ASEN 5050
SPACEFLIGHT DYNAMICS
Conversions, f & g, Orbit Transfers
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 9: Conversions, Orbit Transfers
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Announcements
• Homework #3 is due right now
– You must write your own code.
– For this HW, please turn in your code (preferably in one text/Word/PDF
document)
– After this assignment, you may use Vallado’s code, but if you do you must
give him credit for work done using his code. If you don’t, it’s plagiarism.
• Homework #4 is due Friday 9/26 at 9:00 am
– You’ll also have to turn in your code for this one.
• No Quiz over the weekend! Enjoy your weekend.
• I’ll be at the career fair Monday, so I’m delaying Monday’s office
hours to 2:00.
• Reading: Chapter 6 (SIX, we jumped a few)
Lecture 9: Conversions, Orbit Transfers
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Concept Quiz 7
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Lecture 9: Conversions, Orbit Transfers
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Concept Quiz 7
Lecture 9: Conversions, Orbit Transfers
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Concept Quiz 7
The class is 50-50 split on this!
Talk it over with your neighbor and
convince him/her why you’re right.
Lecture 9: Conversions, Orbit Transfers
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Concept Quiz 7
Lecture 9: Conversions, Orbit Transfers
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Quizzes
• Speaking of quizzes, I had a bug in my grade book
and only the first two quiz scores were shown. As of
now, Quiz 1-6 should be shown.
Lecture 9: Conversions, Orbit Transfers
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Space News
• Sunday: MAVEN arrives at Mars!
• MOI: this Sunday at 19:37 Mountain
• LASP is holding a viewing party
Lecture 9: Conversions, Orbit Transfers
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Space News
Lecture 9: Conversions, Orbit Transfers
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Space News
Lecture 9: Conversions, Orbit Transfers
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Space News
• Then Wednesday: MOM arrives at Mars!
• MOI: Wednesday at 21:00 Mountain
– It will enter occultation at 21:04
– MOI will end at 21:24
– We’ll know if it’s successful around 21:30
Lecture 9: Conversions, Orbit Transfers
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Space News
Lecture 9: Conversions, Orbit Transfers
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Challenge #4
• If you were to plot the position and velocity of a
satellite over time using VNC (Velocity-NormalConormal) coordinates, what would you find?
– Say, an elliptical orbit
V
Lecture 9: Conversions, Orbit Transfers
C
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ASEN 5050
SPACEFLIGHT DYNAMICS
Coordinate Transformations
Prof. Jeffrey S. Parker
University of Colorado - Boulder
Lecture 9: Conversions, Orbit Transfers
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Principal Axis Rotations
Lecture 9: Conversions, Orbit Transfers
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Principal Axis Rotations
rX ¢ = rX cosq + rY sinq
Y¢
rY ¢ = -rX sinq + rY cosq
rZ ¢ = rZ
é cosq
ê
rX ¢Y ¢Z ¢ = ê -sin q
ê 0
ë
sin q
cosq
0
écosq
ê
ROT 2(q ) = ê 0
êë sin q
Lecture 9: Conversions, Orbit Transfers
0 ù
ú
0 ú rXYZ = ROT 3(q )rXYZ
1 úû
0 -sin q ù
ú
1
0 ú
0 cosq úû
Y
q
rX
rXYZ
q
é1
0
ê
ROT1(q ) = ê0 cosq
êë0 -sin q
Equation 3-15
X¢
rY ¢
rX ¢
rY
X
0 ù
ú
sinq ú
cos q úû
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r and v From Orbital Elements
Express the position and velocity in the perifocal system ( xˆ goes
through periapse, zˆ in the direction of h , y perpendicular to xˆ
and zˆ , in the orbit plane.)
Y
rPQW
v PQW
Qˆ
é p cosn ù
ê1 + e cosn ú
ê
ú
ˆ = ê p sinn ú
= r cosn Pˆ + r sinn Q
ê1 + e cosn ú
ê
ú
ê
ú
0
ê
ú
ë
ûPQW
ér cosn - rn sinn ù
= êr sinn + rn cosn ú
ê
ú
0
ëê
ûú
PQW
Lecture 9: Conversions, Orbit Transfers
rn =
r=
y = rsin n
n
m p(1+ ecos n )
h
m
=
=
(1+ ecos n )
r
p
p
rn esin n
m
=
(esin n ),
1+ ecos n
p
rPQW
thus
X
x = r cosn
Pˆ
é
ê
ê
ê
=ê
ê
ê
ê
ê
ë
ù
ú
sin n
ú
p
ú
m
ú
(e + cos n) ú
p
ú
0
ú
ú
û
v PQW
m
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PQW
r and v From Orbital Elements
Now we simply need to rotate into the geocentric equatorial
system. The order of the rotations does matter
Y
Qˆ
é IJK ù
rIJK = ROT 3 ( -W) ROT1 ( -i ) ROT 3 ( -w) rPQW = ê
ú rPQW
PQW
ë
û
é IJK ù
v IJK = ROT 3 ( -W) ROT1 ( -i ) ROT 3 ( -w) v PQW = ê
ú v PQW
ë PQW û
rPQW
n
é cos W cosw - sin W sin w cosi -cos W sin w - sin W cosw cosi sin W sini
é IJK ù ê
ê
ú = ê sin W cosw + cos W sin w cosi -sin W sin w + cos W cosw cosi -cos W sini
ë PQW û ê
sin w sini
cosw sini
cosi
ë
Pˆ
ù
ú
ú
ú
û
Algorithm 10 in book.
