Hadi El Daou, Taavi Salum�ae, Gert Toming and Maarja Kruusmaa

Abstract�This paper studies the modelling, design and

fabrication of a bio-inspired fish-like robot propelled by a

compliant body. The key to the design is the use of a single

motor to actuate the compliant body and to generate thrust. The

robot has the same geometrical properties of a subcarangiform

swimmer with the same length. The design is based on rigid

head and fin linked together with a compliant body. The flexible

part is modelled as a non-uniform cantilever beam actuated by

a concentrated moment. The dynamics of the compliant body

are studied and a relationship between the applied moment and

the resulting motion is derived. A prototype that implements

the proposed approach is built. Experiments on the prototype

are done to identify the model parameters and to validate the

theoretical modelling.

I. INTRODUCTION

Underwater robots provide an engineering tool to practical

applications in marine and military fields, such as monitoring

the environment, harvesting natural resources, undersea

operation, pipe inspection and many other applications.

With millions of years of evolution, aquatic animals, in

particular fish, are very efficient swimmers. This has inspired

scientists to study fish locomotion and build fish-like robots.

MIT�s RoboTuna I and II are the best known bio-inspired

underwater robots [1]. These are tethered robots, mimicking

the thunniform swimmers and use a system of pulleys and

cable tendons actuated by DC-motors. MIT also developed

Robot pike to learn more about the fluid mechanics that fish

use to propel themselves with a purpose to develop small

fish-like autonomous vehicles for reduced energy consumption

and increased operation time [3]. The Vorticity Control

Unmanned Undersea Vehicle (VCUUV) was produced in

Draper Laboratory; it was the first autonomous mission-scale

UUV that utilizes fish-like swimming and manoeuvering

[2]. The University of Essex has developed a series of

autonomous robots G1 to G9 and MT1. The G series have

a multi-motor-multi-joint tail structure, which employs 4

servo motors to drive 4 tail joints separately according to a

predetermined swimming wave sequence [4][5][6][7][8].The

Japenese National Maritime Research Institute developed

many kinds of robotic fish prototypes to increase swimming

efficiency [9].

Most of these designs use rigid links and discrete mechanisms

to achieve fish-like swimming. The complexity of

these systems increases proportionally with the kinematic

This work is supported by European Union 7th Framework program

under FP7-ICT-2007-3 STREP project FILOSE (Robotic FIsh LOcomotion

and SEnsing), www.filose.eu.

Authors are with Center for Biorobotics, Tallinn University

of Technology, Tallinn, Estonia. maarja.kruusmaa,

[email protected]

similarity to fish. An alternative is to use compliant structures;

These bodies can be modelled as dynamically bending

beams [10] whose vibration characteristics are determined by

external and internal forces of the system, which in turn are

related to the geometry, material properties and actuation.

This alternative design concept also has some biological

relevance. The EMG studies of muscle activity of swimming

fish reveal that for swimming at cruising speeds (1 to 2

body lengths per second) fish use mainly anterior muscles

while the posterior part of the body acts like a carrier of the

travelling wave conveying the momentum to the surrounding

fluid [11], [12], [13]. A robotic fish using smart materials for

caudal fin design was developed in Michigan State University

[14] to increase efficiency, focusing on unique physics of

Ionic polymer metal composite (IPMC) materials and its

interaction with the fluid. A subclass of swimmers that

exploits the use of compliant bodies and one servo-motor

for actuation was developed at MIT [15]. It assumes that

a compliant body can be modelled by a cantilever beam

actuated by a single point and studies the dynamics in order

to mimic swimming fish motions.

In this study, the design of a robot with a flexible body

excited by a concentrated moment is studied. It makes use

of the models developed in [16] but differs in many aspects.

In fact, in [16] the mechanical modelling does not take into

account the elasticity of the rigid plate used for actuation and

neglected the hydrodynamic effect on the rigid fin. In this

work, a different approach is used. It takes into account the

non-homogeneity in the material distribution and the effect of

a rigid fin in the end of the compliant body. The dynamics of

the compliant body are studied to find a relationship between

the applied force and the resulting deflections. Moreover in

this study it is believed, that a compliant body cannot be

forced to deform to a random shape but has defined mode

shapes that are determined by the actuation frequencies and

the modal properties of the system.

The objectives of this paper are to:

# study the dynamics of the compliant body and derive the

relationship between the applied forces and the resulting

motion.

# propose a design that implements the proposed approach

and build a fish-like robot prototype.

# identify the model parameters and validate the model

through physical experiments.

The remainder of this paper is organized as follows: In

section II the dynamics of the compliant body are developed.

In section III the prototype of robotic fish is described.

Results from physical experiments on the prototype are

2012 IEEE International Conference on Robotics and Automation

RiverCentre, Saint Paul, Minnesota, USA

May 14-18, 2012

978-1-4673-1405-3/12/$31.00 �2012 IEEE 5340

a

Y

x

L(x,t)

M(t)

l

h(x,t)

fin F (t)

M (t)

fin

Fig. 1. Structural model of the compliant body. a=actuation point, l=length

of the compliant body, M(t)=actuation moment, L(x,t) hydrodynamic distributed

forces, Ffin(t), Mfin(t)=concentrated force and moment resulting

from the hydrodynamic forces acting on the fin and h(x,t)=lateral line

deflection

presented in section IV. Finally, section V discusses the

contributions and future work.

