Neural Network Augmented Identification of Underwater Vehicle Models 2007 Control Engineering...

8/20/2019 Neural Network Augmented Identification of Underwater Vehicle Models 2007 Control Engineering Practice

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Control Engineering Practice 15 (2007) 715–725

Neural network augmented identification of underwater vehicle models

Pepijn W.J. van de Vena,, Tor A. Johansenb, Asgeir J. Sørensenc,Colin Flanagana, Daniel Toala

aDepartment of Electronic and Computer Engineering, University of Limerick, Limerick, Ireland bDepartment of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway

cDepartment of Marine Technology, Norwegian University of Science and Technology, Trondheim, Norway

Received 22 September 2004; accepted 16 November 2005

Available online 4 January 2006

Abstract

In this article the use of neural networks in the identification of models for underwater vehicles is discussed. Rather than using a neural

network in parallel with the known model to account for unmodelled phenomena in a model wide fashion, knowledge regarding the

various parts of the model is used to apply neural networks for those parts of the model that are most uncertain. As an example, the

damping of an underwater vehicle is identified using neural networks. The performance of the neural network based model is

demonstrated in simulations using the neural networks in a feed forward controller. The advantages of online learning are shown in case

of noise impaired measurements and changing dynamics due to a change in toolskid.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Autonomous vehicles; Backpropagation; Marine systems; Neural networks; Nonlinear systems; System identification

1. Introduction

In recent years highly sophisticated nonlinear control

schemes for marine vehicles have been developed and

implemented. Although modelling of marine vehicles is

widely addressed, several parameters still pose uncertain-

ties. This is due to the absence of accurate models to

describe the highly dynamic nature of these hydrodynamic

parameters. Of prime importance in this context is the

dependence of many hydrodynamic parameters and

coefficients on varying velocity regimes, proximity to the

sea bed, sea surface and other structures, just to mention a

few. At present, models are normally only valid for alimited region of operational conditions (Fossen, 2002).

Certain model parameters can be determined analyti-

cally. Other parameters, however, will need to be deter-

mined using numerical methods or identified using (scaled)

model or full scale tests. Both methods can be timeconsuming and expensive. In numerical calculations using

dedicated hydrodynamic software (Faltinsen, 1990), the

vehicle may be divided up into small sections and two

dimensional added mass contributions are calculated for

those sections. Consecutively, an integration over the

whole body yields the three dimensional added mass

parameters. In order to apply this method, which is called

strip theory, the user is required to provide a detailed

description of the vehicle in the form of a CAD drawing.

This part of the modelling process alone can take up

considerable time and, moreover, slender body theory must

be assumed.For bluff bodies other methods must be used. The added

mass parameters can be measured using, e.g., a towing tank

or free decay tests (Ross, Fossen, & Johansen, 2004). Up to

date there are, to the authors’ knowledge, no methods

available to perform those tests for all coupled six degrees

of freedom simultaneously.

Additionally, it should be kept in mind that no means of

online updating of parameters is available from either

method. This possibly even affects the analytically derived

values of parameters that are normally assumed to be

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0967-0661/$ - see front matterr 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.conengprac.2005.11.004

Corresponding author. Tel.: +353 61 234230; fax: +35361 202572.

E-mail addresses: [email protected] (P.W.J. van de Ven),

[email protected] (T.A. Johansen),

[email protected] (A.J. Sørensen), [email protected]

(C. Flanagan), [email protected] (D. Toal).

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constant, as the physical characteristics of the craft may

change from mission to mission. As an illustration of this,

one might think of a remotely operated vehicle (ROV) or

an autonomous underwater vehicle (AUV) whose dy-

namics have been identified. However, due to a change in

toolskid both mass and geometrical characteristics of the

vehicle will change, thus changing the craft’s (hydro)dy-namic behaviour.

