MODELING, ANALYSIS, AND SYNTHESIS OF SERPENTINE LOCOMOTION

WITH A MULTILINK ROBOTIC SNAKE

M. SAITO�, M. FUKAYA

�, AND T. IWASAKI

�Intr oduction

Most currentlyavailablemobile robot mechanismsfor groundoperationshave wheelsdrivenby sometypeof rotationalmotiongeneratorssuchasanengine.Suchwheeledmechanismsarerelativelyefficient,easyto steer, andsuitedfor high-speeddriving on a smoothsurface.They are,however, not effective inruggedenvironmentssuchasroughand/ormuddyterrains.Mobile robotswith legsarebeingactivelyresearched(e.g.[1]–[4]) for severalreasons,includingthefactthatlegsprovidehigherterrainadaptabil-ity thanwheels. Even higherterrainadaptabilitymay be achieved by multilink articulatedrobotsthat“crawl” likesnakes.

Beyond their adaptabilityto the environment,serpentinerobotsoffer a variety of advantagesovermobilerobotswith wheelsor legs. They arerobustto mechanicalfailurebecausethey aremodularandhighly redundant.They mayalsoperformasmanipulatorarmswhenpartof themultilinkedbodyis fixedto a platform. On theotherhand,oneof themaindrawbacksis their poorpower efficiency for surfacelocomotion.Anotheris thedifficulty in analyzingandsynthesizingsnakelike locomotionmechanisms,which arenot assimpleaswheeledmechanisms.While theformeris a fundamentaldrawbackinherentwith thesnakelike locomotion,thelatterpointhasbeenaddressedby many researchers[5]–[15], andwehopeto addanothercontribution to overcomingthedifficulty.

Researchin experimentalbiology (e.g.,[16], [17]) hasrevealedthemechanismsof thefour commonmodes(or gaits)of locomotionin snakes: serpentine,side-winding,concertina,andrectilinear. Fur-theranalysisof snake locomotionwasalsodonefrom biologicalstandpoints[18]–[21] aswell asfrommechanicalperspectives[22], [23]. A successfulrobotic realizationof serpentinelocomotionwasfirstreportedin [24], andtheresearchresultsaresummarizedin [25].

Hirose [25], [26] studiedsnakesandfound that their bodiestake on the so-calledserpenoidcurvewhenthey locomotewith a serpentinegait. Moreover, thekey propertyof snakesin achieving serpen-tine locomotionis thedifferencein thefriction coefficientsfor thetangentialandthenormaldirectionswith respectto the body. In particular, the normalfriction tendsto be muchlarger thanthe tangentialfriction, leadingto avoidanceof sideslipping.Basedon this fact,Hirosebuilt snakelike locomotorsanddevelopeda controlmethodto make themmove to a desireddirectionat a desiredspeed.However, thelocomotorshadwheelsto realizethe directionalfriction andthe control wasopenloop in nature,and�

AutomotiveProductsDivision,(C) Hitachi,Ltd., 2520TakabaHitachinaka,Ibaraki,Japan.�Departmentof ControlEngineering,Tokyo Instituteof Technology, 2-12-1Oookayama,Meguro,Tokyo 152,Japan.�Departmentof MechanicalandAerospaceEngineering,University of Virginia, P.O. Box 400746,Charlottesville,VA

22904-4746,e-mail:iwasaki@virginia.edu.

1

henceadaptabilityto theenvironmentwassomewhatweakened.Nevertheless,thecontrolschemewasvery intuitiveandconstitutedasolid foundationin thestudyof roboticserpentinelocomotion.

Anothernotableresulton the snakelike robotic mechanismhasbeenobtainedby Burdick andcol-leagues[27]–[29]. They studied,with mathematicalrigor andexperimentalvalidation,themechanismof “undulatory locomotion” in which net motion is generatedby the couplingbetweeninternalshapechangesandexternalnonholonomicconstraints.The problemconsideredis kinematicratherthandy-namicin naturedueto thenonholonomicconstraintsarisingfrom idealizationof wheelssuchthat theydo not slip in the directionof rotationalaxis. Otherresultson serpentinelocomotionin the nonholo-nomic settingincludethosedescribedin [30]–[32]. Again, an essentialassumptionin theseresultsisthatserpentinerobotshave wheels,andthusthemainadvantageof serpentinelocomotion,adaptabilityto theenvironment,cannotbefully realized.

In this article,we considera snake robot without wheelsthathassubstantialpotentialfor adaptingto the environmentat the expenseof power efficiency. Our objective is to establisha mathematicalframework for modeling,analysis,andsynthesisof serpentinelocomotionwith amultilink roboticsnake.Wedoapplysomeexistingknowledgefrom biologicalstudyof snakes,but our focuswill beonaroboticrealizationtheoryof snakelike locomotionratherthanthebiomimeticaspectof theproblem.Werestrictour attentionto planarserpentinegait of locomotion,and hencethe more generalthree-dimensionalmotionandothergaitson isotropicfriction environmentareoutof thescopeof this article.

First,wedevelopamathematicalmodelfor anarticulatedsnakerobotconsistingof � rigid links withtorqueactuatorsat ��� joints. Two modelsfor the directionalfriction force actingon eachlink aredevelopedandintegratedinto thetotal equationsof motionfor thesnake robot. We thenshow a modeltransformationthatdecouplesthedynamicsinto theinternalshapemotion(joint torques� joint angles)andtheinertial locomotion(joint angles� inertialpositionandorientation).Thisdecouplingenablesusto concentrateon controlof the inertial locomotionthroughtheshapechange,assumingthat theshapecanreadilybecontrolledasdesiredvia thejoint actuators.

We thenanalyzethe locomotionof the snake robot with serpentinegait from the power efficiencystandpoint.In particular, we investigatetheshapemotion thatachievesthe locomotionat a prescribedspeedwhile requiringthe leastinput power. Basedon theanalysisresults,we determinethe feedbackcontrolarchitectureto realizedesiredlocomotionandproposea methodfor designingsuchcontrollersby suggestinga quasi-linearizinginput transformation.Finally, our control strategy is experimentallyvalidatedby afive-link articulatedsnakerobot.To ourknowledge,this is thefirst researchthataddressesthe problemof modeling,analysis,andfeedbacksynthesisof serpentinelocomotionby an articulatedroboticsnakewithoutwheels.

We usethefollowing notation.Thesetsof � -dimensionalrealvectorsand � ��� realmatricesaredenotedby � ��� and � ������� , respectively. For a matrix � , the transposeis denotedby ��� . For scalars�����! " ! "�#� � , diag$ �����! ! " "�#� � % denotesthe �&�'� diagonalmatrix with ���(�! ! ! )�#� � on the diagonal. Forvectors� and * , �,+ denotesthevectorwith the - th entry �,+. , while �0/ * denotesthevectorwith the - thentry � . / * . . Thederivativeof asignal� $21 % with respectto thetime 1 is denotedby 3� $41 % ( i.e., 3�6587:9;�0/<9 1 ).For a scalar-valuedperiodicsignal � $41 % , ave$ � % is theaveragevalueof � $41 % over a period. Thesignumfunctionis denotedby sgn$>= % , i.e.,sgn$ � % is if �@?BA , A if �C7�A , and �D if �6EFA . For agenericfunctionG $ � % 5 � �H� � � suchassgn, IKJML , and N"O;I , we may let it admit a vector-valuedargumentwith a slightabuseof notation,in which casethe function operateon eachentry of the argumentvector. A similarcommentappliesto functionswith two arguments,e.g.,sat$P= � = % anddzs$P= � = % definedin theAppendix.

