利用水下滑翔機群的海洋行動監測網路技術研發－子計畫一：海洋行動監測網路水下滑翔機群協同式控制系統之研究(2/3)

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行政院國家科學委員會專題研究計畫期中進度報告

利用水下滑翔機群的海洋行動監測網路技術研發--子計畫

一：海洋行動監測網路水下滑翔機群協同式控制系統之研

究(2/3)

期中進度報告(精簡版)

計畫類別：整合型計畫編號： NSC 95-2221-E-002-410- 執行期間： 95 年 08 月 01 日至 96 年 07 月 31 日執行單位：國立臺灣大學工程科學及海洋工程學系暨研究所計畫主持人：郭振華共同主持人：鄭勝文處理方式：期中報告不提供公開查詢

中華民國 96 年 06 月 20 日

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行政院國家科學委員會專題研究計畫期中成果報告

海洋行動監測網路水下滑翔機群協同式

控制系統之研究(2/3)

Cooperative Control System for a Group of Underwater Gliders

in an Ocean Mobil Sensor Network (2/3)

計畫編號：NSC 95-2221-E-002-410

執行期限：95 年 8 月 1 日至 96 年 7 月 31 日

主持人：郭振華

國立臺灣大學工程科學及海洋工程學系

Abstract -- To improve the performance

of a glider in the shallow water operations, we will develop methods to improve the transient behavior of gliders. The accuracy of parameters in a vehicle’s dynamic model strongly affects the dynamic performance of its control system. An optimal input design technique for vehicle parameter estimation is presented in this study. The idea is the combination of a dynamic programming method with a gradient algorithm for the optimal input synthesis. Motion data were analyzed with a least-squares technique so that values of parameters could be estimated. The estimated values are compared with arbitrary input signals that are used in system identification. This algorithm for selecting optimal inputs is found to be efficient and robust to noises. This work validated the analysis used to develop the optimal input design, and

demonstrated the feasibility and practical utility of the optimal input design technique. This report is the mid-term report of the second year.

Keywords: underwater gliders, ocean observation

network, underwater vehicles

摘要正確的重要運動方程式參數為水下滑翔機運動路徑推算及自動控制之必備條件。本報告提出一個用來鑑定水下滑翔機系統參數的模式，藉由一組最佳化輸入函數，控制本計畫所發展的具前後浮力引擎之水下滑翔機，以鑑定出其運動方程式之參數。此報告為第二年之期中報告，內容包含方程式介紹、鑑定方法、以及參數鑑定之主要結果。關鍵詞：水下滑翔機、海洋觀測網路、水下載具 1. Mathematical Model

Dynamic system models are often used in system and control architectures, to enhance the system performance. Identification of system parameters is a well-studied problem. Various effective algebraic and numerical solution techniques have been developed to solve for unknown parameters using dynamic system models [1], [2], [3].

These include techniques based on pseudo-inverses, sliding mode observers, Kalman observers, and others. Kim et al. [4] estimated the hydrodynamic coefficients based on two nonlinear observers, the SMO (Sliding Mode Observer) and the EKF (Extended Kalman Filter). Liu [5] discuss about the estimation of surge motion model of ship and using identification technique to derive the model’s unknown hydrodynamic coefficients. However, the

