1. ASEN 5050 SPACEFLIGHT DYNAMICS Lambert’s Problem
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 16: Lambert's Problem

2. 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, 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

3. Space News
Remember this? Right on!
Lecture 16: Lambert's Problem

4. ASEN 5050 SPACEFLIGHT DYNAMICS Lambert’s Problem
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 16: Lambert's Problem

5. 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

6. Orbit Transfer
We’ll consider orbit transfers in general, though the OD problem is always another application.
Lecture 16: Lambert's Problem

7. 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

8. 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

9. Ellipse
A transfer from r1 to r2 will be on an ellipse, with the central body occupying one focus.
Where’s the 2nd focus? r1 r2
Focus is one of these
Try different a values until you hit your TOF
Lecture 16: Lambert's Problem

10. Lambert’s Problem
Lecture 16: Lambert's Problem

11. Lambert’s Problem
Lecture 16: Lambert's Problem

12. 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:
# revolutions
Transfer Duration
Lecture 16: Lambert's Problem

13. Universal Variables
Simplify
Lecture 16: Lambert's Problem

14. Universal Variables Define Universal Variables:
Lecture 16: Lambert's Problem

15. Universal Variables
Lecture 16: Lambert's Problem

16. Universal Variables Use the trigonometric identity
Lecture 16: Lambert's Problem

17. Universal Variables Now we need somewhere to go
Let’s work on converting this to true anomaly, via:
Lecture 16: Lambert's Problem

18. Universal Variables Multiply by a convenient factoring expression:
Lecture 16: Lambert's Problem

19. Universal Variables
Collect into pieces that can be replaced by true anomaly
Lecture 16: Lambert's Problem

20. Universal Variables Substitute in true anomaly:
Lecture 16: Lambert's Problem

21. Universal Variables
Trig identity again:
Lecture 16: Lambert's Problem

22. Universal Variables
Note:
Lecture 16: Lambert's Problem

23. Universal Variables Use some substitutions:
Lecture 16: Lambert's Problem

24. Universal Variables
Summary:
Lecture 16: Lambert's Problem

25. 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

26. UV Algorithm
Lecture 16: Lambert's Problem

27. UV Algorithm
Bi-section Method
Lecture 16: Lambert's Problem

28. UV Algorithm
Lecture 16: Lambert's Problem

29. A few details on the Universal Variables algorithm
Lecture 16: Lambert's Problem

30. Universal Variables
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
Lecture 16: Lambert's Problem

31. Single-Rev Earth-Venus Type I
-4π
4π2
Lecture 16: Lambert's Problem

32. Single-Rev Earth-Venus Type I
-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
-4π
4π2
Lecture 16: Lambert's Problem

40. Note: Bisection method
Time history of bisection method:
Requires 42 steps to hit a tolerance of 10-5 seconds!
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
Lecture 16: Lambert's Problem

45. 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

46. Note: Newton Raphson Log Method
Note log scale
-4π
4π2
Lecture 16: Lambert's Problem

47. Single-Rev Earth-Venus Type I
-4π
4π2
Lecture 16: Lambert's Problem

48. Single-Rev Earth-Venus Type II
-4π
4π2
Lecture 16: Lambert's Problem

49. Interesting: 10-day transfer
-4π
4π2
Lecture 16: Lambert's Problem

50. Interesting: 950-day transfer
-4π
4π2
Lecture 16: Lambert's Problem

51. Multi-Rev
Seems like it would be better to perform a multi-rev solution over 950 days than a Type II transfer!
Lecture 16: Lambert's Problem

52. A few details
The universal variables construct ψ represents the following transfer types:
Lecture 16: Lambert's Problem

53. Multi-Rev ψ ψ ψ ψ
Lecture 16: Lambert's Problem

54. Earth-Venus in 850 days
Type IV
Lecture 16: Lambert's Problem

55. Earth-Venus in 850 days Heliocentric View Distance to Sun
Lecture 16: Lambert's Problem

56. Earth-Venus in 850 days
Type VI
Lecture 16: Lambert's Problem

57. Earth-Venus in 850 days Heliocentric View Distance to Sun
Lecture 16: Lambert's Problem

58. What about Type III and V?
Type V
Type III
Type VI
Type IV
Lecture 16: Lambert's Problem

59. Earth-Venus in 850 days
Type III
Lecture 16: Lambert's Problem

60. Earth-Venus in 850 days Heliocentric View Distance to Sun
Lecture 16: Lambert's Problem

61. Earth-Venus in 850 days
Type V
Lecture 16: Lambert's Problem

62. Earth-Venus in 850 days Heliocentric View Distance to Sun
Lecture 16: Lambert's Problem

63. Summary 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
Lecture 16: Lambert's Problem

64. Types II - VI
Lecture 16: Lambert's Problem

65. 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, 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