2

Modeling of a Snake Robot

Considerthe snake robot depictedin Fig. 1, whichconsistsof � links connectedthrough �Q� joints.Eachlink is rigid with uniformly distributedmass.Eachjoint is equippedwith a torqueactuator(mo-tor). Thesnake robot is placedon a horizontalsur-face,and we consideronly planarmotion; in thiscase,the mechanicalsystemhas �SRUT degreesoffreedom( �6�V for shape,T for position,and fororientation).

WXWY ZZ Z[ [

[\ \ \ \ \^] ]] ] ] _

_ _ _a` ` ` ` ` `

b � b +x

x

x

x

$ ����� * � %$� + � * + % $ �,cd� * c %

e f�*Tdg chhh hhhi_ _ _ _Kj

Figure1. Snake robot.

Later, wewill developdynamicequationsof motionfor thesnakerobot.To thisend,let usintroducesomenotation.Fix � - * axesin the inertial frameandconsiderthe - th link of thesnake robot. Thelinkis of mass� . , length Tkg . , andmomentof inertia l . $ 7 � . g +. /�m % . Let $ � . � * . % and

b . bethecoordinatesofthecenterof gravity andtheanglebetweenthe link andthe � -axis,respectively. Denoteby � , * , and

bthe � -dimensionalvectorswhose- th entriesare � . , * . , and

b . , respectively. Definen 587 oppq . . . . . .

rtssuSv � �xwy��z �|{ ��� � }~5�7 oppq ��. . . . . . ��

rtssuQv � ��w8��z �|{ ���� 587���� AA ��� � � 587�� =d=d=�^� � v � � � � � 587 ��.�� � � .��� 587 diag$�I#J�L b ���! ! ! )� I#J�L b � % �~� � 5�7 diag$2N"O;I b ���" ! ! "� N"O;I b � %l 587 diag$Pl �(�! ! ! )� l � % � � 587 diag$2� ���! ! " "� � � % ���&5�7 diag$�g �(�! ! ! )� g � %� 587:� n � $ } � z � } � % z � n ���~��5�7 � z � } � $ } � z � } � % z � n �� 5�7 laR ��� � ��� R � � � � � �H� 587 ��� � � � � � � � ��� � �¡587 � ��� � ��� � � � � � �¢ 587:} � z � } � ��£¤587 � z � } � ¢ z � ��¥a5�7 / $ � � � � % � �"¦ 5�7�¥ �In theabove, thesymbols} and

nstandfor the“dif ference”andthe“addition” operators,respectively.

The vector � is a basisof the kernelof } . It canbe shown that the matrices�

and � dependon therelativeangles§ 587¡} b only, andthusaredeterminedby theshapeof thesnake robot.

Considerthefree-bodydiagramfor the - th link depictedin Fig. 2, whereG . and ¨ . aretheforceand

thetorquedueto thefriction betweenthe link andthehorizontalsurface, © . and © . z � aretheconstraintforcesfrom theadjacentlinks, and ª . and ª . z � arethejoint torquesfrom theactuators.Let ¨ v � ��� bethevectorwhose- th entryis ¨ . , andsimilarly for

Gk« � Gk¬ v � � � and © « � © ¬ � ª v � � ��z � , where $ Gd«�­ � Gk¬P­ % arethe( �®� * ) componentsof thefriction forcevector

G . . Finally, we definethe total friction forcevectorG

andthewholepositionvector ¯ byG 587 � Gk«Gk¬ � � ¯ 587 � � * � 3

We shall first develop two friction modelsbasedon a simplified viscousfriction and a Coulombfriction to giveexpressionsfor

Gand ¨ in termsof � , * and

b. Equationsof motionwill thenbederived

usingthefriction models.

°²±³²´µ e°~±³~´h hK¶ hh ·°~±³²´e µ WY x¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸¸ ¸^¹¹¸¸¸¸¸¸¸¸¸¸¸¸¸¸¹¹ b .

$ � . � * . % ] ] ] ] ] º fe» µ

ª .© ¬P­ © «�­G .

© «�­�¼�½ © ¬P­¾¼<½ª . z � ¨ .

Figure2. Freebodydiagramfor the - th link.

*�

� 1¿ 9À¿Á . 3Á .

g .$ � . � * . %

Figure3. Infinitesimalsegmenton the - th link.

Simple Friction Model

Considerthe - th link of thesnake robotdepictedin Fig. 3. Thecoordinatesof theinfinitesimalsegment9À¿ aregivenbyÁ . 7 � � .* . � R � N"O<I b .I#J�L b . � ¿Â Takingthederivativewith respectto thetime,wehave thevelocityvectorin the � - * frame:3Á . 7 � 3� .3* . � R � �6IKJML b .N)O;I b . � ¿ 3b . Thetangentialandnormalcomponentsof thevelocityvector 3Á . aregivenby�ÄÃÅ�Æ ­ÃÅ � ­ � 5�7 � N"O<I b . I#J�L b .�6I#J�L b . N)O;I b . � 3Á . 7 � ÅkÆ ­Å � ­ � R � A¿ 3b . � (1)

where� Å�Æ ­Å � ­ � 5�7 � N"O<I b . I#J�L b .�6I#J�L b . N)O;I b . � � 3� .3* . � (2)

Now assumethatthefriction forceis modeledin thebody( 1 - � ) frameas� 9 G Æ ­9 G � ­ � 7 � �ÈÇ Æ ­ AA Ç � ­ � � ÃÅkÆ ­ÃÅ � ­ � 9 � .where Ç Æ ­ and Ç � ­ are the friction coefficients and 9 � . is the massof the infinitesimal segment(i.e.,9 � . 7 $�� . 9À¿ % / $ÉTdg . % ). Transformingthecoordinates,thefriction forcein theinertial frameis� 9 Gd«�­9 Gk¬P­ � 7 � N"O;I b . �6I#J�L b .IKJML b . N)O;I b . � � 9 G Æ ­9 G � ­ �

4

Integratingover thelink, thetotal friction forceis givenby� Gd«�­Gk¬P­ � 7 �Ê� . � N"O<I b . �6IKJML b .I#J�L b . N"O;I b . � � Ç Æ ­ AA Ç � ­ � � N"O<I b . I#J�L b .�6I#J�L b . N)O;I b . � � 3� .3* . � Finally, thetotal friction torquearoundthecenterof massof thelink is¨ . 7VËB¿�9 G � ­ 7 � � . g +.m Ç � ­ 3b . 7 � Ç � ­ l . 3b .wherethepositivedirectionis counterclockwise.

We now considerthewholesystemof connectedlinks. It canbeverifiedthatthetotal friction forceandtorqueactingon thesystemcanbeexpressedasG 7 �ÊÌ � }ÎÍ ÌÏ�� 3¯¨ 7 � }ÑÐ 3b (3)

where}ÎÍD5�7 � �ÏÆ � AA � � � � � }ÑÐ�587:� � l � Ì � 587 � � � � ������ � � � � (4)�ÏÆÒ587 diag$ Ç Æ ½ �! ! ! )� Ç Æ�Ó % � � � 587 diag$ Ç � ½ �" ! ! "� Ç � Ó % Coulomb Friction Model

As in the previous section,we give the expressionfor the tangentialandnormalvelocity components(1). Assumethatthefriction forceactingon theinfinitesimalsegmentis givenby� 9 G Æ ­9 G � ­ � 7 ��© ��Ô Æ ­ AA Ô � ­ � � sgn$ ÃÅkÆ ­ %

sgn$ ÃÅ � ­ % � 9 � .where Ô Æ ­ and Ô � ­ are,respectively, the Coulombfriction coefficientsin the tangentialandthe normaldirections,© is thegravity constant,and 9 � . is themassof thesegment. Integratingover the link, thetotal friction forceactingon the - th link is asfollows:� G Æ ­G � ­ � 7 �È© ��Ô Æ ­ AA Ô � ­ � ËÑÕ ­z Õ ­ � .Tdg . � sgn$ ÃÅ�Æ ­ %

sgn$ ÃÅ � ­ % � 9À¿; Similarly, thetotal torqueis givenby¨ . 7VËB¿d9 G � ­ 7 � Ô � ­ � . ©Tkg . Ë Õ ­z Õ ­ sgn$ ÃÅ � ­ % ¿d9À¿; UsingLemma1 in theAppendix,wehave� Gd«�­Gk¬P­ � 7 �Ê� . © � N"O<I b . �6IKJML b .I#J�L b . N"O;I b . � ��Ô Æ ­ AA Ô � ­ � sat Ö � ÅkÆ ­Å � ­ � ��� Ag . 3b . �;ר . 7 � T Ô � ­ � . ©Âg . dzs$ Å � ­ � g . 3b . %

