and two vertical tunnel thrusters also equidistant fore and aft of the centre of gravity.

The structure of *ti* for horizontal thrusters is given in (38),

$$t\_i = \begin{bmatrix} 0 & 1 & 0 & -l\_z & 0 & l\_x \end{bmatrix}^T \tag{38}$$

while the structure for vertical thrusters is given in (39).

$$t\_i = \begin{bmatrix} 0 & 0 & 1 & l\_y & -l\_x & 0 \end{bmatrix}^T \tag{39}$$

What can be observed here is that horizontal thrusters provide a sway force as well as a roll and yaw moment, while vertical thrusters provide a heave force as well as roll and pitch moment. In general, horizontal tunnel thrusters are located such that *lz* is zero, and vertical thrusters are located such that *ly* is zero. The result of this choice is that no roll moment is produced by these actuators.

The advantage of tunnel thrusters is that forces and moments can be produced even if the vehicle is stationary with respect to the surrounding water. This greatly increases the manoeuvrability of the vehicle, as control of the vehicle when moving at low speeds is possible. However, there are limitations associated with the use of these actuators. Firstly, these actuators consume more power when activated compared to control surfaces. This is due to force being produced by the thrusters only when the thruster itself is activated. In contrast, control surfaces consume power when the deflection angle is altered, but require very little power to hold the surface in place once the desired angle has been achieved.

Fully Coupled 6 Degree-of-Freedom Control of an Over-Actuated Autonomous Underwater Vehicle 161

A third scheme that is proposed here for the implementation of the control allocation is to break the control allocation problem into two smaller sub-problems, as seen in Figure 2, where the first sub-problem addresses control allocation to the main propeller and control surfaces, and the second addresses control allocation to the tunnel thrusters. Using this type of scheme, the control surfaces can firstly be used to their full extent in order to realise the generalised force as closely as possible. Only after full utilisation of the control surfaces occurs will the tunnel thrusters be introduced to provide forces and moments that the surfaces alone cannot produce. Using this methodology, the low power consumption control surfaces will be used as much as possible, while the higher power consumption tunnel thrusters will be called upon only when required to provide the extra manoeuvring capabilities that they possess. Furthermore, in a situation where accurate manoeuvring is not required, such as traversing from one waypoint to the next with no concern for what trajectory the vehicle follows, the second stage can be disabled such that no tunnel thruster is used, and control is performed entirely by the main propeller and control surfaces. However, if trajectory tracking is desired, the thruster allocation module can still be enabled. This will allow for the situation when the control surfaces provide inadequate force to the vehicle, and therefore the thrusters can assist in providing the extra force required to

**3.3.3 2-Stage scheme** 

maintain the vehicle tracking the desired trajectory.

Fig. 2. 2-Stage Control Allocation Scheme Block Diagram.

control surfaces over tunnel thrusters.

Implementation of this scheme would look somewhat like a 2-stage non-optimal scheme, as seen in Figure 2. The first stage would require the matrix operation (*TK*)*-1τ* for the main propeller and control surfaces in order to obtain as much force required from these actuators. An estimate of the force produced for this particular set of control values would then be calculated such that this force estimate can be subtracted from the total force required. Any residual force requirement would then become the input to the second stage

thrusters in order for these actuators to provide any extra force that the control surfaces cannot deliver alone. Therefore, the computational requirement for this scheme is quite minimal compared to the quadratic programming scheme, yet still heavily biases the use of

*TK* τ

<sup>−</sup> for the tunnel

of the control allocation, which would perform the matrix operation ( ) <sup>1</sup>

Secondly, thruster efficiency is reduced when the vehicle is moving. Under certain conditions, an area of low pressure is produced at the exit of the tunnel, which has the effect of applying a force to the vehicle in the opposite direction to which the water jet from the tunnel thruster is attempting to provide. The result is less total force being applied to the vehicle, and therefore reduced performance when moving at non-zero forward speeds (Palmer et al., 2009).

### **3.3 Allocation methods**

As previously mentioned when reviewing the various actuators for underwater vehicles, if the vehicle is stationary, control surfaces are ineffective while tunnel thrusters are very useful. Conversely, if the vehicle is moving, control surfaces are very efficient in providing force relative to power consumption compared to tunnel thrusters. The role of the control allocation is therefore to find a compromise between all actuators that both applies the desired generalised forces and moments to the vehicle while minimising power consumption. This balancing act allows the vehicle to maintain the manoeuvrability provided by all actuators, while at the same time allowing for as long a mission duration as possible.

#### **3.3.1 Non-optimal scheme**

One of the most straightforward methods for control allocation is application of the inverse of (34), i.e., (40) (Fossen, 2002).

$$
\mu = \left(TK\right)^{-1}\tau\tag{40}
$$

This method is very simple to implement as it consists of a single matrix multiplication. Therefore it is easy and efficient to implement within the computational processing constraints of an AUV. However, due to its simplicity, no attempt is made to minimise power consumption. If the vehicle contains both control surfaces and tunnel thrusters, both of these types of actuators will be utilised equally, even when the vehicle is moving at maximum velocity with respect to the surrounding fluid. However, since it is much more efficient to utilise control surfaces rather than tunnel thrusters while the vehicle is moving with respect to the water, a more intelligent approach is desired for implementing control allocation to minimise power consumption.

### **3.3.2 Quadratic programming**

The limitations of the aforementioned non-optimal scheme can be overcome by formulating a quadratic programming optimisation problem to solve for actuator inputs (Fossen, 2002, Fossen et al., 2009). By introducing a weighting matrix into the problem statement, actuator usage can be biased towards utilising control surfaces over tunnel thrusters. Therefore, the generalised force desired from the control law can be realised by the actuators while minimising power consumption. There are however, limitations associated with this scheme. Firstly, although it is possible to calculate an explicit solution to this problem, in the event of actuator reconfiguration, such as an actuator failure, this explicit solution would need to be recalculated, which can be computationally intensive. Iterative approaches, such as sequential quadratic programming, can be implemented that allow for actuator failures, but this method has the potential to require several iterations of the programming problem be solved at each control sample interval. Again, this can be a computationally intensive task.