II. DYNAMICS OF THE COMPLIANT BODY

Fig.1 shows the structural model of the compliant body.

The �assumed modes� method is used to derive the equations

of motion of the compliant body [17] [18] [19] [20] [21].

It aims at deriving such equations by first discretizing the

kinetic energy, potential energy and the virtual work and

making use of the Lagrange�s equation of motion. The Elastic

deformations are modelled by a finite series:

h(x; t) =

Xn

r=1

'r(x)qr(t) 0 < x < l (1)

where:

# 'r(x): are known trial functions. The eigenfunctions of

a uniform cantilever beam are chosen as trial functions.

# qr(t): are unknown generalized coordinates.

# n: is the order of the expansion.

# l: is the length of the compliant body.

Considering that the passive fin is rigid compared to the

compliant body, its lateral deflection h(x,t) can be expressed

as:

h(x; t) = h(l; t) +

@h(l; t)

@x

(x ?? l) l < x < l + #l

=

Xn

i=1

('i(l)qi(t) + '

0

i(l)qi(t)(x ?? l)) (2)

where #l is the length of the rigid fin.

The external forces acting on the compliant body are: the

time varying moment M(t), the distributed hydrodynamic

forces L(x,t), the concentrated moment Mfin(t) and the

concentrated force Ffin(t). Mfin(t) and Ffin(t) are the

concentrated moment and force resulting from the action of

the hydrodynamic forces on the rigid fin. The hydrodynamic

forces are modelled in terms of added mass and expressed

as:

L(x; t) = D(m(x)_ h(x; t)) # m(x)

@2h

@t2 (3)

where m(x) is the apparent mass of the cross section per unit

length. It is approximated by m(x)=C0#f A(x) where C0 is

a constant that can be determined experimentally, #f is the

fluid density and A(x) is the cross area of a fluid cylinder

surrounding the body at x.

The concentrated force Ffin(t) is:

Ffin(t) = ??

Z l+#l

l

m(x)

@2h

@t2 dx = ??

Xn

i=1

qi(t)[#'i(l)+

'

0

i(l)]

(4)

where:

# =

Z l+#l

l

m(x)dx

=

Z l+#l

l

m(x)(x ?? l)dx (5)

The concentrated moment Mfin(t) is:

Mfin(t) = ??

Z l+#l

l

m(x)

@2h

@t2 (x ?? l)dx

= ??

Xn

i=1

qi(t)[

'i(l) +

0

i(l)] (6)

where:

Z l+#l

l

m(x)(x ?? l)2dx (7)

The kinetic and potential energies can be written as:

T(t) =

1

2

Z l

0

#(x)(

@h(x; t)

@t

)2dx

=

1

2

Xn

i=1

Xn

j=1

q_i(t)q_j(t)

Z l

0

#(x)'i(x)'j(x)dx (8)

V (t) =

1

2

Z l

0

EI(x)(

@2h(x; t)

@x2 )2dx

=

1

2

Xn

i=1

Xn

j=1

qi(t)qj(t)

Z l

0

EI(x)'

00

i (x)'

00

j (x)dx (9)

#(x) and EI(x) are the mass per unit length and the

stiffness at x respectively.

The total virtual work can be expressed as:

#W = #W1 + #W2 + #W3 + #W4

where #W1 is the virtual work of M(t):

#W1 =

Z l

0

M(t)#

��

(x ?? a)#h

0

(x; t)dx =

Xn

j=1

M(t)'

0

j(a)#qj

(10)

#W2 is the virtual work of L(x,t):

#W2 = ??

Z l

0

L(x; t)#h(x; t)dx = ??

Z l

0

m(x)

@2h

@t2 #h(x; t)dx

= ??

Xn

i=1

Xn

j=1

qi(t)

Z l

0

m(x)'i(x)'j(x)dx#qj (11)

5341

#W3 is the virtual work of Ffin:

#W3 =

Z l

0

F(t)#

��

(x ?? l)#h(x; t)dx

= ??

Xn

i=1

Xn

j=1

qi(t)[#'i(l)'j(l) +

'

0

i(l)'j(l)]#qj (12)

#W4 is the virtual work of Mfin(t):

#W4 =

Z l

0

Mfin(t)#

��

(x??l)#h

0

(x; t)dx =

Xn

j=1

Mfin(t)'

0

j(l)#qj

= ??

Xn

i=1

Xn

j=1

qi(t)[

'i(l)'

0

j(l) +

0

i(l)'

0

j(l)]#qj (13)

#��

denotes the Dirac delta function and h

0

(x; t) =

@

@xh(x; t).

The Lagrange�s equations are used to write the equations

of motion of the approximate system:

[M]fq(t)g + [K]fq(t)g = fQ(t)g (14)

where:

mij =

Z l

0

(#(x) + #fA(x))'i(x)'j(x)dx + #'i(l)'j(l)

+

('

0

i(l)'j(l) + 'i(l)'

0

j(l)) +

0

i(l)'

0

j(l)

kij =

Z l

0

EI(x)'

00

i (x)'

00

j (x)dx

and

Qi(t) = ??

