Jianxun WangDepartment of Electrical and

Computer Engineering,

Michigan State University,

East Lansing, MI 48824

e-mail: wangji19@msu.edu

Philip K. McKinleyDepartment of Computer Science

and Engineering,

Michigan State University,

East Lansing, MI 48824

e-mail: mckinley@cse.msu.edu

Xiaobo Tan1

Department of Electrical and

Computer Engineering,

Michigan State University,

East Lansing, MI 48824

e-mail: xbtan@msu.edu

Dynamic Modeling of RoboticFish With a Base-ActuatedFlexible TailIn this paper, we develop a new dynamic model for a robotic fish propelled by a flexibletail actuated at the base. The tail is modeled by multiple rigid segments connected inseries through rotational springs and dampers, and the hydrodynamic force on each seg-ment is evaluated using Lighthill’s large-amplitude elongated-body theory. For compari-son, we also construct a model using linear beam theory to capture the beam dynamics.To assess the accuracy of the models, we conducted experiments with a free-swimmingrobotic fish. The results show that the two models have almost identical predictions whenthe tail undergoes small deformation, but only the proposed multisegment model matchesthe experimental measurement closely for all tail motions, demonstrating its promise inthe optimization and control of tail-actuated robotic fish. [DOI: 10.1115/1.4028056]

1 Introduction

Biomimetic systems have been receiving increasing attentionfrom the robotics community, since natural organisms can provideimportant insights into the theory and design of engineer systems.For example, in the area of aquatic robots, the maneuverabilityand efficiency of live fish [1,2] have motivated significant scien-tific interest over the past two decades in developing, modeling,and controlling robotic fish [3–17]. In addition to providing plat-forms for underwater applications such as environmental monitor-ing [18–20], these robots offer a means to study the behavior oflive fish [21].

Numerous designs of actuation mechanisms have been pro-posed for robotic fish [3,9,10,17,21–33]. A typical approach is touse multiple links actuated separately or jointly to deform thebody itself [3,6,14,17,34,35], which requires multiple actuatorsand/or complex transmission mechanisms. Another class ofdesigns involves an oscillating caudal fin (e.g., [4,9,10,30,36]),sometimes in conjunction with pectoral fins [9]. Among variousdesigns, tail actuation is especially attractive since it is easy torealize, enables both forward swimming and turning maneuvers,and leaves the majority of the body free of moving parts. Thelatter is important when the robot is used in applications such asmobile sensing, where the body space can be maximally used tohouse sensors and electronics [20]. Many tail-actuation designsinvolve rigid, oscillating plates [9,12,13,37]. This design lendsitself to simple construction and tractable analysis. However, ithas been recognized that the flexibility of body and fin structureshas a pronounced impact on the swimming performance of biolog-ical and robotic fish [16,38,39]. Flexible caudal fins can berealized by motor-driven compliant beams or plates [17,36,40],or directly through soft actuation materials such as ionicpolymer–metal composites (IPMCs) [4,7,10,30]. While theseactive materials possess intriguing properties, the thrust they canproduce is still relatively weak and the long-term repeatability oftheir actuation behavior is yet to be established. Therefore, base-actuated soft, passive structures remain a competitive option forflexible tails.

To design and control robotic fish actuated with a flexible tail,it is essential to have a faithful and efficient dynamic model. Mod-eling of a flexible beam attached to a moving base in the air hasbeen studied by a number of researchers [,40–43]. However, inthis work, a major challenge in the modeling lies in properly cap-turing the fluid–structure interactions and the resulting force andmoment on the robot. Most existing work on modeling has dealtwith rigid fins [9,12,13,37,44,45], although modeling of flexiblefins has been conducted recently by several groups [10,17,30,46].In Ref. [17], Alvarado and Youcef Toumi focused on designingcompliant bodies to achieve biomimetic locomotion efficiencyand maneuverability. To demonstrate the results, they also com-pared the experimental results with the linear beam model.Researchers have also developed models that capture the actuationphysics of an anchored IPMC beam and the complex hydrody-namic interactions between IPMC and fluid [10,30]. In Ref. [46],Kopman and Porfiri described a modular biomimetic robotic fishdeveloped for educational activities, along with a modeling frame-work for predicting the robot’s static thrust production. In all thesestudies, a linear Euler–Bernoulli beam model was adopted todescribe the beam dynamics. A disadvantage of this approach isthat, under large oscillations, these models do not accuratelycapture the beam dynamics [47–49] and consequently thehydrodynamics.

