IAA-AAS-DyCoSS2-14-11-06
BOUNDED TRAJECTORIES OF A SPACECRAFT NEAR AN
EQUILIBRIUM POINT OF A BINARY ASTEROID SYSTEM
Pamela Woo* and Arun K. Misra†
With a growing interest in asteroid exploration, combined with the fact that numerous asteroids in nature occur in pairs, it is likely that future missions will include the exploration of binary asteroid systems. Thus, it is useful to study the
motion of a spacecraft in the vicinity of such systems, modelled as the threebody problem. In this paper, the circular restricted full three-body problem is
considered. The zeroth order equations of motion near an equilibrium point are
similar in form to those in the classical case with point-masses or spherical primaries. For most asteroid pairs found in practice, all five equilibrium points are
unstable. However, with selection of appropriate initial conditions, it is possible
to obtain bounded solutions to the zeroth order equations, corresponding to the
Lissajous trajectories near collinear points, and bounded trajectories near noncollinear points. Numerical simulations confirm that when including the additional perturbations due to the asphericity of the asteroid pair, the motion of the
spacecraft is unbounded. Thus, control laws are developed by utilizing an appropriate Lyapunov function, with the solutions to the zeroth order equations as
reference trajectories. These were found to be sufficient to maintain the spacecraft in bounded trajectories.
INTRODUCTION
Ongoing and upcoming dedicated asteroid missions show a growing interest in the exploration
of asteroids for scientific purposes. As a number of near-Earth asteroids occur in pairs, it is likely
that some future missions will include the exploration of binary asteroid systems. For example,
ESA’s proposed AIDA mission targets the binary near-Earth asteroid Didymos.‡ Thus, it is useful
to study the motion of a spacecraft in the vicinity of such systems, modelled as the three-body
problem.
The classical three-body problem, consisting of point-masses, has been studied extensively.
As there is a considerable amount of existing literature on the subject, a few relevant works are
mentioned here. Szebehely’s book is a comprehensive reference, which includes the dynamical
analysis of the system, the study of motion near the equilibrium points, the study of periodic orbits around the primary bodies, and numerical explorations of trajectories.1
*
Ph.D Candidate, Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal
QC, Canada, H3A 0C3.
†
Chair and Thomas Workman Professor, Department of Mechanical Engineering, McGill University, 817 Sherbrooke
Street West, Montreal QC, Canada, H3A 0C3.
‡
http://www.esa.int/Our_Activities/Preparing_for_the_Future/GSP/Asteroid_impact_mission_targets_Didymos
1
A NASA report by Farquhar studied satellite station keeping in the vicinity of the L1 and L2
equilibrium points in the Earth-Moon system.2 He considered nominal trajectories in the plane of
motion. Linear feedback control laws were developed. He found that single-axis control could
provide stability.
Farquhar and Kamel studied quasi-periodic orbits around the L2 point of the Earth-Moon system.3 Through analytical methods, they found that small-amplitude orbits describe a Lissajous
trajectory. For large-amplitude motion, they found that the in-plane and out-of-plane frequencies
are equal, producing halo orbits. Later, Howell and Pernicka developed a numerical iterative process to compute the Lissajous trajectories by patching together the trajectory segments of specified intervals into a continuous path.4
In the current study, the orbital motion of the spacecraft is modelled as the circular restricted
full three-body problem. The problem is “restricted” when the mass of the third body (the spacecraft) is relatively small, such that it does not affect the motion of the primary bodies (the asteroid
pair). The problem is “full” when the shape, size, and mass distribution of both primaries are included in the analysis. The particular case where the primary bodies move in circular mutual orbits is considered. This is the relative equilibrium configuration in a study by Bellerose and
Scheeres on the ellipsoid-sphere system.5 They determined that in one possible configuration, one
of the ellipsoid’s principal axes is aligned with the line that joins the centres of mass of the two
bodies. Woo, Misra, and Keshmiri found that for any arbitrarily shaped bodies, it is always possible to find appropriate initial conditions (on the distance separating the bodies, and their angular
velocity) to obtain circular mutual orbits.6
In their study of the restricted ellipsoid-sphere system, Bellerose and Scheeres first solved for
the two-body problem dynamics, and then substituted it into the three-body problem.7 Woo and
Misra followed a similar approach, using the two-body planar dynamics of the asteroid pair as
prescribed motions in the three-body problem.8 They developed a different formulation to take
into account the asphericity of the primary bodies, such that their model is applicable to general
arbitrarily-shaped bodies.
