Identification modeling of underwater vehicles' nonlinear dynamics based on support vector machines

Ocean Engineering 67 (2013) 68–76

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Identification modeling of underwater vehicles’ nonlinear dynamics based on support vector machines Feng Xu a, Zao-Jian Zou a,b,n, Jian-Chuan Yin a, Jian Cao c a

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China c College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 August 2012 Accepted 23 February 2013 Available online 9 May 2013

An identification method based on support vector machines (SVM) is proposed for modeling nonlinear dynamics of underwater vehicles (UVs), and a typical torpedo-shaped autonomous underwater vehicle (AUV) is employed for the purpose of validation. To obtain the hydrodynamic derivatives of the vehicle and the dynamical models of the thruster and fins, a series of hydrodynamic experiments are conducted by using vertical planar motion mechanism (PMM) and circulating water channel (CWC). Maneuvering simulation is carried out by using the hydrodynamic model obtained from experiments, and SVM is applied to identify the damping terms together with Coriolis and centripetal terms by analyzing the simulation data. By using the identified nonlinear model and experiment-based hydrodynamic model respectively, maneuvering simulations and control applications are implemented. The results are compared to verify the proposed method, and the effectiveness and good generalization performance of SVM in modeling the nonlinear dynamics of UVs are demonstrated. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Underwater vehicle Support vector machines Nonlinear dynamics Maneuvering simulation

1. Introduction It is well known that establishing an accurate dynamic model of underwater vehicles (UVs) is of prime importance for their maneuvering prediction and control application. However, due to the high nonlinear and strong intercoupling hydrodynamic characteristics, this work is extremely difficult and many researchers have devoted themselves to solving this challenging problem. Actually, the key problem is the determination of damping terms, which are generally regarded as the most uncertain part in the dynamic model. With regard to some propeller-driven UVs with a flat body shape, the damping can be simplified as a diagonal matrix including the linear and quadratic terms (Ridao et al., 2004; Tiano et al., 2007). For further simplification, only the linear terms are taken into consideration (Ross et al., 2004). These simplifications are reasonable and convenient for dynamic modeling of UVs mentioned above, but the situation is much more complicated for propeller-fin-driven UVs with streamlined and torpedo-shaped bodies, e.g. NPS ARIES (Marco and Healey, 2000), C-SCOUT (Evans and Nahon, 2004), etc. For such vehicles, more complicated expressions of damping terms are required.

n Corresponding author at: School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China. Tel./fax: þ86 21 34204255. E-mail address: [email protected] (Z.-J. Zou).

0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.02.006

Up to now, there are four commonly-used methods for modeling the UVs’ dynamics including the damping terms in the dynamic model. Captive model test with planar motion mechanism (PMM) or other facilities is regarded as the most reliable one (Bishop and Parkinson, 1970; Rhee et al., 2000). Estimation with empirical formulas is the most practical and convenient one (Evans and Nahon, 2004; Silva et al., 2007), but the precision can not be guaranteed. Numerical calculation based on computational fluid dynamics (CFD) is a modern and powerful modeling method (Sahin et al., 1997; Toxopeus, 2009). System identification (SI) in combination with free-running model test or full scale trial is another powerful modeling method; With the development of SI theory and measurement techniques, this method is widely studied and implemented nowadays. During the last few decades, various modeling methods based on SI technique have been developed and applied for dynamic modeling of surface ships and UVs. The classical algorithms include least square method (LS) (Rhee et al., 1998), extended Kalman filter (EKF) (Abkowitz, 1980; Hwang, 1980; Tiano et al., ¨ ¨ 2007), recursive prediction error method (RPE) (Kallstr om and ˚ ¨ Astr om, 1981; Zhou and Blanke, 1989), maximum likelihood ˚ ¨ ¨ ¨ method (ML) (Astr om and Kallstr om, 1976), as well as their improved ones. These methods are committed to approximating the hydrodynamic derivatives in the mathematical model, but they have inherent defects, such as dependency on initial values, ill-conditioned solutions and simultaneous drift. To overcome these defects, three novel algorithms are proposed to identify