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accuracy/quality of the identified system parameters is a function of both the excitation imposed on the system as well as the measurement noise (sensor noise). The importance of input selection for system identification has been recognized for a long time. Mehra et al. [6] considers the design of optimal inputs for identifying parameters in linear dynamic systems. The criterion used for optimization is the sensitivity of the system output to the unknown parameters as expressed by the weighted trace of the Fisher information matrix. Morelli et al. [7], [8] considered the design of optimal inputs for airplane’s linear model equations form the point of view of Cramér-Rao lower bound and its inverse, the Fisher information matrix. Numerical simulations of three types of sea trials are performed to obtain the sensitivities of motions to hydrodynamic coefficients [9]. Jauberthie et al. [10] presented the design of optimal inputs for aircraft nonlinear controlled dynamic models. Graver et al. [11] described the development of feedback control for autonomous underwater gliders, and derived a nonlinear dynamic model of a nominal glider complete with hydrodynamic forces and coupling between the vehicle and the movable internal mass. Also, Graver et al. [12] identified the model parameters to match the steady glides in new flight test data from the SLOCUM glider.Graver [11] model the underwater glider as a rigid body with fixed wings and tail immersed in a fluid with buoyancy control and controlled internal moving mass. We assign a coordinate frame fixed on the vehicle body to have its origin at the CB and its axes aligned with the principle axes of the ellipsoid. Body axes are illustrated in Fig.1. The different masses and position vectors are illustrated in Fig. 2. ( 1, 2, 3)

T

P P P P

r = r r r denotes the position vector of movable mass. Here the m_h is the uniformly distributed hull mass, m_w is point mass for nonuniform hull mass distribution, and r_w is the position vector from CB to m_w. m_b is the variable mass located at CB. m is the movable point mass. The total mass of the glider is

v h w b m =m +m +m + . m 1 e 2 e 3 e i j k

Fig. 1 Frame assignment on underwater glider

m

P r b m w r h

m

w m

Fig. 2 Glider mass definitions

Ω b v k j i 1 e 2 e 3 e

Fig. 3 Glider position and orientation variables

The position of the glider b=( , , )x y z T is the vector from the origin of the inertial frame to the origin of body frame as shown in Fig. 3. The vehicle moves through the fluid with translational velocity

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1 2 3

( , , )T

v= v v v and angular velocity

1 2 3

( , , )T

Ω = Ω Ω Ω , expressed with respect to the body frame. In this section, we present a mathematical model that describes the longitudinal dynamics of underwater gliders. Following the discussions in Graver [11], the equations of motion for the gliding vehicle restricted to the vertical plane are

1cos 3sin x=v θ +v θ (1) 1sin 3cos y = −v θ +v θ (2) 2 θ= Ω (3) 2 3 1 1 3 1 2 3 3 1 1 3 1 (( ) ( cos sin ) ) P P DL P P P P m m v v mg r J r M r P r P θ θ Ω = − − + + − + (4) 1 3 3 2 3 2 1 1 1 ( sin sin cos ) P o P v m v P m g m L D P θ α α = − Ω − Ω − + − − (5) 3 1 1 2 1 2 3 3 1 (

cos cos sin )

P o P v m v P m m g θ L α D α P = Ω + Ω + − − − (6) 1 1 1 3 2 1 P P P r P v r m = − − Ω (7) 3 3 3 1 2 1 P P P r P v r m = − + Ω (8) where m and ₁ m are the sum of body and ₃

added mass, along the e₁ and e direction. ₃

2

J is the sum of the inertia of stationary mass and added inertia matrix in e₁−e₃

plane. m g presents the weight of the _o glider. P_P₁ and P_P₃ denote linear momentum in body coordinate along the e₁

and e direction. ₃ θ is the pitch angle of glider. Here, α is the angle of attack, D is drag, L is lift and MDL is the viscous

moment as shown in Fig. 4. Lift and drag forces are assumed to act at the glider center of buoyancy. These forces and moment are modeled as

2 0 2 2 0 2 0 ( ) ( ) ( ) L L D D DL M M L K K V D K K V M K K V α α α = + = + = + (9)

where the K s are the hydrodynamic coefficients. The aim for parameter identification is to estimate those constant coefficients. D 1 e

α

θ

V L DL M i j k

Fig. 4 Lift and Drag on the Glider

As shown in Fig. 4, we denote the glider speed V = (v₁2+v₃2), and attack angle

1 3 1 tan (v ) v α ₌ − . 2. Identification Results

In this section, simulation results obtained by using the optimal input design algorithm are presented. A glider with fore and aft buoyancy engines was modeled. The conception of the double buoyancy engines is used to replace the moveable mass in Fig. 2 for shifting the center of gravity. Fig. 5 shows the configuration of the buoyancy engines. The mass m =5kg, ₁ m =70kg, ₃ m =40kg, _h m =1kg, _o

m =9kg, and internal J =12 kg • m₂ 2 are assumed in the simulation.