In the previous part to this study, the authors determined the equations describing the locations
of the equilibrium points in the circular restricted full three-body problem.8 As in the classical
case with point-masses, there are five equilibrium points. However, asphericity of the primary
bodies affects the locations of these points. In this paper, possible trajectories are determined by
considering the zero-th order equations near an equilibrium point. For a set of appropriate initial
conditions, bounded solutions to the zeroth order equations are obtained, where the trajectories
near collinear points differ from those near the noncollinear points. In the general case with arbitrarily-shaped primary bodies, numerical simulations of the full nonlinear equations of motion
show that the trajectories are unbounded. Then, control laws are developed by considering an appropriate Lyapunov function. These are sufficient to maintain bounded trajectories.
CIRCULAR RESTRICTED FULL THREE-BODY PROBLEM
Description of the System
The three-body system under consideration is illustrated in Figure 1, where bodies 1 and 2 are
the asteroid pair, having mass 1 and 2 , respectively, and body 3 is the spacecraft having mass
. Since the problem is restricted, ≪ , . The centre of mass of each primary body is lo-, -, -axes form the inercated at 1 and 2 . Point is the centroid of the asteroid pair. The tial frame. The -, -, -axes form the local rotating frame, with the origin fixed at , the -axis
oriented along the line that joins 1 and 2 . Each body has its own body-fixed -frame, with
2
the origin attached to , for = 1,2. The axes of the body-fixed frames are aligned with the respective body’s principal axes, such that the products of inertia are zero.
The planar rigid body motions of the asteroid pair can be described by four generalized coordinates: the distance between 1 and 2 ; the orientation of the -axis with respect to the
inertial frame; the orientations 1 and 2 of each body with respect to the -axis. The planar motion of the asteroid pair is found by studying the full two body problem,6 then used as prescribed
inputs in the three-body problem.
exThe orbital motion of the spacecraft is described by the position vector = ̂ + ̂ + are unit vectors along the -, -, -axes, respectively.
pressed in the local frame, where , , body 3
Y
X
Y1
C1
Ỹ
X1
O θ
α1
X2
Y2
R
RC
C2
α2
body 2
body 1
X̃
Figure 1. Schematic of the three-body system (not to scale).
The mass ratio is defined as ≡ /! + ". The shape, size, and mass distribution of the
primary bodies can be described by their mass moments of inertia, or by their radii of gyration
about their body-fixed frames: #$$1 , #%%1 , #&&1 , #$$2 , #%%2 , #&&2 . Letting #0 be the characteristic
length of the bodies, the radii of gyration of the primary bodies are nondimensionalized as
($$1 ≡ #$$1 /#0 , (%%1 ≡ #%%1 /#0 , (&&1 ≡ #&&1 /#0 , ($$2 ≡ #$$2 /#0 , (%%2 ≡ #%%2 /#0 , (&&2 ≡
#&&2 /#0 . Letting ) be the characteristic radius of the mutual orbits of the primary bodies, the corresponding mean angular motion is defined as * = +,! + "/) -, where , = 6.6738 ×
10−20 km3kg−1s−1 is the universal gravitational constant. Nondimensional time is defined as
5 ≡ *6. An additional parameter is defined such that 7 ≡ !#8 /)". Assuming that the size of the
mutual orbits is much greater than the size of the primary bodies, 7 ≪ 1.