Z l

0

M(t)#

��

0

(x ?? a)'i(x)dx

To find the response of (14), the eigenvalue problem is

first solved introducing:

q(t) = ae#(t)

This leads to the following characteristic equation:

det(#2M + K) = 0

where #r = ??iwr and wr are the undamped natural

frequencies of the approximate system. To obtain the solution

of (14), the following linear transformation is used:

q(t) = U#(t) (15)

where:

U = [a1 a2 a3 a4 a5 :::::::: an]

Where an is the eigenvector associated with the eigenvalue

#n = ??iwn. The eigenvectors are orthogonal with respect

to the mass and stiffness matrices. They are normalized to

yield:

aTr

:M:as = #rs aTr

:K:as = w2

r#rs

Where #rs is defined as the Kronecker delta.

Introducing (15) in (14) and premultiplying by UT , the

independent modal equations are then obtained:

#(t) + ##(t) = N(t) (16)

in which :

# = diag[w2

1 w2

2 w2

3 w2

4 w2

5 w2

6 ::::: w2n

]

and

N(t) = UTQ(t) (17)

The model must include some damping [22]. It is convenient

to assume proportional damping: a special type

of viscous damping [23]. The proportional damping model

expresses the damping matrix as a linear combination of the

mass and stiffness matrices, that is:

C = #1M + #2K (18)

Where #1 and #2 are constant scalars. The result is that for

the ith mode:

#i(t) + 2#iwni#_i(t) + w2

ni#i(t) = Ni(t) (19)

The solution of (19) can be written by components in the

form of convolution integrals as follows:

#i(t) =

1

wdi

Z t

0

e??#iwni#Ni(t ?? # )sin(wdi# )d# (20)

where wdi =

p

1 ?? #2

i wni is the damped natural angular

frequency.

Finally using (15), the generalized coordinates are calculated.

The motions of the compliant body are then calculated

using (1).

III. PROTOTYPE DESIGN

A prototype that implements the proposed theoretical

approach is built. Its dimensions are acquired from those

of a sub-carangiform swimmer with the same dimensions.

Fig-2, shows the CAD of the prototype; It consists of:

# a compliant body attached to a passive rigid caudal fin.

The length of the compliant body is 0.22 m and that of

the fin is 0.08m;

# a rigid head accommodating the electronics and a

servomotor used to actuate the compliant body. The

servomotor actuates the compliant body by pulling two

cables attached to the rigid plate casted inside the

flexible body;

# an aluminium part connecting the head and the compliant

body.

The dimensions of the robot are chosen to allow it to

accommodate the electronics and the motor and to swim

5342

1

2

3

6

5

7

4

Fig. 2. CAD view of the fish-like robot. 1-Rigid head of the robot; 2-

Servo-motor; 3-Middle part made from aluminum holding the head, the

compliant body and a servo-motor; 4-Steel cables; 5-Actuation plate; 6-

Compliant body; 7-Rigid fin.

freely in the test tank. The Young�s modulus of elasticity

is chosen experimentally. Trials on compliant bodies with

different modulus are performed. The compliant bodies with

a high modulus of elasticity are hard to deform while those

with a low elasticity don�t generate enough thrust.

A compromise solution is a Young�s modulus of 83Kpa.

The flexible part is casted from commercial platinum cure

silicon rubber Dragon Skin

Slacker

distance between the actuation point and the compliant body

base is chosen to be a=0.07m.

IV. EXPERIMENTS AND RESULTS

In this section, the experiments carried out on the compliant

body are described. These experiments aim to:

# estimate the natural frequencies and damping ratios of

the system;

# validate the theoretical modelling;

# measure the average thrusts and velocities as a function

of actuation frequencies.

A. Parameter Identification

A special experimental setup is used. It consists of (see

Fig.3):

# a compliant body attached to a passive rigid caudal fin;

# six metallic markers attached to the compliant body and

used to track its motions;

# a custom torque sensor;

# a servo motor used for actuation.

# a digital camera filming at a rate of 50 frames per

second.

# a water tank.

To estimate the damping ratios, experiments are carried

out on the compliant body in air. The undamped natural

frequencies are calculated using the approach developed

earlier in this paper. Tab-I summarizes the undamped natural

frequencies in air. An expansion series of order n=6 is used.

The servo-motor is controlled to oscillate in the range of

a given interval [??#;+#] for different actuation frequencies.

For each frequency the compliant body is excited for a given

2

3

1

4

Fig. 3. The experimental setup composed of: 1- An compliant body, 2-

Six metallic markers, 3- A custom torque sensor, 4- a servo motor

TABLE I

UNDAMPED NATURAL FREQUENCIES OF THE COMPLIANT BODY IN AIR

fn1[Hz] fn2[Hz] fn3[Hz] fn4[Hz] fn5[Hz] fn6[Hz]

4.0696 11.1351 33.9573 71.1087 813.4664 17556

number of actuation periods and the maximum value of the

torque is recorded. Two trials are performed: In the first

referred to as Exp-1, the compliant body is actuated using

harmonic torques with different actuation frequencies close

to the first undamped natural frequency fn1. The maximum

values of the torques are then drawn as a function of the

actuation frequency as shown in fig.4. The minimum value on

the graph corresponds to f1=3.3 Hz equal to fn1

p

1 ?? 2#2

1

[24]. The first damping ratio is then calculated as #1 = 0:41.