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Fore Ballast Aft Ballast Mass bf m mba C B bf r r_ba Mass

Fig. 5 Fore and Aft Buoyancy Engines

In Fig. 5, m_bf and m_ba are the mass of ballast mass in the fore and aft buoyancy engines, and r_bf,

ba

r present their position vector from CB. Thus, the variable mass of the glider in Fig. 3.2 becomes

b ba bf

m =m +m (10)

Here we assume that the variable m_bf and m_ba

used to turn the center of gravity aside are equivalent to move m in Fig. 5. In addition, we consider the movable mass is constrained in the longitudinal direction e₁. The equation between this replacement becomes 1 ( ) bf bf ba ba w w P w w h w b h w ba bf m r m r m r mr m r m m m m m m m m m + + + ₌ + + + + + + + (11) The position vector r is then derived as, _P₁

1 bf bf ba ba P m r m r r m + = (12) In Eq. (9), K s are unknown parameters.

However, the lift is nearly linear only at low attack angles. Thus the constraint of attack angle was specified by α <20° . To make dynamic programming applicable, the simulation experiment is split into stages. In order to avoid a long computational time, the test is split into four stages. At the initial condition, assume the glider is under steady state: the glider speed V =0.4 m/s, the attack

angle α =1.62D, the pitch angle θ = −27.43D, the angular velocity Ω = . ₂ 0 5 3 4 6 40 50 60 0.1 0.2 0.3 0.4 0.5 drainage absorb Stage 1 Stage 2 Stage 4 Stage 3 Stage 1 Stage 2 Stage 3 Stage 4 Fore Aft Maximum volume change rate(g/sec) Amplitude of input command (kg) Time of stage (sec)

Fig.6 Dynamic Programming Diagram

The simulation variables in this dynamic programming procedure are shown in Fig. 6. All composition of those different variables must be calculated in the optimization process. The variables include the maximum input change rate, the amplitude of square input signal, the time of stage, and the input command choice of absorb or drainage in each stages. Optimal input that makes the cost function minimized can be calculated. The simulation of optimal input is compared with the conventional input. Fig. 7 and Fig. 8 show the optimal input and conventional input signals for the fore and aft buoyancy engines, respectively. This conventional input is used to compare with optimal input, and it is a regular signal which makes the glider ascends and dives, repeatedly. The capacity of each buoyancy engines is assumed 1kg. In these figures, the dash line represents the command input signal and the solid line corresponds to actual input response of the buoyancy engines. The rate of change of the ballast mass is constrained by the buoyancy engine, and this limit is also determined by the dynamic programming principle. It can be seen that the input signals change very slowly and bounded. They could be applied as input signals for real underwater glider system.

Following Eqs. (4)-(6), we can establish the sensitivity differential equation of (Ω , ₂ v , ₁ v )₃ T to hydrodynamic coefficients. Then the output sensitivities were solved from the sensitivity differential equation by Runge-Kutta algorithm of order 4. The quality of the identified parameter can

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be evaluated by the sensitivity analysis of the observation process. Figs.9-26 compare the sensitivities of square inputs and optimal inputs for the parameters. In these figures, the dash-dot line represents the square inputs and the solid line corresponds to optimal inputs, respectively. Obviously, the output sensitivities to K during _D optimal input have larger variation than conventional input.

