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Gaëtan Kerschen
Space Structures &
Systems Lab (S3L)
5B. Orbital Maneuvers
Astrodynamics (AERO0024)
2
Previous Lecture: Coplanar Maneuvers
5.1 INTRODUCTION
5.1.1 Why ?
5.1.2 How ?
5.1.3 How much ?
5.1.4 When ?
5.2 COPLANAR MANEUVERS
5.2.1 One-impulse transfer
5.2.2 Two-impulse transfer
5.2.3 Three-impulse transfer
5.2.4 Nontangential burns
5.2.5 Phasing maneuvers
3
Motivation
Coplanar transfers have wide applicability (e.g., phasing
maneuvers and orbit raising). They alter 4 orbital elements:
1. Semi-major axis.
2. Eccentricity.
3. True anaomaly.
4. Argument of perigee.
Noncoplanar transfers requires applying v out of the orbit
plane and can therefore alter the last two elements:
1. Inclination.
2. RAAN.
4
5. Orbital Maneuvers
5.3 Noncoplanar transfers
5.4 Rendez-vous
5
5. Orbital Maneuvers
5.3 Noncoplanar transfers
5.3.1 Inclination change
6
Gaussian VOP (Lecture 4)
2
2cos(1 )
1 cos
Na ei
e
2
2sin(1 )
sin 1 cos
Na e
i e
ˆ ˆ ˆR R T NT N F e e e
2
5.3.1 Inclination change
7
Where ?
Cranking maneuver: with a single v maneuver, one
wants to change only the inclination of the orbit plane.
Where should we apply this maneuver ?
5.3.1 Inclination change
8
Where ? At an Equator Crossing
The two orbit planes intersect in the equatorial plane. We
can therefore change the inclination at an equator
crossing, by a simple rotation of the velocity vector.
5.3.1 Inclination change
9
ΔV Computation for Unchanged Circular Orbit
5.3.1 Inclination change
∆𝑉 = 𝑉𝑖2 + 𝑉𝑓
2 − 2𝑉𝑖𝑉𝑓𝑐𝑜𝑠𝜃 = 𝑉𝑖 2 1 − 𝑐𝑜𝑠𝜃 = 2𝑉𝑖𝑠𝑖𝑛𝜃
2
∆𝑉 = 2𝑉𝑠𝑖𝑛𝜃
2
10
Inclination Changes Are Very Expensive
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
, degrees
v
(%
of
v)
(24º , 41.6%)
(60º , 100%)
5.3.1 Inclination change
How to minimize ΔV ?
Is this cranking maneuver
the minimum-fuel transfer ?
𝜃
∆
11
Real-Life Examples: Space Shuttle and Hubble
The Space Shuttle is capable of a plane change in orbit of
only about 3º, a maneuver which would exhaust its entire
fuel capacity.
What is Hubble space telescope’s inclination ?
28.46 (HST) vs. 28.35 (KSC)…
5.3.1 Inclination change
12
What’s Unique with this Picture ?
5.3.1 Inclination change
Too costly ! Instead, NASA has
chosen to have another shuttle
ready to lift off to retrieve the
astronauts if needed.
13
Minimum-Fuel Transfer: Bi-Elliptic Transfer
5.3.1 Inclination change
One can always save fuel over a one-impulse maneuver by
using three-impulses: noncoplanar bi-elliptic transfer.
J.E. Prussing, B.A. Conway, Orbital Mechanics, Oxford University Press
14
5. Orbital Maneuvers
5.4 Rendez-vous
5.4.1 Hill – Clohessy – Wiltshire equations
5.4.2 Gemini and Space Shuttle
15
Rendez-vous: Two Successive Phases
1. Long-distance navigation: basic maneuvers bring the
spacecraft in a vicinity (<100 km) of the target in the
same orbit plane.
2. Terminal rendez-vous: the interest is in the relative
motion, i.e. the motion of the maneuvering spacecraft in
relation to the target (spacecraft or neighboring orbit).
Focus of the present discussion.
5.4.1 H-C-W equations
Rescue of Intelsat-6: new
perigee kick motor (STS-49)
ISS & ATV
STS & HST
17
Basic Assumption
The rendez-vous can be solved using the “classical” 2-
body problem. But nonlinear equations of motion have to
be solved and one should resort to numerical methods
(e.g., Lambert’s problem).
Can we use approximate equations considering that
distances are small relative to the dimensions of the orbit ?
Yes ! Use linearization procedures.
5.4.1 H-C-W equations
18
Problem Statement
0r
r
0
0
, 1r
r
r r r
target
(known
orbit)
What is the equation governing the relative motion r ?
A rendez-vous occurs when r =0.
r
chase
5.4.1 H-C-W equations
3r
rr
0 r r r
00 3r
r rr r
?
0 00 03 3 2
0 0 0
3 .
r r r
r r rr r r r
Higher-order terms
neglected
00 3r
r
r
003 2
0 0
3 .
r r
r rr r r
Linear ODE.
