Depth and Heading Control for Autonomous Underwater Vehicle Using Estimated Hydrodynamic Coefficients

Joonyoung Kim, Kihun Kim, Hang S . Choi, Woojae Seong, and Kyu-Yeul Lee Department of Naval Architecture & Ocean Engineering, Seoul National University, Seoul 15 1-742, Korea

Absfract - Depth and heading control of an AUV are considered for the predetermined depth and heading angle. The proposed control algorithm is based on a sliding mode control using estimated hydrodynamic coefficients. The hydrodynamic coefficients are estimated with the help of conventional nonlinear observer techniques such as sliding mode observer and extended Kalman filter. By using the estimated coefficients, a sliding mode controller is constructed for the combined diving and steering maneuver. The simulation results of the proposed control system are compared with those of control system with true coefficients. It is demonstrated that the proposed control system makes the system stable and maintains the desired depth and heading angle with sufficient accuracy.

I. INTRODUCTION

In recent years, intensive efforts are being concerted towards the development of Autonomous Underwater Vehicles (AUVs). In order to design an A W , it is usually necessary to analyze its maneuverability and controllability based on a mathematical model. The mathematical model for most 6 DOF contains hydrodynamic forces and moments expressed in terms of a set of hydrodynamic coefficients. Therefore, it is important to know the true values of these coefficients so as to simulate the performance of the AUV correctly.

The hydrodynamic coefficients may be classified into 3 types; linear damping coefficients, linear inertial force coefficients, and nonlinear damping coefficients. The linear damping coefficient is known to affect the maneuverability of an AUV strongly. Sen [l] examined the influence of various hydrodynamic coefficients on the predicted maneuverability quality of submerged bodies and found that the coefficients of significant effects on the trajectories are the linear damping coefficients. These coefficients are normally obtained by experimental test, numerical analysis or empirical formula. Although Planar Motion Mechanism (PMM) test is the most popular among experimental tests, the measured values are not completely reliable because of experimental difficulties and errors.

Another approach is the observer method that estimates the hydrodynamic coefficients with the help of a model based estimation algorithm. A representative method among observer methods is the Kalman filter, which has been widely used in the estimation of the hydrodynamic coeflficients and

This work was supported by the NRL program from the Ministry of Science & Technology of Korea.

state variables. Hwang [2] estimated the maneuvering coefficients of a ship and identified the dynamic system of a maneuvering ship using extended Kalman filtering technique.

These estimated coefficients are used not only for a mathematical model to analyze AUV’s maneuvering performance but also for a controller model to design AUV’s autopilot. Antonelli et al. [3 J estimated vehicle-manipulator system’s velocity using observer and applied it in tracking control law. Fossen and Blanke [4] designed propeller shaft speed controller by using feedback from the axial water velocity in the propeller disc. Farrell and Clauberg [ 5 ] reported successful control of the Sea Squirt vehicle which used an extended Kalman filter as a parameter estimator with pole placement to design the controller. Yuh [6] has describes the functional form of vehicle dynamic equations of motion, the nature of the loadings, and the use of adaptive control via online parameter identification.

Recently, advanced control techniques have been developed for A W , aimed at improving the capability of tracking desired position and attitude trajectories. Especially, sliding mode control has been successfully applied to AUV because of good robustness for modeling uncertainty, variation from operating condition, and disturbance. Yoerger and Slotine [7] proposed a series of SISO continuous-time controllers by using the sliding mode technique on an underwater vehicle and demonstrated the robustness of their control system by computer simulation in the presence of parameter uncertainties. Cristi et al. [SI proposed an adaptive sliding mode controller for AUVs based on the dominant linear model and the bounds of the nonlinear dynamic perturbations. Healey and Lienard [9] described a 6 DOF model for the maneuvering of an underwater vehicle and designed a sliding mode autopilot for the combined steering, diving, and speed control functions. Lea et al. [ 101 compared the performance of root locus, fuzzy logic, and sliding mode control, and tested using a experimental vehicle. Lee et al. [l 13 designed a discrete- time quasi-sliding mode controller for an AUV in the presence of parameter uncertainties and a long sampling interval.

In this paper, depth and heading control of an AUV are proposed in order to maintain the desired depth and heading angle in a towing tank. The proposed control algorithm represents a sliding mode control using the estimated hydrodynamic Coefficients. The hydrodynamic coefficients are estimated based on the nonlinear observer such as Sliding Mode Observer (SMO) and Extended Kalman Filter (EKF). Because the system to be controlled is highly nonlinear, a

MTS 0-933957-28-9 429

sliding mode control is constructed to compensate the effects of modeling nonlinearity, parameter uncertainty, and disturbance.

