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Near earth objects pose a serious threat to earth. Scientists have proposed several meth-
ods to disrupt or de
ect asteroids. Many ideas were innovated by scientists to de
ect or
destroy asteroids.. The latter is often preferred because disrupting an asteroid increases
the risk of producing new impactors. In this paper gravity tractor concept is studied and
trajectories are designed to reduce the risk of impact of asteroids. This paper proposes an
approach to optimal de
ection of an asteroid with a gravity-tractor spacecraft. The approach
can be split into two parts. First the optimal de
ection trajectory of the asteroid is com-
puted, regardless of the spacecraft dynamics. In the second step through optimal control,
the spacecraft trajectory that would eventually optimally de
ect the asteroid is estimated.
Finally the approach is applied to the de
ection of asteroid 2007VK184, one of the current
asteroids. This approach is both high delity and robust.

INTRODUCTION
1.1 Overview
NEAR-EARTH objects (NEOs) have attracted a lot of attention from astronomers in
recent years. The possibility of an NEO impacting Earth, with dramatic consequences, mo-
tivates scientists and engineers to devise original and innovative ideas to disrupt or de
ect
asteroids. The latter option is often preferred. Aside from political issues arising from de-
ploying nuclear devices for asteroid fragmentation, disrupting an asteroid increases the risk
of producing new impactors, particularly when the NEO is a porous rubble-pile asteroid.
With sucient warning time, de
ection techniques are thus very attractive to avert an im-
pact[1].
The gravity tractor was introduced recently by Lu and Love as a possible means of
asteroid de
ection and asteroid impact mitigation. During close proximity of a spacecraft to
an asteroid, the mutual gravitational acceleration perturbs both objects' dynamics. When
the spacecraft can thrust in a direction that balances the gravitational force exerted by the
asteroid on the spacecraft, the asteroid is accelerated and deviates slightly from its original
path, This can eventually perturb the asteroid orbit, and in the long run will result in ii
signicant de
ection, preventing a potential impact on Earth. One of the main advantages
of this thrusting de
ection technique is that it is relatively independent of the physical prop-
erties of the asteroid, such as its composition, although the shape and gravity held might
be more relevant .The gravity-tractor concept has been the focus of any studies since it was
introduced a few years ago. Most of the studies, however, make major assumptions about
the de
ection strategy and only focus on simple hovering control.
Wie studies the gravity-tractor hovering dynamics. Some of his work also focuses
on the dynamics of solar sails as a gravity tractor. The concept is actually quite attractive,
as the spacecraft in this case has very limited thrust requirement, although maintaining a
non-Keplerian displaced orbit or distant halo-type orbit, with or without solar sail for an
autonomous vehicle, is not straightforward.Fashnestock and Scheeres study the control re-
quirement for the coupled rotational and translational dynamics, in addition to the system
specications[2]. With a rigorous model, this paper points out the diculty of the con-
trol problem.Broschart and Scheeres study the hovering dynamics in me case of scientic
observation of a small asteroid. Although the authors compute the equilibrium condition,
the dynamic in
uence of the sun is neglected. Indeed, during inertial hovering for asteroid
surface mapping, the time of operation is too small for the sun to actually have an in
uence
on the asteroid dynamics. Some work also proposes a method to compute the hovering dy-
namics in the case of non-spherical asteroids.
In this paper[3], entire mission concept is studied from the start of the de
ection
to the end. In particular it is shown that the current literature assumptions limit the
eciency of gravity tractor concept and may lead to non-realizable missions.

Chapter 2
Asteroids
In control theory, sliding mode control, or SMC, is a form of variable structure control
(VSC). It is a nonlinear control method that alters the dynamics of a nonlinear system by
application of a high-frequency switching control. The state-feedback control law is not a
continuous function of time. Instead, it switches from one ies is calledcontinuous structure
to another based on the current position in the state space. Hence, sliding mode control
is a variable structure control method[4]. The multiple control structures are designed so
that trajectories always move toward a switching condition, and so the ultimate trajectory
will not exist entirely within one control structure. Instead, the ultimate trajectory will
slide along the boundaries of the control structures. The motion of the system as it slides
along these boundar a sliding mode and the geometrical locus consisting of the boundaries
is called the sliding (hyper)surface. Asteroids are a class of small Solar System bodies in
orbit around the Sun. The term asteroid was historically applied to any astronomical object
orbiting the Sun that was not observed to have the characteristics of an active comet or
a planet, but it has increasingly come to particularly refer to the small rocky and metallic
bodies of the inner Solar System and out to the orbit of Jupiter. As small objects in the
outer Solar System have begun to be discovered their observed composition diers from the
objects historically termed asteroids. Harbouring predominantly volatiles-based material
similar to comets rather than the more familiar rocky or metallic asteroids, they are often
distinguished from them.
Like most other small Solar System bodies the asteroids are thought to be remnants of plan-
etesimals, material within the young Sun's solar nebula that have not grown large enough to
form planets. The large majority of known asteroids orbit in the main asteroid belt between
the orbits of Mars and Jupiter, however many dierent orbital families exist with signicant
populations including Jupiter Trojans and near-Earth asteroids. Individual asteroids are
categorized by their characteristic spectra, with the majority falling into three main groups:
C-type, S-type, and M-type. These are generally identied with carbon-rich, stony, and
metallic compositions respectively.