JERZY GARUS
Department of Mechanical and Electrical Engineering
Naval University
81-103 Gdynia, ul. Smidowicza 69
POLAND
Abstract: - The paper addresses a method of thrust distribution for an unmanned underwater vehicle. The method is developed on basis of applying of the orthogonal Walsh matrix to decomposition of a configuration matrix describing layout of thrusters in a power transmission system. The proposed solution of thrust allocation is worked out with a view of using it in a control system of the remotely operated vehicle “Ukwial” designed and built for the Polish Navy. The algorithm has been tested for track-keeping control both in faultless work of thrusters and failure one of them. Some computer simulations are provided to demonstrate effectiveness and correctness of the approach.
Key-Words: - Underwater vehicle, control allocation, marine propeller, autopilot
There are various categories of unmanned underwater vehicles. The most often used underwater vehicle is a remotely operated vehicle
(ROV). The ROV is usually connected to a surface ship by a tether, which all communication is wired through. Drag from the tether influences vehicle’s motion and may represent significant disturbances and energy loss. The ROV is equipped in power transmission system and controlled only by thrusters. Simultaneously spatial station-keeping or tracking of the underwater vehicle is difficult task for a human operator and hence supervisory control has been developed toward increased its local intelligence and autonomy.
For the conventional ROVs basic motion is movement in a horizontal plane with some variation due to diving. They operate in crab-wise manner in 4 degree of freedom (DOF) with small roll and pitch angles that can be neglected during normal operations. Therefore, it is purpose full to regard 3-dimensional motion of the vehicle as superposition of two displacements: motion in the horizontal plane and motion in the vertical plane. It allows to divide a vehicle’s power transmission system into two independent subsystems i.e. the subsystem realizing vertical motion and the subsystem responsible for motion in the horizontal plane. A general structure of such a system shows
Fig. 1.
Fig. 1. A structure of power transmission system with
5 thrusters.
The first subsystem usually consists of 1 or 2 thrusters generating driving force acting along a vertical axis. In this subsystem distribution of propulsion is not a complicated task done in such a way that thrust of a propeller or sum of thrusts of propellers given by thrusters is equal to commanded input. The second one consists of at least 3 thrusters assuring surge, sway and yaw motion. The most applied solution is using of 4 thrusters mounted askew in relation to main axes of vehicle’s symmetry (see Fig. 1). Demanded inputs i.e. forces along roll and lateral axes and moment around vertical axis are linear combination of propellers thrusts produced by all subsystem’s thrusters. Hence from operating point of view a control system should have a procedure of power distribution among the thrusters. The procedure

ought to include principles of distribution and determine such allocation of thrusts of the propellers as to obtained values of driving forces and moment are equal to desired input.
The objective of the paper is to present a method of power distribution for the underwater vehicle. The proposed solution assures a proper motion of the vehicle not only in case of correct work of thrusters but also thruster failure.
The paper consists of the following six sections. A short introduction to dynamics and a control system of the underwater vehicle is given in next section. In section 3 a thruster model is discussed. An algorithm of power distribution is presented in Section 4. Section 5 provides results of simulation study. The concluding remarks are given in Section 6. Two appendices contain a brief description of Walsh matrix and the model of the remotely operated vehicle.
The general motion of marine vehicle in 6 DOF can be described by the following vectors [2]:
η v
τ





 x , u ,
X , y , v ,
Y , z , w ,
Z

,
, p ,
K
 q ,
,
 r


T
, M ,
T
N

T
(1) where:

– the position and orientation vector with coordinates in the earth-fixed frame; v – the linear and angular velocity vector with coordinates in the body-fixed frame;

– describes the forces and moments acting on the vehicle in the body-fixed frame.
The nonlinear dynamic equations of motion can be written in form:
M  
C ( v ) v

Dv
 g ( η )
 τ (2) where:
M - inertia matrix (including added mass);
C ( v ) - matrix of Coriolis and centripetal terms
(including added mass);
D ( v ) – hydrodynamic damping and lift matrix; g (

