derived from the synodic period of the two objects, while a Relative Flower ... The Flower Constellations is a recently proposed methodology to design a set of ...
AAS 05-151
Synodic and Relative Flower Constellations with Applications to Planetary Explorations ∗ Daniele Mortari† , Ossama Abdelkhalik‡ , and Christian Bruccoleri§ Texas A&M University, College Station, Texas 77843-3141 Abstract This paper introduces the theory of the Synodic Flower Constellations and of the Relative Flower Constellations, which constitute two novel and alternative methodologies to design Flower Constellations synchronized with the motion of two celestial objects (e.g., two planets) orbiting about the same gravitational mass. These two “objects” can also be natural or artificial satellites (e.g., moons, spacecraft) orbiting about a planet and one of these two objects can also be the central body itself. In particular, a Synodic Flower Constellation is made with orbits that are compatible with a reference frame rotating with a period suitably derived from the synodic period of the two objects, while a Relative Flower Constellation is made of orbits that are, simultaneously, compatible with both the objects rotating reference frames. The latter, however, can be achieved under a very particular condition that is here approximated. The resulting dynamics of these constellations are synchronized with the dynamics of the geometrical positions of the two objects. Potential applications can be to design a Space Network Architecture for planetary communications, to design Solar Global Navigation System, to design small constellations for Surveillance and reconnaissance and well as for space and Earth science.
Introduction John E. Draim in the Breakwell Memorial Lecture given in the 55th International Astronautical Congress specified that “the field of constellation design is still in its infancy, and areas for improvement are almost without limits” [1]. This statement was particularly true when the theory of the Flower Constellations [2] has been introduced. In general, the constellation design is a difficult problem because each orbit has an infinity of choices for six parameters and so for many satellites, the problem is of exceedingly high dimensionality. With the introduction of the Flower Constellations theory, the constellation itself is seen as new object characterized by an axis of symmetry about which the constellation is rotating with prescribed angular velocity. This has been achieved using orbits that are compatible with respect to an assigned rotating reference frame. In particular, when this rotating reference frame is a planet fixed reference frame, then all the satellites follows the same space track in that frame. The beautiful dynamics are obtained by introducing an automatic mechanism, ruled by a set of integer parameters, to distribute the satellites into “admissible locations” which constitute the phasing rule of the Flower Constellations theory. In this way, this new methodology to design satellite constellations has greatly simplified the constellation design problem and, thus, has provided the means to solve an extremely difficult family of problems. ∗ Paper AAS 05-151 of the 15th AAS/AIAA Space Flight Mechanics Meeting, January 23–27, 2005, Copper Mountain, Colorado
1
The Flower Constellations The Flower Constellations is a recently proposed methodology to design a set of satellite constellations which are generally characterized by repeatable space tracks through a proper phasing mechanism. A Flower Constellation, which can be complete or restricted, is presently identified by nine parameters. Five are integer parameters: the number of petals (Np ), the number of sidereal days to repeat the ground track (Nd ), the number of satellites (Ns ), and three integers to rule the phasing (Fn , Fd , and Fh ), and three are the orbit parameters which equals for all satellites if the axis of the constellation coincide with the rotation axis of the planet reference frame. These orbit parameters are the argument of perigee (ω), the orbit inclination (i), and the perigee altitude (hp ). Following the very first presentation [2], some studies and research have been done to understand and to design Flower Constellations [3, 4, 5, 6, 7]. The Flower Constellations present beautiful and interesting dynamical features (see http://flowerconstellations.tamu.edu/) that allow us to explore a wide range of potential applications which include: telecommunications, Earth and deep space observation, global positioning systems [5, 8], and new kind of formation flying schemes. A Flower Constellation can be re-oriented arbitrarily; however, in this case the repeating space track property is associated with a rotating and oriented reference frame. Associated with the theory, the ad-hoc Flower Constellation Visualization and Analysis Tool (FCVAT) software [9] has been developed¶ . The FCVAT not only speed up the design and optimization of the Flower Constellations from weeks to just minutes, but also the easy visibility of heretofore unknown constellations also enables entirely new mission concepts. FCVAT allows one to quickly come up with highly complex satellite formations that no one even knew existed. Among the existing finalized application of the Flower Constellations, there is the Global Navigation Flower Constellation [5, 8] (GNFC). Since a Flower Constellation has the satellites following the same relative trajectory with respect to an Earth rotating reference frame, this has allowed us to design an approximate uniform relative trajectory over a specific geographical region of interest (which can be global) and uniformly distribute (in time) the satellites along the relative trajectory. Based on this idea, the GNFC has been proposed and compared with the existing GPS and GLONASS constellations [5] as well as with the proposed European GalileoSat constellation [8]. GNFC presents better characteristics in terms of attitude and position errors (ADOP and GDOP) and/or presents the same performance using lesser satellites.