F. Xu et al. / Ocean Engineering 67 (2013) 68–76

hydrodynamic derivatives, i.e. frequency domain identification method (Bhattacharyya and Haddara, 2006; Perez and Fossen, 2011), neural network (NN) (Haddara and Wang, 1999; Mahfouz and Haddara, 2003) and support vector machines (SVM) (Luo and Zou, 2009; Zhang and Zou, 2011). However, as two kinds of artificial intelligence algorithms, NN and SVM can not only be used for parametric identification, they are more suitable for nonlinear regression. Moreira and Guedes Soares (2003) presented a maneuvering simulation model for surface ships based on recursive neural network (RNN). van de Ven et al. (2007) applied NN for nonlinear identification of an open frame UV. Rajesh and Bhattacharyya (2008) adopted NN to regress the nonlinear dynamic model of a large tanker. Nevertheless, two defects of NN, i.e. the so-called curse of dimensionality and unsatisfactory generalization ability, present a barrier to its application. In contrast, SVM can perfectly overcome these shortcomings. SVM was first proposed by Vapnik in 1990s (Vapnik, 1995). It has been extensively applied and improved in solving classification and regression problems owing to its favorable performance (Amari and Wu, 1999; Lee and Mangasarian, 2001). Compared with NN, SVM has several merits and demerits. Firstly, unlike NN which is based on empirical risk minimization (ERM), SVM is based on the criteria of structural risk minimization (SRM), so that better generalization ability can be achieved. Secondly, SVM can guarantee a global optimal solution by adopting convex quadratic programming, while NN is apt to fall into local optimization. Thirdly, by making use of kernel function, SVM can easily overcome the curse of dimensionality, but it is much too difficult for NN. The demerits of SVM are its poor on-line capability and inefficiency in handling massive data set, although some improved approaches have been put forward (Platt, 1999; Cauwenberghs and Poggio, 2001). Consequently, SVM is known for its batch processing algorithm and effectiveness in dealing with small sample. There are several types of SVM, i.e. e-SVM, n-SVM, least square-SVM (LS-SVM) etc. Luo and Zou (2009) applied LS-SVM, while Zhang and Zou (2011) adopted e-SVM, to identify Abkowitz model for Mariner class surface ship and gained satisfactory results. In their studies, linear kernel function was selected for off-line parametric identification. This paper makes an effort to apply LS-SVM in identification modeling of UVs’ nonlinear dynamics. SVM with radial basis function (RBF) as kernel function is applied for regression of the nonlinear functions in the dynamic model. A typical torpedoshaped autonomous underwater vehicle (AUV) is employed as study object. Firstly, a series of captive model tests are conducted by using vertical PMM and circulating water channel (CWC) to obtain the hydrodynamic derivatives of the vehicle and the dynamic models of thruster and fins. Then maneuvering motion of the vehicle is simulated by using the dynamic model obtained from model tests. LS-SVM is utilized to analyze the simulation data for identification of the damping terms together with the Coriolis and centripetal terms. Finally, maneuvering simulations and control applications based on the identified model and the experiment-based model are implemented to verify the proposed method.

69

Fig. 1. Coordinate frames for underwater vehicles.

matrix form as follows Mn_ þ CðnÞn þ DðnÞn þ gðZÞ ¼ t

ð1Þ

and the kinematic equations relating the global coordinate frame with the local coordinate frame are

