Gi ven 2 r (t ) , r T r ansf r 2-B od y (t ) , t e, t v (t ) 1 v L amber t Pr...

Astrodynamics 102

Part 1: Lambert+ Algorithm

Gim Der

DerAstrodynamics

July 5, 2013 AMOS

September 11~14, 2011

Launch / Impact Point Prediction

Satellite and Missile Targeting

Mission Planning

Orbit Maneuvers

Station-keeping

Initial Orbit Determination

UCT Cataloging

. . . . . .

: Lambert: v (t ) , v (t ) , Lambert : v (t ) , v (t )

2

1

1 21

2

, inc_to

1 2+Find

Given: r (t ) , r (t ) , t , t

Lambert Problem

For each inc_to:

Single revolution has one solution

Multi-revolution (N) may have

( 2N + 1 ) solutions

r (t )

v (t )2

r (t )1

v (t )1

1

object

at t2

2

2-Body

Transfer

Orbit

Orbit

Sphere /

Spheroid

objectat t

v (t )2

Kepler+Transfer

v (t )1

Analytic Astrodynamics Overview

Astrodynamics 101: Kepler+ Algorithm

Part1: Analytic Prediction Algorithms

Part2: Verifications

Astrodynamics 102: Lambert+ Algorithm

Part1: Analytic Multi-revolution Targeting Algorithms

(Orbit Determination for Radar Data)


Astrodynamics 103: Gauss/Laplace+ Algorithm

Part1: Analytic Angles-only Algorithms

(Orbit Determination for Optical Sensor Data)


Astrodynamics 102: Lambert+ Algorithm

Part 1. Analytic Multi-rev Targeting Algorithms

1. What, Why, How

2. Physics and Mathematics

3. Lambert Algorithm Implementation

4. Analytic Lambert Solutions

5. Applications for SSA

Part 2. Verifications

6. Matter of Reference

7. Numerical Examples

Satellites Rendezvous and Docking

2007

Space Shuttle Discovery and ISS

Chinese Spacecraft and CSS

Radar/Laser Weather/Satellite Tracking

Radar

Beam

Laser

Beam

Satellite

tracking

by Radar

Satellite

tracking

by Laser

Weather

forecast

by Radar

Radar Missile Tracking

2007 2009, . . , 2013

Space Debris Collisions

2007 2009, . . , 2013

2007 2009, . . , 2013

Space Debris and Satellite Growth

What? Satellite rendezvous and docking, weather

forecast, tracking of satellites, missiles

and debris, mission planning, . . . . . , SSA

are many challenging problems

Why? Need accurate, fast and robust utility

algorithms for multiple applications

How? Understand the Physics and Mathematics

of Astrodynamics for analytic orbit

determination using radar and laser data

1. What, Why, How

Solving These Challenging Problems

Requires Analytic Lambert+ Algorithm

Lambert and Lambert+ Algorithms

Lambert (2-Body) solution Lambert+ solution for SSA

(accurate and fast )

Sphere

r (t )

v (t )2

r (t )1

v (t )1

1

objectat t

object

at t2

2

2-Body

Transfer

Orbit

Given: r (t ) , r (t ) , t , t

Find: Lambert: v (t ) , v (t ) ,

2

1

1 21

2

Lambert Problem

For each inc_to:


Multi-revolution (N) may have

( 2N + 1 ) solutions

, inc_to

+1 2Lambert : v (t ) , v (t )

r (t )

v (t )2v (t )

1

object

at t2

2

Kepler+ Transfer

Orbit

Spheroid

2-Body

r (t )1

1

objectat t

( J , J , J , J , J , . .

