genetic algorithm optimization of escape and normal swimming gaits for a hydrodynamical model of...

7/29/2019 Genetic Algorithm Optimization of Escape and Normal Swimming Gaits for a Hydrodynamical Model of Carangifor

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Genetic Algorithm Optimization of Escape and Normal Swimming

Gaits for a Hydrodynamical Model of Carangiform Locomotion

P. D. Kuo

Systems Design Engineering Dept.University of WaterlooWaterloo, ON N2L 3G1

D. Grierson

Civil Engineering Dept.University of WaterlooWaterloo, ON N2L 3G1

Abstract

A hydrodynamic model of a three segment fish is

presented. This model is then optimized using agenetic algorithm subject to two separate criteria;escape and normal swimming behaviour. Withthis optimization, gaits for normal swimmingand escape behaviour are obtained.

1 INTRODUCTIONHuman engineered solutions to underwater propulsiontypically depend on the use of propellers andmaneuvering surfaces. In contrast, natures solutions usechanges in the overall shape of the body of the object inorder to generate thrust and thus propulsion.

It can be argued that the underwater locomotion methodsemployed by nature are superior to human-designeddevices (Mason et al., 2000). One advantage is greatermaneuverability. There exist fish that can perform 180-degree turns within a fraction of a body length ascompared to boats, which typically have very largeturning radii (Mason et al., 1999).

As well, due to the evolution of the so-called escaperesponse (Domenici et al., 1997), fish can acceleratequickly from rest in order to escape from predators. Thisagility also allows them to work in considerably complexhydrodynamic environments such as surf surge andlittoral zones that are currently beyond the capabilities of

propeller driven craft.

It has also been shown by Triantafyllou et al. (1993) thatthe motion of a mechanical tuna significantly reduces itsown drag thus increasing swimming efficiency.

Another advantage is stealth. Human-made craft sufferfrom noisy cavitation created by the propeller. A study byAhlborn et al. (1991) using an artificial fishtail to mimicfishlike swimming concluded that the alternating creationand destruction of the vortices in the wake behind the fishtail provided not only an efficient method of swimming,

but also aided in guarding against detection by predators.

The advantages of improved efficiency, stealth, turningradii, and acceleration could prove to be significant ifapplied to the design of fishlike propulsion systems.

Understanding the dynamics of such systems could provebeneficial in a number of application areas such as

autonomous robots for remote-sensing operations inrivers, littoral zones and at the ocean bottom where robustadaptations to irregular bottom contours, current andsurge are required.

Therefore, obtaining optimum gaits for such devices is animportant undertaking.

2 CARANGIFORM LOCOMOTIONNature's solutions to underwater propulsion are numerous.One such solution is so-called carangiform motion used

by swordfish, tuna, mackerel and salmon. Breder (1926)introduced the term carangiform to members of the

Carangidae family as well as other fish exhibiting similarbody morphology and propulsive efficiency. These fastswimming fish typically have large, high-aspect-ratio tailswith still waves propagating from head to tail as seen inFigure 1.

Figure 1: Carangiform gait (Gray, 1968).

This requires a powerful musculature that generates side-to-side movement of the posterior vertebral column andflexible tail. The hydrodynamics of propulsion with thisside-to-side movement involve the generation of vortices


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within the wake through which water is propelled with thetail also generating lift.

In carangiform locomotion, the amplitude of the stillwaves increases closer to the rear while the fishs body ismostly rigid at the front. Fish classified as usingcarangiform locomotion generally use only the rear third

of the body to move.For a complete treatment of underwater locomotionmethods and analyses please refer to Breder (1926) andDomenici et al. (1997).

3 HYDRODYNAMICAL MODELA model of a three segment fish based on previous work

by Kelly et al. (1998), Mason et al. (1999, 2000) andMorgansen et al. (2001, 2002) has been implemented.This model allows various swimming gaits to be studiedas related to the forces required to generate propulsion.

Three segments model the fish - the body, the peduncle

and the tail. The body segment of the fish can be modeledas a rigid body and is connected to the tail by the

peduncle a slender segment of generally negligiblehydrodynamic influence (Mason et al., 2000).

