Ship replenishment using synchronization control

ELSEVIER

Copyright (Q IFAC Manoeuvring and Control of Marine Craft, Girona, Spain, 2003

IFAC PUBLICATIONS www.elsevier.comllocatelifac

SHIP REPLENISHMENT USING SYNCHRONIZATION CONTROL

Erik Kyrkjebe 1 Kristin Y. Pettersen

Department ofEngineering Cybernetics Norwegian University ofScience and Technology N-7491 Trondheim. Norway E-mail: [email protected]@itk.ntnu.no

Abstract: In this paper synchronization techniques are used to develop a control law for rendezvous control of ships. In particular, recent results on external synchronization in robotics are applied to the problem of rendezvous control of ships. No assumptions on the availability of a dynamic model for the replenished ship are made, and only position/attitude measurements are available for both ships. The synchronization closed loop error dynamics is found to be bounded by the acceleration and velocity of the replenished ship, and the synchronization controller yields semi-global exponential convergence of the closed-loop errors for position keeping, and semi-global uniform ultimate boundedness of the closedloop errors during trajectory tracking.Copyright© 2003 IFAC Keywords: Synchronization, Nonlinear control, Ship control, Observers, Tracking

the supply ship will synchronize itself to the true position of the main ship to increase control accuracy and provide safer operations. Furthermore, with this approach, there is no need for dynamic model information of the ship that is to be replenished, which makes the technique more available to civil applications.

I. INTRODUCTION For centuries, ships have met at sea to exchange fresh water, food and other necessities. More recently, attention has been given to the problem of replenishment at sea for military ships to avoid or shorten port time. The control approaches of these rendezvous use flags and signals to communicate control commands between ships, or some sort of tracking control of both ships in order to maintain trajectories that provide joint motion suitable for replenishment. This paper proposes instead an external synchronization scheme to dynamically control the supply ship on the basis of the true error in position between the two ships, as opposed to using tracking control where the two ships each track a predefined trajectory based on where the main ship is supposed to be. Due to disturbances like wind, waves, and currents, perfect tracking of a predefined desired trajectory is not always possible, and hence strain and tear may be imposed on the replenishment cables, fuel hoses, etc., or the ships may come too close resulting in dangerous situations. Using synchronization control

1

Synchronization is the theory of time conformity between processes, which incorporates both cooperating and coordinated systems. Cooperative synchronization of motion regards all synchronized objects on equal terms, while coordinated, or external, synchronization assumes that one of the systems takes the role of a leader that governs the motion of the others. The synchronization phenomenon was probably first reported by Huygens (1673) who observed that a pair of pendulum clocks hanging from a lightweight beam oscillated with the same frequency. Synchronization has in the last decade attracted a tremendous interest from researchers within physics, dynamical systems, circuit theory, and more lately control theory. The theory of synchronization has been applied in control problems by e.g. (Blekhman, 1988), Nijmeijer (1997), Nijmeijer and Mareels (1997), Blekhman et al. (1997), Hills

Supported by SINTEF Electronics and Cybernetics, Norway.

247

and Jensen (1998), Pogromsky and Nijmeijer (1998), Fradkov et al. (1999), Huijberts et al. (2000), Nijmeijer (200 I), Dong et al. (2002), Rodriguez-Angeles and Nijmeijer (200 I), and Rodriguez-Angeles (2002). The attention ranges from applications as mobile robots and telemedicine to new perspectives in observer and control design. Coordinated control of autonomous marine crafts are treated in Encamacao and Pascoal (200 I), while control approaches for underway replenishment of ships can be found in e.g. Brown and Alvestad (1978) and Dimmick et al. (1978). The rendezvous issue is addressed for unmanned underwater vehicles by Kato and Endo (1989) and White et al. (1996); the latter using fuzzy logic theory. Chen (2003) suggest an increased focus on autopilot design for replenishment operations to reduce risk of collision. Results on the cognate topic of ship maneuvering and formation control has been reported by Skjetne et al. (2002a) and Skjetne et al. (2002b). We show in this paper how recent results on external synchronization control of robotic systems presented in Rodriguez-Angeles and Nijmeijer (200 I) and Rodriguez-Angeles (2002) can be used to develop a control law for rendezvous control of ships. With this approach, we evade the need for a dynamic model of the ship that is to be replenished, and only position measurements of the two ships and the replenishment ship model is needed for control. In particular, nonlinear observers are designed to estimate the velocities and accelerations of the two ships, and these estimates are used in the synchronization controller. The observer-controller scheme provides semi-globally exponentially stable error dynamics for position keeping, and semi-global uniform ultimate boundedness of the error dynamics during trajectory tracking.

