Aerospace and Electronic Systems Magazine July 2018 - 62

Integrated Attitude-Orbit Dynamics and Control of Spacecraft Systems

ORBIT DETERMINATION CONCEPTS
The general method for orbit determination systems is to generate a dynamic model of the orbit using available observations to
improve orbit parameters by the development of differential corrections [16]. Orbit propagation, orbit prediction, and definitive
orbit determination are such of these applications. The orbital state
vector and attitude state vector can be computed onboard, on the
ground, or both. Figure 2 shows all basic areas that are applied in
practice to solve the orbit determination problem.
The differential correction process is applied in calculating
the differences between the measurements and the predicted position from the estimator. The differences are used to compute
a set of corrections to the starting state vector. This method is
used to minimize the differences. The solution is iterated until
the convergence is achieved. The models of the forces that act
on the spacecraft's body are described in the dynamical model.
The forces include both conservative forces, such as gravitational
forces, as well as nonconservative forces, such as aerodynamic
drag. Generally, these models are defined in terms of analytical, semianalytical, or numerical formulas and techniques. The
computational approaches in both batch estimate and sequential
estimate to process the observation data are called statistical estimation technique [16].
Batch methods were used in all early statistical orbit estimation techniques wherewith all observations were used in estimation solution [17]. Generally, in the batch theory, when data arc
decreases, observation errors become more dominant, and for long
data arc lengths, the dominant error sources are dynamic model
errors. Later, the batch technique was slowly replaced with sequential estimation methods, whereby new state estimates were derived
with the addition of each new observation. Kalman filter presents
a sequential methodology that has been used effectively in several on-orbit orbit determination missions with good success [18].
Moreover, hybrid techniques have been used where both batch and
sequential methods are blended.
Orbit determination problem is defined by the equation of motion with respect to an inertial reference frame to represent the
propagation of the motion spacecraft orbit [16]. The equation is
expressed generally as

GM 

r = − 3 r + fG + f NG
r

General orbit determination problem diagram.

techniques. Historically, celestial bodies orbit determination has
attracted some of the best astronomers, physicists, mathematicians,
and aerospace engineers for a solution. After the launch of Sputnik
1, the need for spacecraft orbit determination became crucial.
The development of analytical orbit theory began in the late
1950s and early 1960s with the work of Brower [19] and Kozai
[20]. Brower developed his theory to second-order terms by applying mean orbital elements, and the theory was precise only
for nearly circular orbits or near-equatorial orbits. Kozai's theory
was a first-order method using Lagrange's planetary equations,
and it was easier to understand. Later, Kaula [21], by using
instantaneous orbital elements, conducted an orbital theory in
Keplerian orbital element space. This method allowed handling
third body, resonance, and solid and ocean tidal disturbance effects easily.

ATTITUDE DETERMINATION CONCEPTS
(1)

where G = 6.669 × 10−11m3/kg.s2 is the universal constant of gravi
tation, M = 5.972 × 1024kg is the mass of the Earth, r is spacecraft
state vector, fG is all noncentral gravitational forces, and fNG is all
nongravitational forces acting on the spacecraft. Noncentral gravitational forces include temporal gravity effects due to Earth tides,
general relativity, aerodynamic drag, solar and Earth radiation
pressure, spacecraft thermal radiation, empirically derived forces,
and chemical thrusters torques. In fact, the estimate error will occur due to a combination of effects, such as mathematical errors
in the equation of motion, mathematical errors in the state observation models, random and/or systematic errors in the measurements, and numerical errors in the calculations in the estimation
62

Figure 2.

The total spacecraft angular momentum vector consists of angular momentum of the spacecraft's body and angular momentum of
control actuators [1], [5].The general spacecraft-dynamic model is
given by the following expression
T = hi = h + ω × h and h = hb + hw

(2)

where T is total external torque, h is the angular momentum of the
spacecraft, hb is the angular momentum of the spacecraft body, hw
is the angular momentum of reaction wheels, and ω is spacecraft
body angular velocity.
Attitude determination algorithm refers to the procedure for
gaining an appropriate rotation matrix [22]. By such a rotation ma-

IEEE A&E SYSTEMS MAGAZINE

JULY 2018

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine July 2018