In this work, we take a significant departure from the beammodel approach by approximating a flexible tail with multiplerigid segments connected in series through torsional springs anddampers. A similar multisegment approach has been used tomodel a beam under large deformation in air [49,50] and model aflexible beam in flow sensing [51]. However, the multisegmentbeam model has not been explored in the modeling of a base-actuated flexible beam subject to complex hydrodynamic interac-tions. For comparison, we also present a model based on linearbeam theory, as widely adopted in other studies [10,17,30,46].Note that, for ease of presentation, a tail with uniform height isconsidered in this work. Despite its simple appearance, this casecaptures the key essential challenges (nonlinear beam dynamicsunder large deformation, coupled with hydrodynamic interactions)that would be present for cases with more general tail shapes. Weevaluate the hydrodynamic force on the actuating tail using Light-hill’s large-amplitude elongated-body theory [52], because it has asound balance between fidelity and simplicity and its effectivenessin robotic fish modeling has been demonstrated in our prior work[13,30]. A weakness of Lighthill’s theory is that it neglects the

1Corresponding author.Contributed by the Dynamic Systems Division of ASME for publication in the

JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedJuly 8, 2013; final manuscript received July 17, 2014; published online August 28,2014. Assoc. Editor: Evangelos Papadopoulos.

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effect of vortex wake on the pressure distribution on the body andthus neglects the effect of vortex shedding on the thrust produc-tion [53]. Incorporating such vortex-shedding effects (for exam-ple, using the vortex ring panel method [54]), however, typicallyrequires computational fluid dynamics (CFD) simulation and isnot amenable to the development of a model that is suitable forrobot optimization and control, which is the goal of this paper.

To compare and validate the models, we conduct extensiveexperiments with a robotic fish prototype. We find that both mod-els have similar predictions that are close to the experimentalmeasurements when the flexible beam is under small deforma-tions. However, the discrepancy between the predictions becomesgreater as the beam experiences larger deformations. In particular,we show that the multisegment model is able to predict preciselythe transient trajectory and steady-state speed of the free-swimming robot, as well as the dynamic shape of the tail, under awide range of actuation inputs. In comparison, the linear beammodel-based approach is only able to capture the robot speed andtail shape when the tail undergoes relatively small deformations.A preliminary version of this paper was presented at the 2012ASME Dynamic Systems and Control Conference [55].

The remainder of the paper is organized as follows. The overalldynamic model for a tail-actuated robotic fish is first reviewed inSec. 2, where the hydrodynamic forces generated by the tail areevaluated by Lighthill’s theory. The use of Lighthill’s theoryrequires knowing the tail shape as a function of time, which is themain focus of this paper. The Euler–Bernoulli beam-based modeland the multisegment model for the based-actuated flexible tailare developed in Secs. 3 and 4, respectively. Experimental valida-tion and comparison of the models are presented in Sec. 5. Finally,concluding remarks are provided in Sec. 6.

2 The Overall Dynamic Model for Tail-Actuated

Robotic Fish

The robot is assumed to comprise two parts, a rigid body anda flexible tail. The motion of the robot body is governed byrigid-body dynamics with the added-mass effect incorporated.Lighthill’s large-amplitude elongated-body theory is adopted toevaluate the hydrodynamic forces generated by the flexible tail’smotion. We assume that the height of the tail does not varyabruptly along its length, thus meeting the “elongated body”requirement [52].

2.1 Rigid-Body Dynamics. Figure 1 shows a schemata of therobotic fish, with [X,Y,Z] denoting the inertial coordinates, and[x,y,z] denoting the body-fixed coordinates with unit vectors½x; y; z�. We denote by m and n the unit vectors parallel and per-pendicular to the tail, respectively. We assume that both the body

and the tail are neutrally buoyant, and that the center of gravity ofthe body coincides with the center of geometry at point C. The ve-locity at C expressed in the body-fixed coordinatesVC ¼ ½VCx;VCy;VCz�T comprises surge (VCx), sway (VCy), andheave (VCz) components. In addition, the angular velocityx ¼ ½xx;xy;xz�T comprises roll (xx), pitch (xy), and yaw (xz)expressed in the body-fixed coordinates. We use a to denote thetail deflection angle (the tangential direction of the flexible tail atthe base) with respect to the negative x–axis, and b to denote theangle of attack, formed by the direction of VC with respect to thex–axis. Finally, w denotes the heading angle, formed by the x–axisrelative to the X–axis.

The linear momentum P and angular momentum H of the bodyin the body-fixed coordinates are expressed as

P ¼M � VC þ DT � x (1)

H ¼ D � VC þ J � x (2)

where M and J are the mass and inertia matrices, respectively,and D is the Coriolis and centripetal matrix. For a rigid body in aninviscid fluid, Kirchhoff’s equations of motion in the body-fixedframe are [56,57]

_P ¼ P� xþ F (3)

_H ¼ H� xþ P� VC þM (4)

where F ¼ ½Fx;Fy;Fz�T;M ¼ ½Mx;My;Mz�T denote the externalforces and moments about the body center C, respectively, and“�” denotes the vector product. The assumption of neutral buoy-ancy implies that these forces and moments will only come fromthe hydrodynamic interactions between the robot (including boththe body and the tail) and the fluid. As to be explained in moredetail, these interactions include hydrodynamic forces/momentsdue to the tail motion, which are evaluated with Lighthill’s theory(Sec. 2.2), and the lift and drag on the robot body itself (Sec. 2.3).