5

Assemblingtheaboveexpressionsinto vectorforms,thetotal friction forceandtorquecanbegivenbyG 7 �ÊÌ ��Ø Í sat$|ÌÏ�� 3¯ �ÚÙ 3b %¨ 7 � Ø Ð dzs$|Û � 3¯ �(� 3b % (5)

whereØ ÍD5�7 �ÈÜ Æ ��© AA Ü � ��© � � ÙÝ5�7 � A� � � Ø Ð�7 T Ü � ��© ��� Û � 5�7 � � ��� � � �Ü Æ 587 diag$ Ô Æ ½ �" ! ! "� Ô Æ�Ó % � Ü � 587 diag$ Ô � ½ �" ! ! "� Ô � Ó % Equations of Motion

Applying the first principle to the free-bodydiagramof the - th link in Fig. 2 andassemblinginto the� -link snake robot,weobtaintheequationsof translationalmotion�~Þ�ß7 Gk« R } � © «�²Þ* 7 Gk¬ R } �8© ¬ (6)

andtheequationof rotationalmotionl Þb 7 ¨Î� ��� � n � © « R � � � n � © ¬ R } � ª (7)

Recallthatthesnakerobothas�ÎRàT degreesof freedom,andwehaveusedm � variables( � , * , andb) to

describeits motion. Below, we will developequationsof motion in termsof theabsoluteanglesb

andthepositionof thethecenterof gravityá587 � á «á ¬ � 7 � �È� � � �� � �¡* �

In view of Fig. 1, thevectors� , * , andb

areconstrainedby}a� R n � N"O<I b 7¡A} *âR n � I#J�L b 7�Aã (8)

Fromthedefinitionof á and(8), wehaveä �ß7 � � n � N)O;I bá « � � ä * 7 � � n � I#J�L bá ¬ � � ä 587 � }� �å� / � � Solvingtheseequationsfor � and * , andtakingthetimederivatives,weseethatthetranslationalvelocityis givenby3¯ 7Ý� 3b R � 3á (9)

wherewenotedthatä z � 7 � � z � } �4$ } � z � } � % z � � � 6

Next we decomposethe equationsof translationalmotion (6) into two parts: expressionsfor thecenterof massaccelerationÞá andtheconstraintforces © « and © ¬ . Multiplying (6) with

ä � z � from theleft, wehaveä Þ�ß7 � } Þ�Þá « � 7 � } � z � Gd« R } � z � } �8© «� � Gd« / � � (10)

for the � coordinateandsimilarly for the * coordinate.Solvingfor © « and © ¬ , wehave© « 7 $ } � z � } � % z � $ n � $ � � 3b + R ��� Þb % � } � z � Gd« %© ¬ 7 $ } � z � } � % z � $ n � $ ��� 3b + � � � Þb % � } � z � Gk¬ % (11)

wherewenotedfrom (8) that} Þ�C7 n � $ � � 3b + R ��� Þb % � } Þ* 7 n � $ ��� 3b + � � � Þb % Substituting(11) into (7), andusingthesecondrow of (10),wehave� Þb R � 3b + 7¡} �8ªæR'¨�R � � G�HÞá�7 � � G (12)

Thecompletesetof equationsof motionis now derived: (12), (9), and(3) for thesimplefriction modeland(5) for theCoulombfriction model. In the formercase,all theseequationscanbeassembledintothefollowing:� � AA �ßç � � Þb Þá � R � � 3b +A � R ��è éé �~ê � � 3b 3á � 7 � } �A � ª (13)

where�Êè éé �~ê � 5�7 � }æÐ AA A � R � � �� � � Ì � }ÎÍ Ì � � � � � � (14)

and }ÑÐ and }ÎÍ aredefinedby (4).

Dynamic Decoupling

In this section,we show that thedynamicequationdevelopedin theprevioussectioncanbedecoupledinto two parts:thatis, theshapemotion(joint torques� joint angles)andtheinertial locomotion(jointangles � inertial position andorientation). This decouplingsimplifies the analysisandsynthesisofsnakelike locomotionin thesubsequentsections.

To this end,we first observe somepropertiesof thesnake robotdynamics.It canreadilybeverifiedthatthesystemmatriceshave thefollowing internalstructures:� R � � 7:A3� 7F� 3ë � 3ë �ì (15)

Thesepropertieshave beenfoundfor roboticmanipulatorsandusedfor controldesign(e.g.,[33]), andwill alsobeusefulfor our developmentbelow.

7

Definenew variablesasfollows:� 3§ 3í � 587 � }� � � � 3b Notethat § is thevectorof relativeanglesbetweenadjacentbodiesand 3í canbethoughtof asanaverageangularmomentum.Solvingthis definingequationfor 3b ,3b 7¡£ 3§æR �"¦ 3í (16)

In view of (15),wehave� Þ§ Þí � 7 � } Þb� ��îî Æ $ � 3b % � 7 � } Þb� ��$ � Þb R � 3b + � 3ë � 3b % � 7 � } Þb� ��$ � Þb R � 3b + % �wherewenotedthat� � 3ë � 3b 7 3b � � 3b 7¡Abecause� is skew symmetric. Using theserelationships,the original equationof motion (12) canbetransformedas� } � z �� � � $ � Þb R � 3b + � ¨ï� � � G � } � ª % 7:Aor equivalentlyasÞ§æR } � z � $ � 3b + � ¨ï� � � G % 7 ¢ ªÞí 7 � �2$4¨ÑR � � G % (17)

whereweused� � } � 7:A .We now seethat the torqueinput ª drivestheshape§ of thesnake robot throughthefirst equation

in (17),andthattheshapechange 3§ , whencoupledwith theinertial motion,generatesthefriction forceandtorqueby (3) or (5) with (9) and(16). Thefriction forceandtorquethusgenerateddrive thesnakerobotwith respectto theinertial framethroughthesecondequationin eachof (12)and(17). In thecaseof thesimplefriction model(3), the rationalebehindthis observationmaybemoretransparent,for thedecoupledequationsof motioncanbeconciselyexpressedasfollows:� ¥ AA �ßç � � Þí Þá � R � � �¦ èC�"¦ � �¦ éé � �"¦ ê � � 3í 3á � R � � �¦ èé � � £ 3§ 7:A

Þ§æR } � z � $ � 3b + R è 3b R é 3á % 7 ¢ ª (18)

where 3b is givenby (16).With thedecoupledequationsof motion,it is naturalto developa controlstrategy suchthat:

1. Thejoint torque ª controlstheshape§ of thesnake robot.

2. Theshape§ controlstheinertial behaviorí

and á of thesnake robot.

Sincethe matrix¢

is nonsingular, the control in step1 is easy. In the next section,we examinethedynamicbehavior of thesnake robotrelatedto step2 of theabovecontrolstrategy. Theanalysisresultsfrom thenext sectionwill beusefulfor designingcontrolsystemsto achieve snakelike locomotionin alatersection.