The Synodic Flower Constellations Even though the theory is general, let us introduce the Synodic Flower Constellations for the specific and important example associated with the Earth/Mars systems. Let n1 and n2 be the mean motions of two distinct planets orbiting the Sun (as the Earth and Mars), then Tsyn =
2π | n1 − n2 |
(1)
indicates the synodic period associated with the two planets. This means that, starting with an aligned planet configuration (relative mean anomaly equal to zero) then, after a synodic period, the relative mean anomaly becomes 2π. During the synodic period, the two planets will have the exceeding mean anomaly º ¹ º ¹ n2 n1 2π = n2 Tsyn − 2π (2) ∆α = n1 Tsyn − | n1 − n2 | | n1 − n2 | where bxc indicates the greatest integer lower than x. ¶ Disclosure of FCVAT Software, by Daniele Mortari, Matthew P. Wilkins, and Christian Bruccoleri, signed with the Technology Licensing Office, TAMU 3369, 707 Texas Ave, College Station, TX 77843-3369 in October 15, 2003
2
Mercury 88
Venus 224.7
Earth 365.2
Mars 687
Jupiter 4,331
Saturn 10,747
Uranus 30,589
Neptune 59,800
Pluto 90,588
Table 1: Planets’ Orbital Periods (days) Using the values for the planets orbital periods given in Table 1, the ∆α values associated with any Planet pair are given in Table 2. Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto
Mercury 0.00 231.75 114.29 52.89 7.47 2.97 1.04 0.53 0.35
Venus 231.75 0.00 215.74 174.98 19.70 7.69 2.66 1.36 0.89
Earth 114.29 215.74 0.00 48.55 33.15 12.66 4.35 2.21 1.46
Mars 52.89 174.98 48.55 0.00 67.87 24.58 8.27 4.18 2.75
Jupiter 7.47 19.70 33.15 67.87 0.00 243.01 59.38 28.11 18.076
Saturn 2.97 7.69 12.66 24.58 243.01 0.00 194.99 78.87 48.46
Uranus 1.04 2.66 4.35 8.27 59.38 194.99 0.00 16.98 183.54
Neptune 0.53 1.36 2.21 4.18 28.11 78.87 16.98 0.00 339.23
Pluto 0.35 0.89 1.46 2.75 18.076 48.46 183.54 339.23 0.00
Table 2: ∆α Associated with Planets Pairs (degrees) Let us to introduce a set of reference frames that rotate with the angular velocities defined as ∆α + 2π ω=
Tsyn
Kn Kd
µ¹
º n1 − µ¹ | n1 − n2 | º n2 = n2 − − | n1 − n2 |
= n1 −
¶ Kn | n1 − n2 | = Kd ¶ Kn | n1 − n2 | Kd
(3)
where Kn and Kd can be any integer numbers (even negative), and Kd 6= 0. The reference frames rotating with the angular velocity defined by Eq. (3) are synchronized with the synodic alignment of the two planets. In particular, if Kn = 0, then ω represents the angular velocity of a frame in which the alignment of the two planets occur always at the same angle with respect to the first reference axis. In the general case, the alignment occur after Kd synodic periods during which the reference frame has performed Kn complete rotations (2π) plus the exceeding angle Kd ∆α. Consequently, any orbit compatible with a reference frame rotating at the angular velocity defined by Eq. (3) is synchronized with the relative motion of the two planets. Now, in order to build a constellations that is synchronized with the two planets, we do need to have the spacecraft orbits be distributed accordingly with the displacements of the subsequent planets alignment directions. In other words, if we choose to have the first orbit with the apsidal line aligned with the first planets alignment, then the apsidal line for the second orbit must be located where the planets experience the second (or the third, or the fourth, etc.) alignment, and so on. It is also possible to locate the second satellite half way back from the second alignment (in this way the third satellite will be located at the second alignment) or at one third of the angular distance from the second alignment, and so on. A such satellites orientation distribution implies, therefore, an associated consistent choice of the satellites’ right ascension of the ascending nodes. One of these consistent choices can be to adopt a the sequence Ωk+1 = Ωk + ∆α where Fn and Fd can be any positive integer numbers. 3
Fn Fd
(4)