) - vector of gravitational forces and moments.
Fig. 2 presents a main elements of the vehicle’s control system. The flight planner and trajectory generator provides the desired vehicle position and orientation as functions of time. The autopilot then computes desired vehicle forces and moment by comparing the desired vehicle’s position and orientation with current their estimate based on sensors measurements. The corresponding value of each thruster force is computed in the power distribution module. Then, the desired propeller revolution to each thrusters can be computed by using a mapping from thrust demand to propeller revolution.
Relationship between the vector of forces and moments

acting on the vehicle and the control input of thrusters u is a complicated function depending on density of water, a tunnel length and cross-sectional area, a propeller diameter and revolutions and the vehicle’s velocity vector v :
τ  f
 
(3) where f (  ) is nonlinear function. A detailed analysis of thruster dynamics can be found e.g. in [2,4].
Fig. 2. Block diagram of control system.

In many practical applications the model (3) is approximated by so called an affine model [2], e.g. the system being linear in its input:
τ 
Bu (4) where B is a known non-square constant matrix.
For the affine model u can be computed as: u

B
 τ
(5) where:
B

- pseudoinverse matrix to matrix B ,
B
 
B
T
 

1
.
A condition of usability of the above dependence is proper work of all thrusters. If anyone is nonoperational it can not be used. It is a main disadvantage of the solution based on the equation
(5). To cope with this problem a special algorithms should be implemented in the control system. In the next chapter one from possible approaches is proposed.
Farther in the paper considerations are restricted to motion of the vehicle in the horizontal plane. This limitation results from the construction of the propulsion system. As mentioned in chapter 1, for most of the ROVs it is composed of two subsystems. Only distribution of propulsion in subsystem responsible for motion in the horizontal plane is complex. Usually, this subsystem consists of 4 thrusters layout symmetrical around a gravity centre and it assures linear motion in X and Y axes and rotational motion around Z axis. Such a solution, presented in Fig. 3, requires specialized procedure to allocation of thrust among the thrusters.
Fig. 3. Layout of thrusters in subsystem responsible for
horizontal motion.
Transforming (4) a vector of forces and moment
τ acting on the vehicle in the horizontal plane can be described in function thrusts of the propellers by the following expression [3]:
τ 
TPf (6) where:
τ





,

,


T

1 2 3
 force in X direction,
 direction,

 

T
–
thrusters configuration matrix
T




 t t t
1
2
3








 d
1 moment around Z axis, cos sin sin
 
 
1
 
1
1
 
1


– force in Y
...
...
...
d
4 cos sin sin
 
 
 
4 n
4
 
 – angle between roll axis and direction of i propeller thrust fi ,
4 d i
– distance of the i th
thruster from a centre of





 gravity,
 – angle between lateral axis and the line i connecting the centre of gravity with the i th thruster’s centre of symmetry, f

 f
1
, f
2
,..., f
4

T
– thrust vector,
P
–
diagonal matrix of thrusters’ readiness p ii




0
1

 i th i th thruster thruster off active .
Let us note that elements of the thrusters configuration matrix T are geometry dependent and can be obtained for each vehicle in advance.
The matrix T, f or the ROVs having configuration of thrusters shown in Fig. 3, has the following properties: a) is the row-orthogonal matrix, b) t ij
 t ik
dla i

1 , 3 i j , k

1 , 4 , c) can be written as a product of two matrices: a diagonal matrix Q and row-orthogonal matrix
W f
taking values

1 :
T

Q W f



 q q
2
1
 q
3
 







 w f 1 w w f 2 f 3








 t
11
0
0
0 t
21
0
0
0 t
31



 sign ( T )
(7)
It allows to work out a computationally convenient procedure to calculation of the thrust vector f with applying of the orthogonal Walsh matrix W (see
Appendix A).
The procedure will be regarded for two cases:
1.
all thrusters are operational ( P

