Development of a Shape-Memory-Alloy Actuated Biomimetic Hydrofoil

O.K. Rediniotis‡, L. N. Wilson†, D.C. Lagoudas* and M.M. Khan†

Aerospace Engineering DepartmentTexas A&M University, College Station, Texas 77843

ABSTRACT

The development and testing of a biomimetic active hydrofoil that utilizes Shape

Memory Alloy (SMA) actuator technology is presented. This work is the second stage in the

development of a vehicle that has a skeletal structure similar to that of aquatic animals and SMA

actuators for muscles. The current work describes the development and testing of a six-segment

demonstration vehicle and the control schemes used. Each SMA actuation element consists of a

thin wire that joins together two adjacent vertebrae segments of the hydrofoil skeleton and

induces relative movement of one with respect to the other. Controlled heating and cooling of the

wire sets generates bi-directional rotation of the vertebrae, which in turn causes a change in the

shape of the hydrofoil. Each SMA wire is embedded in an elastic water channel that facilitates

fast active SMA cooling via forced water circulation. This hydrofoil was able to deform to

several shapes mimicking aquatic animal swimming, with controlled oscillation frequencies of

up to 1 Hz, with _ Hz oscillation producing the largest body motion amplitudes.

‡ Associate Professor* Professor,† Research Assistant

INTRODUCTION / BACKGROUND

General

In the area of underwater vehicle design, the development of highly maneuverable

vehicles is presently of interest with their design being based on the undulatory swimming

techniques and anatomic structure of fish. This, and similar pursuits in the field of air vehicles

have triggered the emergence of the science of biomimetics, which is the study of natural

systems in order to improve the design and functionality of synthetic systems.

The study of fish motion dates back to work by Lighthill1 who applied the slender body

theory of hydrodynamics to oscillatory motions of slender fish, resulting in the Elongated Body

Theory (EBT). This study revealed the high propulsive efficiency of fish, a finding that alone

renders the utilization of similar propulsive techniques in man-made vehicles a very attractive

quest. To pursue this idea, several studies have been conducted on the effect of fin appendages,

the dynamics of slender fish, and the propulsion mechanisms of fish motion1-4. A more detailed

literature survey can be found in Garner et al.5 The leading work of the Triantafyllou brothers

and their students should be mentioned here. They studied the fluid mechanics aspects of fish

propulsion and control6 and developed conventionally actuated biomimetic vehicles (RoboTuna7

and Robot Pike8).

Actuation Needs for a Biomimetic Underwater Vehicle

While the work in the existing literature has been crucial in developing an understanding

of the hydrodynamics of fish–like swimming, most of it is limited with regards to the actuation

methods used. The devices used in the previous work utilized conventional systems of gears and

servomotors to provide the actuation power, which leave little interior room in the device for

control systems and payload. In actuator technology, active or “smart” materials have opened

new horizons in terms of actuation simplicity, compactness and miniaturization potential. Shape

Memory Alloy (SMA) actuation is the most suited for the current application. As an illustration

of the benefits of SMA actuation, let us consider an underwater vehicle on the order of 2 meters

in length. Suppose that the vehicle is to be propelled by a caudal-fin equivalent system and the

ratio (caudal-fin trailing-edge excursion)/(vehicle length) is maintained equal to an average value

observed in the propulsion of its aquatic counterparts (around 0.22). For the above data, and

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underwater vehicle speeds between 3 and 7 knots (1.5 m/sec to 3.5 m/sec), caudal-fin oscillation

frequencies will range from 0.7 Hz to 2.5 Hz. The latter, combined with the fact that double-

amplitude trailing-edge excursions will be on the order of 0.5m, renders SMA-based actuation

ideal for the particular application. The high-frequency-response capabilities of piezoelectric and

magnetostrictive materials (in the kHz range) are not required in this application. Further, these

materials are only capable of producing small strains/displacements (small fractions of 1%)

compared to those attained by SMA materials (as high as 8% for one-way trained and 4% for

two-way trained SMAs). Another important consideration is that piezoelectric and

magnetostrictive materials have energy densities that are two to three orders of magnitude

smaller than those of SMAs. Hence, SMAs have the potential to act as compact powerful

actuators, and are excellent candidates for active or “smart” structure applications.

The small size and high force capabilities of SMAs are more likely (compared to

conventional actuators) to result in availability of a much larger percentage of the vehicle

internal volume. This greater internal volume allows for a larger payload for comparably sized

vehicles9,10. Additional SMA advantages include: (a) Simplicity of actuation mechanism. SMAs

can be all-electric devices and can be used as "Direct Drive Linear Actuators" with little or no

additional motion reduction or amplification hardware. These merits permit the realization of

small or even miniature actuation systems in order to overcome space limitations. (b) Silent

actuation. Since no acoustic signature is associated with such a propulsion system, acoustic

delectability will be drastically reduced. (c) Low driving voltages. Nitinol (NiTi) SMAs can be

actuated with very low voltages (10 to 20V), thus requiring very simple power driving hardware.

In contrast, piezoceramic materials typically require voltages on the order of 100V for their

actuation. However, SMA actuation is not free of drawbacks. Some of them, such as actuation

loss and fatigue over repetitive cyclic loading, are in the frontier of SMA research, while others,

such as low energy efficiency and relatively large current requirements, have been less intensely

focused upon.