Z_ ¼ JðZÞn

ð2Þ

where M is the inertia matrix including added mass, C(v) is the matrix of Coriolis and centripetal terms including added mass, D(v) is the damping matrix, gðZÞ is the vector of gravitational forces and moments, t is the vector of control inputs, n is the translational and angular velocity of the vehicle in local frame, n ¼ ½u, v, w, p, q, rT , Z is the position and posture in the global frame, Z ¼ ½x, Z, z, f, y, cT . For the sake of simplicity, external disturbances such as ocean current are not taken into consideration. The detailed definition of each element in Eqs. (1) and (2) and the influence of external environment can be found in Fossen (1994). To precisely determine the mathematical model, one of the key problems is the determination of hydrodynamic derivatives including inertia ones in M and CðnÞ and viscous ones in DðnÞ. With respect to the inertia ones, regression estimation method with empirical formula or theoretical method based on potential theory would generally meet the practical requirements. However, determination of the viscous coefficients in DðnÞ is rather difficult. In order to be more general, the standard motion equations of submarine published by DTNSRDC (Gertler and Hagen, 1967) is employed in this paper, and the six elements of damping terms can be expressed as Eqs. (3)–(8). X ¼ ½X qq q2 þ X rr r 2 þX rp rp½X vr vr þ X wq wq ½X uu u2 þX vv v2 þX ww w2 

ð3Þ

h i   Y ¼  Y p9p9 p9p9 þ Y pq pq þ Y qr qr  Y vq vq þY wp wp þY wr wr " # v 9ðv2 þw2 Þ1=2 99r9  Y r ur þ Y p up þ Y v9r9 9v9 ½Y 0 u2 þ Y v uv þ Y vjvj v9ðv2 þ w2 Þ1=2 9Y vw vw

ð4Þ

  Z ¼  Z pp p2 þ Z rr r 2 þZ rq rq " #   w 9ðv2 þ w2 Þ1=2 99q9  Z vr vr þ Z vp vp  Z q uq þZ w9q9 9w9

2. Mathematical models of underwater vehicles

½Z 0 u2 þ Z w uw þ Z w9w9 w9ðv2 þ w2 Þ1=2 9

To describe the motion characteristics of a UV, two reference frames are adopted, i.e. the global or earth-fixed coordinates ExZz and the local or body-fixed coordinates Oxyz, as shown in Fig. 1. According to Fossen (1994), the nonlinear mathematical model in six degrees of freedom (6-DOF) can be written in a uniform

½Z jwj u9w9 þ Z ww 9wðv2 þ w2 Þ1=2 9Z vv v2

ð5Þ

K ¼ ½K qr qr þ K pq pq þ K p9p9 p9p9 ½K p up þ K r ur þ ½K vq vq þ K wp wp þK wr wr ½K 0 u2 þ K v uv þ K v9v9 v9ðv2 þ w2 Þ1=2 9K vw vw

ð6Þ

70

F. Xu et al. / Ocean Engineering 67 (2013) 68–76

M ¼ ½M pp p2 þ Mrr r 2 þM rq rq þ Mq9q9 q9q9 ½M vr vr þ M vp vp½Mq uq þ M9w9q 9ðv2 þ w2 Þ1=2 9q ½M 0 u2 þM w uw þ M w9w9 w9ðv2 þw2 Þ1=2 9 ½M 9w9 u9w9 þ Mww 9wðv2 þ w2 Þ1=2 9M vv v2

ð7Þ

N ¼ ½N pq pq þN qr qr þN r9r9 r9r9 ½Nwr wr þN wp wpþ Nvq vq ½Np up þ Nr ur þ N9v9r 9ðv2 þw2 Þ1=2 9r ½N0 u2 þ Nv uvþ N v9v9 v9ðv2 þ w2 Þ1=2 9N vw vw

ð8Þ

where Xqq etc. are the viscous hydrodynamic derivatives. There are totally 73 hydrodynamic derivatives in these expressions, and the frequently-used method for acquiring them with relative high accuracy is captive model test by PMM. Another problem for modeling UVs dynamics is solving the thrust of propellers. In the general case, the propeller thrust is calculated by the following formula (Fossen, 1994):

tT ¼ rD4 K T ðJ0 Þ9n9n

ð9Þ

where tT denotes the propeller thrust, r is the fluid density, D and n are the diameter and rotating rate of the propeller, respectively, K T is the thrust coefficient, J 0 ¼ V a =ðnDÞ is the advance number and V a is the advance speed of the propeller. The relevant parameters can be obtained by open water experiment. However, in the practical application, the propellers used for UVs are usually commercial propellers, which are combination of electric machine and propellers. For simplicity, the propeller thrust is reasonably expressed as a function of the input voltage vol and ambient flow velocity ua :

