1. Introduction
Most of the asteroids exist between Mars and Jupiter
orbits called the main-belt, while some of the asteroids
approach and cross the Earth orbit. They are called potentially
hazardous asteroids (PHAs), which have possibility
of impacting the Earth. Many scientists have
studied the PHA orbit propagation and quantified the
hazards of the Earth impact1) .
Several strategies indicate that, in order to prevent
the asteroid collision with the Earth, deflection of the
PHAs by kinetic energy is more effective than fragmentation
of the asteroid body itself2) . Park and the coauthors3)?
5) investigated the optimal magnitude and direction
of velocity increment required for deflecting the
asteroids, but the interplanetary trajectory of the spacecraft
has not been discussed. Ivashkin6) showed that a
spacecraft with eight tons at launch could deflect the
asteroid Toutatis, whose radius is less than 270 m. He
also presented some models for spacecraft impact on the
asteroid, which has a large difference in required momentum
for deflecting the asteroids. Other papers7)?9)
studied the asteroid deflection missions using electric
propulsion or solar sail technology. Izzo10) studied the
asteroid deflection mission for 99942 Apophis that approaches
the Earth in 2029, which states that the asteroid
deflection should be performed at least three years
before the collision to the Earth using a multiple gravity
assist.
A mission, named Don Quijote, proposed by ESA
will investigate the effect of the impulsive asteroid deflection
by using two spacecraft; one will impact on the
asteroid and the other will investigate the asteroid to determine
the collision point and then observe the change
in the asteroid orbit by rendezvousing with the asteroid11)
. The achievements of this mission will be utilized
for future asteroid deflection missions.
The asteroid deflection mission requires orbital parameters,
mass, shapes and the surface construction.
However, it is difficult to determine exact values due
to limitation of measurement accuracy. Therefore, the
asteroid deflection mission should consider uncertainties
of the parameters. Vasile12) studied the optimal trajectory
for asteroid deflection, including the uncertainty of
orbital elements and mass of the asteroids. However, he
didnft consider the uncertainties of the velocity increment
that a spacecraft gives to the asteroid.
This paper investigates the optimal interplanetary
trajectory for deflecting PHAs under the uncertainty of
the velocity increment. The uncertainty of velocity increment
is modeled as a convex model13) under the assumption
that the uncertainties of the magnitude and
the direction of the velocity increment are independently
varied. The optimal trajectory is designed by maximizing
the worst (minimum) value of the Earth closest approach
distance in the uncertain range of the velocity
increment. The effect of the velocity increment uncertainty
is investigated through numerical examples.
Chapter 2 describes the outline of the asteroid deflection
mission treated in the present study. Chapter 3
defines the orbit design problem in consideration with
the uncertainty of velocity increment. Chapter 4 describes
some examples of the optimal trajectories with
and without the velocity increment uncertainty. Comparing
these examples, the importance of considering
the uncertainty of velocity increment is demonstrated.
2. Outline of Asteroid Deflection Mission
The spacecraft of this mission is assumed to follow
a ballistic interplanetary trajectory after leaving the
Earth, and to intercept an asteroid. Then, the spacecraft
gives velocity increment V to the asteroid by perfectly
inelastic impact model, where the mass of spacecraft
m is assumed to be much smaller than that of
the asteroid M (m M) and negligible. The mission
constrains the upper limit to C3, the square root
of escape velocity at the Earth departure, due to the
launcher performance. The spacecraft and the asteroid
orbits are calculated by the patched-conic approximation
in this study, assuming an unperturbed elliptic orbit
around the Sun for the interplanetary trajectory of
the spacecraft and asteroid, and hyperbolic orbit around
the Earth in the vicinity of the Earth for the asteroid.
Collision
point
Sun
Asteroid orbit
Earth
orbit
Earth
Asteroid
2. S/C Asteroid impact
S/C orbit
1. S/C Earth
departure
Original
orbit
New
orbit
3.
4.
DV
New
Original orbit
orbit
S/C
Figure 1. Outline of asteroid deflection mission
Figure 1 shows the strategy of asteroid deflection mission,
which can be detailed as follows.
1. An asteroid is assumed to hit the center of the
Earth.
2. The spacecraft heliocentric orbit is determined from
the Earth departure time tdep and asteroid arrival
time tarr using the Lambert method. C3 can be
calculated from the Earth orbital velocity Vear and
spacecraft Earth departure velocity Vdep as follows:
C3 = |Vdep ? Vear|2 (1)
If multiple-revolution trajectories are obtained, the
trajectory with the smallest value of C3 is adopted.
