ASME PAPER

Yongshun Zhang
Guangjun Liu1
e-mail:gjliu@ryerson.ca
Department of Aerospace Engineering,
Ryerson University,
350Victoria Street, Toronto,ON,M5B2K3,Canada Wireless Swimming Microrobot: Design,Analysis,and Experiments
This paper presents a bidirectional wireless swimming microrobot that has been devel-oped,analyzed,and experimentally tested.The robot is developed based onfin beating propulsion,using giant magnetostrictivefilms for head and tailfins.An innovative drive approach,using separate second order resonance frequencies of the head and tailfins to generate forward and backward thrusts,is proposed and implemented on a bidirectional swimming microrobot prototype.Dynamic model of the proposed microrobot has been derived based on theoretical analysis.Simulation and experimental results have demon-strated the feasibility of the proposed drive approach and design.The developed swim-ming microrobot features a low driving frequency,low power consumption,and a large range of swimming speed in both the forward and backward directions.
͓DOI:10.1115/1.3023137͔
Keywords:swimming microrobot,giant magnetostrictivefilms,resonance frequency, mode shape,bidirectional swimming
1Introduction
Wireless swimming microrobot controlled by magneticfield features noncontact operation,high reliability and safety,and has the potential of entering the blood circulatory system and urinary system of the human body for applications in medical engineering ͓1͔.
Aimed at improving drive efficiency,research on propulsion techniques for swimming microrobots based on biomimetics has attracted extensive research1000interests.Breder͓2͔is among thefirst researchers to classify propulsive modes of swimming fish.Webb and Weihs͓3͔categorized propulsive modes offish into body caudalfin͑BCF͒movements and median pairedfin ͑MPF͒propulsion.Sfakiotakis et al.͓4͔pointed out that many fishes typically swim in the MPF mode for foraging with greater maneuverability,and they can switch to the BCF mode at high speeds and high acceleration rates.Naturally,most studies of ef-ficientfin-based swimming have focused on the BCF mode͓5͔.It was also revealed in Ref.͓4͔thatfin beating is one of the most efficient propulsive modes by considering only the motion of the fish body and itsfin.It is a well-known fact that the highest speed fish has a rigid fore body͑around two-thirds of the whole body͒and a highfin aspect ratio.A high swimming speed can be achieved by tailfin beating even when the amplitude offin beat-ing is very small.This motivated the work in this paper to adopt fin beating for generating propulsive force.
The emergence of new actuators,such as smart materials,has accelerated the development of biomimetic swimming microro-bots.Guo et al.͓6͔developed a swimming microrobot by utilizing two ionic conducting polymerfilm͑ICPF͒actuators with cable. Their robot,swimming in the MPF mode,can move forward,float up and down,and steer to left and right with three degrees of freedom.Kósa et al.͓7͔reported a novel cabled swimming mi-crorobot using an elastic tail with three piezoelectric actuators for propulsion in the BCF mode.
However,swimming microrobots with cable consume exces-sive energy for carrying the cable,especially when the robot moves in environments such as a blood vein.The travel range of cabled robots is limited by traction force among other problems. Wireless drive of a swimming microrobot is a key technique for enhancing its feasibility and reliability in medical engineering ap-plications.Guo et al.͓8͔developed a wireless swimming robot using an electromagnetic actuator for biomedical application,with a tailfin made of copperfilm.The robot adopts the BCF move-ment,achieving high speed swimming in one direction.Mei et al.͓9͔reported a different type of wireless swimming microrobot in the MPF mode with twofins made of a ferromagnetic polymer ͑FMP͒.
The known swimming microrobots are designed mainly for swimming in one direction and can only swim at very low speeds in the opposite direction when driven by an alternating magnetic field at high frequencies,which results in dissymmetrical bidirec-tional swimming speeds and high energy consumption for back-ward swimming.
Giant magnetostrictive thinfilms͑GMFs͒,when used for non-contact drive of a swimming robot,can provide large strain and prompt response͓10͔.Its drive requires only a moderate magnetic field and produces negligible heat.It is also safe to use on the human body.Thus GMF actuators have widespread prospect for applications in thefield of wireless medical microrobots͓11͔. The contributions of this paper include an innovative drive ap-proach,which is based on twofins made of GMFs with separate second order resonance frequencies.Our research has led to the discovery of the relationship between swimming directions and vibrating mode shapes.Based on the proposed drive approach,a bidirectional swimming microrobot prototype is developed.The-oretical analyses on hydrodynamic modeling and experimental re-sults have demonstrated the feasibility of the proposed approach and design.
