experimental validation of a marine propeller thrust estimation scheme , mic-2007!4!2

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Modeling, Identification and Control, Vol. 28, No. 4, 2007, pp. 105112

Experimental Validation of a Marine PropellerThrust Estimation Scheme

Luca Pivano 1 yvind N. Smogeli 2 Tor Arne Johansen 1 Thor Inge Fossen 1

1Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim,

Norway. E-mail: {luca.pivano, tor.arne.johansen, thor.inge.fossen}@itk.ntnu.no

2Department of Marine Technology, Norwegian University of Science and Technology, Trondheim, Norway. E-mail:

[email protected]

Abstract

A thrust estimation scheme for a marine propeller has been experimentally tested in waves and with adevice that simulates the influence of a vessel hull. The scheme is formed by a nonlinear propeller torqueobserver and a mapping to generate the thrust from the observed torque. The mapping includes theestimation of the advance number. This is utilized to improve the performance when the propeller islightly loaded. The advance speed is assumed to be unknown, and only measurements of shaft speed andmotor torque have been used. Accurate results have been obtained in experimental tests.

Keywords: Propulsion, state estimation, nonlinear, observers.

1 Introduction

In the design of vessel control systems, such as Dy-namic Positioning (DP), thruster assisted Position Moor-ing (PM) and autopilot systems, much effort has beenput into the high-level control schemes. More recently,also the issue of local thruster dynamics and controlhas received more attention. For recent references,see for example Bachmayer et al. (2000), Blanke et al.

(2000), Whitcomb and Yoerger (1999), Smogeli et al.(2005), Smogeli (2006) and references therein. Theability to design a good control system is mainly lim-ited by two difficulties: to model the vessels and thepropellers dynamics and to measure the environmen-tal state. For example in severe weather conditionshigh thrust losses due to ventilation, in-and-out-of wa-ter effects and wave-induced water velocities are ex-perienced. There are also losses of thrust due to theinteraction between the vessel hull and the propeller.

Published at the 7th Conference on Manoeuvring and Controlof Marine Craft (MCMC2006), Lisbon, Portugal, September

20-22, 2006.

Recently, observers for monitoring the propeller perfor-mance have been developed and included in new con-trol designs for electrically driven propellers, see Guib-ert et al. (2005) and Smogeli (2006).

All these considerations motivate the developmentof schemes to estimate the propeller thrust because, ingeneral, its measurement is not available. The incor-poration of the estimated thrust in a controller couldimprove the overall control performance. Moreover the

performance monitoring will also be important for im-proving thrust allocation in different working condi-tions of the propeller, from normal to extreme envi-ronmental operating conditions.

The problem of the propeller thrust estimation hasbeen treated in Zhinkin (1989) where full-scale exper-imental results were provided for positive shaft speedand vessel speed in steady-state conditions, in waves,and for slanted inflow. The estimation was based onthe propeller torque measurement and on a linear re-lation between thrust and torque.

Thrust estimation has been also treated in Guib-

ert et al. (2005), where the estimate was computed

ISSN 18901328 c 2007 Norwegian Society of Automatic Control


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Modeling, Identification and Control

from the propeller torque obtained with a Kalman fil-ter where a linear shaft friction torque was considered.The relation between thrust and torque involved anaxial flow velocity model and required the knowledge

of the advance speed, very difficult to measure in realvessel. The scheme was also highly sensitive to hydro-dynamic and mechanical modelling errors. The resultswere presented in a simulation.

In Pivano et al. (2006b) a thrust estimation schemethat works in the four-quadrant plane composed bythe vessel speed and the propeller shaft speed was pro-posed. The scheme involved a nonlinear observer forthe propeller torque and a piecewise linear mapping togenerate the propeller thrust from the observed torque.Accurate results were presented in Pivano et al. (2006b)for open-water tests with the propeller deeply sub-merged. In this paper the scheme presented in Pivanoet al. (2006b) is considered and experimentally testedunder different conditions. The mapping to computethe thrust from the propeller torque has been improvedin order to increase the accuracy when the propeller islightly loaded. Differently from Guibert et al. (2005)the advance speed is assumed to be unknown. Thescheme has been tested in waves to reproduce rough seaconditions and with a device that simulates the influ-ence of a vessel hull. Results show that the estimationscheme provides good estimates in both conditions.

