In this section, the state variables used to specify the 6-DoF motion of a rigid body will be detailed. The translational motion description and the rotational one are separated. The notions introduced in this section will be used in Chapter 4 in order to derive the equations of 6-DoF dynamics for a set of satellites.
2.3.1 Translational Motion
The translational motion of a point in 3-dimensional space is completely defined by six independent parameters. In this work, two different types of parameters are used:
• Inertial position and velocity vectors (rI and vI)
Referring to Figure 2.1, these six parameters (three for the position and three for the velocity) describe, at a given instant, where the satellite is situated and where it will be after an infinitesimal time interval dt with respect to ECIJ200o frame. These elements are very useful for using classical dynamics equation.
• Keplerian elements
This common representation can be visualized in Figure. 2.3. As the previous elements, they define completely an unperturbed orbit followed by a satellite but they allow a simpler description of an orbiting body. The six elements O are usually denoted by the following letters O = (a, e, i, Ω, ω, ν) where
– a is the semi-major axis. It represents the mean between the longest ra and small rp distances of the motion of the satellite about the celestial body, a = ra+r2 p
– e is the eccentricity of the orbit. For elliptical motion it is 0 ≤ e < 1, where 0 denotes circular orbits and 1 parabolic ones
– i is the inclination of the orbit, the angle between the plane of the orbit and the versor ˆZI
– Ω, also called RAAN, is the angle between the plane of the orbit and the versor XˆI
– ω is the argument of perigee and is the angle in the plane of the orbit between the intersection of the plane of the orbit with the equator and the direction of the perigee with respect to the ECIJ2000 frame (denoted as xp in Figure. 2.3)
the vector xp and the position of the satellite
Figure 2.3: Keplerian elements, [2]
We can underline the simplicity of the description of the motion of a satellite when only the central gravity force acts on it. In fact, in this situation the true anomaly ν is the only parameter which is function of time.
2.3.2 Rotational Motion
Rotational motion of a rigid body with respect to an inertial frame, as the trans-lational one, is completely described by a set of six independent parameters. They usually are three angles and the three components of the angular velocity of the body frame with respect to the inertial one. Concerning the angular velocity vec-tor, no additional clarification is required with respect to what has already been stated in the previous sections. Instead, concerning the the other three parameters, a different choice from those stated before has been done in this work. In fact, the description of the attitude of the body through the quaternion representation has been employed. Following the same notation of [16], it is possible to define a
quaternion as a 4-dimensional vector
q =
q1 q2 q3 q4
=
"
esin(θ2) cos(θ2)
#
= q1ˆi + q2j + qˆ 3k + qˆ 4
where esin(θ2) = q1ˆi + q2ˆj + q3kˆ is also called the vector part and cos(θ2) = q4 the scalar one. This comes from the fact that a quaternion simply defines a rotation of intensity given by the value of θ in the direction specified by the vector part e. For completeness, we report the main results of quaternion algebra:
• Norm
The norm of a quaternion is defined as follors
kqk = q
q12+ q22+ q23+ q24
In this work, only unitary quaternions will be considered, i.e. those quater-nions which have the property
kqk = 1
• Conjugate
The conjugate of a quaternion is defined as
q∗ =
−q1
−q2
−q3 q4
• Inverse
The inverse of a quaternion is defined as
q−1= q∗ kqk2
• Multiplication
The multiplication between quaternions is usually denoted by the symbol ⊗
and it is defined by the following
qi⊗ qj =
"
sivj+ sjvi+ vi× vj
sisj− vi· vj
#
where si and sj are the scalar parts, while vi and vj are the vector ones.
• Transformation Matrix
If the quaternion q describes the relative rotation of the frame Fb with respect to Fa, it is possibile to retrieve the rotation matrix from Fa to Fb as
Rab =
q12− q22− q32+ q42 2 (q1q2+ q3q4) 2 (q1q3− q2q4) 2 (q1q2− q3q4) −q12+ q22− q32+ q42 2 (q2q3+ q1q4) 2 (q1q3+ q2q4) 2 (q2q3− q1q4) −q21− q22+ q32+ q42
(2.6)
• Quaternion rate
The time derivative of the quaternion is linked to the angular velocity of the body through the kinematics equation given by the following
˙q = 1
2W (ω)q (2.7)
where
W (ω) =
0 ω3 −ω2 ω1
−ω3 0 ω1 ω2 ω2 −ω1 0 ω3
−ω1 −ω2 −ω3 0
(2.8)
The Equation 2.7 will be used in Chapter when the rotational dynamics equa-tions will be derived.
High-Fidelity Orbital Mechanics Propagation Model
Cooperating satellite formations often require high accuracy in relative dynamics management. In many missions involving this type of architecture, the satellites have to work as a large coordinated system in order to take good data minimizing errors due to non-correct pointing of one or more of the satellites. For these reason an accurate modeling of the GN&C system has to be based on a very accurate model of orbit propagation. The model must account for the main perturbations affecting the orbit in which the formation will be flying. For relative dynamics accelerations in the order of 10−5 ms2 have to be properly described because proved to be of significant importance in contributing to the drift of relative orbits. For our purposes a model of orbit propagation accounting for gravity potential, drag, solar pressure and third-body perturbations, has been created and validated with the General Mission Analysis Tool (GMAT) of NASA [17] (see Appendix A). Next pages will describe what force modeling has been chosen for the orbit propagation model in order to better clarify which is the grade of accuracy of the simulations reported in the next chapters.