# Presentation: 536225

## Libration Orbit Seminar

- (AETD Mini Course)#27
- Presented by
-  David Folta
- Code 595
- Goddard Space Flight Center
- Mission Engineering and Systems Analysis
- October 22nd, 2003

##  Fundamentals of Libration Orbits 

-  What is a libration orbit and how does it works
-  Derive equations of motion and show what happens
-   when you apply the real world environment to these
-   basic concepts.
-  How transfer orbits to these unique regions of space
-   Are achieved
-  Address Common Questions Such as:
-  Stationkeeping Cost
-  Types: Halo Vs. Lissajous
-  Transfers: Direct Vs. Gravity Assist
-  Launch Vehicle Considerations

**AGENDA - I**

## AGENDA - II

-  History, Definitions, and Overview
-  Missions to Co-linear Sun-Earth Points L1 & L2
-  Flight Dynamics Support and Mission Design
-  High Fidelity Perturbation Theory Modeling
-  Flight Dynamics Support – Design, Navigation, Products, Operations
-  Future Challenges and New Technology Applications to
-     Libration Orbit Mission Design
- 	Dynamical System Theory – Manifolds
- 	Control Efforts
- 	Gateways and Servicing
- 	Other Unique Orbits in the Libration Region
- 		-Vertical, Lyapunov, QSO....

**AGENDA - II**

## Coordinate Systems

## Geocentric Initial Coordinate System

**Geocentric Initial Coordinate System**

**The Geocentric Equatorial Inertial System (GEI) has its *****X*****-axis pointing from the Earth towards the first point of Aries (the position of the Sun at the vernal equinox). This direction is the intersection of the Earth's equatorial plane and the ecliptic plane and thus the X-axis lies in both planes. The *****Z*****-axis is parallel to the rotation axis of the Earth and *****Y***** completes the right-handed orthogonal set (Y = Z   X). **

**Vernal Equinox**

## Solar Rotating Coordinate System

**Solar Rotating Coordinate System**

**The solar rotating coordinate system has its +x-axis pointing from the primary body (sun) through the secondary body (Earth).  The +y axis is in the direction of the velocity of the secondary. The +z-axis is perpendicular to the x-y plane. **

**+x-axis**

**+y-axis (****direction of Earth’s velocity****)**

**+x-axis**

**+y-axis**

**+z-axis**

**Sun**

**Earth**

## A libration orbit in GCI and SR Coordinates

**Moon’s orbit**

**Moon’s orbit**

**Transfer orbit**

**Transfer orbit**

**Mission orbit**

**Mission orbit**

** ****Mission to L2**
** ****263 day duration**
** ****Transfer of 110 days**

**L2**

**L2 Position**

**Sun Direction**

***Solar Rotating***

***Geocentric Inertial***

**Earth**

## History of Libration Orbits

## History, Definitions, & Modeling

**Euler**
-  Defined three body problem in work on lunar motion.
-  Proved existence of co-linear points
**Lagrange**
-  Development of equilibrium points

**History, Definitions, & Modeling**

**Poincare**
-  Stability of motion and use of potential functions
-  First to recognize the need for a qualitative approach to three body problem which is unsolvable in closed form
**Jacobi**
-  One exact integral of three body system

_**Definitions & Modeling**_
- Easiest to model the system as the Circular Restricted Three Body Problem (CRTBP) where m1>>m2>>m3
- m1 - primary, m2 - secondary, m3 - body of interest
- motion of Earth about Sun is circular
- motion of m3 is in plane of m1 & m2
- CRTBP can be solved exactly
- Unfortunately, unmodeled forces (solar radiation pressure, other gravitational bodies - Jupiter, etc.) and physical reality (non-circular motion or EM system about sun) cause perturbations

## History, Definitions, & Modeling

**History, Definitions, & Modeling**

## Libration Points

- Equilibrium or libration points represent singularities in the equations of motion where velocity and acceleration components are zero and the forces are balanced
- Viewed in the rotating frame: centrifugal (Coriolis-Type) force balances with gravitational forces of the two primaries
- Libration points are in plane with no Z component. Orbits are mapped to a rotating frame where there are no time dependent forces
- Our system of interest involves the Sun (m1), the Earth-Moon system (m2) and the spacecraft m3
- L1 and L2 distance of 1.5 million km
- L4 and L5 distance of 150. million km

- Where Are They?
- Collinear Points: L1, L2, L3 (unstable)
- Triangular Points: L4, L5 (stable)

**Libration Points**

## Circular Restricted Three Body Problem

## Simplest Method of Constructing a Libration Orbit
Assumptions
Sun, Earth, and Infinitesimally Small S/C Are the 3 Bodies
Earth Moves About Sun in a Perfect Circle
No Other Forces (Gravitational or Otherwise)

Equations of Motion
Rotating Frame
Non-dimensional Coordinates (Sun-Earth Distance =1, Period =1)
Origin at Sun/Earth Barycenter

- Assumptions
- Sun, Earth, and Infinitesimally Small S/C Are the 3 Bodies
- Earth Moves About Sun in a Perfect Circle
- No Other Forces (Gravitational or Otherwise)
- Equations of Motion
- Rotating Frame
- Non-dimensional Coordinates (Sun-Earth Distance =1, Period =1)
- Origin at Sun/Earth Barycenter

**Circular Restricted Three Body Problem (CRTBP) **

## Circular Restricted Three Body Problem (CRTBP) 

** ****The Spacecraft Is Denoted by m**
** ****Primary Bodies of Sun (M****1****) and Earth (M****2****)**
** ****Rotating About Their Common Barycenter Location, *****o*****. **

