modeling of underwater vehicle’s movement dynamics … · modeling of underwater vehicle’s...

Modeling of Underwater Vehicle’s Movement Dynamics Using Neural Networks

BOGDAN ŻAK Department of Mechanical and Electrical Engineering

Naval University of Gdynia 81-919 Gdynia, Smidowicza 69

POLAND email: [email protected]

Abstract: - In the paper the conception of creating the simulation model of underwater vehicle was introduced. There was briefly presented the mathematical model of vehicle using to the simulation. The paper includes also mathematical description of environment disturbance. Moreover in the paper were presented the mathematical description of all components of neural networks using for creating dynamics model. Also paper includes the examples - results of researches. Keywords: Underwater Vehicles, Identification, Modelling, Neural Networks, Simulation. 1. Introduction Through unmanned underwater vehicle will be understood the unmanned object, entirely plunged in water, possessor six degrees of freedom and linked with base through the wire or hydroacoustic channel. In dependence from destination unmanned under-water vehicles have different appearance, configuration and different equipment [4], [7] (see Figure 1). For control of all devices of underwater vehicle respond the system of steering which holds supervision beginning from the propulsion and finishing on switching the video cameras and lighting. In underwater vehicle’s control system the functions of regulator in feedback the most often is realized by operator which is the observing working area on monitor installed in deck console. The control of movement of underwater vehicle happen in three-dimensional space with use the joystick [3], [8], [14]. The movement of underwater vehicle is possible in every direction it means: up and down, forward and back, turn left and right and move to the left and to the right. Across system of switches and manipulators the operator defines the specific work’s parameters of underwater vehicle, after processing through computer they come across as control signals to particular units of underwater vehicle such as propulsions which determining position and parameters of movement of underwater vehicle[6], [9], [15]. The sensors of under-water vehicle make measurement of parameters such as: the depth of immersion, speed bearings etc. and after pass on it,

across computer, to the operator, which gives the information about quality of control [9].

Fig. 1 The example appearance of underwater vehicles

The main problem of creating simulators is to posses the good quality model of dynamics of movement of underwater vehicle. In another hand a problem of modelling of dynamic objects is one of the most important tasks during designing of control systems [10], [11]. Theoretical and practical researches were focused on problems of mathematical modelling and especially on methods of parametric identification were developed [2], [5], [12]. Those methods were very efficient for both linear statically and dynamical objects but not for non-linear ones. In recent years the neural networks have been perceive as very good tools for creating the models of multidimensional, nonlinear objects [13]. The attractive of use the neural networks for modelling objects dynamics is approximation of any curve and to tune the structure basely on experimental and another data [1], [11].

Proceedings of the 11th WSEAS International Conference on Automatic Control, Modelling and Simulation

ISSN: 1790-5117 269 ISBN: 978-960-474-082-6

mailto:[email protected]

2. Mathematical model 2.1 Underwater vehicle During analysis of movement of sailing objects with six degrees of freedom the two coordinate Cartesian systems are defined, which was introduced on figure 2. The one of them is related with sailing object and is called the body-fixed reference system [2], [4], [5]. The movement of body-fixed reference system is described in coordinate system related with the Earth which is called the earth-fixed reference system. It is suggested that orientation of vehicle is described in body-fixed frame meanwhile angular and linear velocities should be described in earth-fixed frame. Magnitudes of movement are described according to SNAME notation (see Table 1). Table1. Notation using to describe movement of sailing objects

DOF Name of movemet Forces and moments

Linear and angular velocity

Positions and Euler angels

1 Surge (motions in the x-direction) X U X

2 Sway (motions in the y-diretion) Y V y

3 Heave (motions in the z-direction) Z W z

4 Roll (rotation about the x-axis) K P φ

5 Pitch (rotation about the y-axis) M Q θ

6 Yaw (rotation about the z-axis) N R ψ

Fig. 2 Body-fixed and earth-fixed reference systems

The non-linear equations of underwater vehicle’s movement treated as rigid body can be written as follows:

(1)