In the second trial, referred to as Exp-2, the compliant body

is actuated using harmonic torques with different actuation

frequencies close to the second undamped natural frequency

fn2. The maximum values of the torques are then drawn

as a function of the actuation frequency as shown in fig.5.

The minimum value on the graph corresponds to f2=9.96

Hz equal to fn2

p

1 ?? 2#2

2 . The second damping ratio is

then calculated as #2 = 0:3. This approach is not applied to

measure the damping ratios for higher frequencies to prevent

damaging the system. Instead #3, #4, #5 and #6 are assumed

to be equal to #2.

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Frequency[Hz]

Maximum Torque[N.m]

f=3.3Hz

Fig. 4. Maximum torque measured in Exp-1 as a function of actuation

frequency.

To estimate the constant C0 defining the added mass used

to model the hydrodynamic forces, experiments are carried

5343

9.5 10 10.5 11 11.5

0.16

0.165

0.17

0.175

0.18

0.185

0.19

0.195

0.2

0.205

Frequency[Hz]

Maximum Torque[N.m]

f=9.96Hz

Fig. 5. Maximum torque measured in Exp-2 as a function of actuation

frequency.

0.5 1 1.5 2 2.5 3 3.5 4

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

Frequency[Hz]

Maximum Torque [N.m]

f=3.37Hz

Fig. 6. Maximum torque measured in Exp-3 as a function of actuation

frequency.

out on the compliant body in water and are referred to as

Exp-3. The same approach applied to measure the damping

ratio is used. The compliant body is excited with actuation

frequencies close to the second undamped natural frequency

in water. The maximum values of the torques are then

drawn as a function of the actuation frequency as shown

in fig. 6. The minimum value on the graph corresponds to

fn2 = p 3:37

1??2#0:312 . The first undamped natural frequency is

small and not easy to identify. Having fn2, C0 is the constant

that makes the calculated and measured second undamped

frequencies equal. In the present case, C0 is equal to 0.8.

Tab-II summarizes the calculated values of the undamped

natural frequencies of the compliant body in water.

TABLE II

UNDAMPED NATURAL FREQUENCIES OF THE COMPLIANT BODY IN

WATER

fn1[Hz] fn2[Hz] fn3[Hz] fn4[Hz] fn5[Hz] fn6[Hz]

0.7924 3.7495 12.2776 33.0006 457.6453 9634

B. Experimental Model Validation

These experiments are carried out to validate the proposed

theoretical modelling. In this framework, the robot is fixed

in a steady position and the compliant body is actuated by

a known torque. The motion of the midline is tracked using

a video-camera filming at a rate of 50 frames/second. The

videos are then processed manually using Matlab. Torques

with different amplitudes and frequencies are applied to

the compliant body. The measured lateral deflections are

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

?0.04

?0.03

?0.02

?0.01

0

0.01

0.02

0.03

0.04

Time[s]

Lateral deflection[m]

Measured and calculated Lateral deflections and errors resulting for f=0.7Hz

Calculated data

Fitted measured data

Absolute error

0 0.1 0.2 0.3 0.4

?0.015

?0.01

?0.005

0

0.005

0.01

0.015

Time[s]

Lateral deflection[m]

Measured and calculated Lateral deflections and errors resulting for f=2.6Hz

Calculated data

Fitted measured data

Absolute error

Fig. 7. Experimental, calculated lateral deflections and absolute errors in

water of the bottom of the compliant body for M(t)= sin(w*t).

then compared to those calculated by the assumed modes

method. The results show that for large deflections (20% of

the compliant body length) the absolute errors between the

measured and calculated motions are relatively small (around

17% of the compliant body length ) . These errors become

more important in the case of small lateral deflections (5% of

the flexible part length) and are around 40% of the compliant

body length. This is because the tracking is done manually

and in the case of small deflection the imprecision becomes

more important. Fig.7 shows the graphs of the calculated

and measured lateral deflection of the midline�s point at the

bottom of the compliant body during one actuation period

with M = sin(wt).

C. Experiments on the Robot

The experimental setup shown in fig.8 is used to measure

the thrust generated by the compliant body while the robot

being held in a static position on a force plate. Experiments

are carried out while the compliant body is actuated with

different frequencies f and amplitudes M0. Fig.9 shows the

average speed and thrust as a function of actuation frequency

for M0=1Nm. One can see that the speed and thrust increase

with the frequency.

V. CONCLUSION AND FUTURE WORK

This paper describes the design and experiments carried

out on a bio-inspired fish-like robot. It brings many contributions

to the field of compliant underwater robots modelling

and control in particular:

# an analytical approach to model the dynamics of robots

with non-homogeneous compliant parts;

5344

1

2

3

x

Fig. 8. Experimental setup used to measure the static thrust. 1-the biomimetic

fish robot; 2-force sensor; 3-metallic plate

0.1

0.2

0.3

Average Thrust[N]

1 1.5 2 2.5 3

0

0.1

0.2

Frequency[Hz]

Velocity [m/s]

Velocity

Thrust

Fig. 9. Average thrust and velocity as function of actuation frequency

# experimental methods to estimate the internal damping

and hydrodynamic forces;

# a model for the effect of adding a passive rigid fin to

the end of the compliant body;

# a prototype for bio-inspired fish like robot.

Future work should address the problem of adding flexible

parts with variable elasticity to the design to force the system

to vibrate near its natural frequencies and to reduce energy

consumption.