8

Analysis of Snakelike Locomotion

Whenthe shape§ of the snake robot changes,the forceG

and the torque ¨ aregenerateddueto thefriction betweenthe robot and the surfaceon which it lies. As a result, the positionof the centerofgravity á andthe angularmomentumof the whole body 3í change.The first equationin (18) showshow thechangein § affectsthemotionvariables 3í and 3á . If we controltheshape§ appropriately, thenthesnake robotwill achieve thedesiredlocomotion.Our objective in this sectionis to find out how tochange§ to achieve thedesiredlocomotionspecifiedby speed,direction,andpowerefficiency.

Thereareat leasttwo approachesto determiningappropriate§ ; oneis to optimize 3§ with respectto awell-definedcriterionsubjectto thedynamicconstraintgivenby thefirst equationin (18); theotheris tosimply mimic themotionof living snakes. Theformerwould bea very challengingapproach,whereasthe latter doesnot guaranteeyielding a motion that is optimal for the robotic snake. Herewe take anapproachthatfits in betweenthetwo.

Wewill restrictourdiscussionto thebiologicallyinspiredserpentinelocomotion[25] (i.e.,themotioninducedby changingtheshapeof thesnake robotsoasto form theserpenoidcurvesobservedin livingsnakes).Thesetof serpenoidcurvesis characterizedby severalparameters.We will examinehow eachparameteraffects the locomotionof our snake robot using the equationof motion developedearlier.Below, wepresentouranalysisresultsfor thecaseof thesimplefriction modelonly. Wealsoperformedthe sameanalysisusing the Coulombfriction model (5), and found that the generaltendency of theresultsis the sameasthe simplefriction case.The analysisresultswill be useful for determiningtheappropriateshapechange 3§ to achieve locomotionin thedesireddirection,at thedesiredspeed,with theoptimalpowerefficiency.

Serpenoid Curve

Let usfirst briefly review theserpenoidcurve. It is defined[25] asfollows:

Definition 1 Considera curve on the � - * planethatpassesthroughtheorigin. It is calleda serpenoidcurveif anarbitrarypoint ( �®� * ) on thecurvecanbeexpressedas� $ ¿ % 7¡Ë^ðñ N)O;Ik$4ò!ó % 9Âô�� *�$ ¿ % 7VËaðñ I#JMLõ$2ò!ó % 9�ô0� ò!ó 5�7�ö N"O<Ik$|÷ ô % R Ç ôfor somescalarsö , ÷ , and Ç , where ¿ is thearclengthfrom theorigin to thepoint.

Figure4 shows severalserpenoidcurvesof unit lengthgeneratedby differentchoicesof theparam-etersö , ÷ , and Ç . We seethateachcurve hasa sinusoidlikeperiodicshape.Theparameterö determinesthe degreeof undulation, ÷ determinesthe numberof periodsin a unit length,and Ç the macroscopiccircularshape.

As hasbeenshown in [25], the curvatureof a serpenoidcurve is a sinusoidalfunction of the arclength.In particular, it canreadilybeverifiedthatø $ ¿ % 587 ùúúû Ö 9 + �9À¿ + × + R Ö 9 + *9ü¿ + × + 7þýÿö ÷0IKJMLõ$�÷ ¿ % � Ç ý� The curvatureis relatedto the muscularforce requiredfor a snake to take on a given shape,andtheabove factsuggeststhat theserpenoidcurve canbeconsidereda naturalshapethata living snake takeson. See[25] for a thoroughdiscussionwith experimentaljustificationof this statement.

9

0 0.2 0.4 0.6−0.3

−0.2

−0.1

0

0.1

0.2

0.3

b = 2π

c = 0

a = π/3 (solid), π/2 (dashed), 2π/3 (dotted)

0 0.1 0.2 0.3 0.4

−0.15

−0.1

−0.05

0

0.05

0.1

0.15 a = π/2

c = 0

b = 2π (solid), 6π (dashed), 10π (dotted)

0 0.1 0.2 0.3 0.4

0

0.05

0.1

0.15

0.2

0.25

0.3 a = π/2

b = 10π

c = 0 (solid), π/2 (dashed), π (dotted)

Figure4. Variousserpenoidcurves.

Now we will give an approximationof the serpenoidcurve in termsof successively connected�segments,eachof which is a straightline with length / � . Let a serpenoidcurve bedescribedby � $ ¿ %and *�$ ¿ % with A��U¿�� . We consider��R� pointson thecurve specifiedby ¿ . 587 - / ��$�- 7UAã�! " ! )� � % .By definition,thevaluesof � $ ¿ . % and *�$ ¿ . % canbeapproximatedby� . 7 .�� � � � N)O;Id$ ö N)O;Ik$ � ÷� % R � Ç� % � * . 7 .�� � � � I#J�Lõ$ ö N"O;Id$ � ÷� % R � Ç� % Theoriginal serpenoidcurve is thenapproximatedby � segmentsconnecting$ � . � * . % ( - 7¡Aã�! ! ! �� � ).

Denotebyb . theangleof the - th segmentmeasuredfrom the � -axis in a counterclockwisemanner.

Then ��� L®$ b . % 7 * . �'* . z �� . � � . z � 7 I#J�L®$ ö N"O;I!$�-É÷ / � % R - Ç / � %N"O<Ik$ ö N"O;I!$�- ÷ / � % R - Ç / � %andhencewehaveb . 7:ö N"O;I!$ -É÷� % R - Ç� Notethattherelativeanglesthatdeterminetheshapeof thediscreteserpenoidcurvearegivenby§ . 5�7 b . � b .� � 7� I#JMLõ$�- ��R � T % R��where 587:ö������ IKJML®$ � T % ����� � � 5�7 ÷� � � 5�7 � Ç� Thus,therelativeangle § . changesin thesinusoidalmanneralongthe“discretearclength” - / � with theamplitude andthebias � . Any two adjacentrelativeangleshave thephasedifference� .

Serpentine Locomotion

Undulatorymotionof a snake canbeimitatedby changingtherelative anglesof thesnake robot in thefollowing manner:§ . $41 % 7� I#J�Lõ$�� 1�R:$2-õ�F % � % R�� � $�- 7 �! ! ! )� �ß�B % (19)

10

where � � , and � arethe parametersthat determinethe shapeof the serpenoidcurve realizedby thesnake robot, and � specifieshow fastthe serpentinewave propagatesalongthe body. Our hypothesisis that theserpentinegait is realizedwhentheshapechangeasin (19) is coupledwith theenvironmentthroughthe directionalfriction property. In this section,we verify by numericalsimulationsthat thishypothesisis indeedcorrect.Anotherobjective hereis to confirmthat thefollowing observation,madeby Hirose[25] for snake robotswith wheels,alsoappliesto our no-wheelsnake robot; Thespeedandthedirectionof theserpentinelocomotionaremainly determinedby � and � , respectively.

We consider(16) andthefirst equationin (18) to describethesnake robotmotionwherethesimplefriction model }ïÍ and }ÑÐ in (4) is usedto definethe matrices è , é , and ê as in (14). The robotparametersarespecifiedas� 7��ã� � . 7 kg � g . 7 m � Ç Æ ­ 7:Aã Hz � Ç � ­ 7 A Hz Theinput 3§ to (18) is generatedto achievetheserpentinemotion(19)asymptotically, andthelocomotionof the snake robot is simulatedwith differentparametervaluesfor ( � � � � � � ). Typical motionsareshown in Figs.5 ( � 7:A ) and6 ( � 7 A deg) wheretheotherparametersarefixedto:S7¡m;A deg � � 7��;A deg � � 7:m rad/s Theinertial ( �®� * ) framehasbeenrotatedafterthesimulationsuchthattheaveragevelocityof thecenterof gravity in thesteadystate(i.e., afterthetransientsdiedout) is directedtowardthepositive � -axis. Ineachfigure, theupperright pictureshows thetracesof thehead(dashedline) andthecenterof gravity(solid line) anda snapshotof thesnake robotat a certaintime. We seethat thesnake robotmovesin astraightline when � 7 A andmovesalonga circle when ���7UA . Thelower left andlower right picturesshow thetimehistoriesof thespeedÅ andtheangularspeed 3ò definedbyÅa5�7 3á « N"O;I òÊR 3á ¬ I#J�L ò � ò 5�7 � �� .�� � b . Notethat ò is theparameterthat representsthedirectionof thewholesnake robotand Å is speedof thecenterof gravity in the ò direction.Both Å and 3ò oscillatewith frequency � in thesteadystate,but theiraveragevaluesconvergeto certainconstantvalues.

