accurate and practical thruster modeling for underwater vehicles

8/8/2019 Accurate and Practical Thruster Modeling for Underwater Vehicles

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Accurate and practical thruster modelingfor underwater vehicles

Jinhyun Kim, Wan Kyun Chung *

Robotics & Bio-Mechatronics Laboratory, Pohang University of Science & Technology (POSTECH),Pohang 790-784, South Korea

Received 17 December 2004; accepted 25 July 2005Available online 26 October 2005

Abstract

The thruster is the crucial factor of an underwater vehicle system, because it is the lowest layer inthe control loop of the system. In this paper, we propose an accurate and practical thrust modeling forunderwater vehicles which considers the effects of ambient ow velocity and angle. In this model,the axial ow velocity of the thruster, which is non-measurable, is represented by ambient owvelocity and propeller shaft velocity. Hence, contrary to previous models, the proposed model ispractical since it uses only measurable states. Next, the whole thrust map is divided into three statesaccording to the state of ambient ow and propeller shaft velocity, and one of the borders of thestates is dened as critical advance ratio (CAR). This classication explains the physicalphenomenon of conventional experimental thrust maps. In addition, the effect of the incoming angleof ambient ow is analyzed, and Critical Incoming Angle (CIA) is also dened to describe the thrustforce states. The proposed model is evaluated by comparing experimental data with numerical modelsimulation data, and it accurately covers overall ow conditions within G 2 N force error. Thecomparison results show that the new model’s matching performance is signicantly better thanconventional models’.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Thrusters; Underwater vehicle; Thrust modeling; Advance ratio; Incoming angle

Ocean Engineering 33 (2006) 566–586

www.elsevier.com/locate/oceaneng

0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.oceaneng.2005.07.008

* Corresponding author. Tel.: C 82 54 279 2172; fax: C 82 54 279 5899.

E-mail addresses: [email protected] (J. Kim), [email protected] (W.K. Chung).

http://www.elsevier.com/locate/oceaneng

http://www.elsevier.com/locate/oceaneng



1. Introduction

Thruster modeling and control are the core of underwater vehicle control andsimulation, because it is the lowest control loop of the system; hence, the system wouldbenet from accurate and practical modeling of the thrusters. In unmanned underwatervehicles, thrusters are generally propellers driven by electrical motors. Therefore, thrustforce is simultaneously affected by motor model, propeller map, and hydrodynamiceffects, and besides, there are many other facts to consider ( Manen and Ossanen, 1988 ),which make the modeling procedure difcult. To resolve the difculties, many thrustermodels have been proposed.

In the classical analysis of thrust force under steady-state bollard pull conditions, apropeller’s steady-state axial thrust ( T ) is modeled proportionally to the signed square of propeller shaft velocity ( U ), T Z c1U jU j (Newman, 1977 ). Yoerger et al. (1990) presenteda one-state model which also contains motor dynamics. To represent the four-quadrantdynamic response of thrusters, Healey et al. (1995) developed a two-state model with thin-foil propeller hydrodynamics using sinusoidal lift and drag functions. This model alsocontains the ambient ow velocity effect, but it was not dealt with thoroughly. InWhitcomb and Yoerger’s works (1999a,b) , the authors executed an experimentalverication and comparison study with previous models, and proposed a model basedthrust controller. In the two-state model, lift and drag were considered as sinusoidal

functions, however, to increase model match with experimental results, Bachmayer et al.(2000) changed it to look-up table based non-sinusoidal functions, and presented a lift anddrag parameter adaptation algorithm ( Bachmayer and Whitcomb, 2003 ). Blanke et al.

Nomenclature

V in voltage controlled input (V)U propeller shaft velocity (rad/s)T propeller thrust (N)Q propeller shaft torque (N m)up axial uid velocity at propeller (m/s)ua ambient axial uid velocity (advance speed) (m/s)u vehicle velocity (m/s)K T thrust coefcient (–) J 0 advance ratio (–) J * critical advance ratio (CAR) (–)

r density of water (kg/m3

)m effective mass of water (kg) D propeller disk diameter (m) Ap propeller disk area (m 2)t thrust deduction number (–)w wake fraction number (–)

J. Kim, W.K. Chung / Ocean Engineering 33 (2006) 566–586 567



(2000) proposed a three-state model which also contains vehicle dynamics. Vehiclevelocity effect was analyzed using non-dimensional propeller parameters, thrustcoefcient and advance ratio. However, in the whole range of the advance ratio, themodel does not match experimental results well.