Abstract�This paper studies the modelling, design and

fabrication of a bio-inspired fish-like robot propelled by a

compliant body. The key to the design is the use of a single

motor to actuate the compliant body and to generate thrust. The

robot has the same geometrical properties of a subcarangiform

swimmer with the same length. The design is based on rigid

head and fin linked together with a compliant body. The flexible

part is modelled as a non-uniform cantilever beam actuated by

a concentrated moment. The dynamics of the compliant body

are studied and a relationship between the applied moment and

the resulting motion is derived. A prototype that implements

the proposed approach is built. Experiments on the prototype

are done to identify the model parameters and to validate the

theoretical modelling.

I. INTRODUCTION

Underwater robots provide an engineering tool to practical

applications in marine and military fields, such as monitoring

the environment, harvesting natural resources, undersea

operation, pipe inspection and many other applications.

With millions of years of evolution, aquatic animals, in

particular fish, are very efficient swimmers. This has inspired

scientists to study fish locomotion and build fish-like robots.

MIT�s RoboTuna I and II are the best known bio-inspired

underwater robots [1]. These are tethered robots, mimicking

the thunniform swimmers and use a system of pulleys and

cable tendons actuated by DC-motors. MIT also developed

Robot pike to learn more about the fluid mechanics that fish

use to propel themselves with a purpose to develop small

fish-like autonomous vehicles for reduced energy consumption

and increased operation time [3]. The Vorticity Control

Unmanned Undersea Vehicle (VCUUV) was produced in

Draper Laboratory; it was the first autonomous mission-scale

UUV that utilizes fish-like swimming and manoeuvering

[2]. The University of Essex has developed a series of

autonomous robots G1 to G9 and MT1. The G series have

a multi-motor-multi-joint tail structure, which employs 4

servo motors to drive 4 tail joints separately according to a

predetermined swimming wave sequence [4][5][6][7][8].The

Japenese National Maritime Research Institute developed

many kinds of robotic fish prototypes to increase swimming

efficiency [9].

Most of these designs use rigid links and discrete mechanisms

to achieve fish-like swimming. The complexity of

these systems increases proportionally with the kinematic

This work is supported by European Union 7th Framework program

under FP7-ICT-2007-3 STREP project FILOSE (Robotic FIsh LOcomotion

and SEnsing), www.filose.eu.

Authors are with Center for Biorobotics, Tallinn University

of Technology, Tallinn, Estonia. maarja.kruusmaa,

[email protected]

similarity to fish. An alternative is to use compliant structures;

These bodies can be modelled as dynamically bending

beams [10] whose vibration characteristics are determined by

external and internal forces of the system, which in turn are

related to the geometry, material properties and actuation.

This alternative design concept also has some biological

relevance. The EMG studies of muscle activity of swimming

fish reveal that for swimming at cruising speeds (1 to 2

body lengths per second) fish use mainly anterior muscles

while the posterior part of the body acts like a carrier of the

travelling wave conveying the momentum to the surrounding

fluid [11], [12], [13]. A robotic fish using smart materials for

caudal fin design was developed in Michigan State University

[14] to increase efficiency, focusing on unique physics of

Ionic polymer metal composite (IPMC) materials and its

interaction with the fluid. A subclass of swimmers that

exploits the use of compliant bodies and one servo-motor

for actuation was developed at MIT [15]. It assumes that

a compliant body can be modelled by a cantilever beam

actuated by a single point and studies the dynamics in order

to mimic swimming fish motions.

In this study, the design of a robot with a flexible body

excited by a concentrated moment is studied. It makes use

of the models developed in [16] but differs in many aspects.

In fact, in [16] the mechanical modelling does not take into

account the elasticity of the rigid plate used for actuation and

neglected the hydrodynamic effect on the rigid fin. In this

work, a different approach is used. It takes into account the

non-homogeneity in the material distribution and the effect of

a rigid fin in the end of the compliant body. The dynamics of

the compliant body are studied to find a relationship between

the applied force and the resulting deflections. Moreover in

this study it is believed, that a compliant body cannot be

forced to deform to a random shape but has defined mode

shapes that are determined by the actuation frequencies and

the modal properties of the system.

The objectives of this paper are to:

# study the dynamics of the compliant body and derive the

relationship between the applied forces and the resulting

motion.

# propose a design that implements the proposed approach

and build a fish-like robot prototype.

# identify the model parameters and validate the model

through physical experiments.

The remainder of this paper is organized as follows: In

section II the dynamics of the compliant body are developed.

In section III the prototype of robotic fish is described.

Results from physical experiments on the prototype are

2012 IEEE International Conference on Robotics and Automation

RiverCentre, Saint Paul, Minnesota, USA

May 14-18, 2012

978-1-4673-1405-3/12/$31.00 �2012 IEEE 5340

a

Y

x

L(x,t)

M(t)

l

h(x,t)

fin F (t)

M (t)

fin

Fig. 1. Structural model of the compliant body. a=actuation point, l=length

of the compliant body, M(t)=actuation moment, L(x,t) hydrodynamic distributed

forces, Ffin(t), Mfin(t)=concentrated force and moment resulting

from the hydrodynamic forces acting on the fin and h(x,t)=lateral line

deflection

presented in section IV. Finally, section V discusses the

contributions and future work.

