On the use of neural network techniques in the analysis of free roll decay curves

ON THE USE OF NEURAL NETWORK TECHNIQUES IN THE

ANALYSIS OF FREE ROLL DECAY CURVES

M.R. Haddara and M . Hinchey

Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John's, Newfoundland, Canada, A I B 3X5

Int. Shipbuild. Progr., 42, no. 430 (1995) pp. 166-178 Received: February 1994

Accepted: September 1994

In this work, neural network techniques are used to analyze free roll decay curves. The nonlinearities in the roll equation are lumped in one nonlinear function in the roll angle and velocity. The technique is then used to identify this general function. This has the advantage of not having to specify the shape of the nonlinearity a priori. The technique has been applied to roll decay curves obtained using numerical simulation and experiments. The agreement between curves predicted using neural network techniques and the actual curves is excellent.

1. Introduction

System Parameter Identification (SPI) is a well known f i t to system behaviour, see Maine and I l i f f [1985]. It assumes a physical model for the system with unknown parameters. A n iteration is used to adjust the parameters such that the model response to some input matches actual response in some sense. A more recent f i t is known as Neural Network Identification (NNI), see for example Moore [1992], Hammerstrom [1993], Lau [1992a] and Sanchez-Sinencio [1992b]. It mimics the human brain and does not use a physical model. Because of this, it should be more robust than SPI.

Parametric identification of the roll motion equation started as the only route for determination of the roll damping parameters, see Froude [1955]. During the past fifteen years, several methods were developed to obtain more accurate estimates for the damping using free and forced roll records, see Dalzell [1978], Mathiesen and Price [1984], Roberts [1985], Bass and Haddara [1988], Haddara and W u [1993]. Recently, the problem of parametric identification was extended to obtain estimates for the restoring moment f r o m measurements of roll motion excited by random waves, see Roberts et al [ 1991 ] , Haddara and W u [ 1993].

problem o f multiple solutions. This is the result of a cancellation effect wherein compensating errors in some o f the coefficients produce a reasonable prediction o f the motion,

A newly developed technique using neural networks provides a means f o r the identification of the nonlinearity in the roll equation without a priori knowledge of the form of the nonlinearity. Once the function has been identified, one can then study the funcrion with the objective o f determining the best f o r m f o r this function. I n this work, we use a neural network technique to identify the damping and restoring moments f r o m the free roll decay curve. The damping moment and the nonlinear part of the restoring moment are lumped together in a single function of the roll angle and velocity. A n expression showing the dependence of the function on the roll angle and velocity is derived using weights obtained from training a neural network to simulate the r o l l response. The technique is applied to both numerically generated and experimentally measured roll data. Excellent agreement is obtained between the actual response and the neural network output.

2. Roll equation of motion

The equation of free roll motion can be written as Bass and Haddara [1988]:

where (p is the roll angle in radians, N is the nonlinear roll damping moment per unit virtual mass moment o f inertia and D is the nonlinear restoring moment per unit virtual mass moment o f inertia. For the purpose of the analysis presented here, we are going to rewrite equation (1) as:

The rationale behind writing the equation in this f o r m is that the natural frequency COQ, can be easily obtained f r o m the roll decay curve. We recognise the fact that the

'^ + N((t>S+D{(p) = 0 (1)

164 On the UseqfNeuralNetworkTedmiques k the Analysis of Free RoU Decay Cunes

natural frequency obtained f r o m the free roll decay curve is the damped nonlinear natural frequency, while Ob is the natural frequency of the linear undamped roll equation. Corrections for this discrepancy can be estimated after the damping level has been obtained, Haddara and W u [1993].