Motion is analyzed only in two dimensions of thehorizontal plane, thus allowing a simplified analysis ofthe thrust generated. The direction of intended travel(longitudinal to the fish body) is referred to as the xdirection. Travel in the lateral direction is referred to astravel in they direction.

Values required for the evaluation of the thrust generatedcan be derived using the Kutta-Joukowski theorem andassuming that the tail hydrofoil is in a quasi-steady

uniform flow with the overall velocity being implied bythe instantaneous velocity of the foils quarter-chord point(Kelly et al., 1998, Mason et al., 1999, 2000, Morgansenet al., 2001, 2002). The drag forces acting on the tailfincan be estimated using Lanchester-Prandtl wing theory.

In total, the model takes into account the followingconsiderations: quasi-static torque generated around themidpoint of the tail, total drag moment acting on the

body, the moment added by mass forces, lift, drag forceacting on the body and fin, and additional mass forces dueto acceleration effects.

Due to simplifications with the model, the special spatialstructure of the wake has been ignored. Vortices shed

from the tailfin are treated as if they are swept away andbecome immediately very distant. Hydrodynamicinteractions between the different components (fin,

peduncle, and body) are ignored. Forces on the peduncleare also ignored.

The model is characterized by ten states: 1 , 2 , x , y , , 1& , 2& , x& , y& , and & . Note that the last five statesare the time derivatives of the first five states. 1 is theangle of the peduncle joint with respect to the longitudinalaxis of the body, 2 is the angle of the tail with respect tothe longitudinal axis of the body, is the position of the

body in the longitudinal direction, y is the position of thebody in the lateral direction and corresponds to theorientation of the body in the inertial reference frame. Allvariables are functions of time t. The equations of motionare:

++++

+++

+++=

)(),(

___

___

2

1

2

1

ammmambf

yamybyfy

xamxbxfxtotal

fLyyxx

fDDL

fDDL

uu

y

xI

&&

&&

&&

&&

&&

where u1 and u2 are the input signals to the system whichrepresent the acceleration of the joint angles with respectto the bodys longitudinal axis,L is the lift forceon the tailfin,Df is the drag force on the tail fin, Db is the drag onthe body,fam is added mass forces due to the acceleration

component of the tail fin,

f is the quasi-static torquegenerated around the midpoint of the tail, b is the totaldrag moment acting on the body, am is the added momentdue to the acceleration component of the tail fin, xm andym are the displacements of the midpoint of the tail in thex andy directions respectively, andItotal is the total inertiaof the system including the inertia of the fluidsurrounding the body and the inertia of the body and tailfin.

It must be stressed that this model is quite simplified butdoes perform relatively well as compared to empiricalresults (Mason et al., 1999, 2000).

4 BEHAVIOUR SELECTIONIn order to optimize the hydrodynamical model fordifferent behaviours, fitness criteria must be selected. Oneway in which to select gaits is to use those which wouldhypothetically lead to a higher survival value for theorganisms which are being modelled. Two behaviourswere chosen: escape gait and normal swimming (mostefficient gait).

An escape gait has survival value for an organism since itmust be able to quickly escape from predators in order toavoid being harmed. The other behaviour, normalswimming, is important due to the fact that an everydaylocomotion gait must be energy efficient. Minimizing

energy use during normal swimming increases survivalvalue since less energy is required and therefore lessenergy needs to be inputted into the system in the form ofcalories.

4.1 NORMAL SWIMMING BEHAVIOURFITNESS FUNCTION

Normal swimming can be characterized as a lowamplitude oscillation having a duration of more than 10s(Domenici et al., 1997). Therefore, the fitness criterion formeasuring normal gait was the distance travelled in 10


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seconds subject to a penalty on inefficient gaits.Efficiency is calculated by relating an input signal to its

power via:

dttpower

=2

)(

Therefore, the fitness criterion was:

inputted

xnormal

power

ntdisplacemeFitness =

The objective is to maximize the fitness subject to apenalty on movement in the lateral direction since onlyforward moving gaits were considered. This is donethrough the implementation of a penalty function which istaken as:

inputted

xtotalnormal

powerntdisplacementdisplacemep =

Therefore, the fitness function to be maximized thenbecomes:

normal

inputted

xnormal p

power

ntdisplacemeFitness =

4.2 ESCAPE BEHAVIOUR FITNESS FUNCTIONThe escape behaviour objective was determined byobtaining the farthest lateral distance travelled within a

given small time period. Fast fish starts are characterizedas an escape response when movement occurs for lessthan 1 second after an escape stimulus has been evoked(Domenici et al., 1997).

