Star Fish2 Nus 021511

Study of Modular Undulating Fin Rays in Bio-inspired Robotic Fish.

Why Fins and not propellers?
Propeller strikes produce greater amount of marine debris,
marine creatures mortality and shallow waters ecosystem
disturbance
Broadband noise have severe acoustic effects on marine wildlife
Maneuvering ships reduce their speed by more than 50 percent,
and their turning radius is at least 10 times larger than
corresponding, value for fish
Fish-like robots are expected to be quieter, more maneuverable
(lesser accidents), and possible more energy efficient (longer
missions) .

Piscine propulsion vs. Propellers
Undulating-finned robot preserve undisturbed condition of its
surroundings for data acquisition and exploration (stealth).
Fish can reverse direction without slowing down and turning
radius only 10 to 30 percent of length of body.
At low speeds, traditional propeller propulsion mechanisms can
have efficiencies below 50%, while energy efficiency of fish
propulsion in nature is estimated to range from 70% to 90%
Able to stay still in currents. Moves through water without
creating ripples and eddies.

Propeller Mechanism
Projection of mass of water in a direction opposite to that of required vessel
motion.
Relation between mass of water acted upon and acceleration imparted to it
should be such that product shall equal estimated resistance of ship, and size
and rate of motion of propelling apparatus
In a motor-driven craft, efficiency is ratio of useful power (thrust times forward
velocity) divided by power expended by motor to drive foil or propeller.
Efficiency is always less than one, because some of motor’s power is wasted
in wayward vortices and other undesirable turbulence as well as heat.
For performance, most important factor is propulsor efficiency at reasonably
high levels of thrust.

Hydrodynamics of Piscine Propulsion
Any object in a flow, like a swimming swordfish, creates a trail of spinning vortices.
Tail of a fish pushes water backward (jet),set a column of moving fluid that includes thrust-producing vortices.
These Jet vortices play Fig 1 central role in generation of thrust, and their optimal formation would increase efficiency tremendously.
Fig 1

Vortices / Strouhal Number
Fluid-dynamics parameter known as Strouhal number to study Vortices.
Product of frequency of vortex formation behind an object in a flow and width of wake, divided by speed of the flow. For a swimming fish, Strouhal number is defined as product of frequency of tail and width of Jet, divided by the speed of fish.
Number indicates, how often vortices are created in wake and how close they are.
Ratio remains constant at about 0.2 for a variety of flow conditions and object shapes.
Analyzed data from flapping foils, indicate that thrust-inducing vortices form optimally when Strouhal number lies between 0.25 and 0.35.
Efficiency should be at a maximum for these values.

Fish Swimming Modes
Classification of fish is based on two main factors
Extend to which propelling process is based on undulatory motion versus oscillatory motion, and
Body structures or fin segments that contribute most in generating propulsion.
Fish swim either by using
Body and/or caudal fin (BCF) locomotion,
Median and/or paired fin (MPF) locomotion,
Combination of both BCF and MPF locomotion .

MPF Propulsion
Swimming types identified by the MPF propulsion
Fig 2

Rajiform Mode
Fin propulsion is generated by passing vertical undulations along wide pectorals with increasing amplitude from anterior part to fin apex and tapers again towards posterior Fig. 3.
Mostly, body of fish is held straight when swimming.
Fig 3

Typical examples are stingrays, skates and mantas, characterized by large, triangular-shaped and flexible pectoral fins
Fig 4

Other Swimming Forms
Propulsion in Amiiform is achieved by undulations of a long-based dorsal fin with body held straight
Propulsion in Gymnotiform is by undulations of a long-based anal fin. Example knifefish does not have dorsal and caudal fins .
Fig 6 Fig 5

Fig. 7 shows fin diagram of any fish, including that of cuttlefish, performing undulations.
Universal Joint that permits two degree of freedom movements at base of each fin ray.
Fig 8 Fig 7

Mechanical Modeling of Undulating Fins
Motion in one degree of freedom from originally two degrees of freedom at base of each fin ray.
A servomotor serves as a muscle producing one degree of freedom at base of each ray. A crank is attached at each servomotor to function as a fin ray.
Fig 9 (a)

To exhibit undulations similar to any undulating fin, each servomotor is programmed so that crank attached to servomotor oscillates based on a sinusoidal function with a specified phase lead or lag defined by β
Fig 9 (b)

Undulating Fin Mechanism Models- Robotic ribbon fin by the Northwestern University Fig 10

Squid-type underwater vehicle by the Osaka University Fig 11

Nanyang Knifefish (NKF-I) robot by the NTU Fig 12

Modeling of Fin Mechanism NTU
Produce undulation motion by virtue of designed crank-slider linkages.
Complete fin mechanism is able to provide various waveform shapes.
Fig 13 Fig 14

Crank Slider Linkages
Fin consists of specified number of servomotors.
Each of them drives a crank that is connected to slider.
Sliders can retract and extend on their own within an allowable length
Fig 15