We further assume that the body is symmetric with respect tothe xz–plane and the tail moves in the xy–plane. Consequently, theheave velocity VCz, the roll rate xz, and the pitch rate xy are allequal to zero, in which case the system has three degrees of free-dom, namely, surge (VCx), sway (VCy), and yaw (xz). We furtherassume that the inertial coupling between the surge, sway, andyaw motions is negligible [10], implying D¼ 0. Under theseassumptions and following Ref. [56], Eqs. (3) and (4) can be sim-plified as

ðmb � X _VCxÞ _VCx ¼ ðmb � Y _VCy

ÞVCyxz þ Fx (5)

ðmb � Y _VCyÞ _VCy ¼ �ðmb � X _VCx

ÞVCxxz þ Fy (6)

ðJbz � N _xzÞ _xz ¼ ðY _VCy

� X _VCxÞVCxVCy þMz (7)

where mb is the mass of the body and Jbz is the inertia of the bodyabout the z–axis. X _VCx

; Y _VCy, and N _xz

are the hydrodynamic deriva-

tives that represent the effect of added mass on the body. Finally,the kinematic equations for the robot are

_X ¼ VCx cos w� VCy sin w (8)

_Y ¼ VCy cos wþ VCx sin w (9)

_w ¼ xz (10)

2.2 Lighthill’s Large-Amplitude Elongated-Body Theory.Lighthill’s theory was originally developed to describe the hydro-dynamic force experienced by a fish swimming in a horizontalplane. In this paper, we apply this theory to the base-actuatedflexible tail of a robotic fish. As illustrated in Fig. 2, the tail isassumed to be inextensible and is parameterized by s, with s¼ 0

Fig. 1 Schematic representation of the robotic fish in planarmotion

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denoting the anterior end and s¼ L denoting the posterior end.The coordinates (X(s,t), Y (s,t)), 0� s� L, denote the trajectoryof each point s on the tail at time t, which could be due to taildynamics or the resulting translational/rotational motion of thewhole fish. A frame of reference is chosen such that the water farfrom the robotic fish is at rest.

Following Ref. [52], given (X(s,t), Y (s,t)), the hydrodynamicreactive force density due to the added-mass effect at each points< L is

fðsÞ ¼fXðsÞfYðsÞ

� �¼ �mw

d

dtðv?nÞ (11)

and at s¼ L, there is a concentrated force

FC ¼FCX

FCY

� �¼ � 1

2mwv2

?mþ mwv?vkn

� �s¼L

(12)

In Eqs. (11) and (12), mw denotes the virtual mass per unit lengthand it can be approximated by 1

4pqwd2, where qw is the density of

water and d is the depth of tail cross section (in Z direction) at s(and thus can vary with s). The two terms in Eq. (12) account forthe pressure force acting on P and the force due to convection ofmomentum out of V across P, respectively, where P is the planeat s¼L that is perpendicular to m, and V is the half-spacebounded by P that includes the tail but excludes the wake [52].As illustrated in Fig. 2, m ¼ ð@X=@s; @Y=@sÞT and n ¼ ð�@Y=@s;@X=@sÞT represent the unit vectors tangential and perpendicular tothe spinal column, respectively, and vk and v? represent the com-ponents of the velocity v ¼ ð@X=@t; @Y=@tÞT at s in m and n direc-tions, respectively,

vk ¼ hv; mi ¼@X

@t

@X

@sþ @Y

@t

@Y

@s(13)

v? ¼ hv; ni ¼ �@X

@t

@Y

@sþ @Y

@t

@X

@s(14)

where h �,� i denotes the inner product of vectors. We note that theactual mass of the tail is negligible (relative to the virtual mass)when the width of the tail is much thinner than the height [52],which is typically true for the tail of a robotic fish. For example,in the model validation part of this work, we use a flexible tailwith a height of 2.5 cm and a thickness of 0.3 mm, which indicatesthat mw is 490.9 g/m. In contrast, the physical tail, which has alength of 8 cm, weighs 0.54 g, resulting in a mass of 6.75 g/m,which is 1.38% of the virtual mass. Hydrodynamic momentsexperienced by the robot body that are due to the tail can be eval-uated accordingly based on fðsÞ and FC.

2.3 Drag and Lift on the Body. As shown in Fig. 1, besidesthe hydrodynamic force and moment transmitted from the tail, the

robot body also experiences drag force FD, lift force FL, and dragmoment MD [9,10,37], which can be represented as

FD ¼1

2qjVCj2SCD (15)

FL ¼1

2qjVCj2SCLb (16)

MD ¼ �CMx2z sgnðxzÞ (17)

where S is a suitably defined reference surface area for the robotbody, and CD, CL, and CM are the drag force coefficient, lift coef-ficient, and drag moment coefficient, respectively.