3
The position vector of the spacecraft is nondimensionalized by dividing it by ), such that:
9!5" ≡
≡ $!5"̂ + %!5"̂ + &!5"
= ̂ + ̂ + )
)
)
)
(1)
In the case where the primary bodies are moving in circular mutual orbits, the local
≡ <; , where ′= is constant.
nondimensional $%&-frame has angular velocity :!5" = ; !5"
Note that ′= and * are not equal. The distance separating 1 and 2 is !5" = = , where = is
constant. This distance is nondimensionalized by a change of variable, such that ?= = )/= . The
primary bodies do not rotate in the local $%&-frame, hence 1 !5" = 0 and 2 !5" = 0.
!9 ;; "@AB + 2D<; E!9 ; "@AB + D<; E9 = F!9"
The vector equation describing the orbital motion of the spacecraft is:
where prime denotes differentiation with respect to 5, and
!9 ; "@AB = $ ; ̂ + % ; ̂ + & ; !9 ;; "@AB = $ ;; ̂ + % ;; ̂ + & ;; (2b)
are the derivatives of 9 in the $%&-frame. Also,
−<; 0
0
0H
0
0
;
−<
0
D<; E = D<; ED<; E = G 0
−<;
0
0
D<; E
0
= G<;
0
(2a)
(2c)
0
0H
0
(2d)
(2e)
and the entries of the vector F!9" are functions of $,%,&, with the details in Appendix A.
The locations of the in-plane equilibrium points are found by setting $ ; = $′′ = 0, % ; = % ;; =
0, & = & ; = & ;; = 0, $ = $< , % = %< in Eq. (2), to obtain the equilibrium conditions:
Equilibrium Points
D<; E9I = F!9I "
where 9I = $= + %= is the position vector of the equilibrium point.
(3)
Motion of the spacecraft near an equilibrium point can be approximated by 9!5" = 9I +
JD9K !5" + 79L !5"E, where J ≪ 1 indicates that the amplitude of motion is much smaller than the
size of the mutual circular orbits, 9K !5" is the motion of the spacecraft if the primary bodies were
both spheres, and 9L !5" are !7" perturbations to the above motion of the spacecraft due to
asphericity of the primary bodies. Substituting this approximate solution into the equations of
motion, the !J 8 " equations reduce to Eq. (3), the equilibrium conditions.
Desired Reference Trajectories
The !J 7 8 " equations are:
where
!9K;; "@AB + 2D<; E!9K; "@AB + D<; E9K = DME9K
9K !5" = $8 !5"̂ + %8 !5"̂ + &8 !5"
4
(4a)
(4b)
and
!9K; "@AB = $8; ̂ + %8; ̂ + &8; !9K;; "@AB = $8;; ̂ + %8;; ̂ + &8;; (4c)
(4d)
−OB + 3Ω@@
3Ω@A
0
DME = N
3Ω@A
−OB + 3ΩAA
0 Q
0
0
−OB
R
1−R
OB = - + #-<
#-<
1−R
R
!1 − R" S− + $< T
R S ? + $< T
?<
<
Ω@@ =
+
U
U
#-<
#-<
R
1−R
ΩAA = V U + U W %<
#-<
#-<
1−R
R
S
+ $< T !1 − R" S− + $< T
?
?
<
<
Ω@A = X
+
Y %<
U
U
#-<
#-<
#-< = Z[
1−R
+ $< \ + %<
?<
#-< = Z[−
(4e)
(4f)
(4g)
(4h)
(4i)
(4j)
R
+ $< \ + %<
?<
(4k)
Note that these !J 7 8 " equations are similar in form to those for the classical problem with
point-masses.