A C or S-shaped bend is normally produced towards thestimulus effectively pushing the fish away from thestimulus (Budick et al., 2000). This bend generally

produces a very quick turn away from the stimulus.Therefore, the following equation was taken tocharacterize the fitness criterion:

yescape ntdisplacemeFitness =

This objective is to maximize the criterion subject to apenalty on the distance travelled in the forward directionvia the following penalty function:

ytotalescape ntdisplacementdisplacemep =

This leads to the following overall fitness function to bemaximized for escape behaviour:

escapeyescape pntdisplacemeFitness =

Note that this fitness function enforces that the fish travel90o as much as possible from the direction that it was

previously travelling. This type of behaviour can beconsidered representative given information regardingescape bends (Domenici et al., 1997).

Note that the direction of the turn as described in thefitness function is not predetermined. A 90o turn to theleft or the right is equally valid. Obtaining an optimumgait for the opposite direction from the resultant gaitobtained through the fitness function is simply a matter ofnegating the amplitudes of the inputted gaits to the model.In this way, optimum turning gaits may be found for turnsin either direction.

With the two fitness functions characterized (escape andnormal swimming), an optimization may now be

performed in order to determine the optimal gaits for eachfitness function.

5 MODEL OPTIMIZATIONPrevious work by Barrett (2002) indicates that geneticalgorithms can be an effective means from which toobtain swimming gaits. However, this previous workinputted gaits obtained by the genetic algorithm directlyinto the experimental apparatus.

This project attempts to obtain optimal gaits underseparate objectives, entirely in simulation(mathematically). The simulation is obtained fromgenerating the requisite equations of motion and then

performing a subsequent optimization.

5.1 GENETIC ALGORITHMSJohn Holland defined genetic algorithms in his seminal

book Adaptation in Natural and Artificial Systems(Holland, 1975). Holland contended that the concepts of

biological evolution could serve as a metaphor forartificial systems. Genetic algorithms use the concept ofsurvival of the fittest by randomly initializing a

population of individuals, where each individual containsthe parameters to a possible solution of a globaloptimization problem.

Each individual of the population is assigned a fitnessvalue that objectively indicates its quality as a solution tothe problem based on its parameters (Holland, 1975).Individuals are then selected to become parents based ontheir fitness to form the next generation of potentialcandidate solutions to the problem. New potentialsolutions are formed from the creation of child solutionsvia the metaphors of genetic crossover and mutation.

The initialization step of the genetic algorithm, wherebyrandom solutions to the problem are generated, effectivelysamples the possible set of solutions. From this initial setof random solutions, the most promising ones are giventhe chance to pass parts of their genetic string into thenext generation. In this way, the overall population moves


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towards the optimum. A thorough treatment of geneticalgorithms may be found in Goldberg (1989), Holland(1975) and Mitchell (1996).

5.2 GENETIC STRING ENCODINGSolutions to this problem were taken to be of the form:

)sin( += t&&

This was assumed to be the optimal form from a separatenon-linear analysis (Morgansen et al., 2001, 2002). Thisallows a parameterization of the solution into a geneticstring representation.

There are six parameters inputted for optimization (threeeach for the peduncle and tail) - the amplitude (),frequency () and phase of oscillation ().

Since the phase angles can be thought of as specifying aphase difference between the peduncle and tail fin, one ofthe phases specified is redundant. Therefore, only one

phase angle was inputted to the system with the otherphase angle always assumed to be 0.

Further, since it is biologically implausible to have thepeduncle and tail joints oscillating at different frequencies(), one of these inputs was eliminated so that both jointsoscillate at the same angular frequency. Thus, the inputsto the model were reduced from six to four parameters.

The parameters in the genetic string can be thought of asforming a 4-D state space of possible combinations ofsolution vectors. Each solution vector can be thought of asrepresenting an individual fish with the associatedswimming gait defined by the parameters of the vector.