A Synodic Flower Constellation is a Flower Constellation made with orbits that are compatible with the rotating reference frames defined by the angular velocities given in Eq. (3) and with orbits apsidal lines distributed accordingly with Eq. (4). An orbit is compatible with respect to a reference frame rotating at the angular velocity ω if its orbital period T satisfies the relationship Tr = Np T = Nd Tω = Nd
2π ω
(5)
where Tr is the repetition period, and Np and Nd are two integer numbers representing the Number of Petals of the SFC and the number of revolutions of the rotating frame within the repetition period, respectively. Equation (5) allows us to evaluate the mean motion n of the compatible orbits belonging to a Synodic Flower Constellation ½ µ¹ º ¶ ¾ 2π Np Np Kn n1 n= = ω= n1 − − | n1 − n2 | = T Nd Nd ½ µ¹ | n1 − n2 | º Kd ¶ ¾ (6) Np n2 Kn = n2 − − | n1 − n2 | Nd | n1 − n2 | Kd Once the orbit period has been quantified, then the orbits apsidal lines are distributed accordingly with Eq. (4) and the satellites are allocated into the “admissible positions” defined by the theory of the Flower Constellations [2]. By considering all the planets orbits be lying on the ecliptic plane, then the “admissible positions” are defined just by evaluating the mean anomaly while the value of the argument of perihelium (that must be identical for all the orbits) constitutes an additional constellation design parameter. 8
SFC with Earth and
x 10
8
Mars; Time = 0.0 days
x 10
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−3
−2
−1
0
1
2
3
−3
SFC with Earth and
−2
−1
Mars; Time = 389.8266 days
0
1
2
8
3 8
x 10
x 10
Figure 1: First Alignment
Figure 2: First Anti-Alignment
In the SFC all the spacecraft follow the same identical relative trajectory in the rotating reference frame and this trajectory rotates in a such a way to tracks the alignment of the two planets. This implies that at every planets alignment the relative trajectory will appear symmetric with respect to the alignment axis. This symmetry is kept also when the planets are anti-aligned. This property becomes very important when designing satellite constellations to serve communications between two planets and can play also a key role in designing GPS-like navigation systems to serve the whole solar system (global) or a sub region of it. In addition, other applications can be identify if the two planets are replaced by two Earth orbiting artificial satellite, or with one satellite and the Moon, or with the Earth and one satellite, etc. 4
8
x 10
SFC with Earth and
8
Mars; Time = 779.6532 days
x 10
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−3
−2
−1
0
1
2
3
−3
SFC with Earth and
−2
−1
Mars; Time = 1169.4798 days
0
1
2
8
x 10
Figure 3: Second Alignment 8
x 10
SFC with Earth and
Figure 4: Second Anti-Alignment 8
Mars; Time = 1559.3064 days
x 10
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−3
−2
−1
0
1
3 8
x 10
2
3
−3
SFC with Earth and
−2
−1
Mars; Time = 1949.133 days
0
1
2
8
3 8
x 10
x 10
Figure 5: Third Alignment
Figure 6: Third Anti-Alignment
Figures 1 through 8 describe the positions of Earth and Mars and a 3 satellite Synodic Flower Constellation. These images are taken at four subsequent alignments (Figs. 1, 3, 5, and 7) and anti-alignments (Figs. 2, 4, 6, and 8) of the two planets. Note the symmetry of the alignment/antialignment axes with respect to the relative trajectory which is common to all three satellites. The simulation parameters producing these figures are: Kn = 1, Kd = 2, Ns = 3, Fn = 3, Fd = 1, Np = 3, Nd = 1, and the eccentricity was chosen to be e = 0.5. This simulation, which has been provided just to describe the dynamics of a Synodic Flower Constellation, is far from being a suitable constellation to accomplish the communication task with Mars. Sub-optimal constellations can be found using genetic algorithm and, for the most interesting solutions, the optimality can be obtained analytically. These further developments are already planned as future researches.