Ex. 2-6
Lecture 9: Conversions, Orbit Transfers
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X
f and g Series
r = xPˆ + yQˆ
v = x˙Pˆ + y˙ Qˆ
Lecture 9: Conversions, Orbit Transfers
r = fro + gv o
v = fro + gv o
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f and g Series
Start by crossing the position vector into the initial velocity vector:
r ´ vo = ( fro + gvo )´ vo
= f (ro ´ vo ) + g(v o ´ vo )
The second term is zero, and the other terms are normal to the plane:
xy˙ o - x˙ o y = fh
xy˙ o - x˙ o y
f =
h
Differentiating this last equation:
˙f = x˙y˙ o - x˙ o y˙
h
Lecture 9: Conversions, Orbit Transfers
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f and g Series
Now cross the initial position vector into the position vector:
ro ´ r = ro ´ ( fro + gv o )
= f (ro ´ ro ) + g(ro ´ vo )
The first term is zero, and the other terms are normal to the plane:
x o y - xy o = gh
x o y - xy o
g=
h
Differentiating this last equation:
x o y˙ - x˙y o
g˙ =
h
Lecture 9: Conversions, Orbit Transfers
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f and g Series
Look at the cross-product:
h = r ´ v = ( fro + gv o )´ ( fro + gv o )
= ff (ro ´ ro ) + fg(ro ´ v o ) + fg(v o ´ ro ) + gg(v o ´ v o )
= fgh - fgh
Which can only be true if:
fg - fg =1
x = r cosn
x=Lecture 9: Conversions, Orbit Transfers
m
p
sin n
A good test!
y = r sin n
y=
m
p
(e + cos n )
22
f and g Series
Which gives:
Lecture 9: Conversions, Orbit Transfers
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f and g Series
So, to summarize, given an initial position and velocity, we can
calculate a future position and velocity given the change in the true
anomaly Dn:
Which you can test using Example 2-4 in the textbook.
Lecture 9: Conversions, Orbit Transfers
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f and g Series: State Transition Matrix
We can re-express our f and g series representation:
r = fro + gv o
v = f˙r + g˙ v
o
o
in terms of a state-variable relationship:
ér ù
X =ê ú
ëv û
é ro ù
éF
Xo = ê ú
X = FX o = ê
ëv o û
ëF˙
é f 0 0ù
ê
ú
where F = ê 0 f 0 ú, etc.,
êë 0 0 f úû
and F is the State Transition Matrix
Lecture 9: Conversions, Orbit Transfers
Gù
Xo
ú
G˙ û
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f and g Series
• What uses do these functions have?
– Given two states, find the time of flight between them.
– Given two states, find an orbit that connects them.
• Big fan of this application.
– Using an iterative technique, such as Newton Raphson, can
determine a future state given a current state and a transfer
time or transfer angle.
Lecture 9: Conversions, Orbit Transfers
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ASEN 5050
SPACEFLIGHT DYNAMICS
Orbital Maneuvers
Prof. Jeffrey S. Parker
University of Colorado - Boulder
Lecture 9: Conversions, Orbit Transfers
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Orbital Maneuvers
• Orbital maneuvers are used to do many things:
– Change a satellite’s orbit
• Size
• Shape
• Orientation
–
–
–
–
–
Change the phase of a satellite in its orbit
Rendezvous and/or proximity operations
Avoid collisions (debris)
Change the satellite’s groundtrack
Etc.
Lecture 9: Conversions, Orbit Transfers
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Terminology
• Coplanar maneuvers: no change to the orbit plane; the
maneuvers can only change a, e, w.
• Impulsive maneuvers: instantaneous change in velocity:
ΔV
– Requires an infinitely powerful engine
• Finite maneuvers: maneuvers that require a duration of
time to achieve
• Ballistic: the trajectory of an object under the effects of
only external forces (no maneuver firings).