**Has constant angular velocity**
**Position Vector of s/c is**
**      **

***o***

**-**

**Circular Restricted Three Body Problem (CRTBP) **

*k*

*Z*

*j*

*Y*

*i*

*X*

*R*

- 

- 

- 

- 

- 

- 

- 

## Circular Restricted Three Body Problem (CRTBP) 

** ****Position Vector of S/C in the rotating frame: ***R=Xi+Yj+Zk*
** ****Inertial Acceleration of S/C is Second Derivative of Position Vector:**
** ****Apply Newton’s 2****nd**** Law:**
** ****Normalize: **Sun-Earth Distance =1
-                         Time, so angular velocity
-                            of primaries =1
-                         Total mass = 1 (= M1/(M1 + M2))
**     ****and substitute**

_**Constructing the Rotating Equations of Motion**_

- 2

- 3

- 2

- 2

- 1

- 3

- 1

- 1

*r*

*r*

*m*

*M*

*G*

*r*

*r*

*m*

*M*

*G*

*R*

*m*

- 

- 

- 

*k*

*Z*

*j*

*Y*

*n*

*X*

*n*

*Y*

*i*

*X*

*n*

*Y*

*n*

*X*

*R*

- 

- 

- 

- 

- 

- 

- 

- )

- 2

- (

- )

- 2

- (

- 2

- 2

*k*

*Z*

*j*

*Y*

*i*

*D*

*X*

*R*

*i*

*D*

*r*

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- )

- (

- 

- 

- 

*k*

*Z*

*j*

*Y*

*i*

*D*

*X*

*R*

*i*

*D*

*r*

- 

- 

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- (

- 2

- 2

- 2

**..      ..         .                     ..         .                   ..**

**  ****_                               _                                _      _**

**..**

**_                    _                  _**

## By using a parameter called a pseudopotential

**Simplified EOM can be written as**

- where

**Circular Restricted Three Body Problem (CRTBP) **

- (z is a simple harmonic)

**Yields**

**Centrifugal potential**

**Gravitational potential**

- 3

- 2

- 2

- 2

- 3

- 1

- 1

- 1

- 2

- )

- (

- )

- (

- 2

*r*

*D*

*X*

*r*

*D*

*X*

*X*

*n*

*Y*

*n*

*X*

- 

- 

- 

- 

- 

- 

- 

**

**

- 3

- 2

- 2

- 3

- 1

- 1

- 2

- 2

*r*

*Y*

*r*

*Y*

*Y*

*n*

*X*

*n*

*Y*

**

**

- 

- 

- 

- 

- 

- 3

- 2

- 2

- 3

- 1

- 1

*r*

*Z*

*r*

*Z*

*Z*

**

**

- 

- 

- 

*X*

*U*

*Y*

*n*

*X*

- 

- 

- 2

*Y*

*U*

*X*

*n*

*Y*

- 

- 

- 2

*Z*

*U*

*Z*

- 

**..        .**

**..        .**

**.. **

**..        .              ..        .            ..**

## Circular Restricted Three Body Problem (CRTBP) 

- Sun-Earth Line

- Orbit

- Venus Orbit

- Sun

- Earth

**Solar Rotating Coordinates System**

**Circular Restricted Three Body Problem (CRTBP) **

**A Typical Motion of a Spacecraft in a CRTB Environment**

- Initial Location,
- ~ 1 a.u.

## Circular Restricted Three Body Problem (CRTBP) 

- Sun-Earth Line

- Orbit-2

- Venus Orbit

- Sun

- Earth

**Solar Rotating Coordinates System**

**Circular Restricted Three Body Problem (CRTBP) **

- Orbit-1

**A Typical Motion of a Spacecraft in a CRTB Environment**

- Initial Location,
- ~ 0.33 a.u.

## Circular Restricted Three Body Problem (CRTBP) 

## 2-D View (Ecliptic Projection)	3-D View of Potential (fixed C)

**2-D View (Ecliptic Projection)****	****3-D View of Potential (fixed C)**

**S**

**E**

**L1**

**L3**

**L5**

**L4**

**L2**

**L1     L2**

**              **_**Jacobi Constant**_
**Helpful in more detailed analysis **
**Plot curves of zero relative velocity**

**Circular Restricted Three Body Problem (CRTBP) **

- C = 2U – (x2 +y2 +z2)

**.     .     .**

## Circular Restricted Three Body Problem (CRTBP) 

**This can be expanded for velocity and acceleration, e.g X =X****0**** + x,  Y =Y****0**** + y,**
**But X****0****, X****0**** , etc. = 0**
** ****The Potential (U****x****,U****y****,U****z****) can also be expansion via Taylor series to get U****xx****,U****yy****,U****zz**** with the libration point terms and off-diagonal terms = 0 **
** ****Yielding;   ****x + 2ny = U****xx   ****,y + 2nx = U****yy  ****, z = U****zz**
** ****A solution of the form  A****1****e************t**** + A****2****e****-************t**** + A****3****e************t**** + A****4****e************t****  is found for x and y**
**which can be simplified to equations in *****sin***** and *****cos*****.**
** ****Permits selection of initial conditions that are bounded (stable) about the co-linear libration point**
** ****Information about the stability (modes) about the libration point**

**Circular Restricted Three Body Problem (CRTBP) **

_**Stability near the libration points**_

**  ****.    .      .   ..   ..     ..**

**.    ..**

**..       .**

**..       .**

**..       **

## Periodic Solutions Can Be Found for Motion About the Collinear Libration Points in the CRTBP
x =    KAy Sin ( xyt+xy )
y =   AY Cos ( xyt+xy )
z =    AZ Sin ( zt+z )

Period of Motion About Libration Points Is 180 Days
      Regardless of Size

There Are Two Types of Motion About Libration Points
Lissajous Orbits  where XY  Z
As a result, the orbit oscillates about the y-axis such that the projection along the sun-line seems to open and close
 Because of this motion, constraints on a solar exclusion zone or eclipse region may be required
Halo orbits
 Special solution of the libration point orbit where XY = Z, or 
       AY is large (e.g. 655,000 km), Az & AY selected, and Phase angles  are selected