To estimate the parameters, Least Squares method is designed using the dynamic model of Eqs. (4) -(6). In these equations, the observations are given by (Ω , ₂ v , ₁ v )₃ T. The hydrodynamic forces and moment in Eqs. (4) -(6) can be written as

1 1 3 3 2 3 2 1 3 3 1 1 2 1 2 3 2 2 3 1 1 3 1 3 3 sin cos 0 cos sin 0 0 0 1 sin cos ( ) ( cos sin ) DL P o P P o P P P P P L D M m v m v P m g P m v m v P m g P J m m v v mg r r r P α α α α θ θ θ θ − ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢₌ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ + Ω + Ω + + − + Ω + Ω + − Ω − − + + + 1 r PP1 P3 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ (13)

The relation between unknown parameters to L , D and M_DL from Eq (9) can also be expressed in matrix form as 0 2 2 0 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 L L D D DL M M K K V V L K V V D K V V M K K α α α ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ₌⎢ ⎥ ⎢ ⎥ ⎢ ⎥ _⎢ _⎥ ⎢ ⎥ ⎢ ⎥ ⎢_⎣ ⎥_⎦ ⎣ _{⎦ ⎢} _⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (14)

Substitute Eqs. (14) into Eq (13), a solution of unknown parameters K s can be identified. The estimated and true parameters are compared in Table 1. This result is calculated from 10 times identification in each case. The error and standard error are used to evaluate the quality of identification.

In Table 1, it is shown that K , _L K have errors _D larger than other parameters. Due to the output sensitivities to K during optimal input are very _D large, the accuracy of K identification in optimal _D input is better than squae input. At the same time of the sensitivity analysis, a number of states of the glider were also computed from Eqs. (3)-(8). Fig. 27 and Fig. 28 show the time history of attack angle and Euler angles. The trajectory of the glider during optimal input and conventional input for parameter identification in the vertical plane can be calculated from Eqs. (1), (.2), and it was presented in Fig. 29. The measurement states including velocities and angular velocity were plotted in Figs. 30-32. Generally speaking, the time history of sensitivity varies with the output response. The sensitivities of angular velocity to parameters are generally very similar between those two inputs. That is because the time histories of angular velocity in the two cases have little difference in Fig. 30. In addition, due to the terms of K , L K and D K in Eq (9) increase M

with attack angle α , the change of attack angle under optimal inputs is more violent than square inputs. The lift in Eq (9) only allows low attack angle. That is the reason why the output sensitivities to K _D are larger than others in the optimal inputs. In a word, although the identified results in Table 1 using optimal inputs only have smaller error to the parameters K_D₀ , K , the accuracy of _D K has _D much higher quality. The standard errors of K_L₀,

0 D

K , K , _D K_M₀ in optimal input were smaller than the square input.

3. Conclusion

In this report we provide a summary on an optimal input design algorithm developed based on sensitivity analysis for underwater vehicle parameter identification. The main contribution of this work is the design method of optimal input which provides better performance for glider parameter identification. The optimization procedure developed in this work provides optimal inputs by minimizing a cost function. Pulse-like inputs were selected utilizing a dynamic programming technique by evaluating output sensitivities to model parameters. A glider

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with fore and aft buoyancy engines was modeled. Successful implementations of a least-squares technique on a Slocum glider confirms the applicability of the parameter identification process.

0 50 100 150 200 0 0.5 1 time (sec) kg

Aft ballast mass

0 50 100 150 200 0 0.5 1 time (sec) kg

Fore ballast mass command

actual

command actual

Fig 7 Optimal Inputs of a Glider

0 50 100 150 200 0 0.5 1 time (sec) kg

Aft ballast mass

command actual 0 50 100 150 200 0 0.5 1 time (sec) kg

Fore ballast mass

command actual

Fig 8 Conventional Inputs of a Glider

0 50 100 150 200 250 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 Time (sec) se n s it iv it y ( K L 0 vs p ) angular velocity VS KL0 Optimal Input Conventional Input