Section I1 describes the nonlinear observers for estimation of the hydrodynamic coefficients. Section 111 presents a sliding mode control for depth and heading control. Section IV shows simulation results. Finally, section V presents the conclusions.

This paper organized as follows:

11. ESTIMATION OF THE HYDRODYNAMIC COEFFICIENTS

The coefficients of significant effects on the dynamic performance of an AUV are found to be the linear damping coefficients. Especially, ten coefficients among the linear damping coefficients are considered as highly sensitive parameters and represented in [ I ] as M,. A4&, N, N&. Ny, Z,, Z,, Y,,, Y, and Y,. In this paper, in order to estimate the sensitive coefficients, the estimate system based on a nonlinear observer is constructed as illustrated in Fig. 1. The nonlinear observer block is composed of SMO and EKF, which is designed based on the AUVs 6 DOF equations of motion. Based on the measured signal of the AUV’s motion, two nonlinear observers are developed for estimating the sensitive coefficients. The AUV block represents the real plant and includes a 6 DOF model of NPS AUV I1 [9]. The value of the sensitive coefficients from this block is used as true value and compared with the estimated ones.

In order to design a nonlinear observer, AUV’s equations of motion are needed. The observer model for this paper includes 6 DOF AUV’s equations of motion and the augmented states for the sensitive coefficients. The observer

Fig. I . Configuration of the estimate system.

Y I I Top

I I

Roar vlow ,

~ l ido vlow = I

I Fig. 2. Coordinate system.

~

430

model describes surge, sway, heave, roll, pitch, and yaw motions and its coordinate system is shown in Fig. 2. The general 6 DOF observer model is as follows:

m[u - v r + wq - x G ( q 2 + r 2 ) + y G ( p q - i )+ z , (pr + q) ]= x m[V+ ur - wp + x , ( p q + i ) - y,(p2 + r 2 ) + zG(qr- b)]= Y

m[W - uq + vp + x, (pr - q) + y , (qr + P) - ZG ( p 2 + q’)] = z 1,j7 + ( I , - I , )qr + I , (pr - q ) - I , (q - r 2 )

Iy4 +U, - I , ) P - I,(qr + P ) + I , ( p q - + 1,(p2 - r 2 ) - m[x,(W - uq + vp) - z,(u - vr + wq)]= M

I,+ + U, - I r ) P 9 - I ,b2 - q 2 ) - IyI(P‘+ 4) + I,(qr - b) + m[x,(V + ur - wp) - yG(u - vr + wq)J= N

- I , ( pq + i ) + m b G (W - uq + vp) - zG (i’ + ur - ~ p ) ] = K

(1)

In the above equation, X, Y, Z, K, M, and N represent the resultant force and moment with respect to x, y, and z axis, respectively, and their detailed expressions are described in [9]. In order to estimate the sensitive coefficients, these coefficients have to be modeled as extra state variables. Consequently, equation (1) is transformed into augmented state-space form.

U

r; w P 4

i e ~

P

i

x+x, Y + Y, z+z, K + K ,

= [MI-{/ M + M , N + N ,

p + qsin + r cos4 tan B qcos4 - r sin4

(qsin 4 + r cos4)sece

0

where Mis the inertia matrix and extra state, 4 denotes,M,, M&, N,, Nh Nv, Z,, Z,, Yh, Y, and Y,, respectively. X,, Y,, Z,, K,, M,, and N, denote the components of the inertial force and moment. The detailed expressions are shown in APPENDIX. Nonlinear observers are designed based on the above (2).

A. Sliding Mode Observer

The Sliding Mode Observer (SMO), which is developed on the basis of the sliding surface concept [12], can set the gain value according to the uncertainty range of the plant model. The SMO is known to be robust under parameter uncertainty and disturbance. Besides, it can be easily applied to the nonlinear system. To estimate the 10 sensitive coefficients, the SMO is designed based on the observer model of (2).

i, = h(i,t) where state variable x represents U, v, w, p , q, r, 4, 0, and v, and additional states 4 represents the sensitive coefficients. The output variables are chosen as U, v, w, p , q, r, 4, 0, and v, respectively. L means the nonlinear gain value, which is determined by satisfying the sliding condition. Detailed derivation of the SMO is described in [13].

B, Extended Kalman Filter

The Extended KaLman Filter (EKF) can estimate the state variables optimally in nonlinear stochastic cases that include the plant perturbation and sensor noise. In particular, unknown inputs or parameters can be estimated by converting them into extra state variables [14]. The EKF is designed utilizing the observer model of (2).

r .-.

i, = h(2,t) where state and output variables are equal to those of the SMO. The gain matrix, K is determined from the Riccati equations [15]. The ten sensitive coefficients, which is represent by 6 , are estimated from (4).

