Added Mass and Drag Determination

422 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 36, NO. 3, JULY 2011

Simultaneous Determination of DragCoefficient and Added Mass

Wai Leung Chan and Taesam Kang

Abstract—The drag coefficient and added mass (hydrodynamicmass) are the essential parameters for the dynamics analysis ofsubmerged objects with mobility such as bio-mimicking fish robotsor underwater vehicles. The shape dependence of these parame-ters makes them difficult to have good theoretical approximationsand the parameters should be determined either numerically or ex-perimentally. Different experiments have been proposed to obtaineither the drag coefficient or added mass. This paper presents anew method to simultaneously determine the drag coefficient andadded mass from a simple and economic experiment and a nu-merical identification procedure. An experiment was carried outto demonstrate the method and the identification error was studiedanalytically and numerically for some experimental uncertainties.

Index Terms—Added mass, drag coefficient, hydrodynamicmass.

I. INTRODUCTION

T HE drag coefficient and added mass (hydrodynamic massor virtual mass) are the essential parameters in the dy-

namic analysis of submerged objects with mobility such as fishrobots or underwater vehicles. The knowledge of these param-eters helps engineers to model the dynamics, determine the en-ergy efficiency, improve existing designs, etc.When an object moves in a liquid media with speed , it ex-

periences a drag force and the drag coefficient is definedas

(1.1)

where is the water density and is the frontal cross-sec-tional area. If the object is accelerated, the surrounding fluid isalso accelerated and the object mass is virtually heavier thanit should be. The virtual increase of mass is called the addedmass or the hydrodynamic mass . Both the drag coefficientand the added mass are the functions of shape and direction ofobject motion. In addition, the drag coefficient also depends onthe Reynolds number. For simple geometries, the drag coeffi-cient can be found in [1] and [2] and the added mass in [3] and[11], for example. For arbitrary shape objects, literature rarelyprovides enough information for estimation; the drag coefficient

Manuscript received August 01, 2009; revised March 09, 2011; acceptedApril 28, 2011. Date of publication June 16, 2011; date of current version July01, 2011. This work was supported by Konkuk University and the Korea Re-search Foundation Grant (KRF—J03303).Associate Editor: L. Whitcomb.The authors are with the Department of Aerospace Information Engineering,

Konkuk University, Seoul 143-701, Korea (e-mail: [email protected];e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/JOE.2011.2151370

and the added mass must be determined numerically [4], [12] orexperimentally in these cases.Different experiments have been designed to determine

the drag coefficient of underwater objects. In constant speedmethod, an object is towed at different constant speeds, bymeasuring the towing force, and the drag coefficient is deter-mined from the plot of force versus the speed squared (e.g., [6],[7], [13], [15], and [17]). This method requires a motion controlunit to pull the object moving at constant speed and forcesensor to measure the towing force. In the deceleration method,the motion of a decelerated object is filmed to determinethe speed. By plotting the inverse of speed versus time, thedrag-coefficient-to-mass ratio (object mass added mass) canbe determined from the slope (e.g., [8] and [5]). This method isapplicable if the added mass is known or estimated. In the droptank method, it is possible to exclude the added mass effectwhen the object is falling, and it reaches the terminal speed in adrop tank apparatus, if the tank is deep enough. In this case, thedrag force is balanced by the object weight in water, and thedrag coefficient can be obtained from (1.1) (e.g., [9] and [10]).In the free decay, the decaying oscillatory motion of an objectin water is recorded and the drag coefficient is determined witha numerical minimization scheme [18].Compared with drag coefficient measurements, there are rel-