II. DYNAMICS OF THE COMPLIANT BODY

Fig.1 shows the structural model of the compliant body.

The �assumed modes� method is used to derive the equations

of motion of the compliant body [17] [18] [19] [20] [21].

It aims at deriving such equations by first discretizing the

kinetic energy, potential energy and the virtual work and

making use of the Lagrange�s equation of motion. The Elastic

deformations are modelled by a finite series:

h(x; t) =

Xn

r=1

'r(x)qr(t) 0 < x < l (1)

where:

# 'r(x): are known trial functions. The eigenfunctions of

a uniform cantilever beam are chosen as trial functions.

# qr(t): are unknown generalized coordinates.

# n: is the order of the expansion.

# l: is the length of the compliant body.

Considering that the passive fin is rigid compared to the

compliant body, its lateral deflection h(x,t) can be expressed

as:

h(x; t) = h(l; t) +

@h(l; t)

@x

(x ?? l) l < x < l + #l

=

Xn

i=1

('i(l)qi(t) + '

0

i(l)qi(t)(x ?? l)) (2)

where #l is the length of the rigid fin.

The external forces acting on the compliant body are: the

time varying moment M(t), the distributed hydrodynamic

forces L(x,t), the concentrated moment Mfin(t) and the

concentrated force Ffin(t). Mfin(t) and Ffin(t) are the

concentrated moment and force resulting from the action of

the hydrodynamic forces on the rigid fin. The hydrodynamic

forces are modelled in terms of added mass and expressed

as:

L(x; t) = D(m(x)_ h(x; t)) # m(x)

@2h

@t2 (3)

where m(x) is the apparent mass of the cross section per unit

length. It is approximated by m(x)=C0#f A(x) where C0 is

a constant that can be determined experimentally, #f is the

fluid density and A(x) is the cross area of a fluid cylinder

surrounding the body at x.

The concentrated force Ffin(t) is:

Ffin(t) = ??

Z l+#l

l

m(x)

@2h

@t2 dx = ??

Xn

i=1

qi(t)[#'i(l)+

'

0

i(l)]

(4)

where:

# =

Z l+#l

l

m(x)dx

=

Z l+#l

l

m(x)(x ?? l)dx (5)

The concentrated moment Mfin(t) is:

Mfin(t) = ??

Z l+#l

l

m(x)

@2h

@t2 (x ?? l)dx

= ??

Xn

i=1

qi(t)[

'i(l) +

0

i(l)] (6)

where:

Z l+#l

l

m(x)(x ?? l)2dx (7)

The kinetic and potential energies can be written as:

T(t) =

1

2

Z l

0

#(x)(

@h(x; t)

@t

)2dx

=

1

2

Xn

i=1

Xn

j=1

q_i(t)q_j(t)

Z l

0

#(x)'i(x)'j(x)dx (8)

V (t) =

1

2

Z l

0

EI(x)(

@2h(x; t)

@x2 )2dx

=

1

2

Xn

i=1

Xn

j=1

qi(t)qj(t)

Z l

0

EI(x)'

00

i (x)'

00

j (x)dx (9)

#(x) and EI(x) are the mass per unit length and the

stiffness at x respectively.

The total virtual work can be expressed as:

#W = #W1 + #W2 + #W3 + #W4

where #W1 is the virtual work of M(t):

#W1 =

Z l

0

M(t)#

��

(x ?? a)#h

0

(x; t)dx =

Xn

j=1

M(t)'

0

j(a)#qj

(10)

#W2 is the virtual work of L(x,t):

#W2 = ??

Z l

0

L(x; t)#h(x; t)dx = ??

Z l

0

m(x)

@2h

@t2 #h(x; t)dx

= ??

Xn

i=1

Xn

j=1

qi(t)

Z l

0

m(x)'i(x)'j(x)dx#qj (11)

5341

#W3 is the virtual work of Ffin:

#W3 =

Z l

0

F(t)#

��

(x ?? l)#h(x; t)dx

= ??

Xn

i=1

Xn

j=1

qi(t)[#'i(l)'j(l) +

'

0

i(l)'j(l)]#qj (12)

#W4 is the virtual work of Mfin(t):

#W4 =

Z l

0

Mfin(t)#

��

(x??l)#h

0

(x; t)dx =

Xn

j=1

Mfin(t)'

0

j(l)#qj

= ??

Xn

i=1

Xn

j=1

qi(t)[

'i(l)'

0

j(l) +

0

i(l)'

0

j(l)]#qj (13)

#��

denotes the Dirac delta function and h

0

(x; t) =

@

@xh(x; t).

The Lagrange�s equations are used to write the equations

of motion of the approximate system:

[M]fq(t)g + [K]fq(t)g = fQ(t)g (14)

where:

mij =

Z l

0

(#(x) + #fA(x))'i(x)'j(x)dx + #'i(l)'j(l)

+

('

0

i(l)'j(l) + 'i(l)'

0

j(l)) +

0

i(l)'

0

j(l)

kij =

Z l

0

EI(x)'

00

i (x)'

00

j (x)dx

and

Qi(t) = ??