In the former studies, there are three major restrictions. First, thruster dynamics aremostly modeled under the bollard pull condition, which means the effects of vehiclevelocity or ambient ow velocity are not considered. However, while the thruster isoperating, naturally, the underwater vehicle system is continuously moving or hoveringagainst the current. In addition, the thrust force would be degraded by up to 30% of bollardoutput due to ambient ow velocity. Therefore, the bollard pull test results are only valid atthe beginning of the operation, and the ambient ow velocity induced by vehiclemovement or current must be taken into consideration. Moreover, non-parallel ambientow effects have received less attention in previous works ( Saunders and Nahon, 2002 ).These are dominant when an underwater vehicle changes its direction, or when an omni-directional underwater vehicle with non-parallel thrusters like ODIN ( Choi et al., 1995 ) isused. Non-parallel ambient ow effects could be modeled simply by multiplying theambient ow by the cosine function, but experimental results have been inconsistent.Second, in the models including the ambient ow effect, the thrust equations are derivedfrom approximations of empirical results without concern for physical and hydro-dynamical analysis. This leads to a lack of consistency in the whole thrust force map,especially, when the directions of thrust force and ambient ow velocity are opposite.Third, most of the previous models contain axial ow velocity of the thruster, because themodels are usually based on Bernoulli’s equation and momentum conservation. However,measuring axial ow velocity is not feasible in real systems, so we cannot apply thoseequations directly to the controller. Hence, in Fossen and Blanke’s work (2000) , theauthors used an observer and estimator for the axial ow velocity. And, Whitcomb andYoerger (1999b) used the desired axial velocity as an actual axial ow velocity for thethrust controller. Those approaches, however, increase the complexity of controller.

To resolve the above restrictions, in this paper, we mainly focus on steady-stateresponse of thrust force considering the effects of ambient ow and its incoming angle, andpropose a new thruster model which has three outstanding features that distinguish it fromother thruster models. First, we dene the axial ow velocity as the linear combination of

ambient ow velocity and propeller shaft velocity, which enables us to precisely t theexperimental results with theoretical ones. The denition of axial ow gives a physicalrelationship between the momentum equation and the non-dimensional representation,which has been widely used to express the relation between ambient ow velocity,propeller shaft velocity, and thrust force. Also, the modeling requires only measurablestates, so it is practically feasible. Second, we divide the whole thrust force map into threestates according to the advance ratio. The three states, equi-, anti-, and vague-directionalstates, explain the discontinuities of the thrust coefcient in the non-dimensional plot.While the former approaches failed to consider anti- and vague-directional states, theproposed model includes all of the ow states. Here, we dene the value of border status

between anti- and vague-directional states as critical advance ratio (CAR) where thepatterns of streamline change sharply. The details will be given in Section 4.1. Third,based on the two above features, we develop the incoming angle effects to thrust force.




Incoming angle means the angle between ambient ow and thruster, which is easilycalculated from vehicle velocity. If the incoming angle is 0 8 , the thrust force coincideswith the equi-directional state, or if the angle is 180 8 , the thrust force coincides with thevague- or anti-directional state according to the advance ratio. It should be pointed out thatthe mid-range of incoming angle cannot be described by a simple trigonometric functionof advance ratio. So we analyze the characteristics of incoming angle, and divide thewhole angle region into the three states above. Also, for the border status among the states,Critical Incoming Angle (CIA) is dened.

This paper is organized as follows. Section 2 describes the experimental setup. InSection 3, the thruster modeling procedure will be explained and a new model for thethruster is derived. Section 4 addresses three uid states with CAR and CIA, and explainsthe physical meanings. Then Section 5 describes the matching results of experiments withthe proposed model simulation, and compares these results with conventional thrustmodels. Finally, concluding remarks will summarize the results.