In order to estimate the sensitive coefficients associated with horizontal and vertical motions, simulation is conducted for combined diving and steering motion of the AUV. The sensitive coefficients from the AUV block in Fig.1 is used as true value and compared with the estimated ones. The estimation performance of the SMO and the EKF is compared when the AUV undergoes combined diving and steering. The motion scenario is as follows: the AUV has the initial speed of 1.832 m/sec and the ruddedelevator angle is applied to 0.35 rad from the start. The rudder and elevator works within -0.4 to 0.4 rad.

Figure 3 compares the estimation results of the SMO and the EKF for the ten sensitive coefficients. The steady-state error is compared in TABLE I. In the figures, thick solid line represents the true value adopted from [9] and dashedholid line represent the SMOEKF result. In general, the EKF shows a good estimation performance, but Y,, Y, and Yv, which are associated with sway motion, have steady- state error. It is well known that the SMO is a robust observer under parameter uncertainty and disturbance, but it has large steady-state error and fluctuation at the transient period. Based on a series of simulation, it is concluded that the EKF estimates the sensitive coefficients with sufficient accuracy. Although the nonlinear observers have been used off-line in order to analyze system identification, those can be implemented online for estimating the state variables and control of an AUV.

TABLE I STEADY-STATE ERROR (Yo)

.................. ..... ......... 9 4.068

0 10 20 30 40 50 60 70 80 Bo 100 Tlme (sec)

4.038

4.04 i! ...............

- EKF

0 10 20 30 40 50 80 70 80 Bo 100 4.044

Time (sec)

4.005 \ - T w 1

.........

nme (sec)

- ............. ...............................

4.015

4 . M 0 10 20 30 40 50 60 70 80 Bo 100 Tlme (sec)

x loJ 1 ,

- 5 1

43 1

4.12 I / ' " r ' ;-True ' 11') - SMO "

........... i - EKF ......... ...... ............................ ' I J

I 4.15' ' ' ' ' ' ' 0 10 20 30 40 50 60 70 80 90 100

nme (sec) 0.06 I

......................................

P

' 0 1 0 2 0 3 0 4 0 0 5 0 7 0 8 0 9 0 1 0 0 Time (sec)

..... ................................. o,06i > _ I - - - i ' . ... ..'..... ' ' ' ' ' ' '

- .. 8 , , , , , , , ! , ' J

Time (sec) 0 10 20 30 40 50 50 70 80 90 100

.................................................

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 4.2

nme (sec)

Fig. 3. Estimation results of the SMO and EKF.

111. CONTROLLER DESIGN

Although an AUV system is difficult to control due to high nonlinearity and motion coupling, sliding mode control has been successfully appiied to underwater vehicles. In this paper, sliding mode control [9] is adopted for an AUV with the uncertainties of system parameters. Especially, when designing a sliding mode controller, the estimated hydrodynamic coefficients in section I1 are applied in the controller model.

It is well known that sliding mode control provides effective and robust ways of controlling uncertain plants by means of a switching control law, which drives the plant's state trajectory onto the sliding surface in the state space. Any system is described as a single input, multi-state equation.

i ( t ) = Ax(t) + bu(t) + sf(t) /e\

x( t ) E R""', A E R""", b E R""' where 6f( t ) is a nonlinear function describing disturbances and unmodelled coupling effects. The sliding surface is defined as

0 = s T z

~

432

where sT represents sliding surface coefficient and 2 the state error, i.e. 2 = x - x d . It is important that the sliding surface is defined such that as the sliding surface tends to zero, the state error also tends to zero. Sliding surface reaches zero in a finite amount of time by the condition.

6. = --r)sgn(a) (7)

where q represents nonlinear switching gain. and (7), we obtain

From (5 )

ST(AX + bU + sf - id) = -Sgn(0) (8)

and control input is determined as follows:

U = -(sTb)-'sTAX + (sTb)-'[-sTSf + S T &

- rlsgn(d1 (9)

If the pair (A,b) is controllable and (s'b) is nonzero, then it may be shown that the sliding surface coeficients are the elements of the left eigenvector of the closed-loop dynamics matrix ( A - b k T ) corresponding to a pole at the origin

S T [ A - b k T ] = 0 (10)

where the linear gain vector, kT is defined (sTb)-'sTA and can be evaluated from standard method such as pole placement. It should be mentioned that one of the eigenvalues of (A - bk') must be specified to be zero. The resulting sliding control law using a 'tunh' function is given as

U = -kTX - (sTb)-*sTSf + ( sTb)- ' sTi (J J )

-q(sTb)-' tanh(a/cD)

where Q, is the boundary layer thickness and it acts as a low-pass filter to remove chattering and noise. The choice of the nonlinear switching gain, and the boundary layer thickness, Q, is selected to eliminate control chattering.

