Geometric Control of Mechanical Systems (Supplements)

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1、FrancescoBulloandAndrewD.LewisGeometricControlofMechanicalSystemsModeling,Analysis,andDesignforSimpleMechanicalControlSystems–SupplementaryMaterial–March22,2006ContentsS1Tangentandcotangentbundlegeometry.................S1S1.1SomethingsHamiltonian....................

2、...........S1S1.1.1Differentialforms...............................S1S1.1.2Symplecticmanifolds...........................S5S1.1.3Hamiltonianvectorfields........................S6S1.2Tangentandcotangentliftsofvectorfields...............S7S1.2.1Moreaboutthetangentlift....

3、..................S7S1.2.2Thecotangentliftofavectorfield................S8S1.2.3Jointpropertiesofthetangentandcotangentlift...S9S1.2.4Thecotangentliftoftheverticallift..............S11S1.2.5ThecanonicalinvolutionofTTQ.................S12S1.2.6Thecanonicalendomorphismo

4、fthetangentbundle.S13S1.3Ehresmannconnectionsinducedbyanaffineconnection....S13S1.3.1Motivatingremarks.............................S13S1.3.2Moreaboutvectorandfiberbundles..............S15S1.3.3Ehresmannconnections.........................S17S1.3.4Linearconnectionsandlin

5、earvectorfieldsonvectorbundles.......................................S18S1.3.5TheEhresmannconnectiononπTM:TM→Massociatedwithasecond-ordervectorfieldonTM..S20S1.3.6TheEhresmannconnectiononπTQ:TQ→QassociatedwithanaffineconnectiononQ..........S21∗S1.3.7TheEhresmannconnectio

6、nonπT∗Q:TQ→QassociatedwithanaffineconnectiononQ..........S23S1.3.8TheEhresmannconnectiononπTTQ:TTQ→TQassociatedwithanaffineconnectiononQ..........S23∗S1.3.9TheEhresmannconnectiononπT∗TQ:TTQ→TQassociatedwithanaffineconnectiononQ..........S25∗S1.3.10RepresentationsofSTandST.

7、..................S26S1.3.11TheSasakimetric..............................S33SiiContentsS2Controllability...........................................S35S2.1Accessibilityandcontrollabilityofsystemswithabasicexternalforce.........................................S35S2.1.1

8、Accessibilityresults.............................S36S2.1.2Controllabilityresults...........................S40S2.2Accessibilityofsy