Observer in an
inertial frame
3 3 00 2
0
3 .1r r
r
r r
00 03 5
0 0
3 .1
r r
r rr r r r
See next page
20
Expression of 𝒓−𝟑
2 2 2
0 0 0 0( ).( ) 2 .r r r r r r r r r
3
23 3 0
0 2
0
2 .1r r
r
r r
Binomial
theorem
3 3 00 2
0
3 .1r r
r
r r
5.4.1 H-C-W equations
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ
rel
rel
x y z
x y z
x y z
n
r i j k
v i j k
a i j k
Ω k
21
Co-Moving Frame
0r
r
r
i
jk
Local vertical
coordinate
frame
Our focus is on a
rotating observer in
the local frame (e.g.,
an ISS astronaut
watching the ATV)
5.4.1 H-C-W equations
003 2
0 0
3 .
r r
r rr r r
20 00 03 2 2
0 0 0
3 . 3 .ˆ ˆ ˆˆ r xn x y z r
r r r
r rr r r i j k i?
22
Assumption of a Circular Orbit
5.4.1 H-C-W equations
𝑣 =𝜇
𝑟0= 𝑟0
= 𝑛 =𝜇
𝑟03
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ
rel
rel
x y z
x y z
x y z
n
r i j k
v i j k
a i j k
Ω k
23
Differentiation in A Rotating Frame
0 02 rel rel
d
dt
Ωr r r Ω Ω r Ω v a r r
5.4.1 H-C-W equations
Circular
Interpretation
of the terms ?
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ
rel
rel
x y z
x y z
x y z
n
r i j k
v i j k
a i j k
Ω k
2 2ˆ ˆ ˆ( 2 ) ( 2 )n x n y x n y n x y z r i j k
Centrifugal Coriolis
2. 2 rel rel r Ω Ω r r Ω v a
24
Finally
5.4.1 H-C-W equations
2 2ˆ ˆ ˆ( 2 ) ( 2 )n x n y x n y n x y z r i j k
20 00 03 2 2
0 0 0
3 . 3 .ˆ ˆ ˆˆ r xn x y z r
r r r
r rr r r i j k i
2
2
3 2 0
2 0
0
x n x n y
y n x
z n z
Hill – Clohessy – Wiltshire
equations: acceleration relative
to a rotating observer fixed in
the moving frame.
25
An Obvious Solution
2
2
3 2 0
2 0
0
x n x n y
y n x
z n z
?
?
?
x
y
z
x=0, y=constant.
z=harmonic motion (the out-of plane motion is
decoupled from the in-plane motion).
Physical interpretation ?
Two spacecraft on the same orbit.
5.4.1 H-C-W equations
26
In Matrix Form
2
2
0 2 0 3 0 0 0
2 0 0 0 0 0 0
0 0 0 0 0 0
x n x n x
y n y y
z z n z
Linear ODEs with constant coefficient: we can solve them !
Gyroscopic damping:
imposes a rotation if
there is a relative velocity
Centrifugal +
gravitational terms: move
away from the target if
not on the same orbit
5.4.1 H-C-W equations
27
Interpretation
Impulse to the left
rotation due to
Coriolis
Forward impulse
backward
motion !
Backward
impulse
forward motion !
5.4.1 H-C-W equations
28
Solution
0 02 0 2 2y n x y n x Cte y n x
2 2 2
0 03 2 0 2 4x n x n y x n x n y n x
0 0
2sin cos 4x A nt B nt y x
n
…
5.4.1 H-C-W equations
29
Solution: Overall
H. Curtis, Orbital Mechanics for Engineering Students, Elsevier.
0 0
0 0
0 0
2 1 cossin0
4 3cos 0 02 cos 1 4sin 3
6 sin 1 0 0
0 0 cossin
0 0
ntnt
n nx nt x x
nt nt nty nt nt y y
n nz nt z z
nt
n
0 0
0 0
0 0
3 sin 0 0 cos 2sin 0
6 cos 1 0 0 2sin 4cos 3 0
0 0 sin 0 0 cos
x n nt x nt nt x
y n nt y nt nt y
z n nt z nt z
( )rr tΦ ( )rv tΦ
( )vr tΦ ( )vv tΦ
( )tr 0 ( )tr 0 ( )t v
( )t v 0 ( )tr 0 ( )t v
5.4.1 H-C-W equations
30
Example: HST and STS
The HST is released from the Space Shuttle, which is in a
circular orbit of 590 km altitude. The relative velocity of
ejection is 0.1 m/s down, 0.04 m/s backwards and 0.02 m/s
to the right. Find the position of HST after 5,10,20 minutes.
