Minimum Energy Trajectories for Techsat 21 Earth Orbiting ClustersObjective and OutlineTechsat 21Equations of MotionsPropulsion Subsystem (Hall Thrusters)Optimal Control TheoryTerminal Conditions (Multi-Spacecraft)Multiple Shooting MethodTolerance SettingTolerance SettingCluster Initialization (1)Cluster Initialization (2)Cluster Re-sizing (1)Cluster Re-sizing (2)Future ConsiderationsConclusionsMinimum Energy Trajectories for Minimum Energy Trajectories for TechsatTechsat21 21 Earth Orbiting ClustersEarth Orbiting ClustersEdmund M. C. KongSSL Graduate Research Assistant Prof David W. MillerDirector, MIT Space Systems LabSpace 2001 Conference & ExpositionAlbuquerqueAugust 28-30, 2001Space Systems LaboratoryMassachusetts Institute of TechnologyObjective and OutlineObjective and OutlineObjective : To determine the optimal trajectories to re-orient a cluster of spacecraftMotivation : To maximize the full potential of a cluster of spacecraft with minimal resourcesPresentation Outline• Techsat 21 Overview• Optimal Control Formulation– Equations of Motions (Dynamics)– Propulsion System (Cost)– LQ Formulation– Terminal Constraints• Results– Tolerance setting– Cluster Initialization– Cluster Re-sizing (Geolocation)• Future Work• ConclusionsSpace Systems LaboratoryMassachusetts Institute of TechnologyTechsatTechsat2121• To explore the technologies required to enable a Distributed Satellite System• Sparse Aperture Space Based Radar• Full operational system of 35 clusters of 8 satellites to provide global coverage• 2003 Flight experiment with 3 spacecraft• Spacecraft will be equipped with Hall ThrustersTechsat 21 Flight ExperimentNumber of Spacecraft : 3Spacecraft Mass : 129.4 kgCluster Size : 500 mOrbital Altitude : 600 kmOrbital Period : 84 mins– 2 large thrusters for orbit raising and de-orbit– 10 micro-thrusters for full three-axis controlGeo-location size : 5000 m* Figure courtesy of AFOSR Techsat21 Research Review (29 Feb - 1 Mar 2000)Space Systems LaboratoryMassachusetts Institute of TechnologyEquations of MotionsEquations of Motions• First order perturbation about natural circular Keplerian orbit• Modified Hill’s Equations:()()()22252 22xyzax c nx ncyay ncxazkz=− − −=+=+&& &&& &&&cos( 1 )21sin( 1 )121cos( )1oooxA nt ssyAntssszAkts=−+=− −−+=−−• Possible trajectory for Techsat 21:-400-2000200400-400-2000200400-400-2000200400Velocity VectorElliptical TrajectoryProjected Circle2x1 EllipseCross AxisZenith-Nadir()222313cos28erefrefJRsir⎡⎤=+⎣⎦where1cs=+()222231cos2erefrefnJ Rkn s ir⎡⎤=++⎣⎦Space Systems LaboratoryMassachusetts Institute of TechnologyPropulsion Subsystem (Hall Thrusters)Propulsion Subsystem (Hall Thrusters)• High specific impulse– low propellant expenditureηmumPe&222=wherem - mass of spacecraft (129.4 kg)u - spacecraft acceleration (m/s)- mass flow rate of propellant (kg/s)- thruster efficiency (%)m&ηBHT-200-X2B Hall ThrusterSpecific Impulse : 1530 sThrust : 10.5 mNMass flow rate : 0.74 mg/sTypical Efficiency : 42%Power Input : 200 W200 W HallThruster ** Figures courtesy of AFOSR Techsat21 Research Review (29 Feb - 1 Mar 2000)• Objective is