Ground-Based Testbed for Replicating the Orbital
Dynamics of a Satellite Cluster in a Gravity Well
David W. Miller
Raymond J. Sedwick
AFRL Distributed Satellite Systems Program
MIT Space Systems Laboratory
Hill’s Equations
F
Governing equations where ‘n’ is orbital frequency in rad/sec:
⎧xśś⎫ ⎡ 0 −2n 0⎤⎧xś ⎫ ⎡−3n 2 0 0 ⎤⎧x ⎫ ⎧a x ⎫
⎥⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎢
⎥⎪ ⎪ ⎢
⎨yśś⎬ + ⎢2n
0 0 ⎥⎨y ⎬ = ⎨a y ⎬
0 0⎥⎨yś ⎬ + ⎢ 0
⎪ ⎪ ⎢
⎪ ⎪
⎥⎪ zś ⎪ ⎢ 0
2 ⎥⎪ ⎪
z
z
0
n
0
0
0
ś
ś
⎩ ⎭ ⎣
⎦⎩ ⎭ ⎣
⎦⎩ ⎭ ⎩a z ⎭
—
—
F
accelerations account for non-central forces (drag, thrust, etc.).
x-axis in zenith, y-axis in frame’s velocity, and z-axis in transverse
directions.
Free orbit solution where ‘A’ and ‘B’ are lengths and ‘α’ and ‘β’ are
phase angles.
+ xo
x = Acos(nt + α)
y = −2A sin(nt + α) − (3/ 2)nx o t + y o
z = Bcos(nt + β)
Closed Cluster Solution
F
There exist free orbits that cause a S/C to follow a closed and periodic
motion with respect to the Hill’s frame as well as other S/C of the same
period.
x = A cos(nt + α)
y = −2A sin(nt + α) + y o
z = Bcos(nt + β)
F
the S/C must follow a two-by-one ellipse in the Hill’s frame’s zenithvelocity plane.
—
F
transverse displacement is independent and oscillatory.
The parameters A, B, α, β, and yo can be selected for each spacecraft in
the cluster.
—
—
—
based upon the projection of some ground track motion.
to allow natural orbital dynamics to most uniquely sweep out aperture
baselines.
to make the array appear “rigid” from some perspective.
Consider a Pendulum in 1-G
F
Parameterize pendulum motion in terms of azimuth (θ) and elevation
(φ) angles:
φ
θ
Dynamics of a Pendulum
F
Define the Lagrangian as the difference between the kinetic and
potential energies:
[
]
1
L = T − V = m (rφś )2 + (rθś sin φ)2 − mgr 1− cos φ
2
F
[
Nonlinear dynamic equations found using Lagrange’s Equation:
d ⎛ ∂L⎞ ∂L
= 0 where q = generalized DOF
⎜ ⎟−
dt ⎝ ∂śq ⎠ ∂q
F
]
Results in the following equations
[φ]:
[θ]:
mr 2φśś − m(rθś )2 sin φcos φ + mgr sin φ = 0
m(r sin φ)2 θśś + 2mr2θś φś sin φcos φ = 0
Perturbed Pendulum Motion
F
F
Perturb motion about a nominal elevation angle and azimuthal angular
rate:
φ = φ o + δφ , θś = θśo + δθś where φ o , θś o = const
Substitute into nonlinear equations and zero higher order terms:
[]
φ :
[θ]:
F
g
δśφś − [θś 2o (cos2 φo − sin2 φo ) − cosφo ]δφ − 2θśo sin φ o cosφoδθś
r
g
2
ś
= (θo cosφ o − )sin φo
r
cosφo ś
ś
ś
ś
δθ + 2θo
δφ = 0
sin φo
Notice that forcing term zeroes about equilibrium motion:
θś 2o =
1 g
cos φo r
Comparison with Hill’s Equations
F
Two DOF Linearized Pendulum Equations:
⎡
0
⎧δ śφś⎫ ⎢
⎨ ⎬=⎢
⎩δ śθś⎭ ⎢ −2 g cos φo
⎢
r sin φ o
⎣
F
⎤
g
2 sin φ o cos φo ⎥ ⎧ ś ⎫ ⎡ g sin 2 φ o
r
⎥ ⎨δφ ⎬ + ⎢ − r cos φ
o
⎥ ⎩δθś ⎭ ⎢
⎢⎣
0
0
⎥
⎦
⎤
0⎥ ⎧δφ ⎫
⎥ ⎨δθ ⎬
⎩ ⎭
0⎥⎦
Evaluated at φo = 64o
⎧δφśś⎫ ⎡ 0
1.8n ⎤⎧δφś ⎫ ⎡ −4.2n2 0⎤ ⎧δφ ⎫
g
⎨ ⎬=⎢
cos φo
⎥⎨ ⎬ + ⎢
⎥ ⎨ ⎬ where n =
r
0⎦ ⎩δθ ⎭
0 ⎦⎩δθś ⎭ ⎣ 0
⎩δθśś⎭ ⎣−2.2n
F
Two DOF Linearized Hill’s Equations:
⎧xśś⎫ ⎡ 0 2n ⎤⎧xś⎫ ⎡3n 2
⎨ ⎬= ⎢
⎥⎨ ⎬ + ⎢
⎩yśś⎭ ⎣−2n 0 ⎦⎩yś⎭ ⎣ 0
0⎤ ⎧x ⎫
⎥⎨ ⎬
0⎦ ⎩y ⎭
General Solutions: Secular & Periodic
F
Pendulum Equations:
δφ = A cos(nρt + α)
+δφo
2A
n(ρ2 − 4)
δθ = −
sin(nρt + α) +
δφot + δθo
ρsin φo
2sin φo
g
sin 2 φ o
where n =
cos φo and ρ = 4 +
cos2 φo
r
F
Hill’s Equations:
+ xo
x = A cos( nt + α)
y = −2A sin( nt + α) − (3 / 2)nx o t + y o
Periodic Solutions
F
Pendulum Equations:
δφ = A cos( nρt + α)
2A
sin( nρt + α) + δθo
δθ = −
ρsin φ o
where n =
F
g
sin 2 φ o
cos φ o and ρ = 4 +
cos 2 φo
r
Hill’s Equations:
x=
A cos(nt + α )
y = −2A sin (nt + α )+ y o
Eigenvalues
F
Pendulum Equations:
g
sin 2 φo
where
n
=
cos φo
s = ±in 4 +
2
r
cos φo
F
Hill’s Equations:
s = ± in
where i = −1
Perturbed Motion About 63 Degree Elevation
Nominal Elevation Angle of 63 Degrees
F
Single pendulum system
—
at 63 degrees elevation, S/C
oscillates slightly less than three
cycles per revolution
1
0.8
0.6
0.4
0.2
0
1
0.5
0
-0.5
y-axis
F
Douple pendulum system
—
—
—
higher elevation S/C moves slower
and falls behind
lower elevation S/C moves faster
and moves ahead
similar to Hill’s equations
-1
-1
-0.5
0
0.5
1
x-axis
Nominal Elevation Angle of 63 Degrees
1
0.8
0.6
0.4
0.2
0
1
0.5
0
-0.5
y-axis
-1
-1
-0.5
0
x-axis
0.5
1
Perturbed Motion at Other Elevation Angles
Nominal Elevation Angle of 25 Degrees
F
Elevation angle of 25 degrees
—
number of oscillations per
revolution decreases with
decreasing nominal elevation angle
1
0.8
0.6
0.4
0.2
0
1
0.5
0
-0.5
0
-0.5
1
0.5
-1 -1
y-axis
x-axis
Nominal Elevation Angle of 45 Degrees
F
Elevation angle of 45 degrees
—
speed increases with increasing
nominal elevation angle
1
0.8
0.6
0.4
0.2
0
1
0.5
0
-0.5
y-axis
-1
-1
-0.5
0
x-axis
0.5
1
Design Parameters
r
(m)
φo
(deg)
śo
nn=
= θθÝ
0
(rad/s)
Circum
(m)
Speed
(m/s)
T
(s)
10
25
45
63
85
25
45
63
85
1.040
1.178
1.470
3.355
0.736
0.833
1.039
2.372
26.55
44.43
55.98
62.59
53.11
88.86
111.97
125.19
4.40
8.33
13.10
33.42
6.22
11.78
18.52
47.26
6.03
5.33
4.27
1.87
8.54
7.54
6.05
2.65
20