atively fewer reports about the experimental measurements ofadded mass. For an object that has a plane of symmetry and theplane is perpendicular to the transverse motion, the added masscan be found by comparing the oscillation frequency in air to thefrequency in liquid media, with the oscillation along the trans-verse motion [11]. Obviously, the method is inadequate if thesymmetry is broken. Lin [7] and Fernandes [13] measured theforce that tows an object at constant acceleration and computedthe added mass with the knowledge of drag coefficient. Small-wold [14] used least square method to numerically identify theadded mass from experimental data. Rosss [16] and Morrison[18] measured the motion data of an object in free decay testsand determined the added mass with system identification.This paper presents a new method to integrate the determina-

tion of the drag coefficient and the added mass of objects withmobility in a liquid media. It is assumed that the drag coeffi-cient is constant over the range of force applied in the experi-ment. The method involves some experimental measurementsfollowed by a numerical identification procedure. The principleof the experiments is to obtain the displacement-time relation-ship of an underwater object and to compare it with a dynamicmodel.The experimental setups for data acquisition are simple and

economic to be constructed, as they do not require expensiveinstruments such as high-speed camera or force sensor, but a

0364-9059/$26.00 © 2011 IEEE

CHAN AND KANG: SIMULTANEOUS DETERMINATION OF DRAG COEFFICIENT AND ADDED MASS 423

Fig. 1. Horizontal setup.

low-cost optical encoder and electronics for timing. The identi-fication procedure does not require the objects to reach the ter-minal speed, and therefore, the setup can be carried out indoorwith limited pool or tank size.In addition to the theoretical explanation, a boxfish robot was

employed to demonstrate the experimental method. The accu-racy of the numerical identification for some experimental un-certainties was studied analytically and numerically. It will beshown that timing accuracy can seriously affect the identifica-tion.

II. THEORETICAL BACKGROUND

The experiment presented in this paper for the drag coefficientand the added mass determination is suitable for small-scalesubmerged objects. The experimental setup is economic andcompact, which makes it feasible for most of the laboratories.For large objects, it is still possible to determine the parameters,if a small-scaled model is applied. The principle of the exper-iment is to record the displacement-time data of an immersedobject, which is towed by a known external force. The dataare then compared with a theoretical model. External weightsare employed for towing and hence a pulley system is requiredto transmit the force in an appropriate direction. Three exper-imental setups are introduced depending on the availability offacilities.

A. Experimental Setups

Fig. 1 shows the horizontal setup, which is carried out in awater pool. An object is connected to an external weight of mass

with a string routed through a pulley system. Orientationand buoyancy adjustments on the object are required so that theobject can immerse at a constant depth below the water surfaceand yield a stable horizontal motion upon external pulling.The object, and hence the external weight, is held fixed until

water is calm and the data acquisition is started. The string willpull the object to cruise straight toward the bottom pulley dueto the falling of the external weight.Fig. 2 shows the vertical setups, which are similar to the drop

tank method. Experiments are carried out in a water tank or apool. The setups can be classified as D-vertical and U-verticalfor the object moving downward and upward, respectively. Sim-ilar to the horizontal setup, an object is connected to an externalweight through a pulley system. The object weight distribution

Fig. 2. D-vertical (left) and U-vertical (right) setups.

should be adjusted so that floating is prohibited and the bodyaxis aligns with the vertical direction for motion.The external weight is held fixed until water is calm and the

data acquisition is started. The object will cruise straight down-ward in D-vertical setup or upward in U-vertical setup when theexternal weight is released.In all setups, an optical encoder is mounted to a pulley. By

timing the events of encoder signal detection, the displacementtime data of a motion trajectory can be obtained for the numer-ical identification. As will be shown in Section V, the accuracyof results (drag coefficient and added mass) highly depends onthe timing precision and thus, high data sampling rate is recom-mended.

