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

SLIDE 1

Goddard Space Flight Center

niversity

Formation Control of the MAXI M L2 Lib ti O bit Mi i

er & Purdue Un

Libration Orbit Mission

David Folta and Kate Hartman dd d l h

ce Flight Cente

NASA Goddard Space Flight Center Kathleen Howell and Belinda Marchand Purdue University

Goddard Spac

Purdue University AIAA/AAS Astrodynamics Specialist Conference and Exhibit P id RI A t 15 18

NASA /

Providence, RI, August 15-18

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

1

SLIDE 2

Goddard Space Flight Center

Agenda

niversity

• MAXIM Introduction
• MAXIM Formation

er & Purdue Un

MAXIM Formation

• Formation Assumptions
• Formation Definition

ce Flight Cente

• Control – Discrete and Continuous
• Results

Goddard Spac

• Summary

NASA / Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

2

SLIDE 3

Goddard Space Flight Center

• Th MAXIM

t f NASA' Bl k H l I i i tili i t f t i

MAXIM Overview

niversity

• The MAXIM concept for NASA's Black Hole Imager mission utilizes interferometric

techniques at the short wavelengths of X-rays

• Very long optical baselines are needed to achieve high-precision angular resolution images

er & Purdue Un ce Flight Cente Goddard Spac NASA / Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

3

SLIDE 4

Goddard Space Flight Center

MAXIM Formation Overview

niversity

• Multiple free-flying spacecraft comprise a sparse aperture providing collecting

area of ~ 1000cm2.

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

ce Flight Cente

• The interference patterns or fringes are observed only if the path lengths are

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

Goddard Spac

spacecraft disturbances driving the need for active control systems.

• 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

NASA /

solar radiation pressure.

• Analysis focuses on amount and duration of the control effort versus science
• bservation requirements as measured in the formation optics plane

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

4

SLIDE 5

Goddard Space Flight Center

MAXIM Formation Assumptions

niversity

MAXIM formation components; Hub (1.3 x 2 meters , 331kg) , Freeflyer (periscope) (1.3 x 2 meters, 304kg) , and the Detector (varying area 1.9 m2 to 5.6 m2 , 619kg)

er & Purdue Un

Optics Plane:

• Hub and Freeflyers form a physical configuration perpendicular to

detector-hub line of sight (LOS) to a target. A i h i l fi i i i d i d

ce Flight Cente

• Associates physical configuration to science requirements derived

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

 I i l f 450 l i d 450 i h

NASA /

 Inertial target of 450 elevation and 450 azimuth  Tolerance of radial distance of a Freeflyer from Hub less than 5 microns  Detector at 20,000km, six freeflyers at the maximum nominal radial

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

5

y distance of 500 meters from the Hub.

SLIDE 6

Goddard Space Flight Center

• MAXIM L libration orbit is a typical mission

MAXIM Halo Orbit

niversity

• MAXIM L2 libration orbit is a typical mission
• Ay = 700,000 km and Az =200,000 km
• Halo orbit computed with a full Ephemeris model

er & Purdue Un

• Halo orbit computed with a full Ephemeris model

 Sun, Earth, Moon point mass  Solar Radiation Pressure

ce Flight Cente

• Hub follows Halo orbit

Goddard Spac NASA /

20,000 km

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

6

SLIDE 7

Goddard Space Flight Center

Th MAXIM h b f i l d h X Y Z i i

MAXIM Frame Definition

niversity

ˆ ˆ ˆ ˆ w C C X S C Y S Z

  

  

The MAXIM hub spacecraft is located at the X,Y,Z origin and the angles ,  provide the alignment toward the target.

er & Purdue Un

ˆ Z ˆ w ˆ u ˆ v

Target

ˆ ˆ ˆ ˆ ˆ w C C X S C Y S Z Z w u Z w

    

    

g

ce Flight Cente

w u

ˆ ˆ ˆ v w u   Direction Cosines for conversion between Optics

Goddard Spac

  ˆ X

Hub S C S C C

    

     

conversion between Optics frame and Inertial Frame

NASA /

X ˆ Y

Hub

I U

C C S S S C C S

      

         

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

7

SLIDE 8

Goddard Space Flight Center

• O

i ti ti t k l b l i f th l l f ti fl i

MAXIM Control Strategies

niversity

• Our investigation takes a global view of the large-scale formation flying

problem.