Finally, by adding the hydrodynamic forces and moments fromthe tail and those directly on the body, we obtain Fx, Fy, and Mz inEqs. (5)–(7) as

Fx ¼ Fhx � FD cos bþ FL sin b (18)

Fy ¼ Fhy � FD sin b� FL cos b (19)

Mz ¼ Mhz þMD (20)

where Fhx, Fhy, and Mhz are the projections of hydrodynamicforces caused by tail motion, and the resulting moment relative tothe center C of the robot body.

3 Dynamic Modeling of a Tail Using Euler–Bernoulli

Beam Theory

The application of Lighthill’s theory requires knowing theshape trajectory of the base-actuated flexible tail. We propose amultisegment approach to the modeling of the tail that undergoeslarge deformations in Sec. 4. But in this section we first develop acomparative approach based on linear beam theory, which hasbeen widely used in the relevant literature. We focus on beammodeling with the robot anchored, as typically adopted in the lit-erature [17,46]. Figure 3 illustrates this approach; the dashed linerepresents a rotating frame, which coincides with the line tangen-tial to the tail at the base, and the blue solid curve represents theshape of the flexible caudal fin. The transverse displacement ofpoints on the flexible tail relative to the rotating base, due to thebeam’s vibration, is given by w(s,t). We note that the tail isassumed to be under small deformation in accordance with theEuler–Bernoulli theory. In other words, the motion direction ofeach point along the flexible tail is perpendicular to the dashedline.

Fig. 2 Illustration of the coordinate system for flexible tail (topview)

Fig. 3 Passive flexible tail modeled by Euler–Bernoulli linearbeam approach

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The tail is excited by the oscillating support at the point A, with

@w1ðs; tÞ@s

� �s¼0

¼ @w2ðs; tÞ@s

� �s¼0

¼ aðtÞ (21)

Here w1(s,t) denotes the beam displacement in the inertial frameand w2(s,t) accounts for the rigid-body (rotating base) motion

w1ðs; tÞ ¼ w2ðs; tÞ þ wðs; tÞw2ðs; tÞ ¼ s � aðtÞ

(22)

The forced underwater vibration of the flexible beam isdescribed by the following equation [46,58]:

@2

@s2EIðsÞ @

2w1ðs; tÞ@s2

� �þ mðsÞ @

2w1ðs; tÞ@t2

¼ pðs; tÞ (23)

where E denotes the Young’s modulus of the flexible tail, Idenotes the area moment of inertia, m(s) denotes the mass of thetail per unit length, and p(s,t) denotes the transverse loading onthe tail which is caused by the interactions between the tail andthe surrounding aquatic environment that can be evaluated byLighthill’s theory. Unlike Refs. [17] and [46], in this work wealso consider distributed viscous damping introduced by the inter-nal resistance opposing the strain velocity [58], which leads to

@2

@s2EIðsÞ @2w1ðs; tÞ

@s2þ j

@3w1ðs; tÞ@s2@t

� �� �þ mðsÞ@

2w1ðs; tÞ@t2

¼ pðs; tÞ

(24)

where j is the stiffness proportionality factor for Rayleighdamping.

Similar to Refs. [12] and [13], we assume that the tail itself hasnegligible mass compared to the added mass effects. Then follow-ing Lighthill’s theory, we obtain

EI@4wðs; tÞ@s4

þ j@5wðs; tÞ@s4@t

� �þ mw

@2wðs; tÞ@t2

¼ Peffðs; tÞ (25)

in which

Peffðs; tÞ ¼ �EI@4w2ðs; tÞ@s4

þ j@5w2ðs; tÞ@s4@t

� �� mw

@2w2ðs; tÞ@t2

represents the effective distributed dynamic loading caused by theprescribed support excitations, and mw is virtual mass per unitlength defined previously, accounting for the added mass effect.Since the first two terms of Peffðs; tÞ are equal to zero asw2ðs; tÞ ¼ s � aðtÞ, the tail beam can be modeled as

EI@4wðs; tÞ@s4

þ j@5wðs; tÞ@s4@t

� �þ mw

@2wðs; tÞ@t2

¼ �mws � €aðtÞ (26)

Using the modal analysis method, we can express the solution ofEq. (26) as the sum of an infinite number of modes

wðs; tÞ ¼X1i¼1

/iðsÞgiðtÞ (27)

where /iðsÞ is the beam shape for the ith mode and gi(t) is the cor-responding generalized coordinate. There is no concentrated forcein the n direction with the anchored body assumption, whichimplies that the boundary conditions for w(s,t) is the same as thosefor a cantilever beam

wð0; tÞ ¼ 0;@w

@sð0; tÞ ¼ 0

EI@2w

@s2ðL; tÞ ¼ 0; EI

@3w

@s3ðL; tÞ ¼ 0

(28)