Lissajous Trajectories Near Collinear Points. For motion near a collinear point, the solutions
to Eq. (4) are the well-known Lissajous trajectories:
$8 !5" = cos `5 + sin `5
−` − <; + OB − 3Ω@@
!5"
D cos `5 + sin `5E
%8
=
9Ωd@A + 4<; `
&8 !5" = cos OB 5 + sin OB 5
where
−Φ + √Δ
`=Z
2
Φ = −2<; − 2OB + 3Ω@@ + 3ΩAA
(5b)
(5c)
(5d)
Δ = 8<; i2OB − 3Ω@@ − 3ΩAA j + 9iΩ@@ − ΩAA j + 36Ωd@A
= $8 !0"
5
(5a)
(5e)
(5f)
(5g)
=
9Ωd@A + 4<; `
1
!0"
k3Ω
$
−
% !0"l
@A 8
2<; `
−` − <; + OB − 3Ω@@ 8
= 3Ω@A − 2<; `
= 3Ω@A + 2<; `
= &8 !0"
&8; !0"
=
OB
$8; !0" = `
`!−` − <; + OB − 3Ω@@ "
; !0"
%8
=
i3Ω@A + 2<; `
j
d
;
9Ω@A + 4< `
(5h)
(5i)
(5j)
(5k)
(5l)
For bounded motion, the initial conditions must satisfy:
(6a)
(6b)
The details on how these solutions were obtained are summarized in Appendix B.
Trajectories Near Unstable Noncollinear Points. From literature on the classical case with
point-masses, stability of L4 depends solely on the mass ratio. In the full three-body problem, the
radii of gyration of the bodies also have an effect of the stability of L4. For most asteroid pairs
found in practice, the mass ratio and radii of gyration are such that the noncollinear point is unstable.
$8 !5" = = mno !
cos p5 + sin p5"
= mno
!5"
! cos p5 + sin p5"
%8
=
;
;
i2< q − 3Ω@A j + 4< p
&8 !5" = cos OB 5 + sin OB 5
For motion near unstable noncollinear points, the solutions to Eq. (4) are:
where
Φ + √Φ − Δ
q=Z
4
−Φ + √Φ − Δ
p=Z
4
= $8 !0"
;
;
= r2< p!q − p − < + OB − 3Ω@@ "
− 2qpi2<; q − 3Ω@A js
m
(7b)
(7c)
(7d)
(7e)
(7f)
t−r!q − p − <; + OB − 3Ω@@ "i2< q
− 3Ω@A j + 4<; qp s$8 !0" − ui2<; q − 3Ω@A j + 4<; p v %8 !0"w
= −r!q − p − <; + OB − 3Ω@@ "i2<; q − 3Ω@A j + 4<; qp s
− r2<; p!q − p − <; + OB − 3Ω@@ " − 2qpi2<; q − 3Ω@A js
= −r!q − p − <; + OB − 3Ω@@ "i2<; q − 3Ω@A j + 4<; qp s
+ r2<; p!q − p − <; + OB − 3Ω@@ " − 2qpi2<; q − 3Ω@A js
6
(7a)
(7g)
(7h)
(7i)
= &8 !0"
&8; !0"
=
OB
$8; !0" = −q
+ p
−q + p
%8; !0" =
i2<; q − 3Ω@A j + 4<; p (7j)
(7k)
For bounded motion, the initial conditions must satisfy:
(8a)
(8b)
Desired Trajectories. Let the desired reference trajectory be 9x !5" = 9I + 9K !5". Making use
of Eqs. (3) and (4), the desired trajectory satisfies:
The details are found in Appendix B.
where
!9x;; "@AB + 2D<; E!9x; "@AB + D<; E9x = F!9I " + DME9K
Numerical Simulations
!9x; "@AB = !9K; "@AB
!9x;; "@AB = !9K;; "@AB
(9a)
(9b)
(9c)
As an example, consider the system where the primary bodies are pear-shaped (modelled as a
truncated ellipsoidal cone) and an ellipsoid, as shown in Figure 2. Let the characteristic lengths be
#0 = 1 km and ) = 10 km. The -, -, -axes dimensions of body 1 are y1 = 1 km, z1 = 0.8
km, {1 = 1.2 km, |1 = 0.6 km. The dimensions of body 2 are y2 = 0.5 km, z2 = 0.4 km,
{2 = 0.3 km. Considering the case where the two bodies are uniform with a density of 2 g/cm3,
the mass ratio is calculated to be = 0.7778. The nondimensional radii of gyration are computed
as ($$1 = 0.3647, (%%1 = 0.4391, (&&1 = 0.5219, ($$2 = 0.2236, (%%2 = 0.2608, (&&2 =
0.2864.