2. Experimental setup

For model verication, we executed experiments using a commercial thruster. In thissection, the experimental setup is briey explained. Fig. 1 shows the schematic diagram of experimental system composed of:

† an industrial PC running RTX q real-time extension,† a Sensoray model 526 PC-104 data acquisition card,† a Tecnadyne model 250 Brushless-DC motor thruster ( Fig. 2), and† a strain gage with Kyowa amplier.

The thruster is composed of a 7 cm diameter 7-blade Nylon propeller, and a 5.1 cmlength Kort nozzle. It has an internal velocity-loop controller operated by an input voltagerange of K 5 to 5 V, and a tachometer to give rotational velocity. The bollard output of thethruster is reported as 5.4 kgf forward and 2.1 kgf reverse.

The thruster was placed in a circulating water channel with a beam where the straingage was attached. To simulate the vehicle velocity, the uid velocity was varied from 0 to1.2 m/s. To verify incoming angle effects, the upper part of the structure was designed to

Fig. 1. Schematic diagram of experimental setup.




change the orientation of thrust by 5 8 increments. The experimental setup is shown inFig. 3. Four strain gages were attached to the beam, and the calibration was executed in airwith known masses. The test results showed linear behavior according to mass variations.

The proposed model was veried by experiments with various forward uid velocitiesin the circulating water channel. Strictly speaking, to consider precisely the effect of ambient ow velocity, we should experiment with a real vehicle and thruster in a longbasin. However, in that case, too many uncertain parameters are involved, so we just didexperiments for various forward uid velocities instead of real vehicle velocities. Also, inthis paper, we assume the forward uid velocity is the same as ambient ow velocity andvehicle velocity. The nomenclature is dened in Nomenclature.

3. Thrust model with motor and uid dynamic

The following section describes existing approaches for thruster modeling and explainsours. First, the motor dynamic model is introduced, and it is veried by propeller shaftvelocity response. Next, the uid model is described with the Bernoulli relation and linearmomentum conservation. Finally, the physical relationship between the uid model andnon-dimensional thrust coefcient is analyzed.

Fig. 2. Tecnadyne model 250 thruster.

Fig. 3. Layout of experimental setup.




3.1. Motor model

In general, the thruster motor and propeller combination possess electro-mechanicaldynamics with voltage-control mode as follows ( Bachmayer et al., 2000 )

_U Z k tV in K f ðUÞK k qQ ; (1)

where f (U ) denotes the frictional effect, Q means shaft torque, and k q is the shaft torquecoefcient. However, in a real system, to measure the shaft torque is not easy, and thevelocity control loop is assumed to compensate for shaft torque effect. Generally, a frictionmodel in mechanical systems is dominated by coulomb and viscous friction. Hence,Eq. (1) can be rewritten as

_U Z k tV in K k f1 U K k f0 sgnðUÞ; (2)

where k f1 and k f0 are viscous and coulomb friction coefcients, respectively. Eq. (2) canalso be expressed using a dead-zone model as follows:

_U Z k t V in Kk f0k t

sgnðUÞ K k f1 U : (3)

In our experimental setup, the propeller shaft velocity is acquired by the tachometeroutput. In contrast to the encoder signal, the tacho output is very noisy, so we cannot use itdirectly. In our experiments, to reduce noise, the Butterworth second-order low pass lterwas used to lter signal as shown in Fig. 4. Fig. 5 shows the dynamic response of motor

Fig. 4. Raw and ltered signal of propeller shaft velocity (3 V input).




T Z T ðU ; upÞ; (5)

where N T and N u are constant coefcients. And, to obtain the four-quadrant thruster

characteristics, they introduced sinusoidal lift and drag coefcients.Fossen and Blanke (2000) proposed a three-state model with vehicle dynamics

_upZ N TT K N u0up K N uðup K uaÞjupj; (6)

mf _u C d 1u C d 2ujuj Z ð1K t ÞT ; (7)

uaZ ð1K wÞu; (8)

T Z T ðU ; upÞ; (9)

where N T , N u , and N u0 are constant coefcients, and, mf is vehicle mass, and d 1 and d 2 aredrag coefcients.