The Relative Flower Constellations This section presents a different approach to design constellations synchronized with the motion of two rotating reference frames as, for instance, with the reference frames rotating at the mean 5
8
x 10
SFC with Earth and
8
Mars; Time = 2338.9596 days
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−3
−2
−1
0
1
2
SFC with Earth and
x 10
3
−3
−2
−1
Mars; Time = 2728.7862 days
0
1
2
8
3 8
x 10
x 10
Figure 7: Fourth Alignment
Figure 8: Fourth Anti-Alignment
motion angular velocities of two planets. The essential idea is to find out which are the conditions for a orbit in order to be compatible with two, distinct, rotating reference frames. To this end, let us consider the two rotating frames having T1 and T2 orbital periods, respectively. The conditions for a third rotating frame to be simultaneously compatible (thus, synchronized) with the first two rotating frames is mathematically described by the two conditions Np1 T = Nd1 T1 Np2 T = Nd2 T2
(7) (8)
where Np1 , Np2 , Nd1 , and Nd2 are all integers, and T represents the synchronized rotating period of the third rotating frame, are simultaneously satisfied. Equations (7) and (8) allow us to derive the necessary condition for the third rotating frame to be simultaneously synchronous with the first two rotating frames. This condition implies that the period ratio T1 Np1 Nd2 = (9) T2 Np2 Nd1 must be rational. By multiplying the first term of Eq. (7) with the first term of Eq. (8), and re-arranging, we obtain the expression for the searched orbit period T 2 = T1 T2
Nd1 Nd2 Np1 Np2
(10)
as the geometrical mean of the two periods provided using Eq. (7) and Eq. (8). Let us looking at this problem from a different perspective. The repetition times of the conditions given in Eqs. (7) and (8) are Tr1 = Np1 T = Nd1 T1
(11)
Tr2 = Np2 T = Nd2 T2
(12)
respectively. This suggests us that the double synchronization can be introduced with the same logic, by finding two integers, K1 and K2 , to satisfy Tr = K1 Tr1 = K2 Tr2 6
(13)
where Tr is the repetition time of the three rotating frames. Unfortunately, the identity (13) yields no additional information. In fact, by substituting Eqs. (11) and (12) into Eq. (13) we obtain K1 Np1 T = K1 Nd1 T1 = K2 Np2 T = K2 Nd2 T2
(14)
By equating the first and third terms, and the second and fourth terms, we obtain K1 Np2 Nd2 T2 = = K2 Np1 Nd1 T1
(15)
from which we derive again the condition given by Eq. (9). As it can easily be proved, by equating the first and fourth terms, and the second and third terms, we obtain K1 Nd2 T2 Np2 T = = K2 Np1 T Nd1 T1
(16)
from which we derive again Eq. (10). This proves that by looking at this problem as done by Eq. (11) through Eq. (14), we add no information. Using the theory of the Flower Constellations, an orbit satisfying Eq. (7) allows us to build a relative trajectory with Np1 petals with respect to the reference frame rotating with an angular 2π velocity equal to the n1 = mean motion. The key parameter to distribute the satellites such T1 Np1 that we obtain this effect is ruled by the ratio. Similarly, an orbit satisfying Eq. (8) allows Nd1 us to build a relative trajectory with Np2 petals with respect to the reference frame rotating with 2π an angular velocity equal to the n2 = mean motion if we distribute the satellites such using the T2 Np2 ration . Nd2
Approximative Solutions for Earth-Planet Systems In general, when T1 and T2 are assigned (e.g. planet orbit periods) or when they are randomly chosen as in simulations, the ratio given by Eq. (9) is not rational. However, an approximate solution can Np1 Nd2 T1 be found by finding the Np1 , Np2 , Nd1 , and Nd2 integers such that approximates . Np2 Nd1 T2 Let us describe how to find out the approximations and the associated issues. The approximated solution is obtained by finding two integers, N and D, such that T1 N Np1 Nd2 ≈ = T2 D Np2 Nd1
