Lib Libration Orbit Mission ti O bit Mi i er & Purdue Un - PowerPoint PPT Presentation

Goddard Space Flight Center niversity Formation Control of the MAXI M L 2 Lib Libration Orbit Mission ti O bit Mi i er & Purdue Un David Folta and Kate Hartman ce Flight Cente NASA Goddard Space Flight Center dd d l h Kathleen Howell and Belinda Marchand Purdue University Purdue University Goddard Spac AIAA/AAS Astrodynamics Specialist Conference and Exhibit P Providence, RI, August 15-18 id RI A t 15 18 NASA / 1 Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

Agenda Goddard Space Flight Center • MAXIM Introduction niversity • MAXIM Formation MAXIM Formation er & Purdue Un • Formation Assumptions • Formation Definition ce Flight Cente • Control – Discrete and Continuous • Results • Summary Goddard Spac NASA / 2 Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

MAXIM Overview Goddard Space Flight Center  Th MAXIM  The MAXIM concept for NASA's Black Hole Imager mission utilizes interferometric t f NASA' Bl k H l I i i tili i t f t i techniques at the short wavelengths of X-rays  Very long optical baselines are needed to achieve high-precision angular resolution images niversity er & Purdue Un ce Flight Cente Goddard Spac NASA / 3 Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

MAXIM Formation Overview Goddard Space Flight Center  Multiple free-flying spacecraft comprise a sparse aperture providing collecting niversity area of ~ 1000cm 2 . er & Purdue Un  Images are generated through interference patterns gathered from the multiple satellites housing the optical elements that form the aperture.  The interference patterns or fringes are observed only if the path lengths are  The interference patterns or fringes are observed only if the path lengths are ce Flight Cente controlled to great precision.  The challenge is to control this path length in the presence of environmental and spacecraft disturbances driving the need for active control systems spacecraft disturbances driving the need for active control systems. Goddard Spac  We focus on the dynamics and control of formation flight in a full ephemeris modeling of the libration orbit to incorporate all gravitational perturbations and solar radiation pressure solar radiation pressure. NASA /  Analysis focuses on amount and duration of the control effort versus science observation requirements as measured in the formation optics plane 4 Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

MAXIM Formation Assumptions Goddard Space Flight Center  MAXIM formation components; Hub (1.3 x 2 meters , 331kg) , Freeflyer (periscope) (1.3 x 2 meters, niversity 304kg) , and the Detector (varying area 1.9 m 2 to 5.6 m 2 , 619kg) er & Purdue Un  Optics Plane: •Hub and Freeflyers form a physical configuration perpendicular to detector-hub line of sight (LOS) to a target. •Associates physical configuration to science requirements derived A i h i l fi i i i d i d ce Flight Cente from a Fourier transform of the image plane, the UV plane.  Observation duration is 100,000 secs Goddard Spac  Controller options: •Off during observation and on to realign and maintain the formation •Continuously on during observations  Inertial target of 45 0 elevation and 45 0 azimuth f 45 0 l  I i l i d 45 0 i h NASA /  Tolerance of radial distance of a Freeflyer from Hub less than 5 microns  Detector at 20,000km, six freeflyers at the maximum nominal radial y distance of 500 meters from the Hub. 5 Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

MAXIM Halo Orbit Goddard Space Flight Center • MAXIM L libration orbit is a typical mission • MAXIM L 2 libration orbit is a typical mission •A y = 700,000 km and A z =200,000 km niversity • Halo orbit computed with a full Ephemeris model • Halo orbit computed with a full Ephemeris model er & Purdue Un  Sun, Earth, Moon point mass  Solar Radiation Pressure ce Flight Cente •Hub follows Halo orbit Goddard Spac NASA / 20,000 km 6 Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

MAXIM Frame Definition Goddard Space Flight Center The MAXIM hub spacecraft is located at the X,Y,Z origin Th MAXIM h b f i l d h X Y Z i i and the angles  ,  provide the alignment toward the niversity target. g  ˆ   ˆ   ˆ ˆ w w C C X C C X S C Y S C Y S Z S Z         er & Purdue Un ˆ  ˆ Z w ˆ v ˆ  ˆ Z u Target ˆ  ˆ Z w ˆ ˆ u u w w ce Flight Cente   ˆ ˆ ˆ v w u Direction Cosines for conversion between Optics conversion between Optics Goddard Spac  frame and Inertial Frame      S C S C C      ˆ Hub Hub     X X NASA /   I U C C S S S C          ˆ  0  C S Y   7 Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

MAXIM Control Strategies Goddard Space Flight Center  Our investigation takes a global view of the large-scale formation flying  O i ti ti t k l b l i f th l l f ti fl i problem. niversity  Previous Research: er & Purdue Un • Near Earth - minimized gravitational perturbation - no close tracking of a reference solution - or use of non-linear (adaptive) 2-body problems • Multi-body y systems y - CRTBP only y or controller effectiveness is ce Flight Cente demonstrated relative to the linear dynamics, not the full nonlinear system - Evolution approximated from the linear dynamics of the integrated lissajous trajectory Goddard Spac • Naturally occurring formations derived from center manifold analysis, as well as a discrete impulsive control approach to maintain a prescribed formation plane NASA /  Continuous control approach Obtain a rough analytical approximation of center manifold motion and determine how continuous optimal control and exact feedback linearization compares, in terms of cost, to the discrete station-keeping li i ti i t f t t th di t t ti k i approach. 8 Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

MAXIM Control Strategies Goddard Space Flight Center • Previous work demonstrates the efficiency and cost effectiveness of both niversity input feedback linearization (IFL) and output feedback linearization (OFL) methods for formation control in the CRTBP methods for formation control in the CRTBP. er & Purdue Un • A linear quadratic regulator (LQR), derived from optimal control theory, yields essentially an identical error response and control acceleration history ce Flight Cente as the IFL approach. • IFL controller is computationally much less intensive and, by comparison, conceptually simple conceptually simple. Goddard Spac • We address the properties of the IFL controller in defining the MAXIM formation control NASA / • Analysis of position deviation of freeflyer or detector wrt Hub • For a comparison, a discrete stationkeeping control approach is devised to force the orientation of the formation plane to remain fixed inertially. f th i t ti f th f ti l t i fi d i ti ll 9 Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

MAXIM Discrete Control Goddard Space Flight Center            •Accuracy of formation maintenance r r A B r      1 0 0   ,     t t           niversity 1 0         v v C D v v 1 0 0 0 •Simple DC can maintain formation er & Purdue Un   •Discrete LQR yields optimal magnitude          1 v B r A r v 0 1 0 0 of differential control impulse •Simple: Target the end state •Simple: Target the end state ce Flight Cente  = STM  = state perturbation Optimal Discrete Simple Discrete  0  0 = Impulsive  V at beginning Impulsive  V at beginning without weights Goddard Spac •Discrete Optimal Control: Optimal Discrete (Q m ) Weighted quadratic of end with weights state error state error NASA / (Q) Weighted quadratic of state deviation along path •Simple has greatest error along path Si l h t t l th 10 Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit