Controlling the position of a remotely operated underwater vehicle

li NOgl~ - ltOUAND

Controlling t h e P o s i t i o n of a R e m o t e l y Operated U n d e r w a t e r Vehicle E d w i n K r e u z e r a n d F e r n a n d o C. P i n t o

Technical University Hamburg-Harburg Ocean Engineering Section H Eissendorfer Strasse 42 D-21073 Hamburg, Germany

Transmitted by F. E. Udwadia

ABSTRACT The major problems encountered in the development of control systems for remotely operated underwater vehicles (ROV) are analyzed. The modeling of the hydrodynamic effects on the vehicle itself and on the umbilical cable is discussed. A robust control based on "sliding modes" is used and the sensor systems that can be employed are presented. Experiments with an underwater double pendulum and with a small, laboratory size, vehicle are used to check the simulation results.

1.

INTRODUCTION

In recent y e a r s r e m o t e l y o p e r a t e d vehicles ( R O V s ) have been built for various purposes. T h e y found a p p l i c a t i o n s ranging from oil prospection a n d inspection of u n d e r w a t e r oil derricks to e x p l o r a t i o n of sea resources. T h e i r d e v e l o p m e n t has been m o t i v a t e d b y t h e high cost of h u m a n divers and t h e e n o r m o u s risk to life w i t h working u n d e r water. Such u n d e r w a t e r vehicles are c o m p l i c a t e d active m e c h a n i s m s o p e r a t i n g in h a z a r d o u s e n v i r o n m e n t s [10]. T h e design of control s y s t e m s to facilitate t h e w o r k of t h e o p e r a t o r needs a s t r o n g knowledge of t h e n a t u r e of t h e h y d r o d y n a m i c forces a c t i n g on the s y s t e m [1]. Thus, R O V s are c h a r a c t e r i z e d b y an extensive i n t e g r a t i o n of concepts from a v a r i e t y of i n d i v i d u a l disciplines.

APPLIED MATHEMATICS AND COMPUTATION 78:175-185 (1996) © Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010

0096-3003/96/$15.00 PII S0096-3003(96)00007-4

176 2.

E. KREUZER AND F. C. PINTO MATHEMATICAL MODEL

The motion of the vehicle gives rise to a pressure distribution over its surfaces, which should be integrated to obtain the hydrodynamic effect of the interaction with the water. The mathematical description of these forces is complicated by the numerous effects present. Especially, the vortex shedding and the forces generated by waves require the use of powerful numerical methods. For low relative velocities, which can be assumed in the case of ROVs, the hydrodynamic forces can, however, be described with sufficient accuracy using the so-called Morison equation [13]:

Fh

=

C~½PwA(v Ivl) + Cm";"

(1)

where F h is the hydrodynamic force itself, Pw is the water density, A expresses a reference area, and v and "~ are the vectors of the relative velocity between fluid and vehicle and of the relative acceleration, respectively. The coefficients Cd and Cm depend on the direction of the relative velocity. Since the acceleration of the water flow is normally very small, the second term on the right-hand side of (1) can be added to the inertia terms of the mass matrix. The coefficient Cm can then be called hydrodynamic added mass. The first term describes the quadratic damping of the water and represents the nonlinear effect of hydrodynamic forces. To include the hydrodynamic effect of the umbilical cable it is modeled as a multibody system, resulting in a chain of rigid cylindrical elements connected through revolute joints, springs, and dampers to account for the stiffness of the cable and the dissipation of energy along it. This model allows a better analysis of the global motion of the umbilical and its influence on the behavior of the vehicle. An experimental set-up consisting of a submerged inverted double pendulum was designed to verify the applicability of (1) for the simulation of the dynamic behavior of the umbilical [5, 7]. The double pendulum represents the simplest model of a small portion of the cable. The lower segment is forced, by means of a servo-motor, to perform a harmonic angular motion. Figure 1 shows a comparison between the simulated and the measured behavior of the pendulum for the case of harmonic excitation of 0.4 Hz. The figure shows the time history of the relative angle ~P2 between the two segments. The dynamic stability was also investigated through symbolic and numerical methods. For more details concerning the stability analysis refer to [5, 81. The good agreement with the experimental results [7] indicates that the Morison equation can be used successfully for the description of the global

177

Controlling the Position of a R O V 0.8

I

measurements

0.6

f=0.4 Hz I

0.4 0.2 0 -0.2 -0.4 -0.6

"0.8

0

'