Use of Shape Memory Alloys as Actuators

SMAs are materials capable of changing their crystallographic structure, due to changes

in temperature and/or stress, from a high temperature austenite to a low temperature martensite.

These changes are reversible and referred to as martensitic phase transformations, which are

diffusionless in nature11. In general, characterization of phase transformation is done by

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transformation temperatures, namely martensite start (Ms), martensite finish (Mf), austenite start

(As) and austenite finish (Af) temperatures. The SMAs used in this work are K type NiTiCu

alloys from Memry Corporation. Nitinol (NiTi) was the first shape memory alloy to be

commercially used and was discovered in 196512. NiTi based SMAs with additional alloying

elements have since been introduced in the market13,14. Alloying with other elements modifies

both mechanical and transformation properties of SMAs. Excess nickel strongly depresses the

transformation temperatures and increases the yield strength of the austenite. Other frequently

used elements are iron and chromium (to lower the transformation temperatures), and copper (to

decrease the hysteresis and lower the deformation stress of the martensite)13. The K type NiTiCu

was selected, because in addition to having low hysteresis which is desirable for an actuator

application, they can be actuated between 40 and 70°C, which can easily be achieved in a lab

environment.

The behavior of SMAs is more complex than many common materials: the stress-strain

relationship is non-linear, hysteretic, exhibits large reversible strains, is strongly temperature

dependent and sensitive to number and sequence of thermomechanical loading cycles. The key

characteristics of SMAs are pseudoelasticity, one-way shape memory effect (OWSME) and two-

way shape memory effect (TWSME)11. The pseudoelastic behavior of SMAs is defined as being

able to induce martensitic phase transformation by a purely mechanical loading (stress induced

martensite), while the material is in austenite phase. The mechanical loading, forces martensitic

variants to reorient (detwin) into a single variant, resulting in relatively large strains and these

strains can be fully recovered upon unloading11,15. This behavior is observed at temperatures

above austenite finish temperatures. Phase transformations are characterized by a change in the

tangent stiffness of the material and provide opportunities for SMAs to be used as damping and

vibration isolation devices16,17,18.

The ability of SMAs to memorize their original configuration after they have been

deformed in the martensitic phase, by heating the alloys above their characteristic transition

temperature (Af) hence recovering large strains, is known as the shape memory effect (SME)11

and is utilized in most SMA actuator applications19-23.

SMAs undergo a change in crystal structure from a parent, cubic austenitic (A) phase to

a number of martensitic (M) variants upon cooling24. The reverse phase change occurs, although

with some hysteresis, upon increasing the temperature above Af. These phase changes are

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accompanied by significant deformations and, when suitably constrained, produce large

actuation forces. Consider an SMA specimen that is first deformed, via appropriate loading, in

the martensitic phase at temperatures less than austenite start As resulting in detwinning of the

multiple variants of martensite into a single variant, and is subsequently unloaded. The specimen

will return to its original shape in the austenitic phase if the temperature is raised above austenite

finish Af. The above process is called one-way shape memory effect (OWSME). If this process is

repeated (that is, if a specimen is deformed in martensite, heated to austenite, and cooled to

martensite), the specimen will develop what is called the two-way shape memory effect

(TWSME). TWSME is not an inherent property possessed by SMAs and can only be acquired by

cyclic loading (training). The SMA specimen in the austenitic phase will expand by the trained

amount when the temperature drops below martensite finish Mf without a load on the specimen.

In other words, formation of detwinned martensite will take place directly from the austenitic

phase without externally applied load. Different training procedures and sequences25-27 are

available in the literature and can be utilized for training SMA actuators.

The K type NiTiCu SMA wires (actuators) used in this work had a diameter of 0.060mm,

260mm in length and they were trained by loading them to 400MPa, undergoing 50 thermal

cycles. Figure 1 shows a typical training cycle, which results in a TWSME of about 3.5%.

Typically, SMA heating, which triggers the martensite-to-austenite (M to A) phase

change, is achieved electrically (Joule resistive heating). The phase change is accompanied by a

significant exchange of heat with the surroundings due to latent heat of phase transformation and

resistive heating. The time rate of the transformation is controlled solely by the time rate of the

heat transfer. The heating rate and thus the transformation rate can be controlled by regulating

the applied voltage and thus the electrical current. To trigger the reverse transformation

(austenite to martensite or A to M) heat removal from the SMA is necessary. Conventional heat-

removal mechanisms are inherently slower than electrical heat addition. Therefore, in a heating-

cooling cycle necessary to complete a full SMA transformation/actuation cycle (M to A and then

A to M), the cooling process usually limits the rapidity of the cycle, i.e., the maximum attainable

frequency response. This problem is addressed in a later section.

Many constitutive models have been developed to describe the unique thermomechanical

behavior of SMAs. Work done on constitutive modeling of SMAs includes phenomenological

models28-34, micromechanical models for polycrystalline SMAs35,36 and empirical models based

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on system identification (ID)37-40. A review of several of these models and different SMA based

smart structure applications is given by Birman41.

