ASEN 5050 SPACEFLIGHT DYNAMICS Lambert s Problem

Transcription

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

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 2

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

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

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 5

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

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 7

8 Lambert s Problem Given: Find: ~R 0 ~ Rf t 0 t f ~V 0 ~ Vf 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

9 Ellipse A transfer from r 1 to r 2 will be on an ellipse, with the central body occupying one focus. Where s the 2 nd focus? Focus is one of these Try different a values until you hit your TOF 2a r 1 r 1 r 2 2a r 2 Lecture 16: Lambert's Problem 9

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

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

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: M = n t M = E e sin E Lecture 16: Lambert's Problem 12

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

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

15 Universal Variables Lecture 16: Lambert's Problem 15

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

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

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

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

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

21 Trig identity again: Universal Variables sin( 2 1 )=sin 2 cos 1 cos 2 sin 1 Lecture 16: Lambert's Problem 21

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

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

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

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 25

26 UV Algorithm Lecture 16: Lambert's Problem 26

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

28 UV Algorithm Lecture 16: Lambert's Problem 28

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

30 Let s first consider our Universal Variables Lambert Solver. Given: R 0, R f, ΔT Universal Variables 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 30

31 Single-Rev Earth-Venus Type I -4π 4π 2 Lecture 16: Lambert's Problem 31

32 Single-Rev Earth-Venus Type I -4π 4π 2 Lecture 16: Lambert's Problem 32

33 Note: Bisection method -4π 4π 2 Lecture 16: Lambert's Problem 33

34 Note: Bisection method -4π 4π 2 Lecture 16: Lambert's Problem 34

35 Note: Bisection method -4π 4π 2 Lecture 16: Lambert's Problem 35

36 Note: Bisection method -4π 4π 2 Lecture 16: Lambert's Problem 36

37 Note: Bisection method -4π 4π 2 Lecture 16: Lambert's Problem 37

38 Note: Bisection method -4π 4π 2 Lecture 16: Lambert's Problem 38

39 Note: Bisection method -4π 4π 2 Lecture 16: Lambert's Problem 39

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

41 Note: Newton Raphson method Lecture 16: Lambert's Problem 41

42 Note: Newton Raphson method Lecture 16: Lambert's Problem 42

43 Note: Newton Raphson method Lecture 16: Lambert's Problem 43

44 Note: Newton Raphson method Lecture 16: Lambert's Problem 44

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 speedup, so robustness may be preferable. Lecture 16: Lambert's Problem 45

46 Note: Newton Raphson Log Method Note log scale -4π 4π 2 Lecture 16: Lambert's Problem 46

47 Single-Rev Earth-Venus Type I -4π 4π 2 Lecture 16: Lambert's Problem 47

48 Single-Rev Earth-Venus Type II -4π 4π 2 Lecture 16: Lambert's Problem 48

49 Interesting: 10-day transfer -4π 4π 2 Lecture 16: Lambert's Problem 49

50 Interesting: 950-day transfer -4π 4π 2 Lecture 16: Lambert's Problem 50

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 51

52 A few details The universal variables construct ψ represents the following transfer types: 8 >< Type of Transfer >: < 0, Hyperbolic =0, Parabolic 0 < < 4 2, 0 revolutions elliptical 4n 2 2 < < 4(n + 1) 2 2, n revolutions elliptical Lecture 16: Lambert's Problem 52

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

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

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

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

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

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

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

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

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

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

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 63

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

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 65