2 Propeller and shaft dynamicsmodeling

A block diagram that represents the system is shownin Figure 1.

Figure 1: Propeller system block diagram.

The shaft dynamics is derived by considering an elec-tric motor attached to a shaft influenced by friction. Itcan be written as:

Jm = Qm Qp Qf(), (1)where Jm is shaft moment of inertia and Qf is the shaftfriction torque which depends on the shaft speed. Inthis paper it will be considered as a Coulomb plus a

linear and nonlinear viscous effect:

Qf() = kf1 arctan

+ kf2 + kf3 arctan(kf4).

(2)

This is motivated by the experimental result of thesystem identification on the shaft friction torque forthe propeller used for the experiments regarded in thispaper (Pivano et al., 2006a). In order to avoid thediscontinuity in zero, the Coulomb effect, usually writ-ten as a sign(), has been replaced by the functionarctan(

) with a small positive . All the coefficients

kfi are constant and positive.

3 Thrust estimation scheme

The scheme implemented to derive the propeller thrust

is shown in the block diagram of Figure 2. The pro-peller shaft speed and the motor torque Qm are as-sumed to be measurable. First a stable observer isdesigned to estimate the propeller load torque Qp. Sec-ond, an estimate of the propeller thrust Tp is computedfrom the estimated propeller torque.

Figure 2: Propeller thrust estimation scheme.

3.1 Propeller torque observer

To derive a stable observer for the propeller torque thefollowing system is considered (Pivano et al., 2006b):

Jm = Qm Qp Qf() + f, (3)Qp = 1

qQp + wq , (4)

where the propeller torque Qp is treated as a time-

varying parameter and modeled as a first order process

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Pivano et al.

with positive time constant q and driven by a boundedrandom noise wq. In (3) a friction modeling error andmeasurement error on Qm are accounted for by f.The following observer with gains L1 and L2 is pro-

posed:

Jm = Qm Qp Qf() + L1(y y), (5)

Qp = 1q

Qp + L2(y y). (6)

The measurement

y = + v (7)

is assumed to be corrupted by an error v. We assumedthat f, v and wq are bounded. With x1 = =x1x1 and x2 = QpQp = x2x2, the error dynamicscan be written as:

x1 =1

Jm

x2 kf1

arctan

x1

arctan

x1

+1

Jm[kf3 (arctan (kf4x1) arctan (kf4 x1))]

+1

Jm[kf2 x1 L1x1 + f L1v] , (8)

x2 = 1q

x2 L2x1 L2v + wq. (9)

Noise and measurement errors can be treated as in-puts, grouped in the vector

u = [u1 u2 u3] = [f v wq]T

.

Proposition 3.1 Suppose that the following assump-tions are satisfied

A1 L1 > kf2A2

1Jm + L2 < 21q

kf2+L1

Jm

.

Then the system of (8) and (9) is input-to-state sta-ble (ISS) with respect u.

Proof Taking the Lyapunov function V(x) := 12

x21 +1

2x22, we can compute its time derivative along the tra-

jectory of the system of (8) and (9):

V = kf2Jm

x21 L1Jm

x21 1

qx22

1

Jmx2x1 +

1

Jmu1x1

2

kf1Jm

arctan

x1

arctan

x1

x1

kf3Jm

(arctan (kf4x1) arctan (kf4 x1)) x1

L1

Jmu2x1

L2x1x2

L2u2x2 + u3x2. (10)

Since [arctan (a) arctan (b)] (ab) 0, we can rewrite(10) as:

V kf2Jm

x21 L1Jm

x21 1

qx22 +

1

Jmu1x1 L1

Jmu2x1

1Jm

x2x1 L2x1x2 L2u2x2 + u3x2. (11)

If u = 0, (11) becomes:

V kf2Jm

x21 L1Jm

x21 1

qx22

1

Jmx2x1 L2x1x2

xTQx, (12)

where x = [x1 x2]T and

Q =

kf2+L1Jm 12

1

Jm+ L2

1

2

1

Jm+ L2

1

q

(13)

If assumptions A1 and A2 hold, Q is positive definite

and the origin of (8) and (9), with u = 0, is globallyexponentially stable (GES), see Khalil (2000), since

V min{Q} x22 0,

where

min{Q} = kf2 + L12Jm

+1

2q

12

kf2 + L1

Jm+

1

q

2+

1

Jm+ L2

2 4 kf2 + L1

Jmq(14)