tT ¼ f ðua ,volÞ

ð10Þ

More accurate thruster model can be found in Kim and Chung (2006). The relationship between the thrust and the input voltage as well as the ambient flow velocity can be obtained by a series of thrust tests. Once input voltage and ambient flow velocity are known, propeller thrust can be immediately calculated by using interpolation algorithm, while in control application, the input voltage can be interpolated by thrust and ambient flow velocity. In the past few decades, fins are widely utilized on UVs to modify the hydrodynamic characteristics for low energy consumption and excellent maneuverability. Unlike the propeller thrust, the fin forces/moments can be gained by using fin coefficients, which can be approximated from lift coefficients C L and drag coefficients C D . Lift and drag forces can be calculated by using the following formulas: 8 < L ¼ 1 C L rAR V 2 2 ð11Þ : D ¼ 12 C D rAR V 2 where L is the lift force, D is the drag force, AR is the side area of contour, V is the stream velocity. Then fin coefficients can be estimated based on Eq. (11) as well as the physical parameters of each fin. The mathematical model of steering gear for the fins is commonly set as (Rhee and Kim, 1999): 8 n < ðdn dÞ=T E ; 9d d9 rT E 9d_ max 9 d_ ¼ ð12Þ n _ : signðd dÞ9d max 9; 9dn d9 4T 9d_ max 9 E

n

where, d is the actuator angle, d is the command angle, T E is the sample interval, 9d_ max 9 is the maximum fin turning rate. 3. Identification modeling using support vector machines A prominent advantage of nonlinear identification is that the identified model is only in relation with the input and output

information, and those hydrodynamic coefficients hard to get are not necessary to be determined. Besides, some higher order terms not included in the mathematical model can also be taken into consideration. Meanwhile, some difficult problems such as multicollinearity and cancellation effect in parametric modeling could be avoided. Consequently, the nonlinear identification model usually owns higher precision. As mentioned in Section 2, the determination of damping forces is a rather complex task, and D(v)v is obviously the most uncertain part in the UVs dynamics. In addition, since the Coriolis and centripetal terms are functions of physical parameters and added mass and vehicle velocities, while damping forces are functions of hydrodynamic derivatives and vehicle velocities, they both can be reasonably considered together. In this section, nonlinear identification of C(v)v and D(v)v based on SVM will be dealt with. LS-SVM shows good performance in solving classification and regression problems and has been widely used in pattern recognition, model regression, optimal control and so on, see e.g. Suykens et al. (2001), Suykens (2009). LS-SVM is a special type of SVM. It chooses the square cost function, which leads to the loss of sparse solution, i.e. all the input samples are support vectors. However, LS-SVM transforms the solution of quadratic optimization problem to a linear system of equations, which greatly simplifies the solution problem. In this paper, LS-SVM is employed for nonlinear regression. With the given training data set ð , YÞ, the feature space representation of LS-SVM is given by Yð

Þ ¼ W T jð

Þ þ bð

A Rm ,Y A RÞ

ð13Þ

where W is a vector in the so-called high dimensional feature space, jðdÞ is the mapping function that maps the input data to a high dimensional feature space. b is a bias for the regression model. The estimation problem is formulated as the following optimization problem: minJðW,eÞ ¼ W,e

n 1 T 1 X W Wþ C e2 2 2 i¼1 i

ð14Þ

subjected to the constraints Y i ¼ W T jð

i Þ þb þ ei ,

i ¼ 1,. . .    n

ð15Þ

where C is the rule factor, e is the regression error and n is the sample number. Lagrange function is defined for the objective function and constraint conditions, LðW,b,e, aÞ ¼ JðW,eÞ

n X

ai fW T jð

i Þ þ bþ ei Y i g

ð16Þ

i¼1

where ai is the Lagrange multiplier. Partial derivative with respect to W,b,e, a gives: 8 n X > > @L > @W ¼ 0-W ¼ ai jð i Þ > > > > i¼1 > > > n > X < @L ¼ 0ai ¼ 0 ð17Þ @b > i¼1 > > > > @L > > @ei ¼ 0-ai ¼ Cei > > > > : @@La ¼ 0-W T jð i Þ þb þ ei Y i ¼ 0 i Substituting the first and third formulas in Eq. (17) into the fourth, subjected to the second, gives 2 ! 3  " ! # 0 1 b 0 4 T 5 ¼ ð18Þ ! 1 a Y 1 OþC I