• 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

systems

• CRTBP
• nly
• r

controller effectiveness is

ce Flight Cente

y y y 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

• ptimal

control and exact feedback li i ti i t f t t th di t t ti k i

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

8

linearization compares, in terms of cost, to the discrete station-keeping approach.

SLIDE 9

Goddard Space Flight Center

MAXIM Control Strategies

niversity

• Previous work demonstrates the efficiency and cost effectiveness of both

input feedback linearization (IFL) and output feedback linearization (OFL) methods for formation control in the CRTBP

er & Purdue Un

methods for formation control in the CRTBP.

• 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

Goddard Spac

conceptually simple.

• 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

f th i t ti f th f ti l t i fi d i ti ll

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

9

force the orientation of the formation plane to remain fixed inertially.

SLIDE 10

Goddard Space Flight Center

MAXIM Discrete Control

niversity

 

1 1 1

, r r r A B t t v v v v C D      

  

                            

• Accuracy of formation maintenance
• Simple DC can maintain formation

er & Purdue Un

 

1 1

v B r A r v   

 

   

• Discrete LQR yields optimal magnitude
• f differential control impulse
• Simple: Target the end state

ce Flight Cente

Simple Discrete Optimal Discrete without weights

• Simple: Target the end state

 = STM  = state perturbation 0 = Impulsive V at beginning

Goddard Spac

Optimal Discrete with weights

0 Impulsive V at beginning

• Discrete Optimal Control:

(Qm) Weighted quadratic of end state error

NASA /

state error (Q) Weighted quadratic of state deviation along path Si l h t t l th

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

10

• Simple has greatest error along path

SLIDE 11

Goddard Space Flight Center

Th i l ti i i th l l ( h i l) di t hil th t l

MAXIM Nominal Motion and Determination of Vehicle Position Relative to Optics-Frame

niversity

1

ˆ

i

HD

r rd 

Freeflyer / Detector The nominal motion is in the local (spherical) coordinates while the control effort is formulated in the inertial focal frame.

er & Purdue Un

1 2 3

ˆ ˆ ˆ

i

HD Ur

rd r C d r d



        

1

ˆ ˆ ˆ ˆ d C C u C S v S w

    

  

y / Kinematics are written as

ce Flight Cente

ˆ v ˆ w

Free Flyer (Di) LOS

1

tan z y 

 

       

1 2 2

tan x y z 

 

           

NOMINAL MOTION:

x 

W

r ˆ v ˆ w

Free Flyer (Di) LOS

1

tan z y 

 

       

1 2 2

tan x y z 

 

           

NOMINAL MOTION:

x 

W

r

x rC C  

Cartesian coordinates to spherical:

Goddard Spac

ˆ u

Hub (H)

 

1 2 3

NOMINAL MOTION: FF : 0 , 0 , 500 m FF : 60 , 0 , 500 m FF : 120 , 0 , 500 m FF : 180 500 m

W W W

r r r r                    

               

z  y  ˆ u

Hub (H)

 

1 2 3

NOMINAL MOTION: FF : 0 , 0 , 500 m FF : 60 , 0 , 500 m FF : 120 , 0 , 500 m FF : 180 500 m

W W W

r r r r                    

               

z  y 

x rC C y rS C z rS

    

    

NASA /

4 5

FF : 180 , 0 , 500 m FF : 240 , 0 ,

W

r         

    6

500 m FF : 300 , 0 , 500 m Detector: 0 , 90 , 20,000 km

W W W

r r r            

        4 5

FF : 180 , 0 , 500 m FF : 240 , 0 ,

W

r         

    6

500 m FF : 300 , 0 , 500 m Detector: 0 , 90 , 20,000 km

W W W

r r r            

       

x rC C r S C r C S y rS C r C C r S S rS r C

           