3. Neural network identification

In a biological network such as that shown in Figure 1, dendrites feed information into each neuron. A t each dendrite/neuron interface, a synapse controls the strength of the signal fed i n . Synapses are basically memory. Each neuron sums its inputs and fires or sends out a signal along its single axon i f the sum is above a certain threshold. The axon branches into dendrites which feed into other neurons. A r t i f i c i a l neural networks try to mimic the biological network. They can be used to form input/output (1/0) maps or fits, see Moore [1992], Hammerstrom [1993], Lau [1992a] and Sanchez-Sinencio [1992b]. A typical mapping net is shown schematically in Figure 2. It consists o f an input layer, a hidden layer and an output layer of neurons. The input and hidden layers each contain one bias neuron with input unity. The summed inputs into hidden layer neurons are processed by a squashing function when they pass through the neuron: the sigmoidal or S shape used by most researchers for this is given by:

Squashing is important for good input/output mapping because it simulates the firing action o f biological neurons. Information flows forward through a net f r o m input to output. For a single input/single output case, the mapping equation for a single neuron is:

(3)

A}(ori

Figure 1. Biological network.

This equation shows how the weights can be used in an I/O plot to scale and shift both horizontally along / and vertically along O the squashing function. In the general case a map is formed by patching together many scaled and shifted squashing functions. This gives the net a local character, something like that o f finite elements, and this is why very good fits are possible. The loss of one or more neurons i n a large net does not destroy the ability of the net: like biological networks, artificial nets are said to degrade gracefully.

For the ship suspension case, the goal is to find the weights WQ and Wj which make the response of the system with the net suspension match the corresponding measured or target responses Oj. Most neural network codes use steepest descent to adjust the weights in an iterative fashion starting f r o m a random distribution. One starts this training by first forming for response the error:

where M is the output f r o m a hidden layer neuron. The goal is to minimise Xe^. For each input/output training pair the squared error is dependent on each WQ and M feeding into it. For small changes in WQ and M, one can write:

G = ON-OT where 0^ = 1, WQM (5)

166 Oniie Use qfNewdNetworkT&Mques in ^Anafysis cfFreeRoEDea^ Curves

N o w M = / [ Z W / / ] - Manipulation gives AM - X9M/3W/ AWj. So, the Ae^ equation becomes: ^

Ae2 = JiA&^ldWo] AWo

+

iaM/9W/

According to steepest descent:

Wo(NEW) = ^^o(OLD) - Kde^ldWo

M^KNEW) = WXOLD) - K{dêldM\ dMIWi.

(8)

When fed many times with each training pair in an input/output training set, this iteration when stable usually backs down the global Xe^ versus weight bowl in hyperspace towards the global minimum. A small value of the parameter /c promotes iteration stability. To avoid convergence to local minima, noise added to the weights could be used to periodically shake up the iteration.

N N I has several drawbacks. For example, N N I is an abstract concept: unlike parameters in SPI, weights i n N N I have no direct physical meaning like mass, say in SPI. Also, steepest descent training is often slow, especially i f started on the large / sections o f the squashing function. This is because the sensitivities used to adjust the weights Wj contain derivatives of M with respect to Wj. M is basically the squashing function. On its large / sections, slopes are very small: thus, weight corrections can be insignificant. Other training methods, such as genetic and reinforcement learning, are supposed to be faster but are not as well developed. More than one hidden layer could have been used for the f i t . However, one is supposed to be sufficient and nets with one train much faster. A very large number of hidden neurons could have been used to improve the f i t . However, too many neurons would give a net which is not good at generalisation.

Figure 3 shows a block diagram of the neural network used for roll motion under study. Inputs into the net were roll angle 0 and roll rate(/> at the last time step. Suspension system force during the step was the only output. The sensitivities used to train the net were determined numerically. There were 6 neurons in the hidden layer.

Figure 2. Neural network.

Figure 3. Block diagram for the neural network used in roll parametric identification.