**x =    KA****y ****Sin ( ********xy****t+********xy ****)**
**y =   A****Y ****Cos ( ********xy****t+********xy ****)**
**z =    A****Z ****Sin ( ********z****t+********z** )
- Period of Motion About Libration Points Is 180 Days
-       Regardless of Size
- There Are Two Types of Motion About Libration Points
_***Lissajous Orbits***_  where ******XY**** ******** ********Z**
- As a result, the orbit oscillates about the y-axis such that the projection along the sun-line seems to open and close
-  Because of this motion, constraints on a solar exclusion zone or eclipse region may be required
_***Halo orbits***_
-  Special solution of the libration point orbit where ******XY**** = ********Z**, or
-        AY is large (e.g. 655,000 km), Az & AY selected, and Phase angles  are selected

**Circular Restricted Three Body Problem (CRTBP) **

## Views of a CRTB Lissajous Orbit

**Views of a CRTB Lissajous Orbit**

**Top**
**(Along z)**

**Side**
**(Along y)**

**Through**
**(along x)**

**Libration Point**

**3-D**

## A Three-Dimensional Orbit.
Periodic 
Rotating Frame
~6 Month Period (Half the Orbital Period of the Primaries)	
Technically, Only Exists in the CRTBP
The In and Out of Plane Frequencies Are Commensurate 
Only Exists for Certain Constrained Values of the Various Amplitudes
Characterized by Z-amplitude, Az

- Periodic
- Rotating Frame
- ~6 Month Period (Half the Orbital Period of the Primaries)
- Technically, Only Exists in the CRTBP
- The In and Out of Plane Frequencies Are Commensurate
- Only Exists for Certain Constrained Values of the Various Amplitudes
- Characterized by Z-amplitude, Az

**Halo Orbit**

## Full Non-linear ODE Has Certain Symmetries
A Given Solution (Either Class) Is Symmetric About the RLP XZ Plane
Trajectory Crosses the XZ Plane Perpendicularly 
    (RLP Velocity = (0, Vy, 0)
Useful for Obtaining a Halo Orbit Numerically Using Differential Corrections
Start at a Point on the RLP XZ Plane With Velocity Perpendicular to That Plane (X, 0, Z, 0, Vy, 0)
Propagate Until One Reaches the XZ Plane Again (~3 Months, Half the Period)
If Velocity Is Perpendicular at This XZ Crossing, Finished.  
If Not, Vary Initial Starting Position on the Xz Plane and the Y-component of Velocity.  Repeat Process

A Given Solution (Either Class), For Practical Purposes, Exist for 650k < Ay < ~1000k (Sun-earth System)
For Ay Less Than This, Obtain “Lissajous” Orbit Where the in and Out of Plane Frequencies Are Not Equal.   

- A Given Solution (Either Class) Is Symmetric About the RLP XZ Plane
- Trajectory Crosses the XZ Plane Perpendicularly
-     (RLP Velocity = (0, Vy, 0)
- Useful for Obtaining a Halo Orbit Numerically Using Differential Corrections
- Start at a Point on the RLP XZ Plane With Velocity Perpendicular to That Plane (X, 0, Z, 0, Vy, 0)
- Propagate Until One Reaches the XZ Plane Again (~3 Months, Half the Period)
- If Velocity Is Perpendicular at This XZ Crossing, Finished.
- If Not, Vary Initial Starting Position on the Xz Plane and the Y-component of Velocity.  Repeat Process
- A Given Solution (Either Class), For Practical Purposes, Exist for 650k < Ay < ~1000k (Sun-earth System)
- For Ay Less Than This, Obtain “Lissajous” Orbit Where the in and Out of Plane Frequencies Are Not Equal.

**Halo Orbit**

## Pattern Repeats Indefinitely (Periodic)
XY Projection Has a Clockwise Pattern for Both Classes
XZ Projection
Class I:  Straight Line with a Upper Left/Lower Right Tilt
Class II:  Straight Line with a Lower Left/Upper Right Tilt
YZ Projection  
Class I:  Clockwise Pattern
Class II:  Counter-Clockwise Pattern

- Pattern Repeats Indefinitely (Periodic)
- XY Projection Has a Clockwise Pattern for Both Classes
- XZ Projection
- Class I:  Straight Line with a Upper Left/Lower Right Tilt
- Class II:  Straight Line with a Lower Left/Upper Right Tilt
- YZ Projection
- Class I:  Clockwise Pattern
- Class II:  Counter-Clockwise Pattern

- Note:  Graphs above are of the _linearized_ solutions for illustrative purposes only.

**Halo Orbit**

## Halo Orbit

**Halo Orbit**

## 
MAP and Triana both Utilize Lissajous Orbits
Lissajous Characteristics
Collapse on themselves (For Triana, a 7 year Cycle)
Can Specify Two amplitudes (e.g., Az and Ay)
‘Almost’ Periodic (In & Out of Plane frequencies differ slightly) 	
Design Issues
Size of Liss (e.g., Az and Ay)
Start Mission at ‘Opening’ Portion of Lissajous
For L1, Keep Sun-Earth-Vehicle (SEV) angle in Desired Range
SEV too small Solar Radiation Interference with Communication Issues
SEV too large  Possible Communication Issues 
For L2, Keep L2-Earth-Vehicle (L2EV) angle in Desired Range
L2EV too small  Earth Eclipse (Power Negative)
L2EV too large  Possible Communication and/or Instrument Issues 
‘Class’ Replaced by ‘Class at Insertion’    
Assume ‘Class’ is defined by the direction (CW or CCW) of the YZ Projection.  
This direction changes over time for a Lissajous orbit