Centripetal matrix also including added mass effects, and D the damping matrix. Paulsen (1996) proved that the Coriolis matrix C can always be represented in terms of the Cristoffel symbols known from robotics, and this representation will be used throughout the paper. The vector g is gravitationallbuoyancy forces and moments, while T is the control torque applied to the ship. The vector 11 represents the earth-fixed position and orientation ofthe ship. We will in the development of the observer-controller scheme assume that the damping is linear.

3. SYNCHRONIZATION OBSERVER-CONTROLLER SCHEME A supply ship is said to be synchronized to the main ship, or controlling ship, if its position/attitude and velocity coincide for all t ~ 0, or asymptotically for t -+ 00. Note that the position vector 11s is synchronized to some offset constant reference 11 r to maintain a position alongside the controlling ship, and the problem is considered as synchronizing 11s to rrm by redefining 11m = rr;ue -11r. The objective is thus to find a control law that stabilizes the error in position and velocity to zero; (e,e) = (0,0). In this section we show how the results for robot synchronization presented in Rodriguez-Angeles and Nijmeijer (200 I) and Rodriguez-Angeles (2002) can be used to develop a ship rendezvous observer-controller scheme that does not require information about the dynamic model of the ship that is to be replenished, only the dynamic model of the supply ship, and which furthermore only requires position and attitude measurements for both ships. The supply ship is synchronized with the main ship through the control law, and is in fact a physical observer of the main ship dynamics. The observer-controller scheme provides semi-globally exponentially stable error dynamics for position keeping, and semi-global uniform ultimate boundedness of the error dynamics during trajectory tracking.

In Section 2, the general ship model in vectorized form is stated, while Section 3 presents the synchronization observer-controller scheme with all states available in 3.1 and only position measurements in 3.2. The closed-loop error dynamics are given in 3.3, while stability is discussed in 3.4. Simulation results are presented in Section 4, and conclusions and comments on future work in Section 5.

3.1 Rendezvous control with position, velocity and acceleration measurements

Assuming that all values of position/attitude, velocity and acceleration of the supply ship and main ship are available for measurement, the following control law gives a stable rendezvous control scheme;

2. SHIP MODELS The general dynamic ship model in vectorial form as given in Fossen (1994) is used throughout this paper, and thus

M; (11;) 11; +

c; (11;, i1;) i]; + D; (11;, i];) i]; +g(11;)

Ts

= Ms (11s) 11m+ Cs (11s,i]s)i]m +Ds (11s, i]s) i]m + g(11s) - Kde - Kpe

(2)

where the synchronization errors are defined as

(I)

e = 11 s - 11m,

= T;

and K d , K p E R

where i E s,m denote the supply ship and the controlling, main, ship respectively. The matrix M is the matrix of inertia and added mass, C the Coriolis and

nxn

e = i]s -

i]m

(3)

are positive definite gain matrices.

The stability proofofthis controller is straight forward using the Lyapunov function

248

v= ~eTMs(Tls)e+~eTKpe

(4)

and stability theorems such as La Salle's theorem and Barbalat's lemma (Khalil, 2002) on the negative semidefinite time derivative

V = _eT (Ds (Tls,Tls) + K,,-)e.

(5)

For any positive definite matrices K p , K,,-, the supply ship synchronizes in the velocity and the offset position to the main ship.