Figure 2. Primary bodies of the example system (not to scale).
When the primary bodies are moving in circular mutual orbits, the reciprocal of the
nondimensional distance is found to be ?= = 1.0040, and the angular velocity of the system is
calculated to be ′= = 1.0079.
7
In the nondimensional $%-plane, the collinear points are located at !$~1 , %~1 " = !0.4021, 0",
!$~2 , %~2 " = !1.2640, 0", !$~3 , %~3 " = !−1.0876, 0". The noncollinear point is located at
i$~4 , %~4 j = !0.2758,0.8619".
Considering an equilibrium point located at i$= , %= j, initial conditions near this equilibrium
point are $0 !0" = 0, %0 !0" = 0.01, &0 !0" = −0.005, &′0 !0" = 0. The initial conditions $′0 !0"
and %′0 !0" are calculated from Eq. (6) for collinear points or from Eq. (8) for unstable
noncollinear points. In this example, motion around L2 and L4 are considered. The values of the
initial conditions in the nondimensional $%-plane are summarized in Table 1.
Table 1. Initial conditions for motion near L2 and L4 of the pear-ellipsoid system.
$!0"
%!0"
&!0"
$ ; !0"
% ; !0"
& ; !0"
L2
1.2640
0.01
-0.005
0.006260
0
0
L4
0.2758
0.8719
-0.005
0.01149
-0.009819
0
The position of the spacecraft as a function of 5 is obtained by numerically solving Eq. (2).
The results are plotted in Figures 3 and 4, for cases where the initial conditions are near the L2
and L4 equilibrium points. Recall that in the nondimensional frame, the distance separating the
primary bodies is near unity. It is apparent that the motions are unbounded, with the spacecraft
escaping the binary asteroid system.
x(τ)
50
0
−50
0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
τ/(2π)
y(τ)
50
0
−50
0
2
4
6
8
10
τ/(2π)
z(τ)
0.2
0.1
0
0
2
4
6
8
10
τ/(2π)
Figure 3. Position of the spacecraft, with initial conditions near L2.
8
x(τ)
50
0
−50
0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
τ/(2π)
y(τ)
50
0
−50
0
2
4
6
8
10
τ/(2π)
z(τ)
0
−0.1
−0.2
0
2
4
6
8
10
τ/(2π)
Figure 4. Position of the spacecraft, with initial conditions near L4.
NONLINEAR CONTROL
Due to the perturbations caused by the asphericity of the asteroid pair, the spacecraft does not
stay in the vicinity of the equilibrium point. The deviation from the desired trajectory, i.e., the
error vector is !5" = 9!5" − 9x !5". Some form of control is required to minimize .
Combining Eqs. (2) and (9), and adding control, the equations of motion are re-written in
terms of  as:
where
!;; "@AB + 2D<; E!; "@AB + D<; E = DME + €
!; "@AB = !9 ; "@AB − !9x; "@AB
!;; "@AB = !9′′"@AB − !9x;; "@AB
and €!5" is a control input vector.
(10a)
(10b)
(10c)
1|
|
1
!"‚ !" − ‚ D<; E
2 |5
|5
2
An appropriate Lyapunov function is chosen to be:
=
such that  is positive definite.