F. Xu et al. / Ocean Engineering 67 (2013) 68–76

! where Y ¼ ½Y 1 ,    ,Y n T , 1 ¼ ½1,    ,1, a ¼ ½a1 ,    , an T , Oij ¼ ! jð i ÞT jð j Þ ¼ Kð i , j Þ, ði,j ¼ 1,    ,nÞ, 0 ¼ ½0,    ,0, Kð i , j Þ is the kernel function. Once Eq. (18) is solved, regression estimation function can be obtained: Yð

Þ¼

n X

ai Kð

i,

Þþb

71

Table 1 Physical parameters of the AUV. Mass in the air (kg) Mass with internal water (kg) Length (m) Diameter (m) Metacenter (m) Inertia of moment about x-axis (kg m2) Inertia of moment about y-axis (kg m2) Inertia of moment about z-axis (kg m2) Design speed (kn)

ð19Þ

i¼1

For the problem of parametric identification, linear kernel function Kð , 0 Þ ¼ ð U 0 Þ is commonly adopted, while RBF 2 kernel function Kð , 0 Þ ¼ expð99 i ci 992 =s2 Þ is usually employed for nonlinear regression, where ci A Rn are chosen kernel centers, s is a chosen constant. ci are usually set as i, see e.g. Suykens et al. (2001); and s is usually tuned together with the rule factor C. In this paper, both s and C are adjusted based on simulated annealing (SA) algorithm according to De Brabanter et al. (2011). Taking the translational and angular velocities as input, and each component of the damping forces together with Coriolis and centripetal terms as output, the block diagram for training the nonlinear model based on LS-SVM is shown in Fig. 2, where the notation denotes ½u, v, w, p, q, rT . Because the six components of C(v)v and D(v)v do not have the same order of magnitude, it is justified to train the six components separately. Besides, LS-SVM is used to deal with the multiple input single output (MISO) problems. Therefore, there are six nonlinear models to be established based on LS-SVM with RBF kernel function.

37 46 1.46 0.214 0.02 0.2437 8.098 8.067 2

Fig. 4. Vehicle model in captive model test.

4. Validations To verify the proposed algorithm in identification of UVs nonlinear dynamics, an AUV developed by the Laboratory of Autonomous Underwater Vehicle at Harbin Engineering University (HEU) is employed. This vehicle possesses a torpedo shape and a couple of horizontal fins as well as a couple of vertical fins. It is propelled by a single through-body thruster, which is driven by Brushless DC Motor. The model is shown in Fig. 3. Some essential physical parameters of this vehicle are listed in Table 1. To obtain the hydrodynamic derivatives and the dynamic models of propeller and fins, a series of captive model tests were

Fig. 2. LS-SVM model for nonlinear identification.

carried out with the vertical PMM and CWC at HEU. The tests consisted of longitudinal resistance tests, oblique towing tests in horizontal and vertical plane, pure yawing and pitching motion, pure heaving and swaying motion. The maximum amplitudes of the PMM tests are 40 mm for pure heaving/swaying motion and 101 for pure yawing/pitching motion; the oscillation frequency is 0.2–1.0 HZ. The full-scale vehicle model in test is shown in Fig. 4. The calculation of hydrodynamic derivatives refers to Bishop and Parkinson (1970), and the results are listed in Table 2. It is worth noting that only a proportion of the total hydrodynamic derivatives can be obtained by regression. Another part can be approximated according to the symmetric characteristics or using empirical formula; the rest are generally assumed to be zeros. The test results of propeller thrust are shown in Fig. 5, while the test results of fin lift and drag coefficients are shown in Fig. 6. Since the shapes and dimensions of the four fins are same, only one pair of lift and drag coefficients are shown. It is not difficult to find out that the tendency of fin lift and drag coefficients changing with fin angle becomes abnormal when the fin angle exceeds 301. Hence the maximum command angle is conservatively set as 251. Approximating the lift and drag coefficients with linear and quadratic regression respectively, fin models can be obtained as follows:

Z ds ¼ Y dr ¼

d2r

F dr,Y

dr

M ds ¼ Ndr ¼

Fig. 3. Torpedo Autonomous Underwater Vehicle (CAD model).