0 0
0 0
0 0
2 1 cossin0
4 3cos 0 02 cos 1 4sin 3
6 sin 1 0 0
0 0 cossin
0 0
ntnt
n nx nt x x
nt nt nty nt nt y y
n nz nt z z
nt
n
(-0.1;-0.04;-0.02)T
3
3986000.0011 rad/s
6968n
5.4.1 H-C-W equations
31
Example: HST and STS
5 minutes: (-33.345 ; -1.473 ; -5.894)T
10 minutes: (-70.933 ; 20.357 ; -11.170)T
20 minutes: (-143.000 ; 137.279 ; -17.766)T
5.4.1 H-C-W equations
32
Example: HST and STS
-200 -150 -100 -50 0 50-500
0
500
1000
1500
2000
2500
3000
x (m)
y
(m
)
5.4.1 H-C-W equations
33
Further Reading on the Web Site
They use Hill – Clohessy –
Wiltshire equations !
5.4.1 H-C-W equations
34
Solution: Overall
0 0
0 0
( ) ( ) ( )
( ) ( ) ( )
rr rv
vr vv
t t t
t t t
r Φ r Φ v
v Φ r Φ v
Assumptions:
1. The two satellites are within a few kms of each other.
2. The target is in a circular orbit.
3. There are no external forces on the interceptor.
Starting from t=0, how to exploit these equations for
rendez-vous after a time tf ?
Linear analog of
Kepler’s equation
5.4.1 H-C-W equations
35
Two-Impulse Rendez-vous Maneuvers
0 00 , and knownt r v
0 00 ( ) ( )rr f rv ft t Φ r Φ v
0Compute for rendez-vous in a time f
t v
1
0 0( ) ( )rv f rr ft t v Φ Φ r
0 0
1
0 0
Compute ( ) ( )
( ) ( ) ( ) ( )
f vr f vv f
vr f vv f rv f rr f
t t
t t t t
v Φ r Φ v
Φ r Φ Φ Φ r
Impose 0f v f f v v
0 0 0 v v v
5.4.1 H-C-W equations
36
Physical Interpretation
The initial velocity change places the vehicle on a
trajectory which will intercept the origin at time tf.
The final velocity change cancels the arrival velocity of
the vehicle at the origin and completes the rendez-vous.
5.4.1 H-C-W equations
37
Example: HST and STS
After 10 minutes, the STS astronauts want to retrieve HST.
Compute the v0+ to rendez-vous in 5 and 15 minutes.
5 minutes: (0.2742 ; 0.0135 ; 0.0359)T
15 minutes: (0.1356 ; 0.0753 ; 0.0082)T
5.4.1 H-C-W equations
38
Real Data: Gemini 10 and Agena 8
A 4-orbit rendez-vous plan similar to that of Gemini 6-7:
O1: no maneuvers to allow spacecraft checkout.
O2: height adjustment at perigee (0.2 m/s for orbital decay)
phase adjustment maneuver at apogee (17 m/s).
O3: switch to a circular orbit in the same plane as Agena
(16 m/s).
O4: terminal phase initiation (10 m/s)
velocity matching (13 m/s)
Total: 56 m/s
NH
NCI
NSR
TPI
TPF
5.4.2 Gemini and Space Shuttle
Nc: nominal corrective burn
Nh: nominal height adjust burn
5.4.2 Gemini and Space Shuttle
40
Real Data: Gemini 10 and Agena 8
5.4.2 Gemini and Space Shuttle
41
Real Data: Gemini 10 and Agena 8
5.4.2 Gemini and Space Shuttle
42
Real Data: Gemini 10 and Agena 8
43
Date Time UTC Name Impulsion Orbite
(m/s) (km x km)
23 15:46:44 MECO ---- 229 x 56
23 16:16:38 OMS-2 70,74 296 x 229
23 18:32:45 NC1 26,06 319 x 296
24 08:50:56 NC2 6,13 319 x 316
24 19:27:09 NC3 0,91 319 x 317
25 07:30:35 NH 7,01 344 x 319
25 08:24:45 NC4 2,74 345 x 327
25 09:55:43 Ti 2,71 347 x 332
25 11:12:45 MC1 0,64 347 x 334
25 11:26:20 MC2 0,64 348 x 334
STS-120 (Rendez-vous with ISS, Oct. 07)
Pour STS123: http://www.obsat.com/sts123tle.htm 5.4.2 Gemini and Space Shuttle
44
STS-120 (Rendez-vous with ISS, Oct. 07)
5.4.2 Gemini and Space Shuttle
45
Maneuvers can find many practical applications and may
drive spacecraft design (fuel consumption).
Focus on impulsive maneuvers (chemical propulsion).
Most important maneuvers in Earth’s orbit have been
described (plane change, circularization, phasing, rendez-
vous).
In Summary
46
Combined maneuvers (e.g., circularization and inclination
change for a GEO satellite).
Fixed V maneuvers.
Other maneuvers
Gaëtan Kerschen
Space Structures &
Systems Lab (S3L)
5. Orbital Maneuvers
Astrodynamics (AERO0024)