- MAP and Triana both Utilize Lissajous Orbits
- Lissajous Characteristics
- Collapse on themselves (For Triana, a 7 year Cycle)
- Can Specify Two amplitudes (e.g., Az and Ay)
- ‘Almost’ Periodic (In & Out of Plane frequencies differ slightly)
- Design Issues
- Size of Liss (e.g., Az and Ay)
- Start Mission at ‘Opening’ Portion of Lissajous
- For L1, Keep Sun-Earth-Vehicle (SEV) angle in Desired Range
- SEV too small Solar Radiation Interference with Communication Issues
- SEV too large  Possible Communication Issues
- For L2, Keep L2-Earth-Vehicle (L2EV) angle in Desired Range
- L2EV too small  Earth Eclipse (Power Negative)
- L2EV too large  Possible Communication and/or Instrument Issues
- ‘Class’ Replaced by ‘Class at Insertion’
- Assume ‘Class’ is defined by the direction (CW or CCW) of the YZ Projection.
- This direction changes over time for a Lissajous orbit

**Lissajous Orbit**

## There are other ‘better’ representations of libration orbits
Elliptical motion of primaries
Richardson – Carey approximation
Analytical with Earth/moon barycenter 
Numerical with lunar effects
Numerical with full ephemeris and 3rd body effects

- Elliptical motion of primaries
- Richardson – Carey approximation
- Analytical with Earth/moon barycenter
- Numerical with lunar effects
- Numerical with full ephemeris and 3rd body effects

**Circular Restricted Three Body Problem (CRTBP) **

## Effects of Perturbations

## High Fidelity Perturbation Theory Modeling

- Numerical Targeting  Methods
- Optimization / Model Upgrades
- Spacecraft Modeling

_**For Simplicity in Libration Orbit Design**_
-   Use 3-body System ( Sun, Earth/moon, Spacecraft)
-        And Restricted Problem With M1>>M2>>M3
-   Use Only Planar Circular Motion in Ecliptic
-   Unmodeled Forces Result in Solutions Which Are Not Exact
_**For Operational Mission Design**_
-   In Mission Design Analysis We Model This System With
-    N-bodies Analytically Determined or Predefined From Database
-    And Numerical Integration With Summation of Accelerations

**Perturbation Effects**

## Perturbation Effects

**Runge-Kutta Nystrom (8/6) & Verner (9/8) **
**12th order vary step Cowell predictor/corrector**
**Adaptive step size control**
_**Force Models for Perturbations / Accelerations**_
**Solar radiation pressure for perturbation and solar sail effects**
**Planetary potential models up to a 100x100 degree and order for all planets and moons **
**Atmospheric model for drag acceleration using Jacchia-Roberts and Planetary unique**
**Solar/lunar/planetary objects via universe selection**
_**Other Perturbations Effect the Orbit by the Addition of Accelerations, Some May Be Un-modeled.**_
** ****Lunar Gravity As a Function of the Monthly Geometry**
** ****Momentum Unloads Via Thrusters**
** ****Outgasing From Instruments and Propulsion System**

**Perturbation Effects**

## Perturbation Effects

**Perturbation Effect Demo**

## Transfer Orbits

## Transfer Orbits

_**Lunar Swingby**_

_**Direct Injection**_

## Comparison of Direct vs. Lunar Gravity assist

** ****	****	**_**Direct**_**	****	****	****	**_**LGA**_
**1- Long launch window****	****	****1-Launch window dependent  ****	****	****	****	****	****  upon number of phasing loops**
**2- Useful for larger y-amplitude      ****	****2-Useful for smaller y-amp**
**3- Launch vehicle performance        ****	****3-More time to correct LV errors **
**    ****need to be corrected soon                **
**4- Higher C3 Required****	**** ****	****4-Change orbit shape w/o**
**5- Straight-forward operations  ****	****   fuel**
**	****	****	****	****	****5-Shadow avoidance concerns**
**	****	****	****	****	****6-Repeat though Radiation zones**

**Transfer Orbits**

## Transfer Orbits – Lunar Gravity Assist

**Lunar Swingby,**
**~1 month after launch**

**3.5 Phasing Loops,**
**~1 week period each**

**Lunar orbit**

**L****2**

**Earth**

**Transfer phase**

**Final mission lissajous orbit**

**Sun-Earth line**

_**Sample DV Budget**_
**Launch Window****	****	****	****16**
**Launch Error Correction****	****	****20**
**Phasing Loops****	****	****	****50**
**Midcourse Corrections(2)****	****	****4**
**Libration Orbit Insertion****	****	****2**
**Stationkeeping****	****	****	****40**
**Momentum Dumps and****	****	****26**
**Other ( calibrations, EOL diposal****	****10**
**            ****finite Burn, cosine loss) ****	**** ------**
**	**** ****	****Total ****	****	****168**

**Launch Opportunities: ~ 14 days/month**
**Daily Launch Window: 20 minutes**
**Launch Vehicle C3: -2.6 km****2****/sec****2**
**Time from Launch to First Nominal Maneuver: **
**                                                                 ****~ 8 days**
**Time from Launch to L2: - 130 days**

**Transfer Orbits – Lunar Gravity Assist**

**An example only!**

## Transfer Orbits – Direct

**Launch Window****	****	****	****16**
**Launch Error Correction****	****	****20**
**Phasing Loops****	****	****	****0**
**Midcourse Corrections(2)****	****	****4**
**Libration Orbit Insertion****	****	****290**
**Stationkeeping****	****	****	****40**
**Momentum Dumps and****	****	****26**
**Other ( calibrations, EOL disposal,****	****10**
**            ****finite Burn, cosine loss) ****	**** ------**
**	**** ****	****Total ****	****	****406**

**Launch Opportunities: ~ ****26**** days/month**
**Daily Launch Window: <120 minutes**
**Launch Vehicle C3: -****0.71**** km****2****/sec****2**
**Time from Launch to First Nominal Maneuver: **
**                                                                 ****~ 110 days**
**Time from Launch to L2: - 110 days**