The above two approaches mainly focus on the dynamic response with axial owvelocity rather than the ambient ow velocity. However, initially, they assumed quasi-stationary ow and used the Bernoulli relations which are valid only for fully developedow, so the dynamic relations are less meaningful. And, to construct thrust controller, theytried to nd a solution to estimate the axial ow velocity, because the axial ow is a non-measurable state. In contrast to the previous methods, we only consider the steady-statemodel and dene axial ow as measurable states as will be shown in the following.

The propeller is represented by an actuator disk which creates across the propeller planea pressure discontinuity of area Ap and axial ow velocity up . The pressure drops to pa justbefore the disk and rises to pb just after and returns to free-stream pressure, pN , in the farwake. To hold the propeller rigid when it is extracting energy from the uid, there must bea leftward thrust force T on its support, as shown in Fig. 7.

If we use the control-volume-horizontal-momentum relation between Sections 1 and 2,

T Z _mðvK uÞ: (10)

Fig. 7. Propeller race contraction; velocity and pressure changes.




A similar relation for a control volume just before and after the disk gives

T Z Apð pb K paÞ: (11)

Equating these two yields the propeller force

T Z Apð pb K paÞZ _mðvK uÞ: (12)

Assuming ideal ow, the pressures can be found by applying the incompressibleBernoulli relation up to the disk

From 1 to a : pN C12

r u2 Z pa C12

r u2p;

From b to 2 : pN C1

2

r v2 Z pb C1

2

r u2p :

(13)

Subtracting these and noting that _mZ r Apup through the propeller, we can substitute for pbK pa in Eq. (12) to obtain

pb K paZ 1

2r ðv2 K u2ÞZ r upðvK uÞ; (14)

or

upZ 1

2ðv C uÞ0 v Z 2up K u: (15)

Finally, the thrust force by the disk can be written in terms of up and u by combiningEqs. (12) and (15) as follows:

T Z 2r Apupðup K uÞ: (16)

Up to this point, the procedures are the same as the previous approaches. Now, wedene the axial ow velocity as

upb k 1u C k 2 DU ; (17)

where k 1 and k 2 are constant. The schematic diagram of the axial ow relation is shown in

Fig. 8. For quasi-stationary ow, the axial ow only depends on ambient ow andpropeller rotational motion. More complex combinations of ambient ow and propeller

Fig. 8. Proposed axial ow model.




velocity are possible, but this linear combination is adequate as will be shown later. Thissomewhat simplied denition gives lots of advantages and physical meanings.

Finally, substituting Eq. (17) to Eq. (16), the proposed thrust model can be derived asfollows:

T Z 2r Apðk 1u C k 2 DUÞðk 1u C k 2 DU K uÞ

Z 2r Apðk 01u2 C k 02uD U C k 03 D2U 2Þ: (18)

This model will be used in the following non-dimensional analysis.

3.3. Non-dimensional analysis

The non-dimensional representation for thrust coefcient has been widely used toexpress the relation between thrust force, propeller shaft velocity and ambient owvelocity as below

K Tð J 0ÞZ T r DU jU j

; (19)

where

J 0 Z u DU

(20)

is the advance ratio. Fig. 9 shows a typical non-dimensional plot found in variousreferences ( Manen and Ossanen, 1988; Blanke et al., 2000 ). In former studies, the non-dimensional relation is only given as an empirical look-up table or simple linearrelationship for the whole non-dimensional map as ( Fossen and Blanke, 2000 )

K Tð J 0ÞZ a 1 J 0 C a 2: (21)

However, as shown in Fig. 9, Eq. (21) cannot accurately describe the characteristics of the thrust coefcient, especially when J 0 ! 0, and, rather than a linear equation, the thrustcoefcient seems to be close to a quadratic equation except for the discontinuity points.Even more, Eq. (21) has no physical relationship with thrust force, but is just a linear

approximation from the gure.The proposed axial ow assumption would give a solution for this. The non-dimensionalization of Eq. (18) is expressed as

Fig. 9. Thrust coefcient as a function of advance ratio and its linear approximation.