When u = 0, (11) can be written as follows:

V min{Q} x22 +1

Jmu |x1|

+|L1|Jm

u |x1| + |L2| u |x2| + u |x2|

min{Q} x22+ u

1 + |L1|

Jm

|x1| + (|L2| + 1) |x2|

(15)

Using the following inequalities for x R2 : x x2 2 x and a |x1| + b |x2| a

2

+ b2

x2 ,

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it is possible to rewrite (15) as:

V min{Q} x2

+ u1 + |L1|Jm 2

+ (|L2| + 1)2

2 x (1 )min{Q} x2 min{Q} x2 (16)

+ u

2

1 + |L1|Jm

2+ (|L2| + 1)2

x

where 0 < < 1. For any x such that

x u

2

1+|L1|Jm

2+ (|L2| + 1)2

min

{Q

}

:= (u)

(17)where () is a class K function, we obtain:

V (1 )min{Q} x2 0 (18)

From Theorem 4.19 of Khalil (2000), the system of(8) and (9) is ISS. Furthermore, the observer error isuniformly ultimately bounded by

supt>t0 (u)

.

3.2 Thrust/torque relationship

The propeller thrust is closely related to the propeller

torque and, in general, the relation is a nonlinear func-tion. The results obtained from experimental test inZhinkin (1989) showed that the relation between thrustand torque is very stable. This allows us to use the pro-peller torque, either measured or estimated, to com-pute the thrust when its measurement is not avail-able. In Pivano et al. (2006b), it was shown thatthe relation between the propeller thrust and torquecould be approximated with a linear piecewise func-tion. The mapping showed good results on reproduc-ing the thrust from the estimated propeller torque dur-ing various experimental tests. The linear relation be-tween thrust and torque used in Pivano et al. (2006b)

may not provide accurate results when the propeller islightly loaded, i.e. working at high values of the ad-vance number J. The advance number J is computedas:

J =2uaD

,

where D is the propeller disc diameter and ua is theadvance speed (the ambient inflow velocity of the waterto the propeller). The advance speed is difficult tomeasure on real vessels and is normally different fromthe vessel speed due to the interaction between the

vessel hull and the propeller. To relate the thrust and

torque, the standard propeller characteristics KT andKQ are considered. From Van Lammeren et al. (1969)we have:

Tp = KT2D4

42 , (19)

Qp =KQ2D5

42. (20)

Figure 3 shows the measured propeller characteris-tics of the propeller considered in this paper.

2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.55

4

3

2

1

0

1

2

3

4

5

J

10KQ( > 0)10KQ( < 0)KT( > 0)KT( < 0)

Figure 3: Propeller characteristics KT and KQ.

Figure 4 (b) shows the ratio between the propellerthrust and torque for positive shaft speed computedfrom the propeller characteristics of Figure 3 as:

TpQp

=KT

KQD. (21)

In Figure 4 (b) we can see that for values of J greaterthan J2, the thrust and torque ratio changes substan-tially and to compute accurately the thrust from thetorque, the values of J need to be known. The pro-cedure employed to estimate the thrust from the pro-peller torque is summarized in the following steps:

Computation of KQ, an estimate of KQ, solving(20) where Qp is used instead ofQp. From Figure4 we can also see that for values ofKQ outside theregion limited by K+Q and K

Q , the ratio between

thrust and torque is basically constant. For thisreason the value of KQ can be set to be equal to

K+Q when KQ computed with (20) is greater than

K+Q and set to KQ if KQ is less than K

Q .

Calculation ofJ, an estimate of the advance num-ber J, inverting the KQ curve using the calcu-

lated KQ. From Figure 4 (a) we can see that it

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Pivano et al.

2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5

2

1

0

1

2

3

4

J

10KQ

(>0)

Zone 1 Zone 2 Zone 3

J1

J2

KQ1

KQ2

(a)

K+Q

KQ

2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.510

0

10

20

30

40

50

J

Tp

/Qp

(>0)

J1

J2

Zone 1 Zone 2 Zone 3

(b)

Figure 4: KQ characteristic and the ratio betweenthrust and torque for > 0.

is not possible to obtain exactly the value of Jaround zero because the KQ curve in not invert-ible. To solve this problem the J axis has beendivided in three zones as shown in Figure 4. Inthe zone 1 (J < J1) and zone 3 (J > J2) theKQ curve is invertible and J can be find accu-rately. When KQ2 KQ KQ1(zone 2) we ap-proximate J with zero. Since in zone 2 the ratio

between thrust and torque does not change con-siderably, this approximation introduces a smallerror in the overall mapping.