**Transfer Orbits – Direct**

**An example only!**

## Map Lissajous Orientation

**  ****The Viewpoint Is Looking From the Earth Towards L2, **
**  ****The L2 Orbit Appears Narrow and Crosses the Sun Line  in the First Example, **
**  ****Change the Gravity Assist Time by Only a Few Hours,  the Orbit Appears Much Wider and Does Not Cross the Sun Line.  **

-   **Timing of the Gravity Assist Affects the Size and Orientation of the Initial L2 Orbit.**
**  ****Orbit Can Be Rotated With Respect to an Earth Viewpoint. **
**  ****Controls How Close the Spacecraft Approaches the Earth's Shadow Without **
**    ****Expending a Lot of Fuel.**

- 100 Limit

## Code 595

- Flight Dynamics Analysis Branch Support

## Flight Dynamics

** ****Support by Mission Phase**

**Applies to all missions**

## L1 Missions

**Earth**

**Lunar Orbit**

**L1 Libration Point**

**L2 Libration Point**

**1.5e6 km**

- _L1 Missions_

- _L2 Missions_

**Libration Missions Support by GSFC**

**	****ISEE-3/ICE(78-03)****	****Halo Orbit, Direct Transfer, L2 Pseudo Orbit, **
**	****	****	****	****Comet Mission****	**
**	****WIND (94-03)****	**** ****	****Multiple Lunar Gravity Assist and Pusedo-L1 Orbit**
**	****SOHO(95-03)****	**** ****	****Large Halo, Direct Transfer **
**	****ACE (97-03)****	**** ****	****Small Amplitude Lissajous, Direct Transfer**
**	****Triana (20??)****	**** ****	****Lissajous Constrained, Direct Transfer**
**	****Genesis (01-03)****	**** ****	****Lissajous, Direct Transfer**

**	****GEOTAIL(1992)****	****	****L2 Pseudo Orbit, Gravity Assist**
**	****MAP(01-03)****	****	****Lissajous Constrained, Gravity Assist**
**	****JWST(2011)****	****	****Large Lissajous, Direct Transfer**
**	****Const-X                                  Lissajous**
**               ****Stellar Imager                       Lissajous**
**               ****SPECS****	****                                Lissajous**

## Code 595 Orbit Personnel 

**with Libration Orbit Flight Experience**

**Dave Folta, ****6-6082****:****	****	****ISEE-3, Geotail, SOHO, ACE, WIND, MAP**
**	****	****	****	****(JWST, ConX, Stellar Imager, MAXIM)****	**
**Mark Beckman, ****6-8866****: ****	****MAP, Triana (JWST, ConX)**
**Mike Mesarch, ****6-9917****: ****	****MAP**
**Steve Cooley, ****6-6653****:****	****	****Triana (FKSI, JPF, RASC,IMDC)**
**Greg Marr, ****6-7749****:  ****	****	****WIND, Triana, MAP, SOHO (FKSI)**
**Karen Richon, ****6-5531****: ****	****MAP**
**Laurie Newman, ****6-3155:****	****MAP, (ConX)**
**Osvaldo Cuevas, ****6-5134:****	****MAP**
**Mark Woodard, ****6-9611:****	****MAP**
**Frank Vaughn, ****6-5551:****	****	****(IMDC)**
**() indicates an upcoming mission, proposal, or research**

**We have over 60 years of C.S. **_**Flight**_** experience**

## Engineering Disciplines Diversity

**Relative Navigation**: 	GPS,Transceivers, RF XLink/Ranging, Optical
**Communications**: 	Interspacecraft, Spacecraft-Ground, Internet
**Maneuvers/Control**:	Linear/NonLinear, Centralized/Decentralized, 				Perturbation models, Unified
**Attitude**:	   	Sensors, Cameras, Point Algorithms, GPS
**Propulsion**:	   	Hydrazine, Bi-Prop, SEP, Solar Sail, PPT,  				MircoThrusters
**Mission Design**: 	Trajectory Design, Dynamical Systems, 					Genetic Algorithms, Optimization
**Autonomy**:		Closed Loop Control, Remote Agents,
- 			Distributed Planning and Scheduling, and
- 			Guidance, Navigation, and Control

**Engineering Disciplines Diversity**

## ISEE-3 / ICE

**ISEE-3 / ICE**

**Earth**

**Lunar Orbit**

**Halo Orbit**

**L1**

**Mission: ****	****	****Investigate Solar-Terrestrial relationships, Solar Wind, Magnetosphere, ****	****	****	****and Cosmic Rays**
**Launch: ****	****	****Sept., 1978, Comet Encounter Sept., 1985**
**Lissajous Orbit: ****	****L1 Libration Halo Orbit, Ax=~175,000km, Ay = 660,000km, Az~ ****	****	****             ****	****120,000km, Class I**
**Spacecraft: ****	****Mass=480Kg, Spin stabilized, **
**Notable:  ****	****	****First Ever Libration Orbiter,  First Ever Comet Encounter**

**Solar-Rotating Coordinates**
**Ecliptic Plane Projection**

## ISEE-3 / ICE

**ISEE-3 / ICE**

## WIND

**Mission: ****	****	****Investigate Solar-Terrestrial relationships, Solar Wind, Magnetosphere**
**Launch: ****	****	****Nov., 1994, Multiple Lunar Gravity Assist**
**Lissajous Orbit: ****	****Originally an L1 Lissajous Constrained Orbit, Ax~10,000km, Ay ~ ****	****	****	****350,000km, Az~ 250,000km, Class I**
**Spacecraft: ****	****Mass=1254Kg, Spin stabilized, **
**Notable:  ****	****	****First Ever Multiple Gravity Assist towards L1**