5.3 GENETIC ALGORITHM (GA)IMPLEMENTATION

Two versions of the genetic algorithm were attempted.The first version used a random initial population,floating point representation, population size of 50, rank-

based selection, crossover with probability 85% andmutation with probability 5%. The second version of thegenetic algorithm used the same parameters as the firstversion except for the use of a crossover blendingoperator instead of the traditional real-valued crossover.

Elitism was also used for both versions of the geneticalgorithm with the percentage of the population retained

set at 15%. Therefore, the top 15% of each generation wasallowed to pass to the next generation without beingsubject to crossover or mutation with the remainingindividuals being created having an 85% probability ofcrossover and a 5% chance of mutation. The convergencecriterion was set so that the GA terminated when the bestindividual found up to that point remained unchanged for25 generations.

Since the genetic algorithm is a stochastic method whichdoes not guarantee finding an optimum and most likelyreturns different answers for each run, it was decided that

10 runs of the algorithm would be performed in order togauge its performance.

6 RESULTS6.1 FIRST VERSION OF GENETIC

ALGORITHM

The parameters for the highest fitnesses found for theescape behaviour were:

Fitness # gens peduncle tail1 4.50 27 0.97 15.80 2.08 2.39

2 7.14 136 2.40 11.90 0.50 1.22

3 12.30 108 2.37 15.20 3.62 2.39

*4 12.40 107 2.39 15.20 3.62 2.39

5 11.20 94 2.35 15.00 3.47 2.40

6 11.08 78 2.35 15.24 3.56 2.36

7 12.26 140 2.39 15.25 3.63 2.39

8 11.40 72 2.39 15.38 3.64 2.39

9 11.18 106 2.40 15.03 3.48 2.37

10 11.79 67 2.39 15.24 3.58 2.39

High 12.40 140 2.40 15.80 3.64 2.40

Low 4.50 27 0.97 11.90 0.50 1.22

Mean 10.52 93.50 2.24 14.92 3.12 2.27

Median 11.30 100.00 2.39 15.22 3.57 2.39

Std

Dev. 2.60 33.90 0.45 1.09 1.03 0.37

* best run

The resultant input of the optimal gait (gait 4) into thehydrodynamical model is shown in Figure 2 with the

progress of the GA run shown in Figure 3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

-4

-3

-2

-1

0

1

2

3

time (s)

x displacement (m)

y displacement (m)

theta (rad)

Figure 2: GA 1st version, optimal escape behaviour.


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0 20 40 60 80 100 1200

2

4

6

8

10

12

14

Generation

Fitness

Best Individual

Population Average

Final Fitness=12.3601a

p=2.3934

W=15.2038

p=3.6241a

t=2.3929

Figure 3: Progress of GA 1st version, escape behaviour.

The gaits found via the GA optimization for the escapebehaviour all have very large amplitudes () at or near themaximum bounds for the amplitude parameters. Thisindicates that for escape behaviour, these two parametersshould be at their respective maximums. Therefore, theupper-bound for these parameters should be set at valueswhich correspond to physical limits for the motion

produced. As well, the angular frequency () was foundto be at approximately the same value for each run

It is noted from Figure 2 that the amount of turningachieved by this gait was very small. This comes as no

surprise due to the assumptions inherent in the model ofthe fish. By modeling only three rigid segments, a quickturn is not easily generated.

Quick escape-type turns in fish are generated through alarge C-shaped bend in the body. Assuming a three-segment fish body makes achieving the forces required inorder to create a sharp turn in the water difficult. As can

be seen in Figure 2, the displacement in the y direction isvery small. It is quickly overtaken by the displacement inthe x direction. This would indicate that either this modelis not a good one in order to study escape type bends orthat the fitness criterion is not appropriate.