to minimize electrical energy required:∫=fottedtPJ• Electrical power required:100 - 200 WHall Thruster *Space Systems LaboratoryMassachusetts Institute of TechnologyOptimal Control TheoryOptimal Control TheoryLinear Quadratic Controller(toto tf)• Linear DynamicsBuAxx +=&• Augmented Cost (Method of Lagrange)dtJTttTafo}[{)(21]xBuAxpRuuu&−++=∫*1-******pBRupApBuAxxTT−=−=+=&&dt-tJTTttTTffTafop}]xBuAxuB]pRuxpApxpu*******δ++δ++]δ+{[+δ−=δ∫&&[[)( )(• First order variation• Quadratic Cost∫=fottedtPJ∫=fottTdtJ Ruuu21)(Boundary Conditions1. x(tf) = xf specifiedterminal statex*(to) = xox*(tf) = xf2. x(tf) freex*(to) = xop*(tf) = 03. x(tf) on the surfacem(x(t)) = 0x*(to) = xom(x*(tf)) = 0∑=∂∂=−kifiiftmt1))](([d)(**xxp0=Space Systems LaboratoryMassachusetts Institute of TechnologyTerminal Conditions (MultiTerminal Conditions (Multi--Spacecraft)Spacecraft)Phasing Condition (Cluster):,21cosNio ijjiCR==−θ∑5NiNijiijyymCzz+=⎡⎤ ⎡⎤=−−⎢⎥ ⎢⎥⎣⎦ ⎣⎦∑()γ−γ=−⎥⎦⎤⎢⎣⎡γγ+γ+⎥⎦⎤⎢⎣⎡=sincos1sin2/5cossin2221zxmRzxRymoo()γ−γ=−⎥⎦⎤⎢⎣⎡γγ+γ+⎥⎦⎤⎢⎣⎡=sincos1sin2/5cossin4223zxmnRzxnRymoo&&&&&For each spacecraft (Roprojection on y-z plane):• Position Conditions• Velocity Conditions• Tying Condition()γ+γ= cossin5zxym&()γ+γ+γ− sin25cossin2onRzxy&&61**1p() d[ (x())]xNifi fimtt−=∂−=∂∑N-th Condition (Total of 6N conditions)for i = 1, 2, …, N-1where4 spacecraft example:C1= 4.35C2= 3.42C3= 1.67Space Systems LaboratoryMassachusetts Institute of TechnologyMultiple Shooting MethodMultiple Shooting MethodSolving two point boundary value problemstm-1(t1,s1)(t2,s2)(t3,s3)(tm-1,sm-1)(tm,sm)xttmt3t2t1tm-1(t1,s1)(tm,sm)xttmt3t2t1Simple shooting methodMultiple shooting method• Guess states at tkand compare the integrated states at tk+1with states at tk+1• Numerically more stable• Computationally expensive• Guess the missing states at toand compare the integrated states at tfwith terminal constraints• Numerically unstable - errors are amplified due to integrationSpace Systems LaboratoryMassachusetts Institute of TechnologyTolerance SettingTolerance Setting(a)(c)Tolerance Set at 10-3Space Systems LaboratoryMassachusetts Institute of TechnologyTolerance Set at 10-3(a)(c)(e)Tolerance SettingTolerance SettingSpace Systems LaboratoryMassachusetts Institute of TechnologyCluster Initialization (1)Cluster Initialization (1)• Cluster initialization from Hill’s origin to Ro= 250 m• In general, average energy required are similar for different N spacecraft clusters• Slight differences in energy requirements are due to the more stringent constrains placed on phasing the array (eg. E2sc< E3sc)• Average energy required decay rapidly as a function of initialization timeSpace Systems LaboratoryMassachusetts Institute of Technology200 WCluster Initialization (2)Cluster Initialization (2)Power History for cluster of 3 S/C• Peak power required is below Techsat 21 maximum (200 W) for initialization periods greater than 0.2 period• ∆V required asymptotes to ~ 0.45 m/s• Recommend initialization time of 1 period due to significant ∆V savings (67%)Space Systems LaboratoryMassachusetts Institute of TechnologyCluster ReCluster Re--sizing