To overcome the above mentioned problems, neural

networks can be used as they offer a means of parameter

identification without the necessity of detailed model

knowledge and with the possibility of online updates. As

a result they can identify the parameters of interest over the

full region of operation and can adapt for changing

circumstances. Neural networks have several advantages

over other nonlinear identification methods. They can

represent a function with high accuracy in a smooth

fashion without the necessity of extensive amounts of

memory. The latter ability to compress data drastically

requires long training times, but in offline training this is

generally not an issue. Model evaluation, on the contrary,

is fast. Furthermore, neural networks have an aptitude for

dealing with noisy training signals while still obtaining

accurate results. The application of neural networks for

control and modelling (Narendra & Parthasarathy, 1990)

has been given considerable attention in recent years. In

Van de Ven, Flanagan, and Toal (2003) three classes of the

application of neural networks to the task of modelling and

control of underwater vehicles were identified: (i) combined

control and learning, CCL, (ii) separate control and learning,

SCL and (iii) augmented control , AC . In the first class,

CCL, a neural network is used in series with the craft.While controlling the craft it learns to do so better and

better. Examples of this approach can be found in

Akkizidis and Roberts (1998), Farrell, Goldenthal, and

Govindarajan (1990), Guo, Chiu, and Wang (1995), Kim

and Yuh (2001), Labonte (2002), Seube (1991), Venugopal,

Sudhakar, and Pandya (1992), Wang, Lee, and Yuh

(1999a, 1999b), Wang, Lee, and Yuh (2000), Wang and

Lee (2002), Yuh (1990), Yuh and Lakshmi (1993) and Yuh

(1994). In SCL, neural networks are trained outside the

loop. Once satisfactory performance has been obtained

they are used in the control loop, either as a process model

or directly as a controller. The application of SCL is

described in Fujii and Ura (1990), Ishii, Ura, and Fujii

(1994), Ishii, Fujii, and Ura (1995), Ishii and Ura (2000)

and Ura, Fujii, Nose, and Kuroda (1990). The third group,

AC, contains all strategies that apply neural networks to

augment the performance of conventional methods in some

way. Examples of this strategy can be found in Campa,

Sharma, Calise, and Innocenti (2000), Kodogiannis,

Lisboa, and Lucas (1996), Li, Lee, and Lee (2002), Pollini,

Innocenti, and Nasuti (1997) and Yamamoto (1995).

All three approaches have their vices and virtues. For

CCL the fact that learning is performed with the latest

data, and thus circumstances, is a clear advantage.

However, finding an inverse model of the craft, which is

necessary in this strategy, might be hard or even

impossible. This becomes even more of an issue when

considering that control and identification both take place

in the same loop. The time for the identification process is

thus limited by the time between control commands. In

SCL this problem is circumvented by identifying the

necessary parameters outside the control loop. An initial(possibly conventional) controller, or a previously synthe-

sised controller, can be used during this learning process,

thus allowing longer time for learning. This, however, often

leads to a more complicated, and thus more expensive

architecture. As both these strategies exclusively make use

of neural networks, stability is a major issue. This stability

issue can be alleviated by applying AC. In AC neural

networks are used to improve the performance of a

controller, whereas minimum, but stable, control perfor-

mance is guaranteed by a conventional linear or nonlinear

feedback controller.

No matter what strategy is used, in the literature

describing CCL, SCL or AC architectures, the neural

networks are always used in parallel with a known part of

the craft dynamics or controller (Campa et al., 2000; Li et

al., 2002; Pollini et al., 1997; Yamamoto, 1995). Or, if no

knowledge is assumed regarding model or controller, the

network is used to represent the whole model or controller

(Comoglio & Pandya, 1992; Ishii & Ura, 2000; Kodogian-

nis et al., 1996; Venugopal et al., 1992). Rather than using a

neural network in such an overall approximating fashion,

in this article the use of neural networks to model specific

parameters in the model is proposed. Although this

approach is not new, see e.g. Psichogios and Ungar

(1992) and Thompson and Kramer (1994), little attentionhas previously been paid to it in the underwater vehicle

literature.

Before delving into the identification of the underwater

vehicle dynamics using neural networks, in Section 2 a brief

overview of marine craft dynamics is given. The (hydro)

dynamical model is introduced and the parameters

playing a role are discussed. Then, in Section 3, the use

of neural networks for identification is proposed. It is

argued why they should not be used in an overall

approximating fashion, but should rather be used to

approximate certain parameters in the model. To illustrate

this divide and conquer approach, a method to identify

the hydrodynamic damping with neural networks is

presented. With the presented identification method

simulations are performed in Section 4. The resulting

parameters are compared to parameters identified using

a least squares identification method in Section 5. Finally,

in Section 6 the findings are summarised and conclusions

are drawn.

