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 August 28-30, 2001 Space Systems Laboratory Massachusetts Institute of Technology 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 Space Systems Laboratory Massachusetts Institute of Technology 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 Geo-location size : 5000 m * Figure courtesy of AFOSR Techsat21 Research Review (29 Feb - 1 Mar 2000) Space Systems Laboratory Massachusetts Institute of Technology Equations of Motions ? First order perturbation about natural circular Keplerian orbit ? Modified Hill’s Equations: () ( ) () 22 2 52 2 2 x y z ax c nx ncy ay ncx azkz =? ? ? =+ =+ && & && & && cos( 1 ) 21 sin( 1 ) 1 21 cos( ) 1 o o o xA nt s s y Ants s s zAkt 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 e ni t h -N ad i r () 2 2 2 3 13cos2 8 e ref ref JR si r ?? =+ ?? where 1cs= + () 2 2 2 2 3 1cos 2 e ref ref nJ R kn s i r ?? =++ ?? Space Systems Laboratory Massachusetts Institute of Technology Propulsion Subsystem (Hall Thrusters) ? High specific impulse – low propellant expenditure ηm um P e & 2 22 = 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 o t t e dtPJ ? Electrical power required: 100 - 200 W Hall Thruster * Space Systems Laboratory Massachusetts Institute of Technology Optimal Control Theory Linear Quadratic Controller (t o to t f ) ? Linear Dynamics BuAxx += & ? Augmented Cost (Method of Lagrange) dt J T t t T a f o }[ {)( 2 1 ]xBuAxp Ruuu & ?++ = ∫ *1-* ** *** pBRu pAp BuAxx T T ?= ?= += & & dt- tJ TT t t TT ff T a f o p}]xBuAx uB]pRu xpApxpu *** ** ** δ++ δ++ ]δ+{[+δ?=δ ∫ & & [ [ )( )( ? First order variation ? Quadratic Cost ∫ = f o t t e dtPJ ∫ = f o t t T dtJ Ruuu 2 1 )( Boundary Conditions 1. x(t f ) = x f specified terminal state x * (t o ) = x o x * (t f ) = x f 2. x(t f ) free x * (t o ) = x o p * (t f ) = 0 3. x(t f ) on the surface m(x(t)) = 0 x * (t o ) = x o m(x * (t f )) = 0 ∑ = ? ? =? k i f i if t m t 1 ))](([d)( ** x x p 0= Space Systems Laboratory Massachusetts Institute of Technology Terminal Conditions (Multi-Spacecraft) Phasing Condition (Cluster): , 21cos N io ij ji CR = = ?θ ∑ 5 N i Ni ji ij yy m C zz + = ?? ?? =?? ?? ?? ?? ?? ∑ () γ?γ= ? ? ? ? ? ? ? γ γ+γ + ? ? ? ? ? ? = sincos 1 sin2/5 cossin 2 22 1 zxm R zx R y m oo () γ?γ= ? ? ? ? ? ? ? γ γ+γ + ? ? ? ? ? ? = sincos 1 sin2/5 cossin 4 22 3 zxm nR zx nR y m oo && &&& For each spacecraft (R o projection on y-z plane): ? Position Conditions ? Velocity Conditions ? Tying Condition ()γ+γ= cossin 5 zxym & γ+γ+γ? sin 2 5 cossin 2 o nRzxy && 61 ** 1 p () d[ (x())] x N i fi f i m tt ? = ? ?= ? ∑ N-th Condition (Total of 6N conditions) for i = 1, 2, …, N-1 where 4 spacecraft example: C 1 = 4.35 C 2 = 3.42 C 3 = 1.67 Space Systems Laboratory Massachusetts Institute of Technology Multiple Shooting Method Solving two point boundary value problems t m-1 (t 1 ,s 1 ) (t 2 ,s 2 ) (t 3 ,s 3 ) (t m-1 ,s m-1 ) (t m ,s m ) x t t m t 3 t 2 t 1t m-1 (t 1 ,s 1 ) (t m ,s m ) x t t m t 3 t 2 t 1 Simple shooting method Multiple shooting method ? Guess states at t k and compare the integrated states at t k+1 with states at t k+1 ? Numerically more stable ? Computationally expensive ? Guess the missing states at t o and compare the integrated states at t f with terminal constraints ? Numerically unstable - errors are amplified due to integration Space Systems Laboratory Massachusetts Institute of Technology Tolerance Setting (a) (c) Tolerance Set at 10 -3 Space Systems Laboratory Massachusetts Institute of Technology Tolerance Set at 10 -3 (a) (c) (e) Tolerance Setting Space Systems Laboratory Massachusetts Institute of Technology Cluster Initialization (1) ? Cluster initialization from Hill’s origin to R o = 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. E 2sc < E 3sc ) ? Average energy required decay rapidly as a function of initialization time Space Systems Laboratory Massachusetts Institute of Technology 200 W 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 Laboratory Massachusetts Institute of Technology Cluster Re-sizing (1) -3000 -1500 0 1500 3000 -3000 -1500 0 1500 3000 -1500 -750 0 750 1500 Velocity vector (m) Cross Axis (m) Zeni t h - N adi r ( m ) Reconfigure 500m 5km Radar Mode Geolocation Mode * Figure courtesy of AFOSR Techsat21 Research Review (29 Feb - 1 Mar 2000) Optimal Cluster Re-sizing ? Objective of Techsat 21 Geo- location mission is to provide 10-50 m geo-location accuracy ? Geo-location accuracy is inversely proportional to size of cluster ? Re-size cluster to an elliptical trajectory of 2.5 km to achieve approximately 10 m ground resolution ? Example application is to quickly locate a lost pilot (Time critical mission) Space Systems Laboratory Massachusetts Institute of Technology Cluster Re-sizing (2) 200 W ? Minimum re-sizing time of 0.5 periods (48 mins) is required for Techsat 21 geo-location ? Maximum size of 1250 m can be attained if re-sizing time of 30 minutes is allowed Maximum Power Required For Geo-location Corresponding ?V Required ? Minimum ?V of 10 m/s is required to perform Techsat 21 geo-location operation (25% of total ?V budgeted) ? Significant ?V savings can be achieved by increasing re-sizing time to at least 1 period (97 mins) Space Systems Laboratory Massachusetts Institute of Technology Future Considerations ? Solutions obtained are only guaranteed to be local minimum - not global optimum ? Must check for minimum energy trajectories ? Plume contamination due to thruster firings ? Penalize thruster firings at other spacecraft ? Penalize firings in the plane of elliptical trajectory -3000 -1500 0 1500 3000 -3000 -1500 0 1500 3000 -1500 -750 0 750 1500 Velocity vector (m) Cross Axis (m) Zeni t h - N adi r ( m ) -3000 -1500 0 1500 3000 -3000 -1500 0 1500 3000 -1500 -750 0 750 1500 Velocity vector (m) Cross Axis (m) Z e n i th -N a d ir (m ) Space Systems Laboratory Massachusetts Institute of Technology Conclusions ? Techsat 21 Cluster initialization – achievable even with a short initialization time – recommend an initialization time of at least 1 period due to significant ?V savings ? Techsat 21 Geo-location problem – a minimum re-orientation time of at least 1 period – extremely high ?V expenditure operation ? Developed a tool to – determine minimum energy trajectories – evaluate minimum resources required for cluster re-configuration – size power subsystem for propulsion -3000 -1500 0 1500 3000-3000 -1500 0 1500 3000 -1500 -750 0 750 1500 Velocity vector (m) Cross Axis (m) Z e n i t h -N a d i r (m ) Reconfigure 500m 5km Radar Mode Geolocation Mode