I ),
2.
one from thrusters is off due to a fault
( P

I ).
4.1. Algorithm for all thrusters active
Let us denote:
τ z

  z 1
,
 z 2
,
 z 3

T
- required input vector,

f

 f
1
, f
2
,..., f
4

T
- thrust vector necessary to generate input vector
 z
.
Substituting (7) into equation (6) gives:
τ z

QW f
Pf (8)
Multiplying both sides of (6) by Q

1 we obtain:
Q

1 τ z

W
Hence, substituting: f
Pf
S



Q
0

1 τ z



(9)









0 z 1

 q
11 z q
22 z
2 q
33
3








(10)
W
 and regarding

 w
W
0 f

 where w
T
0


1





1
1




(11)
 1 
P

I the equation (7) can be written in form:
S

Wf (12)
Finally, taking into account the Walsh matrix properties that:
W

1 
W
T
and W

1
W

1
I n where n=dimW the following formula to compute of thrust vector f is obtained: f

1
4
WS

1
4
W


Q
0

1 τ z

 (13)
4.2.
Algorithm for the non-operational thruster
Let denote as above the required input vector by
τ  the thrust vector by f and assume that the k th z thruster is off. It means f k

0 and element p kk

0 . The other elements of f will be calculated using the formula (13).
Defining: f '

 f
1
P '
W f

'

,..., diag
 f
 w f 1 k

1 p
11
,...,
, f k
,..., w

1 f k p k
,...

1

1
, f k w
4

1 f k

T
,

1
, p k

1
,..., k

1 w
,..., f 4
 p
44
 the expression (13) can be written in a form:
τ z

Q W f '
P ' f ' (14)
Since and
Q , W ' i P ' are square matrices dim 3

3
P '

I f
, the thrust vector f ' can be calculated by a formula: f '

W f '

1
Q

1 τ z
(15)
Hence values of the thrust vector f are obtained as follows: f

 f
1
' ,..., f ' k

1
, 0 , f k
' ,...
f m
'

T
(16)
The described method of thrust allocation is worked out to be implemented in a control system of the ROV called “Ukwiał”. The vehicle is operational used on board of polish minesweepers.
It is an open frame robot controllable in 4 DOF, being 1.5 m long and having a propulsion system consisting of 6 thrusters. Displacement in the horizontal plane is done by means of 4 thrusters which can generate force up to

750 N assuring speed up to

1.2 m/s and

0.6 m/s consequently in
X and Y direction.
A simulation study has been performed for the vehicle’s model of dynamics written in
Appendix A. Simulation experiments were made for tracking control under interaction of sea current disturbance (speed 0.3 m/s, direction 135 0 ).
Maximum velocities were also determined by length of tether. The vehicle was assumed to follow the trajectory beginning from position and orientation (10 m, 10 m, 0 0 ), passing target waypoints (10 m, 90 m, 90 0 ), (30 m, 90 m, 0 0 ), (30 m, 10 m, 270 0 ), (60 m, 10 m, 0 0 ) and ending at
(60 m, 90 m, 90 0 ). The autopilot calculated command signals
 z 1

X ,
 z 2

Y and
 z 3

N .
The power distribution module processed the signals and gave required hydrodynamic thrusts using formulas 12 or 16 in depend on state of thrusters.
The inputs, desired and real outputs and tracking errors are shown in Fig. 4 and Fig. 5. The first figure is for proper work of all thrusters, the second one for the fault of the 3 rd thruster. It can be seen that the failure of a single thrusters has a small influence on accuracy of the vehicle’s motion
(position and orientation). In both cases a tracking error is on the same level. The examples demonstrate the ability of proposed method of power distribution to cope with a such type of unserviceability of power transmission system.

Fig. 5. The vehicle’s position and orientation /d – desired, r – real/, deviation from position and orientation, inputs and thrusts of propellers for trackkeeping and all thrusters operational.
The paper presented a method of power distribution for unmanned underwater vehicle. The proposed solution is based on the affine model of the thrusters and decomposition of the thruster configuration matrix. It makes the method simple and useful for practical usage.