2. ROV kinematics, dynamics and hydrodynamics

In Fossen (2002) and Sørensen and Ronæss (2000) it was

shown that the nonlinear dynamic equations of motion of a

marine vehicle in six degrees of freedom can be expressed in

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vector notation as

M_m þ Cðm Þm þ Dðm Þm þ gðgÞ ¼ s, (1)

with the kinematic equation

_g ¼ JðgÞm . (2)

The matrix JðgÞ converts velocity in a body fixed frame, m ,to velocity in an earth fixed frame, _g. The interpretation of

the used symbols is as follows:

g position and orientation of the vehicle in the

Earth-fixed frame

m linear and angular velocity of the vehicle in the

body-fixed frame

_m linear and angular acceleration of the vehicle in the

body-fixed frame

M inertia matrix including added mass

Cðm Þ matrix consisting of Coriolis and centripetal terms

Dðm Þ matrix consisting of damping or drag terms

gðgÞ vector of restoring forces and moments due togravity and buoyancy

s vector of control forces and moments

In (1) it is assumed that no water current is present.

Introducing the latter with velocity m c results in the added

mass contribution to the Coriolis matrix and the damping

matrix to be a function of the relative velocity, m r, defined as

m r ¼ m m c ¼ ½u v w p q rT ½uc vc 0 0 0 0

T, (3)

where u; v; w; p; q and r are the craft’s velocity compo-

nents in six degrees of freedom in a body-fixed frame and ucand vc are the velocity components of the surrounding

water in a horizontal plane. For brevity, and without loss

of generality, in this article the current velocity is assumed

to be zero. A detailed derivation of the nonlinear equations

of motion can be found in Fossen (2002). Below a small

summary of the model is given.

In the matrix M two inertial components are accounted

for,

M ¼ MRB þ MA. (4)

The rigid body inertial matrix, MRB , represents the mass

and inertia terms. Added mass is accounted for by the

matrix MA.

For the matrix Cðm Þ, a similar discourse can be held.Both the Coriolis and the centripetal forces are functions of

the rigid body mass and added mass and the velocity, m :

Cðm Þ ¼ CRB ðm Þ þ CAðm Þ. (5)

CRB ðm Þ accounts for the rigid body while CAðm Þ accounts for

the added mass.

In the damping matrix, Dðm Þ, four terms are combined:

Dðm Þ ¼ DP þ DS ðm Þ þ DW þ DM ðm Þ, (6)

where DP is the potential damping (relevant when

operating in the wave zone), DS ðm Þ the linear and quadratic

skin friction, DW the wave drift damping and DM ðm Þ the

damping due to vortex shedding.

Accurate calculation of these phenomena is difficult.

Hence, often the damping is approximated by a diagonal

matrix containing the linear and quadratic damping terms

according to

Dðm Þ ¼ diagfX u; Y v; Z w; K p; M q; N rg

diagfX ujuj

juj; Y vjvj

jvj; Z wjwj

jwj; K pj pj

j pj,

M qjqjjqj; N rjrjjrjg, ð7Þ

where X u, Y v, Z w, K p, M q and N r are the linear terms, and

X ujuj, Y vjvj, Z wjwj, K pj pj, M qjqj, N rjrj the quadratic terms of

the damping in six degrees of freedom. Although (7) is

a good approximation for decoupled motion, for man-

oeuvres involving movements along and about several

body axes at a time, such simple models might prove to be

insufficient.

3. System identification using artificial neural networks

In this paper neural networks will be used to identify thenonlinear and coupled damping in an underwater vehicle.

Neural networks can be applied both as control plant

models and as controllers. A clear advantage of using a

neural network is that strong assumptions regarding the

damping, as discussed in Section 2, are not made. Higher

order terms and coupling between various degrees of

freedom can be taken into account by the neural network.

Normally the neural network is used in parallel with

conventional models or controllers in a switching or

output-blending fashion. In this case, however, the neural

network makes no, or only partial, use of the available a

priori knowledge. Due to looking at the neural network as

some nonlinear mapping between the plant’s input data

and output data, knowledge regarding the dependence

between parameters is lost. This might lead to an

unnecessarily complicated function to be learned by the

neural network. To illustrate this the process dynamics of

Eq. (1) are expressed in state-space form. First Eq. (1) is re-

ordered:

_m ¼ M1½s Cðm Þm Dðm Þm gðgÞ. (8)

Taking the state vector to be m and the inputs to the system

as sðtÞ and gðtÞ, (8) can be written in state-space form:

_m ¼ U½m ðtÞ; sðtÞ; gðtÞ,

y ¼ W½m ðtÞ, (9)

with U½m ðtÞ; sðtÞ; gðtÞ the right hand side of (8) and W½m ðtÞ

simply m ðtÞ.