3.2.3. Main ship state observer The estimated main ship velocity and acceleration values ~m' ilm are not available through direct measurement, and must be reconstructed from the position/attitude and error estimates. To compensate for the lack of a dynamic model, the velocity and acceleration values for the main ship are reconstructed based on information of the supply ship and the synchronization closed-loop system. Estimates for T/m, iJm, flm are given as

-

iim = iis-e i)m =iJs-e

3.2 Rendezvous control with position and attitude measurements only If, on the other hand, the velocity and acceleration of the two ships are not available for measurement, but only the position and attitude of the two ships are measured, the control law (2) is not feasible. In order to overc~m~ th~ pr~blem, observers that provides estimates iJSI fl s' iJm' flm are designed. The synchronization controller 't's will depend on estimated values for velocities and accelerations, but on position/attitude measurements. The feedback control law of (2) is rewritten as

't's

=Ms(Tls)f)m+Cs(Tls'~s)~m +Os (Tls' ~s) ~m + g(Tls) - Kd~ -

f)m =

-;-

~

:t (~s-~)r

= -

(Ms (T/s)-I

+ L e2) e+ Lp2Tis

3.3 Synchronization closed-loop error dynamics To simplify the stability investigation of the closedloop error dynamics, an assumption on the positive definiteness of Lel, Le2, Lpl' L p2 is made. The gain matrices are chosen as symmetric matrices, and Le! = Lpl and Le2 = Lp2.

(6)

Kpe.

3.3.1. Coordinate transformation To simplify the calculations, a coordinate transformation is introduced as

Tim =

L -

Tis

!r ~ = -Ms (Tls)-I [Cs (Tls' ~s) ~

( 12)

= iJs - Lp! Tis

and (13)

(7)

Kd~+ Kpe] + Le2e

It follows directly from (12) that the estimation error of the main ship is

where Le!, Le2 are positive definite gain matrices, and the estimated position/attitude and velocity synchronization errors are defined as

e = e-e,

e- Tis

1Jm = ~- ~s - LplTim

de =e+ ele

+Os (Tls' ~s) ~+

(11 )

where the last relation stems from (7) and (9).

3.2.1. Synchronization error observer The estimated values for the errors e, e can be obtained through a full state nonlinear Luenberger observer d~

~

(14)

and that the error between the main and supply ship trajectories is

(8)

Note that the observer (7) and (8) introduces an extra correcting term in ~ Lele that yields faster performance during transients, but have some negative effects on noise sensitivity.

(15)

e= -

3.3.2. Synchronization error dynamics To obtain the synchronization closed-loop error dynamics, we go through the steps of the error dynamics of the synchronization (e, e), the estimation synchronization error fe,~) and the estimation position and velocity error (Tis, ~ s) before applying the coordinate transformation of(12) and (13). The synchronization error dynamics using (6) in (I) is

3.2.2. Supply ship state observer The estir1!.ated supply ship position and velocity values iis, iJs is found using the full state nonlinear observer d _ -;_ ~Tls = 11s + L p!11s (9) di~s = -Ms (T/s)-I [Cs (T/s, ~s

F

+Ds (11s' ~s) ~ +

Ms (11s) e+ Cs (11s, i)s)e + Ds (T/s, i)s) e + K,,-e +Kpe = Ms (11s) (ilm - iim) - Kd (~-

Kd~+ Kpe] + Lp2Ti

+Cs (TlSI ~s)

where L pl , L p 2 are positive definite gain matrices, and the estimated supply ship position/attitude and velocity errors are defined as

-11s = 11s - -T/s,

iJs = iJs - i)s'

-

~m -

Cs (T/s, ~s) ~m

e)

(16)

+Ds (11S1 T/s) 11 m- Os (11s,11s) 11m

(10)

while the properties of (3), (8), (10) and (11) yields

249

i1m - T1m = e - T1s ilm - ilm = e- ils

- -

~

-::-"

(17)

d(77)

T1 m-T1m = dt e-T1s . The Coriolis matrix C (T1, il) does not have a unique definition, and several definitions can be found in Fossen (1994). Paulsen (1996) showed that the matrix C (TI , il) can be represented in terms ofthe well known Christoffel symbols from robotics, and thus it can be shown that the following equality holds

C(T1,x)y = C(T1,Y)x.

(18)

From (16) we can now write

Cs (1}s, ~s)

~m-Cs (1}s, ils) ilm = Cs (1}s, ils) ~

+Cs(1}s,~s)~s-2Cs(T1~ils~~s

(19)

+Cs (T1s,1}s) e-Cs (11s, 11s)e The shorthand notation of C

= Cs (1}s, ils)

and

C=

Cs ( 1}s, ~s) is introduced for simplicity. The property of (18) does not hold for the damping matrix D (11, il),

and the discussion is here limited to linear damping to simplify the stability investigation. The damping term Ds (1}s, ils)ilm reduces to Dsilm, and the resulting error dynamics is given using (17) as

where C is rewritten in trrms of the coordinate transformation of (12) as Cs ~ 1}s, rys + L pl Tis) . The acceleration of the controlling ship, iim, appears in (24) as a non-vanishing disturbance, and the origin of the closed-loop error space is no longer an equilibrium for (23-24). The closed-loop error is at best ultimately bounded by the controlling ship acceleration.