Sun , Moon , Drag , . . )32 4 3122

add perturbations analytically

2. Physics

Physics and Mathematics

Five SSA Features of Lambert+

Classical Lambert+ Lambert

Inclination indication of transfer orbit,

inc_to, specified NO/maybe YES

Bounded independent variable NO/maybe YES

Robust multi-revolution capability NO/maybe YES

Accurate iterative method YES YES

Perturbations compliant and analytic NO YES

Inclination Indication of Transfer Orbit

Choice of additional input

for Lambert problem:

Better for

multi-rev

transfer

method

or transfer

motion

direction

Inclination

Indication of

Transfer orbit,

Inc_to

vs.

(Escobal) (Others)

r1

r2

h = r x r21 k = (0, 0, 1)

direct transfer orbit

(inclination <= 90 )

r1

r2

h = r x r21

k = (0, 0, 1)

retrograde transfer orbit

(inclination > 90 )

r1

r2

h = r x r21 k = (0, 0, 1)

oo

o = 2

= k h = k h

o = <

o = 2

o = <

r1

r2

h = r x r21

Inclination indication

of transfer orbit,

inc_to = 1, posigrade

= 1, retrograde

in-plane

in-plane

out-of-plane

out-of-plane

= cos< o

1r r

2

1r r

2[ ]

Bounded Independent Variable

Solution is

guaranteed

Uncertainties in: 1. Initial guess of x

2. Convergence

x

Poor choice of independent variable:

1. x is unbounded

2. Slope d/dx is unfit for most

iterative method

= is giveno

x

Good choice of independent variable:

1. x is bounded

2. Slope d/dx is fit for many

iterative methods

2 solutions

1 solution

0 solution

Slope

2 solutions?o o

Bounded x Unbounded x

(d/dx)

non-dimensional

transfer time

Robust Multi-revolution Capability

Choice of bounded x, gives a

“vertical U”, allowing A be found easily.

Then with given, the number of

solutions, 0, 1, or 2 can be determined.

Wrong choice of x leads

to difficulties of finding A,

on a “horizontal U”

Multi-revolution, N > 0, needs to determine the minimum time point, A

(If x is the semi-

major axis, then

it is unbounded)

x

2 solutions

1 solution

0 solutionx

# of solutions?

+1-1 0 0

x x

Bounded Unbounded

N > 0

N > 0

A A ?

ooo = is given

flat curve

(If x is the “path

parameter”, then

it is bounded)

Vertical U

Horizontal U

0

Accurate Laguerre Iterative Method

Solve

with

where , , are known, and

Vary as needed n = 2, 3, . . . ,

A micro-second slower, but convergence assured

Perturbations Compliant Analytic Lambert+

Astrodynamics 101

Vinti

(J2, J3, J4

included)

Astrodynamics 102

Lambert+ (J2, J3, J4

and other

perturbations))

Classical Lambert

(2-Body) +

Targeting

by Kepler+

Kepler+ (J2, J3, J4

and other

perturbations))

Classical Kepler

(2-Body)

3. Analytic Lambert Algorithms

Lambert Algorithmic

Implementations

Classical Lambert Algorithm

r (t )2

v (t )

r (t )1

v (t )1

object at t1

object at t2 2

2-Body

Transfer

Orbit

Spherical

Earth

Equations of Motion (2-Body)

1. Classical and Universal Lambert Equations

ii+1 i

22ii i i i

i

n F (x )for i 1, 2, ..

F (x )F (x ) (n 1) F (x ) n (n 1) F (x )F (x )

F (x )

x x

2. Implementation by Laguerre Iterative Equation

Lambert Algorithm

3aF (a ) = [ ( sin ) sin ) ] t 0

F ( x ) = ( x ) y ) 0

(single revolution)

(fixed n)

2

2 3

d=

d t r

rr

Given: r (t ) , r (t ) , t , t , inc_to

Find: v (t ) , v (t )

2

1

1 21

2

Lambert Problem

inc_to given: one solution

if not given: two solutions

Note:

Multi-revolution Lambert Algorithm

References: Sun, F.T., “On The Minimum Time Traj . . ”, AAS 79-164

Der, G. J., “The Superior Lambert Algorithm”, AMOS,2011

Lambert's Equation for multi-revolutions (Sun)

Iterative Equation (Laguerre)

F ( x ) = (x ) y ) N 0

ii+1 i

22ii i i i

i

n F (x )for i 1, 2, ..