The derivative of  with respect to 5 is found to be:
|
= !−D<; E + DME + €"‚ i!; "@AB + D<; Ej
|5
To ensure a stable system, the function |/|5 is forced to be negative definite by letting:
9
(11)
(12)
|
‚
(13)
= i!; "@AB + D<; Ej DƒEi!; "@AB + D<; Ej
|5
for some positive definite 3 × 3 matrix DƒE. Comparing Eqs. (12) and (13) and choosing DƒE to
€ = −DƒEi!9 ; "@AB − !9x; "@AB j − D„E!9 − 9x "
be symmetric, the control input vector should have the form:
with
ƒ
0
0
0 H
DƒE = G 0 ƒ
0
0 ƒ-„ „ 0
D„E = G„ „ 0 H
0
0 „--
(14a)
(14b)
(14c)
where ƒ11 , ƒ22 , ƒ33 > 0 are gains to be tuned, and the entries of D„E are computed for each equilibrium point as:
Numerical Simulations
„ = <; − OB + 3Ω@@
„ = −ƒ <; + 3Ω@A
„ = ƒ + 3Ω@A
„ = <; − OB + 3ΩAA
„-- = −OB
(14d)
(14e)
(14f)
(14g)
(14h)
Returning to the numerical example of Section 2.4, a control input is added, where the gains
are chosen to be ƒ1 = ƒ2 = ƒ3 = 20. For the L2 and L4 equilibrium points, the values of the
gains „11 , „12 , „21 , „22 , „33 are calculated from Eq. (14) and summarized in Table 2.
Table 2. Calculated control gains for motion near L2 and L4 of the pear-ellipsoid system.
„
„
„
„
„--
L2
5.2837
-20.1573
20.1573
-1.1182
-2.1340
L4
0.7620
-19.4247
20.8899
2.2842
-1.0146
The position of the spacecraft is plotted in Figures 5 and 6, for cases where the initial conditions are near the L2 and L4 equilibrium points. The motion remains bounded. The trajectories of
the spacecraft near L2 and L4 are given in Figures 7 and 8, respectively, where the blue solid
curve is the actual trajectory, the green dotted curve is the desired reference trajectory, the red star
is the location of the equilibrium point, the blue circle indicates the initial position of the spacecraft. Results show that with nonlinear control, the spacecraft can track a reference trajectory in
the vicinity of an equilibrium point.
The control efforts are shown in Figures 9 and 10, for motion near L2 and L4, respectively.
10
1.268
x(τ)
1.266
1.264
1.262
1.26
0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
τ/(2π)
0.01
y(τ)
0.005
0
−0.005
−0.01
0
2
4
6
8
10
τ/(2π)
−3
x 10
z(τ)
5
0
−5
0
2
4
6
8
10
τ/(2π)
Figure 5. Position of the spacecraft with control, with initial conditions near L2.
0.282
x(τ)
0.28
0.278
0.276
0.274
0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
τ/(2π)
y(τ)
0.87
0.865
0.86
0
2
4
6
8
10
τ/(2π)
−3
x 10
z(τ)
5
0
−5
0
2
4
6
8
10
τ/(2π)
Figure 6. Position of the spacecraft with control, with initial conditions near L4.
11
0.01
−3
5
z
0.005
0
y
actual
desired
L4
initial
x 10
0
−5
−0.005
0.01
0
−0.01
1.26
1.262
1.264
1.266
1.268
y
x
−3
6
6
4
2
2
0
0
−2
−2
−4
−4
z
4
1.26
1.262
1.264
1.266
1.268
x
−3
x 10
−6
1.26
−0.01
1.262
1.264
1.266
x 10
−6
−0.01
1.268
−0.005
x
0
0.005
0.01
y
Figure 7. Controlled motion of the spacecraft near L2 in the nondimensional local frame.
0.872
−3
actual
desired
L4
initial
x 10
0.87
5
0.868
y
z
0.866
0
0.864
−5
0.862
0.87
0.86
0.865
0.858
0.274
0.276
0.278
0.28
0.86
0.282
x
−3
z
6
0.274
0.86
0.865
0.278
0.28
0.282
x
−3
x 10
6
4
4
2
2
0
0
−2
−2
−4
−4
−6
0.274
y
0.276
0.276
0.278
0.28
−6
0.282
x
x 10
0.87
y
Figure 8. Controlled motion of the spacecraft near L4 in the nondimensional local frame.