Z l

0

M(t)#

��

0

(x ?? a)'i(x)dx

To find the response of (14), the eigenvalue problem is

first solved introducing:

q(t) = ae#(t)

This leads to the following characteristic equation:

det(#2M + K) = 0

where #r = ??iwr and wr are the undamped natural

frequencies of the approximate system. To obtain the solution

of (14), the following linear transformation is used:

q(t) = U#(t) (15)

where:

U = [a1 a2 a3 a4 a5 :::::::: an]

Where an is the eigenvector associated with the eigenvalue

#n = ??iwn. The eigenvectors are orthogonal with respect

to the mass and stiffness matrices. They are normalized to

yield:

aTr

:M:as = #rs aTr

:K:as = w2

r#rs

Where #rs is defined as the Kronecker delta.

Introducing (15) in (14) and premultiplying by UT , the

independent modal equations are then obtained:

#(t) + ##(t) = N(t) (16)

in which :

# = diag[w2

1 w2

2 w2

3 w2

4 w2

5 w2

6 ::::: w2n

]

and

N(t) = UTQ(t) (17)

The model must include some damping [22]. It is convenient

to assume proportional damping: a special type

of viscous damping [23]. The proportional damping model

expresses the damping matrix as a linear combination of the

mass and stiffness matrices, that is:

C = #1M + #2K (18)

Where #1 and #2 are constant scalars. The result is that for

the ith mode:

#i(t) + 2#iwni#_i(t) + w2

ni#i(t) = Ni(t) (19)

The solution of (19) can be written by components in the

form of convolution integrals as follows:

#i(t) =

1

wdi

Z t

0

e??#iwni#Ni(t ?? # )sin(wdi# )d# (20)

where wdi =

p

1 ?? #2

i wni is the damped natural angular

frequency.

Finally using (15), the generalized coordinates are calculated.

The motions of the compliant body are then calculated

using (1).

III. PROTOTYPE DESIGN

A prototype that implements the proposed theoretical

approach is built. Its dimensions are acquired from those

of a sub-carangiform swimmer with the same dimensions.

Fig-2, shows the CAD of the prototype; It consists of:

# a compliant body attached to a passive rigid caudal fin.

The length of the compliant body is 0.22 m and that of

the fin is 0.08m;

# a rigid head accommodating the electronics and a

servomotor used to actuate the compliant body. The

servomotor actuates the compliant body by pulling two

cables attached to the rigid plate casted inside the

flexible body;

# an aluminium part connecting the head and the compliant

body.

The dimensions of the robot are chosen to allow it to

accommodate the electronics and the motor and to swim

5342

1

2

3

6

5

7

4

Fig. 2. CAD view of the fish-like robot. 1-Rigid head of the robot; 2-

Servo-motor; 3-Middle part made from aluminum holding the head, the

compliant body and a servo-motor; 4-Steel cables; 5-Actuation plate; 6-

Compliant body; 7-Rigid fin.

freely in the test tank. The Young�s modulus of elasticity

is chosen experimentally. Trials on compliant bodies with

different modulus are performed. The compliant bodies with

a high modulus of elasticity are hard to deform while those

with a low elasticity don�t generate enough thrust.

A compromise solution is a Young�s modulus of 83Kpa.

The flexible part is casted from commercial platinum cure

silicon rubber Dragon Skin

Slacker

distance between the actuation point and the compliant body

base is chosen to be a=0.07m.

IV. EXPERIMENTS AND RESULTS

In this section, the experiments carried out on the compliant

body are described. These experiments aim to:

# estimate the natural frequencies and damping ratios of

the system;

# validate the theoretical modelling;

# measure the average thrusts and velocities as a function

of actuation frequencies.

A. Parameter Identification

A special experimental setup is used. It consists of (see

Fig.3):

# a compliant body attached to a passive rigid caudal fin;

# six metallic markers attached to the compliant body and

used to track its motions;

# a custom torque sensor;

# a servo motor used for actuation.

# a digital camera filming at a rate of 50 frames per

second.

# a water tank.

To estimate the damping ratios, experiments are carried

out on the compliant body in air. The undamped natural

frequencies are calculated using the approach developed

earlier in this paper. Tab-I summarizes the undamped natural

frequencies in air. An expansion series of order n=6 is used.

The servo-motor is controlled to oscillate in the range of

a given interval [??#;+#] for different actuation frequencies.

For each frequency the compliant body is excited for a given

2

3

1

4

Fig. 3. The experimental setup composed of: 1- An compliant body, 2-

Six metallic markers, 3- A custom torque sensor, 4- a servo motor

TABLE I

UNDAMPED NATURAL FREQUENCIES OF THE COMPLIANT BODY IN AIR

fn1[Hz] fn2[Hz] fn3[Hz] fn4[Hz] fn5[Hz] fn6[Hz]

4.0696 11.1351 33.9573 71.1087 813.4664 17556

number of actuation periods and the maximum value of the

torque is recorded. Two trials are performed: In the first

referred to as Exp-1, the compliant body is actuated using

harmonic torques with different actuation frequencies close

to the first undamped natural frequency fn1. The maximum

values of the torques are then drawn as a function of the

actuation frequency as shown in fig.4. The minimum value on

the graph corresponds to f1=3.3 Hz equal to fn1

p

1 ?? 2#2

1

[24]. The first damping ratio is then calculated as #1 = 0:41.

In the second trial, referred to as Exp-2, the compliant body

is actuated using harmonic torques with different actuation

frequencies close to the second undamped natural frequency

fn2. The maximum values of the torques are then drawn

as a function of the actuation frequency as shown in fig.5.