                         

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

11

Detector Detector

z rS r C

 

  

SLIDE 12

Goddard Space Flight Center

MAXIM IFL Controller Development

Control of Equations of Motion (EOM) in E h i F W E h (P )

niversity

• EOM for Freeflyer/detector
• EOM for Hub

 

   

 

2 2 2 2 2 2

,

i i i i

D P D P D P D I I I I I I P H P H P H I I

r f r r u t f      

Ephemeris Frame Wrt Earth (P2)

er & Purdue Un

• EOM for Hub
• Controller is selected as type of response as a critical damped

 

2 2 2

,

P H P H P H I I I I I

r f r r 

ce Flight Cente

• Control in the local frame
• Controller eliminates system dynamics terms yields critical response

l

   

    



   

i i i

D D HD U U I U I U I U I I I

r C f C u t C f u t        

Goddard Spac

control

   

 

 

   

* 2 *

2

i i i i

D D HD HD U I U I n U n U

u t C f r r r r            

   

 

   

i i i i

D D HD HD I I U U I I I U I I

r f u t C r f u t         

NASA /

• Once control determined in optics frame, rotate into inertial frame for

controller

 

 

 

I I I U I I

f f

    



   

i i

D D I U I

u t C u t  

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

12

(Note: Full state feedback for IFL and no constraints)

SLIDE 13

Goddard Space Flight Center

MAXIM Freeflyer Placement

Freeflyers at a maximum 500 meters from hub evenly spaced

niversity

Freeflyers at a maximum 500 meters from hub evenly spaced in azimuth at 60 degrees

er & Purdue Un

Optics Plane View Inertial View

ce Flight Cente Goddard Spac NASA / Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

13

SLIDE 14

Goddard Space Flight Center

MAXIM Maintenance – Thrust Profiles

niversity

180 d IFL i l Detector < 7 mN

er & Purdue Un

• 180 day IFL continuous control

Freeflyer ~ tenths of N

ce Flight Cente

Freeflyer tenths of N

Goddard Spac

Th t P fil ti l

NASA /

Thrust Profiles proportional to spacecraft mass, e.g. 2:1

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

14

SLIDE 15

Goddard Space Flight Center

MAXIM Maintenance and Recovery

• Maintenance for 1 day

niversity

y

• Control off during observation of 100,000 seconds
• Increase in radial errors of detector and freeflyer
• Recovery back to original positions in ½ day

er & Purdue Un

W W

r r 

 W W

r r 



Recovery back to original positions in ½ day

 Error growth is not

ce Flight Cente Nominal Radial Vector in UVW Coordinates Actual Radial Vector in UVW Coordinates Nominal Radial Vector in UVW Coordinates Actual Radial Vector in UVW Coordinates

linear  P k f 15 k

Goddard Spac Thrusters off = 100,000 sec Thrusters off = 100,000 sec

 Peak error of 15 km for detector

NASA /

 Peak errors range from 300mm to 550mm for freeflyer

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

15

for freeflyer

SLIDE 16

Goddard Space Flight Center

MAXIM Maintenance and Recovery

Deviation in the Optics Plane During Observation With Control Off

niversity

Observation With Control Off Detector Vertical Scale: u:15 km to 0 km Freeflyer Vertical Scale: +/ 400 mm

er & Purdue Un

v: +/- 5 km w:+/- 5 km Vertical Scale: +/- 400 mm In all 3 components ( u,v,w)

ce Flight Cente Goddard Spac NASA / Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

16

SLIDE 17

Goddard Space Flight Center

MAXIM Maintenance and Recovery

Freeflyer Errors As Pointing Errors (Arc-seconds)

niversity

  



  



y g ( )

er & Purdue Un

Nominal Actual Nominal Actual

Azimuthal angle () maximum ~120

ce Flight Cente Goddard Spac

  



Nominal Actual

  



Nominal Actual

Out-of-plane ()