2Prototype Swimming Microrobot
A developed bidirectional swimming robot prototype is shown in Figs.1͑a͒and1͑b͒with the robot in a transparent pipe segment filled with liquid.The robot hull is built using a10mm long,8 mm wide,and8mm high vesicant polyvinyl rosin,and two GMF actuators͑fins͒with different structures at the head and tail.One actuator is made of magnetostrictive amorphous T b0.28D y0.72F e1.93
1Corresponding author.
Contributed by the Dynamic Systems,Measurement,and Control Division of
ASME for publication in the J OURNAL OF D YNAMIC S YSTEMS,M EASUREMENT,AND C ON-
TROL.Manuscript received February27,2007;final manuscript received August5,
2008;published online December5,2008.Assoc.Editor:Nader Jalili.
Journal of Dynamic Systems,Measurement,and Control JANUARY2009,Vol.131/011004-1
Copyright©2009by ASME
and S m F e1.95thinfilms,sputtered on the opposite sides of a poly-imidefilm substrate with a thickness of35␮m,and the other actuator is sputtered on the opposite sides of a copperfilm sub-strate with a thickness of50␮m.Both substrates have a length of 45mm and a width of8mm.
When a magneticfield is applied along the longitudinal direc-tion͑along the pipe axis,as shown in Fig.1͑b͒͒,the positive magnetostrictive T b0.28D y0.72F e1.93film expands and the negative magnetostrictive S m F e1.95film contracts.Hence the actuators de-flect downwards.When an alternating magneticfield with variable frequency is applied along the longitudinal direction,both the front and rearfins vibrate in response to the alternating magnetic field.Therefore,a propulsive force is generated from the interac-tions between the vibratingfins and the liquid.
In general,the vibration deflection curve of a GMF cantilever is mainly comprised of the anterior three mode shapes.Our experi-mentation shows that the direction and magnitude of the propul-sive force generated by a vibratingfin influid vary with the vi-bration mode shape.Based on this observation,the second order resonance frequencies of the twofins are separated by design.As shown by analysis and experiments later in this paper,the two second order resonance frequencies lead to the maximum forward and backward driving speeds,respectively.Bidirectional swim-ming of the robot is realized when it is driven between the two second order resonance frequencies of the fore and rearfins.
3Dynamic Model
In order to study the dynamic characteristics of the swimming microrobot and its control,the dynamic behavior of GMFfins in a time variant oscillating magneticfield is analyzed.An analytical relationship is established between the driving frequency of the magneticfield and the propulsive force that is generated as a result of interaction between the vibrating GMFfins and the liq-uid.
The dynamics of the GMFfins is quite complicated,due to the nonlinearity in the magnetostrictive coefficient.In this paper a dynamic cantilever model of the GMFfins is developed based on force oscillation theory.In this model,an equivalent external drive moment is converted from internal magnetostrictive stresses based on the equivalent internal moment effectiveness in any section of the compound GMF cantilever.Under the excitation of a moderate magneticfield,an equivalent external magnetostrictive moment can describe the dynamic behavior effectively in terms of mode shape and amplitude of vibration.
3.1Magnetic Coils.For the GMFfin actuation,a direct sine wave current is selected as follows:
I=0.5I0͑1+sin␻t͒͑1͒where I0is the amplitude and␻is the angular velocity of the electric current.
The magneticfield H generated by the electric current along the axis of coils is governed by
H˙=
NI˙R1
2l1͑R2−R1͒ͩ␤ln␣+
ͱ␣2+␤2
1+ͱ1+␤2−
␥ln
␣+ͱ␣2+␥2
1+ͱ1+␥2
ͪ͑2͒
where N is the number of circles of the two coils,R1is the internal diameter of the coils,R2is the external diameter of the coils,␣=R2/R1,␤=͑l1+0.5c͒/R1,␥=0.5c/R1,l1is the length of one coil, and c is the distance between the two coils.