Next we fix and � asbeforeandrun the samesimulationsfor differentvaluesof � and � . Theaveragevaluesof thespeedÅ andtheangularspeed 3ò , denotedby ave$ Å % andave$ 3ò % , arethenobtainedasfunctionsof � and � andareplottedin Figs.7 and8. Thevalueof ave$ Å % is thehighestat the lowerright cornerof Fig. 7, whereasthevalueof ave$ 3ò % is thehighestat theupperright cornerof Fig. 8. Weseethatwhen � is small,ave$ Å % is determinedby � andis insensitiveto � , whereasave$ 3ò % is determinedby � andis insensitive to � . Thus, � and � arethedominantparametersfor thespeedandthedirectionof theserpentinelocomotion,respectively.

Optimally Efficient Motion

In this section,weconsidertheserpentinemotionalonga straightline ( � 7:A ) andshow by simulationswhatchoicesof theparameters( � � � � ) aremost“efficient.” Let usfirst formally definetheefficiencyof thesnake robotlocomotion.Multiplying (13)by � 3b � 3á ��� from theleft, we have thefollowing powerequation:� � R � Õ 7 � .

11

0 2 4 6 8−40

−20

0

20

40

Time [s]

φ [d

eg]

0 2 4 6 8−1

0

1

2

3

4

5

Time [s]

Spe

ed [m

/s]

0 2 4 6 8−0.4

−0.2

0

0.2

0.4

Time [s]

Ang

ular

Spe

ed [r

ad/s

]

0 10 20 30 40−3

−2

−1

0

1

2

3

Head:dashedCG:solid

x [m]

y [m

]

Figure5. Serpentinelocomotion ( � 7:A deg).

0 2 4 6 8

−20

−10

0

10

20

30

40

Time [s]

φ [d

eg]

0 5 10 15−1

0

1

2

3

4

5

Time [s]

Spe

ed [m

/s]

0 5 10 15

−0.5

−0.4

−0.3

−0.2

−0.1

0

Time [s]

Ang

ular

Spe

ed [r

ad/s

]

−10 0 10−25

−20

−15

−10

−5

0

5

Head:dashed

CG:solid

x [m]

y [m

]

Figure6. Serpentinelocomotion ( � 7 A deg).

1 2 3 4 50

10

20

30

40

50

60

ω [rad/s]

γ [d

eg]

Average Speed

Figure 7. Contour plot of the averagespeedasa functionof � and � .

1 2 3 4 50

10

20

30

40

50

60

ω [rad/s]

γ [d

eg]

Average Angular Speed

Figure8. Contourplot of theaveragean-gularspeedasa functionof � and � .

with � � 5�7 99 1 � T 3b � � 3b R � T 3á � 3á�� Kinetic power� Õ 5�7 � 3b 3á � � �âè éé �²ê � � 3b 3á � Power loss� . 5�7 3§ � ª Inputpower

wherewenotedfrom (15) that99 1 $ T 3b � � 3b % 7 3b � � Þb R T 3b � 3� 3b7 3b � � Þb R T 3b � $ � 3ë R 3ë � � % 3b7 3b � � Þb R 3b � � 3b + 12

Fromthepower equation,we seethatpartof the input power

� . from the joint actuatoris convertedtothekinetic power

� � andtherestis lost as

� Õ dueto thefriction. In fact, thepower lossis essentialforthesnake locomotion,sinceit is thefriction forcethatgeneratestheforwardacceleration.In thesteadystate,theaveragekineticenergy is constant,andhencetheaveragekineticpowerave$ � � % is zero.In thiscase,on average,all theinput power

� . suppliedby thejoint actuatorsis consumedasthepower loss

� Õdueto thefriction.

Theobjectiveof thissectionis to find theoptimallyefficientmotionwithin theframework of serpen-tine locomotion.Moreprecisely, we investigatefor whatchoicesof theparameters( � � � � ) theaveragepower lossave$ � Õ % is minimalwhile keepingaprescribedaveragespeedave$ Å % .

Gridding the parameterspaceas �7 mã���ü�� ã�! ! " ��� ;A deg, � 7 �ã� kT � "! �! ! " �� "! A deg, and � 7Aã $#ü� � $#ü�! ! ! ��%# rad/s,thequantitiesave$ � Õ % andave$ Å % arecomputedvia simulationsfor eachparam-etertriplet ( � � � � ) andareplottedin Fig. 9 (left). The figure clearly shows the paretooptimal curvecorrespondingto theminimumpowerwith fixedvaluesof thespeed.Extractingthepointson thecurve,onecanfind the optimal relationshipbetweenthe averageforward speedave$ Å % andthe frequency �(Fig. 9 (right)). As seenin thefigure,ave$ Å % and � areroughlyproportionalto eachother. Theparetooptimal pointsareplotted in the ( � � � � ) spacetogetherwith its projectionsonto the threeplanesinFig. 10. Note that theparetooptimalphase-shift� is independentof theothervariablesandis �'& 7 �;Adeg.

Next werunthesamesimulationswith differentnumbersof links � to seehow theoptimallyefficientparametersdependupon � . For eachvalueof � 7:m� ! ! �� A , figuressimilar to Fig. 10canbedrawn, andthe qualitative propertiesof the optimal parametersarefound not to dependupon � . The meanvalue(over � � �)(ã *# ) of the optimal is found to be insensitive to � , whereasthat of � significantlychangeswith � , asshown in Fig. 11. It seemsthattheparetooptimalvalueof � is essentiallydeterminedby thenumberof links only andis roughlyequalto T,+ / � (thesolid curve in Fig. 11). This meansthatthesnake robotformsjust oneperiodof theserpenoidcurvewhenthepowerefficiency is optimized.

Finally, we run thesamesimulationswith differentvaluesof thefriction coefficient Ç Æ . Therelation-shipbetweenthemeanvalueof theoptimal andtheratio Ç Æ|/ Ç � is shown in Fig. 12. Weseethat is anincreasingfunctionof Ç Æ|/ Ç � . Thusthesnake robotshouldundulatewith largeramplitudewhentheratioÇ Æ|/ Ç � is larger(i.e.,whenthesnake robotis morelikely to slip in thenormaldirection).

In summary, we have foundthefollowing propertiesof theparetooptimalparameters:- � is a linearfunctionof theforwardspeed.-. is anincreasingfunctionof theratio Ç Æ|/ Ç � of thefriction coefficients.- � is adecreasingfunctionof � ; roughly, � 7 T,+ / � .

Theseobservationsareusefulfor (heuristically)determiningtheoptimalcontrollaw for thesnakerobot.

Synthesisof Snakelike Locomotion

Control Ar chitecture

Recall from the analysisin the previous sectionthat the parameters� and � in the serpentinemotiondeterminethespeedanddirectionof thesnake robot,respectively. On theotherhand,we foundthatthevaluesof and � thatgiveoptimalefficiency aredeterminedby thefriction coefficientsandthenumber

13

0 2 4 60

50

100

150

200

250

300

350

Speed [m/s]In

put P

ower

[J/s

]

0 2 4 6 80

1

2

3

4

5

ω [r

ad/s

]

Speed [m/s]

Figure9. Speed/power trade-off.