The mode shape /iðsÞ takes the form

/iðsÞ ¼ ðcos bis� cosh bisÞ � diðsin bis� sinh bisÞ (29)

where bi can be obtained by solving

1þ cos biL cosh biL ¼ 0

and

di ¼cos biLþ cosh biL

sin biLþ sinh biL

With the damping ratio for the ith mode

ni ¼jxi

2

where xi is the natural frequency for the ith mode,

xi ¼ b2i

ffiffiffiffiffiffiffiEI

mw

r(30)

gi(t) can be solved from

d2giðtÞdt2

þ 2nixidgiðtÞ

dtþ x2

i giðtÞ ¼QiðtÞMi

; i ¼ 1; 2… (31)

where

QiðtÞ ¼ðL

0

/iðsÞPeffðs; tÞds (32)

and

Mi ¼ðL

0

/iðsÞ2mwds (33)

When the tail shaft oscillates sinusoidally, we can obtain a closed-form solution for gi(t). Consider in particular the following formfor the base angle:

aðtÞ ¼ a0 þ aA sinðxatþ /aÞ (34)

where a0, aA, xa, and /a denote the bias, amplitude, frequency,and initial phase of the tail beat, respectively. Then, we have

giðtÞ ¼mwaAx2

a

ðL

0

s/iðsÞds

MiZixasinðxatþ /a þ /iÞ (35)

where

Zi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2xiniÞ2 þ

1

x2aðx2

a � x2i Þ

2

s

is the magnitude of the impedance function and

/i ¼ arctan2xaxini

x2a � x2

i

� �

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is the phase lag of the oscillation relative to a(t). Using Eqs. (11)and (12), the total hydrodynamic force acting on the tail is then

F ¼ðL

0

fðsÞds� 1

2mwv2

?m

� �s¼L

(36)

where fðsÞ ¼ �mw½ðs€aþ ð@2w=@t2ÞÞn� _aðs _aþ ð@w=@tÞÞm�. Thecorresponding moment relative to the center C of the robot bodyis

M ¼ðL

0

rCs � fðsÞdsþ rCL � � 1

2mwv2

?m

� �s¼L

� �(37)

where rCs and rCL denote the vectors from the body center C tothe point s and L on the tail, respectively.

4 Dynamic Modeling of a Tail Using

Multisegment Approximation

A fundamental underlying assumption in establishing the modeldescribed in Sec. 1 is that the flexible tail is under small deforma-tion. However, based upon observation of the dynamic shape ofthe tail, this assumption does not always hold, especially when thetail undergoes large angular displacement and/or high-frequencyoscillations. In this section, we propose a novel model by repre-senting the tail as multiple rigid elements connected in seriesthrough torsional springs and dampers, to capture the large defor-mation of the beam. N rigid segments, with equal length of l, areused to represent the tail, as illustrated in Fig. 4. Each segment islinked with its neighboring segments through joints modeled by atorsional spring KS and a viscous damper KD.

Following Ref. [49], we can evaluate the stiffness of each tor-sional spring as:

KS ¼Edh3

12l(38)

with h denoting the thickness of tail. KD can be evaluated asKD¼ jKS, where j is the proportional constant as defined inthe Sec. 1. Following Lighthill’s large-amplitude elongated-bodytheory, we need to compute the motion of every point along thetail over time, in order to evaluate the tail actuation-inducedhydrodynamic force. Consequently, we need to know the jointangles ai, made by the ith link with respect to the negative x-axis(as illustrated in Fig. 4), so that Eqs. (11) and (12) can be appliedto evaluate the hydrodynamic forces on the tail section. The dis-placement rsi

of every point si on the ith segment in the inertialframe can be described as

rsi¼ l �

Xi�1

k¼1

mk

" #þ simi (39)

then the velocity perpendicular to the ith segment is

vsi? ¼ l �Xi�1

k¼1

_ak cosðak � aiÞ" #

þ si _ai (40)

Therefore, the force density acting on the ith segment can beevaluated as

fsi¼�mw

d

dtðvsi?niÞ

¼�mw lXi�1

k¼1

€ak cosðai�akÞ½ � _ak sinðai�akÞð _ai� _akÞ�þsi€ai

( )ni

þmw _ai lXi�1

k¼1

_ak cosðai�akÞ½ �þ _aisi

( )mi (41)

The total reactive force Fi on the ith segment, and the momentMi relative to point Ai�1 can be evaluated using

Fi ¼ðl

fsids; Mi ¼

ðl

simi � fsids (42)

Defining FAi and MAi to be the force and the moment exerted bythe (iþ 1)th segment on the ith segment, respectively, we canexpress the interactions between adjacent segments as