NqruyurppxmpqIIrI

MpuqxqruzmrpIIqI

KurpzpuqymqrIIpIZprqyqrpxqpzpuqm

Yrqpxpqrzpryurpm

Xqprzrpqyrqxqrum

GGzyz

GGzxy

GGyzx

GGG

GGG

GGG

=+−−+−+−+

=+−−+−+−+

=+−−+−+−+=++−++−+−

=++−++−+−

=++−++−+−

)]()([)(

)]()([)(

)]()([)()]()()([

)]()()([

)]()()([

22

22

22

ωυωυ

υωωυ

ωυυωυω

ωυ

ωυ

&&&

&&&

&&&

&&&

&&&

&&&

where: m – vehicle mass; – moments of inertia in relation to

symmetry axis of vehicle; zyx III ,,

– co-ordinates of centre of gravity. GGG zyx ,,The general representation of equation of movement in body-fixed frame can be written as:

τη =+++ )()()( gvvDvvCvM & (2) where: η – vector of state; τ – vector of control; – vector of velocities; v M – the matrix of vehicle masses and added water

masses; – the matrix of centripetal and Coriolis forces; )(vC – the matrix of hydrodynamic dumping; )(vD )(ηg – the matrix of restoring moments and forces. The creating of mathematical model of underwater vehicle is the complex problem. It is difficulty to delimitate or calculate many parameters, which has to be well-known to solve the equations of movement. It is possible to reduce the number of parameters making the certain assumptions related with vehicle’s construction such as: the symmetry of vehicle in different surfaces, the position of centre of gravity and centre of uplift pressure, and suitable selection of reference system origin. 2.2 Environment disturbance To the basic environment’s disturbances, which are considered at analysis of sailing objects belongs:

- waviness; - wind; - underwater currents.

At examination of objects entirely plunged such underwater robots it is possible to omit influence of wind on object, meanwhile waviness matters only to the depth of 10 meters. For requirements of simulation only underwater currents disturbances will be considered.


ISSN: 1790-5117 270 ISBN: 978-960-474-082-6

Forces and moments induced by water currents are taken into consideration in equations of dynamics of movement at assumption that the equations of movement can be represented in form of relative velocities:

cr vvv −= (3) where: – is the vector of un-rotary

velocities of currents in body-fixed reference frame.

Tcccc vuv ]0,0,0,,,[ ω=

It should be assumed, that the vector of velocity of water current in earth-fixed reference frame will be recorded as: . From that we can calculate component occurrent in body-fixed frame using Euler’s theory.

],,[ Ec

Ec

Ec vu ω

We found that the velocity of current in body-fixed frame is constant or slow-changed so that fulfils:

vvv rc &&& =⇒= 0 (4) From this the equation of motion (2) take form:

τη =+++ )()()( gvvDvvCvM rrrr& (5) 3. Neural network for modeling Because the processes taking place during movement of underwater vehicle have dynamic character, neuronal modeling such cases requires to use special solutions [1], [3], [10], [12]. One of the possibilities is to use the recurrent neural networks as Hopfield’s networks or the simplest backpropagation neural network with recurrent connections. It the last case the effects of dynamics behaviour is obtain by widening entrance vector with the information of value of model’s output in previous moments. In last investigations in aim of receipt of dynamic character of neural network the dynamics becomes introduced to the neuron in such way so output of neuron depends from his internal states. The interesting solutions is to model the weights of neuron by discreet operational transmitation G . In such case the equation of neuron accepts form:

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

=

−N

iii kxzGFky

1

1 )()()( (6)

where: - index of discreet time; - numer of inputs; - linear discreet operational transmitation.

k NG The taking into account the way of introduction of dynamics in equation, every neuron in neural dynamic network reproduces the past values of signals having at command two signals: entrance signal and output signal. The network for modelling dynamics of unmanned under-water vehicle it was built from neurons which heve one output and many inputs (see Figure 3).

Fig. 3 The model of dynamic neuron

In block of summation follows the calculation of sum of weighed information using the dependencies as follow:

∑=

=N

iii xwk

1)(ϕ (7)

Calculated weighted sum is processed in filter block which can be the linear dynamic system of any grade. This structure can be describe with following differential equation:

)(...)1(1)(0

)(...)1(1)(

nknbkbkb

nknakak

−++−++

+−−−−−=

ϕϕϕ

γγγ (8)

where: ϕ – input of filter block; γ – output of filter block; – the coefficient of backward and forward connections (see Figure 4).

ba,

Fig. 4 The model of filter block

The output signal of neuron which is output signal of activation block is calculated from equation:

