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ASEN 5050 SPACEFLIGHT DYNAMICS Groundtracks, Prox Ops

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ASEN 5050 SPACEFLIGHT DYNAMICS Groundtracks, Prox Ops Prof. Jeffrey S. Parker University of Colorado Boulder Lecture 14: Groundtracks, ProxOps * – PowerPoint PPT presentation

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Title: ASEN 5050 SPACEFLIGHT DYNAMICS Groundtracks, Prox Ops


1
ASEN 5050 SPACEFLIGHT DYNAMICS Groundtracks, Prox
Ops
  • Prof. Jeffrey S. Parker
  • University of Colorado Boulder

2
Announcements
  • Homework 3 is due again this Friday
  • CAETE by Friday 10/10
  • You can recover ½ of the points you lost, or get
    a 90, whichever is higher.
  • No Concept Quiz 11 yet if I get one out Ill
    email you!
  • 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.
  • Reading Chapter 6, 7.6

3
Space News
  • Earth has a new quasi-satellite 2014 OL339
  • Its in a quasi-stable orbit that extends out to
    the Sun-Earth L4 and L5 points. A horseshoe
    orbit.

4
HW
  • Use your resources to check your answers.
  • Its just so much easier to grade correct answers
    ?

5
Quiz 10
6
Quiz 10
7
Quiz 10
8
Quiz 10
Tricky if you think about a circular orbit.
9
Quiz 10
10
ASEN 5050 SPACEFLIGHT DYNAMICS Groundtracks
  • Prof. Jeffrey S. Parker
  • University of Colorado Boulder

11
Groundtracks
12
Groundtracks
13
Groundtracks
  • Circular orbits are line (symmetric about lines
    of longitude) and hinge (symmetric WRT equator)
    symmetric.
  • Can determine the orbital period from groundtracks

14
Groundtracks
15
Groundtracks
  • Effects of Eccentricity and Argument of Perigee

Note were not changing orbital period, so they
all start and stop at the same place!
16
Groundtracks
  • Effects of Inclination

17
Special Groundtracks
  • Repeating Groundtrack
  • Orbital period is selected to make sure the
    groundtracks repeat after some specified duration
    of time (1 day, 15 days, whatever)
  • Geosynchronous
  • Remains tied to a certain area of the Earths
    surface.
  • Molniya
  • 12-hour orbits that swap between USAs and
    Russias landmasses
  • Sun-synchronous
  • Travels across the ground at the same rate as the
    terminator

18
Molniya
19
Sun Synchronous Orbits
20
Sun Synchronous Orbits
21
ASEN 5050 SPACEFLIGHT DYNAMICS Prox Ops
  • Prof. Jeffrey S. Parker
  • University of Colorado Boulder

22
Prox Ops on the Ground
  • Earth-based hypothetical analog to prox ops
  • Flat
  • Horizontal
  • Frictionless
  • If you want to travel 1000 m in 1000 seconds
  • ?V1 1 m/s, ?V2 1 m/s
  • If you want to travel 1000 m in 100 seconds
  • ?V1 10 m/s, ?V2 10 m/s
  • If you want to travel 10 m in 1 second
  • ?V1 10 m/s, ?V2 10 m/s

23
In-Space Prox Ops
  • Over very short timescales, the linear
    assumption holds just fine.
  • E.g., if you want to close the distance to an
    object over 10 seconds.
  • Over long timescales, this breaks down
  • We saw this with circular rendezvous trajectories
  • To close the distance to an object over 1
    revolutions, you burn the opposite direction!

24
In-Space Prox Ops
  • Say you want to close a distance of 100 meters,
    where the target is in front of us in our orbit.
  • Lets consider what would happen if we just
    executed a maneuver in the along-track direction.
  • What happens?
  • Depends on how much ?V we burn!
  • If we burn a lot then well close the distance
    really quickly.
  • If we dont then it could take more than 10
    minutes to arrive at the destination, and the
    destination will have moved nonlinearly!