Computation ofKT, an estimate ofKT, using thepropeller characteristics and J .

Calculation of the thrust with (19) where KT isused instead of KT.

A block diagram that shows the procedure is pre-sented in Figure 5.

Figure 5: Thrust estimation block diagram.

4 Experimental results

4.1 Setup

The experiments were performed at the MCLab( www.itk.ntnu.no/marinkyb/MCLab/ ), an experimen-tal laboratory located at NTNU (Trondheim, Norway).The basin, 6.45 m wide, 40 m long and 1.5 m deep, isequipped with a 6DOF towing carriage that can reacha maximum speed of 2 m/s and with a wave generatorable to generate waves up to 0.3 m.

The tests have been performed on a four bladed pro-peller with a diameter of 0.25 m. A metallic grid hasbeen placed upstream of the propeller in order to re-duce the speed of the inflow to the propeller disc. Inthis way we could simulate the presence of the vesselhull. A sketch of the setup is shown in Figure 6.

Figure 6: Sketch of the experimental setup.

Some tests were performed in order to measure thestandard propeller characteristics shown in Figure 3and to measure the four-quadrant propeller character-istics CT and CQ, plotted as a function of the advanceangle . The advance angle is computed with thefour-quadrant inverse tangent function as:

= arctan2 (ua, 0.7R) , (22)

where R is the propeller disc radius. The four-quadrantthrust and torque coefficients are computed from VanLammeren et al. (1969) as:

CT =Tp

1

2V2r A0

, (23)

CQ =Qp

1

2V2r A0D

, (24)

where A0 is the propeller disc area, is the water den-sity, D is the propeller diameter and Vr is the relativeadvance velocity:

V2r = u2a + (0.7R)

2. (25)

The four-quadrant characteristics of to the propeller

considered in this paper is shown in Figure 7.

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0 50 100 150 200 250 300 3502.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

[deg]

CT,10CQ

Fourier CTFourier 10CQmeas CTmeas 10CQ

Figure 7: Propeller four-quadrant open-watercharacteristics.

Some tests were performed with different profilesof towing carriage speed and various types of motortorque: square, sinusoidal and triangular waves of dif-ferent amplitudes and frequencies. This was done us-ing the built-in torque controller of the motor driver.Other tests have been performed using the built-in ve-locity controller enabling control of the propeller shaftspeed. To perform tests in rough sea conditions, regu-lar waves of 0.2 m amplitude have been generated with

the wave maker. At the same time the propeller wasmoving in a sinusoidal vertical motion to simulate thevertical oscillation that occurs in rough sea due to ves-sel motion and waves.

4.2 Friction Torque

The friction torque has been modeled as the static func-tion of (2). Figure 8 shows the friction torque com-puted from measurements and the model which hasbeen used in the observer. For the propeller tested,the losses due to the friction torque are quite high com-pare to a full scale propeller, where losses are usually

less than 6%.

4.3 Results

The thrust estimation scheme has been validated withthe observer gains L1 and L2 reported in Table 1.

Figure 9 shows results from a test where the car-riage speed varies from 1 m/s to -1 m/s while the shaftspeed assumes positive and negative values. Both theestimated thrust Tp and torque Qp are very accurate.In Figure 9, the thrust computed through (22), (23)and (25) with the four-quadrant characteristic CT de-

picted in Figure 7 is reported. In (25) the speed of

80 60 40 20 0 20 40 60 802

1.5

1

0.5

0

0.5

1

1.5

2

[rad/s]

Qf

[Nm]

measured Qfmodeled Qf

Figure 8: Friction torque: computed from measure-ments and a static approximation.