B. External Weight Selection

The function of the external weight is to pull the object formotion with a known external force. The weight should be se-lected so that the string force acting on the object yields the ter-minal speed, which is within the object swimming or propulsioncapability.If the object of length can travel at a terminal speed, where is the terminal body length speed, drag force

acting on the object should be

(2.1)

For most fish robot or underwater vehicle designs, the massdensity is close to that of water, and mass can be approxi-mated as , thus

(2.2)

Assuming that the object’s terminal speed is larger than 0.5 bodylength per second and the drag coefficient is 0.5,then drag force is approximated as

0.125 mL (2.3)

If the pool or the tank in the experiment is large enough suchthat the object can move at the terminal speed, the string forceshould balance the drag force, i.e.,

(2.4)


From the setups in Figs. 1 and 2, assuming that the stringmass is negligible and the total pulley friction is constant,the string force can be expressed as

horizontalfor D-vertical

U-vertical

(2.5)

where is the buoyancy of the object. Therefore, if the pulleyfriction is negligible, the minimum/maximum external weight

in the experiment can be obtained by substituting (2.3)and (2.5) into (2.4) as

0.125 mL horizontal0.125 mL for D-vertical0.125 mL U-vertical

(2.6)

The expression of external weight in (2.6) is just an estimateto find the required minimum or maximum external weight.Testing must be carried out to check whether the pulley frictionis significant and to make the fine adjustment of the weight. InD-vertical setup, either object mass or buoyancy shouldbe adjusted so that is a positive quantity.Due to the existence of pulley friction, a single motion trajec-

tory is not enough to determine both the drag coefficient and theadded mass from the second-order Newtonian equation of mo-tion. It is necessary to have several different motion trajectoriesby using different external weights. Once the minimum or max-imum external weight of a setup is available, the externalweight can be expressed as


U-vertical(2.7)

where , called additional weights, are some positiveweights leading to different motion trajectories.

C. Equation of Motion and Solution

It is assumed that the air drag on the external weight is negli-gible, the total pulley friction is constant, the string is inelastic,and its mass is negligible for the experimental setups as shownin Figs. 1 and 2. The equation of motion can be written in theform

(2.8)

where

(2.9)


U-vertical(2.10)

Term is the added mass and is the inertia contributionfrom the pulley system, which can be expressed as

(2.11)

where is the rotational inertia and is the shortest distancebetween the string and the center of the th pulley.Solving (2.8) with the initial conditions ,

the solutions for the displacement and speed are

(2.12)

(2.13)

(2.14)

(2.15)

Note that, from (2.12) and (2.13), time and speed can bewritten as the explicit functions of displacement as

(2.16)

(2.17)

and these two expressions will be more useful than (2.12) and(2.13) in the numerical identification.

D. Numerical Identification Procedure

The numerical identification procedure utilizes models(2.14)–(2.17) to generate two equations; one equation is non-linear while the other equation is linear. Experimental dataare substituted into these two equations to extract useful in-formation for the determination of drag coefficient , addedmass , and pulley friction . Since the linear equation is afunction of the additional weight different motiontrajectories (due to different ) are required in the identifi-cation procedure.The nonlinear equation is obtained by considering the fol-

lowing quantity:

(2.18)

where , and with arethree distinct displacement-time points within a motion trajec-tory. The substitution of in (2.16) into (2.18) yields

(2.19)

Nonlinear equation (2.19) indicates that can be solved byNewton’s method with the knowledge of other parameters inthe equation. Therefore, by picking three different displacementpoints , and hence the value of , in an experimentalmotion trajectory, the corresponding value of can be com-puted. One should note that there will be different valuesof determined from (2.19) for different additional weights

.With the definition of the quantity , the linear equa-

tion can be obtained by taking the square of in (2.17) andusing the definitions of and in (2.14) and (2.15) as

(2.20)

(2.21)


Linear equation (2.20) suggests that, if a fixed displacementpoint is selected for all experimental motion trajecto-ries (due to different ), has a linear relationship with

. In each motion trajectory, speed can be computedby the linear regression of some nearest experimental displace-ment-time points; and the values of in (2.20) can be computedwith the value of determined from nonlinear equation (2.19).From the plot of versus , the drag coefficient can be de-termined from slope in (2.21), the added mass from (2.14) andthe pulley friction from the -intercept in (2.21) as

