麻省理工学院：《Satellite Engineering》Lecture 12 Objective and Outline

Minimum Energy Trajectories for Techsat 21 Earth Orbiting Clusters Edmund M C Kong SSL Graduate Research Assistant Prof david w. miller Director, MIT Space Systems Lab Space 2001 Conference& Exposition Albuquerque August28-30,2001

Minimum Energy Trajectories for Minimum Energy Trajectories for Techsat Techsat 21 Earth Orbiting Clusters Earth Orbiting Clusters Edmund M. C. Kong SSL Graduate Research Assistant Prof David W. Miller Director, MIT Space Systems Lab Space 2001 Conference & Expositi o n Albuquerque August 28-30, 2001

Objective and outline Objective To determine the optimal trajectories to re orient a cluster of spacecraft Motivation To maximize the full potential of a cluster of spacecraft with minimal resources Presentation Outline · Techsat21 Overview Results Optimal control Formulation Tolerance setting Equations of Motions Cluster Initialization Dynamics) Cluster Re-sizing Propulsion System(Cost) (Geolocation) LQ Formulation Future Work Terminal Constraints Conclusions Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Objective and Outline Objective and Outline Objective : To determine the optimal trajectories to re￾orient a cluster of spacecraft Motivation : To maximize the full potential of a cluster of spacecraft with minimal resources Presentation 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 • Conclusions

Techsat 21 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 Techsat 21 Flight Experiment Spacecraft will be equipped with Hall Thrusters Number of Spacecraft 3 Spacecraft Mass :1294kg 2 large thrusters for orbit raising and de-orbit Cluster Size :500m 10 micro-thrusters for full three. Orbital Altitude :600km axis control Orbital Period 84 mins Figure courtesy of AFOSR Techsat21 Geo-ocation size Research Review(29 Feb-1 Mar 2000 5000m Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Techsat Techsat 21 • 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 Thrusters Techsat 21 Flight Experiment Number of Spacecraft : 3 Spacecraft Mass : 129.4 kg Cluster Size : 500 m Orbital Altitude : 600 km Orbital Period : 84 mins – 2 large thrusters for orbit raising and de-orbit – 10 micro-thrusters for full three￾axis control * Figure courtesy of AFOSR Techsat21 Geo-location size : 5000 m Research Review (29 Feb - 1 Mar 2000)

Equations of Motions First order perturbation about natural circular Keplerian orbit Modified Hill's Equations 0r=-152-2)nx-2(nc)y (velocity vector) a,=j+2(nc)i a =z+k where 3E[1+3cs(2 √1+s re 3nJR- k=nv1+s+ Possible trajectory for Techsat 21: 200+-----i----i-Zx1 Ellipse i cOS Eliptical Trajectory x=A cos(ntv1-s) 0 2√1+s R-200 A sin(nt S 400 ------ Projected Ci 400 2√1+s A coS(kt -200 S Cross axis 400400 Velocity Vector Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Equations of Motions Equations of Motions • First order perturbation about natural circular Keplerian orbit • Modified Hill’s Equations: ( ) ( ) ( ) 2 2 2 5 2 2 2 x y z a x c n x nc y a y nc x a z k z = − − − = + = + && & && & && cos( 1 ) 2 1 sin( 1 ) 1 2 1 cos( ) 1 o o o x A nt s s y A nt s s s z A kt s = − + = − − − + = − − • Possible trajectory for Techsat 21: -400 -200 0 200 -400 400 -200 0 200 400 -400 -200 0 200 400 Velocity Vector Elliptical Trajectory Projected Circle 2x1 Ellipse Cross Axis Z eni t h-Nad i r ( ) 2 2 2 3 1 3cos 2 8 e ref ref J R s i r = + ⎡ ⎤ ⎣ ⎦ where c s = 1+ ( ) 2 2 2 2 3 1 cos 2 e ref ref nJ R k n s i r = + + ⎡ ⎤ ⎣ ⎦

Propulsion Subsystem (Hall Thrusters)// High specific impulse low propellant expenditure Electrical power required e 2mn where 200 W Hall 100-200W m-mass of spacecraft (129. 4 kg) Thruster Hall thruster u -spacecraft acceleration(m/s) BHT-200-X2B Hal Thruster m -mass flow rate of propellant (kg/s)Specific Impulse :1530s n- thruster efficiency (% Thrust :10.5mN Mass flow rate 0.74mg/s Objective is to minimize electrical Typical Efficiency :42% energy required: Power Input :200W Pdt Figures courtesy of AFoSR Techsat21 Research Review(29 Feb-1 Mar 2000) Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Propulsion Subsystem (Hall Thrusters) Propulsion Subsystem (Hall Thrusters) • High specific impulse – low propellant expenditure mη m u Pe 2 & 2 2 = where m - mass of spacecraft (129.4 kg) u - spacecraft acceleration (m/s) - mass flow rate of propellant (kg/s) - thruster efficiency (%) m & η BHT-200-X2B Hall Thruster Specific Impulse : 1530 s Thrust : 10.5 mN Mass flow rate : 0.74 mg/s Typical Efficiency : 42% Power Input : 200 W 200 W Hall Thruster * * Figures courtesy of AFOSR Techsat21 Research Review (29 Feb - 1 Mar 2000) • Objective is to minimize electrical energy required: ∫ = f ott J Pedt • Electrical power required: 100 - 200 W Hall Thruster *

