Evolving Gait Control of Physically Based Simulated Robots

16 Pages·2012·0.38 MB·English

Evolving Gait Control of Physically Based Simulated Robots
MiltonRobertoHeinen1
FernandoSantosOs(cid:243)rio2
Abstract: ThispaperdescribestheLegGenSystem,usedtoautomaticallycreate
andcontrolstablegaitsforleggedrobotsintoaphysicallybasedsimulationenviron-
ment. Inourapproach,thegaitisde(cid:2)nedusingthreedifferentmethods: a(cid:2)nitestate
machine based on robot’s leg joint angles sequences; a locus based gait de(cid:2)ning the
endpoint(paw)trajectoryandusinginversekinematicstodeterminejointangles;and
throughahalfellipsecyclicfunctionusedtode(cid:2)neeachendpointtrajectory. Thepa-
rametersusedtocontroltherobotthroughthesemethodsareoptimizedusingGenetic
Algorithms. The model validation was performed by several experiments realized
withdifferentcon(cid:2)gurationsoffourandsixleggedrobots,simulatedusingtheODE
physical simulation engine. A comparison between these different robot con(cid:2)gura-
tionsandcontrolmethodswasrealized,andthebestsolutionwasselectedinorderto
helpustobuildaphysicalleggedrobot. Theresultsalsoshowedthatitispossibleto
generatestablegaitsusingGeneticAlgorithmsinaef(cid:2)cientmanner,usingthesethree
differentmethods.
1 Introduction
Theautonomousmobilerobotshasbeenattractingtheattentionofagreatnumberof
researchers,duetothechallengethatthisnewresearchdomainproposes:makethesesystems
capableofintelligentreasoningandabletointeractwiththeenvironmenttheyareinsertedin,
through sensor’s perception (infrared, sonar, bumpers, gyroscopes, etc) and motor’s action
planning and execution [10, 17]. At the present time, the most part of mobile robots use
wheels for locomotion, what does this task easy to control and ef(cid:2)cient in terms of energy
consumption, but they have some disadvantages since they have problems to move across
irregular surfaces and to cross borders and edges [18]. So, in order to make mobile robots
better adapted to human environments and to irregular surfaces, they must be able to walk
and/ortohaveasimilarlocomotionmechanismusedbythehumansandanimals,thatis,they
shouldhavealeggedlocomotionmechanism[1].
However,thedevelopmentofleggedrobotscapabletomoveinirregularsurfacesisa
quite dif(cid:2)cult task, that needs the con(cid:2)guration of many gait parameters [28]. The manual
con(cid:2)gurationoftheseparametersdemandsalotofeffortandspenttimeofahumanspecialist,
1InformaticsInstitute,UFRGS,CEP91501-970
{[email protected]}
2AppliedComputing,UNISINOS,CEP93022-000
{[email protected]}
EvolvingGaitControlofPhysicallyBasedSimulatedRobots
and the obtained results are usually suboptimal and speci(cid:2)c for one robot architecture [5].
Thus,itisinterestingtogeneratetherobotgaitcon(cid:2)gurationinanautomaticmanner,using
MachineLearningtechniques[23]toperformthistask.
OneoftheseMachineLearningtechniquesthataremostadaptedforthisspeci(cid:2)ctask
are the Genetic Algorithms (GA) [8, 22]. This is a reasonable choice because according to
theEvolution’sTheory[6],thelocomotionmechanismsoflifeformsresultedfromthenat-
ural evolution, what makes the use of Genetic Algorithms a natural solution since they are
biologically inspired and can generate biologically plausible solutions. From the computa-
tionalpointofview,theGeneticAlgorithmsarealsoverywelladaptedfortheautomaticgait
con(cid:2)gurationofleggedrobots,because:(a)theyuseamulti-criterionoptimizationmethodto
searchsolutionsinthecon(cid:2)gurationspace,thatmeansinourspeci(cid:2)ccase,theyarecapable
to optimize not only the gait velocity, but also the stability and even other gait parameters;
(b)theydon’tneedlocalinformationfortheerrorminimization,northegradientcalculation,
what is very important for the gait parameters generation and optimization, since it is very
dif(cid:2)culttohaveavailablesomeaprioritrainingdataforsupervisedlearning;(c)ifcorrectly
used,theGeneticAlgorithmsarecapabletoavoidlocalminima[22].