2

3

4

5 t[8]

6

FIC. 1. Simulated and measured behavior.

motion of the umbilical. Higher-order effects such as vortex shedding and structural energy dissipation along the cable are indeed not considered with this approach [3]. The modeling of the hydrodynamic forces that act on the ROV's hull can be accomplished in the same manner [2]. In this case, however, it is not possible to calculate the necessary coefficients of (1). They must be measured experimentally. To obtain the values of the coefficients Ca a wind tunnel test was performed. The ROV model, which is described in Section 4, was fixed to a platform that was capable of measuring the forces and moments acting on the vehicle in the three spatial directions. These values were related to the frontal area of the vehicle furnishing the desired values for the damping coefficients. The coefficients are highly dependent on the actual geometry of the ROV. Different configurations of the working tools and packages mounted on the vehicle could lead to different coefficients. The control system must cope with these variations, which are to occur even during normal operations of the ROV [15]. 3.

ROBUST POSITION CONTROL

The pilot of the ROV is responsible for its positioning relative to a structure. This task is difficult by the lack of good position information

178

E. KREUZER AND F. C. PINTO

under water. However, the effectivity of the operation of a ROV can be improved if its positioning could be assisted by a control system. The design of the control must take the variations on the system parameters p into account. Among other effects the hydrodynamic forces acting on the vehicle can vary as a consequence of different configurations of the system. The efficiency of the thrusters may change during their life. During the completion of tasks the inertial characteristics of the ROV can be modified by picking tools or parts of the structure which are to be transported. To avoid collisions with the underwater structures it is very important to guarantee that the ROV possesses a similar dynamic behavior in all these situations. This implies a robust control system. A way of achieving robustness [14, 16] is through the use of sliding-mode [11] or variable structure controllers. The dynamics of the umbilical cable may be treated as an external disturbance of the system. If the cable forces are bounded the required robustness may not be affected to a great extent. For most ROVs the pitch and roll motions are stabilized through the inherent hydrostatic characteristics of the construction itself. The control system should deal only with the depth z, the cartesian positions x and y, and with the yaw angle ~. In general the uncontrolled angles for roll and pitch motions remain small and the depth can be decoupled from the other coordinates. The set of second-order equations of motion can be transformed in the usual state-space representation: ~[ -- F ( X , p )

+ U(X,c).

(2)

For the case of a ROV, F(X, p) is a nonlinear function of the state-vector X, representing all mechanical and hydrodynamic effects on the ROV. The vector U(X, c) of the control forces has also a strong nonlinear character due to the control law based on sliding modes. The approach of a multivariable sliding mode controller is to confine the dynamics of the controlled system in a high-dimensional manifold. The control parameters c describe this manifold in the state space. Their choice must Mso guarantee the stability of the motion; i.e., once the system achieves the manifold it should converge to the desired position, sliding along it. The simplest definition for the manifold S = 0 is a linear one:

x=Cx + (3) =

+

Controlling the Position of a ROV

179

It should be noted that Sx, by, and Sa must be defined using the coordinates x and y projected in a local coordinate system, fixed to the ROV. Ex, Ey, and Ea are the position errors in this local system according to: E~= +(x~-

x)cos(a) + (y~-

y)sin(a),

Ey = - ( x ~ -

x)sin(a) + (Ys-

y)cos(a),

(4)

The index s indicates the values of the set points. Assuming a position control instead of a trajectory control the desired velocities upon reaching the final position should all be equal to zero, so that V~ and Vy are the linear velocities, expressed also in the local system, and Va is the angular velocity of the yaw motion. The control forces in the vector U(x, c), which are needed to obtain the sliding mode on the manifold, are of nonlinear nature. A discontinuous control law as in the case of the so-called bang-bang controllers is a common choice. The major disadvantage is due to the high frequencies that are necessary on the actuators. This problem can be minimized if the discontinuity is smoothed by the use of a function of the form: 2 arctan(psi) Fi =

,

(5)

7r

where the index i represents x, y, and a from (3). The discontinuous character of the force law in the sliding mode control can then be controlled through the parameter p. With the smoothing effect due to the use of the arctan function, the high frequencies on the control forces are also smoothed, improving the performance of the actuators. The generalized forces determined by (3) and (5) are then transformed through a linear combination in the actual thrusts of the propellers. To design the controller, the control forces can be substituted into (2), which can be linearized in the vicinity of the set points for the position. Requiring that the real part of the eigenvalues of the linearized system be negative the domain where a stable motion can be achieved may be determined as a function of the parameters cx, cy, and Q and of the characteristics of the thrusters. This leads to the conclusion that all coefficients must be negative for the behavior of the system to be stable once in the manifold.