DEVELOPMENT OF THE SIX-SEGMENT HYDROFOIL

General Design Principles

The present work is motivated by the effort to develop biomimetic underwater vehicles

that share the agility, hydrodynamic efficiency and stealth of aquatic animals. Several

biomimetic principles are incorporated in the conceptual design shown in figure 2. Since the

detailed vehicle design is discussed in a later section, the purpose of figure 2 is to illustrate the

biomimetic principles of operation of the vehicle (i.e. the use of a spine and SMAs as muscles) in

conjunction with specific, basic design choices, as explained below. In this design, the

combinations of SMA actuators and springs function as muscles and constitute the mechanical

equivalent of the fish myotome. Either side of an individual SMA actuator (wire) or spring, in

the longitudinal direction, is attached to the skeletal structure of the vehicle. The SMA wires are

two-way trained. In figure 2, the SMA wires in the austenitic phase (and thus contracted) are

drawn with thicker lines to distinguish them from the wires in the martensitic phase (and thus

extended). The springs introduce a stress bias for restoring purposes. Figure 2 illustrates how

selective SMA actuation causes body shape changes, by rotating the skeletal vertebrae.

Adaptive Actuation Control Of The SMA Actuators

In a first-generation, single-segment hydrofoil5, the SMA actuators were totally exposed

to the ambient medium. It is obvious that in the final vehicle design this would have to be

avoided, since the wide range of ambient temperatures encountered by the vehicle would have

significant adverse effects on the SMA control. In warm water, the SMA cooling rate would not

be sufficient and in cold water the electrical energy required for actuating the SMAs would be

exorbitant. Moreover, the corrosive properties of seawater would significantly reduce the useful

life of the SMA actuators. However, it was also obvious that in order to achieve SMA actuation

frequencies high enough to generate propulsive and maneuvering forces for a vehicle on the

order of 1 m long, forced cooling of the SMAs had to be used. It was therefore decided, in the

second-generation vehicle, to thermally insulate the SMA actuators from the ambient water

environment and embed them inside the vehicle. It was also decided to embed them inside

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closed-system channels through which the cooling medium would circulate, the temperature of

which would be controlled.

In situations like the one encountered here, i.e. electrical heating for the M to A

transformation and forced cooling for the A to M transformation, one difficulty imposed by the

forced convection is that temperature measurements of the SMA cannot be achieved via

thermocouples. A thermocouple is usually attached to the SMA wire/strip with a small “ball” of

thermal paste that electrically, but not thermally, insulates the thermocouple bead from the wire.

As discussed earlier, due to the strong heat transfer caused by the active convection of the

cooling medium, the thermal paste dissipates the heat from the wire much faster than it would in

a pure conduction environment, and the resulting temperature measurement is not indicative of

the actual temperature of the SMA.

In this work we have pursued actuation control schemes that rely only on actuator

displacement feedback and do not require temperature feedback. The control approach, has been

studied numerically and experimentally by our group for the control of SMA wire actuators, and

has been extensively described in work done by Webb39 and Webb et al.40,42 Only general

principles are reiterated here for completeness. Background work and earlier generations of the

control technique can be found in references 37, 38 and 43.

The control approach relies on hysteresis compensation and, in order to improve the

performance of compensation, adaptive hysteresis models for on-line tuning and identification

have been utilized. Furthermore, closed-loop compensation is considered since it has the

advantage of accounting for hysteresis nonlinearities, while providing feedback to correct the

hysteresis model on-line to accurately predict the actuator response. Webb created a

parameterized hysteresis model based on modification37,38 of the Preisach model45 concept which

can be adaptively updated for both identification and closed-loop compensation. The model is

referred to as the adaptive KP hysteresis model.

The present technique implements an adaptive KP model for feedback compensation. The

adaptive hysteresis feedback compensation accounts for any change in the "envelope of

hysteretic behavior" for an SMA actuator, since work done by Miyazaki15, Bo and Lagoudas29

has shown that cyclic deformation of SMAs can cause accumulation of microscopic residual

stresses and hence change the "envelope of hysteretic behavior". In feedback compensation, the

tracking error is not directly used in the control law. Rather, it is used to update (identify on-line)

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the hysteresis model ()H

to account for discrepancies between the model and the actual input-

output relationship. The inverse maps a desired reference trajectory into a temperature signal.

When temperature can be measured, an adaptive thermal model46,40 then commands a current to

control the temperature of the SMA to follow the desired temperature. For cases where

temperature measurements are unavailable, integration of a simplified thermal model (based only

on rough estimates of the thermal parameters) is used in place of actual temperature

measurements.

The control algorithm used for a single-wire test setup can best be described by referring

to the schematic in figure 3. A given reference displacement signal, r(t), is mapped into a

reference temperature, Tr, via the hysteresis model inverse, H-1. Equations 1 and 2 represent the

adaptive thermal model. Equation 1 calculates the required heating, u, to force T̂ to track Tr,

while Equation 2 is integrated to determine T̂ . The current signal is output to the SMA wire as

i u= . Note that the current is limited by strict bounds on u, in that u is constrained to lie in the

interval [0, imax2], where imax is a pre-defined safety limit on the current. The lower limit on u

essentially means that the fastest cooling that can be achieved is when the current to the SMA

wire is turned off. a bm m= are gains that determine how fast $T will approach Tr. a and b are

physical parameters that depend on the convection and resistance of the wire, respectively. These

parameters only have to be within 30% of the actual values.ˆ mmTabuTrabb-=-+

(1)

Tmmm rTT ba +-= ˆ&̂ (2)

To adaptively update the hysteresis model to represent the y versus $T relationship, the

estimate of the displacement output, $y , is generated from H(T). The estimation error is

calculated using the error between the actual displacement, y, and $y . The hysteresis model

parameters are updated using the gradient adaptive law from section 3.1.2 of Reference 39, in

turn updating the inverse model.