F dr,X

X ds ds ¼ X dr dr ¼

¼

F dr,N

dr

¼

rAR l1 d2r V 2 d2r

rAR l2 dr V 2 dr ¼

¼ 0:2517V 2

rAR l2 dr lV 2 dr

¼ 0:0038V 2

¼ 0:1528V 2

where dr and ds denote the angles of the vertical and horizontal fin, respectively. F dr,X , F dr,Y and F dr,N are forces and moment generated by vertical fins in the surge, sway and yaw motions, respectively. l1 and l2 are regression parameters of drag and lift

72

F. Xu et al. / Ocean Engineering 67 (2013) 68–76

Table 2 Hydrodynamic derivatives obtained by captive model tests. Tests

Nondimensional hydrodynamic derivatives

Longitudinal resistance

X 0u9u9

Oblique towing

 5.9e  3 Y 0v9v9

Pure heaving Pure swaying Pure yawing Pure pitching

Z 0w9w9

1.67e  1 Z 0w  4.272e  2 Y 0v  4.496e  2 Y 0r 2.216e  2 Z 0q

Z 0w_  3.08e  2 Y 0v_  3.075e  2 Y 0r_ 5.677e  3 Z 0q_

 1.3e  1 M 0w 1.028e  2 N 0v  9.378e  3 N 0r  1.168e  2 M 0q

M 0w_  1.035e  3 N 0v_ 1.06e  3 N 0r_ 2.976e  3 M 0q_

 1.709e  2

 5.495e  3

 1.021e  2

3.107e  3

35

propeller thrust (N)

30

ua = -1.5 m/s ua = -1.0 m/s

25

ua = -0.5 m/s ua = 0 ua = 0.5 m/s

20

ua = 1.0 m/s

15

ua = 1.5 m/s

10 Fig. 7. State-space representation of the nonlinear dynamic equations.

5 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

input voltage (v) Fig. 5. Propeller thrust versus input voltage and surge velocity.

1.4 1.2

drag coefficients lift coefficients

coefficients

1 0.8 0.6 0.4 0.2 0 0

5

10

15 20 25 fin angle (deg)

30

35

40

Fig. 6. Fin lift and drag coefficients versus fin angle.

model from experiments. The simulations in this paper are all based on zero initial conditions except a 5 m initial submerged depth. The solution of the dynamic model is shown in Fig. 7. It is worth noting that the actuator including input voltage and fin angles for maneuvering simulation will considerably affect the modeling results. Since the general range of operating voltage is from 3.5 v to 4.5 v, the input voltage for modeling is set as the average value, i.e. 4.0 v, and kept unchanged. Like parametric identification, the determination of horizontal and vertical fin angles refers to sensitivity analysis of the hydrodynamic derivatives. Yeo and Rhee (2006) evaluated the sensitivity of three submersibles with different appendages by using a direct method with the help of genetic algorithm (GA), and pointed out that bang–bang type fin angles give maximum sensitivity, while sensitivity-maximizing inputs increase the precision of the approximated hydrodynamic derivatives. Therefore, it is reasonable to believe that more accurate nonlinear models can be established by adopting bang-bang style fin angles. The magnitudes of the fin angles are set as 81. Using the simulation data, the nonlinear regression models can be built according to Section 3. The parameters used for training the SVM models are tuned with SA and listed in Table 3. To verify the identified model, two cases, i.e. maneuvering simulation and control application, are implemented; and comparison between the identified nonlinear models and the experiment-based model is made. Eq. (1) can be rewritten as ~ nÞn þgðZÞ ¼ t Mn_ þ C~ ðnÞn þ Dð

coefficients, respectively. l is the distance from the center of vertical fin to the origin of the local frame. In practical application, the samples for nonlinear modeling can be collected from full-scale test, while in this paper, they are obtained by maneuvering simulation based on the mathematical