Optimal Control Theory Linear dynamics Quadratic Cost X=AX+Bu Pdt口J(u)= 25 Rudt Augmented Cost First order variation (Method of Lagrange) (u)=-p()x+{[pA+p8x U KU LuR+p Bou +p'[Ax+Bu-x刘t +[Ax Bu-x plat=0 Boundary conditions inear Quadratic Controller 1. x(tr)=Xr specified x(to)=Xo to terminal stat (t)=X A*, 2. x(ta) free x(tb)=X。 Ax +Bu p(t)=0 3. x(t) on the surface X (to)=Xo Ap m(x()=0 ()=2dm(x( RB p (x(t)=0 Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Optimal Control Theory Optimal Control Theory Linear Quadratic Controller (to to tf) • Linear Dynamics x & = Ax + Bu • Augmented Cost (Method of Lagrange) dt J T t t T a f o [ } ( ) {21 p Ax Bu x] u u Ru + + − & = ∫ * -1 * * * * * * u R B p p A p x Ax Bu T T = − = − = + & & - dt J t T T t t T T f f T a f o Ax Bu x ] p} u R p B] u u p x p A p x * * * * * * * + + δ + + δ δ = − δ + {[ + ]δ ∫ & & [ [ ( ) ( ) • First order variation • Quadratic Cost ∫ = f ott e J Pdt ∫ = f ott T J u u Rudt 21 ( ) Boundary Conditions 1. x(tf) = xf specified terminal state x * (to) = xo x * (tf) = xf 2. x(tf) free x*(to) = xo p *(tf) = 0 3. x(tf) on the surface m(x(t)) = 0 x *(to) = xo m(x*(tf)) = 0 ∑ = ∂ ∂ − = k i f i f i t m t 1 ( ) d [ ( ( ))] * * x x p = 0

Terminal Conditions(Multi-Spacecrafyrnk For each spacecraft (Ro projection on y-z plane) 300 c Position Conditions c xsiny+ =COSY R (5/2] siny m2 =xcosy-2sIny Velocity Conditions 100 y_xsin+=cosy np/x (5/2川 InR siny」 m4 =xcosy-2siny 30902001000 00200300 Where Velocity Vector (m) · Tying Condition ms=j(xsiny +coSY C=√2R∑ vI-cos 0 for j=1.2.N-1 ylrsiny+EcosY)+anR siny 4 spacecraft example C1=435C2=342C3=167 Phasing Condition(Cluster) N-th Condition(Total of 6N conditions) ∑ 6N-1 p(1)=∑d4(x(t) Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Terminal Conditions (Multi Terminal Conditions (Multi-Spacecraft) Spacecraft) Phasing Condition (Cluster): 2 1 , c o s N i o i j j i C R = = ∑ − θ 5 N N i i j i i j y y m C z z + = ⎡ ⎤ ⎡ ⎤ = − − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ∑ ( ) = γ − γ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ γ γ + γ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = cos sin 1 5 / 2 sin sin cos 2 2 2 1 m x z R x z R y m o o ( ) = γ − γ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ γ γ + γ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = cos sin 1 5 / 2 sin sin cos 4 2 2 3 m x z nR x z nR y m o o & & & & & For each spacecraft ( R o projection on y-z plane): • Position Conditions • Velocity Conditions • Tying Condition = ( ) sin γ + cos γ 5 m y& x z − ( ) γ + γ + sin γ 2 5 sin cos 2 o y x& z& nR 6 1 * * 1 p ( ) d [ ( x ( ) )] x N i f i f i m t t − = ∂ − = ∂ ∑ N-th Condition (Total of 6 N conditions) for i = 1, 2, …, N-1 where 4 spacecraft example: C1 = 4.35 C 2 = 3.42 C 3 = 1.67

Multiple shooting Method Solving two point boundary value problems (tm,sm) (t3,S3) (tm-lsm-i.m? 二一 t t2 t3 m-1 m-1 Simple shooting method Multiple shooting method Guess the missing states at t · Guess states at t, and and compare the integrated compare the integrated states states at t with terminal at tk+, with states at tKy constraints Numerically more stable Numerically unstable-errors Computationally expensive are amplified due to integration Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Multiple Shooting Method Multiple Shooting Method Solving two point boundary value problems tm-1 (t1,s1) (t2,s2) (t3,s3) (tm-1,sm-1) (tm,sm) x t t t t1 t2 t3 m m-1 (t1,s1) (tm,sm) x t t t1 t2 t3 m Simple shooting method Multiple shooting method • Guess states at tk and compare the integrated states at tk+1 with states at tk+1 • Numerically more stable • Computationally expensive • Guess the missing states at to and compare the integrated states at tf with terminal constraints • Numerically unstable - errors are amplified due to integration

Tolerance Setting Tol= 3.8e-002 Normalized Energy= 0.3342(a) Normalized Energy vs Convergence Tolerance 03344 0.5 03342 0334 Cross Axis(m) Velocity vector a)尝0338 b Tol =1.3e-002 Normalized Energy =0.3336(c) 03336 e 0.3334 Convergence Tolerance Tolerance Set at 10-3 Cross Axis(m) Velocity vector(m) Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Tolerance Setting Tolerance Setting (a) (c) Tolerance Set at 10-3

Tolerance Setting Ne Tol= 3.8e-002 Normalized Energy= 0.3342(a) e tolerance Tol= 2.5e-003 Normalized Energy 0 3335(e) 0.5 .5 b Cross Axis(m) Velocity vector )(e ss Axis(m) 11 Velocity vector(m) Tol =1.3e-002 Normalized Energy =0.3336(c) c)0336 d 0.3334 10 10 10 Convergence Tolerance Tolerance set at 10 Cross Axis(m) Velocity vector(m) Space Systems Laboratory Massachusetts Institute of Technology

Space Systems Laboratory Massachusetts Institute of Technology Tolerance Set at 10-3 (a) (c) (e) Tolerance Setting Tolerance Setting