(2.22)

(2.23)


U-vertical(2.24)

Note that, due to measurement uncertainty and external dis-turbance, different motion trajectory may yield different valueof the added mass. The added mass determined from the numer-ical identification procedure in this paper is the average value ofthe added mass obtained from (2.14) for different motion trajec-tories (i.e., different ).If the first slit of the encoder, which is to invoke the first

encoder signal, is positioned initially at , the time data willsuffer an offset, which should be accounted for the verificationof model (2.12). Once the values of , the drag coefficient, theadded mass, and friction are obtained, the value of can be com-puted through (2.15). With (2.16), the time offset due to theinitial displacement is

(2.25)

Therefore, the identification procedure can also account for thetiming offset in measurement due to the initial displacement.

III. EXPERIMENTS

A robotic boxfish, as shown in Fig. 3, was employed todemonstrate the drag coefficient and added mass determinationusing the horizontal setup as shown in Fig. 1. The boxfishweight was adjusted so that it could stay at a constant depth of3 cm below the water surface. The experiment was carried outin a circular inflatable pool with the diameter of 1.5 m and thedepth of 25 cm.An optical encoder circuit was mounted to a pulley in the

setup and recorded motion as shown in Fig. 4. In this exper-iment, the configuration of the encoder plate is illustrated inFig. 5. When the external weight falls, the string will bring thepulley and hence the encoder plate to rotate. Light sensor is trig-gered when a slit passes through the horizontal dotted line anda microcontroller will record the time of the event.With the knowledge of the encoder resolution and radius, the

discrete displacement data are

(3.1)

Fig. 3. The ostraciiform fish robot.

Fig. 4. The horizontal setup with an encoder circuit attached to a pulley.

where is the number of signal detections since the begin-ning of motion; 0.7854 cm is the distance interval trav-eled by the string between two consecutive signal detections;and the initial displacement is the displacement of the objectbefore the first signal detection. As is shown in Section II-D,three distinct displacement-time points in a motion trajectoryare required for numerical identification; and the displacementcan be represented by integer , such that

, and with . Table I lists theinformation of the setup.Seven external weights of 10–40 g with a step of

5 g, i.e., 10 g and 0, 5, 10, , 30 g, were usedand 80 data were recorded in all motion trajectories.

IV. RESULTS

The experimental data (down-sampled discrete points) forthe displacement versus time are plotted in Fig. 6 for differentexternal weights. Three displacement points, 19.08 cm

42.64 cm , and 58.35 cmwere chosen to compute using (2.19) in every mo-

tion trajectory.To compute the values of in linear equation (2.20),

59.13 cm was selected as the displacement point inlinear equation (2.20) and the corresponding speed was deter-mined using linear regression with the nearest six points. Fig. 7plots versus according to (2.20). From the slope, -in-tercept, and expressions (2.22)–(2.24), the determined drag co-efficient, the added mass, and the pulley friction are 0.70 g,619 g, and 32.8 mN respectively. From these values, fittingcurves for the displacement-time relationships are computed


Fig. 5. Configuration of an encoder plate before the start of a motion trajectory.

TABLE IINFORMATION FOR THE HORIZONTAL SETUP

Fig. 6. Experimental data (discrete points) and the curve fitting (solid lines)using model (2.12) for different external weights .

using model (2.12) and plotted in Fig. 6. It can be seen that theexperimental data agree well with the model.