4. Results and discussion

The technique described i n this paper can be used to predict the damping and restoring characteristics for a ship or a model using the data obtained f r o m a free roll decay experiment. I t is assumed that roll motion is described by equation (2). Thus, the damping moment and the nonlinear part of the restoring moment (per unit virtual moment of inertia) o f the ship are both lumped together in one function F(0,^). The network shown in Figure 3 can be used to identify the function F(0,^). The function is expressed as a nonlinear function o f the roll angle and velocity using two sets o f weights obtained f r o m the neural network algorithm as

168 On the Use cfNeural Network Tedmiques in die Analysis of Free RoUDecay Curves

F(U)=j;^Wo^'^Mk

k=0

(9) 2

where W[, Wo are the synaptic weights to the hidden and output layers respectively. Xi, / = 1, 2 are the inputs to the input layer. The weights WP^ and Wo^^^ equal the thresholds applied to input and the hidden layers. is the output o f the k-th neuron in the hidden layer.

Having obtained the two sets of weights (Wj ,Wo) for a ship or a model, one can use equation (9) to evaluate the roll damping moment and the nonlinear component o f the restoring moment f o r the ship or model. This can be used in generating simulations or predicting rolling motion responses for the ship. The function F(0,0) can also be used to identify the damping and restoring moment parameters. For example, the linear damping coefficient can be obtained by plotting as a function o f ^, and evaluating its first derivative at the origin. Other parameters can be obtained using regression techniques.

Numerical results were obtained using three sets of data. The first set was obtained using a numerical simulation technique. Appendix I gives the numerical model used in the generation of this roll decay curve. Figure 4 shows a comparison between the predicted curve obtained using the set of weights (Wi,Wo) generated using the neural network technique and the original curve used in the training of the network. The second and third sets were obtained experimentally for a model o f a small fishing vessel, see Haddara et al [1994]. These were used to train the neural network shown in Figure 3 to predict the function Figures 5 and 6 show the comparison between the predicted and the actual free roll decay curves, i n these cases. Table 1 gives details o f ship models used i n the experiment. Table 2 shows the weights obtained f o r the three cases. These results were obtained after 8000 iterations and took about ten minutes on a U N I X computer.

Mass (kg) 69.5 G M ( m ) 0.0451

li\ 0.2024

H2 -1.8402

Figure 4. Comparison between an simulation of a free roll decay curve and predicted curve obtained using the neural network algorithm.

170 Oniie Use cfNeurdNetwork Tedmiques in ^ Analysis of Free RoUD&xiy Curves

Table 2.

Weights for Figure 4 • Weights for Figure 5 Weights for Figure 6

Wo Wo Wo -0.2163337143870948 -0.9869155213955854 -.94564 -8.4832577928796171E-02 1.449339143282372 .914207 -0.2418008971019680 4.2417993439725077E-04 .745197 -0.4728612962382309 -0.7660371501831307 -1.10717 -0.5247821957436652 -0.3142954498319718 -.42457 -6.5523716509715349E-02 -0.7196355230608023 -1.4144 Wi Wi Wi -1.7063125547816169E-02 -0.7776201151795272 -.17934 9.9780607195839421E-02 -4.900165947885665 -4.76333 -0.1000541295507047 1.445221419397757 1.193271 -7.6145590185524747E-02 -0.3949891538912471 .349404 -1.4741938815798402E-02 -2.503964716290204 -2.27513 -9.98283 14992486789E-02 0.3445763879986841 -.27677 -6.1030773958165082E-O3 -0.821145001 1250645 -1.60623 0.1242844406009797 -4.007945180023325 ^ . 3 0 2 5 7 -9.891763 967085223 7E-02 1.795805941174089 1.798825 9.8326016389495258E-02 -1.674559193786484 -2.10721 0.3286897589254794 5.026113455707549 4.409446 -9.2803074788323309E-02 -4.171572649984453 -4.99372 0.1229006610393272 -1.679016623875262 -2.12594 0.3815690575225337 5.057957670317249 4.462557 -8.6336936483750612E-02 -3.812636679001168 -4.55494

It was found that roll angle and velocity are needed as inputs to the network in order that we get good agreement between the initial roll angle record and the network output. In the numerical simulation case, both roll angle and velocity were generated digitally and were fed into the network. However, in the case of experimental data only roll angle history was available, so the roll velocity record was generated using numerical differentiation. The agreement between the actual roll angle input and the predicted roll angle record is seen to be excellent in all cases.

