ASEN 5050SPACEFLIGHT DYNAMICS

Lambert’s Problem

Prof. Jeffrey S. Parker

University of Colorado – Boulder

Lecture 16: Lambert's Problem 1

Announcements• Quiz after this lecture.

• THIS WEDNESDAY will be STK LAB 2!– Wed morning lecture canceled– Alan will be in Visions Wed 9-10– Alan will be in ITLL 2B10 Fri 2-3– STK Lab 2 will be due 10/17, right when the mid-term exam starts.

• Homework #5 is due next Friday 10/10– CAETE by Friday 10/17

• Homework #6 will be due Friday 10/17– CAETE by Friday 10/24– Solutions will be available in class on 10/17 and online by 10/24. If you turn in HW6 early,

email me and I’ll send you the solutions to check your work.

• Mid-term Exam will be handed out Friday, 10/17 and will be due Wed 10/22. (CAETE 10/29)

– Take-home. Open book, open notes.– Once you start the exam you have to be finished within 24 hours.– It should take 2-3 hours.

Lecture 16: Lambert's Problem 2

Space News

Lecture 16: Lambert's Problem 3

• Remember this? Right on!

ASEN 5050SPACEFLIGHT DYNAMICS

Lambert’s Problem

Prof. Jeffrey S. Parker

University of Colorado – Boulder

Lecture 16: Lambert's Problem 4

Lambert’s Problem

• Lambert’s Problem has been formulated for several applications:

– Orbit determination. Given two observations of a satellite/asteroid/comet at two different times, what is the orbit of the object?

• Passive object and all observations are in the same orbit.

– Satellite transfer. How do you construct a transfer orbit that connects one position vector to another position vector at different times?

• Transfers between any two orbits about the Earth, Sun, or other body.

Lecture 16: Lambert's Problem 5

Orbit Transfer• We’ll consider orbit transfers in general, though the OD

problem is always another application.

Lecture 16: Lambert's Problem 6

Orbit Transfer

Orbit Transfer True Anomaly Change

“Short Way” Δν < 180°

“Long Way” Δν > 180°

Hohmann Transfer (assuming coplanar) Δν = 180°

Type I 0° < Δν < 180°

Type II 180° < Δν < 360°

Type III 360° < Δν < 540°

Type IV 540° < Δν < 720°

… …

Lecture 16: Lambert's Problem 7

Lambert’s Problem

• Given:

• Find:

• Numerous solutions available.– Some are robust, some are fast, a few are both – Some handle parabolic and hyperbolic solutions as well as

elliptical solutions– All solutions require some sort of iteration or expansion to build

a transfer, typically finding the semi-major axis that achieves an orbit with the desired Δt.

Lecture 16: Lambert's Problem 8

Ellipse

Lecture 16: Lambert's Problem 9

r1

r2

A transfer from r1 to r2 will be on an ellipse, with the central body occupying one focus.

Where’s the 2nd focus?

Focus is one of these

Try different a values until you hit your TOF

Lecture 16: Lambert's Problem

Lambert’s Problem

10

Lecture 16: Lambert's Problem

Lambert’s Problem

11

Universal Variables

• A very clear, robust, and straightforward solution.– There are a few faster solutions, but this one is pretty clean.

• Begin with the general form of Kepler’s equation:

Lecture 16: Lambert's Problem

Transfe

r Dur

ation

# rev

olutio

ns

12

Universal Variables

• Simplify

Lecture 16: Lambert's Problem 13

Universal Variables

• Define Universal Variables:

Lecture 16: Lambert's Problem 14

Universal Variables

Lecture 16: Lambert's Problem 15

Universal Variables

• Use the trigonometric identity

Lecture 16: Lambert's Problem 16

Universal Variables

• Now we need somewhere to go

• Let’s work on converting this to true anomaly, via:

Lecture 16: Lambert's Problem 17

Universal Variables

• Multiply

by a convenient factoring expression:

Lecture 16: Lambert's Problem 18

Universal Variables

• Collect into pieces that can be replaced by true anomaly

Lecture 16: Lambert's Problem 19

Universal Variables

• Substitute in true anomaly:

Lecture 16: Lambert's Problem 20

Universal Variables

• Trig identity again:

Lecture 16: Lambert's Problem 21

Universal Variables

• Note:

Lecture 16: Lambert's Problem 22

Universal Variables

• Use some substitutions:

Lecture 16: Lambert's Problem 23

Universal Variables

• Summary:

Lecture 16: Lambert's Problem 24

Universal Variables

• Once you have expressions for y, A, etc., use the f and g series (remember those!?) to convert to r and v!

Lecture 16: Lambert's Problem 25

UV Algorithm

Lecture 16: Lambert's Problem 26

UV Algorithm

Lecture 16: Lambert's Problem 27

Bi-section Method

UV Algorithm

Lecture 16: Lambert's Problem 28

A few details on the Universal Variables algorithm

Lecture 16: Lambert's Problem 29

• Let’s first consider our Universal Variables Lambert Solver.