3.4 Stability The stability properties of the closed-loop error dynamics are investigated to assure stability of the observer-controller scheme, and convergence of the supply ship position and velocity to the main ship motion.

Theorem 1. Consider the ship model (I), the controller (6) and the observers (7) and (9). Under the assumption that the signals ilm (t) and fim (t) are bounded, that there exists bounds V M and AM such that i.e. sup lIilm (t)1I = VM < 00 I

suplliim(t)1I

d+

(~m - ~s) + Kd~+

t

(20)

(C-2C)~s+ce-C~+Ds~-Ds~s

xT

Together with the observer defined in (7), the estimation synchronization error dynamics reduce to d_ 7 _ =M~1}s)

_I

-

(-

) -;-

[-Kpe+ C-2C 11 s

(21)

-Dsilsl- L e2e

eT

~T

eT

~~ Ti;]

(26)

The bounds on the velocity and acceleration of the main ship can be established based on knowledge of the desired trajectory for the main ship during replenishment, or by the limitation imposed by the maximum acceleration and velocity given by the propulsion system. The boundedness assumption of the acceleration and velocity thus has a clear physical interpretation in marine control systems. The bounds on the minimum eigenvalues Kp,m, Kd,m, Lpl,m, L p2,m of the gain matrices can be determined by a gain scheduling procedure. The first part of the theorem clearly states that the synchronization closed-loop errors are bounded under a trajectory following scheme, where the main ship tracks a trajectory during replenishment. The second part ofthe theorem states the convergence of the closed-loop errors during position keeping. A sketch of the proof is presented in the following.

and the supply ship error dynamics from (9) yields d _ 7 _

11 1}s = 1} -

= [ eT

is semi-globally uniformly ultimately bounded when (ilm, iim) # (0,0). Furthermore, if the main ship achieve steady state (ilm, iim) = (0,0) after ts ~ to, we have semi-global exponential convergence of the synchronization closed-loop error x for t ~ t s. <>

-e =e-Lele

'dt!1-;-11s

~~

00,

and that the minimum eigenvalues ofthe gain matrices K p,Kd, Lpl, L p2 are chosen to satisfy a set of lower bounds, then the synchronization closed-loop error

Ms (1}s)e+Ce+ Dse+ Kde+ Kpe = Ms (1}s) d t

= AM <

L p1 11s

d/ = M~1}s)-1 [-2K pe+ (C-2C) ~s (22) -DsilsJ - 2L e2e+ Lp2Tis - ilm. Regarding the coordinate transformations of (12) and (13), the synchronization closed-loop error dynamics are

Ms (1}srTI + c1f + D;r/+ KdTi + KpTf = M (T1s) Lp1 rym + DsLpl Tim + C(Ti - L pl Tim) - - ('" _)) (23) +C ( L pl 1}m 1}s+ L pl1}s +Kd (rys + Lpl (Tim + Tis)) - KpTim and

~m= -M (1}s)-1 Kp(Tim + Tis)-Lplrym -Lp2Tim - ilm 1)s= -M(1)s)-IK p (Tim+Tis)-L plrys (24) +M(1}s)-1 (C-2C-Ds) (rys+Lp1Tis) -Lp2(Tim + Tis)

3.4.1. Stability results candidate function

Considering as a Lyapunov

V(y) = yTp(y)y with the vector y defined as

250

(27)

T _ [-'-T -T ",T -T ",T -T] 11 11 1I m 1Im 11. 11.

Y -

p.(Y)

= [M.(1I.) ~M·(1I·)]Eo ~M. (11.) K p + ~K.t

P2(Y)

=

P3(Y) =

I [ I [

J,4)

I] I]

1+1I11mll p 1 + IITimll I L 2 J,4)

'lb 1+IITi.1I 1+ IlTi.1I I L p2 'lb

(28)

(29)

-...