3. The spacecraft impacts on the asteroid with the
asteroid arrival velocity Varr. Then the velocity
increment V added to the asteroid can be approximated
as follows:
V = m
M + m ?
m
M
Varr (2)
4. The velocity increment V makes the asteroid orbit
changed, and hence the asteroid orbital period
is increased as shown in Figure 1.
5. The asteroid flies by the Earth delayed from the
original orbit, and the collision with the Earth is
avoided. Figure 2 shows the hyperbolic trajectory
of the asteroid approaching near the Earth, and
d
periapsis
V\
Asymptote
parallel
Hyperbolic orbit
of asteroid
Earth
d\
Figure 2. Hyperbolic trajectories of the asteroid in avoiding
the collision to the Earth
the Earth closest approach distance d can be obtained
from the following equation.
d = e
|V|
(
1 + |V|4d2
2e
? 1
)
(3)
where e is a gravity constant of the Earth, V
is the asteroid relative velocity with respect to the
Earth, and d is the distance from the Earth center
to the asteroid incoming asymptote.
3. Orbit Design Problem under Uncertainties
3.1. Uncertainty model of velocity increment
It is difficult to estimate the velocity increment that the
spacecraft gives to the PHA at the time of collision exactly
because shape, surface material property and mass
distribution of the PHA are not known exactly. The velocity
increment is considered to be varied from the case
of perfectly inelastic collision, that is, the magnitude
and the direction are considered to have uncertainties.
In this research, the quantity of the magnitude and the
direction change of the velocity increment are modeled
through independent uncertainty parameters.
The uncertainty parameter of the velocity increment
magnitude is denoted as that represents the ratio between
the effective momentum and the momentum in
the case of perfectly inelastic collision. The range of the
magnitude uncertainty parameter is determined from
the average and the standard deviation of V magnitude
(ʃ, Ѓ).
The uncertainty parameters of the direction change
of the velocity increment are defined as d and d that
describe the changes of vertical and parallel component
of the velocity increment direction as shown in Figure 3.
The velocity increment under the uncertainty, V , is
Xsc
Ysc
Zsc
q
dq
df
DV
DV'
|DV'|=a|DV|
g
Figure 3. V uncertainty model
then described as follows:
V = |V |
?
??
cos d cos( + d)
cos d sin( + d)
sin d
?
??
(4)
where is the angle between Xsc axis of the spacecraftcenter
reference frame and V as shown in Figure 3.
Considering the impact of the spacecraft with the asteroid,
uncertainty parameters of V direction d and
d are dependent on each other. It is natural to assume
V spreading as a conic. Therefore, the following relationship
is assumed:
d2 + d2 ? 2 (5)
where is the upper limit of the uncertainty range that
is determined from the standard deviation Ѓ. Therefore,
the uncertainty range of the velocity increment
(, d, d) is modeled by a convex hull as shown in
Figure 4. The effect of considering uncertainty of the
velocity increment is evaluated by the worst value d corresponding
to the minimum closest approach distance
between the Earth and the asteroid. The worst value
for the prescribed interplanetary trajectory determined
from the Earth departure time and the asteroid arrival
time is evaluated by the following optimization problem:
Minimize : d = d (, d, d) (6)
subject to : ʃ ? 3Ѓ ? ? ʃ + 3Ѓ
d2 + d2 ? 2
If d is represented as a convex function in term of the
uncertainty parameters, the minimum value of d lies
along the boundary of the uncertainty convex hull. That
is called a convex model13) that the minimum value is
easier to find. In this study, the worst value is determined
analytically without searching in the convex
hull14) .
3.2. Optimal trajectory design problem under
uncertainty
In the present spacecraft trajectory optimization, the
Earth departure time tdep and asteroid arrival time tarr
d
d
m
m+3Ѓ
m-3Ѓ
3Ѓ
Uncertainty convex of
velocity increments
Figure 4. Convex model of uncertainty of the velocity
increment
are chosen as design variables in order to maximize the
worst value of the Earth closest approach distance d
under C3 constraint, that imposes an upper limit C3u
for the mission feasibility. Then, the optimal trajectory
problem is formulated as follows:
Maximize : d (tdep, tarr) (7)
subject to : C3 ? C3u
tl
dep ? tdep ? tu
dep
tl
arr ? tarr ? tu
arr
where, tl
dep, tu
dep, tl
arr and tu
arr are the lower and upper
limits of the Earth departure date and the asteroid arrival
date, respectively. Considering the problems (6),
the optimal trajectory problem with uncertainty is formulated
as max-min problem.