A. Depth Control

linearized diving system dynamics are developed as follows: In order to design a controller in the vertical plane, the

P P - ( I ~ --L~M,)~=(-L~uM~)~ 2 2 - (zG we + ( - L u P 3 2 - M6)SS

2 e = q

z = -ue In (12), the values of Gq and Gh are taken from the estimated ones in section 11. Then the dynamic model for

depth control yields the state equation as

The sliding surface^ is defined as

os = 28.18?+14.378h-Z (14)

when the poles are placed at [0 -0.25 -0.261. Finally, the depth control law is determined as

S, = 2.3531q + 0.00628 - 0.16982, (15) + 2.3 tanh(a, /4).

For implementation of the above depth control law an AUV has to be equipped with the pitch rate, pitch angle, and depth sensors.

B. Heading Control

follows: The linearized steering system dynamics are given as

P P s 2 2 2 2

- (- L4Nv)i -f- ( I , - - L N , )i = (E L3ui,)v+ (E L%ir)r

@ = r (16)

In (16), the values of ~ , ~ , f & , $ v , $ , , a n d $&are taken from the estimated ones. Then the dynamic model for heading control yields the state equation as

[;]=[I:;:: 1:;:; ! ] [ j + [ - o ; 5 2 / 5 r . 0.145

(17)

The values to place the poles of the steering system at [O -0.41 -0,421 become

(18) CT, = 0.15V + 1.657 + @ and the heading control law is as follows:

6, = 0.5260~ + 0.1621r + 4.3465~, (19) + 1.5 -(ar / 0.05)

In order to implement the above heading control law, it is necessary to measure the signals of v, Y , and v/ .

IV. SIMULATION RESULTS

Numerical simulations have been performed in order to show the effectiveness of the proposed control system. The simulation program, which is developed using MATLAB 6.0 with SIMULINK 4.0 environment, is shown in Fig. 4. The controller block is composed of a sliding mode controller for the depth and heading control. The AUV data of input and output as the true plant are taken from NPS AUV I1 [9]. The depth and heading controls are simulated with the full nonlinear equation and the sliding mode controller developed in (15) and (19). The responses of control law using the estimated hydrodynamic coefficients are compared with those of control law with true coefficients.

Figure 5 shows the desired depth, tracking trajectory, and other controlled variables for the depth control simulation. The desired depth is given 1 m down from the initial depth during the first 50 secs, and then returning back to the initial depth after 50 sec. From these figures, the performance of the control law using estimated coefficients is similar to that of the control law with true coefficients. Especially, the proposed control system has accurate tracking performance and almost no drift in sway direction as can be seen from Fig. 5 (d). Also, sliding mode control shows the robustness on the presence of parameter uncertainty.

Figure 6 shows the response of the heading control. The heading control simulations are performed together with depth control in order to prevent the vertical motion occurring from the coupling. In order to follow the desired path, we define the line of sight [9] in terms of a desired heading angle. The proposed heading control follows the desired heading angle and is compared with the control law with true coefficients. The desired path is chosen 1 m towards y-direction during the first 50 secs, and then returning to the initial position. The proposed control law follows the desired path accurately similar to the depth control law.

I i

433

(c) Pltch angle

E. 4 5 - Desired Depth

- Estimate -1

-4 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

@) Elmtorangle (d) Horizontal dedation

P ii -20 -0.1

o io 20 30 40 50 60 70 80 90 100 -O20 1 0 2 0 3 0 4 0 5 6 6 0 7 0 8 0 9 0 1 0 0 Time (sac) nme (sec)

Fig. 5. Simulation results of depth control.

(a) Hodzontal trajactoty

Desired Path

- Estimate

(a) Hodzontal trajactoty

DeSiredPath 11

I 0 10 20 30 40 50 60 70 80 90 100

(b) Rudderangle 40

(c) Yaw angle

-5 0 10 20 30 40 50 60 70 80 90 100

(d) Vertical de\latlm 0.01

0.m -

O’

Fig. 6. Simulation results of heading control.

