A Bio Inspired Compliant Robotic Fish Engineering Essay

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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)
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#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.

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