T r D4U 2

Zp

2k 01

u DU

2C k 02

u DU

C k 03 : (22)

And, the quadratic thrust coefcient relation is obtained as following:K Tð J 0ÞZ

p

2½k 01 J 20 C k 02 J 0 C k 03 : (23)

Hence, contrary to other models, the axial ow denition of Eq. (17) gives anappropriate relationship between the thrust force equation and non-dimensional plot sincethe derivation was done by physical laws. Also, Eq. (23) can explain the characteristics of the quadratic equation of the thrust coefcient. From this phenomenon, we can perceivethat the axial ow denition in Eq. (17) is reasonable. The coefcients of quadraticequations could be changed depending on hardware characteristics. However, there is still

a question of the discontinuities of thrust coefcient in Fig. 9 which has not been answeredyet by existing models. This problem will be addressed in the following section.

4. Thrust force with ambient ow model

If ambient ow varies, thrust force changes even with the same propeller shaft velocity,which means that the ambient ow disturbs the ow state under the bollard pull condition.Flow state is determined by a complex relation between propeller shaft velocity, ambientow velocity and its incoming angle. This will be shown in the following subsections.

4.1. Flow state classication using CAR

In this section, we dene three different ow states according to the value of advanceratio and the condition of axial ow. To distinguish them, we introduce critical advanceratio (CAR), J *.

The three states are as below:

† Equi-directional state

J 0 O 0; (24)

upZ k 1u C k 2 DU O 0: (25)

† Anti-directional state

J * ! J 0 ! 0; (26)

upZ k 1u C k 2 DU O 0: (27)

† Vague-directional state

J * O J 0; (28)




upZ k 1u C k 2 DU ! 0: (29)

Fig. 10 shows the ow states schematically.

Equi-directional state: The equi-directional state occurs when the ambient owdirection and axial ow direction coincide. In this state, if the ambient ow velocityincreases, the pressure difference decreases. Hence the thrust force reduces, and thestreamline evolves as a general form ( Fig. 10(a)).

Anti-directional state: The anti-directional state happens when the ambient ow andaxial ow direction are opposite. However, the axial ow can thrust out the ambient ow,hence the streamline can be built as sink and source. The Bernoulli equation can be appliedand the thrust equation is still valid but the coefcients are different from those of the equi-directional state. Also, the thrust force rises as the ambient ow velocity increases,because the pressure difference increases ( Fig. 10(b)).

Vague-directional state: In the vague-directional state, the axial ow cannot be welldened. The axial ow velocity cannot thrust out the ambient ow, hence the direction of axial ow is not obvious. This ambiguous motion disturbs the ow, so the thrust forcereduces. In this case, we cannot guess the form of the streamline, so the thrust relationcannot be applied. However, the experimental results show the proposed thrust relation is

still valid in this state ( Fig. 10(c)).Former studies did not consider the anti- and vague-directional states, however they canbe observed frequently when a vehicle tries to stop or reverse direction. The CAR dividesbetween the anti- and vague-directional states as shown in Fig. 11 . It would be one of theimportant characteristics of a thruster. At this CAR point, the ambient ow and propellerrotational motion are kept in equilibrium. Hence, to increase the efciency of the thrusterin the reverse thrust mode, an advance ratio value larger than the CAR is preferable, asshown in Fig. 12.

4.2. Effects of incoming angle on thrust force

In this section, the incoming angle effect on thrust force is analyzed. Fig. 13 shows thedenition of incoming angle. Naturally, if the angle between ambient ow and thrust force

Fig. 10. Three ow states: (a) equi-directional state; (b) anti-directional state; and (c) vague-directional state.




is non-parallel, the thrust force varies with the incoming angle. Basically, by multiplyingambient ow velocity by the cosine of the incoming angle, the thrust force can be derivedfrom Eq. (23). In that case, however, the calculated thrust force will not coincide withexperimental results except at 0 and 180 8 , which shows that the incoming angle andambient ow velocity have another relationship. Hence, we develop the relationship basedon experiments, and Fig. 14 shows the result.

In Fig. 14, the whole range of angles is also divided into three state regions as denoted

in the previous subsection. And, we dene the borders of the regions as Critical IncomingAngles (CIA) which have the following mathematical relationship

q*1 ðuÞZ p = 2K a 1u; (30)

q*2 ðU ; uÞZ a 2uðU K b2ÞC c2 C q*

1 ; (31)

Fig. 12. Thrust force as a function of ambient ow velocity.