Fig. 1 shows the corresponding block diagram. Assum-

ing that one has partial knowledge regarding the function

U, a neural network can be used to model the unknown

part of the system in parallel with the known part of the

system (in an ‘‘all-in-one’’ fashion) as depicted in Fig. 2. In

this approach one assumes U ¼ UM þ Û where UM corresponds to the known part of U and Û to the unknown

part approximated by the neural network. If the same

assumption is made for the matrices M1, Cðm Þ, Dðm Þ and

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gðgÞ, (8) can be written as

_m ¼ M1M fs CM ðm Þm DM ðm Þm gM ðgÞg

þ M1M f Ĉðm Þm D̂ðm Þm ĝðgÞg

þ M̂1

fs CM ðm Þm Ĉðm Þm

DM ðm Þm D̂ðm Þm gM ðgÞ ĝðgÞg. ð10Þ

The last three lines of (10) represent Û. These terms will be

estimated by the neural network.

The proposed method in which various model para-

meters are identified with their own neural network is

shown in Fig. 3. The neural networks in Fig. 3 model

NN 1 ¼ M̂1

; NN 2 ¼ Ĉ þ D̂; NN 3 ¼ ĝðgÞ, (11)

which is a considerably easier task, demonstrating that the

‘‘all-in-one’’ neural network mapping is unnecessarily

complicated.

Another advantage of the approach depicted in Fig. 3 is

that use can be made of known structural properties during

the learning stage. Examples of features are symmetry of

the mass matrix, independence of the rigid body mass

matrix on the velocity, m , and known coupling between

certain degrees of freedom. To benefit from the mentioned

advantages, the alternative configuration of the neural

networks, as shown in Fig. 3, might prove worthwhile and

is therefore used in this study.

In most practical cases, all parameters, apart from the

damping parameters, can be calculated with sufficient

accuracy. The contribution of potential damping can be

calculated accurately through the application of potential

theory. However, without towing tank experiments, the

other contributors to the damping cannot be estimated

accurately. Therefore, it is assumed that only the dampingmatrix Dðm Þ is unknown and hence, initially, not accounted

for. In this case the dynamic equation for the reference

system or process plant model (ppm) and the initial

approximate or control plant model (cpm), respectively,

can be written as

_m ppm ¼ M1½s Cðm ppmÞm ppm

Dðm ppmÞm ppm gðg ppmÞ, ð12Þ

_m cpm ¼ M1½s Cðm cpmÞm cpm gðgcpmÞ. (13)

With the ppm, a simulation is performed in which the craft

is controlled by a human operator through joystickcommands. During the simulation one-step-ahead predic-

tions are computed from the cpm. The influence of the

damping matrix can then be computed as follows. At every

time step one assumes both systems have the same state

vector: m ppm ¼ m cpm ¼ m . Substituting m for m ppm and m cpm in

Eqs. (12) and (13), the difference between Eqs. (12) and

(13) becomes the damping matrix Dðm Þ, multiplied by the

state vector m and the inverse of the mass matrix, M1. The

product Dðm Þm can thus be calculated as

Dðm Þm ¼ M½_m cpm _m ppm. (14)

Eq. (14) will be used for training of the neural networks

and hence is the identification model.

The neural networks are initially trained offline using the

Levenberg–Marquardt algorithm. To prevent problems

during the learning stage, rather than using one neural

network to represent the damping, six neural networks are

used. This is done for two reasons. First of all it should be

noted that the six elements of the vector Dðm Þm do not have

to be of the same order of magnitude. Especially the

elements pertaining to degrees of freedom that cannot be

actuated will in general have a small magnitude. As back

propagation learning relies on feeding back an error signal

which will generally be larger for larger output values, the

speed of learning can drastically change from output

ARTICLE IN PRESS

M-1C+D

Φ

∫

g()

+

+

−

−.

Fig. 1. State-space representation of the nonlinear dynamic equations of

motion.