• Given: R0, Rf, ΔT

• Find the value of ψ that yields a minimum-energy transfer with the proper transfer duration.– ψ is a function of e – it captures the orbital shape.– χ requires ψ and a. So we’re ultimately testing the orbit.

• Applied to building a Type I transfer

Universal Variables

Lecture 16: Lambert's Problem 30

Single-Rev Earth-Venus Type I

-4π 4π2

Lecture 16: Lambert's Problem 31

Single-Rev Earth-Venus Type I

-4π 4π2

Lecture 16: Lambert's Problem 32

Note: Bisection method

-4π 4π2

Lecture 16: Lambert's Problem 33

Note: Bisection method

-4π 4π2

Lecture 16: Lambert's Problem 34

Note: Bisection method

-4π 4π2

Lecture 16: Lambert's Problem 35

Note: Bisection method

-4π 4π2

Lecture 16: Lambert's Problem 36

Note: Bisection method

-4π 4π2

Lecture 16: Lambert's Problem 37

Note: Bisection method

-4π 4π2

Lecture 16: Lambert's Problem 38

Note: Bisection method

-4π 4π2

Lecture 16: Lambert's Problem 39

Note: Bisection method

• Time history of bisection method:

• Requires 42 steps to hit a tolerance of 10-5 seconds!

Lecture 16: Lambert's Problem 40

Note: Newton Raphson method

Lecture 16: Lambert's Problem 41

Note: Newton Raphson method

Lecture 16: Lambert's Problem 42

Note: Newton Raphson method

Lecture 16: Lambert's Problem 43

Note: Newton Raphson method

Lecture 16: Lambert's Problem 44

Note: Newton Raphson method• Time history of Newton

Raphson method:

• Requires 6 steps to hit a tolerance of 10-5 seconds!

• Note: This CAN break in certain circumstances.

• With current computers, this isn’t a HUGE speed-up, so robustness may be preferable.

Lecture 16: Lambert's Problem 45

Note: Newton Raphson Log Method

-4π 4π2

Note log scale

Lecture 16: Lambert's Problem 46

Single-Rev Earth-Venus Type I

-4π 4π2

Lecture 16: Lambert's Problem 47

Single-Rev Earth-Venus Type II

-4π 4π2

Lecture 16: Lambert's Problem 48

Interesting: 10-day transfer

-4π 4π2

Lecture 16: Lambert's Problem 49

Interesting: 950-day transfer

-4π 4π2

Lecture 16: Lambert's Problem 50

• Seems like it would be better to perform a multi-rev solution over 950 days than a Type II transfer!

Multi-Rev

Lecture 16: Lambert's Problem 51

• The universal variables construct ψ represents the following transfer types:

A few details

Lecture 16: Lambert's Problem 52

Multi-Rev

ψ

ψ ψ

ψ

Lecture 16: Lambert's Problem 53

Earth-Venus in 850 days

Type IV

Lecture 16: Lambert's Problem 54

Earth-Venus in 850 days

Heliocentric View Distance to Sun

Lecture 16: Lambert's Problem 55

Earth-Venus in 850 days

Type VI

Lecture 16: Lambert's Problem 56

Earth-Venus in 850 days

Heliocentric View Distance to Sun

Lecture 16: Lambert's Problem 57

What about Type III and V?

Type IV

Type IIIType V

Type VI

Lecture 16: Lambert's Problem 58

Earth-Venus in 850 days

Type III

Lecture 16: Lambert's Problem 59

Earth-Venus in 850 days

Heliocentric View Distance to Sun

Lecture 16: Lambert's Problem 60

Earth-Venus in 850 days

Type V

Lecture 16: Lambert's Problem 61

Earth-Venus in 850 days

Heliocentric View Distance to Sun

Lecture 16: Lambert's Problem 62

• The bisection method requires modifications for

multi-rev.

• Also requires modifications for odd- and even-type

transfers.

• Newton Raphson is very fast, but not as robust.

• If you’re interested in surveying numerous revolution

combinations then it may be just as well to use the

bisection method to improve robustness

Summary

Lecture 16: Lambert's Problem 63

Types II - VI

Lecture 16: Lambert's Problem 64

Announcements• Quiz after this lecture.

• THIS WEDNESDAY will be STK LAB 2!– Wed morning lecture canceled– Alan will be in Visions Wed 9-10– Alan will be in ITLL 2B10 Fri 2-3– STK Lab 2 will be due 10/17, right when the mid-term exam starts.

• Homework #5 is due next Friday 10/10– CAETE by Friday 10/17

• Homework #6 will be due Friday 10/17– CAETE by Friday 10/24– Solutions will be available in class on 10/17 and online by 10/24. If you turn in HW6 early,

email me and I’ll send you the solutions to check your work.

• Mid-term Exam will be handed out Friday, 10/17 and will be due Wed 10/22. (CAETE 10/29)

– Take-home. Open book, open notes.– Once you start the exam you have to be finished within 24 hours.– It should take 2-3 hours.

Lecture 16: Lambert's Problem 65