V. EFFECTS OF INITIAL DISPLACEMENT UNCERTAINTYAND TIME PRECISION ERROR

In the experiment, the initial displacement is not the sameand there exists an initial displacement uncertainty in everymotion trajectory. Besides the displacement uncertainty, thetiming precision will also affect the results as determined in

Fig. 7. versus at 58.35 cm.

the numerical procedure. For a timer with time unit ,precision error , the magnitude of which is always less thana timing unit , is defined as

(5.1)

where and are the experimental time measured (with finiteprecision) and the actual time (with infinite precision), respec-tively. Therefore, the precision errors in the time data are alwaysless than 1 ms for the experiment in Section III, for example.If the timing device starts before the motion trajectory, the

starting time will be different for every motion trajectory andhence, the time precision error will be different even though twomotion trajectories have the same experimental parameters.The initial displacement uncertainty affects both values of

in (2.19) and of in (2.20) directly, while the time precisionerror leads to the speed error and the uncertainty of throughthe error of defined in (2.18). The error of will finally affectthe calculation of the drag coefficient and the added mass.

A. Analytical Calculation of Errors

To study the effects of initial displacement uncertainty andtime precision error analytically, the error of results is computedby the first-order error approximation.The time precision error first leads the error of . From the

definition of in (2.18), we have

(5.2)

where to is the time precisionerror on the actual time data due to the measurement.Therefore, the first-order change in , denoted , due to thetime precision error is bounded by

(5.3)


where

(5.4)

The error of , together with the initial displacement uncer-tainty, results in the error of upon solving (2.19). Taking thefirst-order approximation of (2.19), the error of can be approx-imated as

(5.5)

where

(5.6)

The error of , the initial displacement uncertainty, and thetime precision error will finally contribute to the calculationerror of in (2.20) as follows:

(5.7)

where is the speed error generated in the linear regressiondue to the truncation of time data. With the error of for eachmotion trajectory due to different , the least square errorof slope in (2.20) is

(5.8)

Therefore, the first-order changes in the drag coefficient from(2.22) and the added mass from (2.23) are

(5.9)

(5.10)

The error of the drag coefficient is proportional to the ac-tual value of the drag coefficient with a factor , whilethe error of the added mass is approximately proportional tothe quantity , with a larger factor of

. Therefore, the maximum percentageerror of the drag coefficient is always less than that of the addedmass.Given three displacement points , and with

, these displacement points should be chosen carefullyto reduce the error of the drag coefficient and the added mass inthe numerical identification. Point should be large enough toreduce the percentage error of the displacement due to . As

TABLE IISIMULATION PARAMETERS

can be seen from (5.5) and (5.6), quantityand hence the error of will diverge as close to zero.From (5.3), points and should be chosen at a large dis-

tance from point to increase the denominatorsand and reduce the error in . On the other hand,points and cannot be too close since is close to 1 andleads to a singularity problem of solving (2.19).The value of in (2.20) should be evaluated at a larger dis-

placement point to reduce the calculation error. From (5.7),the error of due to the first two terms is reduced when a largervalue of is applied. The acceleration is also smaller in thissituation; hence, the error of speed, computed by linear regres-sion, can be reduced.

B. Numerical Simulations

The initial displacement uncertainty and time precision errorcan affect the results as seen in the previous first-order errorcalculations. To find the reliability of the experimental resultsas computed in Section IV with respect to this uncertaintyand measurement error, numerical simulations were executedto compute the error range of the results. Table II lists theparameters that simulate the experiment in Section III. It isassumed that the drag coefficient is 0.7, the added mass is 620g, and the pulley friction is 33 mN. In addition to the horizontalsetup, the same parameters are also applied in the D-verticaland U-vertical simulations with the buoyancy of1096 g 10.74 N.Every computational experiment (for each setup) simulates

seven motion trajectories, which use different external weights( 10–40 g with a step of 5 g). In each

simulated motion trajectory, 80 data of theoretical displacementand time are first generated using

(5.11)

(5.12)

where values of and are calculated from (2.14) and (2.15),respectively; and time data use model (2.16) with the sub-stitution of from (5.11) into . Initial displacement isassumed to be a uniformly distributed random number between0.13 and 0.33 cm (i.e., average 0.23 cm) and is different forevery simulated trajectory. Since the timer is running before the