Fig 9 Time history of sensitivity. (KL0 to angular

velocity Ω ) 0 50 100 150 200 250 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 Time (sec) se n s it iv it y ( K L 0 vs v1 ) v1 VS KL0 Optimal Input Conventional Input

Fig 10 Time history of sensitivity. (K_L₀ to velocity

1 v ) 0 50 100 150 200 250 -10 -8 -6 -4 -2 0 2 4x 10 -3 Time (sec) s e n s it iv it y ( K L 0 v s v 3 ) v3 VS KL0 Optimal Input Conventional Input

Fig.11 Time history of sensitivity. (K_L₀ to velocity

3 v ) 0 50 100 150 200 250 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1x 10 -3 Time (sec) s e n s it iv it y ( K L v s p) angular velocity VS KL Optimal Input Conventional Input

Fig. 12 Time history of sensitivity. (K to angular _L velocity Ω )

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0 50 100 150 200 250 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5x 10 -3 Time (sec) s e n s it iv it y ( K L v s v 1 ) v1 VS KL Optimal Input Conventional Input

Fig. 13 Time history of sensitivity. (K to velocity _L

1 v ) 0 50 100 150 200 250 -4 -3 -2 -1 0 1 2x 10 -4 Time (sec) s e n s it iv it y ( K L v s v 3 ) v3 VS KL Optimal Input Conventional Input

Fig. 14 Time history of sensitivity. (K to velocity _L

3 v ) 0 50 100 150 200 250 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Time (sec) s e n s it iv it y ( K D 0 v s p ) angular velocity VS KD0 Optimal Input Conventional Input

Fig. 15 Time history of sensitivity. (K_D₀ to angular velocity Ω ) 0 50 100 150 200 250 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Time (sec) s e n s it iv it y ( K D 0 v s v 1 ) v1 VS KD0 Optimal Input Conventional Input

Fig.16 Time history of sensitivity. (K_D₀ to velocity

1 v ) 0 50 100 150 200 250 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 Time (sec) s e n s it iv it y ( K D 0 vs v3 ) v3 VS KD0 Optimal Input Conventional Input

Fig. 17 Time history of sensitivity. (K_D₀ to velocity

3 v ) 0 50 100 150 200 250 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2x 10 -4 Time (sec) s e ns it iv it y ( K D v s p ) angular velocity VS KD Optimal Input Conventional Input

Fig. 18 Time history of sensitivity. (K to angular D

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0 50 100 150 200 250 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0x 10 -3 Time (sec) se n s it iv it y ( K D vs v 1 ) v1 VS KD Optimal Input Conventional Input

Fig. 19 Time history of sensitivity. (K to velocity _D

1 v ) 0 50 100 150 200 250 -0.5 0 0.5 1 1.5 2 2.5 3x 10 -4 Time (sec) s e ns it iv it y ( K D v s v 3 ) v3 VS KD Optimal Input Conventional Input

Fig. 20 Time history of sensitivity. (K to velocity D 3 v ) 0 50 100 150 200 250 -6 -4 -2 0 2 4 6x 10 -3 Time (sec) s e n s it iv it y ( K M 0 v s p ) angular velocity VS KM0 Optimal Input Conventional Input

Fig. 21 Time history of sensitivity. (K_M₀ to angular velocity Ω ) 0 50 100 150 200 250 -14 -12 -10 -8 -6 -4 -2 0 2x 10 -3 Time (sec) s e n s it iv it y ( K M 0 v s v 1 ) v1 VS KM0 Optimal Input Conventional Input

Fig. 22 Time history of sensitivity. (KM0 to velocity 1 v ) 0 50 100 150 200 250 -1.5 -1 -0.5 0 0.5 1 1.5x 10 -3 Time (sec) se n s it iv it y ( K M 0 vs v3 ) v3 VS KM0 Optimal Input Conventional Input