Fig. 6. The vehicle’s position and orientation /d – desired, r – real/, deviation from position and orientation, inputs and thrusts of propellers for track-keeping and the 3 rd thruster off.
The nonlinear model of the vehicle “Ukwiał” was applied for computer simulations. The investigations were carried out for full efficient power transmission system and the system with a thrusters being off due to a fault. The obtained results for tracking control system show the proposed algorithms enable to control the vehicle in the horizontal plane with high accuracy in both cases.
The main advantage of the approach is its flexibility with regard to the construction of the vehicle’s power transmission system and number of thrusters. The developed algorithms of power distribution are of a general character and can be successfully applied for all types of the ROVs.
Walsh matrix
The Walsh matrix is a square, orthogonal matrix having all elements equal to

1 . The forms of the
Walsh matrices W
N
for N =2, N =4 and N =8 are given below:
W
2



1
1

1
1


W
4



1

 1


1
1




1
1
1


1
 1

1


1


1


1
1
1
1


1
1
1
1


1

1


1
1


W
8 



1
1
1
1
1
1
1
1




1
1
1
1
1
1
1
1




1
1
1
1
1
1
1
1



1
1
1
1
1
1
1
1





1
1
1
1
1
1
1
1





1
1
1
1
1
1
1
1



1
1
1




1


1 
1

1


1


The Walsh matrix W is convenient to generate using of its connection with Hadamard matrix H .
The main advantage of Hadamard matrix H is possibility to be generated in recursive way by means of the following dependence:
H
2 N



H
H
N
N

H
N
H
N

 where H
1

.
For example for N =4 the matrix has the form:
H
4






1
1
1
1


1
1
1
1


1
1
1
1


1

1


1


1 
An algorithm of transformation the H
N
matrix into the W
N
matrix can be found e.g. in work [1].

The ROV model
The following ROV model of dynamics has been used in the computer simulation:
M
 diag

99 .
0 ,
D
 
 diag

10 .
0 ,
108 .
5 ,
0 .
0 ,
126 .
5 ,
0 .
0 , 0 .
223
8 .
2 32 .
9 ,
1 .
918 ,
29 .
1
1 .
603


 diag

227


.
18 u , 405 .
41 v ,
3 .
212 p ,
478 .
03 w
14 .
002 q , 12 .
937 r



C
 











0
0
0
0
26 .
0
28 .
0 v w g
 

0
0
0
26 .
0 w
0

18 .
5 u
0
0
0

28 .
0 v
18 .
5 u
0
0

26 .
0 w
28 .
0 v
0

5 .
9 r
6 .
8 q











17 .
0 sin(

17 .
0 cos(

17 .
0 cos(

)
) sin(
)

) cos(

)
279

.
2
279
 sin(
.
2
 cos(

)

) sin( cos(

)

) cos(

)
0










26 .
0 w
0

18 .
5 u
5 .
9 r
0

1 .
3 p
The values of thrusters configuration matrix T corresponding to the Fig. 3 are as follows:

18

1
28 .
0 v
.
6
.
0
3
0
.
5
8 p u q









T


 


0
0
.
.
875
485
0 .
332


0
0
0
.
875
.
485
.
332


0
0
0
.
.
875
485
.
332


0
0
0
.
.
875
.
485
332

 


References:
[1] Ahmed N., Rao K.R: Orthogonal Transforms for Digital Signal Processing , Springer-Verlag,
Berlin 1975.
[2] Fossen T.I.: Guidance and Control of Ocean
Vehicles, John Wiley and Sons, Chichester
1994.
[3] Garus J.: Fault Tolerant Control of Remotely
Operated Vehicle, Proc. of the Ninth IEEE Int.
Conference on Methods and Models in
Automation and Robotics, Miedzyzdroje
(Poland), 2003, vol. I , pp. 217-221.
[4] Healey A.J., Rock S.M., Cody S., Miles D.,
Brown J.P.: Toward an Improved
Understanding of Thruster Dynamics for
Underwater Vehicles, IEEE Journal of Oceanic
Engineering , No 20, 1995, pp. 354-361.