Aerospace and Electronic Systems Magazine July 2018 - 64

Integrated Attitude-Orbit Dynamics and Control of Spacecraft Systems
n

σ =  aibiT ri
i =1

n

(13)

B =  ai bi riT

(14)

S = B + BT

(15)

i =1

n

z =  ai bi × ri

(16)

S − σ I3 z 
k=
T
σ 
 z

(17)

i =1

where I3 is a third-order identity matrix. Now, the solution becomes
qˆ  arg max qT Kq and qT q  1, q   q

(18)

KALMAN FILTER CONCEPTS
The Kalman filter has not only been applied in aerospace, but
it has also been implemented in nuclear power plant instrumentation, demographic modeling, manufacturing, and computer
vision applications. The main advantages of the Kalman filter
are its ability to provide the quality of the estimate (i.e., the
variance), its relatively low complexity, and its real-time execution without storing observations or past estimates. On the other
hand, its main disadvantage is that it provides accurate results
only for Gaussian noise and linear models. Thermal vibrations
of atoms in conductors, poor illumination, and transmission
such as electronic circuit noise generate measurement noises.
However, Gaussian noise is not able to guarantee a realistic environment behavior, but by referring to the Central Limit Theorem of probability theory, the Gaussian noise is an appropriate assumption [35]. For Gaussian noise models with limited
nonlinearity, EKF is appropriate [36]. For non-Gaussian noise
and nonlinear models, particle filtering is the most appropriate
approach, since it is able to provide arbitrarily posterior probability distribution [31].

q

This problem is the same as the eigenvalue decomposition
problem. One important point in q-method is that if the scalar part
of the corresponding quaternion vector is negative, the sign of the
quaternion vector should be changed.
Unlike the TRIAD method, the q-method is able to utilize an
arbitrary number of measurement pairs. Moreover, the q-method
determines the attitude optimally. However, since the q-method applies eigenvalue decomposition, it is, therefore, highly demanding
in terms of computation compared with the TRIAD method.

ATTITUDE ESTIMATION CONCEPTS
Spacecraft attitude estimation method consists of two main processes: (1) orientation estimation of the spacecraft from sensor
measurements and known reference frames and (2) filtering the
noises of the measurements. The filtering of the noises is achieved
by combining the models and measurements, which can be done
by using different ways [3], [29]. The Kalman filter and its derivatives, such as extended Kalman filter (EKF), are well-known realtime attitude estimators of spacecraft. However, recently several
alternatives for the Kalman filter are introduced, such as particle
filters and unscented filters [30], [31]. The quaternion estimator
(QUEST) algorithm, published in 1981, and was a solution to the
Wahba problem. The QUEST algorithm has the advantage of being able to utilize more than two measurement sets to provide an
optimal estimation. Since then, numerous other algorithms have
been proposed [32]. The QUEST technique uses the point-by-point
solution of the spacecraft attitude [33]. A more thorough review of
the estimation techniques can be found in the book by Simon [34].
The next section presents a brief review of the Kalman filter technique and shows that its methods and applications are developing
in the spacecraft industry.
64

INTEGRATED ORBIT AND ATTITUDE DYNAMICS AND
CONTROL
Since precise (near-actual) mathematical models of the spacecraft
motion enhance the control system performance, therefore, integrated attitude and orbit dynamics is one of the most intriguing
topics in the field of aerospace.
Large Space Structures (LSS) are defined as bodies with considerable extension [37]. Space agencies are rapidly developing
LSS for most advanced missions such as deep-space human habitats, astronomical observatories or staging infrastructures. In these
structures, the effects of the gravity gradient over the mass distribution must be included in the dynamics formulation. Therefore,
the gravitational force must be computed including the effect of a
distributed mass and, this leads the orbital and attitude dynamics
to mutually interact [38]. Similarly, for these structures, the area
of the exposed surfaces to the solar radiation pressure is greater
than other cases. Therefore, the solar radiation pressure should be
included in the model as well. Developing the capability to control and estimate the attitude motion in more complex dynamical
environments benefits to improve the safety of docking to space
stations, pointing accuracy of the telescopes or reconstruction of
the orientation history of captured asteroids [39].
The gravitational parameter, the orbital angular velocity, and
orbital radius are the main factors in the coupling of attitude and
orbit dynamics. For a spacecraft with a high moment of inertia,
due to strong nonlinear disturbances, integrated attitude and orbit
dynamics model is essential to design an accurate orbital trajectory [40]. Moreover, these coupling terms are also more apparent in spacecraft that have constraints on their thrust direction.
There are a few studies in the field of integrated attitude and orbit
dynamics, and most of them are conducted by Pan, Wong, and
Kapila [41], [42].

IEEE A&E SYSTEMS MAGAZINE

JULY 2018



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