3.2Equivalent Drive Moment.In Refs.͓12,13͔,the dynamic relationship between the magneticfield and magnetostrictive co-efficient␭is determined by interpolating experimental curves as follows:
␭͑H͒=L͓1−exp͑−aH b͔͒+kH͑3͒where the parameters L,a,b,and k are optimized tofit experi-mental data.This formula is used to calculate an internal magne-tostrictive stress in thefilm,depending on the magneticfield.For a membrane,the even stress␴m is
␴m=Ϯ␭E1/͑1+␥p͒͑4͒where E1is Young’s modulus of the giant magnetostrictive mate-rial͑GMM͒and␥p is Poisson’s coefficient of the GMM.The minus sign in Eq.͑4͒represents compressive stress inside the T b0.28D y0.72F e1.93thin layer when it extends,while the positive sign in Eq.͑4͒represents the tensile stress inside the S m F e1.95thin layer when it contracts in the magneticfield along the longitudinal direction.
A classical bending theory for a trimorph cantilever is appli-cable to the magnetostrictive thin-film cantilever,as shown in Fig. 2͑a͒,in order to obtain its equivalent external drive moment by stress analysis inside the thin-film cantilever.
Assume that a neutral layer is a section that overpasses the x axis.The neutral axis can be found as follows͓14͔:
͵
͑n−1͒h
nh
E͑z͒zdz=0͑5͒
where z is the coordinate along the axis and E͑z͒is Young’s modulus as a function of thickness.As indicated in Fig.2͑a͒,h is the thickness of the whole GMF cantilever.We define n͑0Ͻn
Ͻ1͒to satisfy the condition that the upper and the lower surfaces Fig.1A swimming microrobot prototype and test setup:…a…prototype and…b…test setup
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are located at z =nh and z =͑n −1͒h ,respectively.With the as-sumption that T b 0.28D y 0.72F e 1.93and S m F e 1.95have the same Young’s modulus,the neutral layer locates in the middle,i.e.,n =0.5.
Stress distribution of the cantilever in a section a −a Јis shown in Fig.2͑b ͒when it pressive stress ␴m of T b 0.28D y 0.72F e 1.93can be regarded as constant when it expands because the thickness of a sputtered film is quite thin,while the tensile stress −␴m of S m F e 1.95can also be regarded as constant when it contracts for the same reason.
Internal stress above the neutral layer is tensile,while internal stress below the neutral layer is compressive.The equivalent drive moment of the trimorph cantilever is described as
M =
͵
−0.5h
0.5h
E ͑z ͒␭͑z ͒zBdz ͑6͒
where B is the width of the cantilever and ␭͑z ͒is strain as a function of thickness.By integrating Eq.͑6͒,the equivalent drive moment is determined as
M =␴m Bh 1͑h 1+h 2͒−
BE 26E 1h 2
2
␴m ͑7͒
where E 2is Young’s modulus of the substrate,h 1is thickness of
the giant magnetostrictive film,and h 2is thickness of the sub-strate.Denote ␦=Bh 1͑h 1+h 2͒−͑BE 2/6E 1͒h 22.Equation ͑7͒can be rewritten as
M =␦␴m
The moment caused by internal stress at any section across the trimorph cantilever is denoted by M Ј,as shown in Fig.2͑b ͒.Its internal moment effect on the trimorph cantilever is exactly the same as that caused by an external moment M ,as if the same value of M Јwere applied to the free end of the cantilever.3.3Dynamic Model of GMF Fins.Based on the cantilever model under the disturbing moment M ͑t ͒exerted to the free end of the cantilever,as shown in Fig.2͑b ͒,the dynamic characteris-tics of the GMF fin are analyzed.