Lecture 9: Conversions, Orbit Transfers
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Orbital Maneuvers
Tangential Burns: in velocity/anti-velocity vector direction
- Do not change velocity orientation, just magnitude
- Do not change flight path angle
Lecture 9: Conversions, Orbit Transfers
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Orbital Maneuvers
Nontangential: plane changes, orbit rotations
Lecture 9: Conversions, Orbit Transfers
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Orbital Maneuvers
Hohmann Transfer – Walter Hohmann (1880-1945) showed
minimum energy transfer between two orbits used two
tangential burns.
Lecture 9: Conversions, Orbit Transfers
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Hohmann Transfer
(math, compute DV and DT)
Lecture 9: Conversions, Orbit Transfers
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Hohmann Transfer
atrans
rinitial + rfinal
=
2
v initial =
v final =
m
rinitial
m
rfinal
Dv a = v transA - v initial
Dv total = Dv a + Dv b
t trans
3
Ptrans
atrans
=
=p
2
m
v transA =
v transB =
2m
rinitial
2m
rfinal
-
m
atrans
m
atrans
Dv b = v final - v transB
(Algorithm 36, Example 6-1)
Can also be done using elliptical orbits, but must start at apogee
or perigee to be a minimum energy transfer.
Lecture 9: Conversions, Orbit Transfers
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Hohmann Transfer
• We just argued that the Hohmann Transfer is (usually) the
most energy-efficient orbital transfer.
• Why?
– Consider Elliptical—Elliptical transfer
– Tangential Burns
– Energy efficiency considerations
2
V
m
E=
2 R
dE
=V
Þ
dV
Lecture 9: Conversions, Orbit Transfers
DE » V × DV
V is highest at perigee
DV 2
DE = V × DV +
2
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Energy Changes
Lecture 9: Conversions, Orbit Transfers
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Orbital Maneuvers
Bi-elliptic Transfer – Uses two Hohmann transfers. Can save Dv
in some cases. rb must be greater than rfinal, but can otherwise be
optimized.
Lecture 9: Conversions, Orbit Transfers
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Bi-elliptic Transfer
atrans 1
rinitial + rb
=
2
v initial =
v trans 1 B =
v trans 2 C =
atrans 2
m
v trans 1 A =
rinitial
2m
rb
-
2m
rfinal
m
v trans 2 B =
atrans 1
-
m
v final =
atrans 2
Dv a = v trans 1 A - v initial
Dv c = v final - v trans 2 C
t trans = p
3
atrans
1
m
rb + rfinal
=
2
+p
2m
rinitial
2m
rb
-
-
m
atrans 1
m
atrans 2
m
rfinal
Dv b = v trans 2 B - v trans 1 B
Dv total = Dv a + Dv b + Dv c
3
atrans
2
(Algorithm 37, Example 6-2)
m
Much longer flight times for bi-elliptic transfer, but sometimes less energy.
Lecture 9: Conversions, Orbit Transfers
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Hohmann vs Bi-elliptic
Lecture 9: Conversions, Orbit Transfers
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One-Tangent Burns
Lecture 9: Conversions, Orbit Transfers
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Orbit Transfer Comparison
Lecture 9: Conversions, Orbit Transfers
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Changing Orbital Elements
•
•
•
•
•
•
Δa  Hohmann Transfer
Δe  Hohmann Transfer
Δi  Plane Change
ΔΩ  Plane Change
Δω  Coplanar Transfer
Δν  Phasing/Rendezvous (later discussion)
Lecture 9: Conversions, Orbit Transfers
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Changing Inclination
• Δi  Plane Change
• Inclination-Only Change vs. Free Inclination Change
Lecture 9: Conversions, Orbit Transfers
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Changing Inclination
• Let’s start with circular orbits
Vf
V0
Lecture 9: Conversions, Orbit Transfers
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Changing Inclination
• Let’s start with circular orbits
Vf
V0
Lecture 9: Conversions, Orbit Transfers
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Changing Inclination
• Let’s start with circular orbits
Are these vectors the
same length?
Vf
Δi
V0
What’s the ΔV?
Is this more expensive
in a low orbit or a high
orbit?
Lecture 9: Conversions, Orbit Transfers
46
Changing Inclination
• More general inclination-only maneuvers
Where do you perform
the maneuver?
How do V0 and Vf
compare?
What about the FPA?
Line of Nodes
Lecture 9: Conversions, Orbit Transfers
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Changing Inclination
• More general inclination-only maneuvers
Lecture 9: Conversions, Orbit Transfers
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Changing The Node
Lecture 9: Conversions, Orbit Transfers
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Changing Argument of Perigee
Lecture 9: Conversions, Orbit Transfers
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