4.2 Dealing with the Step Steering Effects

　

 According to the former assumptions and the linearization condition, the equations do not take nonlinear effects into consideration. However, the full-scale tests are zigzag steering and the rudder deflection was acutely changing during steering. This suggests dealing with the sample input data by eliminating the step steering phase, which does not satisfy the linearized equations.

　

 To testify the proposed method and also evaluate the assumptions, a special kind of NN training uses the data, among which the steering phase is eliminated. Under such condition, the assumptions and Eq. (2) can be fully satisfied.

　

 Using the special trained NN, the whole time serial simulation is calculated. Fig.5 and Fig.6 show the results.

　

 Because the special data does not include step steering, it was treated as mild maneuvering and linear, when the NN learns. But the data of simulation do include acute steering, that means each equation of Eq. (2) has to add one steering induced term, which is usually called nonlinear term.

　

Therefore, Eq. (2) becomes:

　

　

 From the figures, jumps emerge and the time serials of jumping appearance are well coincident with the measured time serials as shown by the arrows.

　

4.3 Simulation of the Test4 by Using the Test3 Trained NN

　

 Because there are too many factors to decide the barge train maneuvering motion, it is possible that the jumps are produced by external disturbance. This subsection simulates the upstream Test4 using the special trained NN based on the measured data of Test3. Identically, the time serials of jumps of Test4 simulation are also coincident with the measured time serials, as shown in Fig.7 and Fig.8.

　

 Thus it can be claimed that the jumps are just because of the nonlinear effects induced by rudder steering.

　

4.4 The NN Training Including the Data of Steering Phase

　

 The nonlinear effects are discussed in previous sections, which seem very troublesome. But, what's fortunate is that the trouble can be smoothed away by the NNRM. In fact, it is really a good helper for nonlinear dealing. It can be summarized as follows: you give linear data to it, you can get good result; you give nonlinear data to it, you still get good result automatically with nonlinear effects included.

　

 So we give the nonlinear data to train the model by restoring the steering phase. During the same full scale test (we choose the Test3), the data at the middle range of time is chosen as sample data sets, and the rest data is used for validation and prediction of the barge train motion.

　

 The simulated motion, i.e. the sway velocity v' and the yaw rate r are shown in Fig.9 and Fig.10. As we expected, the jumps disappeared. The results show satisfactory even with the acute rudder deflections included.

　

4.5 Generalization of NNRM

　

 To further confirm our reasoning and to fully understand the NN recursive work, the NNRM is generalized by simulating Test4 while using the well trained NN model based on the sample data of Test3. Fig.11 and Fig.12 show the results.

　

 To predict all conditions the testing barge train met during maneuvering motion in inland waterways, the NNRM is generalized to both upstream voyage and downstream voyage. Hereby, the data obtained when the barge train went downstream are treated as the sample data to predict its maneuvering motion upstream.

　

 Although the upstream and downstream are quite different in external disturbance, the results display somewhat satisfactory. It can be seen that the error exists almost through the simulation time. This can be possibly explained by the full cover of the training data all through the time serials.

　

 The error is probably produced by the following sources: a) the data record error; b) the error s accumulation on time; c) the difference between upstream and downstream; d) other effects, because the hill-scale tests were made under strong current and in restricted, bumpy grounded and shallow waterway.

　

4.6 Trajectory Track Using the NNRM

　

 This subsection is trying to simulate the trajectory point using the proposed NNRM, which may be able to track the actual trail of the barge train. Fig.13 shows the simulated result compared with the measured one. It can be seen that the tracking result has some drift.

　

 In Section 2, the equation deduction is based on the assumption that the force change in x direction is small. But in fact, the force in x direction is not constant due to some effects, such as the river current during the test. The prior assumptions are just for deduction of the NNRM. The NN training has taken some uncertain factors, but not the force change, into account. This error source may explain the trajectory drift shown in Fig.13. It seems that the trajectory of barge trains in inland waterways is not easy to precisely track. But from what is shown in Fig.13, using the proposed method, the track's trend is satisfactory.