ð20Þ

~ nÞn denotes the nonlinear model which is estabwhere C~ ðnÞn þ Dð lished by SVM regression and will be called SVM model hereafter. Then the solution of the dynamic model can be executed with SVM model via flow diagram shown in Fig. 8.

F. Xu et al. / Ocean Engineering 67 (2013) 68–76

4.1. Case 1 maneuvering simulations with SVM model Maneuvers operated for verification of SVM model are dives from 5 m to approximately 15 m while the vehicle follows a circular path in horizontal plane. In the diving maneuvers, fin angle is firstly executed to 51 or 101 at the descending stage, and Table 3 Tuned parameters for SVM with SA. C(v)vþ D(v)v

C

s

Surge Sway Heave Roll Pitch Yaw

9.183e þ 6 2.528e þ 7 9.665e þ 5 1.316e þ 5 2.747e þ 5 2.081e þ5

1.0934 0.0922 0.0578 0.0321 0.0698 0.0136

73

then changed to  51 or 101 when the submerged depth almost reaches 15 m. At the same time, two kinds of thrust mode are selected, i.e. 3.5 v and 4.5 v input voltages. The simulation results are shown in Figs. 9 and 10, in which the results by experimentbased model (EB model for short) and SVM model are compared. It is easy to find out that the simulation results are all in good agreements. In comparison, the simulation results with 3.5 v input voltage are better than those with 4.5 v input voltage. This is due to the effect of the chosen input voltage for modeling. Consequently, the choice of input voltage will also considerably affect the accuracy and the generalization performance of SVM model, and the key of the problem is whether the in–out samples for identification modeling are able to reflect the characteristics of the actual dynamic model.

4.2. Case 2 control application with SVM model

Fig. 8. State-space representation of the nonlinear dynamic equations with SVM model.

To further investigate the performance of SVM model with variable input voltage and fin angles, the nonlinear model is applied in control application. The control targets contain surge velocity control, heading control, and depth control. The control strategy employed in this section is the so-called S plane controller, which is developed by combination of fuzzy controller and PID controller. S plane controller was firstly proposed by Liu and Xu (2001), and has been widely developed in parametric optimization (Sun et al., 2011). It has been employed in practical application for control of UVs due to its simple structure, less input and high effectiveness. The control strategy of S plane

70

18 EB model

δ r = 5°

60

SVM model

16 14

50

δ s = ±5 °

ζ (m)

η(m)

12 40 30

10

δ r = 10°

8

20

6

10 0 -40

δ s = ±10 °

4 -20

0

ξ(m)

20

2 0

40

50

100 t(s)

150

200

Fig. 9. Motion trajectories with 3.5 v input voltage. (a) Trajectories in horizontal plane and (b) time histories of submerged depth.

18

70

δr = 5°

60

12

ζ(m)

40

η(m)

SVM model

δ s = ±5 °

14

50

30

δr = 10°

20

10 8 6

10

4

0

2

-10 -40

EB model

16

-20

0

ξ(m)

20

40

0 0

δ s = ±10 °

20

40 t(s)

60

80

Fig. 10. Motion trajectories with 4.5 v input voltage. (a) Trajectories in horizontal plane and (b) time histories of submerged depth.

74

F. Xu et al. / Ocean Engineering 67 (2013) 68–76

5

Table 4 Control parameters.

EB model 4.8

k1

k2

k3

Surge velocity control Heading control Depth control

0.8 0.9 2.3

0.1 1.7 2.4

0.02 0.0024 0.00006

1.6

SVM model

4.6 input voltage(v)

Parameters

4.4 4.2 4

1.4

3.8 1.2

3.6 u(m/s)

1 Target velocity 0.8

3.4 0

EB model

10

SVM model 0.6

20 t(s)

30

40

Fig. 12. Input voltage for surge velocity control.