**WIND**

**Solar-Rotating Coordinates**
**Ecliptic Plane Projection**

## SOHO

**SOHO**

**Mission: ****	****	****Investigate Solar-Terrestrial relationships, Solar Wind, Magnetosphere, ****	****	****	****and Cosmic Rays**
**Launch: ****	****	****December 1995, Direct Transfer**
**Lissajous Orbit: ****	****L1 large lissajous Orbit, Ay~ 666,000km, Ax~206,000km, Ay~ 120,000km, ****	****	****Class I**
**Spacecraft: ****	****Mass=1863Kg, Three-Axis Stabilized, Sun Pointing**
**Notable:  ****	****	****First Solar Observatory, Three-axis Stabilized. **

**Earth**

**Lunar Orbit**

**Sun**

**Solar-Rotating Coordinates**
**Ecliptic Plane Projection**

**L1**

## ACE

**ACE**

**Mission: ****	****	****Investigate low-energy solar and high-energy galactic particles and Solar ****	****	****	****Wind**
**Launch: ****	****	****August, 1997, Direct Transfer**
**Lissajous Orbit: ****	****L1 Lissajous Constrained Orbit, Ay~ <264,000, Ax~tbd km, Az~157,000km, ****	****	****Class I**
**Spacecraft: ****	****Mass=757Kg, Three Axis Stabilized, ‘Solar’ Pointing**
**Notable:  ****	****	****First Constrained Transfer Orbit**

## MAP

**Mission: ****	****	****Produce an Accurate Full-sky Map  of the Cosmic Microwave Background **
** ****	****	****Temperature Fluctuations (Anisotropy)**
**Launch: ****	****	****Fall 2000, Gravity Assist Transfer**
**Lissajous Orbit: ****	****L2 Lissajous Constrained Orbit Ay~ 264,000km, Ax~tbd, Ay~ 264,000km, **
**	****	****Class II**
**Spacecraft: ****	****Mass=818Kg, Three Axis Stabilized,**
**Notable:  ****	****	****First Gravity Assisted Constrained L2 lissajous Orbit; Map-earth Vector ****	****	****	****Remains Between   0.5******** and 10° off the Sun-earth Vector to Satisfy ****	****	****	****Communications Requirements While Avoiding Eclipses**

**Solar-Rotating Coordinates**
**Ecliptic Plane Projection**

## Triana

**Mission:  ****	****	****Full Time Observations of the Whole Earth, Reflected and Absorbed Solar ****	****	****Energy**
**Launch:  ****	****	****200?, Direct Transfer**
**Lissajous Orbit: ****	****L1 Lissajous Constrained Ay~ 264,000km, Ax~81,000km, Az~ ****	****	****	****	****148,000km, Class I**
**Spacecraft: ****	****Mass~500Kg, Three Axis Stabilized, ‘Small- Explorer - lite’**
**Notable:  ****	****	****First Constrained L1 Libration Orbit design using Manifolds**

**Solar-Rotating Coordinates**
**Ecliptic Plane Projection**

## JWST

**Mission:  ****	****	****JWST Is Part of Origins Program.  Designed to Be the Successor to the ****	****	****	****Hubble Space Telescope.  NGST Observations in the Infrared Part of the ****	****	****	****Spectrum. **
**Launch: ****	****	****~2011, Direct Transfer**
**Lissajous Orbit: ****	****L2 large lissajous, Ay~ 294,000km, Ax~800,000km, Az~ 131,000km, Class I ****	****	****or II**
**Spacecraft: ****	****Mass~6000kg, Three Axis Stabilized, ‘Star’ Pointing**
**Notable: ****	****	****Observations in the Infrared Part of the Spectrum.  Important That the ****	****	****	****Telescope Be Kept at Low Temperatures, ~3****0****K.   Large Solar Shade/Solar Sail**

**Earth**

**Lunar Orbit**

**Sun**

**L2**

**Solar-Rotating Coordinates**
**Ecliptic Plane Projection**

## Launch Vehicle Issues

## Launch Vehicle and C3 Parameter

** ****What Is C3?  – It’s a Computation of Energy That a Launch**
**   ****Vehicle Manufacturer Uses to Estimate the LV Payload**
**   ****Capability to the Mission Orbit**
** ****Simply It Can Be Computed as the Quantity of**
**   ****C3 ~ (******** ****Earth**** / Semi-major axis ),   or as**
**   ****C3 ~ ( Vel****po****2**** + Vel****inj****2****) - Escape Velocity****2**
** ****Comparison**
**   ****Need a C3 of 0.0 km****2****/sec****2**** to Escape the Earth/Moon System**
**   ****C3 to Reach Moon ~ -2.0 km****2****/sec****2**** **
**   ****C3 for Direct Transfer ~ -0.6 km****2****/sec****2**** **
** ****The Injection Velocity From an Upper Stage Is Roughly 3.2 km/s**

## Taken from the ksc web site:

**Taken from the ksc web site:**
**http://elvperf.ksc.nasa.gov**

**D4**

**A5**

**A3**

**D2**

**T**

**Launch Vehicle and C3 Parameter**

## Stationkeeping

** ****Because of the instabilities of the Lagrangian points, stationkeeping is required to maintain an orbit about L2 for an extended period of time**
** ****At it’s simplest, a stationkeeping maneuver is **
**performed at an XZ-plane crossing to provide for a complete revolution about L2 (motion at right is from A********B********C********D********E)**
** ****For example:**
** ****a maneuver is performed at point A to ensure a one period revolution to point C**
** ****then a maneuver is performed at point B to ensure a one period revolution to point D**
** ****Typical stationkeeping costs are anywhere from **_***1 - 4 m/s per year***_** (depending on the requirements)**

## Stationkeeping

**      ****of Perturbations and Constraints**
**  ****Accelerations From Mis/un-modeled Perturbations**
**      ****Spacecraft Material  I.E. Solar Shades/arrays**
**      ****Un-calibrated Propulsion System**
**      ****Momentum Dumps**
**      ****Fidelity of Environmental Models**
**  ****Amount and Quality of Tracking Data Accuracy of Velocity**
**      ****Needed for Maneuver Planning**
**  ****Constraints**
**      ****Shadow Avoidance**
**      ****Communication / Antenna Limits**
**  ****Rule of Thumb Is Allocating 4 M/s Per Year of Operations**