)]([)( kgFky sγ= (9) where: F – the non-linear function of activation; – the coefficient of inclination of function of activation.

sg

The main aim of teaching algorithm is to determine the values of every neuron parameters such as the values of weights , value of filter parameters and , and the coefficient of inclination of activation function . Based on data the set of the pair of input and output patterns using the general methods of neural network teaching it is possible to calculate the values of all parameters.

w a bsg

In research the method of changeable metrics is used for neural networks parameters modification. Accepted function of detination has the following form:


ISSN: 1790-5117 271 ISBN: 978-960-474-082-6

( ) ( )[{ 2

21 kykyEJ d −= ] } (10)

where: E – operator of expected value, – desired response of neural network, – actual response of neural network.

)(kyd

)(ky

Correction of network parameter is done according to the formula:

( ) ( ) ( )( )[ ] ( ){ }kJkHkk υηηυυ ∇−+=+ −11 (11) where: υ - the generalized network parameter, η - learning rate, H - hessian in the point of last solution,

- value of gradient. (kJυ∇ ) This equation describes the Newton’s algorithm of optimization. This formula is strictly theoretical because it required to assure the positive determinateness of hessina in ever step of iteration. Because of that in practical implementation the approximation of hessian value is used. In research the approximation according to the equation of Broydena-Goldfarba-Fletchera-Shanno which process of matrix’s values update describes by recursive formula. Using the following notation: ( ) ( ) ( )( ) ( ) ( )( ) ( )( )[ ] 1

11

−=

−∇−∇=−−=

kHkV

kJkJkrkkks

υ

υυ

υυ

this equation can be written as:

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

( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( )krks

kskrkVkVkrks

krksksks

krkskrkVkrkVkV

T

TT

T

T

T

T

11

111

−+−−

+⎥⎥⎦

⎤

⎢⎢⎣

⎡ −++−=

At the begining the value of V is equal 1.

Fig. 5 The teaching of neural network for modeling of

underwater-vehicle dynamics To build the neural network which will simulate dynamic of unmanned underwater vehicle the dynamic

neurons where used. The structure of proposed neural network is similar to static one way multi layer neural network but in place of static neurons the dynamic neurons where used (see Figure 6).

Fig. 6 Structure of neuronal model of unmanned under-water vehicle Input coset consists of elements which task is to transmission the inputs signals to the first hidden coset of neural network. The number of neurons in input coset of neural networks is determined by control vector size, and number of neurons in output coset of network in equal to size of observed state vector. The hidden coset configuration was specified experimentally. The connections between neurons were made each other to each other between neighboring cosets without recurrent connections and connections which skip any coset because of presented neural networks feature. 4. The results of research The simulating investigations were conducted for underwater vehicle about mass 110 kg and dimensions 1500[mm] x 750[mm] x 750[mm]. Simulating model of this vehicle was built as neural model from introduced mathematical model peaceably. For verification of neural model of underwater vehicle the simulation research were made. The control vector for simulation has form as:

],,[ NZX τττ (12) where:

Xτ - force along X axis of vehicle;

Zτ - force along Z axis of vehicle;

Nτ - rotational moment around Z axis of vehicle. The values of each components of control vector were setup through the deflection of manipulator’s


ISSN: 1790-5117 272 ISBN: 978-960-474-082-6

sticks. The state vector accepted for research composed of linear and angular velocities and has form as:

],,,,,[ rqpwvu (13) where: u - linear velocity along X axis; v - linear velocity along Y axis; w - linear velocity along Z axis; p - angular velocity around X axis; q - angular velocity around Y axis; r - angular velocity around Z axis. Based on such measured state vector it is possible to calculate the rest parameters of motion of underwater vehicle such as position and an angle of inclination around the main axis of vehicle.

Fig. 7 Trajectory of underwater vehicle

Fig. 8 The values of control vector during simulation

Fig. 9 The values of linear velocities

Fig. 10 The values of angular velocities

During research the many test using the neural model of vehicle dynamics along the presented above methods were made. All parameters described the motion of vehicle and the neural model state, during simulations were registered for further analysis. The choosed research results are presented below on figures. In the research the operator gave the signals extortionary by


ISSN: 1790-5117 273 ISBN: 978-960-474-082-6

Joystic giving the same the trajectories of movement of underwater vehicle. Realized trajectory in time of simulation be recorded in three-dimensional space, extortionary signals generated through engines as well as co-ordinates of state this is in all progressive speeds three axises, angles of inclination as well as angular speed the regard of coordinate axis. It the chosen results of simulating investigations were introduced (see Figures 7 to 10). On the figures the bleck lines means values measured for object, blue lines means values calculated from neural network based model.

