Dynamics of Satellite Formation Utilizing the Perturbed Restricted Three-Body Problem ()
1. Introduction
Parallel to the start of the space programs, the study of spacecraft relative dynamics became one of the most important aspects in designing and analyzing space missions. Several authors were interested in this study to analyze rendezvous and docking of two spacecrafts in addition to maintenance of spacecraft formation. The most famous models of relative motion, Clohessy-Wiltshire and Tschauner-Hempel, have been used to analyze relative guidance, navigation and control systems. However, these two models assumed that the spacecraft relative distance is significantly small compared with its position w.r.t the centre of mass of the primary. Moreover, they assume pure Kepler motion (e.g. two-body problem without including any perturbations) [1] [2]. Afterwards, different models have been constructed to overcome the limitations of these two models [3] - [8].
In 2019, Giovanni Franzini and Mario Innocenti studied the relative motion dynamics using the classical restricted three-body problem (unperturbed problem). They found that a three-body scenario is more suitable than a two-body scenario to describe the dynamics of satellite relative motion and more efficient for studying the relative guidance and navigation system [9]. Recently, many authors are interested in modelling the formation dynamics employing the restricted three-body problem and studied the relative motion around the libration points [10] [11] [12] [13] [14].
The current study aims to get a more accurate formulation of the relative motion employing the perturbed restricted three-body problem. The dynamical model assumes that the primaries are radiating and oblate spheroids considering only zonal harmonics. Based on constructing recurrence formulas and applying the Lie series approach, the model is solved numerically. Finally, the numerical application is performed assuming a circular three-dimensional problem.
2. Formulation of the Problem
2.1. The Perturbed Restricted Three-Body Problem
The restricted three-body problem is one of the most famous dynamical modellings of celestial systems. Extensive studies were performed using a variety of methods assuming that the primaries are spheres and only their gravitational attraction is considered [15] - [20]. For a more accurate model, additional perturbing forces such as oblateness and radiation pressure are considered. The gravitational potential “
” of a massive body is given by [21] [22] [23]:
(1)
G is the universal gravitational constant,
is the mean radius of each body,
is the latitude of the infinitesimal body,
is the separation between every two bodies and
is the dimensionless coefficient which represents the non-spherical components of the potential. The Legendre polynomials “
” of degree n is given by:
The gravitational force “
” exerted by the body is:
(2)
where
denotes the vector differential operator. Apart from the gravitational potential, if the body is radiating, then it exerts a radiation force “
” in the opposite direction of its gravitational force. Consequently, the total force “
” is:
(3)
where
is the radiation factor
. Substitute (1) and (2) into (3):
(4)
where
Consider a system of three bodies that have masses
,
and
such that the masses of primaries
and
is the mass of the chief satellite in a formation. Let
is a sidereal (inertial) coordinate system with the origin lies in the centre of mass of the two primaries, the motion of each body is given by:
where
,
and
and
. For the restricted case of three bodies, the primaries are not affected by the gravitational influence of the satellite. Assuming that the formation is revolving around the smaller primary, the equation of motion of the chief satellite is (see Figure 1):
Figure 1. Restricted three-body problem in an inertial frame.
As clear in Figure 1,
is the relative position of the chief w.r.t. the smaller primary and
, then
(5)
Equation (5) can be normalized by assuming that the gravitational constant
, the distance between the two primaries is unity and the sum of their masses is unity. Let the normalized mass of the small primary parameter be
, then that of the big primary is “
”. Consequently, the chief equation of motion will be:
(6)
Similarly, the motion of a deputy satellite of the formation will be:
(7)
Let
is a synodic coordinate system with an origin that lies in the centre of mass of the smaller primary which is defined as follows:
,
,
where
is the angular momentum of the smaller primary w.r.t. the bigprimary. Let this frame rotates with angular velocity
w.r.t the inertial frame
. Then, the equation of motion of the chief w.r.t. smaller primary is:
(8)
Similarly, the motion of the deputy satellite will be:
(9)
where
2.2. The Relative Motion in Perturbed Restricted Three Body Problem
To describe the relative motion of the deputy w.r.t. the chief, let
is a Local Vertical Local Horizontal frame “LVLH” with an origin that lies in the chief centre of mass and is defined as (As is clear in Figure 2):
,
,
(10)
where
is the chief angular momentum w.r.t. the second primary. The position of the deputy w.r.t. the smaller primary is given by (see Figure 2):
(11)
Let
rotates with angular velocity
w.r.t. the inertial frame
, then
(12)
Introducing (6) and (7) into (12), then
Figure 2. Local vertical local horizontal frame w.r.t. synodic frame.