∧Φ

ΦM

NN

∫

+

+.

Fig. 2. State-space representation of the nonlinear dynamic equations

with neural network in parallel, to model unknown parameters.

MMC M +D M

NN 1 NN 2

NN 3

∫

g M ()

++

+

+

+

+

-

-

--1

.

Fig. 3. State-space representation of the nonlinear dynamic equations

with several neural networks to model unknown parameters.



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element to output element. As a result parts of the neural

network will learn faster than other parts. This again might

result in overtraining of certain parts (i.e. the neural

network might learn to represent the noise in the training

data rather than the actual information). On the other

hand, parts of the neural network will be ‘‘undertrained’’,

again resulting in poor performance. For offline trainingthe above described problem can be circumvented by

normalising the data. However, for online training there is

no guarantee that this normalisation still holds. The second

reason is closely related to the first reason: in general, it is

impossible to guarantee the same rate of learning for

different output nodes in a neural network. The straight-

forward way to tackle the mentioned problems is to apply

six neural networks, with each one output neuron, to the

identification problem.

If improvements are expected the neural networks are

further trained online, using a standard back propagation

algorithm with a momentum term.

4. Simulations

In this section the identification method described in

Section 3 will be applied to an underwater craft in a

simulation study. This open frame underwater craft,

named Tethra, is currently being developed within the

Mobile & Marine Robotics Group of the University of

Limerick (Molnar, Toal, Flanagan, & Hayes, 2004). To

demonstrate the advantages of neural networks for coupled

and nonlinear damping characteristics, the damping matrix

inhibits significant off-diagonal elements and nonlineari-

ties. In the simulations the following assumptions aremade:

Acceleration and velocity of the vehicle can bemeasured.

Only the damping matrix is unknown.

As described in Section 3, training data are derived from a

simulation in which the craft is controlled by a human

operator. Use was made of the Matlab virtual reality

toolbox to provide an interface between human and

computer.

Fig. 4 gives an impression of the virtual-world interface.Joystick commands are interpreted in the block called

‘‘Hand Control Unit’’. These joystick commands, which

represent excitations in four degrees of freedom, are then

converted to thruster commands in the block ‘‘Control

Allocator’’. Time delays and saturation of the thrusters are

described by the block ‘‘Propulsion System’’ and the

vehicle dynamics are described in the block ‘‘ROV model’’.

Interfacing with the Virtual Reality toolbox is provided by

the block ‘‘Virtual World Interface’’.

In the Matlab Virtual World environment the University

of Limerick test pool was drawn to obtain similar

circumstances in simulation and future testing. Fig. 5 shows

the vehicle in the pool at the beginning of a simulation.After gathering the training data, the neural networks

are trained to represent the damping matrix. The neural

networks have 12 input neurons, 5 neurons in one hidden

layer and 1 linear output neuron. The hidden layer neurons

use a hyperbolic tangent activation function. Both the

hidden and the output layers use bias parameters. Rather

than only supplying the six degrees of freedom velocity to

the neural networks, the magnitude of the velocity vector in

six degrees of freedom was supplied as well, resulting in an

input layer width of 12 neurons. In this way, the a priori

knowledge that the damping is a function of the magnitude

of the velocity can be explicitly used in the neural network

identification.

Initially the neural networks are trained offline with the

Levenberg–Marquardt algorithm. This algorithm was

implemented in Matlab. The offline training, with a batch

of N ¼ 18 000 data samples, is stopped after 50 cycles, after

which the mean square error between network output, yðk Þ,

and desired output, yd ðk Þ:

E ¼XN k ¼1

ð yd ðk Þ yðk ÞÞ2, (15)

is smaller than 0.1% of the average magnitude of the

desired output for all six degrees of freedom. General-

isation of the neural networks is then tested with a test set

consisting of 4500 samples, which yields comparable mean

square errors.

ARTICLE IN PRESS

Hand

Control

Unit

Control

Allocator

Propulsion

System ROV model

Virtual

World

Interface

Fig. 4. Block diagram of the simulation environment in Matlab.

Fig. 5. The University of Limerick in-house vehicle in the virtual pool at

the start of a simulation.



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After the identification of Dðm Þm in (14) has been

performed, the neural networks can be used in the model

as shown in Fig. 6. The control plant model now becomes

_m cpm ¼ M1½s Cðm Þm gðgÞ D̂ðm Þm , (16)

where D̂ðm Þm indicates that this is the neural network

approximation of Dðm Þm .