F (x )F (x ) (n 1) F (x ) n (n 1) F (x )F (x )

F (x )

x x

2-Body Lambert Algorithm

v (t )

object at t2

2

Transfer

Orbit 2

v (t )1

r (t )1

r (t )2

Transfer

Orbit 1

v (t )1

v (t )2

Sphere

object at t1

2

1

1 21

2

Lambert Problem


Find: v (t ) , v (t )

Transfer orbit 1: inc_to = 1, i < 90o

Transfer orbit 2: inc_to = 1, i > 90o

Multi- revolutions (N)

may have ( 2N + 1 ) solutions

for each inc_to and a given

Targeting to Lambert+

Celestial Pole

Final Keplertrajectory

k

ji

Final position at t

Central

body

v2

Given: r , r , t = t t , Computed v , v (Lambert) 1 2 t1 t22 1

Find: v , v (Lambert )v1 v2

1

2r

v2

v1 vt1

vt2

v1

2

1

C

A

B

C

A

InitialKepler

Lambert

Step 1

Step 2Initialposition at t

A

vv1

vv2

v1

FinalKepler

r2

r2

Lamberttrajectory

+

+

+

+

v2

r

+ 2

Targeting by

Kepler at t

4. Analytic Lambert Solutions

Lambert Solutions

and

Orbit Determination

Single Revolution Lambert Solutions

Sun, F.T., AAS Paper 79-164,

“On the minimum time trajectory and

multiple solutions of Lambert problem”

0 < < , 0 < < 1





v (t )

object at t2

2

Transfer

Orbit 2

v (t )1

r (t )1

r (t )2

Transfer

Orbit 1

v (t )1

v (t )2

Sphere

object at t1N

orm

aliz

ed T

ime

Elliptic Orbits

Path parameter,1 < x < 1

x

High Path Low Path

Parabolic Orbits

Path parameter, x = 1

Hyperbolic Orbits

Path parameter, x > 1

3

3

1/

2

N = 0

ME: Minimum Energy

( G

ive

n tim

e d

iffe

ren

ce )

( Unknown to be solved for )

( Single revolution, N = 0, 0 < < 360 )o

ME

Pat

h l

ine

t

m

]

Multi- Revolution Lambert Solutions

Sun, F.T., AAS Paper 79-164,

“On the minimum time trajectory and

multiple solutions of Lambert problem”



Multi- revolutions (N)

may have ( 2N + 1 ) solutions


0 < < , 0 < < 1

Nor

mal

ized

Tim

e

Elliptic Orbits

Path parameter,1 < x < 1

x

High Path Low Path

Parabolic Orbits

Path parameter, x = 1

Hyperbolic Orbits

Path parameter, x > 1

3

3

( Multi revolution, N > 0, > 360 )

o

1/

2

N = 1

N = 0

N = 2

ME: Minimum Energy

ME

Pat

h l

ine

t

m

]

( G

ive

n tim

e d

iffe

ren

ce )

( x = Unknown to be solved for )

N = Revolution number

v (t )

object at t2

2

Transfer

Orbit 2

v (t )1

r (t )1

r (t )2

Transfer

Orbit 1

v (t )1

v (t )2

Sphere

object at t1

2-Body vs. Lambert+ Solutions


Find: v (t ) , v (t )