4. Results and Discussions
4.1. Mission Conditions
The asteroid is assumed to impact the Earth on January
1, 2060 at the ascending node. The asteroid shape is
assumed to be a perfect sphere with 150 m diameter
and 5.3 Mton mass, whose average density is 3.0 g/cm3.
This orbit is determined referring to 3361 Orpheus that
has average orbital parameters for PHAs. The orbital
elements of the fictitious asteroid for the present study
and Orpheus are listed in Table 1. The upper limit of
C3 is determined as 50 km2/s2 corresponding to the
capability of the average of HII-A and M-V launchers.
The mass of spacecraft is assumed to be 1.0 ton when
it impacts with the asteroid. The average and standard
deviation of uncertainty parameters are listed in Table 2.
4.2. Optimal trajectories without uncertainty
At first, the optimal spacecraft trajectory to maximize
the closest approach distance without uncertainty of
V (asteroid velocity increment at spacecraft impact)
Table 1. Orbital elements of asteroids
Fictitious asteroid Orpheus
a [AU] 1.283 1.209
e 0.3226 0.3226
i [deg] 2.683 2.683
[deg] 100.4 189.7
[deg] 301.6 301.6
0 [deg] 58.42 346.0
Epoch 1/1/2060 9/13/2000
Table 2. Assumptions of fictitious asteroid, spacecraft,
V
Earth collision date of the asteroid 1/1/2060
Asteroid diameter [m] 150
Asteroid density [g/cm3] 3.00
Asteroid mass [ton] 5.30 ~106
Spacecraft mass [ton] 1.00
C3 limit at Earth [km2/s2] 50.0
Average of V magnitude ʃ 0.60
Standard deviation of V 0.10
magnitude Ѓ
Average of V direction ʃ [deg] 0.00
Standard deviation of V 10.0
direction Ѓ [deg]
is obtained to investigate the effect of the spacecraft arrival
time at the asteroid on the asteroid deflection.
We treat the gnominal V h case where the uncertain
parameter is set to (, ) = (ʃ, 0). Under
this deterministic condition, the closest approach distance
is known to change periodically with respect to
the asteroid arrival date of the spacecraft, and that local
maximum value of the approach distance is reduced
as the asteroid arrival date is delayed.3)?5) The closest
approach distance is evaluated assuming the range
of the Earth departure date [1/1/2008, 1/1/2012] and
the asteroid arrival date [1/1/2010, 1/1/2014], where
the flight time is smaller than six years. The closest
approach distance distribution is illustrated in Figure 5,
and the asteroid periapsis dates in the range, 6/22/2010,
12/6/2011 and 5/19/2013 are indicated by dashed lines.
It is found that the local maximum values of the closest
approach distance are taken at these asteroid periapsis
dates. This result is identical to Park and coauthorsf
research.3)?5)The global maximum closest approach distance
22492.06 km is obtained at (tdep, tarr) = (7/16/2010,
12/6/2011) as a deterministic optimum design. The optimum
trajectory is illustrated in Figure 6.
Earth departure date
Asteroid arrival date
[km]
20000
15000
10000
5000
01/01/2008 01/01/2009 01/01/2010 01/01/2011 01/01/2012
01/01/2010
01/01/2011
01/01/2012
01/01/2013
01/01/2014
06/22/2010
12/06/2011
05/19/2013
Optimal point
Figure 5. Distribution of closest approach distance
-1.5
-1.0
-0.5
0.0
0.5
1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
Z [AU]
X [AU]
Y [AU]
S/C
Asteroid arrival
Earth departure (12/6/2011)
(7/16/2010)
Collition point
Fictitious
asteroid
Earth
Figure 6. Optimal trajectory for nominal V case
4.3. Optimal trajectories under uncertainty
Optimal interplanetary trajectories are now designed
considering the velocity increment uncertainties (gworst
V h case), and the effects of the uncertainties of the
velocity increment on the closest approach distance are
investigated. From the deterministic results illustrated
in section 4.2, the range of the Earth departure date and
the asteroid date are selected as [1/1/2008, 3/30/2008]
and [6/1/2010, 8/29/2010].
Figures 7 and 8 illustrate the distribution of closest
approach distances with and without uncertainties. The
cross points (sign g+h) in the figures indicate the optimal
Earth departure date and asteroid arrival date of
the spacecraft. Figure 9 illustrates the optimal trajectories
corresponding to the cross points with and without
uncertainties. These figures indicate that the distribution
of the closest approach distance with uncertainty is
different from that without uncertainty.