050

100150

0

50

0

2

4

β [deg]α [deg]

ω [r

ad/s

]

0 50 100 1500

20

40

60

80

β [deg]

α [d

eg]

0 50 100 1500

1

2

3

4

5

β [deg]

ω [r

ad/s

]

0 20 40 60 800

1

2

3

4

5

α [deg]

ω [r

ad/s

]

Figure10. Pareto-optimalparameters.

2 4 6 8 100

50

100

150

n

β [d

eg]

Figure11. Phaseshift � vs. the numberof links � .

0 0.2 0.4 0.6 0.820

30

40

50

60

70

α [d

eg]

ct / c

n

Figure12. Undulationangle vs. thera-tio of friction coefficients Ç Æ�/ Ç � .

14

of links, respectively. Hencewefix and � anduse� and � to controlthespeedandthedirectionof thesnakerobot. In thesimulationsshown below, weconsiderthefive-link snake robot( � 7�# ) with � . 7 kg and g . 7 m, andwefix and � to thefollowing values:S7 + T � � 7 T,+� Figure13 shows the block diagramof the basiccontrol system. The block labeled“Snake Robot” isgivenby (13). Thecontrollermainlyconsistsof inner/outerloops.Theouter-loopcontrollers�0/ $ ¿ % and�01 $ ¿ % are,respectively, for speedanddirectioncontrolsandgeneratethecommandmotionvariable § �throughthetime-varyingtransformation(representedby

ä)§ �. 5872 IKJML®$3� 1�R:$�-®�B % � % R4�

to achieve thedesiredmotion Å � and ò � . The inner-loop controller �65 $ ¿ % generatesthetorque ª sothattheactualmotion § follows thecommandsignal § � .

77

7ä �01�6588

�0/� 98

99 9 9

SnakeRobot

:9 ;

;§ �

�Å �ò �

ª § ò Å�

��

Figure13. Controlsystemconfiguration.

In the first-rounddesign,we tried PID controllersfor �0/ $ ¿ % , �01 $ ¿ % , and �65 $ ¿ % andfound that thedesignwasnotsuccessful.In particular, wehaddifficulty tuningthePID parametersto yield satisfactorymotioncontrolperformance,especiallyfor speedcontrol. Hencewe decidedto usePID controllersfor�01 $ ¿ % and �65 $ ¿ % only andto design�0/ $ ¿ % in amoresophisticatedmanner. Weexplain theprocedurefordesigning�0/ $ ¿ % in thenext section.

SpeedControl System

In this section,we developa speedcontrol system�0/ $ ¿ % within the control architectureintroducedintheprevioussection.In thesequel,we usethesimplefriction modelof thesnake robotwith Ç Æ ­ 7þAü Hz and Ç � ­ 7 A Hz for control design. The morerealisticCoulombfriction modelwill be usedforsimulationsto validateour resultafterthecontroldesign.

To examinehow thevariable� affectsthespeedŠ, wehavechanged��$21 % in asinusoidalmannerandobserved Š$21 % by simulation. As a result,we found that Š$21 % is alsoa sinusoidlike signal,but with an

15

increasingamplitudedivergingto infinity. Thusthemappingfrom � to Å is notbounded-input,bounded-output(BIBO) stable,andthis factmotivatedusto consideranew controlinput thatgivesaBIBO stablemappingandhencewouldbemoresuitablefor speedcontrol.

Herewe introducethefollowing variable 3< asanew controlinput:3<^5�7 9=<9 1 7 3�ì1õR>� � <�587 � 1 (20)

Notethat 3< is the“effective frequency” of thesinusoidalsignal IKJML®$3� 1 % with time-varyingfrequency � .If 3< $21 % is sinusoidal,so is thesignal < $ 7 � 1 % . Hence��$41 % approacheszeroasthe time goesto infinity,eventhough 3< $41 % is asinusoidalsignalwith afixedamplitude.Thusweexpectthatwhen 3< is sinusoidal,theresultingspeedÅ would not divergeandthemappingfrom 3< to Å would beBIBO stable.We foundby simulationthatthis is indeedthecase.

We have run simulationsfor sinusoidalinputs 3< usingthesimplefriction model. In particular, 3< ischosenas3<æ72? IKJMLõ$A@ 1 %for fixedvaluesof @ and ? . Thevalueof ? is setto 1 rad/s.Wehavetriedothervaluesaswell andfoundthattheresultsgivenbelow areinsensitiveto ? . Themotionvariable 3§ . is calculatedby3§ . $41 % 7� N"O;I!$ < $21 % R¡$�-õ�F % � % 3< $21 %andis appliedastheinput to thesnakerobotmodel(18). Notethattheinitial value < $ A % is equalto zeroby definition.

Typical speedresponsesare plotted in Fig. 14 for the case @ 7 Aã yA and in Fig. 15 for the case@ 7B# . FromFig. 14 we seethattheoutputsignal Å containshigh-frequency harmonics,but thelargelydominantpart is thesinusoidwith thesamefrequency astheinput. Whentheinput frequency is higher(seeFig. 15), a similar commentappliesaswell, with a notabledifferencebeingthat thephaselag andtheamplitudeattenuationof theoutput Å aremoreapparent.Thismeansthat themappingfrom 3< to Å isalmostlinear! Thus,choosingtheeffectivefrequency 3< astheinput,thehighly nonlinearrobotdynamicscanbequasi-linearized.

0 100 200 300 400 500 600−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

η [r

ad/s

]

0 100 200 300 400 500 600−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

v [m

/s]

Figure14. SpeedresponseÅ for sinusoidalin-put 3< with @ 7¡Aü 8A .

0 1 2 3 4 5 6 7−1

−0.5

0

0.5

1

Time [s]

η [r

ad/s

]

0 1 2 3 4 5 6 7−0.5

0

0.5

Time [s]

v [m

/s]

Figure15. SpeedresponseÅ for sinusoidalin-put 3< with @ 7C# .

Repeatingthe simulationfor differentvaluesof @ , we obtaina “frequency response”of the snakerobot from the input 3< to the output Å , wherewe ignorethe higherfrequency harmonicsof theoutput

16

10−2

10−1

100

101

102

−40

−30

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−10

0

10

Mag

nitu

de [d

B]

magnitude responseapploximation

10−2

10−1

100

101

102

−100

−80

−60

−40

−20

0

Pha

se [d

eg]

Frequency [rad/s]

phase responseapploximation

Figure16. Frequency response: 3< � Å ( � 7A ).

10−2

10−1

100

101

102

−50

−40

−30

−20

−10

0

10

Mag

nitu

de [d

B]

↓ γ = 0

γ = 0.5 →

10−2

10−1

100

101

102

−100

−80

−60

−40

−20

0

Pha

se [d

eg]

Frequency [rad/s]

↓ γ = 0

γ = 0.5 →

Figure17. Approximatedfrequency responses:3< � Å ( AD� � ��Aã $# ).signal.Thegainandphaseplotsthusobtainedareshown by thesolid curvesin Fig. 16. Fromthefigureweseethatthemappingfrom 3< to Å maybeapproximatedby afirst-ordertransferfunctiongivenbyE $ ¿ % 7 T ! # GF¿ R&T !ÂTH! �whosefrequency responseis plottedby the dashedlines in Fig. 16. Thus, introducinga new controlinput 3< , we mayapproximatethedynamicsof thesnake robotby a simplefirst-ordertransferfunction.Clearly, suchapproximationshouldbeusefulfor speedcontroldesign.