The GA criterion for efficient swimming was also run 10

times in order to gauge the performance of the algorithm.The optimal results of the GA runs for the efficientswimming criterion were:


2 215.00 80 0.06 15.90 3.06 0.06

*3 560.00 75 0.01 16.00 3.15 0.01

4 348.00 67 0.05 15.60 2.95 0.02

5 536.00 170 0.02 16.00 3.06 0.016 555.00 92 0.01 15.80 3.03 0.01

7 544.00 169 0.00 15.90 2.93 0.00

8 293.00 70 0.04 15.80 3.08 0.04

9 273.00 118 0.05 15.90 3.08 0.04

10 343.00 89 0.04 15.90 3.07 0.03

High 560.00 170 0.06 16.00 3.15 0.06

Low 215.00 67 0.00 15.60 2.93 0.00

Mean 415.60 102.50 0.03 15.86 3.04 0.02

Median 418.50 90.50 0.03 15.90 3.06 0.01

Std

Dev. 134.21 38.22 0.02 0.12 0.07 0.02

* best run

Note that the values indicating the amplitude of thegaits produced at the joints is significantly smaller thanthose found for the escape behaviour. This is as expectedsince large amplitude gaits were penalized by the GA

because of their energy requirements. We can see that thegaits found are all small amplitude, high frequency gaitsas was expected. As well, as can be seen in Figure 4, theswimming trajectory of the optimal gait (gait 3) is in theforward direction with little or no movement in the lateral(y) direction as expected. The progress of the GAoptimization for the optimal run is shown in Figure 5.

0 1 2 3 4 5 6 7 8 9 10-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Time (s)

x displacement (m)

y displacement (m)

theta (rad)

Fitness=559.7922a

p=0.0097163

W=15.9655

p=3.1458a

t=0.0067468

Figure 4: GA 1st version, optimal normal gait.


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0 10 20 30 40 50 60 70 80-100

0

100

200

300

400

500

600

Generation

Fitness

Best Individual

Population A verage


p=0.0097163

W=15.9655

p=3.1458a

t=0.0067468

Figure 5: Progress of GA 1st version, normal gait.

6.2 SECOND VERSION OF GENETICALGORITHM USING CROSSOVER

BLENDING

A second version of the GA optimization added the use ofa blended crossover operator in an attempt to obtain evenhigher fitness solutions. The previous crossover operatorcuts the solution chromosome only at the boundaries

between solution parameters. This creates a child that ismade up strictly of parts of each parent solution.

The blended crossover operator adds another parameter

that is chosen randomly. Instead of directly cutting thesolution chromosome at the boundaries betweenparameters, a site within one of the parameters (betweenboundaries) may be chosen.

All parameters other than the one cut are made up of thetwo parent solutions much in the same way as the

previous crossover described. However, the solutionparameter of the child at the crossover point is composedof a randomly chosen weighted sum of the two parents forthis given parameter. Therefore, the child solution iscomposed of the first parent up to the crossover point, alinear combination of the first and second parent at thecrossover point, and then the rest of the second parentmaking up the child solution (Mitchell, 1996).

It was hoped that such a crossover technique would addmore exploratory type characteristics to this real-valuedGA. By combining weighted sums of a single parameterin each child solution, we are effectively adding newgenetic material into the search. This can also be thoughtof as a form of crossover with a directed mutation(Mitchell, 1996).

The gaits obtained for the escape behaviour were asfollows:

Fitness # gens peduncle tail*1 12.53 198 2.383 15.166 3.674 2.391

2 6.71 198 2.390 12.047 0.337 1.108

3 11.17 99 2.286 15.156 3.584 2.346

4 7.07 222 2.398 11.941 0.481 1.181

5 7.04 137 2.339 11.902 0.529 1.2376 11.94 208 2.371 15.145 3.547 2.399

7 11.90 258 2.288 15.178 3.566 2.392

8 12.47 146 2.398 15.202 3.673 2.388

9 5.94 58 0.546 15.269 3.588 1.936

10 8.32 36 2.075 15.426 3.637 2.256

High 12.53 258.00 2.40 15.43 3.67 2.40

Low 5.94 36.00 0.55 11.90 0.34 1.11

Mean 9.51 156.00 2.15 14.24 2.66 1.96

Median 9.74 172.00 2.36 15.16 3.57 2.30

Std

Dev. 2.72 73.56 0.57 1.58 1.53 0.56

* best run

The resultant input of the optimal gait (gait 1) into thehydrodynamical model is shown in Figure 6 with the

progress of the GA run shown in Figure 7.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

-4

-3

-2

-1

0

1

2

3

4

Time (s)

x displacement (m)

y displacement (m)

theta (rad)

Fitness=12.5337a

p=2.3835

W=15.166

p=3.6741a

t=2.3906

Figure 6: GA 2nd

version optimal escape behaviour.