the towing carriage u has been used instead of the un-known advance speed ua. When the carriage speed ispositive the computed thrust is lower than the mea-sured one because, due to the metallic grid, the ad-vance speed is lower than the carriage speed. Whenthe carriage speed is negative, the thrust computedwith the propeller characteristic is about the averageof the measured thrust because the advance speed isequal to vessel speed. When the vessel travels back-

wards the inlet water flow is not affected by the gridwhich is placed upstream of the propeller.Figure 10 shows the result of a test performed in reg-

ular waves with height 0.20 m and the propeller movedalong its vertical axis with a sinusoidal motion. Fig-ure 10 (c) shows the vertical displacement along thepropeller vertical axis that points upwards as shown inFigure 6. The propeller shaft speed has been kept con-stant at 38rad/s. A drop of thrust and torque occurswhen the propeller rotates close to the water surfacesince the load decreases due to ventilation. The oscil-lations of torque, due to waves that disturb the inflowto the propeller, are well reproduced by the estimate.The estimated thrust is not as accurate as for the testwithout waves but the drop is properly captured andthe estimation error is small. The results show thateven in this extreme case, the estimates provided arequite accurate.

5 Conclusion

In this paper, a thrust estimation scheme for marinepropellers was experimentally tested in waves to re-produce rough sea conditions and with a device that

simulates the presence of a vessel hull. The scheme in-

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Pivano et al.

Table 1: Observer parameters

Parameter Value Parameter ValueJm[Kgm2] 5.84 103 1 106kf1 1.8

102 Q [s] 10

kf2 1.29 102 L1 3.5kf3 6.96 101 L2 1/Jmkf4 8.03 101

cluded a nonlinear observer to estimate the propellertorque and a mapping to compute the thrust from theobserved torque. The advance speed was assumed tobe unknown and only measurements of shaft speed andmotor torque were used. Good Experimental resultsshowed good performance in terms of accuracy of thethrust estimate.

6 Acknowledgment

The authors acknowledge Professor Sverre Steen forvaluable suggestions and discussions. The NorwegianResearch Council is acknowledged as the main sponsorof this project.

References

Bachmayer, R., Whitcomb, L. L., and Grosenbaugh,M. A. An accurate four-quadrant nonlinear dynam-ical model for marine thrusters: Theory and exper-imental validation. IEEE Journal of Oceanic Engi-neering, 2000. 25(1):146159.

Blanke, M., Lindegaard, K., and Fossen, T. I. Dynamicmodel for thrust generation of marine propellers. In5th IFAC Conference of Manoeuvring and Controlof Marine craft (MCMC). Aalborg, Denmark, 2000pages 363368.

Guibert, C., Foulon, E., At-Ahmed, N., and Loron,

L. Thrust control of electric marine thrusters. In31nd Annual Conference of IEEE Industrial Elec-tronics Society. IECON 2005. Raleigh, North Car-olina, USA, 2005 .

Khalil, H. K. Nonlinear Systems. Prentice Hall, thirdedition, 2000.

Pivano, L., Fossen, T. I., and Johansen, T. A. Nonlin-ear model identification of a marine propeller overfour-quadrant operations. In 14th IFAC Symposiumon System Identification, SYSID. Newcastle, Aus-

tralia, 2006a .

Pivano, L., Smogeli, . N., Johansen, T. A., and Fos-sen, T. I. Marine propeller thrust estimation in four-quadrant operations. In 45th IEEE Conference onDecision and Control. San Diego, CA, USA, 2006b .

Smogeli, . N. Control of Marine Propellers: FromNormal to Extreme Conditions. Ph.D. thesis, De-partment of Marine Technology, Norwegian Univer-sity of Science and Technology (NTNU), Trondheim,Norway, 2006.

Smogeli, . N., Ruth, E., and Sorensen, A. J. Ex-perimental validation of power and torque thrustercontrol. In IEEE 13th Mediterranean Conference onControl and Automation (MED05). Cyprus, 2005pages 1506 1511.

Van Lammeren, W. P. A., Manen, J. D. V., and Oost-

erveld, M. W. C. The Wageningen B-Screw Series.Transactions of SNAME, 1969.

Whitcomb, L. L. and Yoerger, D. Developement, com-parison, and preliminary experimental validation ofnonlinear dynamic thruster models. IEEE Journalof Oceanic Engineering, 1999. 24(4):481494.

Zhinkin, V. B. Determination of the screw propellerthrust when the torque or shaft power is known. InFourth international symposium on practical designof ships and mobile units. Bulgaria, 1989 .

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Figure 9: Experimental results with the vessel inmotion.

Figure 10: Experimental results with waves and con-stant shaft speed.

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