Utilizing a normalized coordinate system ͓15͔,the typical vi-bration deflection curve under the disturbing moment M ͑t ͒can be described as
h ͑f ,x ,t ͒=
͚i =1
ϱ
C i X i
͑x ͒␾i
͑t ͒
where ␾i ͑t ͒is a function of time,X i ͑x ͒is an eigenfunction,and C i is a normalization ing the method of virtual work,a typical differential equation of force oscillation driven by the dis-turbing moment M ͑t ͒can be described by
␾¨i +c eq m ␾˙i +p i
2␾i =M ͑t ͒X il
Јm
,i =1,2,3, (8)
where m is the mass per unit length,c eq is the equivalent viscous
damping coefficient,p i is the natural frequency ͑p i =␤i 2ͱEI /m ͒,
X il is the eigenfunction at x =l m ,and l m is the length of the GMF ing the boundary conditions of the cantilever,its eigenfunction is obtained as
X i ͑x ͒=ch ␤i x −cos ␤i x +␰i ͑sh ␤i x −sin ␤i x ͒
where ␰i =−͑sh ␤i l m −sin ␤i l m ͒/͑ch ␤i l m +cos ␤i l m ͒and ␤i =͑i −1
2͒␲/l m .
Under the normalization condition that ͐0l m C i X i 2dx =1,the nor-malization constant can be derived as C i =ͱ1/l m .Denote K
=1
2I 0RN ͑E 1/͑1+␥p ͒͒␦,where R =͑R 1/2l 1͑R 2−R 1͒͒͑␤ln ͑␣
+ͱ␣2+␤2͒/͑1+ͱ1+␤2͒−␥ln ͑␣+ͱ␣2
+␥2͒/͑1+ͱ1+␥2͒͒and the damping coefficient ␰=c eq /2p i m .After simple algebraic opera-tions,Eq.͑8͒can be expressed more conveniently as
␾¨i +2␰p i ␾˙i +p i
2␾i =KX il Јm sin ͑␻t ͒+KX il
Јm
͑9͒
Using Duhamel’s integral,the function ␾i can be determined as
␾i =e −␰p i t ͑c 1sin q i t +c 2cos q i t ͒+␾sin ͑␻t −␺͒+
KX il
Јmp i 2
where the constants c 1and c 2can be obtained using initial condi-tions as
c 1=␾
˙i 0+␰p i ␾i 0q i
,
c 2=␾i 0
Hence the steady-state solution of the force oscillation is
␾i =␾sin ͑␻t −␺͒+
KX il
Јmp i 2
where ␺=arctan ͑2␰p i ␻/͑p i 2−␻2
͒͒,
q i =ͱ1−␰2p i ,and ␾=KX il Ј/m /͑ͱ͑p i 2−␻2͒2+͑2␰p i ␻͒2
͒.
Finally the steady-state deflection of the vibrating cantilever caused by moment M ͑t ͒is
h ͑f ,x ,t ͒=
͚i =1
ϱͫC i ͩ
X i ␾sin ͑␻t −␺͒+X i
KX il
Јmp i 2
ͪͬ
͑10͒
where f =␻/2␲.
Vibration mode shapes inside liquid are the same as those in air.However,because of a bigger damping coefficient inside liquid,the amplitude of vibration is smaller in liquid than that in air.In the following hydrodynamic modeling,vibration deflection ex-pressed in Eq.͑10͒is utilized to simulate swimming characteris-tics of the robot in liquid,and the mass of beam is considered to be constant.
3.4Modeling of the Propulsion and Swimming Speed.In the modeling of the mean swimming speed,propulsions generated by the caudal fin and the drag force against the robot hull are analyzed separately.When the robot swims smoothly at a certain speed excited by the alternating magnetic field with a certain fre-quency,the propulsion generated by the fins is equal to the drag on the robot.The intersectional point of the mean propulsion force curve and the drag force curve corresponds to the mean swimming speed of the robot.
When the robot is swimming,the shape of a caudal fin and its motion are shown in Fig.3,where V is the mean swimming speed,and ␣is an angle between the tangent of the fin and the negative direction of the x axis.
This model must be used with the following
hypotheses.
Fig.2Stress analysis of a GMF fin and its equivalent moment
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JANUARY 2009,Vol.131/011004-3
1.Thefluid is incompressible and disturbanceflow around ro-
botfins and pipe wall is negligible.
2.The thickness of the robotfin is negligible compared with its
length.
3.The mean steady-state speed of the robot relative to the
fluid,after the thrust and drag balance has been reached,is low and constant.
Afish swimming modeling method͑so-called added mass method͒is available in literature͓3͔,which is established under the condition that the momentum of the liquid is taken into ac-count and the viscosity of the liquid is ignored.