It should be noted that, unlike the other blocks, the

block with caption D̂ðm Þm should not be interpreted as a

multiplication of the inputs and the block’s argument, but

rather as a mapping from the input, m , to an output, D̂ðm Þm .

To test the accuracy of the identification process, the

trained neural networks are used in a feedforward

controller. From Eq. (16) it can be seen that if the

parameters M, C, g and m are known, the neural networks

representing Dðm Þm can be used to calculate a thrust force, s,

such that a wanted acceleration, _m is obtained:

s ¼ M _m cpm þ Cðm Þm þ gðgÞ þ D̂ðm Þm . (17)

Eq. (17) is used as the feedforward control law. The

performance of this feedforward controller is evaluated by

comparing the resulting path to a reference path made with

a feedforward controller that does have full knowledge of

the vehicle’s dynamics. To probe the neural network

representation of the damping for cross-coupling the

followed trajectory incorporates movement in several

degrees of freedom at the same time. A dive from 1.5 m

depth to 3:5 m depth is performed while the craft is

following a circular path in the x – y plane.

In the first computer experiment, Section 4.1, the

damping matrix will be approximated offline using velocity

and acceleration data assuming zero measurement noise.

Then, in Section 4.2, noise is added to the setup. Online

learning will be used to decrease the state prediction

error. Finally, in Section 4.3, the beneficial influence of

online learning on changing vehicle dynamics will be

demonstrated.

4.1. Case Study I: drag estimation using noise free signals

Identification of the drag under ideal circumstances, i.e.

without sensor noise, is shown in Figs. 7a and b. Fig. 7a

shows the path followed in the horizontal plane while

Fig. 7b shows the depth profile versus time of the dive.

As the two trajectories are on top of each other it can beconcluded that identification of the damping with ideal

measurements is highly accurate. Fig. 7 shows that the

horizontal projection of the trajectory is not a circle. This is

due to the fact that neither in the reference feedforward

controller nor in the neural network feedforward con-

troller, the thruster dynamics have been incorporated.

4.2. Case Study II: drag estimation under noisy conditions

To investigate the ability of the neural network to

perform estimation of the damping using noisy training

ARTICLE IN PRESS

∧M-1

∧C

∧D ()

.

∫

g M ()

+

+

-

-

-

Fig. 6. Model of vehicle with the damping effects modelled by a neural

network.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(a) x

Y

0 5 10 15 20 25 30 35 40-3.5

-3

-2.5

-2

-1.5

-1

t [s]

z [ m ]

(b)

Fig. 7. Path followed by ppm (solid line) and cpm (with damping

identified using neural networks) (dotted line): (a) horizontal projection;

(b) depth profile versus time.



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signals, typical noise is added to the measurements. For the

linear accelerations a zero mean noise sequence with an

amplitude of 0:5 m s2 is added. The angular accelerations

are summed with a noise component of 0:05rads2. The

velocity measurements are assumed to be impeded with a

zero mean noise sequence, the amplitude of which is equal

to 0.1% of the magnitude of the velocity. It should benoted that in the simulations no noise prefiltering is

assumed. This would make matters considerably easier

for the neural network. Performance of the neural net-

works is shown in Figs. 8a and b.

Clearly, the added noise affects the prediction abilities of

the model. The prediction of the right control force, s, can

be improved by continuing the learning process online. The

result of this experiment is shown in Figs. 9a and b,

demonstrating that online learning improves the predic-

tions considerably. However, in this case the neural

network does not necessarily represent the damping any

more. Online learning results in a better local estimate of

the damping as the neural network is now trained with

larger weight on the latest data points. The globalestimation accuracy may suffer from this, unless a

considerable amount of the state space is visited and thus

used in the online learning process.