(17)
which implies the error | T1 /T2 − N/D |. Now, in order to find the searched four integer parameters, the factors of N and D must be evaluated. Let us show how to proceed by explaining the simplest case when N and D are both primes. In this case we have just the four possibilities given in Table 3. Associated with each one possibility we have three estimations of the orbital period s N N Nd1 Nd2 d1 d2 T¯(1) = T1 , T¯(2) = T2 , and T¯(3) = T1 T2 Np1 Np2 Np1 Np2 which, in theory, should be equals. Therefore, the relative percentage error ¯ 100 ¯¯ ¯ δ = ¯(3) ¯ T¯(1) − T¯(2) ¯ T 7
(18)
(19)
# 1 2 3 4
Np1 1 1 N N
Nd2 N N 1 1
Np2 1 D 1 D
Nd1 D 1 D 1
Table 3: Combinations for N and D primes quantifies how good the quartet of integers is to be an approximated solution. Note that, since T¯(3) represents the geometrical mean of T¯(1) and T¯(2) , its value will always be between T¯(1) and T¯(2) . Moreover, note that min(δ) necessarily implies min(| T1 /T2 − N/D |). From a mathematical point of view the Relative Flower Constellation approximated solution can be mathematically re-casted as follows: ¯ ¯ ¯ Nd1 Nd2 ¯¯ ¯ Given T1 and T2 , find the positive integers Np1 , Np2 , Nd1 , and Nd2 , minimizing ¯ T1 − T2 . Np1 Np2 ¯ In general, N and D, are not primes and, therefore, they must be decomposed into a product of prime factors Pk . Therefore, in general, we have Y n(k) Y d(k) N = Np1 Nd2 = Pk and D = Np2 Nd1 = Pk (20) k
k
and, there will be a set of integer quartets associated with every possible two subset prime products Y n (k) Y n (k) Np1 = Pk 1 and Nd2 = Pk 2 (21) k
and Np2 =
Y
k
d (k)
Pk 1
and
Nd1 =
k
Y
d (k)
Pk 2
(22)
k
For instance, if N = 50 = 21 · 52 , then we have P1 = 2, n(1) = 1, P2 = 5, and n(2) = 2; and the associated possibilities for Np1 and Nd2 are given in Table 4. Using an easy computer program it is possible to identify all the possible combinations of two integers (e.g. Np1 and Nd2 ) whose product is an assigned integer (e.g. N ). Np1 Nd2
1 50
50 1
2 25
25 2
5 10
10 5
Table 4: Example for N = 50 Let us apply the theory to the interesting example where T1 and T2 are the Earth and Mars orbital periods, respectively. In this case, for instance, the T1 /T2 ratio has the numerical value T1 365.2 Np1 Nd2 = ≈ 0.5315866 ≈ T2 687.0 Np2 Nd1
(23)
Table 5 lists the results obtained to design an Earth-Mars Relative Flower Constellation. For the Earth-Jupiter Relative Flower Constellation, Table 6 provides the values of the Relative Flower Constellation design parameters. Once the Np1 , Np2 , Nd1 , and Nd2 integer parameters have been found, then it is important to distribute the spacecrafts such that they all belong to the same relative trajectory in both the rotating reference systems. This problem is solved by the Flower Constellation theory, which demonstrates 8
N 59 42 67 25 50 25 51 17 34 26 35 8 16 9 9 15 7 14
Factors [59] [2, 3, 7] [67] [52 ] [2, 52 ] [52 ] [3, 17] [17] [2, 17] [2, 13] [5, 7] [23 ] [24 ] [32 ] [32 ] [3, 5] [7] [2, 7]
D 111 79 126 47 94 47 96 32 64 49 66 15 30 17 17 28 13 27
Factors [3, 37] [79] [2, 32 , 7] [47] [2, 47] [47] [25 , 3] [25 ] [26 ] [72 ] [2, 3, 11] [3, 5] [2, 3, 5] [17] [17] [22 , 7] [13] [33 ]
Np1 /Nd2 1/1 79/14 7/1 1/1 2/1 1/5 96/17 8/17 8/17 49/13 6/5 5/2 5/2 1/9 17/3 7/5 1/1 3/2
Np2 /Nd2 59/111 3/1 67/18 25/47 50/47 5/47 3/1 1/4 2/8 2/1 7/11 4/3 8/6 1/17 3/1 3/4 7/13 7/9
δ 0.010361 0.011091 0.029986 0.061737 0.061737 0.061737 0.063342 0.063342 0.063342 0.18346 0.24175 0.32805 0.32805 0.40996 0.40996 0.77349 1.285 2.4891