180

E. KREUZER AND F. C. PINTO

The global stability of the system m a y be investigated with the use of a quadratical Ljapunov function such as L(S) = S 2. The manifold is global stable if

dL d--t- ~< 0

(6)

on the region of the state space t h a t is of interest. This consideration places another constraint on the parameters c. The use of small absolute values for the coefficients m a y ir~crease the region where (6) holds. It leads, however, to a very slow system dynamics once in the manifold. On the other hand high values for cx, c~, and ca increase the necessary power of the actuators in order to achieve the sliding condition. Coefficients on the range - 0 . 3 to - 3 are a good compromise. Nevertheless, the analysis of the variation of the kinetic energy of the system shows t h a t the dynamic behavior of the system is attracted to the manifold even if (6) does not hold. Figure 2 shows the behavior of the system in the state space for different initial conditions and for one degree of freedom, e.g. the depth control. Figure 3 shows the time history for the motion of the vehicle from the origin to the point with coordinates x~ = 2 and Ys = 3 with orientation ~ = a r c t a n ( y J x ~ ) . The parameters of the controller are c i = - 0 . 5 . The R O V has a mass of 40 kg and the thrusters, see Section 4, are capable of

0.4

0.2

0 o

-0.2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Position [m] FIG. 2.

Dynamics of the system for different initial conditions in the depth control.

181

Controlling the Position of a R O V i.°o...~o ....

,

/

2.5

Y~

.//'//

/

,,//

g.

/

1.5

0.5

0

I

I

I

!

I,

5

I0

15 t (sl

20

25

FIG. 3.

30

Simulation of the ROV's motion.

exerting 5N each in the x-direction and 7N in the y-direction. The actual thrust in x is, however, limited by the necessity of simultaneously applying a moment about the z-axis. The robustness of the control system can be seen in Fig. 4 where the actual trajectory on the x-y-plane is shown for different values of the mass of the vehicle. The behavior of the ROV is almost the same in the three cases although the mass varies from 20 to 80 kg. The dynamics on the manifold is shown in Fig. 5 for the coordinate x. In Fig. 6 the yaw motion of the vehicle is drawn to show the effect of the control on the angular position.

4.

EXPERIMENTAL MODEL

A small experimental ROV has been developed at the Ocean Engineering Section II of the Technical University Hamburg-Harburg; see Fig. 7. This vehicle was designed to operate in the wave channel of the department serving as a test bed for different kinds of propulsion, sensor, and control systems. The vehicle possesses two mechanical arms, with 6 degrees of freedom, which are intended to be used as position sensors in the proximity of a structure [4, 6, 9]. The joint positions can be measured with special poten-

182

E. KREUZER AND F. C. PINTO 3.5 3

M=2Okg M--4Okg

/.

- .......

'~80kg

........

2.5 2

e

.(.'Y'

Set-Point

1.5 1

0.5 0 -0.5 ............................ -0.5 0 0.5 1

i................

1.5 x [m]

2

2.5

3

3.5

FIG. 4. Trajectoriesin the ~:y-plane.

0.2 0.18

]

M--2Okg - M=4Okg

~

. . . . . .

0.16 0.14 a :' 0.12 0.1 >,

0.08

0.06 0.04 0.02 0

0

0.5

1 1.5 Position [m]

2

2.5

FIG. 5. Dynamicsof the system in the manifold for motion in the x-direction.

183

Controlling the Position of a R O V

Set-Poinf

X

>

FIG. 6.

Yaw motion.

tiometers [12]. The propulsion is achieved by eight electrical motors with propellers arranged as in Fig. 7. A buoyancy chamber guarantees the static stability of the upward position. Sensors for the angular velocity and for the accelerations of the ROV are placed inside it. Figure 8 shows the time history of the x and y coordinates of the ROV. The ROV is driven by the control system to the desired position, where it

~

Buoyancy C h a m b e r

Umblllcal Thrusters

FIG. 7.

CAD-model of the ROV.