**Stationkeeping**

## New Orbits and Future Challanges

## New Orbits and Future Challenges

**        ****Biased Orbits when using large sun shades**
**        ****Minimal Fuel**
**        ****Frequent momentum unloads**
**        ****Constrained communications**
**        ****Shadow restrictions**
**        ****Very small amplitudes (<10000km)**
**        ****Limited thruster directions**
**        ****Use of external libration orbits to L****4****, L****5**
**        ****L****3**** Co-linear orbits**
**        ****Reorientation to different planes and classes**
**        ****Earth-Moon libration orbits**
**        ****Low thrust transfers**
**        ****Solar sail applications**
**        ****Continuous control to reference trajectories **
**        ****Quasi-stationary orbits **
**        ****Human exploration**
**        ****Servicing of resources in libration orbits**

**New Orbits and Future Challenges**

## Dynamical System Application:

** ****Together With Differential Corrections, the Use of Invariant Manifolds**
**      ****Provides an Efficient Method to Obtain Transfers Between Primaries**
**      ****to Halo Orbits and/or Between Halo to Halo**
** ****Invariant Manifolds Can Be Used As a First Guess for Numerical Analysis**

**Dynamical System Application:**
** ****Invariant Manifolds**

## Numerical Systems	    Dynamical Systems

-  Limited Set of Initial
-     Conditions
-  Perturbation Theory
-  Single Trajectory
-  Intuitive DC Process

-  Qualitative Assessments
-  Global Solutions
-  Time Saver / Trust Results
-  Robust
-  Helps in choosing numerical
-     methods
-    (e.g., Hamiltonian =>
-     Symplectic Integration Schemes?)

**Dynamical System Application:**
** ****Invariant Manifolds**

## MAP

**MAP**

-  MAP Manifold and Earth Access
-  Manifold Generated Starting
-   with Lissajous Orbit

-  Numerical Propagation
-  Trajectory Generated Starting
-   with Manifold States

- Starting Point

## JWST Manifold

**JWST Manifold**

-  NGST Manifold and Earth Access
-  Manifold Generated Starting
-   with Halo Orbit

-  Numerical Propagation
-  Trajectory Generated Starting
-    with Manifold States

- Starting Point

## Periodic Orbits Near L1

- Continuation of Periodic Orbits Around Lagrange
- Points and AUTO2000
- The Three-Body Problem and Space Mission Design
- Randy Paenroth
- Applied and Computational Mathematics
- California Institute of Technology

**Periodic Orbits Near L1**

## Vertical Orbits

- Continuation of Periodic Orbits Around Lagrange
- Points and AUTO2000
- The Three-Body Problem and Space Mission Design
- Randy Paenroth
- Applied and Computational Mathematics
- California Institute of Technology

## Vertical Orbits - Small

**Vertical Orbits - Small**

## Vertical Orbits - Large

**Vertical Orbits - Large**

## Use of Libration Orbits to Reach Other Locations

- Notes:
- Launch C3 = -0.59 on
- Total mission duration = 375 days
- Total mission V = +/- 80m/s assuming mothership is placed on middle trajectory,   if mothership  remains in libration orbit, then vs range from  10m/s to 160 m/s
- (Ignore lines from end of trajectory to maneuver location)

- L2 Point

- Sun-Earth Line

- Lunar Orb it

- Blue represents the trajectory in the
- Geotail region ( 0 to 20 deg)

- V  increments of 10m/s

**Use of Libration Orbits to Reach Other Locations**

- Trajectories derived from a standard
- libration orbit

## Use of Libration Orbits to Reach Other Locations

- Notes:
- Launch C3 = -0.71 on
- Total Mission Duration = 372 Days
- Total DV = 2.75 km/s

- Earth-Sun L2 Point

- Sun-Earth Line

- V1 = 1.3 km/s
- At day number 142

- V2 = 1.45 km/s
- At day number 250

**Use of Libration Orbits to Reach Other Locations**

## Rotating Coordinate Frames

**Rotating Coordinate Frames**

**Sun-Earth   and   Earth Moon**

-  X is towards smaller body (sun to Earth)
-  Y is along smaller body velocity
-  Z is out ot Ecliptic plane
-  Co-linear Unstable locations at L1,
-     L2, and L3
-  Stable locations at L4 and L5

##  Cis-lunar Transfer to EML1 

-  Cis-lunar Transfer to EML1
-  Transfer to EML2 and return to EML2
-  Return to EML2

-  Shown in Earth-Moon Rotating Coordinate System
-  LEO ~ 186-km
-     Injection C3 ~ -2.13km2/s2
-  EML1
-   Insertion V~ 432 m/s
-   Transfer time ~ 5.3 days
-  EML1 to EML2
-   Transfer V ~ 0.2 m/s
-   Transfer time ~ 9 days
-  Return EML2
-    Insertion V~50 m/s

**Transfer in Earth-Moon Region from LEO**

**X**

**Y**

**z**

**Z**

## Earth - Moon Libration Orbits

**Earth - Moon Libration Orbits**

## References

- Many Code 595 documents and papers* *
*Howell*, Purdue University, Any of her paper, articles.
*Simo, Gomez*, *Masdemont*, “Dynamics and Mission Design Near Libration Points”, VolIII, Advanced methods for Colinear Points, 2001
*Farquhar*, R. W., “The Control and Use of Libration-Point Satellites”, NASA Technical Report, NASA TR R-346, September 1970
*Roy*, A. E., _Orbital Motion_, Adam Hilger Ltd., Bristol, 1982
*Szebehely*, V., _Theory of Orbits:  The Restricted Problem of Three Bodies_, Academic Press, New York, 1967
*Gurzadyan*, G.A., _Theory of Interplanetary Flights, Gordon and Breach_, 1996