TABLE IIICONFIDENCE LEVELS OF COMPUTATIONAL RESULTS WITHIN SOME SPECIFIC

RANGE OF ERRORS; TIMING UNIT 1 ms

TABLE IVCONFIDENCE LEVELS OF COMPUTATIONAL RESULTS WITHIN SOME SPECIFIC

RANGE OF ERRORS, PERFECT TIMING

start of a motion trajectory, the time is not equal to zero whenthe experiment starts, but at a starting time , which is a uni-formly distributed random number between 0 and 1 s.To generate experimental displacement and time , it is

assumed that initial displacement is distributed uniformly be-tween 0.13 and 0.33 cm with average 0.23 cm for all trajectoriesand the timing unit 1 ms. Hence

(5.13)

(5.14)

where denotes the flooring of to an integer. From thesegenerated experimental data and , the drag coefficientand the added mass in every computational experiment arecomputed using the numerical identification as described inSection II-D.Table III summarizes the simulation results of 10 000 com-

putational experiments. It can be seen that the drag coefficienterrors are almost less than 2.5% of the theoretical valuein all setups. The added mass errors are larger for the same con-fidence interval compared with the drag coefficient; neverthe-less, all the errors are less than 10% of the theoretical values.From the simulation results, it can be claimed that, due to the ini-tial displacement uncertainty and time precision error, the per-centage errors of the drag coefficient and the added mass areless than 2.5% and 5.0% of the theoretical values (and 620 g) respectively, with 99% confidence interval.

C. Precision of Time Data

To see the effect of time precision error, two computationalexperiments were repeated using the parameters in Table II, ex-cept the timing accuracy.For the case of the perfect timing (no truncation error, i.e.,

), Table IV lists the confidence levels of the results.Comparing the results with Table III, the initial displacementuncertainty has small effects on the error of the results.If the timing unit is changed to 10 ms, then

(5.15)

TABLE VCONFIDENCE LEVELS OF COMPUTATIONAL RESULTS WITHIN SOME SPECIFIC

RANGE OF ERRORS; TIMING UNIT 10 ms

The confidence intervals of the results are summarized inTable V. It can be seen that, with the poor timing accuracy(larger precision error) compared to the cases in Table III, theerror ranges of both the drag coefficient and the added massare increased for a given confidence interval. The percentageerror of the drag coefficient is within 10% of the theoreticalvalue with 95% confidence interval, while the added mass is nolonger reliable.Note that the confidence interval of the drag coefficient within

a specific range of error is always larger than that of the addedmass, which is agreed with conclusion drawn in (5.9) and (5.10).This means that the determined drag coefficient is always morereliable that the added mass.

D. Nonlinear Least Square Curve Fitting

The numerical identification presented in Section II-D is de-signed for the experiments in this paper specifically. The non-linear least square curve fitting is an option to find the valuesof and simultaneously. However, this method is no longerreliable when the initial displacement uncertainty exists.To compare the numerical identification presented in this

paper and the nonlinear least square method, computationalexperiments, using Matlab nonlinear least square curve fittingfunction lsqcurvefit for finding of and , was carried out withthe same parameters used in Table II. Due to the existence ofthe initial displacement , and noting that the time offset canbe computed by (2.25), model (2.16) should be modified as

(5.16)

and nonlinear function can be written as

(5.17)

For each external weight, nonlinear function (5.17) togetherwith the generated experimental data are applied in the non-linear least square method to find the values of and . Similarto the identification procedure presented in this paper, the valuesof and are then substituted into (2.20) to plot versus ,and the drag coefficient and the added mass are calculated using(2.22) and (2.23).


TABLE VICONFIDENCE INTERVAL OF COMPUTATIONAL RESULTS (USING NONLINEAR

LEAST SQUARE CURVE FITTING) WITHIN SOME SPECIFIC RANGEOF ERROR; TIMING UNIT 1 ms

Ten thousand computational experiments were carried outand the results are summarized in Table VI. Comparing withTable III, the numerical identification method presented in thispaper is superior if the initial displacement uncertainty exists.

