车轮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ϕvx=(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車軸T1mx&21Mv2v21J&21I&2=+(x+y)+ϕ+θ(3)車輪2222Fig.1Wheel-typeInversePendulumFig.2CoordinatePotentialenergyU=Mglcosθ(4)RelationshipofcoordinatesParametersofthesystemx=rϕ,x&=rϕ&(5)Massofwheelsm=0.061[kg]EquationofmotionMomentofinertiaofwheels2.7910−5[kg

3、m2]{mMr2J}&&Mlr&&Mlr&2D&&(6)J=×⋅(+)+ϕ+(cosθ)θ−(sinθ)θ+(ϕ−θ)=τMl2I&&Mlr&&Mlr&&MglD&&(7)Radiusofwheelr=0.029[m](+)θ+(cosθ)ϕ−(sinθ)ϕθ−sinθ−(ϕ−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)(+)+ϕ+θ+(ϕ−θ)=τCoefficientofviscousdampingD=1.0×10−3[Nm/(rad/s)]Ml2I&&Mlr&&MglD&&(10)(+)θ+ϕ−θ−(ϕ−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)τϕθx&=Ax−BKx=(A−BK)x(20){}222(m+M)r+J(Ml+I)−(Mlr)Characteristicequation22Mglrϕ&&=−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){}−(&−&)+τDϕθ(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{}