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1、OPTIMALCONTROLAPPLICATIONSANDMETHODSOptim.ControlAppl.Meth.(2009)PublishedonlineinWileyInterScience(www.interscience.wiley.com).DOI:10.1002/oca.912DiscretemechanicsandoptimalcontrolforconstrainedsystemsS.Leyendecker1,∗,†,S.Ober-Bl¨obaum2,J.E.Marsden2andM.Ortiz31AeronauticsandControlandDyn
2、amicalSystems,CaliforniaInstituteofTechnology,1200E.CaliforniaBoulevard,MailCode107-81,Pasadena,CA91125,U.S.A.2ControlandDynamicalSystems,CaliforniaInstituteofTechnology,1200E.CaliforniaBoulevard,MailCode205-45,Pasadena,CA91125,U.S.A.3AeronauticsandMechanicalEngineering,CaliforniaInstitut
3、eofTechnology,1200E.CaliforniaBoulevard,MailCode205-45,Pasadena,CA91125,U.S.A.SUMMARYTheequationsofmotionofacontrolledmechanicalsystemsubjecttoholonomicconstraintsmaybeformulatedintermsofthestatesandcontrolsbyapplyingaconstrainedversionoftheLagrange-d’Alembertprinciple.Thispaperderivesast
4、ructure-preservingschemefortheoptimalcontrolofsuchsystemsusing,asoneofthekeyingredients,adiscreteanalogueofthatprinciple.Thispropertyisinheritedwhenthesystemisreducedtoitsminimaldimensionbythediscretenullspacemethod.Togetherwithinitialandfinalconditionsontheconfigurationandconjugatemomentum
5、,thereduceddiscreteequationsserveasnonlinearequalityconstraintsfortheminimizationofagivenobjectivefunctional.Thealgorithmyieldsasequenceofdiscreteconfigurationstogetherwithasequenceofactuatingforces,optimallyguidingthesystemfromtheinitialtothedesiredfinalstate.Inparticular,fortheoptimalcont
6、rolofmultibodysystems,aforceformulationconsistentwiththejointconstraintsisintroduced.Thisenablesonetoprovetheconsistencyoftheevolutionofmomentummaps.Usingatwo-linkpendulum,themethodiscomparedwithexistingmethods.Further,itisappliedtoasatellitereorientationmaneuverandabiomotionproblem.Copyr
7、ight2009JohnWiley&Sons,Ltd.Received8August2008;Revised9June2009;Accepted27July2009KEYWORDS:optimalcontrol;constrainedsystems;multibodydynamics;discretevariationalmechanics1.INTRODUCTIONfortheaccurate,robustandefficienttimeintegrationofconstraineddynamicalsystems(inpa
