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