FAi¼ Fiþ1 þ FAiþ1

(43)

MAi¼Miþ1 þMAiþ1

þ lmi � FAiþ1(44)

For the last segment of the tail, as illustrated in Fig. 4, the reactiveforce and moment are

FAN¼ � 1

2mw l

XN

k¼1

_ak cosðaN � akÞ" #2

mN

þ mw lXN

k¼1

_ak cosðaN � akÞ" #

lXN

k¼1

_ak sinðaN � akÞ" #

nN

(45)

MAN¼ 0 (46)

Defining MðSþDÞi the moment produced by the spring and damperat joint Ai, then the moment balance equation implies, fori¼ 1, 2, …, N� 1,

MAi¼MðSþDÞi (47)

where

MðSþDÞi ¼ ½KSðaiþ1 � aiÞ þ KDð _aiþ1 � _aiÞ�z (48)

Equation (47) has (N� 1) scalar equations involving (N� 1)unknown variables a2,� � �,aN, which is solvable.

FA0is the force that the flexible tail exerts on the robotic fish

body, which can be written as FA0¼ Fhx � xþ Fhy � y, where Fhx

and Fhy are the components of FA along x and y, respectively. Themoment to the center of the body caused by the oscillation of thetail can be evaluated as

MC ¼MA0� cx� FA0

(49)

It is clear that MC is along the z direction, which we denote asMhz.Fig. 4 Passive flexible tail modeled by multiple rigid segments

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5 Experimental Model Validation

To evaluate the dynamic models described in Secs. 3 and 4, wehave conducted experiments with the free-swimming robotic fishprototype shown in Fig. 5. The robot has a simple mechanismfor actuation, and it satisfies most of the assumptions used in themodeling work, which facilitates model validation. In particular,the height of the tail does not change abruptly along the lengthdirection, thus meeting the “elongated body” requirement. Thethickness of the tail is much less than its height, so that the actualmass of the tail is much less than the virtual mass, as assumed inSec. 2.2. Through a chain transmission mechanism, a servomotor(HS-5085MG from Hitec) is able to control the angular positionof the tail shaft and thus the tail deflection angle a, i.e., sinusoidalmotion, accurately. On the other hand, in the modeling work, weassume that the robot is anchored, as typically adopted in theliterature. This assumption, of course, does not hold fully duringfree-swimming experiments, which might explain the slight dis-crepancies between the model predictions and experimental mea-surement in those experiments. The tail was a rectangular plasticslice, which was 8 cm long, 2.5 cm wide (high), and 0.3 mm thick.

5.1 Parameter Identification. The same experimental proto-type (with a different tail) was used in Ref. [12] to validate adynamic model for a robotic fish with a rigid tail. The followingparameters for the robot were identified in Ref. [12]: c¼ 0.07 m,

mb¼ 0.311 kg, q¼ 1000 kg/m3, S¼ 0.0108 m2, Jbz¼mb(2c)2/12.The added masses and inertias are calculated by approximatingthe robot body as a prolate spheroid [10,57]: �X _VCx

¼ 0:0621 kg;�Y _VCy

¼ 0:2299 kg, and �N _xz¼ 1:0413� 10�4 kg �m2.

The drag and lift coefficients, CD, CL, and CM, are also identi-fied empirically for the robotic fish with a rigid tail and then usedin validating the flexible tail models. In particular, we have tuned

Fig. 5 A free-swimming robotic fish prototype used for modelvalidation

Fig. 6 Experimental setup for measuring the Young’s modulusof the flexible tail

Fig. 7 Experimental results for measuring the Young’s modu-lus of the flexible tail

Fig. 8 Displacement of the tail tip generated by the modelusing different number of rigid segments

Fig. 9 Computation time needed to simulate the model usingdifferent numbers of rigid segments

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these parameters to match the turning radius, turning period, andthe average of jbj obtained in simulation, with the experimentalmeasurements recorded for a particular tail beat pattern. We setamplitude aA¼ 13.6 deg, frequency xa¼ 1.8p rad/s (0.9 Hz) andevaluate the suitable drag and lift coefficients for the tail-beatbias a0 equal to 20 deg, 30 deg, and 40 deg as Ref. [13]. Usingleast-square-error fitting, we obtain CD¼ 0.276, CL¼ 4.5, andCM¼ 7.4� 10�4 kg �m2, when a0¼ 0.