Theaboveapproximatedmodelis valid only for thecase� 7¡A in theserpentinemotion§ . $41 % 7� I#J�Lõ$�� 1�R:$2-õ�F % � % R�� Recallthat � is usedto control thedirectionof thesnake locomotion,andwe needto considerthecasewhere � is nonzeroaswell to allow for thedirectionalcontrol. Herewe considerthevaluesof � suchthat ý � ýI�ÝAü *# .

Sincethe locomotionof thesnakerobotis “symmetric” in � , we considerthe interval AJ� � ��Aü *#only. For eachvalueof � within theinterval,weobtainedthefrequency responsefrom 3< to Å asdescribedabove. The responsesareapproximatedby first-ordertransferfunctionsandplotted in Fig. 17. Thetransferfunctionfor each� is givenbyE $ ¿ % 7 ÷¿ R öwherethe parametersö and ÷ vary with � , asshown in Fig. 18. We seethat the valuesof ö and ÷ areapproximatelythesamefor eachfixed � andvary in the interval Aã $#K��öML ÷ ��m . Thuswe have thefollowing linearsystemmodelwith anuncertainparameterN :E�O $ ¿ % 7 ö¿ R ö � ö^587:ö ñ R'N öã��� ý N ýP� whereö ñ 5�7 F # and öã� 5�7 T # . Thismodelis suitablefor robustcontroldesign.

We designa controllersuchthat the speedÅ of the snake robot closelyfollows a given commandsignal Å � . In particular, we considertheblock diagramin Fig. 19 anddesign Q / $ ¿ % thatminimizesanupperboundon theworst-case� + gain(see,e.g.,[34], [35]) from Å � to ¯ 5�7 �<¯ � ¯ + ��� , wheretheworst

17

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

a

b

↓

γ = 0

← γ = 0.5

Figure18. Theparametersö and ÷ for differentvaluesof � .

7�ð

Q / $ ¿ %�

ERO $ ¿ %: : :;: ;

; ;;�Å

Š��

¯ � ¯ +3<

ÅFigure19. Speedcontrolsystem.

caseis takenover all possibletime-varyingparametricuncertaintiesNü$21 % v �¾�� � � . Theparameter�

istheweighton thecontrolsignal 3< , to bespecifiedlater.

Thegeneralizedplantfor thiscontrolsystemis givenbyoppppppppq3Å 3�¯ ñ¯ �¯ +*rtssssssssu 7

oppppppppq� ö ñ A öã� A ö ñ�� A A A�� A A A A A A AA A A A ��� A A A

rtssssssssuoppppppqÅ�áÅ �3<rtssssssu � á�7 N�¯ ñ

Approximatingthe worst-case� + gain by the scaled�TS

norm [36], [37], we obtaina controller thatminimizesthescaled

�DSnormasfollows:Q / $ ¿ % 7:Aã ! m TUF�R Aã V ;A Tü¿

wherethedesignparameteris chosento be� 7 by trial anderror. Notethat thecontrollerturnedout

to have theproportional-integralstructure.

18

Simulation Results

Wenow havea totalcontrolsystemfor thesnakerobotin which themappingfrom $ Å � � Å % and $4ò � � ò %to § � in Fig. 13 is replacedby<æ7 ¿ Q / $ ¿ % $ Å � � Å % � � $ ¿ % 7��01 $ ¿ % $2ò � � ò % � § � 72 IKJML®$ < R:$2-õ�F % � % R�� Thecontrollers�01 $ ¿ % and �65 $ ¿ % arechosento havethePID structureandtheirparameterstunedby trialanderror. The controller Q / $ ¿ % is the onedesignedin theprevious sectionvia a scaled

�DSsynthesis

method.In our simulation,theparametersof thesnake robotarechosenasg . 7 m � � . 7 kg � $2- 7 �! ! ! �� � %

ThecommandvelocitysignalsÅ � $41 % 5�7 m/s� ò � $21 % 587 + T rad

areappliedwhenthesnake robotis initially straight(b . $ A % 7:A ) andat rest( 3b . $ A % 7:A , 3á $ A % 7:A ).

First,we run asimulationfor thesimplefriction modelwith thefollowing parameters:Ç Æ ­ 7�Aã � Ç � ­ 7 Aã� $�- 7 �! " ! )� � %�01 $ ¿ % 7 �D � �65 $ ¿ % 7 $> A;A R A¿ R� A;¿ % ç The result is shown in Fig. 20 wherethedashedlines in the speedÅ anddirection ò plots indicatethecommandedsignalsandthesolidlinesaretheactualresponsesof thesnakerobot.Thetransientresponseof thespeedmaynot seemadequate;however, it is a resultof our tuningthecontroller �01 $ ¿ % suchthatthesnake robotstartsmoving with boththedesiredspeedandthedesireddirectionassoonaspossible.Whencontrollinga snake robot, it doesnot make senseto have it move at theright speedin thewrongdirection.Thus,in ourview, theresponsesin Fig. 20aresatisfactory.

Next we runasimulationfor theCoulombfriction modelwith thefollowing parameters:Ô Æ ­ 7:Aã � Ô � ­ 7 Aã� $�- 7 �! ! ! )� � %�01 $ ¿ % 7 � Aã ym=#ü� �65 $ ¿ % 7 $ #<A;A R m;A¿ R� A;¿ % ç Theresultis shown in Fig. 21. We seethat thespeedanddirectionresponsesaremoreoscillatorythanthe caseof thesimplefriction model,but theeffect of theseoscillationson theactualtrajectoryis notsubstantial.

19

0 5 10 15−1

0

1

2

3

Time [s]

Spe

ed [m

/s]

speed desired speed

0 5 10 150

20

40

60

80

100

120

Time [s]

Dire

ctio

n [d

eg]

direction desired direction

Figure 20. Simulationresult (simple frictionmodel).

0 5 10 15 20 250

0.5

1

1.5

Time [s]

Spe

ed [m

/s]

speed desired speed

0 5 10 15 20 250

20

40

60

80

100

120

140

Time [s]

Dire

ctio

n [d

eg]

direction desired direction

Figure21. Simulationresult(Coulombfrictionmodel).

Experimental Validation

In this section,wevalidateour controldesignstrategy througha laboratoryexperiment.

SystemConfiguration

The experimentaltestbedmainly consistsof five components—asnake robot, motor drivers, a CPUboard,aCCD camera,andapersonalcomputer(PC)(seeFig. 22).

Thesnake robotis simplyaconnectionof five rigid links with four jointswith totalweightof 1.8kg(Fig. 23). Eachlink is madeof aluminumof length0.15m andhasedgesat the bottomthat generatelargerfriction forcesin thenormaldirectionthanin thetangentialdirection(Fig. 24).

Mountedon eachjoint is a 1.7 W DC motor (RH–5A HarmonicDrive; Linx) with an embeddedencoder(28,000pulsesperrevolution). ThemotordriversandtheCPUboardaremountedon thesnakerobot.Two voltages(5 V for logic circuitson themotordriversandtheCPUboardand12 V for drivingthemotor)aresuppliedfrom outsidethroughcables.

TheCCD camera(CV-10BX; JAI) collects30 imagespersecondandis usedto detectthepositionsof two markedpointsonthefirst link of thesnakerobot.ThePCreceivestheimagedatafrom thecameraevery 33 msandcomputesthecoordinates$ ����� * � % of themiddlepoint andtheabsoluteangle

b � of thefirst link. This imageprocessingtakesabout60 msandhenceapproximatelyoneout of two imagedatapointsfrom theCCD camerais discarded.Informationon thepositionandorientationof thefirst link issentfrom thePCto theCPUboardthroughanRS-232Cserialcable.

TheCPUboardcalculatesthecommandtorque W (moreprecisely, a quantityto beusedasthedutyratio W of the PWM signal,asexplainedbelow) usingthe velocity commandsignal ( Å � � ò � ), the posi-tion/orientationinformationfrom thePC,andtherelative angleinformationfrom thecounter, all basedon thecontrolrule previously discussedwhere ª is replacedby W . Thesamplingrateof theCPUboardis 10 ms.