In this control methodology, the thermal parameters remain constant. Although it may be

possible to adaptively update the thermal parameters to represent the heating and convection

more accurately for a time-varying cooling environment, in this paper the burden of accounting

for these variations lies solely with the adaptive hysteresis model. Reasonable estimates of the

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thermal parameters serve to ensure that $T remains in the defined input region of the hysteresis

model39.

An experimental setup was used to test the SMA control theory. The details of the setup

can be found in reference 42. A vertical test stand provided support for an SMA wire while

allowing easy access to the wire. The wire was embedded in a cooling channel. The SMA wire

was connected to power leads. The bottom end of the wire was firmly attachment to the test

stand, while its top end was attached to the lever arm by a kevlar string in a manner that allowed

wire lengths of 1 to 14 inches (2.54 – 35.56 cm) to be accommodated in the frame. A spring

attached to the opposite end of the lever arm both held the SMA wire in tension and provided a

restoring force for the SMA. The lever arm magnified the SMA displacement by a factor of 4,

which was then measured using a Celesco string-potentiometer.

We have applied our adaptive control scheme to the water-cooled SMA actuators of this

experimental setup and have achieved very good tracking for actuation frequencies as high as

20Hz. Figure 4 presents the control results for the case of using water as the convection medium.

The tracking is very good and actuation frequencies as high as 20Hz could be achieved.

Although the adaptive controller worked well with one channel, controlling

simultaneously all six muscles of the six-segment vehicle placed stringent computational

demands on the hardware, which could not meet the sampling rates required for the adaptive

controller to work properly. A simpler control routine was therefore used, which was a basic

Proportional-Integral controller. We felt, however, that it was important to report the

development of the adaptive controller herein, since it: (a) clearly counters the perception that

SMA actuators are low-frequency-response actuators (order of 1 Hz) and should not be used in

applications where higher actuation frequencies are required and (b) with regard to the specific

biomimetic application, it shows that practically the entire spectrum of swimming frequencies

encountered in acquatic animals can be reproduced via SMAs. For the Proportional-Integral

controller, equations 3-5 show how the control value is determined, where i is the particular

segment, and KP[i] and KI[i] are the control gains. The gains were determined experimentally.

][][][ iAirefiE -= (3)

Ú= dtiEiE ][][int (4)

][][][][][ int iEiKIiEiKPiP ⋅+⋅= (5)

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The instantaneous angle, A[i] is compared to the reference signal ref[i]. The control, P[i],

determines how much power needs to be sent to the wire to get it to match the reference, that is

to make the error E[i] equal zero. This controller is computationally simple and allowed the

simultaneous control of all six muscles, although it could not track high frequency reference

signals, like the adaptive controller discussed earlier.

Hydrofoil Design

The profile shape of the hydrofoil was based on a NACA 0009 airfoil section with a 30-

inch (76.2-cm) chord and minor modifications in the aft portion of the profile. There are three

main components to the biomimetic hydrofoil and each one resembles its natural counterpart, a)

the aluminum backbone and rib structure, b) the SMA muscles, and c) the skin.

The Ribs and Backbone

The ribs are seven, 0.25-in. (0.64-cm) thick aluminum plates. These ribs support all the

weight and forces of the hydrofoil. The seven ribs are connected with six sets of two hinges

(vertebrae). The vertebrae are machined aluminum, with bearings for the pivot points to ensure

smooth motion between the ribs. Figure 5 shows the skeletal structure of the hydrofoil. Mounted

on one vertebra of each pair is a position encoder (on the extended shafts in figure 5). The

encoder gives the relative angle between adjacent ribs. By combining all the relative angles, the

shape of the hydrofoil can be reproduced.

SMA Actuators (Muscles)

Just like in biological muscles, the SMA muscle has two parts that pull in opposing

directions. The opposing force is provided by springs. Figure 6 shows how one SMA muscle is

laid out. By referring to figures 2 and 6, the process through which the SMA-spring pairs impart

relative movement of adjacent vertebrae can be easily understood. When an SMA wire is

subjected to an electrical current, Joule resistive heating causes the SMA actuator to contract and

thereby rotate the adjacent rib in one direction. If the current is then removed the actuator cools

and returns to its original length allowing the spring to rotate the vertebrae in the opposite, for

TWSME trained wires, as described earlier and shown in Figure 1. This process is performed by

each segment to achieve a desired shape. The SMA wires run parallel to the ribs, i.e. in the

spanwise direction. This re-orientation allowed the wires to be much longer. For fixed strain,

longer wires meant more displacement/rotation of each segment. The different moment arms for

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the springs and the SMA wires allowed use of smaller springs. Moreover, placing the SMAs at

small moment arms translated the linear SMA displacement to larger angular displacements of

the vertebrae. The SMAs were placed on alternating sides of the hydrofoil so that, when

deflecting all the segments in the same direction, only half the wires are deflected, which means

less current is being drawn from the power supply.

The SMA wires are internal to the hydrofoil; this allows for active cooling independent of

the surrounding environment. To accomplish this, each wire is embedded in a cooling channel.