Fig. 11. Thrust coefcient as a function of advance ratio and critical advance ratio (CAR).

Fig. 13. Incoming angle of ambient ow.




where a 1 , a 2 , b2 , and c2 are all positive constants. And, q*i ðuÞand q*

2 ðU ; uÞare the rst andsecond CIA, respectively. Theoretical reasons have not been developed to explain the CIAequations, but empirical results give a physical insight and the above equations can becorrelated to experiments. The equi-directional region and anti-directional region aredifferentiated with the rst CIA. The rst CIA only depends on the ambient ow velocity.At the rst CIA, the thrust coefcient is the same as the thrust coefcient with no ambientow velocity. The second CIA separates the anti-directional region and the vague-directional region. The second CIA depends not only on ambient ow angle but also on

propeller shaft velocity. From Eqs. (30) and (31), the three regions shift to the left as theambient ow velocity increases.

Now, we derive the incoming angle effect on the thrust force as following:

K aT Z K 0T C f að J 0; qÞ; (32)

T Z K aT r D4U jU j; (33)

where K 0T Z K T ð J 0 Z 0Þ, and

f a Z

ðK 0T K K

CT Þ sin

qq*

1

p

20@ 1AK 124 35; 0% q% q*1 ;

K a J 0 sinqK q*

1p K q*

1

p

20@

1A

; q*1 ! q% q*

2 ;

K a J 0 sinq*

2 K q*1

p K q*1

p

20@

1A

K ðK KT K K 0T Þ24

35

cosqK q*

2p K q*

2

p

20@

1A

C ðK KT K K 0T Þ; q*2 ! q% p :

8>>>>>>>>>>>><>>>>>>>>>>>>: (34)

In Eq. (34), K a is a constant which has to be acquired by experiments. K CT Z K T ð J 0Þand

K KT Z K T ðK J 0Þ. Eq. (32) coincides with Eq. (23) at 0 and 180 8 . Hence Eq. (32) contains all

of the effects of ambient ow and implies that the total thrust force is composed of thrust

Fig. 14. Thrust force as a function of incoming angle.




force with bollard pull condition, K 0T , and additional force induced by ambient owvelocity and its incoming angle, f a( J 0 , q).

5. Experimental results

To verify the proposed model, rstly, we operated the thruster under various ambientow velocities: G 1.2, G 1.0, G 0.8, G 0.6, G 0.4, and 0 m/s with a zero degree incomingangle. Then, for 0.4, 0.6 and 0.8 m/s ambient ow velocities, the thruster was tilted at 5 8

increments from 0 to 180 8 to change incoming angle. For simplicity, we only considercases where U O 0.

Fig. 16. Comparison results of experiment and simulation by the proposed model: (a) thrust force—experimentand (b) thrust force—simulation.

Fig. 15. Experimental thrust coefcient as a function of advance ratio.




Fig. 15 shows the experimental thrust coefcient plot. As shown in the gure, the thrustcoefcient is divided into three parts. The difference between Figs. 11 and 15 comes from

hardware characteristics, for example, shape of duct, pitch ratio, etc. Hence thecoefcients of quadratic equations should be identied from experimental results.In Fig. 16, the experimental thrust forces are compared with simulation results of the

proposed model with an input voltage range from 1.5 to 4.5 V. Both results are verysimilar except at some localized points. The deviation could be caused by the thruster notbeing located in sufciently deep water due to the restriction of the experimentalenvironment. Thus, the anti- and vague-directional response could have been disturbed byspouting water.

Fig. 18. Thrust force matching error: (a) by the proposed model and (b) by the conventional model.

Fig. 19. Thrust force matching error comparison: (a) maximum error and (b) average error.

Fig. 17. Comparison results of experiment and simulation: (a) K 1.2 m/s ambient ow velocity; (b) K 0.8 m/sambient ow velocity; (c) K 0.4 m/s ambient ow velocity; (d) 0.0 m/s ambient ow velocity; (e) 0.4 m/s ambientow velocity; (f) 0.8 m/s ambient ow velocity; and, (g) 1.2 m/s ambient ow velocity.