ThemotordriversreceivethetorquecommandW fromtheCPUboardandthengeneratePWMvoltagesignalswith theduty ratio W . for themotoron the - th joints,where W . ( - 7 �" ! ! �� �Ä�Ý ) is the - th entryof thevector W .

20

Figure22. Systemconfiguration.

Figure23. Top view of thesnakerobot. Figure24. The“belly” of thesnake robot.

Modeling and Control Design

Thefirst stepin our controldesignis themodelingof thesnake robot. Insteadof taking the frequencyresponsesfrom 3< to Å , asin theprevioussection,we chooseto modelthesystemby stepresponsesforsimplicity. Anotherdifferencefrom the simulationcaseis that the minor feedbackloop for the shapecontrolis implementedfirst asfollows:�65 $ ¿ % 7 T A R F ¿ R&T ¿andthenstepresponsesaretaken for thesnake robotwith theminor feedback loop. Fig. 25 shows theresponsesÅ whenthestepinputsof magnitude+ /H� rad areappliedat time 1 7X� s to 3< with differentvaluesof � . We have alsoobserved stepresponseswith different input magnitudes,andfound, as inthesimulationstudy, that theresponsesnormalizedby theinput magnitudesareinsensitive to theinputmagnitudes.The threeresponsesfor � 7 A are obtainedthroughthreedifferent experiments. Theresponsesarefoundto beoscillatory, but we go on to obtaina crudeapproximationof thesystemfrom

21

3< radto Å m/sasfollows:E�O $ ¿ % 7:Aã yA;m !ÂTÑ� ö¿ R ö � ö^587:ö ñ RYN öã��� ý N ýP� whereö ñ 5�7�Aã $#U# and öã� 5�7�Aã V(=# . Thestepresponsesof

ERO $ ¿ % for N 7CZ arealsoshown in Fig. 25.

5 10 15 20 25 30 35−1

0

1

2

3

4

Time [s]

v [c

m/s

]approximation(a=0.1) experiment(γ =0) experiment(γ =0) experiment( γ =0) experiment(γ =0.1)approximation(a=1)

Figure25. Stepresponses.

Basedon this linearuncertainsystemmodel,we designeda controller Q / $ ¿ % rad/masdescribedintheprevioussectionwith thedesignparameter

� 7:Aã asfollows:Q / $ ¿ % 7 T #ü $#<m R �ã $��( T¿ The parametersof the PID controller �01 $ ¿ % are tunedthroughactualexperiments,anda suitable

controlleris foundto bea simpleconstantgain:�01 $ ¿ % 7 � Aã Thecontrol systemthusobtainedis discretizedvia Tustin transformation(e.g.,[38]) with thesam-

pling rateof 10 msandimplementedon theCPUboard.

Experimental Results

An experimentis conductedunderthefollowing conditions:Thestepvelocity commandsÅ � $41 % 5�7:Aü 8A m �;A m/sò � $21 % 587¡A rad

areappliedwhenthesnake robot is at restandits body is straightat anangleof about25 deg from thedesireddirectionof locomotion.

Figure26 shows the motion picturesof thesnake robot wherethewhite andblack arrows, respec-tively, indicatethe initial anddesireddirectionsof the snake motion. Fig. 27 shows the time historiesof the speedÅ of thecenterof gravity andthe direction ò of the whole body. We seethat thespeedÅsatisfactorily tracksthedesiredspeed.Thedirectionparameterò is oscillatory, but its averageis approx-imatelyzero,meaningthatthesnakerobotis moving towardthedesireddirectiononaverage.To justifythisclaim,themotionof thesnakerobotis shown onthe � - * planeusingtheexperimentaldata(Fig.28).The snake robot movesfrom right to left in the figure. We seethat the directionof the snake robot is

22

1 7 [

1 7C#[

1 7 T A[

1 7 T #[

1 7:m;A[

1 7¡m=#[

1 7�(ÂA[

1 7�#<A[

1 72�;A[

1 7 F AFigure26. Locomotionof thesnakerobot.

23

10 20 30 40 50 60

−2

−1

0

1

2

3

4

Time [s]

v [c

m/s

]v desired v

10 20 30 40 50 60

−40

−20

0

20

40

Time [s]

Dire

ctio

n [d

eg]

direction desired direction

Figure 27. Speedand direction of the snakerobot.

−80 −60 −40 −20 0 20 40 60 80

−100

−80

−60

−40

−20

0

← Desired Direction

t = 1

↓

t = 8

↓

t = 74

↓

t = 28↑t = 46

↑

x [cm]

y [c

m]

Figure28. Snake motionfrom theexperimen-tal data.

not alignedwith thedesireddirectionat thebeginning,but therobotsoonbeginsmoving in thedesireddirection.

In summary, we concludethatour controlstrategy yieldeda reasonablevelocity controlsystemforthelocomotionof thesnake robot.

Conclusion

Wehavedescribedasnakerobotwithoutwheelsanddevelopedits modelbasedonthedirectionalfrictioncoefficients. After the model transformationthat decouplesthe inertial locomotionfrom the internalshapemotion,anoptimallyefficientserpentinelocomotionis investigated.Basedontheanalysisresults,a velocity controlschemeis proposedusinga quasi-linearizinginput transformation,andits validity isdemonstratedvia a laboratoryexperimentwith afive-link serpentinerobot.

The resultsreportedin this article focusedon the planarserpentinegait that is suitedfor locomo-tion in anenvironmentwherefriction with directionalpreferencecanberealized.While amathematicalframework is establishedfor modeling,analysis,andsynthesisof theserpentinegait, tremendousadapt-ability of snakelikemechanismto ruggedenvironmentshasnotyetbeenfully exploited.Studiesof othergaits remainto be doneto make a snake robot capableof, for example,adaptingto isotropic frictionenvironmentsand/orperformingthreedimensionalmotion.

Acknowledgments: The authorsgratefully acknowledgehelpful discussionswith B.D.O. Anderson,S. Hirose,T. Mita, P. Prautsch,H. Yamauchi,andR. Nakashima.This researchis supportedby TheGrant-in-Aidfor Center-Of-ExcellenceResearchProjectof SuperMechano-Systemsby TheMinistry ofEducation,Science,SportandCulturein Japan.

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Appendix

SomeIntegral Formulae

Lemma 1 Letscalars \ , ] , and ^`_�a begiven.Thefollowing hold:bc ^6dTef e sgngA\`hY]jiGkmlnipo satgq\sr�]j^tkb^%uvd ef e sgngA\ph']ji,kmi"lIi`o dzsgq\wr�]%^xkwhere functionssatgzy{r|yk anddzsgzy}rxyk aredefinedby

satg�~�r���k0�$o����� ����b g�~M_B� ����k~��,� gAa���� ~��P��� ����k� b g�~M� � � ����ka g�~�o�a=k r dzsg�~�r���k0�*o ��� �� b � g�~�����k u gq��_B� ~���ka g%� ���=��� ~���kg�~�����k u � b gq��� � � ~���k��

PROOF. It is straightforwardto verify theresultby calculatingtheintegral for differentcases.Hencetheproof is omitted.

Notethatsatg�y}r|y*k anddzsgzy{r|yk arethesaturationandthe“deadzonesaturation”functionsthatcanbeplottedasin Fig. 29. It is intuitively clearandeasyto verify that both of thesefunctionsbecomethesignumfunctionin extremecases:

satg�~�r�a=k�o sgng�~�kjr dzsgqa�r���k�o sgng���k �

-|y|

|y| x

sat(x,y)

-1

1

-1

1

-|x|

|x| y

dzs(x,y)

Figure29. Functionssatgzy*k anddzsgzy*k .

25

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