Each cooling channel is a parallel branch off a manifold running down the middle of the

hydrofoil. Elastic silicon tubing is used as the cooling channels to allow for the change in SMA

length during actuation. The SMA wire terminates into a brass connector, which is then

connected to the rib by a piece of kevlar string. The kevlar string connects to an eyebolt to allow

adjustment to the initial angle of the segment. Figure 7 shows the hydrofoil with all the SMA

muscles, kevlar tendons, cooling tubes, power-wires, and position encoders before the skin is

added. A zoom-in view shows specific structural details. Amongst else, it shows how the SMA

attaches to the skeleton via a kevlar tendon. In addition to physically terminating the SMA wire,

the brass connectors also act as both electrical terminals and cooling connections. Because of the

multi-functionality of the brass connector, fewer connectors are needed and more space is

available inside the model.

The Skin

The skin is what pushes against the water, and allows the hydrofoil to generate thrust and

maneuver. An innovative approach was taken in the design and assembly of the skin. Contrary to

the biomimetic vehicles of MIT (Robotuna and Robot Pike) the skin is not continuous, but is

rather segmented. By allowing the interior of the vehicle to be flooded (thus allowing the vehicle

specific gravity to approximate that of actual aquatic animals, which is close to 1), there was no

need to place demanding water-sealing properties on the skin. This allowed us to segment the

skin, which resulted in the skin presenting no additional resistance to the motion of the spine.

Moreover, the segmented skin allows for the implementation of a biomimetic flow control

scheme often employed by certain species. This consists of fluid injection into the boundary

layer as the vehicle surface transitions from concave to convex, thus transitioning from favorable

to adverse streamwise pressure gradients. This is when (in adverse pressure gradients) the

boundary layer is most susceptible to separation and momentum injection is needed to prevent it.

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The segmentation of the skin will allow the high-pressure on the concave vehicle surface to

direct fluid towards the low-pressure convex surface, resulting in momentum injection into the

separation-susceptible boundary layer on the convex side. The skin is set up as six scales on each

side, with nose and tail sections added at either end. Each scale is bent such that it remains in

contact with the adjacent overlapped scale through the entire range of skeletal deflections. The

scales are not water tight, nor do they cover the top or bottom of the hydrofoil. Figure 8, a picture

of the fully assembled hydrofoil, shows how the scales stay in contact with each other.

Experimental Setup

The six-segment hydrofoil was mounted on a modular setup that could, with little

modification, be used both for in-air and water-tunnel experiments. The hydrofoil was attached

to the setup via low-friction linear bearings to allow for force measurements.

Potentiometers/encoders were mounted to the hinges between each pair of ribs, for a total of six,

and were numbered one to six, with angle 1 referring to the relative angle of the segment

attached to the tail and angle 6 referring to the relative angle of the segment attached to the nose

of the hydrofoil. The encoders were calibrated over a ±20° and the encoder specifications

combined with the specifications of the data-acquisition system resulted in angle measurement

resolution and uncertainty of 0.012° and 0.04°, respectively. Custom-designed electronics was

used to split the power from a single power supply into six individually controllable signals that

controlled the six SMA muscles of the hydrofoil. The water tunnel used was the Aerospace

Engineering water tunnel, with a 2’x3’x6’ (0.61m x 0.91m x 1.83m) glass test section and

variable flow velocities up to 3 ft/s (0.91 m/s).

RESULTS AND DISCUSSION

In-Air Actuation

The preliminary goal of the in-air actuation tests was to track various types of reference

signals. The two main reference signals used were step inputs and sine waves. Step input control

was used in one of two ways, either one segment at a time was controlled, or all the channels

were deflected to the same angle. The individual control was normally used to bring the

hydrofoil from the initial “no-power” position, to the starting, or zero-deflection, position. A

zigzag shape of the initial position is a result of SMA wires being on alternating sides of the

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hydrofoil. Each segment starts with the SMA at its maximum length, which is defined as a

negative deflection for that segment. Using the step input to control all the segments at one time

allows the hydrofoil to be moved to the maximum possible deflection (in the same direction).

The resultant shape is a crescent moon and will be used for turning and maneuvering. Figure 9

shows the maximum deflection shape. In the figure each segment is deflected to 8°, resulting in a

total angle of almost 50° between head and tail. Figures 10 shows the performance of the SMA

controller stepping up to the crescent moon shape. The first type of sine wave used to maneuver

the hydrofoil was a simple sine wave passed through the segments according to equation 6

below, where i is the segment number, Lmean is the mean angle value (usually 0°), Lampl is the

sine amplitude, w is the angular frequency and t is the time. The resulting shape is a ripple

moving down the hydrofoil.

)]6/(2sin[5.0][ itiref amplmean +L+L= wp (6)

The second type of sine wave more closely resembles how fish swim. The nose of the hydrofoil

is held at or close to zero and the tail is deflected side to side. The magnitude of the deflections

(all in the same direction) is given by equation 6 without the i/6 phase shift. Figures 11 and 12

show plots of the deflections of different segments for different sinusoidal reference signals. _

Hz was the frequency that allowed the maximum deflection and gave the best performance.

When the frequency was raised above _ Hz, the oscillations would often become unstable. The

instability was caused by the momentum of the tail carrying it past the desired deflection. The

center of gravity of the tail has a long moment arm about the mounting rib. When the angle of

the segment next to the rib has a large error, the control input is large. The large control input

causes a large force from the SMA that causes the segment to overshoot the desired angle, and

this process continued with a growing error magnitude. Within only a few cycles, the instability

would grow to as large as ±20°, with a frequency greater than 1 Hz. Figure 13 shows how

quickly, and violently, an instability can form. The trailing off of the angle in figure 13 is the

result of the power being shut off.

