**3.3.3 2-Stage scheme**

160 Autonomous Underwater Vehicles

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

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,

One of the most straightforward methods for control allocation is application of the inverse

( ) <sup>1</sup> *u TK*

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

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.

τ

<sup>−</sup> = (40)

while at the same time allowing for as long a mission duration as possible.

(Palmer et al., 2009).

**3.3 Allocation methods** 

**3.3.1 Non-optimal scheme** 

of (34), i.e., (40) (Fossen, 2002).

allocation to minimise power consumption.

**3.3.2 Quadratic programming** 

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 maintain the vehicle tracking the desired trajectory.

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

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 of the control allocation, which would perform the matrix operation ( ) <sup>1</sup> *TK* τ <sup>−</sup> for the tunnel 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 control surfaces over tunnel thrusters.

Fully Coupled 6 Degree-of-Freedom Control of an Over-Actuated Autonomous Underwater Vehicle 163

the body are set to 0m. Hence, there is no roll moment applied to the vehicle when these thrusters are activated. The thrusters introduced to the model are based on the 70mm IntegratedThrusterTM produced by TSL Technology Ltd. This particular device can provide a maximum thrust of 42N, and due to its compact size, is well suited to this particular application. Also, the propulsion unit has been altered within the model. The simulation model used contains a propulsion unit based on the Tecnadyne Model 540 thruster. This device is able to provide approximately 93.2N of thrust and, with a propeller diameter of

The trajectory that the vehicle is asked to follow here consists of a series of unit step inputs applied to each DoF. The translational DoFs experience a step input of 1 metre whereas the rotational DoFs experience a step input of 1 radian, or approximately 60°. All inputs are applied for a period of 20 seconds, such that both transient and steady state behaviour can be observed. These unit step inputs excite the vehicle in all combinations of DoFs, from a single DoF, through to all DoFs being excited at once. This simulation assumes no water current, and therefore no water flow over the vehicle when it is stationary. Hence, roll cannot be compensated for when stationary and for this reason this simulation does not excite the roll component of the model. All other DoFs are excited,

Due to the way the aforementioned trajectory is supplied to the vehicle model, no guidance system is present. The step inputs will be applied at set times regardless of the state of the vehicle, and therefore observation of the control performance can be observed without influence from unnecessary systems. Therefore total execution time is constant for all simulations. Performance metrics used here in evaluating each control system is the accumulated absolute error between the desired translation/rotation and actual translation/rotation for each DoF. By looking at the following plots, observations can be made regarding such time-domain properties as rise time, settling time and percentage

Figures 4-13 show the desired and actual trajectories for each individual DoF when this complex set of manoeuvres is applied to both the uncoupled SMC and the coupled SMC. Figures 4, 6, 8, 10 and 12 show the complete trajectory for each DoF, while Figures 5, 7, 9, 11 and 13 show these same trajectories with the focus being on the last 300 seconds of the mission. This latter section of the mission is when multiple DoFs, particularly the rotational

Figure 4 shows the surge motion of the vehicle for the two different control systems. As can be seen, both systems exhibit desired properties for the first 180 seconds. During this period, all manoeuvring is exciting only the translational DoFs which the main propeller and tunnel thrusters can handle independently. After 180 seconds, the other DoFs are also excited, and the effect of this combined motion produces significant overshoots, especially for the uncoupled system, within the surge motion of the vehicle. This is more easily seen in Figure 5 where the larger overshoots can be seen for the uncoupled system compared to the

The sway motion of the AUV is shown in Figure 6. Minor overshooting is observed for both systems, especially when rotational DoFs are excited in combination with the translational DoFs. This can be observed in Figure 7. However, when the sway motion is excited,

15.2cm, is therefore well suited for this application.

however.

overshoot.

coupled system.

DoFs, are excited simultaneously.

convergence to the desired set point is observed.