However,the vibration amplitude of the microrobotfin is small, and the angle␣,as shown in Fig.3,is small too.As a result, although the tangential force coefficient is small,the tangential fluid force in the swimming direction is relatively large and has to be taken into consideration in the modeling.The modeling of the normal and tangentialfluid forces is subject to small Reynolds number conditions,as the microrobot has a small size or swims in liquid with high viscosity.Therefore the added mass method can-not be applied directly to model the microrobot.
As for a solid object,the normal and tangentialfluid forces on thefin are modeled as
F n=−1
2
C n␳S V nʈV nʈ,F t=−12C t␳S V tʈV tʈ
where␳is the density of thefluid,V n is the normal speed of the fin surface relative to liquid,V t is the tangential speed of thefin surface relative to liquid,C n is the normal drag coefficient and C t is the tangential force coefficient,and S is the wet area of thefin. Drag coefficient mainly depends on the Reynolds number,the shape of the moving body,and theflow condition.The Reynolds number characterizes the relative speed V,the length of the mov-ing body l,and the kinematic viscosity␯of thefluid,and is determined as
Re=Vl/␯
For the normal drag coefficient C n,under the assumption that theflow is stable,using existing experimental results in Ref.͓16͔, for l m/B=45/8=5.6,where l m and B are the length and width of thefin,respectively,C n=1.2.
The tangential force coefficient depends on the Reynolds num-ber andflow conditions͓16͔.When ReϽ5ϫ105,the boundary layer on theflat surface appears as laminarflow,and the tangential force coefficient C t is described as
C t=1.372ͱ1Re
Obviously the deflection offin beating may arouse unstable flow.Deriving an accurate hydrodynamic model is so complicated that some assumptions have to be made to simplify the model.For the smallfin vibration,its disturbance to theflow is ignored,and the liquidflow is regarded as laminar.
Next we discuss the method for propulsive force calculation,which stems from basic analysis offluid forces applied normally and tangentially on a small segment of thefin,and the total forces in the axial and lateral directions can be obtained from the integral along the entire length of thefin.
The shape of the caudalfin is defined by the functions b1͑x͒and b2͑x͒.The function h͑f,x,t͒describes the lateral displacement in the xz plane with respect to time.The total width of thefin is given as b͑x͒=b1͑x͒−b2͑x͒.
According to resultant motion theorem,the absolute velocity of fin is described as
V a=V+V Z͑11͒where V is the robot swimming or embroiled velocity and V Z is the lateral or relative velocity.
Suppose n and t are unit vectors normal and tangential to the deflection curve of thefin,respectively,in the coordinate system Oxz shown in Fig.3.The speeds of thefin along axes x and z are
V x=U,V z=
ץh
ץt
͑12͒
Decomposing the speeds in Eq.͑12͒in the directions of n and t yields
V n=V X sin͑␣͒+V Z cos͑␣͒
V t=−V X cos͑␣͒+V Z sin͑␣͒͑13͒For small vibrations,␣=tan͑␣͒=−ץh/ץx.Thus the tangential force dF t and normal force dF n on a small segment of thefin surface are described,respectively,as follows:
dF t=−1
2
C t␳V tʈVʈt ds,dF n=−12C n␳V nʈV nʈds
The forces dF t and dF n in the coordinate system Oxz are
dF x=−dF t cos͑␣͒+dF n sin͑␣͒
dF z=dF t sin͑␣͒−dF n cos͑␣͒͑14͒Integrating the above equations along the whole length l m yields F x͑f,t͒=͵−l m012␳b͑x͒͑C t V tʈV tʈ−C n V nʈV nʈtan͑␣͒͒dx͑15͒F z͑f,t͒=−͵−l m012␳b͑x͒͑C t V tʈV tʈtan͑␣͒+C n V nʈV nʈ͒dx͑16͒
The second order mode shape is taken as an example to eluci-date how to apply the derived model to calculate propulsion.The propulsion generated by a vibrating caudalfin,as formulated in Eq.͑15͒,is a periodic function.Thus it is only necessary to ana-lyze propulsion in one complete cycle.For convenience of analy-sis,the cycle is divided into four parts,and analysis of the pro-pulsion in each quarter of the cycle is conducted.
