Minimum Energy Trajectories for Techsat 21 Earth Orbiting Clusters

Transcription

1 Minimum Energy Trajectories for Techsat 1 Earth Orbiting Clusters Edmund M. C. Kong SSL Graduate Research Assistant Prof David W. Miller Director, MIT Space Systems Lab Space 1 Conference & Exposition Albuquerque August 8-3, 1

2 Objective and Outline Objective : To determine the optimal trajectories to reorient a cluster of spacecraft Motivation : To maximize the full potential of a cluster of spacecraft with minimal resources Presentation Outline Techsat 1 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

3 Techsat 1 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 3 Flight experiment with 3 spacecraft Spacecraft will be equipped with Hall Thrusters large thrusters for orbit raising and de-orbit 1 micro-thrusters for full threeaxis control Figure courtesy of AFOSR Techsat1 Research Review (9 Feb - 1 Mar ) Techsat 1 Flight Experiment Number of Spacecraft : 3 Spacecraft Mass : 19.4 kg Cluster Size : 5 m Orbital Altitude : 6 km Orbital Period : 84 mins Geo-location size : 5 m

4 Equations of Motions First order perturbation about natural circular Keplerian orbit Modified Hill s Equations: a = && x 5c n x nc y& x ( ) ( ) ( ) a = && y+ nc x& y = && + az z k z where 3JR e s = 1+ 3cos ( iref ) 8r ref 3nJ Re k = n 1+ s + cos i r Possible trajectory for Techsat 1: x = A cos( nt 1 s) o ( ref ) 1+ s y = Ao sin( nt 1 s) 1 s 1+ s z = Ao cos( kt) 1 s ref c = 1+ s Zenith-Nadir Cross Axis x1 Ellipse Elliptical Trajectory Projected Circle Velocity Vector -4

5 Propulsion Subsystem (Hall Thrusters) High specific impulse low propellant expenditure Electrical power required: where P e = m u & mη m - mass of spacecraft (19.4 kg) u m& η - spacecraft acceleration (m/s) - mass flow rate of propellant (kg/s) - thruster efficiency (%) Objective is to minimize electrical energy required: J = f t t o P e dt W Hall Thruster BHT--XB Hall Thruster Specific Impulse Thrust Mass flow rate : 153 s : 1.5 mn :.74 mg/s Typical Efficiency : 4% Power Input 1 - W Hall Thruster : W Figures courtesy of AFOSR Techsat1 Research Review (9 Feb - 1 Mar )

6 Linear Dynamics x & = Ax + Bu Augmented Cost (Method of Lagrange) J t f 1 a( u) = { t o + p T u Optimal Control Theory T Ru [ Ax + Bu x] & } dt Quadratic Cost J δj a = f t t o P e dt First order variation T ( u) = p ( t ) δx + [ u T + [ Ax f R + p f T + + Bu -x& 1 t f T J ( u) = u Rudt to t t o f B] δu {[ p T ] δp} dt A + p& = T ]δx Boundary Conditions 1. x(t f ) = x f specified x (t o ) = x o terminal state x (t f ) = x f. x(t f ) free x (t o ) = x o p (t f ) = 3. x(t f ) on the surface x (t o ) = x o k m(x(t)) = m p ( t f ) = di[ m(x i= 1 x (t f )) = i ( x ( t f ))] Linear Quadratic Controller (t o to t f ) x& p& u = Ax = A = R T -1 + Bu p B T p

7 Terminal Conditions (Multi-Spacecraft) For each spacecraft (R o projection on y-z plane): Position Conditions y xsin γ + z cos γ m1 = + ( ) Ro 5/ Ro sin γ m = x cos γ z sin γ m m Velocity Conditions 3 4 y& x& sin γ + z& cos γ = + ( ) nro 5/ nro sin γ = x& cos γ z& sin γ Tying Condition m 5 = y& xsin γ + z cos γ y x& sin γ + z& cos γ Phasing Condition (Cluster): N y y m5n+ i= C = z z j i i j i 1 1 ( ) 5 ( ) + sin γ nr o where C N = R 1 cosθ i o i, j j= i m p ( ) = d [ (x ( ))] 6N 1 i tf i tf i= 1 x for i = 1,,, N-1 4 spacecraft example: C 1 = 4.35 C = 3.4 C 3 = 1.67 N-th Condition (Total of 6N conditions)

8 Multiple Shooting Method Solving two point boundary value problems x x (t m,s m ) (t 3,s 3 ) (t m-1,s m-1 ) (t m,s m ) (t 1,s 1 ) (t 1,s 1 ) (t,s ) t 1 t t 3 t m-1 t m t t 1 t t 3 t m-1 t m t Simple shooting method 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 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

9 Tolerance Setting (a) (c) Tolerance Set at 1-3

10 Tolerance Setting (a) (c) (e) Tolerance Set at 1-3

11 Cluster Initialization (1) Cluster initialization from Hill s origin to R o = 5 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 sc < E 3sc ) Average energy required decay rapidly as a function of initialization time

12 W Cluster Initialization () Power History for cluster of 3 S/C Peak power required is below Techsat 1 maximum ( W) for initialization periods greater than. period V required asymptotes to ~.45 m/s Recommend initialization time of 1 period due to significant V savings (67%)

13 Objective of Techsat 1 Geolocation mission is to provide 1-5 m geo-location accuracy Cluster Re-sizing (1) 5m Reconfigure 5km Geo-location accuracy is inversely proportional to size of cluster Radar Mode Geolocation Mode Figure courtesy of AFOSR Techsat1 Research Review (9 Feb - 1 Mar ) Re-size cluster to an elliptical trajectory of.5 km to achieve approximately 1 m ground resolution Example application is to quickly locate a lost pilot (Time critical mission) Zenith-Nadir (m) Cross Axis (m) Velocity vector (m) Optimal Cluster Re-sizing -3

14 Cluster Re-sizing () Maximum Power Required For Geo-location Corresponding V Required W Minimum re-sizing time of.5 periods (48 mins) is required for Techsat 1 geo-location Maximum size of 15 m can be attained if re-sizing time of 3 minutes is allowed Minimum V of 1 m/s is required to perform Techsat 1 geo-location operation (5% of total V budgeted) Significant V savings can be achieved by increasing re-sizing time to at least 1 period (97 mins)

15 Future Considerations Solutions obtained are only guaranteed to be local minimum - not global optimum Must check for minimum energy trajectories Zenith-Nadir (m) Cross Axis (m) Velocity vector (m) -3 Zenith-Nadir (m) Plume contamination due to thruster firings Cross Axis (m) Velocity vector (m) -3 Penalize thruster firings at other spacecraft Penalize firings in the plane of elliptical trajectory

16 Developed a tool to Conclusions determine minimum energy trajectories evaluate minimum resources required for cluster re-configuration size power subsystem for propulsion 15-3 Techsat 1 Cluster initialization Zenith-Nadir (m) Cross Axis (m) -3 achievable even with a short initialization time recommend an initialization time of at least 1 period due to significant V savings Velocity vector (m) Techsat 1 Geo-location problem a minimum re-orientation time of at least 1 period extremely high V expenditure operation 5m Radar Mode 5km Reconfigure Geolocation Mode