For measuring the Young’s modulus, we setup the experimentshown in Fig. 6, and evaluate E using

E ¼ L3bPL

3IbwL

(50)

where Lb and Ib are the length and the area of moment inertia ofthe testing beam, respectively. PL is the load at the tip end, and wL

is the end’s displacement. A plastic beam (the same material asthe flexible tail) is clamped at the upper surface of a rectangularblock, which is fixed on a piston that can move up and down alongthe stand. A custom LabVIEW (2011 SP1) virtual instrumentgraphical user interface (GUI) is developed to perform the dataacquisition through a dSPACE system (RTI 1104, dSPACE). Theforce exerted on the beam is captured by a load cell (GS-10,Transducer Techniques) with a custom-made amplifier circuit.The displacement of the beam tip is measured with a laser sensor(OADM 20I6441/S14F, Baumer Electric). Prior to experiment,the system is calibrated with a weight applied to the load cell.Three sets of data are collected to calculate the Young’s modulus,and in each set, a least square error method is adopted to approxi-mate the slope between the force and displacement. Figure 7shows results for three tests, indicating that E � [1.41, 1.51] GPa.In this work, we take the average E¼ 1.48 GPa.

When modeling a tail using the multisegment approximation,the number of rigid segments affects both modeling accuracy andcomputational complexity [51]. Specifically, a higher number ofelements results in a more accurate model, but is also morecomputation-intensive. Figure 8 shows the simulated responses ofthe beam tip displacement (relative to the x axis along the body)to a sinusoidal base excitation, when different numbers of seg-ments are used to model the flexible beam. The properties of thebeam used in the simulation are same as those of the beam identi-fied from the experiments. It can be observed that, the beam tipdisplacements gradually converge to each other when the numberof segments increases. Figure 9 shows the running time neededfor the simulation illustrated in Fig. 8, which was conducted withMATLAB/SIMULINK on a desktop PC (Dell Vostro 460 with 3.1 GHzIntel i5-2400 central processing unit (CPU) and 4 GB memory).In particular, for all cases, the fundamental sampling time inSIMULINK is fixed at 0.000167 s, the simulation time is set to be 10 sin total, and the elapsed CPU time is obtained using the MATLAB

macro cputime. We can see that the computation time increasesrapidly with the number of rigid segments used in the simulation.A five-segment approximation achieves a sound tradeoff betweenthe modeling accuracy and computational efficiency, and there-fore is adopted in this study.

According to Eq. (39), we have KS¼ 5.2� 10�3 N�m that isused in the model using multisegment approximation. The valuesof j are identified empirically by fitting the data of forward speedversus beating frequency for both models. In particular, we choosethe value of j such that the simulated steady-state speed of the

Fig. 10 Comparison between model predictions and experi-mental measurement of the speed versus tail-beat frequency.The amplitude is fixed at 13.6 deg

Fig. 11 Comparison between model predictions and experi-mental measurement of the speed versus tail-beat frequency,with the new tail. The amplitude is fixed at 13.6 deg

Fig. 12 Comparison between multisegment model predictions and experimental measurement for forward swimming (includ-ing transients): (a) time trajectory of X-coordinate of the robot; (b) time trajectory of the Y-coordinate of the robot; (c) path ofthe robot in the XY-plane. For both the experiment and simulations, the amplitude and frequency of the tail beat are fixed at13.6 deg and 0.9 Hz, respectively.

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robot, when a0¼ 0 deg, matches the experimental results when thetail beats at 0.45 Hz, 0.9 Hz, and 1.35 Hz, respectively (for whichcases the flexible tail undergoes relatively small deformations).This results in j¼ 0.0846 s for the Euler–Bernoulli beam modeland j¼ 0.0731 s for the multisegment approximation model. Wetake the average j¼ 0.079 s, which leads to KD¼ 4.1� 10�3

N�m�s, using KD¼jKS. These parameters are then applied in thesimulation of all other cases.

5.2 Model Verification. Figure 10 compares the two modelsand experimental measurements in terms of the steady-state speedof the robotic fish with respect to the actuation frequency. The

trends are similar for the two model predictions. In particular, therobot’s speed increases with the frequency up to a threshold value(1.35 Hz in the simulated case) and then starts to drop. While thepredictions from both models match the experiments relativelywell (due to the tuning process for parameter j as described in theprevious subsection 5.1), the discrepancy between the predictionsbecomes larger as the actuation frequency increases. The flexibletail is subjected to the added mass effect as explained in Eq. (11).Under a high-frequency excitation, the lateral hydrodynamic load-ing on the tail increases and causes larger deformation of the tail.By using linear Euler–Bernoulli beam theory, we assume that thetail is under small deformation, and thus the motion direction ofeach point along the flexible tail is perpendicular to the dashedline in Fig. 3. The latter assumption no longer holds when theexcitation frequency gets high due to the large deformation, whichexplains the poor prediction performance of the linear beammodel at relatively high frequencies. On the other hand, the modelusing the multisegment approximation matches the experimentalmeasurement closely throughout the actuation frequency rangeused in the experiments.