In the meantime,the second order mode shape curve is divided into three sections along the x axis direction:OA,AB,and BC. Points O and B are called nodes,which are the intersectional points of the mode shape and the x axis in coordinate system Oxz. Nodes remain stationary duringfin vibration.Because the moving trend of one section is different from the others,a mode shape has to be divided into several sections for convenience of propulsion analysis.For the second order mode shape,motion tendency of three sections is shown in Fig.4in thefirst quarter of a cycle. According to the motion tendency of each section,the propul-sive force in the x axis is obtained using Eq.͑15͒in thefirst quarter of the cycle,and propulsions in the other three quarters of the cycle are obtained similarly.The average propulsion in the whole cycle is taken as the mean propulsion of the second order mode shape.Propulsion of other mode shapes can be obtained
in Fig.3Motion and propulsive force analysis of afin
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the same way.
When the mean propulsive force is positive,it acts as a push force;otherwise,it acts as a drag force.The above analysis pro-cess leads to the mean propulsive force ͓10͔
F ¯͑f ,V ͒=
͵
T
F x ͑f ,t ͒dt T
͑17͒
where T is the period of cycles.The drag force on the robot hull is determined by
F R =1
2C n
Ј␳S b V 2͑18͒
where V is the swimming speed of the robot,S b is the wetted area
of the robot hull body,and C n Јis the drag coefficient of the robot hull.
The drag coefficient C n Јof the robot hull is obtained in the same way as that for the fin.For h 1/b 1=8/8=1,C n Ј=1.15,where h 1and b 1are the height and the width of the robot hull,respectively.The intersectional point of the mean propulsive force curve and drag force curve corresponds to the mean swimming speed.Thus by combining Eqs.͑17͒and ͑18͒,the robot mean swimming speed can be expressed as
V ͑f ͒=
ͱ
2
͵
T
F x ͑f ,t ͒dt
TC n
Ј␳S b ͑19͒
where T is the period of the applied alternating magnetic field.
4Bidirectional Motion Control
Bidirectional control is important for applications of swimming
robots.A new drive and control method based on double reso-nance frequencies is developed after exploring the relationship between the vibration mode shape and the mean propulsive force.4.1Motion Analysis.A simulated mean propulsive force curve using Eq.͑15͒at I =0.5A for the fin with GMFs sputtered on the opposite sides of a 35␮m polyimide film is as shown in Fig.5.
The vibration deflection of a GMF fin is mainly comprised of the anterior three order resonance modes.It is well known that the amplitude of vibration reaches the maximum when the driving frequency is equal to the first order resonance frequency.
However,the magnitude and direction of mean propulsive force,as confirmed by theoretical calculation and experiments,are determined mainly by the mode shape.For instance,the mean propulsive force remains small at the first order mode although the vibration amplitude reaches the maximum,and the mean propul-sive force reaches the maximum at the second order mode,while the mean propulsive force becomes negative at the third order mode,indicating that the vibrating fin at the third order mode
generates backward propulsion.According to Eq.͑19͒,the simu-lated mean speed curve versus the driving frequencies at three different magnetic field strengths for the same GMF actuator as the caudal fin is shown in Fig.6.
The mean speed curve demonstrates similar characteristics as that of the mean propulsive force,i.e.,the mean speed remains small at the first order resonance or natural frequency,and it reaches the maximum at the second order resonance mode,while the mean speed is negative at the third one,indicating that the robot swims backward.The mean speed curve is in agreement with our previous experimental results.
According to the relationship between the mean propulsive force and natural frequencies shown in Fig.6,the adverse mean swimming speed is small when driven at the third order resonance mode frequency,which is usually so high that it arouses high impedance inside coils,resulting in aberrance and attenuation of magnetic field.Thus it is not an efficient way to implement ad-verse swimming control.
The characteristics that maximum mean swimming speed is reached at the second order natural frequency can be utilized
to
Fig.4Motion tendency of three sections in the first quarter of a
cycle
Fig.5Mean propulsive force varies with driving
frequency
Fig.6Mean swimming speed varies with driving frequency
Journal of Dynamic Systems,Measurement,and Control
JANUARY 2009,Vol.131/011004-5
realize bidirectional swimming by installing two GMF actuators with different second order natural frequencies at the front and at the rear of a robot.If the interference between the two actuators is ignored,the swimming speed of this bidirectional swimming ro-bot can be obtained as follows from Eq.͑19͒:
V C ͑f ͒=
ͱ2
͵
T
F 1x ͑f ,t ͒dt
TC n
Ј␳S b −
ͱ2
͵
T
F 2x ͑f ,t ͒dt
TC n
Ј␳S b ͑20͒
where V c ͑f ͒is the speed of the double actuator swimming robot.