Fig. 23 Time history of sensitivity. (K_M₀ to velocity

3 v ) 0 50 100 150 200 250 -6 -4 -2 0 2 4 6x 10 -3 Time (sec) s e n s it iv it y ( K M vs p ) angular velocity VS KM Optimal Input Conventional Input

Fig. 24 Time history of sensitivity. (K to angular _M velocity Ω )

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0 50 100 150 200 250 -12 -10 -8 -6 -4 -2 0 2x 10 -3 Time (sec) se n s it iv it y ( K M vs v 1 ) v1 VS KM Optimal Input Conventional Input

Fig. 25 Time history of sensitivity. (K to velocity _M

1 v ) 0 50 100 150 200 250 -1.5 -1 -0.5 0 0.5 1 1.5x 10 -3 Time (sec) se n s it iv it y ( K M vs v3 ) v3 VS KM Optimal Input Conventional Input

Fig. 26 Time history of sensitivity. (K to velocity _M

3 v ) 0 50 100 150 200 250 -10 -5 0 5 10 15 20 Time (sec) deg re e attack angle Optimal Input Conventional Input

Fig. 27 Time history of attack angle.

0 50 100 150 200 250 -60 -50 -40 -30 -20 -10 0 10 20 Time (sec) degr ee

The pitch angle of Glider

Optimal Input Conventional Input

Fig. 28 Time history of Euler angle.

0 20 40 60 80 100 120 140 -90 -80 -70 -60 -50 -40 -30 -20 m m

The trajectory of Glider

Fig. 29 The 2D Trajectory of the Glider for Parameter Identification. 0 50 100 150 200 250 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Time (sec) ang ular v eloc it y ( rad/ s )

The angular velocity of Glider

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0 50 100 150 200 250 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Time (sec) v e lo c ity ( m /s )

The velocity of Glider(v1) Optimal Input

Conventional Input

Fig. 31 Time history of velocity (v ). ₁

0 50 100 150 200 250 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Time (sec) v e lo c ity ( m /s )

The velocity of Glider(v3)

Fig. 32 Time history of velocity (v ). ₃

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[6] R. K. Mehra, “Optimal Inputs for Linear System Identification,” IEEE Transactions on Automatic Control, vol. AC-19, No.3, pp. 192-200, June 1974.

[7] E. A. Morelli, “Flight Test Validation of Optimal Input Design and Comparison to Conventional Inputs,” AIAA paper 97-3711, AIAA Atmospheric Flight Mechanics Conference, New Orleans, Louisiana, August 1997.

[8] E. A. Morelli, “Practical Input Optimization for Aircraft Parameter Estimation Experiments”, Sc.D. Dissertation, George Washington University JIAFS, July 1990.

[9] Yeo, Dong Jin, Rhee, Key Pyo, “Sensitivity Analysis of Submersibles’ Manoeuvrability and Its Application to the Design of Actuator Inputs.” Ocean Engineering, vol. 33, No.17-18, 2006. [10] C. Jauberthiea, “Optimal Input Design for

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[12] J. G. Graver, “Underwater Glider Model Parameter Identification”, Proc. 13th Int. Symp. on Unmanned Untethered Submersible Technology (UUST), August 2003.

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Table 1 Identification for Parameters of a Glider Parameter True value Identificatio n with Optimal Input Standard Error Error Identificatio n with Conventiona l Input Standard Error Error KL0 0 0.4979 0.001356 0.4979 0.25182 0.000432 0.25182 KL 132.5 113.57 0.683944 18.93001 117.2841 0.675519 15.21587 KD0 2.15 2.14177 0.00000321 0.00823 2.01355 0.0000835 0.13645 KD 25 27.95711 0.592654 2.95711 42.56143 2.821345 17.56143 KM0 0 -0.13341 0.0000926 0.13341 -0.07062 0.000215 0.07062 KM -100 -94.9997 0.070849 5.00035 -95.8451 0.026218 4.15486 Avg. Error 4.587835 6.231842 Cost Function 21.0126 87.1188