4.3. Case Study III: drag estimation for time-varying

dynamics

In this simulation the suitability of neural network based

identification methods for changing toolskids is demon-

strated. The AUV from Case Study II is equipped with a

60 kg heavy torpedo shaped sidescan sonar. As a result

several matrices, which were previously assumed to be

ARTICLE IN PRESS

-2 -1.5 1 0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

Y

(a)

0 5 10 15 20 25 30 35 40

-4

-3.5

-3

-2.5

-2

-1.5

-1

t [s]

z [ m

]

(b)

Fig. 8. Path followed by vehicle with damping approximated from noisy

measurements (ppm solid line, cpm dotted line): (a) horizontal projection;


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

X

Y

(a)

0 5 10 15 20 25 30 35 40

-3.5

-3

-2.5

-2

-1.5

-1

t [s]

z [ m ]

(b)


measurements and online updating (ppm solid line, cpm dotted line):

(a) horizontal projection; (b) depth profile versus time.



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known, will change drastically. However, changes in the

mass matrix, M (and thus the Coriolis matrix, Cðm Þ) and

the restoring forces, gðgÞ, are assumed to be known.

Changes in the damping matrix will be assumed to be

unknown and online learning is performed to adapt to

these changes. Figs. 10a and b show the trajectory for the

process plant model and the control plant model using theneural network to model damping, but without online

learning.

Clearly, the change in damping, which amounts to

parameters changing up to a factor 10, has a detrimental

effect on the performance of the feedforward controller.

Again online learning can be applied to perform an

update of the neural network knowledge regarding the

damping. The trajectories resulting from such an online

update while controlling the craft are shown in Figs. 11a

and b. Although the neural network feedforward controller

does not follow the reference path perfectly, a considerable

increase in performance is observed. Again the correspond-

ing changes in the neural network estimate of the dampingwill show local improvements while the approximation

might be worse globally. However, while operating the

vehicle further the state space visited will gradually become

a representative set. If online learning is applied the neural

networks will thus learn to represent the new damping

eventually.

5. Comparison with parameters obtained with a classical

model with linear and quadratic terms

To investigate the identification performance obtained

with the proposed identification method further, a classical

ARTICLE IN PRESS

-2 -1 0 1 2 3 4

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

X

Y

(a)

0 5 10 15 20 25 30 35 40

-3.5

-3

-2.5

-2

-1.5

-1

t [s]

z [ m ]

(b)


measurements and unknown change in damping (ppm solid line, cpm

dotted line): (a) horizontal projection; (b) depth profile versus time.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

X

Y

(a)

0 5 10 15 20 25 30 35 40

-3.5

-3

-2.5

-2

-1.5

-1

t [s]

z [ m ]

(b)

Fig. 11. Path followed by vehicle with unknown change in damping and

online learning (ppm solid line, cpm dotted line): (a) horizontal projection;




9/11

model with a linear and quadratic damping term is used to

identify the damping parameters.

D ¼ Dl þ diagðDqÞdiagðjm jÞ, (18)

where diagðDqÞ is a diagonal matrix with the six

quadratic damping terms on the main diagonal. With the

data obtained from Eq. (14), a least squares (LS)

approximation of the linear and quadratic damping

matrices is calculated. The same matrices are then extracted

from the neural network by presenting all values for m ,

obtained from the simulation, to the neural network. From

m and the neural network output, Dm , a linear andquadratic damping matrix can be calculated, again using

an LS algorithm. Table 1 presents the thus obtained

matrices.

To compare the accuracy of the two identification

methods, three error functions are defined. The criteria

used are the maximum relative error of the elements on the

main diagonal of the matrices:

E Diag ¼ maxi

abs Aestði ; i Þ AModel ði ; i Þ

AModel ði ; i Þ

,

AModel ði ; i Þa0, ð19Þ

the maximum relative error of those off-diagonal elementsthat are non-zero in the model matrix:

E Off -diag ¼ maxi ; j

abs Aestði ; j Þ AModel ði ; j Þ

AModel ði ; j Þ

,

i a j ; AModel ði ; j Þa0, ð20Þ

and the maximum error of those elements that are zero in

the model matrix, relative to the largest element in the

matrix:

E Zero-elements ¼ maxi ; j

abs Aestði ; j Þ

maxðAModel Þ

,

AModel ði ;

j Þ ¼ 0, ð21Þ

where maxðAÞ denotes the value of the largest element inmatrix A. Table 2 lists the errors thus obtained for the

newly presented algorithm and the LS algorithm.