3




As the ambient ow velocity decreases, the thrust force initially increases until theCAR and decreases after crossing the CAR. Fig. 17 shows the result of the propeller shaftvelocity versus the thruster force under various ambient ow velocities. The resultsindicate that the proposed thruster model provides high accuracy in the whole range of thrust force map. As shown in Fig. 17(a)–(c), the thrust forces are modeled using twoquadratic equations due to the CAR.

To highlight the performance of the proposed model, we compare the results with thoseof the conventional model described in Eq. (21). The comparison results are shown inFigs. 18 and 19 . These gures show that our results are signicantly better than theconventional model in the anti- and vague-directional regions.

Fig. 20 shows the matching results between experiment and simulation with themultiplication of cosine function to ambient ow. As shown in the gure, the matchingerrors increase as the ambient ow velocity goes fast. Hence, this model makes the controlperformance of underwater vehicles to be poor.

Fig. 21(a), (c) and (e) shows the thrust force comparison between experiment andsimulation as a function of incoming angle. The errors of matching, as shown in Fig. 21(b),

Fig. 20. Comparison results with the multiplication of cosine function to ambient ow: (b) matching error with0.4 m/s ambient ow velocity; matching error with 0.6 m/s ambient ow velocity; and (f) matching error with0.8 m/s ambient ow velocity.




(d) and (f), are mostly within G 2 N. Note that the maximum force of the thrust is up to

50 N.From the matching results with ambient ow velocities and incoming angles, we can

say that our initial denition of axial ow is valid, and the proposed model shows good

Fig. 21. Comparison results of experiment and simulation with incoming angle: (a) 0.4 m/s ambient ow velocity;(b) matching error with 0.4 m/s ambient ow velocity; (c) 0.6 m/s ambient ow velocity; (d) matching error with0.6 m/s ambient ow velocity; (e) 0.8 m/s ambient ow velocity; and (f) matching error with 0.8 m/s ambientow velocity.




agreement with experimental results under various ambient ow velocities and incomingangles.

6. Conclusion

In this paper, we proposed a new model of thrust force. First, we dene the axial ow asa linear combination of the ambient ow and propeller shaft velocity which are bothmeasurable. In contrast to the previous models, the proposed model does not use the axialow velocity which cannot be measured in real systems, but only uses measurable states,which shows the practical applicability of the proposed model. The quadratic thrustcoefcient relation derived using the denition of the axial ow shows good matching withexperimental results.

Next, three states, the equi-, anti-, and vague-directional states, are dened according toadvance ratio and axial ow state. The discontinuities of the thrust coefcient in the non-dimensional plot can be explained by those states. Although they have not been treatedprevious to this study, the anti- and vague-directional states occur frequently when avehicle stops or reverses direction. The anti- and vague-directional states are classied byCAR (critical advance ratio), which can be used to tune the efciency of the thruster.

Finally, the incoming angle effects to the thrust force, which are dominant inturning motions or for omni-directional underwater vehicles, are analyzed and CIA

(critical incoming angle) was used to dene equi-, anti-, and vague-directionalregions.

The matching results between simulation with experimental results show excellentcorrelation with only G 2 N error in the entire space of thrust force under various ambientow velocities and incoming angles. The results are also compared with conventionalthrust models, and the matching performance with the proposed model is several timesbetter than those of conventional linear ones.

Acknowledgements

The authors would like to thank J. Han who designed and built the thrust experimentalenvironment and helped the experiments. They wish to thank him for his generouscontributions.

This research was supported in part by the International Cooperation ResearchProgramm (“Development of Intelligent Underwater Vehicle/Manipulator and Its ControlArchitecture”, M6-0302-00-0009-03-A01-00-004-00) of the Ministry of Science &Technology, Korea, and supported in part by the Ministry of Education and Human

Resources Development (MOE) and the Ministry of Commerce, Industry & Energy(MOCIE) through the fostering project of the Industrial-Academic Cooperation CenteredUniversity.




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accurate and practical thruster modeling for underwater vehicles

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