12
2
x−acceleration [m/s ]
−9
x 10
1
0.5
0
−0.5
−1
0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
τ/(2π)
y−acceleration [m/s2]
−10
x 10
5
0
−5
0
2
4
6
8
10
τ/(2π)
z−acceleration [m/s2]
−10
x 10
5
0
−5
0
2
4
6
8
10
τ/(2π)
Figure 9. Control effort for motion of the spacecraft near L2.
x−acceleration [m/s2]
−11
x 10
10
5
0
0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
τ/(2π)
y−acceleration [m/s2]
−11
x 10
5
0
−5
0
2
4
6
8
10
τ/(2π)
z−acceleration [m/s2]
−11
x 10
15
10
5
0
0
2
4
6
8
10
τ/(2π)
Figure 10. Control effort for motion of the spacecraft near L4.
13
CONCLUSIONS
In this paper, the orbital motion of a spacecraft in the vicinity of an unstable equilibrium point
in a binary asteroid system was studied, modelled as the circular restricted full three-body problem. First, by considering zeroth order equations near an equilibrium point, the requirements on
the initial conditions were determined to ensure bounded motion, and closed trajectories were
computed near unstable collinear and noncollinear points. As an example, a system with pearshaped and ellipsoidal primary bodies was considered. Through numerical simulations of the full
nonlinear equations of motion, the spacecraft was found to escape the binary asteroid system rather quickly. This can be attributed to the unstable nature of the equilibrium points. With the
choice of an appropriate Lyapunov function, the control input vector was computed. It was found
to be sufficient in maintaining the spacecraft in closed trajectories near the equilibrium points.
These trajectories would be useful for the observation of the asteroid bodies. These results are
also applicable to binary asteroid systems with very low eccentricity.
APPENDIX A
Consider the irregularly shaped primary bodies of the three-body system illustrated in Figure
1. In the particular case where the primary bodies move in circular mutual orbits, the equations of
motion of the third body is described by Eq. (2a), where the components of F!9" = † !9"̂ +
are:
†2 !9" + †3 !9"
†1 !9" = −R V
1
#313
+
−
3
2#513
5
#213
7 ‡3(2$$1 + (2%%1 + (2&&1
1−R
k[
?=
− ! 1 − R" V
+
−
3
2#523
5
#223
1
1−R
+ $\ (2$$1 + %2 (2%%1 + &2 (2&&1 lˆW [
+ $\
?=
2
#323
7 ‡3(2$$2 + (2%%2 + (2&&2
k[−
R
?=
2
+ $\ (2$$2 + %2 (2%%2 + &2 (2&&2 lˆW [−
14
R
?=
+ $\
(A1)
†2 !9" = −R V
1
#313
+
−
3
2#513
5
#213
7 ‡(2$$1 + 3(2%%1 + (2&&1
1−R
k[
?=
+ $\ (2$$1 + %2 (2%%1 + &2 (2&&1 lˆW %
1
− ! 1 − R" V 3
#23
+
−
3
2#523
5
#223
2
(A2)
7 ‡(2$$2 + 3(2%%2 + (2&&2
k[−
R
?=
2
+ $\ (2$$2 + %2 (2%%2 + &2 (2&&2 lˆW %
1
†- !9" = −R V #-
3
U 7 k(@@ + (AA + 3(BB
2#
5 1−R
− k[
+ $\ (@@
+ % (AA
+ & (BB
lˆW &
?<
#1
− !1 − R" V #3
+ U 7 ‡(@@
+ (AA
+ 3(BB
2#
5
R
− k[− + $\ (@@
+ % (AA
+ & (BB
lˆW &
?<
#+
where
#- = ‰S ‹ + $T + % + & ; #- = ‰S− ‹ + $T + % + & mŠ
Œ
Š
Œ
15
(A3)
(A4)
It is noted that in Eq. (4), the out-of-plane &-motion is decoupled from the in-plane motion of
the system, leading to the equation:
APPENDIX B
with solution:
&8;; + OB &8 = 0
&8 !5" = &8 !0" cos OB 5 +
where &0 !0" and &′0 !0" are initial conditions.
