SIMULATION TECHNIQUES IN BAYESIAN STOCHASTIC SHIP GUIDANCE SYSTEM
Toshio Iseki (Tokyo University of Mercantile Marine, Japan)
Daisuke Terada (M.O. Marine Consulting, Ltd., Japan)
Abstract: Simulation techniques for estimating ship response in waves are proposed. The techniques are developed for on-board marine guidance system in heavy weather operations. To simplify the system, the hull of a ship is regarded as a wave sensor and time history data of ship motions is analyzed to estimate directional wave spectra. Using the estimated spectra, ship response to all possible course and speed changes can be simulated based on short-term prediction techniques from the field of seakeeping research. Information about ship response can then be predicted before a decision to change course or speed is made.
Many on-board monitoring systems have been developed in the recent past. Most of these systems give short-term predictions about ship response based on knowledge from seakeeping research under the assumption that the seaway can be modeled by stationary stochastic processes. The authors also developed a guidance system for ship operators . This system can simulate changes in the ship response under several operating conditions. Thus, mariners can obtain future information about ship response before deciding to change course or speed.
In these procedures, the assumption of stationary stochastic processes is applied to the seaway, but not to ship response because ship response also depends on ship maneuvers. Ship response is strongly affected by changes in the encounter angle and frequency of waves. Therefore, on-board guidance systems need to be an online system that can deal with nonstationary stochastic processes and estimate the cross spectra of ship motions in real-time. For this, the authors introduced time-varying vector autoregressive (TVVAR) coefficient modeling into the cross spectrum analysis . TVVAR modeling was originally applied to analysis of the time-history data of earthquakes that exhibited strong nonstationarity [3,4]. Using the TVVAR modeling, instantaneous cross spectra of ship motions were shown to be able to be calculated during course alterations.
A recursive form of the Bayesian modeling algorithm was also proposed . This algorithm is then used to transform the original guidance system developed by the authors into an online system. In the original Bayesian modeling algorithm, the majority of computational time is consumed by a nest of iterative calculations that are used to solve non-linear equations and investigate the optimum hyperparameter. The hyperparameter controls the trade-off between a good fit to the data and the smoothness of parameter change in the model. In the proposed method, convergence of iterative calculations of the hyperparameter is not achieved at every time step, but is achieved gradually over several time steps.
In this paper, the simulation techniques for the Bayesian stochastic ship guidance system are described in detail and the results of on-board tests are shown. The tests were carried out under nonstationary conditions in order to examine the reliability of the guidance system. Finally, ship response is shown to be able to be predicted based on estimated directional wave spectra.
2. BASIC IDEA OF SIMULATION FOR ON-BOARD GUIDANCE SYSTEM
Based on short-term prediction techniques from the field of seakeeping research, the ship motion in waves and longitudinal bending stress at a certain point on the ship's hull can be evaluated without direct measurements. This means that ship motions can be predicted before a decision is made to change the ship course or speed. The procedure of prediction is based on the assumption of the linear response and illustrated in Fig.1. The most crucial problem is acquisition of the directional wave spectrum. In this system, the directional wave spectrum is estimated by two stages regarding the ship's hull as a wave sensor. In the first stage, instantaneous cross spectra of ship motions are estimated using the TVVAR modeling. In the second stage, the directional wave spectrum is estimated by using the instantaneous cross spectra according to the recursive Bayesian modeling algorithm.
Fig.1 Basic idea of the guidance system.
After estimating directional wave spectra, ship responses can be evaluated using suitably estimated transfer functions.
In this study, transfer functions were calculated using the New Strip Method (N.S.M) in which added masses and damping coefficients are calculated by the direct boundary element method known as the "close fit method", and wave exciting forces are evaluated by a sum of the Froude-Krylov forces, and diffraction forces that are approximately calculated based on the concept of relative motion. Using more accurate method, the significance of the system will be improved.
3. INSTANTANEOUS CROSS SPECTRUM ANALYSIS USING TVVAR MODEL
The TVVAR model for a nonstationary k-variate time series is defined by Akaike et al , Kitagawa & Akaike :
where ε(n) is a k -variate, normally distributed, white noise sequence with mean zero and covariance matrix V. D(n) is the coefficient matrix of instantaneous responses and Bl(n) is the k x k time-varying autoregressive coefficient matrix for the l-lag component. Assuming that elements of ε(n) are independent of each other, the fitting problem of Εq. 1 can be transformed into k independent fitting problems of the following univariate models:
The main advantage of using the TVVAR modeling procedure with instantaneous responses is the improvement in efficiency of calculations.
Assuming bijl(n) are random variables, a Gaussian smoothness prior distribution can be introduced to estimate the time-varying autoregressive coefficients more stably. The prior distribution is the stochastic constraint that controls the smoothness of changes of the coefficients bijl(n). The form of the prior distribution can be expressed as follows on the assumption that the q-th order difference of bijl(n) equals the normal white noise vijl(n) with mean zero and unknown variance τ2ijl
▽qbijl(n) = vijl(n), (i,j = l,2,・・・,k,l = O,1,・・・,p) (3)
where ▽q is the q-th order difference operator defined by the following equations:
▽bijl(n) = bijl(n) - bijl(n - 1)
▽qbijl(n) = ▽q-1（▽bijl(n)）
Therefore, the unknown variance τ2ijl can be treated as the hyper-parameter that controls the trade-off between a good fit to the data and the smoothness of parameter change in the model.
The instantaneous cross spectrum matrix is defined using the estimated TVVAR coefficient matrix as follows:
P(f,n) = B((f,n)-1VB*(f,n)-T (4)
where B(f,n) is the Fourier transform of Bl(n) and is calculated using the assumption that BO(n) = D(n) - I
The state-space representation of the TVVAR model (Εq. 2) and the stochastically perturbed difference Εq. 3 are expressed as
x(n) = Fx(n - 1) +Gw(n) (6)
yi(n) = H(n)x(n) + εi(n), (i = 1,2,・・・,k)
x(n) = (bil0(n),・・・,bi(i-1)0(n),
w(n) = (Vi10(n),・・・,Vi(i-1)0(n),
H(n) = (y1(n),・・・,yi-1(n),
and F, G are identity matrices.
Given the observations yi(n), (i = 1,・・・,k) and the initial conditions x(O | 0) and V(0 | 0) , the TYVAR coefficients x(n) can be evaluated using the Kalnian filter algorithm, whereas the parameters σi2 and τi2 can be estimated by maximizing the log-likelihood function as
where θi = (τi2,σi2)