ThemaingoalofthispaperistodescribetheLegGenSystem[12,13,11].Thissystem
iscapabletoautomaticallyevolvethegaitcontrolofphysicallybasedsimulatedleggedrobots
usingGeneticAlgorithms. Thispaperisstructuredasfollows: TheSection3describesthe
Genetic Algorithms and the GAlib software library adopted in our system; The Section 4
theuseofaphysicalsimulationengine,theSection5describessomepossiblealternativesto
leggedrobotcon(cid:2)guration;TheSection6describestheLegGensystem,andtherobotsused
inthesimulations; TheSection 7describestheaccomplished experiments andtheobtained
results;andtheSection8providessome(cid:2)nalconclusionsandfutureperspectives.
2 RelatedWorks
Control of locomotion in legged robots is a challenging multidimensional control
problem [7, 1]. It requires the speci(cid:2)cation and coordination of motions in all robots’ legs
while considering factors such as stability and surface friction [19]. This is a research area
whichhasobvioustieswiththecontrolofanimallocomotion,anditisasuitabletasktouse
toexplorethisissue[27]. Ithasbeenaresearchareaforaconsiderableperiodoftime,from
the (cid:2)rst truly independent legged robots like the Phony Pony built by Frank and McGhee
[21],whereeachjointwascontrolledbyasimple(cid:2)nitestatemachine,totheverysuccessful
algorithmiccontrolofbipedsandquadrupedsbyRaibert[26].
Lewis[20]evolvedcontrollersforahexapodrobot,wherethecontrollerwasevaluated
on a robot which learn to walk inspired on insect-like gaits. After a staged evolution, its
behavior was shaped toward the (cid:2)nal goal of walking. Bongard [2] evolved the parameters
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of a dynamic neural network to control various types of simulated robots. Busch [4] used
genetic programming to evolve the control parameters of several robot types. Jacob [16],
ontheotherhand,usedreinforcementlearningtocontrolasimulatedtetrapodrobot. Reeve
[27]evolvedtheparametersofvariousneuralnetworkmodelsusinggeneticalgorithms. The
neuralnetworkswereusedforthegaitcontroloftetrapodrobots.
Inthemostpartoftheseapproachesdescribedabove,the(cid:2)tnessfunctionusedwasthe
distancetraveledbytherobotinaprede(cid:2)nedamountoftime. Althoughthis(cid:2)tnessfunction
is largely used, it may hinder the evolution of more stable gaits [9]. In our approach, we
use in the (cid:2)tness function, beyond distance traveled, sensorial information (gyroscope and
bumpers)toguaranteestableandfastgaits.
3 GeneticAlgorithms
GeneticAlgorithmsareoptimizationmethodsofstochasticspacestatesearchbasedon
theDarwin’sNaturalEvolutionTheory[6],thatareproposedinthe60sbyJohnHolland[15].
They work with a population of initial solutions, called chromosomes, which are evolved
through several operations during a certain number of generations, usually reaching a well
optimized solution, and preserving the best individuals according to a speci(cid:2)c evaluation
criterion. Inordertoaccomplishthis, ineachgenerationthechromosomesareindividually
evaluated using a function that measures its performance, called (cid:2)tness function [22]. The
chromosomeswiththebest(cid:2)tnessvaluesareselectedtogeneratethenextgenerationapplying
thecrossoverandmutationoperations.Thus,eachnewgenerationtendstoadaptandimprove
thequalityofsolutions,untilweobtainasolutionthatsatis(cid:2)esaspeci(cid:2)cobjective.
TheGeneticAlgorithmsimplementationusedinoursystemwasbasedontheGAlib
software library3, developed by Matthew Wall of Massachusetts Institute of Technology
(MIT). GAlib was selected as it is one of the most complete, ef(cid:2)cient and well known li-
brariesforGeneticAlgorithmssimulation,andalsoitisafreeandopensourceC++library.
IntheLegGenSystem,aGeneticAlgorithmasdescribedbyGoldberginhisbook[8]
wasused,anda(cid:3)oatingpointtypegenomewasadopted. Inordertoreducethesearchspace,
alleleswereusedtolimitgeneratedvaluesonlytopossiblevaluesforeachparameter.