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0 20 40 60 80 100 120 140 160 180 200

2

4

6

8

10

12

14

Generation

ness

Best Individual

Population Average


p=2.3835

W=15.166

p=3.6741a

t=2.3906

Figure 7: Progress of GA 2nd version, escape behaviour.

A slightly higher fitness was obtained using thismodification. However, note that the results of the runsusing blended crossover were not significantly differentfrom those of the previous algorithm version for escape

behaviour.

This is to be expected since for the previous version, mostparameters were at their upper bounds. Since this will alsolikely be the case for any other algorithm modification, itis not surprising that the results of the runs were verysimilar to those obtained in the previous version.

For the normal swimming gaits, the following gaits wereobtained using the blended crossover operator:


2 254.00 92 0.064 15.900 3.040 0.043

3 307.00 81 0.038 15.800 3.400 0.034

4 513.00 116 0.016 15.900 3.040 0.010

5 605.00 89 0.012 15.900 3.010 0.004

6 433.35 143 0.035 15.987 3.049 0.011

7 376.48 78 0.045 15.949 2.876 0.015

*8 635.61 54 0.010 15.990 3.115 0.003

9 477.16 166 0.027 15.974 2.990 0.010

10 486.83 266 0.025 15.978 3.037 0.011

High 635.61 266.00 0.06 15.99 3.40 0.04

Low 254.00 54.00 0.01 15.80 2.88 0.00

Mean 460.74 126.20 0.03 15.93 3.06 0.02

Median 482.00 104.00 0.03 15.92 3.04 0.01

Std

Dev. 121.42 63.45 0.02 0.06 0.13 0.01

* best run

Note that the fitness measure is generally higher for theblended crossover operator than for the previousalgorithm version. This may be explained by the moreexploratory aspects of this crossover operator. Based onthe data obtained, it would appear that the blendedcrossover operator is superior to the previous standardcrossover operator given the efficient gait criterion.

The result of the optimal gait (gait 8) when inputted intothe model are shown in Figure 8 with the progress of theGA optimization shown in Figure 9. Notice that thetrajectory ofx,y and are similar to those presented inFigure 4.

0 1 2 3 4 5 6 7 8 9 10-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

Time (s)

x displacement (m)

y displacement (m)

theta (rad)

Fitness=635.6125a

p=0.0095125

W=15.9898

p=3.1148a

t=0.0032824

Figure 8: GA 2nd version optimal normal gait.

0 10 20 30 40 50 60-100

0

100

200

300

400

500

600

700

Generation

Fitness

Best Individual

Population Average


p=0.0095125

W=15.9898

p=3.1148a

t=0.0032824

Figure 9: Progress of GA 2nd version, normal gait.


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Results from both fitness criteria would suggest that theblended crossover operator does in general provide higherfitness solutions than the previous crossover operator.

7 CONCLUSIONSBy creating a hydrodynamical model of a fish, andsuitable fitness functions for the different criteria ofinterest, genetic algorithms may be used as a method forobtaining gaits for underwater locomotion.

As well, for this specific application of the geneticalgorithm, the crossover blending operator gives higherfitness solutions than the traditional real-valued crossoveroperator. This is most likely due to the operatorsincreased ability to explore the search space by combiningvalues of the two parents to form a new parameter at thecut location.

Future Work

Future work will focus on generating an extendedhydrodynamic model with more segments than the threecurrently implemented. This should allow more effectivesharp turning motions to be generated. As well, thisallows comparisons between the two models showing theefficacy of modeling carangiform motion using threesegment models.

Further modifications to the genetic algorithm will also bestudied such as the effects of population size, elitism anddifferent convergence criteria.

Acknowledgments

The authors would like to acknowledge Chris Eliasmithfor his help in formulating the hydrodynamical model ofthe fish.

References

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genetic algorithm optimization of escape and normal swimming gaits for a hydrodynamical model of...

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