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1、OptimizationandstabilizationoftrajectoriesforconstraineddynamicalsystemsMichaelPosa1,ScottKuindersma2,andRussTedrake1Abstract—Contactconstraints,suchasthosebetweenafootandthegroundorahandandanobject,areinherentinmanyrobotictasks.Theseconstraintsdefineamanifoldoffeasiblestates;whilewellunderstood
2、mathematically,theyposenumericalchallengestomanyalgorithmsforplanningandcontrollingwhole-bodydynamicmotions.Inthispaper,wepresentanapproachtothesynthesisandstabilizationofcomplextrajectoriesforbothfully-actuatedandunderactuatedrobotssubjecttocontactconstraints.Weintroduceatrajectoryoptimization
3、algorithm(DIRCON)thatextendsthedirectcol-locationmethod,naturallyincorporatingmanifoldconstraintstoproduceanominaltrajectorywiththird-orderintegrationaccuracy—acriticalfeatureforachievingreliabletrackingcontrol.Weadapttheclassicaltime-varyinglinearquadraticFig.1.AnAtlasrobotclimbingastepwithaid
4、ofahand.Thealgorithmsregulatortoproducealocalcost-to-gointhemanifoldtangentpresentedheresynthesizeandstabilizedynamic,multi-contacttrajectories.plane.Finally,wedescendthecost-to-gousingaquadraticprogramthatincorporatesunilateralfrictionandtorquecon-manifolds.WeintroduceDIRCON,aplanningalgorithm
5、forstraints.Thisapproachisdemonstratedonthreecomplexconstraineddynamicalsystemswiththird-orderintegrationwalkingandclimbinglocomotionexamplesinsimulation.accuracy;itmaintainstheadvantagesofdirectcollocation,I.INTRODUCTIONtherebyenablingmorereliablestabilizationascomparedwithManyofthefundamental
6、problemsinrobotics,suchasexistingmethods.Theresultsarethencombinedwithrecentlocomotionandmanipulation,involvetherobotmakingandadvancesinhumanoidcontrol,leveragingquadraticprogram-breakingcontactwithitsenvironment.Thelastfewyearsming,toincorporateconstraintssuchasinputsaturationshaveseenrapidadv
7、ancesinmotionplanningtechniques,andfrictionlimitsintothefeedbackpolicy[19],[18].Thesebasedontrajectoryoptimization,tosynthesizelocallyopti-threeelementsprovideanend-to-endrecipeforgeneratingmaltrajectorieswhichmakeandbreakcontact,