To further compare the two models under different conditions,a second experiment has been conducted with another tail ofdifferent dimensions, which is 2.34 cm high and 9.8 cm long.Following Eq. (39) and the linear relationship KD¼ jKS, the val-ues of KS and KD are updated from the original KS¼ 5.2� 10�3

N�m and KD¼ 4.1� 10�3 N�m�s to KS¼ 4.0� 10�3 N�m andKD¼ 3.13 � 10�3 N�m�s, respectively. Figure 11 shows the com-parison between the two models and the experimental measure-ments for the robot with the new tail. Consistent with the resultsin Fig. 10, one can see that the model using the multisegmentapproximation matches the experimental measurement closely forall actuation frequencies, while the linear beam model performswell only for low-frequency actuation.

Fig. 13 Experimental setup to capture the dynamic shape of aflexible tail actuated at the base (top view).

Fig. 14 Comparison between experimental measurement of the time-dependent tail shape with model predictions. The tailbeats at 0.4 Hz with 0 deg bias and 14 deg amplitude. The black solid line, blue dashed line with circles and the red dashed-dotted line imply the experimental measurement, predictions from multisegment model and Euler–Bernoulli beam model,respectively.

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To further assess the proposed multisegment model, we con-duct an experiment involving transients in forward motion,where the robot started from at rest. Figures 12(a) and 12(b)compare the model predictions of the X/Y-coordinate time-trajectories of the robot with the experimental measurements,and Fig. 12(c) compares the predicted and actual robot paths inthe XY-plane. It can be seen that the proposed model is able tocapture well both the transient and steady-state behaviors of therobot.

We have conducted additional experiments to compare thetime-dependent shape of the flexible tail with those predicted bythe models. As can be seen in Fig. 13, the free swimming robotshown in Fig. 5 is fixed by a bracket and set to oscillate the tail. ACasio Exilim EX-FH25 high-speed camera with a frame rate of120 frames/s is used to record the tail’s motion from above.

Figures 14 and 15 compare the measured time-dependent tailshape and those predicted by the two models for 0.4 Hz and0.9 Hz, respectively. To save space, we show every ninth framefor half a period of the tail oscillation. It can be seen that whenthe tail beats at the relative low-frequency, 0.4 Hz, both modelsproduce very close approximations of the tail shape. However,at 0.9 Hz, when the beam deformation is bigger, the multiseg-ment modeling approach produces much more precise predic-tions of the beam shape than the linear beam theory-basedapproach.

6 Conclusion

In this study, we have developed a model for robotic fishwith a base-actuated flexible tail. The tail is modeled as multi-ple rigid segments connected by springs and dampers, andLighthill’s elongated-body theory is used to evaluate the tail-generated hydrodynamic forces. For comparison, we have also

constructed a model using linear beam theory. We compare pre-dictions of both models to experimental results with a roboticfish, in terms of steady-state cruising speeds and dynamic tailshapes under different actuation frequencies. From these results,we conclude that when the tail is excited under a relativelylow-frequency, and consequently experiences small deformation,both models produce similar predictions that are close to experi-mental measurements. However, when the actuation frequencyincreases, the two models differ, and the model using multiseg-ment approximation is able to predict much better the robotspeed and the tail shape. Additional experimental results alsoindicate that the proposed model is capable of capturing thetransient dynamics of the robot.

In summary, the work presented in this paper provides a com-putationally efficient and accurate model for capturing large taildeformation and the resulting hydrodynamic force for a tail-actuated robotic fish. The model will facilitate effective tail designoptimization and controller development for such robots. Whileone could also use nonlinear beam models and CFD to achievefaithful modeling of the tail, the latter approach would be muchmore computationally expensive, and would be difficult to inte-grate with the robot dynamics for controller design.

This work can be extended in several directions. First,although this paper has focused on the case of a rectangular tailfor ease of presentation, the mathematical derivation itself isgeneral and the approach can be extended to tails of othershapes (e.g., trapezoidal) following a similar treatment, wherethe properties of each segment (mass, inertia) and each joint(spring and damper constants) will depend on the local shape.Second, we are particularly interested in using the proposedmodel to understand the effect of tail shape and stiffness prop-erties on the robot’s locomotion performance, and exploit suchunderstanding for design optimization. Finally, we will utilize

Fig. 15 Comparison between experimental measurement of the time-dependent tail shape with model predictions. The tailbeats at 0.9 Hz with 0 deg bias and 13.6 deg amplitude. The black solid line, blue dashed line with circles, and the red dashed-dotted line imply the experimental measurement, predictions from multisegment model and Euler–Bernoulli beam model,respectively.

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the proposed model to design controllers for flexible tail-actuated robotic fish.

Acknowledgment

This work has been supported in part by NSF Grant(Nos. CNS-0751155, CCF-0820220, DBI-0939454, IIS-0916720,CNS-0915855, ECCS-1050236, ECCS-1029683, CNS-1059373,CCF-1331852, and IIS-1319602). The authors would like to thankProfessor Songlin Chen and Tony Clark for their constructive dis-cussions in this work.

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