184

E. KREUZER AND F. C. PINTO 0.55 0.5 0.45

i

0.4 0.35 0.3 0.25

I

I

I

I

I

I

I

I

190

200

210

220

230

240

250

260

270

t [S]

FIc. 8. Measured x and y coordinatesof the ROV.

stays at rest. The set point is changed for approximately 15 seconds and is then finally changed back to its original value. The vehicle follows the determined positions. The effects of the umbilical cable slow down the whole motion. This is due to the relatively high stiffness of the cable. The necessity of transmitting enough power for the electrical motors does not allow the use of a cable with a very small diameter. The precision of positioning is, however, of about 0.05 m for each direction.

5.

CONCLUSIONS

A mathematical description of the hydrodynamic effects of the water on the system ROV and umbilical is achieved using the Morison equation. Experiments with an underwater double pendulum have confirmed the applicability of the proposed modeling for the analysis of the global motion of the umbilical. The respective coefficients for a model ROV were measured in a wind tunnel. Robust control of the position relative to a structure is achieved with the use of a sliding mode controller. The design takes into account the local and global stability of the system. A manipulator is used as a position sensor in the region close to a structure under water. A model ROV was constructed and used as a test bed for the sensor and control systems described.

Controlling the Position of a ROV

185

REFERENCES 1 L. Bevilacqua, W. Kleczka, and E. Kreuzer, On the mathematical modelling of ROV's, in Proceedings of the Symposium on Robot Control, Vienna, pp. 595-598, 1991. 2 D. Lewis, J. Lipscomb, and P. Thompson, The Simulation of Remotely Operated Undemvater Vehicles, ROV '84, Marine Technology Society, 1984. 3 M. Every and M. Davbies, Predictions on the Drag and Performance of Umbilical Cables, ROV '84, Marine Technology Society, 1984. 4 L. Hsu, R. Costa, and J. Cunha, Medi~o de Posi~o Pela Estrat~gia "TautWire" Para o Controle de Posi§~o de Um VOR, Relatdrio interno C O P P E / U F R J , Rio de Janeiro, 1990. 5 W. Kleczka, E. Kreuzer, and F. Pinto, Analytic-Numeric Study of a Submerged Double Pendulum, ASME International Symposium on Flow-Induced Vibrations & Noise, Anaheim, 1992. 6 F. Pinto and E. Kreuzer, Uma Compara~8o entre Sistemas de Sensores Para Um Veiculo Submarino, F6rum sobre VOR na Pertobr£s, Rio de Janeiro, 1993. 7 W. Kleczka, E. Kreuzer, and F. Pinto, Experimental and Analytical Investigations of a Submerged Double Pendulum, 1st European Nonlinear Oscillations Conference, Hamburg, 1993. 8 W. Kleczka, Symbolmanipulationsmethoden zur Analyse nichtlinearer dynamischen Systeme am Beispiel Fluid-gekoppelter Strukturen, Fortschritt-Berichte der VDI-Zeitschriften, Reihe 11, Nr. 213, VDLVerlag, Diisseldorf, 1994. 9 E. Kreuzer and F. Pinto, Sensing the Position of a Remotely Operated Underwater Vehicle, 10th CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators, Gdansk, 1994. 10 S. Polomsky, Rechnergest~ttztes Entwerfen yon ferngesteuerten Unterwasserfahrzeugen, VDI Fortschritt-Berichte, Reihe 20, Nr. 48, VDI Verlag, Diisseldorf, 1991. 11 V. Utkin, Sliding Modes in Control Optimization, Springer-Verlag, Berlin, 1992. 12 O. Fiedler, StrSmungs- und Durehflussmesstechnik, R. Oldenburg Verlag, MSnchen, 1992. 13 J . N . Newman, Marine Hydrodynamics, 4th ed., MIT Press, Cambridge, MA, 1982. 14 H. Suzuki and K. Yoshida, Trajectory tracking control of a ROV for lifting objects, in Proceedings, 1st International Offshore and Polar Eng. Conference Edinburgh, 1991. 15 P. Sayer and C. Miller, The hydrodynamics of ROVs carrying work packages, in Proceedings, 1st International Offshore and Polar Eng. Conference Edinburgh, 1991. 16 I. Yamamoto and T. Nagamatu, A control system design of a tethered underwater vehicle, in Proceedings, 4th International Offshore and Polar Eng. Conference, Vol. 2, Osaka, 1994.