Kinematics Diagram
Rotational Axis is perpendicular to Longitudinal wave direction
With slider, Amplitude of fin motion can be varied.
Fig 16 Fig 17

Kinematics Modeling of Fin rays
Modeled as a ruled surface in 3-D space.
Fin baseline is directrix of the ruled surface, while fin ray is generatrix.
Fig 18

Undulation can be generated through a sequential oscillating of generatrix on ruled surface.
Given by ruled-surface expression Fig 18.
P (r, s, t) = b (s, t) + r.d (s) .c (s, t) , 0 ≤ r ≤ 1
b (s, t) is fin base curve that describes change of respective fin
ray’s starting point along fin base,
c (s, t) is a time-varying vector overlapping fin ray at y = s
d (s) is length of fin ray at y = s,
t is time
r is normalized
… (1)

In order to model multi-degree-of-freedom propulsor / undulating fin motion function used :
x is amplitude of undulating wave,
y-axis is centre line of wave,
c 0 is profile of undulating fin at initial or starting point.
c 1 is linear wave amplitude envelope
c 2 is quadratic wave amplitude envelope,
α is wave frequency
k = 2 π / λ is wave number
… (2)

Discrete Model of Sinusoidal waveforms
Crank’s root point at horizontal line shows position of each respective servomotor.
Lines away from root points represent respective cranks (example, ac and db)
Lines connecting tip points of cranks form resulting fin wave, which is pushing away water to provide locomotion.
Basic sinusoidal function, formed by roots of fin rays oscillating at same frequency but out of phase .
Fig 19

Waveform of Fin Rays generated by Servomotors and Inter-Connected Sliders
R is crank length,
L is distance between two servomotors,
S is length of slider,
S min minimum (when slides are fully retracted),
S max maximum (when sliders are fully extended),
θ 1 angular position of a crank attached to 1st servomotor,
θ 2 angular position of a crank attached to 2nd servomotor.
Fig 20

Basic Layout for the Waveform
A is amplitude of a resulting sinusoidal movement,
x is vertical height from crank end to reference line,
θ i is angular position of a crank attached to i th servomotor, where i = 1, 2,. . . ,m, where m is total number of motors
Fig 21

To provide an arbitrary and predictable fin wave (a fin profile), with arbitrary amplitudes along fin, at each individual segment by controlling actuator angles, θ n .
Joint variables θ n are related to wave amplitude, crank length and phase angle γ n
Substitution yields
… (3) … (4) … (5)

Joint Control Law to drive Servomotors
Since γ n rotates reciprocately, we can assume that it subjects to following function phase angle γ 2 is only variable
Substituting general expression for angular position θ n (t) of a crank attached to n th servos at time ‘t’ can be written in nonlinear form as
n is servomotor number along longitudinal axis,
α is fin undulating frequency,
β = γ n - γ n1 is phase difference (difference of the two adjacent
phase angles),
A n is amplitude of n th undulating fin crank
… (6) … (7)

Use fin segment to generate a sinusoidal wave, say, a harmonic wave.
Resulting wave can x be interpreted as vibrations of infinite successive particles.
λ is wavelength of whole undulating fin
Distal end point of crank, B, acts as oscillating particle to generate harmonic wave described above.
Independent variable y will then be discretized as
… (8) … (9)

Wave equation in discrete form associated to distal end of individual cranks as
If crank rotates at a constant speed, projection of point B on x axis satisfies a harmonic vibration equation, which provides a harmonic wave along undulating fin segments.
Control law for a single motor in one vibration cycle can be simply expressed by linear equations as
where
in which θ max is extreme position of crank, which is determined by waveform amplitude A
… (12) … (10) … (11)

Parametric Study of Workspace
Workspace: Space generated by motion of two cranks with parameters defined Fig 22.
Parameter study is concerned with fin design to provide an arbitrary sinusoidal movement with amplitudes as larger as possible with phase range of 30–90 deg
All crank angles in series are in the range of -90 to +90 ; Three designated positions have been marked
Fig22

Slider moves smoothly within working area, position P1,
Slider reaches end of track when it hits blue lines, position P2 (left-right boundaries),
Slider gets disconnected when hitting red lines (up-down boundaries), position P3.
Fig 23

Effect of Phase Difference β on fin waveforms
Assuming L = 70 mm, R = 60 mm, and A = 41 mm.
For a suitable fin design, phase difference β should be between 30 deg and 90 deg to generate a sinusoidal locomotion.
Fig 24 Illustrates workspace and wave profiles of fin segments in terms of various phase differences β
Motion with phase difference, β = 10°. Not a sinusoidal movement of fin locomotion
Fig24(a)

Workspace and Waveform of the respective Fin segments
Motion with phase difference, β = 30°. Producing a sine wave of 180°, suitable for fin locomotion
Motion with phase difference, β = 90°. Producing one and a half unsmooth sine wave.
Motion with phase difference, β = 120°. The wave produced is not a sinusoidal movement
Fig24(d) Fig24(c) Fig24(b)