2

1

1 21

2

Lambert Problem

Spherical

Earth

2-Body/Keplerian

Trajectory

Good: 2-Body Lambert solution

Spheroidal

Earth

Better: Targeting by Kepler+

Lambert+ solution

Kepler+ Trajectory

(J , J , J + N Body ,

+ J , J , Drag , . .) 2 43

31 32

_

r (t )2

v (t )

r (t )1

v (t )1

1

objectat t

object

at t2

2

Orbit

Sphere /

Spheroid

2-BodyKepler+

TransferOrbit

2-Body Lambert+

Lambert+Transfer

2-Body

Lambert Algorithm Developers

2. Sun/Der Lambert algorithm applies to multi-rev and rarely has singluarity

(simple theory + straightforward implementation = speed, accuracy, robustness)+

Theories/

Formulations

Implementations/

Iterative Methods

Gauss

Battin

Shepperd

Gooding

Klumpp

Two Equations

and

Two Unknowns

One Equation

and

One Unknown

Newton,

Halley,

and Others

Laguerre

and Modified

Laguerre

Lambert

Gauss

Battin

Lancaster/Blanchard

Godal

Vinh

Sun

Everyone

(almost)

Conway

Der

1. Most (over 90%) Lambert algorithms apply to zero rev and have singularities

Lambert Algorithm Characteristics:

Orbit Determination and Prediction/Propagation

SP is accurate, but

slow for SSA

Analytic algorithms in OD and P/P

need to process 100,000+ Objects

in less than 12 hours with accuracy

of 10 km to centimeters for SSA

SGP4 needs TLE

conversion

(not efficient for SSA)

Kepler+ is accurate

and fast for SSA

1

3

2

Radars /Opticalsensors

RawObservationdata

Orbitalelement set

r and v

(Estimated initial)positin andvelocity vectors

Prediction /

Propagation

SGP4 Kepler+Numerical

Integration

Osc2Mean

Ephemerides

Close approach(miss distance)

Rise/SetSite visibilityOthers . . . . . .

Un

suit

ab

le f

or r

eal-

tim

eau

tom

atic

pro

cess

ing

1 32

(Estimated future/past)position and velocity vectors

o

Osculating OrbitalElements at t

o o[ r (t ), v (t )]

r

v

[ r , v ]

Initial OrbitDetermination

DifferentialCorrection

ProcessedObservationdata

Batch / KF

Astrodynamics102: Lambert /103: Gauss/Laplace

SPAstro 101:Kepler

Object at t

Futurelook angles( pointing prediction )

DeterminationOrbit

TLEconversiondifficulties

Singularitydifficulties

5. Applications

SSA and Other

Applications

Applications of Analytic Algorithms(1)

Missile Launch- and impact-Point Predictions

BM – Impact Point

Predictions

• Numerical solutions

possible but too late

for countermeasures

• Analytic Lambert+ (speed & accuracy)

A Few Minutes Too Late for any Intercept

Applications of Analytic Algorithms(2) UNCLASSIFIED

SF

Example: I = J = K = 100

Correlation combination

= I J K = 1,000,000

Solve by New

angles-only algorithm:

90+ % of

objects to

catalog

Takes a few seconds for

a million combinations

Correlating

3 detected

ranges

3 computed

ranges =

UCT processing

I

J K

SF

?

Multi-sensor Multi-object UCT Cataloging using Radar Data

Fence or Radar

Applications of Analytic Algorithms(3) UNCLASSIFIED

Multi-sensor Multi-object UCT Cataloging using Optical Sensor Data

Applications of Analytic Algorithms(4)

Solution: New Analytic Astrodynamics algorithms

Problem: SSA

Next

Astrodynamics 102

Part 2: Verifications (Please download iOrbit:

http://derastrodynamics.com/index.php?main_page=index&cPath=1_7

and run lam for Astrodynamics 102 Verifications)


Numerical Example 1

Rev # Path, x v_eci (t1) (km/s) v_eci (t2) (km/s)

N = 0

Output: lambert2 converged to the correct 2-Body solution

inc_to = 1, (inclination of transfer orbit < 90 deg.), t1 = 0., t2 = 1618.50 (seconds)

Input:

r_eci (t1) (km) r_eci (t2) (km)

ICBM transfer orbit , Single revolution 2-Body solution

computed inclination of transfer orbit = 67.895 deg., transfer angle = 296.368 deg.