V. CONCLUSIONS

A sliding mode control using the estimated hydrodynamic coefficients is proposed in this paper to maintain the desired depth and heading angle. The hydrodynamic coefficients are estimated based on the nonlinear observer such as SMO and EKF. Especially, the EKF has a good estimation performance and estimates the coefficients with sufficient accuracy. Using the estimated coefficients, a sliding mode controller is designed for the diving and steering maneuver. The control system using estimated hydrodynamic coefficients is compared with the control system with true coefficient. It is demonstrated that the proposed control system #is stable and follows the desired depth and path accurately. It means that the sliding mode control shows the robustness under parameter uncertainties. The proposed estimation method is believed to reduce the PMM test for measuring the hydrodynamic coefficients. In

addition, the proposed control system makes the AUV stable and controllable in the presence of parameter uncertainties and external disturbances.

REFERENCES

D. Sen, “A study on sensitivity of maneuverability performance on the hydrodynamic coefficients for submerged bodies,” J. ofship Research, vol. 44, no. 3, pp. 186- 196, Sept. 2000. W. Y. Hwang, “Application of system identification to ship maneuvering,” MIT Ph.D. Thesis, 1980. G. Antonelli, F. Caccavale, S . Chiaverini, and L. Villani, “Tracking control for underwater vehicle- manipulator systems with velocity estimation,” IEEE J. Oceanic Eng., vol. 25, no. 3, pp. 399-413, July 2000. T. I. Fossen, and M. Blanke, “Nonlinear output feedback control of underwater vehicle’ propellers

434

using feedback from estimated axial flow velocity,” IEEE J. Oceanic Eng., vol. 25, no. 2, pp. 241-255, April 2000.

J. Farrell, and B. Clauberg, “Issues in the implementation of an indirect adaptive control system,” IEEE A Oceanic Eng., vol. 18, no. 3, pp. 31 1- 318, July 1993. J. Yuh, “Modeling and control of underwater vehicles,” IEEE Trans. Syst,, Man, Cybern., vol. 20, pp. 1475- 1483, 1990 D. R. Yoerger, and J. J. E. Slotine, “Robust trajectory control of underwater vehicles,” IEEE J. Oceanic Eng., vol. OE-10, no. 4, pp. 462-470, 1985. R. Cristi, F. A. Papoulias, and A. J. Healey, “Adaptive sliding mode control of autonomous underwater vehicles in the dive plane,” IEEE J. Oceanic Eng., vol. 15, no. 3, pp. 152-160, July 1990.

A. J. Healey, and D. Lienard, “Multivariable sliding mode control for autonomous diving and steering of unmanned underwater vehicles,” IEEE J. Oceanic Eng., vol. 18, no. 3. pp. 327-339, 1993.

[lo] R. K. Lea, R. Allen, and S . L. Meny, “A comparative study of control techniques for an underwater flight

vehicle, ” International J of System Science, vol. 30, no. 9, pp. 947-964, 1999.

[ l l ] P. M. Lee, S. W. Hong, Y. K. lim, C. M. Lee, B. H. Jeon, and J. W. Park, “Discrete-time quasi-sliding mode control of an autonomous underwater vehicle,” IEEE A Oceanic Eng., vol. 24, no. 3, pp. 388-395, July 1999.

[12] J. J. E. Slotine, J. K. Hedrick, and E. A. Misawa, “On sliding observers for nonlinear systems,” ASME J. of Qnamics, Measurement, and Control, vol. 109, pp.

[13] R. A. Masmoudi, and J. K. Hedrick, “Estimation of vehicle shaft torque using nonlinear observers,” ASME J. of Dynamics, Measurement, and Control, vol. 114,

[14] L. R. Ray, “Stochastic decision and control parameters for IVHS,” 1995 ASME IMECE Advanced Automotive Technologies, pp. 114-1 18, 1995.

[15] M. Boutayeb, H. Rafaralahy, and M. Darouach, “Convergence analysis of the extended Kalman filter used as an observer for nonlinear deterministic discrete-time systems,” IEEE Trans. Autom. Control, vol. 42, no. 4, pp. 581-586, 1997.

245-252, 1987.

pp. 394-400, 1992.

APPENDIX

M =

0 n - - L X , P 3 2 0 ~ - - L ~ Y , P

2 0 0

0 -mzG - - L ~ K + P 2

m G 0

P 2

0 - - L4N, ... ...

0 0

0

P m - - L ~ Z , 2 0 I , - - LsKb

P 2

- mG - - L4Yp

0

P 2

- f 2 L4Mw - Iv

0 - I , - ~ L ~ N , 2 ... ...

013x6

m G 0

0 --L4T P : - f L4Zg 0

-1, 2

I)’ - - L M g 2 -1,

-1, 2

2

2 - I , --L5K, P i 0,,,3

I , - - L N , P 5

P s

... ... ... ... ’ ‘13x13

435