4 MobileRobotSimulation
In order to do more realistic mobile robots simulation, several elements of the real
world should be present in the simulated model, doing the simulated bodies to behave in a
3theGAlibisavailablefordownloadinthesitehttp://www.lancet.mit.edu/ga/
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similarwayrelatedtothereality. Especially,itisnecessarythattherobotsuffersfrominsta-
bilityandfallsdownifbadlypositionedandcontrolled,andalsoitshouldinteractandcollide
against the environment objects in a realistic manner [24]. To accomplish that, it is neces-
sarytomodelthephysicslawsinthesimulationenvironment(e.g. gravity, inertia, friction,
collision). Nowadays, severalphysicssimulationtoolsexistusedfortheimplementationof
physics laws in simulations. After study different possibilities, we chose a widely adopted
freeopensourcephysicssimulationlibrary,calledOpenDynamicsEngine-ODE4.ODEis
asoftwarelibraryforthesimulationofarticulatedrigidbodiesdynamics. Withthissoftware
library, it’s possible to make autonomous mobile and legged robots simulations with great
physical realism. In ODE, several rigid bodies can be created and connected through dif-
ferent types of joints. To move bodies using ODE, it’s possible to apply forces or torques
directlytothebody,oritispossibletoactivateandcontrolangularmotors.Anangularmotor
isasimulationelementthatcanbeconnectedtotwoarticulatedbodies, whichhaveseveral
controlparameterslikeaxis,angularvelocityandmaximumforce. Withtheseelements,it’s
possible to reproduce articulations present in real robots, humans or animals, with a high
precisionlevel[24].
5 GaitGeneration
Inleggedrobots,thegaitcontrolcanbegeneratedinseveraldifferentways.Oneofthe
mostsimplewaystocontroltherobotjointsisusingaFiniteStateMachine(FSM),inwhich
isde(cid:2)nedforeachstatethedurationofthestateandtheactivationlevel(force/velocity)for
eachjoint. TheTable1illustratesanexampleofaFSMtableusedtocontrolanarticulated
mobile robot. Since the most of the robots are symmetrical (and great part of living forms
too), only one half of the robot’s states need to be learned - the other half is obtained by
switchingtheleftlegsvalueswithrightlegsvalues.
Table1.ExampleofaFSMtable
State1 State2 State3 State4
Duration 0.05 0.10 0.55 0.80
JointA 1.25 2.45 3.00 0.95
JointB 2.70 0.15 0.70 1.95
JointC 1.15 1.65 0.30 2.70
The main disadvantage of this approach described above is that the robot control is
accomplished without any feedback from the external world, that is, no sensor data is used
bytherobotcontroller. Butinarealrobot,smalldifferencesintheactuatorsbehavior,inthe
4AlibraryODEisfreeandavailablefordownloadinthesitehttp://openode.sourcefourge.net
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batterylevelcharge, thefrictioninthejointsorexternalobstaclescollisionmaychangethe
effectivevelocityandresponseofthejoints,andthustherobotmaynotworkasexpected.
An alternative approach was developed in this work, using a FSM table similar to
theabovedescribedmethod,butinsteadofdeterminingthedurationandthevelocityofeach
state,itisdeterminedforeachstateandforeachrobotjointtheir(cid:2)nalexpectedanglescon(cid:2)g-
uration. Inthisway,thecontrollerneedstocontinuallyreadthejointsanglestate,inorderto
checkifthejointmotoraccomplishedthetask. Realrobotsdothisusingsensors(encoders)
tocontroltheactualangleattainedbythejoints[1]. So, inthisapproachthegaitcontrolis
accomplished in the following way: initially the controller verify if the joints have already
reached the expected angles. The joints that do not have reached them are moved (activate
motors),andwhenallthejointshavereachedtheirrespectiveangles,theFSMpassestothe
following state. If some of the joints have not reached the speci(cid:2)ed angles after a certain
limited time, the state is advanced independently of this. In a future version of the system,
we are planning to treat thissituation more carefully, because the leg can be blocked by an
obstacleandtherobotcanbedamagedinthiscase.
Tosynchronizethemovements,itisimportantthatalljointscanreachtheirrespective
anglesatalmostthesametime.Thisispossiblewiththeapplicationofaspeci(cid:2)cjointangular
velocityadjustvalueforeachjoint,calculatedbytheequation:
Vij =ki((cid:11)ij (cid:0)(cid:11)ij(cid:0)1) (1)
whereV isthevelocityadjustappliedtothemotorjointiinthejstate,(cid:11) isthejointangle
ij ij
iinthejstate,(cid:11)ij(cid:0)1isthejointangleiinj(cid:0)1state,andkiisaconstantoftheistate,used
tocontrolthesetvelocity. Thek isaparameterofthegaitcontrolthatisalsooptimizedby
theGeneticAlgorithm. Theotherparametersarethejointsanglesforeachstate. Toreduce
the search space, the Genetic Algorithm only generates values between the maximum and
minimumacceptedvaluesforeachspeci(cid:2)cparameter.
