Closed-Loop Control of Close Orbits Around Asteroids
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Closed-Loop Control of Close Orbits Around Asteroids
Mauricio Guelman
∗
Technion — Israel Institute of Technology, 32000 Haifa, Israel
DOI: 10.2514/1.G000158
The purpose of this work is to develop a simple control law to perform orbit transfer about a small rotating celestial
body. The celestial body is assumed to be rotating about a principal axis, with constant rotational velocity along the
largest moment of inertia. An analysis of the spacecraft natural trajectories is performed in the body-fixed coordinate
system. A three-dimensional closed-loop guidance law is defined and analyzed, enabling the determination of the
guidance constants to assure convergence to any desired orbit about the irregular celestial body. The particular case
of synchronous orbits is further studied, enabling determination of the required guidance constants to implement
stable synchronous orbits. Numerical results are presented for representative cases.
I. Introduction
S
MALL bodies, such as asteroids and comets, have been of
increasing interest due both to their scientific importance and
possible practical applications. There is a clear interest to understand
the primal constituents and dynamic processes of the solar system.
The possible “mining” of asteroids [1] for exotic materials, as well as
the placement of transmitters on selected asteroids to provide an
independent navigational capability for interplanetary flight, were
proposed [2]. There are a number of ongoing and future missions to
these bodies. NASA is currently investigating a conceptual robotic
mission to collect a small near-Earth asteroid and transport it to an
orbit in the Earth–moon system [3,4]. A common feature of many of
these missions is a phase of close orbiting these bodies.
A substantial number of papers that examine the orbital dynamics
of satellites around asteroids have been published [5–8]. Asteroids
and comets usually have irregular shapes and this leads to the
complicated orbital dynamics in comparison with approximately
spherical bodies such as the Earth. When close to a small rotating
body, the spacecraft must contend with its irregular gravity field. To
take into account this irregular gravitational field, use is made of the
gravitational potential, dependent upon the shape of the asteroids and
the distribution of mass inside the asteroid [9,10]. For these irregular
rotating bodies, it was shown by numerical computations that orbits
exist that are quite unstable and may even crash onto the asteroid
surface in a relative short period of time.
Although there is an increasing interest in missions to asteroids and
comets, the necessity and importance of orbital control when close to
the small solar system bodies becomes a critical factor of those
missions success. A number of papers that examine the orbital control
of satellites around asteroids have been published [11–16].
The aim of this work is to develop a simple three-dimensional
closed-loop guidance control to achieve stable orbits with respect to
rotating celestial bodies. The guidance law assumes the spacecraft to
be capable of determining the position and velocity components with
respect to the central body. This guidance law is a development of a
guidance law previously employed for orbit transfer about a central
body in the equatorial plane [11].
This guidance law will be applied for the case of a rotating,
irregularly shaped body. Because the main drive of this work is orbital
control, the MacCullagh’s approximation [17] will be employed to
represent the celestial body gravitational potential due to its ana-
lytical simplicity. In this approximation, the first three terms in the
power series expansion of the gravitational potential are directly
related to the inertia properties of the attracting body.
The paper first summarizes a dynamics analysis to provide a
framework for the control system design. A control law for three-
dimensional orbit transfer is then presented and analyzed. Computer
simulations are carried out to illustrate the effectiveness of the control
law. Finally, the case of orbit transfer to synchronous orbits is studied.
II. Open Loop Trajector ies
Let us consider the case of a spacecraft S under the gravitational
influence of an irregularly shaped central body in pure rotation about
the axis of maximum moment of inertia, as shown in Fig. 1. The
gravitational force acting on the spacecraft is defined by f
∂V
∂r
,
where V is the gravitational potential of the central body.
Let A, B, and C be the moments of inertia of the central body about
axes x, y, and z, with A < B < C. The gravitational potential V of the
central body, when considering only the first three terms in the power
series expansion, is given by the MacCullagh approximation [17],
V
Gm
r
G
2r
3
A B C − 3I (1)
where G is the universal gravitational constant, m is the central body
mass,
I
1
r
2
Ax
2
By
2
Cz
2
(2)
and r x
2
y
2
z
2
1∕2
.
The gravitational force is given by
∂V
∂r
−
μ
r
3
r f
p
(3)
where μ Gm is the central body gravitational constant and f
p
is the
perturbation specific force vector defined by
f
p
−
3G
2r
5
A B C − 5Ir −
3G
r
5
Ax By Cz (4)
The perturbation components in the central body-fixed frame are
given by
f
x
−
ν
1
x
r
5
(5)
f
y
−
ν
2
y
r
5
(6)
f
z
−
ν
3
z
r
5
(7)
Received 15 July 2013; revision received 25 October 2013; accepted for
publication 10 January 2014; published online 23 April 2014. Copyright ©
2013 by the American Institute of Aeronautics and Astronautics, Inc. All
rights reserved. Copies of this paper may be made for personal or internal use,
on condition that the copier pay the $10.00 per-copy fee to the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include
the code 1533-3884/14 and $10.00 in correspondence with the CCC.
*Emeritus Professor, Faculty of Aerospace Engineering, Technion City.
854
JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS
Vol. 38, No. 5, May 2015
Downloaded by NANJING UNIV OF AERONAUTICS on May 26, 2015 | http://arc.aiaa.org | DOI: 10.2514/1.G000158
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