25
In-Space Prox Ops
  • Say you want to close a distance of 100 meters,
    where the target is in front of us in our orbit.
  • Lets consider what would happen if we just
    executed a maneuver in the along-track direction.
  • What happens?

26
In-Space Prox Ops
  • What if the target is in some off-axis direction?
  • Certainly the Space Shuttle and other vehicles
    have had to do some interesting maneuvers to
    operate in the vicinity of the ISS!
  • They dont want to hurry for many reasons.
  • They must have a good understanding of
    spaceflight dynamics!

27
CW / Hill Equations
  • Enter Clohessy/Wiltshire (1960)
  • And Hill (1878)
  • (notice the timeline here when did Sputnik
    launch? Gemini was in need of this!)

28
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
29
Coordinate Systems
  • Satellite Coordinate System (RSW) --
    (Radial-Transverse-Normal)

30
Coordinate Transformations
  • To convert between IJK and PQW
  • To convert between PQW and RSW
  • Thus, RSW ? IJK is

31
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
  • Use RSW coordinate system (may be different from
    NASA)
  • Target satellite has two-body motion
  • The interceptor is allowed to have thrusting
  • Then
  • So,

32
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
  • Need more information to solve this
  • Consider the oblique triangle
  • We can use the cosine law to find

33
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
  • Now,
  • If is small relative to then
  • Simplify using binomial series (1x)n 1 nx
    n(n-1)x2/2!

34
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
  • Substituting in

35
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
Recall
Coriolis Acceleration
We have this
Centripetal Acceleration
We want this
Acceleration due to a changing frame rate (zero
for circular orbits)
36
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
0 for circular orbit
37
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
  • If we assume circular motion,
  • Thus,
  • Assume 0 (good for impulsive DV maneuvers,
    not for continuous thrust targeting).

CW or Hills Equations
38
Assumptions
  • Please Take Note
  • Weve assumed a lot of things
  • Weve assumed that the relative distance is very
    small
  • Weve assumed circular orbits
  • These equations do fall off as you break these
    assumptions!

39
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
  • The above equations can be solved (see book
    Algorithm 48) leaving

So, given of
interceptor, can compute of
interceptor at future time.
40
Applications
  • What can we do with these equations?
  • Estimate where the satellite will go after
    executing a small maneuver.
  • Rendezvous and prox ops!
  • Examples. First from the book.

41
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
42
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
43
Hubbles Drift from Shuttle
  • RSW Coordinate Frame

44
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
45
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
46
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
  • We can also determine DV needed for rendezvous.
    Given x0, y0, z0, we want to determine
    necessary to make xyz0. Set first 3
    equations to zero, and solve for .
  • Assumptions
  • Satellites only a few km apart
  • Target in circular orbit
  • No external forces (drag, etc.)

47
Clohessy-Wiltshire (CW) Equations (Hills
Equations)
48
Comparing Hill to Keplerian Propagation
Deviation 100 meters 1 cm/s
Position Error 1.4 meters/day
49
Comparing Hill to High-Fidelity, e0.15
Generally returns to 0.0 near one apsis
Position Error 9.5 km/day
50
Comparing Hill to High-Fidelity, e0.73
Generally returns to 0.0
Position Error 4.2 km/day
51
Analyzing Prox Ops
52
Analyzing Prox Ops
  • What happens if you just change x0?
  • Radial displacement?

53
Analyzing Prox Ops
54
Analyzing Prox Ops
  • What happens if you just change vx0?
  • Radial velocity change?

55
Analyzing Prox Ops
56
Analyzing Prox Ops
  • What happens if you just change vy0?
  • Tangential velocity change?

57
Analyzing Prox Ops
58
Analyzing Prox Ops
  • What happens if you just change vz0?
  • Out-of-plane velocity change?

59
Analyzing Prox Ops
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