Resulted workspace of the selected fin dimension with five different phase differences (30–90 deg).
Fig 25

Effect of Amplitude on Fin Waveforms
Different amplitude trends of crank series will generate different
sinusoidal waveforms.
Change in amplitude (with same phase difference) will have a
significant effect on waveform, i.e. larger amplitude will lead to
bigger waveform, and vice versa.
Maximum allowable amplitude will depend on allowable
workspace

Fin sinusoidal waves vs Amplitudes
Maximum allowable amplitude will depend on allowable workspace
Fig26(a) Fig26(b) Fig26(c)

Effect of sliders length on Fin Waveforms
Length of slider S between two cranks is changeable by combination of two independent joint angles.
Length of slider S can be derived Fig 20 as follows:
in which
… 13 … 14 … 15

S in terms of two crank angles, range of angles θ 1
and θ 2 is between -90 and +90 deg.
Larger crank angle will produce higher amplitude.
Increase of crank angle will however decrease allowable phase
difference, which constrains performance of whole fin.
Devise a method to balance both of them.
… 16

Workspace Area Ratio
Area ratio is introduced to compare different workspaces for a comprehensible solution
Q is a constant (Q = 180) and P (P = min {P1, P2}) is the smaller length to determine the working square area (P2).
Fig 27

With area ratio η , find out a set of parameters of single fin segment by investigating different workspaces Table 1.
Ratio implies how much usable area of workspace, limits set as P1, P2 and P3
Maximum ranges can be obtained for a given fin waveform in terms of joint angles
Table 1

Specifications of NKF-II .
Fin dimensions obtained in parametric study,
L = 70 mm and R = 60mm, used for all cranks.
Fish robot comprises of three individual modules:
Buoyancy tank module,
Motor compartment module,
Undulating fin module
Control for buoyancy tank and undulating fins require μ P one BasicX-24p μ P while undulating fins required another BasicX-24p μ P and one servomotor controller – Servo 8 T.
Power source for two systems consists of two separate 7.2 V 3A-batteries in order to prevent problems in transmission of data and signals between microprocessors.

Fully Assembled Nanyang Knifefish II (NKF-II) Fig 28 Fig 29 Nanyang Knifefish II Detailed View of crank’s dimension

Features
Modular and scalable design, various types of bio-mimetic can be modeled by different number and arrangement of undulating fin(s).
Modular concept enables to easily and conveniently construct various bio-mimetic fish robots swimming by fin undulations in different forms, while re-configurable assembly allows us to construct fish robots in different forms
Approximation of Sinusoidal (Fin) wave with adjustable amplitudes.
Able to attach single or multiple sets of fin spines to any position of fish body
.

Limitations /Improvements Requ ired
Scenarios introducing Errors
Distal end point of cranks doesn’t perfectly fit a sinusoidal wave at a specified time
Connecting segment between two cranks is a straight and rigid link, which is unable to resemble a sinusoidal curve.
Actual path of distal point of crank is a circular arc, which is not exactly same as desired vertical path (dash line) shown.
Fig 29

Basis of Assumptions are not given:
Larger amplitude of fin motion provides higher thrust force
in workspace q = 180
In following computation, crank length and distance
between servomotors are selected as R = 60 mm and L =
70 mm

Exploring Research Areas
Understand impact of design parameters to robot’s performance and efficiency.
Investigating different undulating layouts depicted for more energy efficiency.
Hydrodynamics incorporated for efficient swimming and a better controlling of fish robot response to external disturbances.
Improve design of existing prototype for compactness, easier waterproofing, energy-saving fin motion, etc.
Useful body/fin materials, effective payload capability, communication, and team coordination of fish robot

Study of Swarm Intelligence and other Control strategies can provide insight in designing bio-mimetic underwater robots for complex group coordination tasks.
Hybrid driving system with smart materials and servomotors for Actuation,
Passive or elastic components (EAP) could be a way to optimize energy application of flexible fin materials.

Videos
Northwestern Uni Knifefish
Osaka Uni SquidFish
NTU RoboFish 1
NTU RoboFish 2

References
Low K H: Modelling and parametric study of modular undulating fin rays for fish robots . Mechanism and Machine Theory , 2009, 44(3): pp.615–632.
Low K H and Willy A: Development and Initial Experiments of NTU Robotic Fish with Modular Fins Proceedings of 2005 IEEE International Conference on Mechatronics and Automation ( ICMA2005 ), Niagara Falls, Canada, Jul-Aug 2005, pp. 958-963.
Jing-Fa, T., & Chung-Wen, L. (2002). Study on the resistance and propulsion performance of biomimetic autonomous underwater vehicle. Proceedings of the 2002 International Symposium on Underwater Technology ,pp. 167-171.
Biomemetic Robotic Swimming www.stanford.edu/~hoffert/projects/robots/robots.pdf

Star Fish2 Nus 021511

by Bhuneshwar Prasad

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