ARO309 - Astronautics and
Spacecraft Design
Winter 2014
Try Lam
CalPoly Pomona Aerospace Engineering
Two-Body Dynamics:
Orbits in 3D
Chapter 4
Introductions
• So far we have focus on the orbital mechanics
of a spacecraft in 2D.
• In this Chapter we will now move to 3D and
express orbits using all 6 orbital elements
Geocentric Equatorial Frame
r = X2 +Y 2 + Z2
d = sin -1 ( Z /r)
ì
æ
ö
-1 X /r
(Y /r > 0)
÷
ïï cos ç
è cosd ø
í
a=
ï2p - cos-1æç X /r ö÷ (Y /r £ 0)
ïî
è cosd ø
Orbital Elements
• Classical Orbital Elements are:
a = semi-major axis (or h or ε)
e = eccentricity
i = inclination
Ω = longitude of ascending node
ω = argument of periapsis
θ = true anomaly
Orbital Elements
v r = r× v /r
iˆ
h=r ´v= X
ˆj
Y
kˆ
Z
Vx
Vy
Vz
iˆ
n = kˆ ´ h = 0
hx
ˆj
0
hy
kˆ
1
hz
Orbital Elements
æ hz ö
i = cos ç ÷
èhø
-1
ì
æ
ö
-1 n× e
(ez ³ 0)
÷
ïï cos ç
è ne ø
í
w=
ï2p - cos-1æç n× e ö÷ (e < 0)
z
ïî
è ne ø
ù
1 éæ 2 m ö
e = êçv - ÷r - rv r vú
m ëè
rø
û
and
e = e× e
or
h 2 æ 2 2m ö
e = 1+ 2 çv - ÷
m è
r ø
Orbital Elements
ì
æ 1 æ h 2 öö
ì
æ
ö
-1
-1 e× r
ï
cos
(v r ³ 0)
ç ç -1÷÷
(v r ³ 0) ï
÷
ïï cos ç
è er ø
è e è mr øø
q =í
=í
æ
ö
æ æ 2 öö
ï2p - cos-1ç e× r ÷ (v < 0) ï
-1 1 h
r
ïî
è er ø
ï2p - cos çè e çè mr -1÷ø÷ø (v r < 0)
î
Coordinate Transformation
• Answers the question of “what are the parameters in
another coordinate frame”
y
y’
Q
x
z
x’
Transformation
(or direction cosine)
matrix
é1 0 0ù
ê
ú
T
[Q] [Q] = [1] = ê0 1 0ú
êë0 0 1úû
z’
Q is a orthogonal transformation matrix
[Q][Q] = [1]
T
Coordinate Transformation
Where
And
[ x'] = [Q][ x ]
T
[ x ] = [Q] [ x']
éQ11 Q12 Q13 ù é iˆ / × iˆ iˆ / ×
ê
ú ê ˆ/ ˆ ˆ/
[Q] = êQ21 Q22 Q23 ú = ê j × i j ×
êëQ31 Q32 Q33 úû êëkˆ / × iˆ kˆ / ×
iˆ / × kˆ ù
ú
/
ˆ
ˆj × k ú
ˆj kˆ / × kˆ ú
û
ˆj
ˆj
Where is made up of rotations about the axis {a, b, or c} by the angle
{θd, θe, and θf}
[
]
[Q] = [ Ra (q d )][ Rb (q e )] Rc (q f )
3rd rotation
2nd rotation
1st rotation
Coordinate Transformation
For example the Euler angle sequence for rotation is the 3-1-3 rotation
[Q] = [ R3 (g )][R1(b)][R3 (a )]
(0 £ a < 360°) (0 £ b £ 180°) (0 £ g < 360°)
where you rotate by the angle α along the 3rd axis (usually z-axis), then by
β along the 1st axis, and then by γ along the 3rd axis.
[Q] 313
é -sin a cos b sin g + cos a cos g
ê
= ê-sin a cos b cos g - cos a sin g
êë
sin a sin b
cos a cos b sin g + sin a cos g
cos a cos b cos g - sin a sin g
-cos a sin b
Thus, the angles can be found from elements of Q
Q31
tan a =
-Q32
cos b = Q33
Q13
tan g =
Q23
sin b sin g ù
ú
sin b cos g ú
cos b úû
Coordinate Transformation
Classic Euler Sequence
from xyz to x’y’z’
Coordinate Transformation
For example the Yaw-Pitch-Roll sequence for rotation is the 1-2-3 rotation
[Q] = [ R1(g )][ R2 ( b)][ R3 (a )]
(0 £ a < 360°) ( -90 < b < 90°) (0 £ g < 360°)
where you rotate by the angle α along the 3rd axis (usually z-axis), then by β
along the 2nd axis, and then by γ along the 1st axis.