This instability occurrence, however, carries an important message. In retrospect, the

main reason for the instability is the fact that the actuation frequency for which it was observed

matched the basic resonant harmonic of the hydrofoil dynamical system. This, in turn, implies

that system resonance can be exploited to generate drastic hydrofoil maneuvers, without

expending additional power. If as observed, under resonance each segment can deflect by angles

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as high as 20°, a six-segment hydrofoil can achieve shapes with total deflection angles between

the nose and the tail as high as 120°. In fact, this behavior has been clearly captured in video,

albeit accidentally at the time. This behavior is currently being capitalized upon by designing the

system’s dynamics such that its resonant frequencies coincide with the actuation frequencies for

fast turning maneuvers.

In-Water Actuation

The goal of the water tunnel actuation is to deflect the segments in a manner that

generates thrust. The exact deflection scheme was optimized via numerical modeling. We

initially searched the biology literature on swimming waveforms of fish. The waveform for

saithe was selected for further study. The swimming performance of this waveform was then

analyzed using an unsteady potential method numerical code that we have developed. This code

also integrates the body dynamics in it. The amplitudes and phases in the present work are based

on these waveforms. Both simple sinusoidal and tail-only deflection patterns were tested. Figure

14 is a photograph of the traveling sine wave through the body of the hydrofoil. The damping

from the apparent mass of the water eliminated the violent instabilities seen for in-air actuation.

This is expected, since the system’s resonant frequencies have changed due to its coupling with

the ambient medium, with the medium’s density having a first-order effect. Figure 15 shows how

the water environment allows large angular jumps of the same segment, without the instabilities.

While the violent instabilities associated with the tail were eliminated, a previously unseen

coupling between two segments (4 and 5, with segment one being the one closest to the tail)

caused the nose to oscillate. The oscillation could be removed for the tail-only actuation, but not

for the traveling wave actuation. For the tail only actuation, the control gains, KP and KI, for

channels corresponding to segments four and five were set at 10% the normal value. This change

increased the time it took to bring the hydrofoil to the zero-deflection starting position, but

eliminated the oscillations. This was tried with the sine wave actuation, but the oscillations were

still present. This coupling is not yet understood, although its source most likely pertains to the

system’s dynamics. One possibility is that the oscillations were associated with the large mass of

water “trapped” in the nose section.

For the force data, negative force indicates drag while positive force indicates thrust.

Figure 16 is a plot of the force measurements for the traveling-wave type of actuation in the

15

water tunnel. The signal from the load cell was noisy, but it was clear that when the hydrofoil

was actuated, positive thrust was generated. However, the produced thrust was insufficient to

overcome the drag for the tested, free stream velocity (approximately 30cm/s). Comparing figure

16 with results obtained for the tail-only type of actuation (figure 17) in the water tunnel, it

appears that the tail-only actuation was able to generate more force than the traveling sine wave

actuation scheme. The force generated by the tail-only actuation was not substantially larger than

that of the traveling-wave actuation, but the deflection angles needed to produce comparable

thrust magnitudes were only half of those required by the traveling-wave method. This agrees

with observation of aquatic animals, in that most of the propulsive force is generated from

deflection of the aft part of their body.

The force generated by the swimming hydrofoil was small because the tail was too

flexible. The forces from the water apparent mass deformed the tail to non-optimal caudal fin

configurations. Figure 18 is a series of snapshots taken during one actuation cycle that shows tail

deformation during actuation. As the tail deformed, the tip never moved more than 4 inches

(10.16 cm). A fish of comparable size has tip deflections on the order of 7 or 8 inches (17.78 or

20.32 cm). The compliant tail absorbed the force that should have been applied to the water. The

tail will have to be redesigned to generate the thrust needed to propel the vehicle along with

further testing at different free stream velocities to draw accurate conclusions.

CONCLUSIONS

An experimental investigation was conducted to investigate the feasibility of using Shape

Memory Alloys as artificial muscles to actuate a biomimetic hydrofoil. First, a novel adaptive

control algorithm for SMAs was developed. The control routine was a combination of an open-

loop model with a predetermined hysteresis model and a closed-loop controlled with error

feedback. The error feedback portion of the control updated the hysteresis model. The controller

performed very well on a single wire setup, allowing an SMA to be controlled and tracked at

frequencies up to 20 Hz.

A six-segment biomimetic hydrofoil with SMAs as its muscles, was developed and

tested. Several of the hydrofoil features were:

16

• The TWSME trained SMA actuators/muscles were placed internally and embedded in active

cooling channels to increase actuation frequency response and eliminate dependence on

environmental conditions.

• Internal position encoders allowed the exact hydrofoil shape to be known without external

measurements.

• Antagonistic SMA pairs, used in an earlier vehicle generation, were replaced by TWSME

trained SMA wire-spring combinations to increase the life of the SMA wires and decrease

the antagonistic force requirements (needed to overcome internal frictions only).

• The SMA actuators ran along the ribs of the hydrofoil (spanwise), allowing longer SMAs and

therefore higher rotations of the vertebrae.

• The same mounting apparatus was used for both in-air and water tunnel tests and a load cell

was added to the mount to measure thrust and drag during the water tunnel tests.

