XU Feng

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: hljxufeng@163.com

ZOU Zao-jian

School of Naval Architecture, Ocean and Civil Engineering and State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

YIN Jian-chuan

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

CAO Jian

College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

(Received March 5, 2012, Revised April 22, 2012)

PARAMETRIC IDENTIFICATION AND SENSITIVITY ANALYSIS FOR AUTONOMOUS UNDERWATER VEHICLES IN DIVING PLANE*

XU Feng

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: hljxufeng@163.com

ZOU Zao-jian

School of Naval Architecture, Ocean and Civil Engineering and State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

YIN Jian-chuan

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

CAO Jian

College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

(Received March 5, 2012, Revised April 22, 2012)

The inherent strongly nonlinear and coupling performance of the Autonomous Underwater Vehicles (AUV), maneuvering motion in the diving plane determines its difficulty in parametric identification. The motion parameters in diving plane are obtained by executing the Zigzag-like motion based on a mathematical model of maneuvering motion. A separate identification method is put forward for parametric identification by investigating the motion equations. Support vector machine is proposed to estimate the hydrodynamic derivatives by analyzing the data of surge, heave and pitch motions. Compared with the standard coefficients, the identified parameters show the validation of the proposed identification method. Sensitivity analysis based on numerical simulation demonstrates that poor sensitive derivative gives bad estimation results. Finally the motion simulation is implemented based on the dominant sensitive derivatives to verify the reconstructed model.

parametric identification, Autonomous Underwater Vehicles (AUVs), support vector machine, sensitivity analysis

Introduction

Autonomous Underwater Vehicles (AUVs) are intelligent robots employed to carry out predefined underwater tasks. They play an irreplaceable role in the exploitation and utilization of ocean resources due to its non-substitutable superiority for risky and tiring undersea work. In recent years, AUVs are widely used for oceanographic survey, target detection, underwater rescue, mine hunting and so on. Take REMUS for example, it has successfully collected the density, sali-nity, temperature, water quality and other useful information[1]. To realize energy efficient AUV with low resistance, low energy consumption and excellent maneuverability, the researchers devoted themselves to improving the performance of the AUV modular, and conventional only-propeller- driven pattern is replaced by propeller-fin-driven model[2]. Consequently, the research on maneuvering performance of AUV with horizontal fins in diving plane is more significant, because the maneuverability in the horizontal plane is more or less the same as surface craft which has been extensively developed.

In the research on AUV?s maneuverability in diving plane, the pivotal problem is precisely determining the hydrodynamic coefficients. For calculation of inertia coefficients, the common methods are captive model tests by Planar Motion Mechanism (PMM), regression estimation method with empirical formulaand calculation method using potential theory. There are four frequently-used methods for calculation of viscous coefficients, which are empirical formula method, captive model test, Computational Fluid Dynamics (CFD) method based on viscous flow and method of free-running model or full-scale tests in combination with system identification. With the development of experimental measurement techniques and system identification methods, the forth method has been more and more widely employed.

In general, there are several commonly-used methods of system identification for underwater vehicles. The traditional techniques include the Least Square (LS) method[3], Maximum Likelihood (ML) method[4], and Extended Kalman Filter (EKF)[5], while the modern ones are those using artificial intelligent algorithm including Neural Network (NN)[6]and Support Vector Machines (SVM)[7]. With regard to the traditional methods, the approximation results would be sensitive to the initial value, which is unfavorable for maneuvering prediction. Two serious defects, i.e., bad generalization and so-called curse of dimensionality, restrict the application of NN, SVM was put forward based on statistical analysis in 1990s, and originally used to deal with pattern recognition and classification problem, but it performs very well in function regression. It is based on the law of minimization of structural risk, considering both empirical risk and level of confidence, which consequently improves its generalization performance. SVM contains the LS-SVM, ε support vector machines (ε-SVM), ν support vector machines (v-SVM) and so on. It has been successfully applied to surface vehicles and satisfactory results have been gained[8,9].