(13)
where
The angular velocity of “
” w.r.t “I” is given by:
(14)
where
and
are angular velocities of
w.r.t.
and
w.r.t. I respectively. To determine
and
, a simple scheme based on the time derivatives of LVLH w.r.t. the synodic frame “
” [9] [24]:
,
,
(15)
Multiply (15) by the relative unit vector as follows:
By summing up the previous equations we get
(16)
Considering (10), the time derivative of the unit vectors of the LVLH frame is:
(17)
Substitute from (17), into (16), then:
and
(18)
where
and the jerk
can be obtained by Equation (8) and its differentiation. Equation (13) along with (14) and (18) represent the equation of relative motion of the deputy w.r.t. the chief in the frame of the perturbed restricted three-body problem. It noted that the equation of relative motion is a nonlinear 2nd order differential equation with time-varying parameters which can be simplified assuming the circular case of the restricted three-body problem.
3. The Relative Motion in the Circular Restricted Three-Body Problem
Assuming that the two primaries revolve in a circular orbit around their common centre of mass, then the following simplifications will be considered [10]:
,
,
,
,
Consequently, the angular velocity and acceleration of the LVLH frame w.r.t. the inertial frame is simplified as follows:
For more simplifications, assume that both primaries are radiating and only the second zonal harmonic is considered. Then
Let
Under these assumptions, the equation of relative motion (13) will be reduced to:
(19)
where
and
are the components of the angular velocity
and
.
4. Solution Algorithm
Power series approaches are widely used to solve different celestial mechanics problems. Many authors depend on that algorithm to find an approximate solution for their problems [25] [26]. The Lie-integration method is one of the most famous power series algorithms that can be applied to find both displacement and velocity components of the deputy satellite. The method is outlined in the following three steps [16] [17] [26]:
Step I: Construction of the Lie operator
Let
Then
In general, the components of the equation of relative motion can be rewritten in a matrix form as:
where
,
and
are 6 × 1 matrices defined as follows:
and
With
. Based on (18),
, then
However,
is defined as:
where is 6 × 6 matrix given by
The Lie-Operator is defined as [16]:
For explicit time variables, then
For
, assuming that
,
and
then
and
(20)
Step II: Construction of the recurrence relations for each variable
Applying the Lie operator Equation (20) on
, we obtain that
1) The recurrence formulas for
Generally,
(21)
a) The recurrence formulas for
The higher powers of
and
are computed as
and
(22)
The recurrence relation of the rest of the elements of
is zero.
b) The recurrence formulas for
By definition of
, its noted that
then
and
then
then
then
(23)
Then
(24)
Then
(25)
The recurrence relation of the rest of the elements of
is zero.
2) The recurrence formulas for
(26)
The higher powers of
and
are computed as
,
and
(27)
The recurrence relation of the rest of the elements of
is zero.
Step III: Find the Lie-series solution
The solution is given by:
Then
(28)
5. Numerical Application
Consider a circular restricted three-body problem in the Earth-Moon system where the period of the Moon about the Earth is 27.23 days. The parameters of the primaries, the properties of the chief orbit and the initial condition of the deputy in the LVLH frame are tabulated in Table 1 and Table 2.
The calculations have been made using 10−2 step size and five terms calculations over 24 hours. To assess the motion of the deputy employing the concepts of the restricted three-body problem, the solution algorithm is applied for both the unperturbed (classical) and the perturbed cases. A set of curves represent the relative motion of the deputy satellite w.r.t. the chief in both cases.
As is clear in Figures 3-5, in all panes of motion there is a significant difference in the components of the deputy position between the classical and perturbed cases. Consequently, the magnitude of its relative position vector is changed as is clear in Figure 6 where the solid curve represents the perturbed motion and the dotted curve represents the unperturbed motion and their difference is represented in Figure 7.
Table 1. The parameters of the primaries.
Table 2. Initial Conditions of the chief and deputy satellites.
Figure 3. Motion of the deputy satellite w.r.t. the chief in x-y plane for both the unperturbed and perturbed cases of the three-body problem.
Figure 4. Motion of the deputy satellite w.r.t. the chief in x-z plane for both the unperturbed and perturbed cases of the three-body problem.
Figure 5. Motion of the deputy satellite w.r.t. the chief in z-y plane for both the unperturbed and perturbed cases of the three-body problem.
Figure 6. Comparison between the deputy relative motion for the unperturbed and perturbed cases of the three-body problem.
Figure 7. Relative distance difference between the unperturbed and perturbed cases.
6. Conclusion
The motion of the deputy satellite w.r.t. the chief of the formation is modelled in frame of the perturbed restricted three-body problem using the LVLH frame. The model is a more accurate formulation compared with the previous work where it considers the effects of radiation of both primaries in addition to their oblateness second zonal harmonic. The system is simplified assuming the circular problem and solved numerically using the Lie series approach. The solution is tested using suitable initial conditions and applied for both the classical and perturbed restricted three-body problems. By comparing the two cases, the results show that there is a significant difference in the deputy relative distance. Consequently, the model will be suitable for a more accurate study of the different space mission issues (e.g. satellite rendezvous and control).