The biomimetic hydrofoil could be controlled either by setting each individual segment to

a particular static angle, or by dynamically deforming the hydrofoil as a whole to a prescribed

shape history. The latter method was the primary means of actuation. The two shapes histories

primarily used were: a) a traveling sine wave which produced a ripple through the spine of the

hydrofoil, and b) a tail-only actuation. The second method of actuation generated the most thrust

and more closely resembled the actuation of aquatic animals.

The force generated by the swimming hydrofoil for the tested free stream velocity was

small because the tail was too flexible. The forces from the actuation deformed the tail instead

of generating thrust.

The work demonstrated the potential of SMAs as artificial muscles in biomimetic

applications and has produced controllable SMA actuators at significantly higher actuation

frequencies than previously considered. It was also shown that by coupling the actuation

frequencies to the system’s (hydrofoil’s) resonant characteristics, dramatic maneuvers can be

produced with no additional power expense.

ACKNOWLEDGEMENTS

This work was supported by the Office of Naval Research and Aeroprobe Inc. through

the STTR program, under contract N00014-98-C-0061. The authors would like to thank the

technical monitor of the project, Dr. Teresa McMullen, for her support. Special thanks also go to

17

Dr. Glenn Webb for his help with the actuator controls and Mr. Richard Allen for his help with

the experimental setups.

REFERENCES

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14. Proceedings of Engineering Aspects of Shape Memory Alloys (Lansing, MI). 1988

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21. Rediniotis, O.K., Lagoudas, D.C., Garner, L. and Wilson, N. “Experiments and Analysis ofan Active Hydrofoil with SMA Actuators,” AIAA Paper No. 98-0102, 36th AIAA AerospaceSciences Meeting, Reno, Nevada, January 1998.

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actuated by shape memory alloy wires”, Shape Memory Alloy Material Structures, 1997,

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26. Miller, D. A. and Lagoudas, D. C, 2000. "Influence of cold work and heat on the shape

memory effect and plastic strain development of NiTi wire actuators". Material Science and

Engineering. As submitted.

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27. Stalmans, A.V, van Humbeeck, J., and Delaey, L, 1991, Training and the two way memory

effect in copper based shape memory alloys, European Symp. on Martensiteic

Transformation and Shape Memory Properties, J. Physique Coll. IV, 1, C4, 403-8.

28. Lagoudas, D. C., Bo, Z., and Qidwai, M. A., "A unified thermodynamic constitutive model

for SMA and finite element analysis of active metal matrix composites ", Mechanics of

Composite Materials and Structures, vol. 3, 153, 1996.

29. Bo, Z. and Lagoudas, D. C., "Thermomechanical modeling of polycrystalline SMAs under

cyclic loading, Part I-IV" International Journal of Engineering Science, vol. 37 1999.

30. Brinson, L. C., "One-dimensional constitutive behavior of shape memory alloys:

Thermodynamical derivation with non-constant material functions" Journal of Intelligent

Material Systems and Structures, vol. 4(2), 229, 1993.

31. Liang, C. and Rogers, C., "One dimensional thermomechanical constitutive relations for

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32. Tanaka, K., "A thermomechanical sketch of shape memory effect: One-dimensional tensile

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33. Sato, Y. and Tanaka, K., "Estimation of energy dissipation in alloys due to stress-induced

martensitic transformation." Res Mechanica, vol. 23, 381, 1986.

34. Boyd, J. and Lagoudas, D. C., Internaltional Journal of Plasticity, vol. 12, 805, 1996.

35. Patoor, E., Eberhardt, A., and Berveiller, M., "Potential pseudoelastic et plasticite de

transformation martensitique dans les mono-et polycristaux metalliques." Acta Metall 35(11),

2779, 1987.

36. Falk, F., "Pseudoelastic stress strain curves of polycrystalline shape memory alloys

calculated from single crystal data" International Journal of Engineering Science, 27, 277,

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37. Banks, H., Kurdila, A. and Webb, G. 1997. “Identification of Hysteretic Control Influence

Operators Representing Smart Actuators, Part (i): Formulation,” Technical Report CRSC-

TR96-14, Center for Research in Scientific Computation, North Carolina State University,

Raleigh, NC.

20

38. Banks, H., Kurdila, A. and Webb, G. 1997. “Modeling and Identification of Hysteresis in

Active Material Actuators, Part (ii): Convergent Approximations,” Journal of Intelligent

Material Systems and Structures, 8(6).

39. Webb, G. V., “Adaptive Identification and Compensation for a Class of Hysteresis

Operators,” Ph.D. Dissertation, Dept. of Aerospace Engineering, Texas A&M Univ., College

Station, TX, 1998.

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Control and Dynamics.

21

List of Figures

Figure 1. Typical training cycle for 0.060mm diameter NiTiCu SMA wire under 400MPa

applied load.

Figure 2. Schematic of Design Principles of Biomimetic Hydrofoil.

Figure 3. SMA Control Scheme.

Figure 4. SMA Actuator Control with Forced Water Convection.

Figure 5. Vertebrae-Like Backbone of Biomimetic Hydrofoil.

Figure 6. Schematic of SMA Muscle Placement in Biomimetic Hydrofoil.

Figure 7. Photograph of Hydrofoil Skeleton with SMA Muscles, Encoders and Cooling

Tubes Added.

Figure 8. Fully Assembled Biomimetic Hydrofoil.

Figure 9. Maximum Static Deflection of Hydrofoil.