The minimum value on the graph corresponds to f2=9.96

Hz equal to fn2

p

1 ?? 2#2

2 . The second damping ratio is

then calculated as #2 = 0:3. This approach is not applied to

measure the damping ratios for higher frequencies to prevent

damaging the system. Instead #3, #4, #5 and #6 are assumed

to be equal to #2.

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Frequency[Hz]

Maximum Torque[N.m]

f=3.3Hz

Fig. 4. Maximum torque measured in Exp-1 as a function of actuation

frequency.

To estimate the constant C0 defining the added mass used

to model the hydrodynamic forces, experiments are carried

5343

9.5 10 10.5 11 11.5

0.16

0.165

0.17

0.175

0.18

0.185

0.19

0.195

0.2

0.205

Frequency[Hz]

Maximum Torque[N.m]

f=9.96Hz

Fig. 5. Maximum torque measured in Exp-2 as a function of actuation

frequency.

0.5 1 1.5 2 2.5 3 3.5 4

0.38

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

Frequency[Hz]

Maximum Torque [N.m]

f=3.37Hz

Fig. 6. Maximum torque measured in Exp-3 as a function of actuation

frequency.

out on the compliant body in water and are referred to as

Exp-3. The same approach applied to measure the damping

ratio is used. The compliant body is excited with actuation

frequencies close to the second undamped natural frequency

in water. The maximum values of the torques are then

drawn as a function of the actuation frequency as shown

in fig. 6. The minimum value on the graph corresponds to

fn2 = p 3:37

1??2#0:312 . The first undamped natural frequency is

small and not easy to identify. Having fn2, C0 is the constant

that makes the calculated and measured second undamped

frequencies equal. In the present case, C0 is equal to 0.8.

Tab-II summarizes the calculated values of the undamped

natural frequencies of the compliant body in water.

TABLE II

UNDAMPED NATURAL FREQUENCIES OF THE COMPLIANT BODY IN

WATER

fn1[Hz] fn2[Hz] fn3[Hz] fn4[Hz] fn5[Hz] fn6[Hz]

0.7924 3.7495 12.2776 33.0006 457.6453 9634

B. Experimental Model Validation

These experiments are carried out to validate the proposed

theoretical modelling. In this framework, the robot is fixed

in a steady position and the compliant body is actuated by

a known torque. The motion of the midline is tracked using

a video-camera filming at a rate of 50 frames/second. The

videos are then processed manually using Matlab. Torques

with different amplitudes and frequencies are applied to

the compliant body. The measured lateral deflections are

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

?0.04

?0.03

?0.02

?0.01

0

0.01

0.02

0.03

0.04

Time[s]

Lateral deflection[m]

Measured and calculated Lateral deflections and errors resulting for f=0.7Hz

Calculated data

Fitted measured data

Absolute error

0 0.1 0.2 0.3 0.4

?0.015

?0.01

?0.005

0

0.005

0.01

0.015

Time[s]

Lateral deflection[m]

Measured and calculated Lateral deflections and errors resulting for f=2.6Hz

Calculated data

Fitted measured data

Absolute error

Fig. 7. Experimental, calculated lateral deflections and absolute errors in

water of the bottom of the compliant body for M(t)= sin(w*t).

then compared to those calculated by the assumed modes

method. The results show that for large deflections (20% of

the compliant body length) the absolute errors between the

measured and calculated motions are relatively small (around

17% of the compliant body length ) . These errors become

more important in the case of small lateral deflections (5% of

the flexible part length) and are around 40% of the compliant

body length. This is because the tracking is done manually

and in the case of small deflection the imprecision becomes

more important. Fig.7 shows the graphs of the calculated

and measured lateral deflection of the midline�s point at the

bottom of the compliant body during one actuation period

with M = sin(wt).

C. Experiments on the Robot

The experimental setup shown in fig.8 is used to measure

the thrust generated by the compliant body while the robot

being held in a static position on a force plate. Experiments

are carried out while the compliant body is actuated with

different frequencies f and amplitudes M0. Fig.9 shows the

average speed and thrust as a function of actuation frequency

for M0=1Nm. One can see that the speed and thrust increase

with the frequency.

V. CONCLUSION AND FUTURE WORK

This paper describes the design and experiments carried

out on a bio-inspired fish-like robot. It brings many contributions

to the field of compliant underwater robots modelling

and control in particular:

# an analytical approach to model the dynamics of robots

with non-homogeneous compliant parts;

5344

1

2

3

x

Fig. 8. Experimental setup used to measure the static thrust. 1-the biomimetic

fish robot; 2-force sensor; 3-metallic plate

0.1

0.2

0.3

Average Thrust[N]

1 1.5 2 2.5 3

0

0.1

0.2

Frequency[Hz]

Velocity [m/s]

Velocity

Thrust

Fig. 9. Average thrust and velocity as function of actuation frequency

# experimental methods to estimate the internal damping

and hydrodynamic forces;

# a model for the effect of adding a passive rigid fin to

the end of the compliant body;

# a prototype for bio-inspired fish like robot.

Future work should address the problem of adding flexible

parts with variable elasticity to the design to force the system

to vibrate near its natural frequencies and to reduce energy

consumption.