In fact, even though the identification method is commendable, it is still difficult to obtain high-precision parameters for some complicated maneuvering motion models. On one hand, the sampling information including the sample size and the input excitation will influence the results, on the other hand, the characteristic of the model will also take effect due to the sensitivity of the system parameters. Sensitivity analysis is helpful for model development, model validation and reduction of uncertainty[10]. Rhee and Kim pointed out that the accuracy of the identification results is closely related to the sensitivity of the parameters[11], that is, more sensitive derivative gives more efficient estimation result. There are many different methods of sensitivity analysis. In this paper, a simple calculation process is adopted to measure the sensitivity of the viscous hydrodynamic derivatives. After that, the mathematical model for the AUV?s diving motion is reconstructed.

This paper applies the LS-SVM method to identify the viscous hydrodynamic derivatives in the mathematical model of AUV?s diving motion[12]. Zigzag-like simulation is carried out based on the mathematical model. Separate identification method is proposed to identify the hydrodynamic derivatives for surge, heave and pitch motions respectively. Sensitivity analysis is implemented to explain some poor results and the numerical simulation is carried out by using dominant sensitive coefficients.

Fig.1 Global coordinates and local coordinates

1. Mathematical model for AUV’s diving motion

To describe the motion characteristics for AUV, the coordinate system is established, as shown in Fig.1, where E-ξηζ denotes the global or inertial coordinate frames, while G-xyz represents the local or body-fixed frames.

Generally, we should consider all the six degrees of freedom for the motion of an AUV, but only the maneuvering motion in the diving plane is considered here under the condition of weak maneuvering[13]. In addition, the environment surrounding the AUV, including ocean current, wall effect, wave interaction near the free surface and so on, is always extremely complicated[14,15]. Here, infinite deep and unbounded flow field is assumed, and the environment disturbance is ignored.

Under these assumptions, the mathematical model of the AUV maneuvering motion in the vertical plane, with the origin of the local coordinate system being located at the center of gravity, is described as follows[12]: where m is the vehicle mass, Iyis the moment of inertia about the y axis,zBisthe vertical coordinate of the center of buoyancy,ρ is the water density, L is the length of the vehicle, B is the displacement,,etc. are the hydrodynamic derivatives, δsis the horizontal fin angle, XTis the propeller thrust, u, w, q and their differential coefficients are linear and angular velocity and acceleration, and θ is the pitch angle.

The transformation of the translational velocities between the global and the local frames is described as

whereξis the trajectory about the x axis in global frames, and ζis the diving depth inglobal frames.

It is not difficult to find that the maneuvering model is strongly nonlinear and coupling. To carry out the numerical simulation, Eqs.(1) should be transformed to a new format, and the differential terms are represented as follows

where

2. Least square support vector machines

The LS-SVM used in this paper is an improved type of SVM which owns standard and improved descriptions. Because of the choice of square loss function, it loses the sparse solution feature. But it does not heavily influence the accuracy of the results, in contrast, it transfers the solution of quadratic optimization problem to solving a linear system of equations, which immensely simplifies the computational quantity.

The feature space representation of LS-SVM can be described as

Then the optimization problem is

subject to,

where C is rule factor, and e is regression error.

The following Lagrangian function is defined for target function and constraint conditions

where αiis Lagrangian multiplier.

Taking the partial derivatives of L with respect tow, b, e, α, we have

Table 1 Main model parameters

Substituting the first and third formulas to the forth and subject to the second, gives

where y=［y1,…,yn］T,1=［1,…,1］T,α=［α1,…,and K(xi， xj) iskernel function. The regression estimation function can be obtained once Eq.(9) is solved

In consideration of the problem on parametric identification, the linear kernel function K(x,x＇)= (xgx＇) is commonly adopted. Rewrite Eq.(10) as

are identified parameters.

3. Numerical simulation and parametric identification

The physical parameters of the AUV for numerical simulation are shown in Table 1[12].

3.1 Numerical simulation

Th e forth -ord er Runge-Kut ta method is emp loyedtorealizethemaneuveringsimulationbasedon

Fig.2 Time histories of horizontal fin angle and pitch angle

Fig.3Motion trajectory

Eq.(3) and the motion parameters can be acquired. The Runge-Kutta algorithm can be expressed as follows:

Table 2 Identified parameters of surge motion (10-3)

Table 3 Identified parameters of heave motion (10-3)

Table 4 Identified parameters of pitch motion(10-3)

where n denotes time series, xn=［u,w,q,θ,ζ］is the state vector, ki(i=1,…4)is the intermediate variable, and δn+1/2=0.5(δn+δn+1), h is sample int erval.