Figure 10. Vertebra Response During Commanded Stepping Up to Maximum Deflection.

Figure 11. Vertebra Response During Commanded Traveling Sine Wave: a) Segment 1, b)

Segment 3 (Solid Lines are Reference, Dashed Lines are Actual Angles).

Figure 12. Vertebra Response During Tail-Only Actuation: a) Segment 1, b) Segment 3

(Solid Lines are Reference, Dashed Lines are Actual Angles).

Figure 13. Instabilities in Hydrofoil In-Air Actuation.

Figure 14. Hydrofoil in Water - Traveling Sine Wave.

Figure 15. Actuation Instability Disappeared in Water Actuation.

Figure 16. Plot of Force for Traveling Sine Wave for 30cm/sec free stream velocity.

Figure 17. Plot of Force for Flapping Tail Actuation for 30cm/sec free stream velocity.

Figure 18. Photographs of Tail Positions during One Actuation Cycle.

22

0 20 40 60 80 100 120

-0.5

-0.4

-0.3

-0.2

-0.1

0

Temperature(deg C)

Stra

in(%

)

~ 3.

5% T

WSM

E

0 20 40 60 80 100 120

-0.5

-0.4

-0.3

-0.2

-0.1

0

Temperature(deg C)

Stra

in(%

)

~ 3.

5% T

WSM

E

Figure 1, Rediniotis et al.

23

SMA in the Austenitic

Phase

SMA in the Martensitic

Phase

Spring

Vertebra

Figure 2, Rediniotis et al.

24

SMA Wire

$y$T

$& $T T u= - +a b

r(t)Tr

+ -y

+-H

-1H

e (t)H

u i

am - ab

bm

b

z

Figure 3, Rediniotis et al.

25

14.00 14.20 14.40 14.60Time (sec)

0.10

0.20

0.30

Dis

plac

emen

t (in

)

20 Hz

12.00 13.00 14.00 15.00 16.00Time (sec)

0.00

0.20

0.40

Def

lect

ion

(in)

47.00 47.20 47.40 47.60 47.80 48.00Time (sec)

0.00

0.20

0.40

Def

lect

ion

(i n)

2 Hz

10HzReference

Actual

Figure 4, Rediniotis et al.

26

Figure 5, Rediniotis et al.

27

Restoring Spring

Connecting Wire (Non-SMA)

SMA Wires

Connecting String

Brass Connectors

Top ViewSide View

Figure 6, Rediniotis et al.

28

Brass ConnectorsKevlar StringPower WiresEyeboltSMA wires incooling channels

Figure 7, Rediniotis et al.

29

Figure 8, Rediniotis et al.

30

Figure 9, Rediniotis et al.

31

64.00 68.00 72.00 76.00 80.00Time (sec)

-4.00

0.00

4.00

8.00

12.00

Ang

le (d

eg)

Reference

Segment 1

Segment 3

Figure 10, Rediniotis et al.

32

80.00 82.00 84.00 86.00 88.00Time (sec)

-8.00

-4.00

0.00

4.00

8.00

Ang

le (d

eg)

80.00 82.00 84.00 86.00 88.00Time (sec)

-8.00

-4.00

0.00

4.00

8.00A

ngle

(deg

)

Figure 11, Rediniotis et al.

(a)

(b)

33

a)b)

20.00 22.00 24.00 26.00 28.00Time (sec)

-4.00

-2.00

0.00

2.00

4.00

Angle

(deg

)

20.00 22.00 24.00 26.00 28.00Time (sec)

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

Angle

(deg

)

Figure 12, Rediniotis et al.

(a)

(b)

34

56.00 58.00 60.00 62.00Time (sec)

-10.00

-5.00

0.00

5.00

10.00

Ang

le (d

eg)

Reference

Angle 3

Figure 13, Rediniotis et al.

35

Figure 14, Rediniotis et al.

36

0.00 10.00 20.00 30.00Time (sec)

-12.00

-8.00

-4.00

0.00

4.00

8.00

Ang

le (d

eg)

Reference

Angle 3

Figure 15, Rediniotis et al.

37

100.00 110.00 120.00 130.00 140.00 150.00Time (sec)

-8.00

-4.00

0.00

4.00

8.00

An

gle

(deg

)

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20T

hrus

t(lb

f)

Angle 1

Thrust

100.00 110.00 120.00 130.00 140.00 150.00Time (sec)

-8.00

-4.00

0.00

4.00

8.00

An

gle

(deg

)

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20T

hrus

t(lb

f)

Angle 1

Thrust

Figure 16, Rediniotis et al.

38

0.00 20. 00 4 0.00 60.0 0 80 .00

T im e (s e c )

- 12.0 0

- 8.00

- 4.00

0 .00

4 .00

8 .00

An

gle

(d

eg

)

- 0 .30

- 0.20

- 0.10

0.00

0.10

0.20

0.30

0.40

0.50T

hrus

t(l

bf)

A n g le 1

Thrust

0.00 20. 00 4 0.00 60.0 0 80 .00

T im e (s e c )

- 12.0 0

- 8.00

- 4.00

0 .00

4 .00

8 .00

An

gle

(d

eg

)

- 0 .30

- 0.20

- 0.10

0.00

0.10

0.20

0.30

0.40

0.50T

hrus

t(l

bf)

A n g le 1

Thrust

Figure 17, Rediniotis et al.

39

Figure 18, Rediniotis et al.