Table 2 shows that the classical model yields less accurate

estimates than the newly proposed identification method

using neural networks even though the same LS algorithm

has been used to distill the damping parameters from the

information contained in the neural networks. Two reasons

can be identified. Firstly, an LS algorithm yields the best

linear unbiased estimate, only if the noise is uncorrelated. In

this case the velocity measurements were impeded with a

noise component proportional to the magnitude of the

velocity. A certain correlation thus exists. Due to general-

isation, the neural network may have filtered this noise

component. Secondly, during data collection, a certain

amount of outliers is collected due to the craft being exposed

to boundary conditions, being the walls, floor and surface of

the pool. The sudden and stepwise changes in velocity,

occurring at these boundary surfaces, will affect the classical

model directly, whereas the neural network again suppresses

the influence of such outliers due to generalisation.

6. Conclusions

In this work, the use of neural networks for

the identification of underwater vehicle dynamics was

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Table 1

Comparison of damping matrices identified with classical model and neural network identification scheme

Model LS NN

Dl 116 0 0 0 15 0

0 116 0 15 0 0

0 0 116 0 0 00 15 0 26 0 0

15 0 0 0 25 0

0 0 0 0 0 26

0

BBBBBBBB@

1

CCCCCCCCA

112 0 0 8 27 1

1 128 0 31 1 1

1 1 115 4 1 00 12 1 22 2 0

12 1 3 3 19 0

0 5 0 0 5 22

0

BBBBBBBB@

1

CCCCCCCCA

113 1 0 7 17 1

1 128 0 18 1 1

1 1 115 4 1 00 12 1 25 2 0

12 1 3 2 22 0

0 6 0 0 5 29

0

BBBBBBBB@

1

CCCCCCCCA

Dq 100 0 0 0 0 0

0 100 0 0 0 0

0 0 100 0 0 0

0 0 0 30 0 0

0 0 0 0 30 0

0 0 0 0 0 30

0BBBBBBBB@

1CCCCCCCCA

103 0 0 0 0 0

0 98 0 0 0 0

0 0 101 0 0 0

0 0 0 49 0 0

0 0 0 0 39 0

0 0 0 0 0 32

0BBBBBBBB@

1CCCCCCCCA

102 0 0 0 0 0

0 98 0 0 0 0

0 0 101 0 0 0

0 0 0 36 0 0

0 0 0 0 25 0

0 0 0 0 0 26

0BBBBBBBB@

1CCCCCCCCA

Table 2

Errors in estimated damping matrices for NN algorithm and LS algorithm

Dl Dq

NN LS NN LS

E Diag (%) 12 24 20 63

E Off -diag (%) 20 107 NA NA

E Zero-elements (%) 6 6 NA NA



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discussed. Neural networks are appealing candidates for

innovative identification methods because of their aptitude

for learning nonlinear mappings from input and output

data and because of the possibility to perform online

updates. Most literature sources describe the use of neural

networks in parallel with a mathematical representation of

the model, or even on its own. Learning thus becomes adifficult task as, in general, the neural network will have to

represent a complicated nonlinear function. In this work it

is proposed to use several neural networks to identify

different parts of the model separately. The advantages are

that the learning process is significantly simplified and

knowledge regarding the parameters and the model can

easily be incorporated. This procedure is illustrated in a

computer experiment in which the hydrodynamic damping

is identified. During these simulations, initially data are

gathered while a human operator controls the craft. The

data are then used to train the neural networks to represent

the damping. The accuracy of the identified damping is

tested by using the neural networks in a feedforward

control loop. To test the robustness of the neural networks,

noise and changing dynamics are introduced. Online

learning is used to improve the neural network perfor-

mance if necessary. The resulting trajectories are compared

to reference trajectories, showing that the damping has

been identified accurately with the neural networks.

Additionally, the newly proposed identification method is

compared to a standard offline least squares identification

algorithm and found to yield better estimates of the

damping parameters. As for the simulations a model as

proposed by Fossen (1994) was used, which is generally

regarded to be a truthful model, and as noise wasintroduced in the simulations, the authors expect the

proposed method to be applicable to real world problems

with noisy measurements. Such experiments are part of

planned future work.

Acknowledgements

This work was supported by the EC Research Directo-

rates through a project at the Marie Curie Training Site,

CyberMar at NTNU (HPMT-CT-2001-00382), and by the

Irish Research Council for Science, Engineering and

Technology: funded by the National Development Plan.The first author would also like to express his gratitude

towards the Irish Marine Institute for their financial

support. The reviewers are thanked for their valuable

comments.

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