## BACKUP

## Linear Analysis

**Sample Oscillating and Stable Modes**

-  Continuous-time Eigenvalues and Eigenvectors Associated With Libration Orbit
-  Matlab Integration of Eq. Of Motion for a Sample CTBR Case, No External
-       Forces
-  Shows Unstable and Stable Modes which are useful in StationKeeping Estimation

**Halo Orbit**

***Linear Analysis***

## Stationkeeping Analysis

- Cross-section  of the Non-linear Potential

***Stationkeeping Analysis***

-  _Station Keeping Possibilities_:Minimize
-     Unstable Mode Growth by Vs
-     or Through ‘Solar Sail’ Concept
-  Use Vs in X-Axis Direction (MAP)
-  Use Vs in X or Y-Axis Direction (NGST)
-   MAP Example: (20m2 area)
-   - Average Acceleration ~ 0.2 ****m/s2
-   - Attitude Varies Acceleration by 0.9 nm/s2
-   - Applied in Y-axis Direction Yields
-       0.3**0.9 nm/s2 ~1 km/2
-   - Attitude with 0.250 control ~ 0.5 nm/s2
-                ~ 0.6 km/
-  NGST Application with >100m2 area

## Map Transfer Trajectory

***Map Transfer Trajectory***

- _Solar - Rotating Ecliptic Projection_

- Viewed from NEP

- Viewed from -y Axis

-   Utilizes Three Phasing Loops for Launch Window
-   Gravity Assist to minimize Lissajous Insertion Requirements
-   Lissajous Orbit Orientation

## Targeting and Optimization

_**Targeting Method**_**:   ****	****Differential Corrector**
-   Based on Newton-Raphson Method
-   Truncated Taylor series solved numerically
-   Sensitivity matrix for >1 variable (x & y ) and goal (A & B)
-    Gauss-Jordan Elimination used for inversion
-   Perturbations , x and y are user defined
-   Finite Difference used to compute derivatives

## B-Plane and Floating Endpoint Targeting

-  **Analytical method to determine B-Plane targets to achieve desire gravitational assist since B-Plane will change slightly with each spacecraft trajectory  update**
** ****Floating Endpoint converts B-Plane targets to Keplerian targets, e.g. B.T and B.R become persilene distance and  lunar inclination**

***B-Plane and Floating Endpoint Targeting***

## Triana L1 Mission design

** ****Manifold Design for the Triana Mission**
** ****L1 Mission, lissajous Orbit ( ~MAP Size)**

## Slide 81

## Direct Transfer Example

**Sun Direction**

**Lunar Orbit**

**Solar Rotating Coordinates / Ecliptic Plane Projection**

**Injection C3 = -0.642 km****2****/sec****2**
**Injection ********V = 3.093 km/sec**

** ****Target the anti-Sun right ascension and declinations at the appropriate launch epoch**
** ****Target the solar rotating coordinate system velocity of the Sun- Earth libration x-z **
**plane crossing conditions to achieve a quasi-libration orbit, L2 x-axis velocity ~ 0               **
** ****Vary the launch injection C3 and parking orbital parameters: **** ********, ********, i, and parking orbit coast  **
** ****Re-target conditions via addition of deterministic ********Vs to achieve the correct orientation and Lissajous pattern size with respect to the ecliptic plane.  **
** ****Defines transfer injection and L2 insertion maneuvers.**

- +x

- +y

***Direct Transfer Example***

## Lunar Gravity Assist Example

**encounter epoch**
**Target the lunar B-Plane condition to **
**achieve a gravity assist and a Sun-Earth **
** ****libration x-z plane crossing**
**Target a x-z plane crossing velocity which **
**yields a second x-z plane crossing,  **
**target to a one period revolution at L2**
**Re-target lunar B-plane conditions to **
**achieve the correct orientation of the **
**Lissajous pattern with respect to the **
**ecliptic plane.  **
**Vary the launch injection Launch C3 and parking orbital parameters: **** **
**         ********, ********, i, and parking orbit coast **

***Lunar Gravity Assist Example***

**Lunar Swingby**

**SUN**

**Lunar Orbit**

**L2**

**Solar Rotating - Ecliptic Plane Projection**

**Constrained Lissajous Orbit with Three Phasing Loops **

## F(u) = [ Wi [ (Zi(u) - Xi )/Si])Pi ] P     Where W= weight,  S=scale factor, and  P = power

_***Accelerated Projected Gradient Method***_
-  Compute gradient and Hessian using central differencing or BFGS algorithm for
-      search direction
-  Line search for minimum assuming quadratic and cubic fits, the step size comes
-      from the gradient and the Hessian
-  Objective function calculated at minimum values
-  Yields dimensional minimizer, used as initial point for next iteration
-  Procedure repeated until gradient falls below given value

***Targeting and Optimization***

***Objective Function:***
- Generated by taking the sum of the squares of the differences between the achieved and targeted final states.  The formulation produces an objective function with a minimum at the zero of the function
- F(u) = [ Wi [ (Zi(u) - Xi )/Si])Pi ] P     Where W= weight,  S=scale factor, and  P = power

- i=1

- n

## L2 Manifold Generation Process

- Transfer

- Manifold

- Monodromy

- Liss

- Orbit

- in

- Spline

- Fixed

- pts

- .

- And Stable/Unstable

- Manifold

- Approx

- .

- 1-D Manifold

- Transfer

- Trajectory

**Universe/User Control and **
**Patch Pts**** **

**Output / Intermediate **
**Data**

- Lissajous

***L2 Manifold Generation Process***

-  **Generate Lissajous of Interest**
** ****Compute Monodromy Matrix**
**     ****And Eigenvalues/Eigenvectors**
**     ****For Half Manifold of Interest**
** ****Globalize the Stable Manifold**
** ****Use Manifold Information for a**
**    ****Differential Corrector Step**
**    ****To Achieve Mission Constraints.**