The effect of the uncertainty of the V direction is
discussed here. The components of the asteroid velocity
direction on the spacecraft impact with and without
uncertainties are described as follows.
P Vast = |V | cos (8)
V ast = |V | cos( + 3Ѓ) (9)
Earth departure date
Asteroid arrival date
[km]
21000
20000
19000
18000
17000
16000
15000
14000
13000
12000
01/01/2008 01/31/2008 03/01/2008
06/01/2010
07/01/2010
08/01/2010
03/30/2008
08/29/2010
21000
20000
19000
18000
17000
16000
15000
14000
13000
12000
21000
20000
Figure 7. Closest approach distance distribution for nominal
condition
Earth departure date
Asteroid arrival date
[km]
6500
6000
5500
5000
4500
4000
3500
3000
2500
2000
01/01/2008 01/31/2008 03/01/2008
06/01/2010
07/01/2010
08/01/2010
03/30/2008
08/29/2010
6500
6000
5500
5000
4500
4000
3500
3000
2500
6500
6000
Figure 8. Closest approach distance distribution for
worst condition
where P Vast is the component of the asteroid velocity
direction of the nominal V (deterministic) case, V ast
is that of the worst V (uncertain) case, and is an
angle between the spacecraft velocity and the asteroid
velocity vectors. If the |V | takes a constant value, the
ratio between P Vast and V ast indicates the uncertainty
effect of V direction. Hence, the effects of the uncertainty
to the V and the closest approach distance can
be described to .
To investigate the effect of the optimal trajectories
due to the standard deviation of V direction uncertainty
on the optimal trajectories, the optimal departure
date and arrival date are calculated in the range
of Ѓ = [0, 10] deg. The optimal departure dates and
arrival dates of corresponding uncertainty range, Ѓ =
[0, 10] deg is illustrated in Figure 10 that also shows
the distribution of an angle between the spacecraft and
the asteroid velocities, . It is found that the optimal
departure and arrival dates are shifted to the smaller
angle of , as the standard deviation of the velocity
increment direction is larger. Figure 11 illustrates the
ratio of the closest approach distance with and without
uncertainties. This distribution is similar to that of
-1.5
-1.0
-0.5
0.0
0.5
1.0 -1.5
-1.0
-0.5
0.0
0.5
1.0
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
Z [AU]
X [AU]
Y [AU]
Asteroid arrival
(S/C worst opt)
Earth departure
(S/C worst opt)
S/C
Asteroid arrival
(S/C nominal opt)
Earth departure
(S/C nominal opt)
Figure 9. Optimal trajectories under V uncertainty
Earth departure date
Asteroid arrival date
[deg]
20
18
16
14
12
10
8
6
4
2
01/01/2008 01/31/2008 03/01/2008
06/01/2010
07/01/2010
08/01/2010
03/30/2008
08/29/2010
20
18
14
12
10
8
6
4
2
10
8
6
4
14
12
10
2
nominal
worst sg=10
Figure 10. distribution
in comparison with Figure 10. Therefore, the effect
of V uncertainty decreases, as the angle between the
spacecraft velocity and the asteroid velocity vector decreases.
That is, the optimum trajectory under V uncertainty
is selected as that gives a small angle between
the asteroid and spacecraft velocities at the collision.
5. Conclusion
This paper describes the optimal interplanetary trajectories
for deflecting asteroids, including the uncertainty
of velocity increment of the asteroid due to the
spacecraft impact into account. The uncertainty of velocity
increment is modeled using a convex model, where
its magnitude and direction are independently varied.
Comparing the optimal trajectories between those with
and without the velocity increment uncertainties, the
importance of including the uncertainty for the asteroid
deflection mission design is demonstrated. It is found
that the optimal trajectories of the spacecraft are sensitive
to the uncertainty of the velocity increment direction.
The effect of velocity increment direction uncertainty
depends on the angle between the velocity vector
between the spacecraft and the asteroid.
Earth departure date
Asteroid arrival date
0.32
0.31
0.30
0.28
0.26
0.24
0.22
0.20
0.18
0.16
0.14
01/01/2008 01/31/2008 03/01/2008 03/30/2008
06/01/2010
07/01/2010
08/01/2010
08/29/2010
0.32
0.31
0.30
0.28
0.26
0.24
0.22
0.20
0.18
0.16
0.31
0.30
0.28
Figure 11. Distribution of closest approach distance ratio
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