(B1)
&8; !0"
sin OB 5
OB
(B2)
Assuming solutions of the form $0 !5" = =#5 and %0 !5" = =#5 , for some amplitudes and
and eigenvalue #, the equations for the in-plane $%-motion can be re-written as the following
eigenvalue problem:
1
Vu
0
−<; + OB − 3Ω@@
0 −2<;
0 v# +  ;
Ž# + k
2<
0
1
−3Ω@A
0
=t w
0
−3Ω@A
;
lW t w
−< + OB − 3ΩAA
# d − i−2<; − 2ΩB + 3Ω@@ + 3ΩAA j# (B3)
The characteristic equation is:
The roots are:
where
+ !<; − OB + 3Ω@@ "i<; − OB + 3ΩAA j − 9Ωd@A = 0
# =
Φ ∓ √Δ
2
Φ = −2<; − 2OB + 3Ω@@ + 3ΩAA
(B4)
(B5a)
Δ = 8<; i2OB − 3Ω@@ − 3ΩAA j + 9iΩ@@ − ΩAA j + 36Ωd@A
(B5b)
(B5c)
In the case of collinear points, it is found that Δ > 0 and √Δ > Φ, such Eq. (B4) has two
purely imaginary roots and two purely real roots:
B.1. Motion Near Collinear Points
#, = ±Z
−Φ + √Δ
= ±`
2
Φ + √Δ
#-,d = ±Z
2
(B6a)
(B6b)
For each eigenvalue #’ , Eq. (B3) is utilized to find the corresponding amplitude ratio ’ /’ ,
for ’ = 1, … ,4.
For bounded motion, 3 = 4 = 3 = 4 = 0 is required, such that the initial conditions
must satisfy Eq. (6).
16
In the case where the noncollinear point is unstable, it is found that Δ < 0, such that the roots
of Eq. (B4) are purely complex:
B.2. Motion Near Unstable Noncollinear Points
#, = −q ± p
#-,d = q ± p
where q and p are given in Eq. (7d) and (7e).
(B7a)
(B7b)
Similarly, for each eigenvalue #’ , Eq. (B3) is utilized to find the corresponding amplitude ratio
’ /’ , for ’ = 1, … ,4.
Again, for bounded motion, 3 = 4 = 3 = 4 = 0 is required, such that the initial conditions satisfy Eq. (8).
REFERENCES
1
Szebehely, V., Theory of Orbits: The Restricted Problem of Three Bodies, Academic Press Inc.: New York, USA
(1967).
2
Farquhar, R.W., “The control and use of libration-point satellites,” Report number NASA TR R346 (1970).
3
Farquhar, R.W., Kamel, A.A., “Quasi-periodic orbits about the translunar libration point,” Celestial Mechanics 7
(1973) 458–473.
4
Howell, K.C., Pernicka, H.J., “Numerical determination of Lissajous trajectories in the restricted three-body problem,” Celestial Mechanics 41 (1988) 107–124.
5
Bellerose, J., Scheeres, D.J., “Energy and stability in the full two body problem,” Celestial Mechanics and Dynamical
Astronomy 100 (2008) 63–91.
6
Woo, P., Misra, A.K., Keshmiri, M., “On the planar motion in the full two-body problem with inertial symmetry,”
Celestial Mechanics and Dynamical Astronomy 117 (2013) 263–277.
7
Bellerose, J., Scheeres, D.J., “General dynamics in the restricted full three body problem,” Acta Astronautica 60
(2008) 563–576.
8
Woo, P., Misra, A.K., “Dynamics of a spacecraft in the vicinity of binary asteroids,” 64th International Astronautical
Congress, Beijing, China (2013) Paper number IAC-13-C1.9.3.
17