The second possible approach we implemented uses a (cid:147)locus based gait(cid:148) to control
therobotjoints[28].Inthisapproach,insteadofusingaFSMtodeterminevelocityorangles
ofeachjoint,thepositionsofeachendpoint((cid:147)foot(cid:148)or(cid:147)paw(cid:148))aredeterminedandspeci(cid:2)ed
by spatial coordinates in the x, y and z axis. In order to generate the gait, the controller
needs to calculate the inverse kinematics and then use the calculated joints angles to move
the robot. The advantage of this technique is that the search space can be reduced, even
if the legs are composed by several segments. The main disadvantage is that we need a
fast and ef(cid:2)cient way to calculate the inverse kinematics. This also requires some speci(cid:2)c
knowledgeabouttherobotstructureandaboutthedirectkinematicsmodelimplemented. In
our implementation, the direct kinematics was calculated using the R Statistical Software5
andtheinversekinematicswascalculatedinrealtimeusingthePowell’smethod[3,25].
5TheRSoftwareisafreestatisticalsoftwareavailablefordownloadinthesitehttp://www.r-project.org/
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Anotherwayweusedtocontroltherobotgaitwastomodelthetrajectoryofendpoints
using acyclical function, asforexample ahalfellipse. The Figure 1, reproduced from[9],
illustratesthissituation.
Figure1.Exampleofahalfellipsetrajectory[9]
Withthetrajectoryde(cid:2)nedbyahalfellipse,theonlyparametersthatneedtobeopti-
mizedbytheGeneticAlgorithmarethepositionofeachleg’sellipsecenter(x,y andz axis
coordinates),theheightandwidthofeachellipse,thetotaltimecycledurationandthespe-
ci(cid:2)ctimedurationofthecycleduringwhichthelegistouchingtheground. Theadvantageof
thisapproachisthattheoreticallyitreducesthesearchspace, simplifyingthelearningtask.
Themaindisadvantagesare: (a)thehalfellipserestrictsalotthepossiblerobotmovements,
hinderinggaitinirregularsurfaces;(b)theinversekinematicsalsoneedstobecalculatedin
afastandef(cid:2)cientway.
6 ProposedSystem
TheLegGenSystemwasdevelopedtoaccomplishthegaitcontrolofsimulatedlegged
robotsinanautomaticway[12,13,11,14,14]. ItwasimplementedusingtheC++program-
minglanguageandthefreesoftwarelibrariesODEandGAlib.TheLegGenSystemreadstwo
con(cid:2)guration(cid:2)les,onedescribingtherobotformatanddimensionsandtheother(cid:2)ledescrib-
ing the simulation parameters. Table 2 shows the parameters used by the LegGen System,
withthevaluesusedinthesimulations(describedbelowinSection7). TheCrossover,Mu-
tation,PopulationsizeandNumberofgenerationsparametersareuseddirectlybytheGAlib
software. The Number of states parameter is the number of FSM states, the Time of walk
parameteristhetimeofeachindividualwalkduringthe(cid:2)tnessevaluation, andtheVelocity
minandVelocitymaxarethekintervalgeneratedbytheGeneticAlgorithm.
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Table2.ParametersoftheLegGenSystem
Par-ID Parameter Value
1 Uniformcrossover 0.60
2 Mutation 0.05
3 Populationsize 50
4 Numberofgenerations 100
5 Numberofstates 4
6 Timeofwalk 60
7 Velocitymin 0.0
8 Velocitymax 1.0
TheLegGenSystemworksasfollows: initiallythe(cid:2)ledescribingtherobotisloaded,
andtherobotiscreatedintheODEenvironmentaccordingto(cid:2)lespeci(cid:2)cations. Afterthis,
the system parameters are loaded (Table 2), and the Genetic Algorithm is initialized and
executeduntilthenumberofgenerationsisreached. Theevaluationofeachchromosomeis
realizedinthefollowingway:
(cid:15) Therobotisplacedinthestartingpositionandorientation;
(cid:15) ThegenomeisreadandtherobotcontrolFSMtablevaluesareset;
(cid:15) Thephysicalsimulationisexecutedduringaprede(cid:2)nedtime;
(cid:15) Gaitinformationandsensordataarecapturedduringeachphysicalsimulation;
(cid:15) FitnessiscalculatedandreturnedtotheGAlib.