0.4 0.2

60 Desired yaw angle

0 0

10

20

30

50

40

EB model SVM model

t(s)

40

Fig. 11. Control results of surge velocity.

ψ(°)

30 controller can be expressed as 2:0 1:0 þ k3 m¼ _ 1:0 þ expðk1 ek2 eÞ

Z

20 edt

ð21Þ

where m is the output of the controller, e and e_ are the inputs, i.e., the difference and the changing rate of the difference between the expected value and the actual value, k1 , k2 and k3 are the control parameters corresponding to the proportional, differential and integral terms in PID controller, respectively. It is worth noting that the input and output of the controller must go through the normalization process according to practical situation. Control parameters are manually adjusted and given in Table 4. The control of surge velocity is to investigate the generalization performance of SVM model with variable input voltage. The results compared with those of the EB model are shown in Fig. 11, where the solid straight line is the target velocity, which is set as maximum design velocity, i.e. 1.5 m/s. Fig. 12 shows the corresponding input voltage. It can be seen that the control results based on the two models are in good agreement. The small discrepancies between the SVM model and the EB model are in the overshoot of the control results and the stable input voltage. These discrepancies are not only due to the difference in the two models for surge motion, but also due to the differences and coupling effects existing in the two models for motions in other degrees of freedom. Heading control is to investigate the SVM model with variable vertical fin angels, while depth control is to check the SVM model with variable horizontal fin angles. The input voltage is set as 3.5 v, and the results compared with those of the EB model are shown in Figs. 13–16. The solid line in Fig. 13 is the desired yaw angle, which is set as 501 in the first 40 s and zero in the last 40 s, while the solid line in Fig. 15 is the desired depth, which is set as 10 m in the first 40 s and 5 m in the last 40 s. These figures show that the SVM model is effective for control application on one

10 0 -10 0

20

40 t(s)

60

80

Fig. 13. Control results of heading.

hand, and demonstrate that the SVM model is practicable for variable fin angles in a large extent on the other hand.

5. Conclusions The mathematical model of a torpedo AUV is established based on captive model tests, and the simulation results are applied to identify the nonlinear dynamics of the AUV based on SVM method. Maneuvering simulations and control applications are implemented to verify the proposed method, and satisfactory results are obtained. Several conclusions can be drawn from this study. The dynamic model established by using captive model tests in this paper is applicable. Nonlinear identification of damping terms together with Coriolis and centripetal terms is feasible. It is commendable for UVs maneuvering prediction and control application. As an advanced artificial intelligence algorithm, SVM is suitable for nonlinear function regression, and the simulation results show its high generalization performance.

F. Xu et al. / Ocean Engineering 67 (2013) 68–76

Actuator fin angle type for identification modeling can be determined with the help of sensitivity analysis, and the bangbang type is appropriate. The input voltage for identification modeling is of great significance because it has great effect on the accuracy and the generalization performance of the identified model. The future work will be concentrated on identification modeling of UVs with environmental disturbances, and application of SVM model in full-scale trails. Besides, on-line algorithm and parametric optimization of SVM will be also an important part of the future work.

15 EB model SVM model

10

5 δr (°)

75

0

-5

Acknowledgments

-10

-15 0

20

40 t (s)

60

80

Fig. 14. Vertical fin angles for heading control.

This work is supported by the National Natural Science Foundation of China (Grant nos. 50979060 and 51079031). The authors would like to express their sincere thanks to the Laboratory of Autonomous Underwater Vehicle at Harbin Engineering University for providing the model test data. References

11 Desired depth EB model

10

SVM model

ζ(m)

9 8 7 6 5 4 0

20

40 t(s)

60

80

Fig. 15. Control results of depth.

20 EB model 15

SVM model

10

δs(°)

5 0 -5 -10 -15 -20 0

20

40 t(s)

60

Fig. 16. Horizontal fin angles for depth control.

80

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