The experiment design is very important to system identification, and the purpose is to collect experimentaldata with sufficient information. Accordingto the system identification theory, pseudo random binary codes will be the optimal excitation, but it is difficult for practicalooperation. Therefore, a zigzag-like maneuver with 5 horizontal finoangle is implemented, and the rudder rate is set as 2.5/s. In the simulation test, the original position of the AUV is set at (0,0,100). The sample interval is 0.5 s and the total time history is 120 s. With zero initial state and the propeller thrust fixed at 120 N, the numerical simulation results are shown in Figs.2 and 3. Here, only the variations of the horizontal fin angle, position and attitude are given.

3.2 Parametric identification

On account of the huge difference between the equations, Eq.(3) cann ot be rewritten as a matrix of the unity form like surface vehicle[16]. To rewrite Eq.(3) as a matrix of the unity, several virtual input terms have to be constructed, which will increase the trouble in parametric identification. Therefore, the model must be identified separately, i.e., the motions including surge, heave and pitch will be dealt with all alone. Normally the propeller thrust XTis constant, so it will be processed as a bias instead of an input.

where ai(i=0,…,3), bi(i=0,…,6), ci(i=0,…,6) are parameters including hydrodynamic derivatives to be identified. Let

The inputs can be determined as

Then the outputs can be written as

Once A, B and C are solved, the hydrodynamic coefficients can be gained immediately. According to the principle of parameter identifiability, inertial hydrodynamic coefficients can be obtainedby theoretical calculation or tests rather than by system identification.

In the identification process, the rule factorin the LS-SVM is4set as a bigger positive constant, which is C=3.0×10, because it does not influence the results for system identification to a large extent, while it must be checked for nonlinear function regression. All the samples are chosen, and the sample size is 240. Then the surge, heave and pitch motions are identified by using the proposed method. The estimation results are listed in Tables 2, 3 and 4. Here, Cu(k)is the coefficient of u(k) and zBB is that of sinθ. Both of them are known parameters before the identification process, but they are also listed as a verification of the results.

By surveying the relative errors in these tables it can be seen that most of the hydrodynamic derivatives are identified well, but two coefficients,and, are obtained with a great error that exceeds 5%. For this situation, the multicollinearity is originally thought to be the cause of the ill results, and additional excitation method is adopted to eliminate the errors, but t he results are not satisfactory. On this count, sensitivity analysis method is applied to analyze the identification results. Since the identification results are in good agreement with the original value for surge motion, only the heave motion and pitch motion are considered.

4. Sensitivity analysis of hydrodynamic derivatives

Sensitivity analysis is an integral part of model development and involves analytical examination of input parameters to aid in model validation and provide guidance for future research. In this section, it is employed to evaluate the identifiability of the parameters. That is, the identifiability of a parameter relates to its contribution to the system. If a coefficient has no effect on the system, it is impossible to approximate the coefficient from the input and output data, while it b ecomesmore identifiable if it has significant effect on the system[11].

Table 5 Sensitivity analysis of the viscous hydrodynamic coefficients

There are several methods for confirming the parameters? sensitivity, such as Hwang?s method[17]and its improved ones,objectivefunctionsmethod. The target is to determine the ratio betweenthe relative change of the model coefficients and the relative change of the model output. To access the sensitivity of the viscoushydrodynamic derivatives in the nonlinear maneuvering motion equations for the AUV in the diving plane, a simple study via numerical simulation is performed, and the following sensitivity cost function is defined[18]

5. Conclusion

By analyzing the zigzag-like motion simulation for AUV in the diving plane, the LS-SVM has been applied to identify the viscous hydrodynamic derivatives with separate identification method. The proposed method is verified by comparing the identification results with the standard coefficients. Then, sensitivity analysis has b een implemented to explain some ill results, and it demonstrates that higher identification error is in accordance with lower sensitivity. Finally, the maneuvering model is reconstructed through the sensitivity analysis and is proved to be effective.

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10.1016/S1001-6058(11)60299-0

* Project supported by the National Natural Science Foundation of China (Grant Nos. 50979060, 51079031).

Biography: Xu Feng (1985-), Male, Ph. D. Candidate

ZOU Zao-jian,

E-mail: zjzou@sjtu.edu.cn