NASA /

p ( ) Maximum ~120

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

17

SLIDE 18

Goddard Space Flight Center

MAXIM Maintenance, Observation, and Recovery

Three day simulation with maintenance 1 day, 100 000 sec observation and ½ days recovery

niversity

100,000 sec observation, and ½ days recovery Recovery: Detector required 1N F fl i d < 15 N Maintenance: Detector required 3e-3 N F fl i d < 0 05 N

er & Purdue Un

Freeflyers required < 15N Freeflyers required < 0.05N

Detector R P fil

ce Flight Cente

3 mN  3 mN  3 mN  3 mN 

Recovery Profile

Goddard Spac

Freeflyers

NASA /

A B C

 

O 0 05 N 

 

O 0 05 N 

A B C

 

O 0 05 N 

 

O 0 05 N 

Freeflyers

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

18

 

O 0.05 N 

 

O 0.05 N 

 

O 0.05 N 

 

O 0.05 N 

SLIDE 19

Goddard Space Flight Center

MAXIM Reorientation

i i d i i di i

niversity

• 90 degrees rotation about the z-axis
• Target initially along the inertial x-axis
• x-axis reoriented into y-axis direction
• Elevation angle set to zero

er & Purdue Un

Initial Orientation of Optics Plane

ce Flight Cente

ˆ ˆ || w X

Goddard Spac

1 2 3 4

NOMINAL MOTION: FF : 0 , 0 , 500 m FF : 60 , 0 , 500 m FF : 120 , 0 , 500 m FF : 180 , 0 , 500 m

W W W W

r r r r                    

               

Final Orientation of Optics Plane

NASA /

ˆ ˆ || w Y

4 5

: 80 , 0 , 500 FF : 240 , 0 ,

W

     

    6

500 m FF : 300 , 0 , 500 m Detector: 0 , 90 , 20,000 km

W W W

r r r            

       

p

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

19

|| w Y

  



  



Initial Formation Orientation: , 0 ,0 Target Formation Orientation: , 90 ,0      

   

SLIDE 20

Goddard Space Flight Center

MAXIM Reorientation

• 7 day Simulation

niversity

y

• Detector ~ 1.5 N
• Freeflyer ~ 2.5 N

er & Purdue Un

ˆ ˆ || w X ˆ ˆ T t || Y ˆ ˆ || w X ˆ ˆ T t || Y

Thrust Levels Freeflyer Displacement in Inertial Frame

V ti l S l +/ 0 5 K

ce Flight Cente

|| w X Target: || w Y

Reconfiguration Tim e Increased to 7 days to reduce Detector S/C Control Thrust

|| w X Target: || w Y

Reconfiguration Tim e Increased to 7 days to reduce Detector S/C Control Thrust

Vertical Scale +/- 0.5 Km Detector

Goddard Spac

Freeflyer

NASA /

Freeflyer

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

20

SLIDE 21

Goddard Space Flight Center

• Two Approaches Discrete and Continuous Were Investigated for the Control of

Summary

niversity

• Two Approaches, Discrete and Continuous, Were Investigated for the Control of

the Maxim Formation.

• Simple or Optimal Discrete or by Input Feedback Linearization (IFL) Control.

 Di t C t l A h C ti Ti I t l Eff t

er & Purdue Un

 Discrete Control Approaches Continuous Time Interval Effort.  IFL Continuous Control Combines the Effect of Annihilating the Environmental Dynamics While Adding a Specific User-defined Critically Damped Response

ce Flight Cente

p p

• The Total Maintenance Control Effort Requires

Detector Thrust Level that Ranges From 4 mN to 7 mN Freeflyer Thrust Levels of 0.1 N to 0.3 N.

Goddard Spac

y  

• Formation Recovery

Detector Thrust Less than 1 N Freeflyers Less than 15 N

NASA /

Freeflyers Less than 15 N

• These Efforts Do Not Include Navigation or Maneuver Errors or Navigation

Measurement Updates.

Folta, Hartman, Howell, Marchand AIAA/AAS Astrodynamics Specialist Conference and Exhibit

21

• The Challenge Is Propulsion System Implementation and Required Power Levels

as Current Propulsion Technology Can Meet Minimum Thrust Levels