[Q]123
é
cos a cos b
ê
= êcos a sin b sin g - sin a cos g
êëcos a sin b cos g + sin a sin g
sin a cos b
sin a sin b sin g + cos a cos g
sin a sin b cos g - cos a sin g
Thus, the angles can be found from elements of Q
Q12
tan a =
Q11
sin b = -Q13
Q23
tan g =
Q33
-sin b ù
ú
cos b sin g ú
cos b cos g úû
Coordinate Transformation
Yaw, Pitch, and Roll
Sequence from xyz to
x’y’z’
Transformation between Geocentric
Equatorial and Perifocal Frame
Transferring between pqw frame and xyz
{r} pqw
ìcosq ü
ï
h /m ï
í sin q ý
=
1+ ecosq ï
ï
0
î
þ
{v} pqw
ì -sin q ü
ï
mï
= íe + cos q ý
hï
ï
0
î
þ
2
Transformation from geocentric equatorial to perifocal frame
Transformation between Geocentric
Equatorial and Perifocal Frame
Transformation from perifocal to geocentric equatorial frame is then
Therefore
Perturbation to Orbits
Oblateness
• Planets are not perfect spheres
Req - Rpole
oblateness =
Req
Perturbation to Orbits
Oblateness
˙r˙ = - m3 r + p
r
p = pr uˆ r + pt uˆ t + ph hˆ
æ R ö2
pr = -1.5 2 J 2 ç ÷ 1 - 3sin 2 (i) sin 2 (w + q )
r èrø
m
[
æ R ö2 2
pt = -1.5 2 J 2 ç ÷ sin (i) sin 2 (2(w + q ))
r èrø
m
æ R ö2
ph = -1.5 2 J 2 ç ÷ sin(2i) sin 2 (w + q )
r èrø
m
]
Perturbation to Orbits
Oblateness
Perturbation to Orbits
Oblateness
Perturbation to Orbits
Oblateness
Sun-Synchronous Orbits
Orbits where the orbit plane is at a fix angle α from the Sun-planet line
Thus the orbit plane must rotate
360° per year (365.25 days) or
0.9856°/day
Finding State of S/C w/Oblateness
• Given: Initial State Vector
• Find: State after Δt assuming oblateness (J2)
• Steps finding updated state at a future Δt assuming
perturbation
1.
2.
3.
4.
Compute the orbital elements of the state
Find the orbit period, T, and mean motion, n
Find the eccentric anomaly
Calculate time since periapsis passage, t, using Kepler’s equation
Me = nt = E - esin E
Finding State of S/C w/Oblateness
5.
6.
Calculate new time as tf = t + Δt
Find the number of orbit periods elapsed since original periapsis passage
n p = t f /T
7.
Find the time since periapsis passage for the final orbit
[
( )] T
torbit _ n = n p - floor n p
8.
Find the new mean anomaly for orbit n
M e ) orbit _ n = n t orbit _ n
9.
Use Newton’s method and Kepler’s equation to find the Eccentric
anomaly (See slide 57)
Finding State of S/C w/Oblateness
10. Find the new true anomaly
tan
q orbit _ n
2
E orbit _ n
1+ e
=
tan
1- e
2
11. Find position and velocity in the perifocal frame
{r} pqw
{v} pqw
ìcosq ü
ï
h /m ï
í sin q ý
=
1+ ecosq ï
ï
0
î
þ
ì -sin q ü
ï
mï
= íe + cos q ý
hï
ï
î 0 þ
2
Finding State of S/C w/Oblateness
12. Compute the rate of the ascending node
13. Compute the new ascending node for orbit n
14. Find the argument of periapsis rate
15. Find the new argument of periapsis
Finding State of S/C w/Oblateness
16. Compute the transformation matrix [Q] using the inclination, the
UPDATED argument of periapsis, and the UPDATED longitude of
ascending node
17. Find the r and v in the geocentric frame
Ground Tracks
Projection of a satellite’s orbit on the planet’s surface
Ground Tracks
Projection of a satellite’s orbit on the planet’s surface
w E »15.04 deg/hr
Ground Tracks
Projection of a satellite’s orbit on the planet’s surface
Ground Tracks
reveal the orbit
period
Ground Tracks
reveal the orbit
inclination
i = LATmax or min
If the argument of perispais, ω, is zero, then the shape
below and above the equator are the same.