--o~----=o---.;:;ooo----,:;;-.:.

-,oo~----=~~,oo~~,..----:!. -to!

Fig. 1. Position [m] and yaw angle [degrees] for main and supply ship.

....--.--_ ...

where Eo, ~ ,J,4) ,'lb are posItive constants. P (y) is positive definite if Kd,m > ~M.,M (11.) where the latter term defines the largest eigenvalue of M. (11.) , and if L p2 ,m > max{JLJ,io}. Since the off-diagonal terms in P 2 and P3 are clearly bounded, there exist constants Pm and PM such that

...

(30)

-.

-..o);------;::;---,.;;,oo:;--~,___--;i

,

The negative definiteness of V (y) can be investigated by introducing the norm of Y, denoted YN, and the upper bound of if (y) is thus given by

if (y) ~ IIYNII ( ao - QN mIIYnll + a211YN1I2)

-...

-,oo~____=~~,oo=--~o--_=!

-;

(31)

-.

where ao and a2 are positive scalars. If the assumptions of Theorem I are satisfied, then QN is positive definite. Since a2 depends on Lp,M, the bound for the closed-loop errors can be made arbitrarily small by a proper choice of Lp,M, which yields local uniform ultimate boundedness ofy. Moreover, the region of attraction for the closed-loop error vector of(26) can be expanded by similar choices, and the synchronization closed-loop errors are thus semi-globally uniformly ultimately bounded, and the ultimate bound can be made arbitrarily small by a proper choice of K p,m and Lpl,m (Rodriguez-Angeles, 2002).

-;

o

'f ..;

-..o);-----;:;,-------;;;;,oo,----.:..-------.i. -I-I

Fig. 2. Synchronization control errors e, observer errors e, ~ (bottom).

M"

=

0] 1.1274 0 0 1.8902 -0.0744 [ o -0.0744 0.1278

0

C"

If position keeping during replenishment is considered, and the dynamic positioning system of the main ship is able to achieve steady state in finite time, then iJm (t) = 0 and iim (I) = 0 for lE (1.,00) where I. 2: to. Under the assumptions of Theorem 1, the control law of (6) and the observers (7) and (9) yield semiglobal exponential convergence ofthe synchronization closed-loop error x for t 2: t•.

D"

e (top) and

0 0

o

[ -C/3

-C]3

0.0414 [

o o

-1.8902v" + 0.0744r"] 1.1274u"

0

0 0] 0.1775 -0.0141 -0.1073 0.0568

VI" - sin VI" 0] sin VI" cos VI" 0 o 0 1

COS

J" (11") = [

where v" = [v",u",r"f and 11" = [x",y", VI"] are related through iJ" = J" (1]") v". Synchronization tracking simulations are shown in Figure I, and the control and observer errors in Figure 2. The observer is started with no a priori knowledge of the states. The uniform ultimate boundedness of Theorem I is illustrated during the main ship maneuvering phase, while the semi-global exponential convergence of the closed-loop errors is observed after the main ship has achieved steady state behavior.

4. SIMULATION RESULTS The synchronization controller and the nonlinear observers were simulated as a 3DOF nonlinear maneuvering problem on a nonlinear supply vessel given in Fossen and Berge (1997) with mass m = 6.4· 10 6 (kg) and the nonand length 76.2 (m). Here 11 = [x,y, dimensional matrices used are given in Bis-scaling as