Figure 5. Comparison between an experimental free roll decay curve and predicted curve obtained using the neural network algorithm.

Figure 6. Comparison between an experimental free roll decay curve and predicted curve obtained using the neural network algorithm.

172 Onthe Use of Neural NetworicTedmiques in ée Analysis ofFreeRoUDecay Curves

This technique has great potential in advancing the state of the art of roll motion study. We have demonstrated in this work that one can predict with great accuracy, general functional form for the damping and restoring moments in the roll equation f r o m the roll decay curve. There are several points being pursued by the authors in the meantime. These include extending the method to the case of forced roll equation excited by regular and random waves, and expressing the general function F(0,0) in terms o f the roll angle and roll velocity in a way which w i l l be physically meaningful. A method can then be used for estimating ship's transverse stability at sea.

References

1. Bass, D . W . and Haddara, M.R., 1988, "Non-Linear Models of Ship Roll Damping Moment", International Shipbuilding Progress, vol. 35, no. 401, pp. 5-24.

2. Dalzell, J.F., 1978, " A Note on the Form of Ship Roll damping". Journal of Ship Research, vol. 22, no. 3, pp. 178-185.

3. Froude, W., 1955, "The Papers of W i l l i a m Froude", The Inst, o f Naval Arch., London.

4. Haddara, M . R . and W u , X . , 1993, "Parameter Identification of Nonlinear RolHng Motion in Random Seas", International Shipbuilding Progress, vol. 40, no. 423, pp. 247-260.

5. Haddara, M . R . , Wishahy, M . and W u , X . , 1994, "Assessment o f Ship's Transverse Stability at Sea", to be published in Ocean Engineering.

6. Hammerstrom, D., 1993, "Working with Neural Networks", IEEE Spectrum, July 1993.

7. Hinchey, M . , 1993, "Neural Network Fits to Ship Data", Proceedings o f the 12th International Conference on Offshore Mechanics and Arctic Engineering, vol. I , Offshore Technology, Glasgow, Scotland, pp. 263-266.

8. Lau, C. ed., 1992a, "Neural Networks: Theoretical Foundations and Analysis", published by IEEE press.

9. Maine, R.E. and I l i f f , K . W . , 1985, "SPI: Identification of Dynamic, Systems: Theory and Formulation", N A S A RP 1138.

10. Mathiesen, J.B. and Price, W.G., 1984, "Estimation of Ship Roll damping Coefficients", Trans, of the Royal Inst, of Naval Arch., pp. 295-307.

11. Moore, K., 1992, "Artificial Neural Networks", IEEE Potentials.

12. Roberts, J.B., 1985, "Estimation of Nonlinear Ship Roll Damping From Free-Decay data". Journal of Ship Research, vol. 29, no. 2, pp. 127-138.

Adcmov/ledgemenlt

This work has been financially supported by funding f r o m the Natural Science and Engineering Research Council of Canada.

A P P E N D I X I

The equarion o f motion used in generating the numerical free roll decay curve is given by:

^ + 2C(Oo[<^ + e<^] + (OQ\<P + ^ I < / ^ + ^2^] = 0 ( I - l )

where ^ = 0 . 0 1 1 6 , Oo = 3.4468 rad/sec, (1-2) £ = 3 . 2 0 4 , ^ 1 = 0 . 1 4 8 , ;U2 =-1.5676 Nomenclature / ( • ) Squashing function.

F{(p,^) A function comprising the roll damping moment and the nonlinear part of the

restoring moment.

ƒ Input to a single neuron net.

Mk Input to the k-ih. neuron in a hidden layer.

O Output of a single neuron net.

ON Network output •: OT Target output

174 On^e Use cfNeurdNdwoHcTedmiquesiniie Analysis ofFree RoU Decay Curves

WJ Synaptic weights applied to inputs to the neurons in the hidden layer. WjB Threshold applied to neurons in the hidden layer.

Wo Synaptic weights applied to inputs to the neurons in the output layer. WoB Threshold applied to neurons in the output layer.