ThewayastheFSMtablevaluesaresetbythegenomedependsofthegaitgeneration
method used. The LegGen System implements three methods for the gait generation: (a)
FMSanglestable;(b)locusbasedgait;(c)halfellipsecontroller. The(cid:2)tnessevaluationuses
thefollowingsensorialinformationthatmustbecalculated:(a)thedistanceDcoveredbythe
robot; (b) instability measure G; and (c) average number of endpoints touching the ground
B. ThecovereddistanceDisgivenbytheequation:
D =Px1(cid:0)Px0 (2)
whereDisthedistancetraveledbytherobotinthexaxis(forwardwalkfollowingastraight
line),Px0istherobotstartpositionandPx1isthe(cid:2)nalrobotpositioninthexaxis.
The instability measure is calculated using the robot position variations in the x, y
and z axis. These variations are collected during the physical simulation, simulating a gy-
roscope/accelerometer sensor, which is a sensor present in some modern robots [7]. The
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instabilitymeasureG(Gyro)isthencalculatedbythefollowingequation[9]:
N (x (cid:0)x )2+ N (y (cid:0)x )2+ N (z (cid:0)x )2
G= i=1 i x i=1 i y i=1 i z (3)
s N
P P P
where N is the number of sample readings, x , y and z are the data collected by the
i i i
simulated gyroscope in the time i, and x , x and x are the gyroscope reading means,
x y z
calculatedbytheequation:
x = Ni=1xi; x = Ni=1yi; x = Ni=1zi (4)
x N y N z N
P P P
To obtain the average number of endpoints touching the ground, we simulate touch
sensorsthatimitatetheoperationofbumpers[7]placedundereachpawoftherobot. During
the simulation, the number of endpoints touching the ground iscollected, and at the end of
eachphysicalsimulationtheaveragenumberBiscalculatedusingtheequation:
N
b
B = i=1 i (5)
N
P
whereN isthenumberofsamplescollectedandb isthenumberofendpointstouchingthe
i
groundatthetimei. After(cid:2)nishedthesensorialinformationprocessing,the(cid:2)tnessfunction
F isthencalculatedthroughtheequation:
D
F = (6)
1+G+(B(cid:0)L=2)2
whereListhenumberofrobotlegs. Analyzingthe(cid:2)tnessfunction, weseethatB reaches
its best value when the robot maintains half of its endpoints touching the ground, what is
desirable when the gait used is the trot. In this way, the best solutions have (B (cid:0)L=2)2
closetozero,sothisparameterwillhaveastrongin(cid:3)uenceinthepopulationevaluationand
evolution. Relatedtotheotherparameters,theindividualbetterquali(cid:2)edwillbetheonethat
has the best relationship between velocity and stability, so the best solutions are those that
movesfast,butwithoutlosingthestability[9].
Duringthesimulation,ifallpawsoftherobotleavethegroundatsametimeformore
thanonesecond,thesimulationofthisindividualisimmediatelystopped,becausethisrobot
probablyfelldown,andthereforeitisnotnecessarytocontinuethephysicalsimulationuntil
theprede(cid:2)nedendtime.
6.1 ModeledRobots
According tothe documentation, computational complexity when using the ODE li-
brary is O(n2), where n is the amount of bodies present in the simulated physical world.
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Thus,inordertomaintainthesimulationspeedinanacceptablerate,weshouldusefewand
simpleobjects.Forthisreason,allthesimulatedrobotsweremodeledwithsimpleobjects,as
rectanglesandcylinders, andtheyhaveonlythenecessaryarticulationstoperformthegait.
Inordertokeepourrobotprojectsimple,thejointsusedintherobotslegsjustmovearound
the z axis of the robot (the same axis of our knees), and the simulations just used robots
walking in a straight line. In the near future, we plan to extend our system to accept more
complexrobotmodelsandjoints.
