車(chē)輪型倒立振子安定化制御en

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1、OtherpropertiesWheel-typeInversePendulumMicrocomputerAKI-H8/3664RotaryencoderBTE030，90[p/r]GyrosensorENC-03J，0.67[mV/(deg/s)]DCmotorRE-260ジャイロセンサGearratio29.8９Ｖ電池１．５Ｖ電池×２Gg:gravityI,Mθlτ:torquev:velocityofcenterofgravityマイクロコンピュータd基板?vx=(x+lsinθ)=x&+(lcosθ)θ&(1)dtロータリエンコーダdDCモータvy=(lcos

2、θ)=?(lsinθ)θ&(2)rJ,mdt減速ギアxKineticenergy車(chē)軸T1mx&21Mv2v21J&21I&2=+(x+y)+?+θ(3)車(chē)輪2222Fig.1Wheel-typeInversePendulumFig.2CoordinatePotentialenergyU=Mglcosθ(4)RelationshipofcoordinatesParametersofthesystemx=r?,x&=r?&(5)Massofwheelsm=0.061[kg]EquationofmotionMomentofinertiaofwheels2.7910?5[kg

3、m2]{mMr2J}&&Mlr&&Mlr&2D&&(6)J=×?(+)+?+(cosθ)θ?(sinθ)θ+(??θ)=τMl2I&&Mlr&&Mlr&&MglD&&(7)Radiusofwheelr=0.029[m](+)θ+(cosθ)??(sinθ)?θ?sinθ?(??θ)=?τAssumingMassofbodyM=0.421[kg]?32sinθ?θ，cosθ?1,(8)MomentofinertiaofbodyI=2.95×10[kg?m]&2and&,wecanobtainlinearizedequations.andneglectingnonline

4、artermslikeθθθDistancebetweencenterofwheelandgravityl=0.119[m]{mMr2J}&&Mlr&&D&&(9)(+)+?+θ+(??θ)=τCoefficientofviscousdampingD=1.0×10?3[Nm/(rad/s)]Ml2I&&Mlr&&MglD&&(10)(+)θ+??θ?(??θ)=?τStateEquationStatefeedback（Poleplacement）{2}Controlinput(m+M)r+J(Mgl)θ&&=θ{}(m+M)r2+J(Ml2+I)?(Mlr)2τ=?[

5、k1k2k3k4]x=?Kx(19)Closedloopsystem{2}?(m+M)r+J+Mlr{}?D(&?&)(11)τ?θx&=Ax?BKx=(A?BK)x(20){}222(m+M)r+J(Ml+I)?(Mlr)Characteristicequation22Mglr?&&=?θs4+(ck+dk)s3+(a+ck+dk)s2{}(m+M)r2+J(Ml2+I)?(Mlr)22413(21)?(ad?bc)k4s?(ad?bc)k3=02Mlr+(Ml+I){}?(&?&)+τD?θ(12)Ifdesiredpolesofthesystemareexpre

6、ssedaspi,weobtainthefollowingequation{}222(m+M)r+J(Ml+I)?(Mlr)43s?(p1+p2+p3+p4)sByusingfollowingexpressions2{2}+(p1p2+p2p3+p3p4+p4p1+p1p3+p2p4)s(22)(m+M)r+J(Mgl)a={}222?(p1p2p3+p2p3p4+p3p4p1+p4p1p2)s(m+M)r+J(Ml+I)?(Mlr)+p1p2p3p4=022Mglrb=?WecandeterminethefeedbackcoefficientKbycomparing

7、coefficientsofEqs.(21)and(22).{}222(m+M)r+J(Ml+I)?(Mlr)Polesofuncontrolledsystemare?14.1798,-2.1268,0,9.3.Hereisanunstablepole.{2}(m+M)r+J+Mlrc=?Tomakethesystemstable,wedesignpolesasfollows{}222(m+M)r+J(Ml+I)?(Mlr)p1,p2,p3,p4=?3±0.1j,?15±0.1j.(23)2Mlr+(Ml+I)d=(13)Thenweobtain{}