*

*

Numerical Example 2


N = 0

Output: lambert2 converged to the correct 2-Body solution

inc_to = 1, (inclination of transfer orbit > 90 deg.), t1 = 0., t2 = 1618.50 (seconds)

Input:


ICBM transfer orbit , Single revolution 2-Body solution


*

*

Numerical Example 3


N = 0

N = 1

N = 1

Output: lambert2 converged to correct 2-Body solutions

inc_to = 1, (inclination of transfer orbit < 90 deg.), t1 = 0., t2 = 36000. (seconds)

Input:


High Earth Orbit (Molniya transfer orbit) , Multi- revolution 2-Body solutions

*

*

* inc_to as input dictates the inclination of the transfer orbit.

Also allows multi-rev solutions better grouping, as all solutions (N = 0, 1) have the same inclination

and transfer angle.

computed inclination of the three transfer orbits = 63.388 deg., transfer angle = 44.705 deg.

Numerical Example 4


N = 0

N = 1

N = 1

Output: lambert2 converged to correct 2-Body solutions

inc_to = 1, (inclination of transfer orbit > 90 deg.), t1 = 0., t2 = 36000. (seconds)

Input:


High Earth Orbit (Molniya transfer orbit) , Multi- revolution 2-Body solutions

*

* computed inclination of the three transfer orbits = 116.612 deg., transfer angle = 315.295 deg.

* inc_to as input dictates the inclination of the transfer orbit.

Also allows multi-rev solutions better grouping, as all solutions (N = 0, 1) have the same inclination

and transfer angle.

Numerical Example 5

Algs. Rev# v_eci (t1) (km/s) v_eci (t2) (km/s)

2-Body N =0

Vinti

_targeting

N = 0

Lambert+ N = 0

Output:

inc_to = 1, (inclination of transfer orbit < 90 deg.), t1 = 0., t2 = 1618.50 (seconds)

Input:


ICBM transfer orbit , Single revolution 2-Body, Vinti targeting, Kepler+ targeting solutions

*

* computed inclination of transfer orbit = 67.895 deg., transfer angle = 296.368 deg.

clock1 = at t1 (needed for Lambert+)

Numerical Example 6


2-Body N = 0

Vinti

_targeting

N = 0

Lambert+ N = 0

Output:

inc_to = 1, (inclination of transfer orbit > 90 deg.), t1 = 0., t2 = 1618.50 (seconds)



*

*

Input:


ICBM transfer orbit , Single revolution 2-Body, Vinti targeting, Kepler+ targeting solutions

Numerical Example 7


2-Body, N = 0

Vinti_targ N = 0

Lambert+ N = 0

2-Body, N = 1

Vinti_targ N = 1

Lambert+ N = 1

2-Body, N = 1

Vinti_targ N = 1

Lambert+ N = 1

Output:

inc_to = 1, (inclination of transfer orbit < 90 deg.), t1 = 0., t2 = 36000. (seconds)

Input:




Molniya transfer orbit , Multi- revolution 2-Body, Vinti targeting, Kepler+ targeting solutions

Numerical Example 8


2-Body, N = 0

Vinti_targ N = 0

Lambert+ N = 0

2-Body, N = 1

Vinti_targ N = 1

Lambert+ N = 1

2-Body, N = 1

Vinti_targ N = 1

Lambert+ N = 1

Output:

inc_to = -1, (inclination of transfer orbit > 90 deg.), t1 = 0., t2 = 36000. (seconds)

Input:



Molniya transfer orbit , Multi- revolution 2-Body, Vinti targeting, Kepler+ targeting solutions


Lambert+ Verifications

Astrodynamics 102

Part 2: Verifications (Please download iOrbit:


and run lam for Astrodynamics 102

for more Lambert+ Verifications)


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