Thus we can control the speed of the swimming robot in two directions continuously by adjusting the driving frequency within the two second order natural frequencies of the two GMF actua-tors,featuring low driving frequencies and a wide speed range,as illustrated in Fig.7,where f 2is the second order natural frequency of one GMF actuator,f 2Јis the second order natural frequency of the other actuator,and f 0is the critical driving frequency,corre-sponding to the frequency of zero speed.
Ideally,for symmetrical bidirectional control of the robot,the following are satisfied by design:
V c ͑f 2͒=−V c ͑f 2Ј͒f 2
Ј−f 0=f 0−f 2͑21͒
The two GMF actuators of the bidirectional robot can be made by
sputtering T b 0.28D y 0.72F e 1.93and S m F e 1.95on the opposite sides of two different film substrates in the same length and width,whose second order natural frequencies could be adjusted by the thick-ness of sputtered T b 0.28D y 0.72F e 1.93and S m F e 1.95films,or the sub-strate film,to satisfy the design requirements defined in Eq.͑21͒.A bidirectional robot prototype has been built to verify the drive principle of using double resonance frequencies.One actuator with a polyimide film substrate has a second order natural fre-quency of f 12=21Hz,while the other actuator with a copper film substrate has a second order natural frequency of f 22=82Hz.Speed simulation curves using bidirectional drive robot model ͑20͒are shown in Fig.8,indicating that the bidirectional drive can be realized between the driving frequencies of 21Hz and 82Hz.4.2Mechanical Efficiency.Based on the model for calculat-ing the propulsion and mean swimming speed,the mechanical drive efficiency ͓17͔of this bidirectional robot prototype is studied.
Using Eqs.͑15͒and ͑20͒,the power of the robot with two fins in the direction of the x axis can be described as
P x =͑F x 1͑f ,t ͒−F x 2͑f ,t ͒͒V c ͑f ͒
Combining Eqs.͑10͒and ͑16͒,the power of the head fin and that of the tail fin of the robot in direction of axis z are P z 1=−
͵
−l m 1
12␳b ͑x ͒͑C t V t ʈV t ʈtan ͑␣͒+C n V n ʈV n ʈ͒ץh 1͑f ,x ,t ͒ץt
dx P z 2=−
͵
−l m 2
12␳b ͑x ͒͑C t V t ʈV t ʈtan ͑␣͒+C n V n ʈV n ʈ͒ץh 2͑f ,x ,t ͒ץt
dx The total power of the robot with two fins in the direction of the
z axis is
P z =P z 1+P z 2
͉P x ͉+͉P z ͉is the total consumed power,while ͉P x ͉is the useful power.Thus the mechanical efficiency is
␩=
͉P x ͉͉P x ͉+͉P z ͉
͑22͒
According to Eq.͑23͒,the mechanical efficiency of the robot with regard to different driving frequencies can be obtained;the absolute values of P x and P z in Eq.͑22͒keep the driving effi-ciency of both forward and backward swimming positive.
However,at the critical driving frequency,P x is zero,i.e.,the robot model reaches nought driving frequency and stays motion-less.The simulated mechanical efficiency curve versus the driving frequency is shown in Fig.9,which features a horn shape.
The highest drive efficiency is less than 0.2.This is determined by both the physical feature and the structure of GMF actuators.If the amplitude of vibration reflection becomes bigger,the robot has higher efficiency.Obviously,when the size of the robot hull is kept constant,the robot with a smaller fin length has lower effi-ciency.When the robot has a constant caudal fin length,a smaller hull size leads to higher drive
efficiency.
Fig.7Velocity varies with second order resonance
frequency
Fig.8Bidirectional drive speed versus driving frequency for
three different strengths of magnetic
field
Fig.9Swimming efficiency of robot versus excitation
frequencies
011004-6/Vol.131,JANUARY 2009Transactions of the ASME。