In the following section, we define the geometry of the system being considered.
In
Section 3, the nonlinear equations of motion are presented, and in Section 4, the equations
are linearized.
2. System Description
The two-spacecraft array being considered is depicted in Figure 2.1.
The
X
,
Y
,
Z
coordi-
nate frame represents a global, non-rotating frame whose origin lies at the center of mass
of the two-spacecraft array.
The first spacecraft, denoted as “spacecraft A,” lies at coordi-
nates
r
,
φ
,
ψ
. Since the global frame’s origin coincides with the array’s center of mass, and
we are considering the two spacecraft to be identical in mass and geometry, the second
spacecraft, denoted as “spacecraft B,” lies at coordinates
r
,
φ
+
π
,
ψ
(or equivalently
r
,
φ
,
ψ
+
π
).
r
Z
Y
φ
ψ
e
ˆ
r
e
ˆ
φ
e
ˆ
ψ
e
ˆ
x
e
ˆ
y
e
ˆ
z
X
Spacecraft A
Spacecraft B
Figure 2.1
Geometry of Two-Spacecraft Array
While the
X
,
Y
,
Z
frame represents a global frame, the
r
,
φ
,
ψ
frame represents a local
frame whose origin lies at the center of mass of spacecraft A. The
r
,
φ
,
ψ
frame is not
fixed to the body in that it does not rotate or “tilt” with the spacecraft.
Notice the
e
ˆ
r
vec-
System Description
2