Computational Intelligence - November 2013 - 42

represented by X i0 = [x i0, y i0, z i0, xo i0, yo i0, zo i0] T . As shown in Fig. 3,
the core of satellite formation reconfiguration based on twoimpulse control is to choose the impulse time vector t i and the
input values u i for each satellite i, i = 1, 2, ..., n, satisfying collision avoidance constraints and minimizing the overall fuel cost.
In the problem formulation, t i = [t 1i , t 2i ], 0 # t 1i < t 2i # T,
represents the two impulse time instants of the i th satellite, T
is the periodic time of the desired flying-around orbit, and
the cor responding input value is represented by
u i = [U 1i , U 2i ], where U ij, j = 1, 2, is the input vector of the
ith satellite at time t ij . Assume that the initial state of the ith
satellite in the desired formation configuration is denoted by
X id = [x id, y id, z id, xo id, yo id, zo id] T , which is on a thrust-free orbit
and is designed based on various space missions. To regulate
satellite i from the initial state X i0 to the desired state X id, the
final configuration constraint should hold [39]

L

Figure 3 Formation reconfiguration of satellites.

U (T - t 2i ) {U (t 2i - t 1i ) [U (t 1i ) X i0 + BU 1i ] + BU 2i } = X id , (6)

Eq. (3), which is often utilized to keep the formation configuration without additional control input once it is formed [4]
Z
2yo
yo
] x (t) = xo 0 sin (~t) + c - 3x 0 - 0 m cos (~t) + 2 c 2x 0 + 0 m
~
~
~
]
2yo 0
o0
]
2
x
] y (t) = 2 c 3x 0 + ~ m sin (~t) + ~ cos (~t)
(3)
[
o
]
- 3 (2~x 0 + yo 0) t + c y 0 - 2x 0 m
]
~
]
o0
z
] z (t) = n sin (~t) + z 0 cos (~t)
\
where X 0 = [x 0, y 0, z 0, xo 0, yo 0, zo 0] T represents the initial state of
relative motion at t 0 . The two commonly used formations are
shown in Fig. 2.
theorem 1 [39]
The trajectory (See Eq. (3)) of the relative motion is a closed
ellipse, whose geometric center is (0, 0, 0), if the following initial condition holds
yo 0 =-2~x 0
'
y 0 = 2xo 0 /~

(4)

theorem 2 [39]
The trajectory (See Eq. (3)) of the relative motion is a closed
circle, with its geometric center O = (0, 0, 0) and its radius
r = 2 x 20 + (xo 0 /~) 2 , if the initial condition satisfies
Z y 0 = 2xo 0 /~
]]
yo 0 =-2~x 0
[ 2
2
]] z 0 = 3x 0
2
2
\ zo 0 = 3xo 0

where the state transition-matrix U (t) of system (see Eq. (2))
without control is obtained from matrix A and is expressed by
f11 f12
E
U ( t) = ;
f21 f22
4 - 3 cos ~t 0 0
f11 = >6 (- ~t + sin ~t) 1 0 H,
0
0 cos ~t
0
2 (1 - cos ~t)
sin ~t
f12 = 1 >2 (- 1 + cos ~t) - 3~t + 4 sin ~t 0 H,
~
0
0
sin ~t
0
0
3 sin ~t
0 H,
f21 = >6 (- 1 + cos ~t) 1
0 - sin ~t
0
cos ~t
2 sin ~t
0
0 H.
f22 = >2 sin ~t - 3 + 4 cos ~t
0
0
cos ~t
Given the fact that X id is on the thrust-free orbit with the
periodic time T , we have
U (T) X id = X id .

(7)

Combine Eqs.(6) and (7) and the constraint is finally written as:
U (t 2i ) X i0 + U (t 2i - t 1i ) BU i1 + BU i2 = 0 .

(8)

Therefore, the control inputs are calculated from Eq. (8) for
each t i = [t 1i , t 2i ], 0 # t 1i < t 2i # T [39]
(5)

)

U 1i = - f -121 (t i2 - t i1) (f11 (t i2) Dr0 + f12 (t i2) Dv 0)
U i2 = - (f21 (t i2) Dr0 + f22 (t i2) Dv 0 + f22 (t i2 - t i1) U i1)

(9)

B. Two-Impulse Control Based Formation Reconfiguration

Formation reconfiguration is the mapping from an old formation to a new formation, which specifies an optimal satellite
arrangement for the current space mission. Consider there are n
satellites in the formation and the initial state of the i th satellite is

42

IEEE ComputatIonal IntEllIgEnCE magazInE | noVEmBER 2013

As a critical requirement of satellite reconfiguration maneuver
involving multiple satellites, collision avoidance constraints are
considered in this paper, where every pair of satellites has to be
at least a specified distance apart from each other during the



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