Vlf,

251

5. CONCLUSIONS

tor dynamics. Proc. 36th IEEE Con! on Decision and ControlS, 4237 - 4242. Fradkov, A.L., H. Nijmeijer and Yu.A. Pogromsky (1999). Adaptive observer based synchronization. pp. 417-435. Hills, J. W. and IF. Jensen (1998). Telepresence technology in medicine: Principles and applications. Proceedings ofthe IEEE 86,569-580. Huijberts, H.J.C., H. Nijmeijer and R. Willems (2000). Regulation and controlled synchronization for complex dynamical systems. Int. 1. Robust Nonlinear Control 10,363-377. Huygens, C (1673). Horoloquium Oscilatorium. Paris, France. Kato, N. and M. Endo (1989). Guidance and control of unmanned, untethered submersible for rendezvous and docking with underwater station. OCEANS '89. Proceedings 3,804-809. Khalil, H.K. (2002). Nonlinear Systems. 3rd ed.. Prentice Hall. New Jersey. Nijmeijer, H. (1997). On synchronization of chaotic systems. Proc. 36th Conf. Decision and Control pp. 384-388. Nijmeijer, H. (2001). A dynamical control view on synchronization. Physica D 154 pp. 219-228. Nijmeijer, H. and I.M.Y. Mareels (1997). An observer looks at synchronization. IEEE Trans. Circuits Syst. 144 pp. 882-890. Paulsen, M.J. (1996). Nonlinear Control of Marine Vehicles Using Only Position and Attitude Measurements. PhD thesis. The Norwegian University of Science and Technology. Trondheim, Norway. Pogromsky, Yu.A. and H. Nijmeijer (1998). Observer based robust synchronization of dynamical systems. Int. 1. Bifurc. Chaos 8 pp. 2243-2254. Rodriguez-Angeles, A. (2002). Synchronization of Mechanical Systems. PhD thesis. Eindhoven University of Technology. The Netherlands. Rodriguez-Angeles, A. and H. Nijmeijer (2001). Coordination of two robot manipUlators based on position measurements only. International JournalofContro174,1311-1323. Skjetne, R., A.R. Teel and P.v. Kokotovic (2002a). Stabilization of sets parametrized by a single variable: Application to ship maneuvering. Proc. 15th Int. Symp. Mathematical Theory of Networks and Systems. Skjetne, R., S. Moi and T.I. Fossen (2002b). Nonlinear formation control of marine craft. Proc. IEEE Conf. on Decision and Control pp. 1699-1704. White, K.A., S.M. Smith, K. Ganesan, D. Kronen, G.J.S. Rae and R.M. Langenbach (1996). Performance results of a fuzzy behavioral altitude flight controller and rendezvous and docking of an autonomous underwater vehicles with fuzzy control. Proceedings of the 1996 Symposium on Autonomous Underwater Vehicle Technology, AUV'96 pp. 117-124.

An observer-controller scheme was developed for ship rendezvous control based on recent results on robot synchronization by Rodriguez-Angeles (2002). The proposed observer-controller scheme does only require knowledge of the dynamic model of the supply ship, while the dynamics of the ship that is to be replenished is not required. Furthermore, only position measurements of both ships are needed, as observers have been developed that estimate the velocities and accelerations of both ships based on the position measurements and the dynamic model of the supply ship only. The closed-loop error dynamics were shown to be semi-global exponential uniformly ultimately bounded for stationary position keeping, and semiglobal uniformly ultimately bounded for trajectory following during replenishment.

In the development of the control scheme it was assumed that only linear damping was present. Future work aims at including higher order terms in the damping matrix, and furthermore on experimental verification of the results.

6. REFERENCES Blekhman, 1.1. (1988). Synchronization in Science and Technology. ASME Press Translations. New York. Blekhman, 1.1., A.L. Fradkov, H. Nijmeijer and Yu.A. Pogromsky (1997). On self-synchronization and controlled synchronization. Systems and Control Letter 31, 299-305. Brown, S.H. and R. Alvestad (1978). Simulation of maneuvering control during underway replenishment. Journal ofHydronautics 12(3), 109-117. Chen, H. (2003). Probabilistic Evaluation of FPSOTanker Collision in Tandem Offloading Operation. PhD thesis. The Department of Marine Technology, Norwegian University of Science and Technology. Trondheim, Norway. Dimmick, J.G., R. Alvestad and S.H. Brown (1978). Two-block romeo (simulation of ship steering control for underway replenishment). 28th IEEE Vehicular Technology Conference pp. 382-389. Dong, S., H.N. Dong and S.K. Tso (2002). Tracking stabilization of differential mobile robots using adaptive synchronized control. Proc. IEEE Conf. Robotics and Aut. pp. 2638-2643. Encamacao, P. and A. Pascoal (200 I). Combined trajectory tracking and path following: An application to the coordinated control of autonomous marine craft. Proc. 40th IEEE Conference on Decision and Control 1, 964 - 969. Fossen, T.I. (1994). Guidance and Control of Ocean Vehicles. John Wiley Ltd. Fossen, T.!. and S.P. Berge (1997). Nonlinear vectorial backstepping design for global exponential tracking of marine vessels in the presence of actua-

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