Fig. 11 The values of square average error

5. Conclusion The results of simulations shows, possibilities of the presented neural network for modelling of underwater vehicle dynamics. Researches show that neural model is very good adjustment to the objects taking as coefficient the square average error. Model based on neural network is much more better then calculated using common methods. After teaching neural network has possibilities to generalize the knowledge such as produce the correct outputs in new not presented before inputs. Such behaviour can’t be reached using numerical methods. The influence of environment a speciality the underwater currents if information about them is presented to the networks are very good taking into account during reconstruction of objects state. This method can be used to the time varying identification.

References: [1] P.Li.K. Connally, G.W. Irwin, Prediction and

Simulation error. Based Perceptron Training: Solution Space Analysis and a Novel Combined Training Scheme. Neuro Computing, Vol. 70, 4-6, January 2007, ISSN: 0925-2312, pp.819-827.

[2] T.I. Fossen, Guidance and control of ocean vehicles. John Wiley & Sons, Chichester,1994.

[3] J. Garus, J. Małecki, B. Żak, Application of Network Hopfield’s for Modelling of Motion Ship,

XIII Native Control Conference, Poland, 1999, pp. 211-214.

[4] M. Gerler, G. Hagen, Standard equations of motion for submarine simulation. Technical Report, Defense Technical Information Center document #A653861, David Taylor naval Ship Research and Development, 1967.

[5] K. Goheen, E. Jefferys, The application of alternative modeling techniques to ROV dynamics. Proc. of the IEEE International Conference on Robotics and Automation, 1990, pp.1302–1309,

[6] D.J. Klein, P.K. Bettalle, B.I. Triplett, K.A. Morgansen, 2008. Autonomous underwater multivehicle control with limited communication: Theory and experiment. In Proceedings of the Second IFAC Workshop on Navigation, Guidance and Control of Underwater Vehicles, 2008.

[7] A.E. Kunkel, K.A. Morgansen, Buoyancy control of an autonomous fin-actuated underwater vehicle. In Proceedings of the AIAA Region VI Student Conference, April. Winner of Undergraduate Third Place Prize, 2006, pp. 1168-1176.

[8] K.A. Morgansen, B. I. Triplett, D.J. Klein, Geometric methods for modeling and control of free-swimming fin-actuated underwater vehicles. IEEE Transactions on Robotics, 23(6), 2007, pp. 1184-1199.

[9] K.A. Morgansen, T.M. La Fond, J.X. Zhang, Agile maneuvering for fin-actuated underwater vehicles. In Proceedings of the Second International Symposium on Communications, Control and Signal Processing, 2006, pp. 586-594.

[10] W.T. Miller, R.S. Sutton, P.J. Werbos, Neural Networks for Control. The MIT Press, Cambridge, MA, 1990.

[11] K.S. Narendra, K. Parthsarathy, Identyfication and control of dynamical systems using neural networks. IEEE Trans Neural Networks, Vol.1, No.1, 1990, pp. 4-27.

[12] J.Li.K. Peng, G.W. Irwin, A novel continuous forward algorithm for RBF neural modeling. IEEE Transactions on Automatic Control, Vol. 52, No. 1, January 2007, pp. 117-122,

[13] B.Li.K. Pizzileo, G.W. Irwin, A fast fuzzy neural modelling method for nonlinear dynamic systems. International Symposium on Neural Networks, June 2007, Nanjing, China.

[14] The Society of Naval Architects and Marine Engineers: Nomenclature for treating the motion of submerged body throught a fluid. Technical and Research Bulletin No. 1-5, 2005.

[15] B. Żak, J. Małecki, Z.Kitowski, Modelling of ship’s motion using artificial neural networks. World Scientific and Engineering Society Press, Inc., MA 01923 USA, 2001, pp.298-303.


ISSN: 1790-5117 274 ISBN: 978-960-474-082-6

modeling of underwater vehicle’s movement dynamics … · modeling of underwater vehicle’s...

Documents