VI. DISCUSSIONS

A. Optical Encoder Versus High-Speed Camera

High-speed camera is a popular tool to capture motion data,but it is highly recommended to use optical encoder for all ex-perimental setups presented in this paper for the following rea-sons.The high-speed camera may require proper lighting for ex-

posure. To reduce the error of displacement measurement dueto a view angle, the setup may require a movable mounting tofollow the object or external weight during motion. The tedioussetups make high-speed camera an unfavorable option.As is shown in the error analysis and simulations, the timing

accuracy can seriously affect the results. High-speed cameraswith higher frame rate would be more expensive and requirelarger memory for picture storage. Optical encoder method, onthe other hand, can record the data at high rate even with low-cost electronics, and it requires less memory for storage since afloating-point datum (time datum), instead of a picture, is savedfor every captured moment.The optical encoder method does not require data post-

processing, since the time data are recorded directly and thedisplacement data are automatically known if the encoderradius and resolution are known. Therefore, the displace-ment-time data are immediately available after measurements.On the other hand, image processing is required to extract thedisplacement information if high-speed camera is applied, andthis could be time consuming.

B. Constraint Function for Finding the Value of

It is possible to generate other constraint functions to find thevalue of instead of using (2.19). Nevertheless, the computationof in (2.18) can effectively eliminate any time offset, whichis a finite amount of time for an object to travel the initial offset,

which is undetermined. Therefore, expression (2.19) is free ofthe time offset problem.

VII. CONCLUSION

In this paper, a newmethod is presented to determine the dragcoefficient and the added mass of some submerged objects si-multaneously. Three simple and economic experimental setupsare introduced for data acquisition. It is assumed that the dragcoefficient is constant over the range of force applied in the ex-periments and a numerical identification procedure is presentedto calculate the drag coefficient and the added mass from theexperimental data. A boxfish was employed to demonstrate themethod. From the experimental data, the drag coefficient is 0.7and the added mass is 619 g. Simulations were executed to ex-plore the effect of initial displacement uncertainty and time pre-cision error; the results show that the drag coefficient and theadded mass determined by the numerical identification proce-dure have small percentage of error compared with the theoret-ical values. The time precision error is critical to the reliabilityof the results and high timing accuracy can reduce the error ofthe results.

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Wai Leung Chan received the B.S. degree in physicsfrom the Hong Kong University of Science and Tech-nology, Hong Kong, in 1997 and the M.Sc. and Ph.D.degrees from the University of California San Diego,La Jolla, in 2002 and 2005, respectively. His Ph.D.research was about the minimal mass design, control,and experimental study of the deployable structurescalled tensegrity.After graduating from his Ph.D. program, he was

a Systems Engineer in an engineering company inSan Diego, CA. The projects involved the deploy-

ment and vibration isolation of a space antenna, and energy harvesting under-

water vehicle. He is currently working as a Researcher in the Department ofAerospace Information Engineering, Konkuk University, Seoul, Korea, and hisresearch includes bio-mimicking robots, swimming study, and experiments onautonomous fish robots.

Taesam Kang received the B.S., M.S., and Ph.D. de-grees from the Department of Control and Instrumen-tation Engineering, Seoul National University, Seoul,Korea, in 1986, 1988, and 1992, respectively.He was an Associate Professor at Hoseo Uni-

versity, Asan, Korea, from 1994 to 2001. SinceSeptember 2001, he has been teaching controlsystem design and its applications as a facultymember in the Department of Aerospace Informa-tion Engineering, Konkuk University, Seoul, Korea.His research area includes sensor design and signal

processing, control theory, and applications for aerospace, marine, and groundsystems.

Added Mass and Drag Determination

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