Severalrobottypesweredevelopedandtested,beforewede(cid:2)nedthe(cid:2)nalfourmain
modelspresentedhere,thatareshowninFigure2. TheFigure2(a)model,calledHexaL3J,
(a) (b) (c) (d)
Figure2.Robotmodelsusedinthesimulations
havesixlegsandthreepartsperleg. Thepawsarewiderthantheremaininglegs,inwayto
givealargesupporttotherobot. TheFigure2(b)model, calledTetraL3J,issimilartothe
previousmodel,butithasjustfourlegs. BothmodelsinFigure2(a)and2(b)havethepaws
(cid:2)naljointanglesautomaticallycalculatedusingdirectkinematics,insuchamannerasthese
pawsarealwaysparalleltotheground. ThemodelofFigure2(c),calledHexaL2J,issimilar
to the Figure 2(a) model, but it has just two articulations per leg, in other words, it doesn’t
havepaws. Thus,inthismodelallthejointanglesarecalculatedbytheGeneticAlgorithm,
withoutusingdirectkinematics.Atlast,themodelofFigure2(d),calledTetraL2J,issimilar
tothepreviousone,withtwoarticulationsperlegandnopaws,butithasjustfourlegs. The
Table3showsthedimensionsoftherobotsinmeters. Thesimulatedrobotsdimensionsare
approximatelythedimensionsofadog.
TheuseofpawsasshowedinFigures2(a)and2(b)modelswasdesignedtoallowa
morestablewalk,mainlywhendynamicstabilitywasused. Therobotjointshavemaximum
and minimum joint angle limits similar to horses and dogs, but these animals have more
articulatedmembersthantheimplementedinourmodels.
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Table3.Dimensionsofthesimulatedrobots
Body Thighandshin Paw
Robot x y z x y z x y z
HexaL3J 0.80 0.15 0.30 0.05 0.15 0.05 0.08 0.05 0.09
TetraL3J 0.45 0.15 0.25 0.05 0.15 0.05 0.08 0.05 0.09
HexaL2J 0.80 0.15 0.30 0.05 0.15 0.05 - - -
TetraL2J 0.45 0.15 0.25 0.05 0.15 0.05 - - -
7 Results
In order to determine the best robot model to build, several experiments were con-
ducted. The(cid:2)rstexperimentsaimedtodiscoverthebestrobotcon(cid:2)gurations,includingthe
numberofrobotlegs,andthenumberofsegmentsperleg. Ourintentionwastoconstructa
robotusingthesmallestamountofarticulations,inordertosimplifythehardwareandreduce
the robot building costs. Other tests were conducted to discover the most suitable method
to be applied in the robot gait control. Several tests were made using a FSM angles table
(Angles),alocusbasedgait(Locus)andthehalfellipsemodeling(Ellipse). Thegaitmethod
adoptedinourexperimentswasmainlythetrot(twolegsareliftatthesametime),butsome
otherexperimentsweremadeusingfourleggedrobotsinawalkgait(justonelegpertimeis
movedawayfromtheground). TheTable4showstheresultsobtainedintheaccomplished
experiments. Each type of experiment present in this table was repeated ten times using
different random seeds, and the mean and standard deviation values relative to the (cid:2)tness
functionandsensorsinformationobtainedfromtheseexperimentswerecalculated.
The (cid:2)rst column indicates the experiment identi(cid:2)cation, the (cid:2)fth and sixth columns
show, respectively, the mean and the standard deviation of the (cid:2)tness (F), the seventh and
theeighthcolumnsshowthemeanandthestandarddeviationofthedistancecoveredbythe
robot (D) in meters and the last two columns show the mean and the standard deviation of
therobotinstabilitymeasure(G). TheFigure3showstheboxplotgraphicofallexperiments,
theFigure4(a)showstheboxplot oftheTetraL3Jexperiments(Exp3, 7, 9and11)andthe
Figure4(b)showstheboxplotoftheTetraL2Jexperiments(Exp8,8,10and12).
From the observed results presented in Table 4 and the (cid:2)tness distributions of the
Figure3,wecanreachtothefollowingconclusions:
(cid:15) Sixleggedrobotsarefasterthanfourleggedrobots;
(cid:15) Sixleggedrobotswithoutpawsarealittlebitfasterthanequivalentrobotswithpaws;
(cid:15) Fourleggedrobotswithoutpawsdidnotachieveasatisfactorydisplacement;
(cid:15) Althoughnotsigni(cid:2)cantstatistically,theexperimentsusingtheFSManglestablewere
alittlebitmoreef(cid:2)cientthantheothers;
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