Table 5: Design Parameters for a Earth-Mars Relative Flower Constellation (T1 /T2 = 0.53159) that, in order to distribute the all the spacecrafts into positions that are, simultaneously, admissible with both the rotating reference frames, the apsidal lines between two subsequent spacecraft orbits must obey to the relationship Np1 Nd2 − Np2 Nd1 ∆Ω = 2π L (24) Nd1 Nd2 where the integer parameter L can be set a priori. As example, Figures 9 through 16 provide the dynamics of a 3-spacecraft Relative Flower Constellation for the Earth-Jupiter systems that use Nd1 = 4, Np1 = 1, Nd2 = 1, and Np2 = 3. The figures represent pictures of the whole system (including the Earth and Jupiter orbits and positions), taken at subsequent specified times. Note how the spacecraft belong, simultaneously, to the relative trajectories in Earth and Jupiter rotating reference frames. 8
8
Earth−Jupiter RFC with Nd1/Np1=4/1, and Nd2/Np2=1/3; Time = 0.0 days
x 10
6
6
4
4
2
2
0
0
−2
−2
−4
−4
−6
−6
−8
−6
−4
−2
0
2
4
Earth−Jupiter RFC with Nd1/Np1=4/1, and Nd2/Np2=1/3; Time = 175.1407 days
x 10
6
8
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
8
0.8
1 9
x 10
x 10
Figure 9: EJ-RFC t = 0 days
Figure 10: EJ-RFC t = 175 days 9
Np1 Nd2 7 6 5 4 5 3 6 1 3 4 7 7
Factors [7] [2, 3] [5] [22 ] [5] [3] [2, 3] [1] [3] [22 ] [7] [7]
Np2 Nd1 83 71 59 47 60 36 72 12 36 48 84 84
Factors [83] [71] [59] [47] [22 , 3, 5] [22 , 32 ] [23 , 32 ] [22 , 3] [22 , 32 ] [24 , 3] [22 , 3, 7] [22 , 3, 7]
Np1 /Nd2 1/7 71/3 1/1 1/2 6/5 9/1 9/2 4/1 6/1 8/2 6/1 12/1
100 δ/T¯(3) 0.017813 0.21882 0.50091 0.92554 1.1798 1.1798 1.1798 1.1798 1.1798 1.1798 1.1798 1.1798
Np2 /Nd2 1/83 2/1 5/59 2/47 1/10 3/4 3/8 1/3 3/6 2/6 7/14 7/7
Table 6: Design Parameters for a Earth-Jupiter Relative Flower Constellation (T1 /T2 = 0.084322) 8
8
Earth−Jupiter RFC with Nd1/Np1=4/1, and Nd2/Np2=1/3; Time = 357.5789 days
x 10
6
6
4
4
2
2
0
0
−2
−2
−4
−4
−6
−6
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Earth−Jupiter RFC with Nd1/Np1=4/1, and Nd2/Np2=1/3; Time = 540.0171 days
x 10
0.8
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
9
Figure 11: EJ-RFC t = 357 days
0.8
1 9
x 10
x 10
Figure 12: EJ-RFC t = 540 days
Conclusion This paper describe the theory of the Synodic Flower Constellations and the Relative Flower Constellations. These new constellations are built using compatible orbits in synodic rotating reference frames and orbits that are, simultaneously, compatible with two rotating reference frames, respectively. In other words, these constellations are designed to be synchronized with two orbiting objects (planets, moons, spacecraft, etc.). The Synodic Flower Constellations and the Relative Flower Constellations can be adopted and be suitable in many applications. In particular, in order to support future planetary exploration missions, the most important applications are: 1. Space Network Architectures for Planetary communications. Interplanetary communications are presently performed by means of single-hop links. In this simple architecture there is one node at the exploration planet (e.g. Mars) and one node at the Earth (specifically, the antennae of the NASA Deep Space Network). This simple architecture presents two severe constraints: it requires direct visibility (and hence limited duration operation) and it does not tolerate the node failure. In order to avoid these critical constraints and to improve the 10
8
8
Earth−Jupiter RFC with Nd1/Np1=4/1, and Nd2/Np2=1/3; Time = 722.4553 days
x 10
6
6
4
4
2
2
0
0
−2
−2
−4
−4
−6
−6
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Earth−Jupiter RFC with Nd1/Np1=4/1, and Nd2/Np2=1/3; Time = 904.8935 days
x 10
0.8
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
9
x 10
Figure 13: EJ-RFC t = 722 days 8
Figure 14: EJ-RFC t = 904 days 8
Earth−Jupiter RFC with Nd1/Np1=4/1, and Nd2/Np2=1/3; Time = 1087.3317 days
x 10
6
4
4
2
2
0
0
−2
−2
−4
−4
−6
−6
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Earth−Jupiter RFC with Nd1/Np1=4/1, and Nd2/Np2=1/3; Time = 1269.7699 days
x 10
6
−1
1 9
x 10
0.8
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
9
Figure 15: EJ-RFC t = 1087 days
0.8
1 9
x 10
x 10
Figure 16: EJ-RFC t = 1269 days
communications necessary for human planetary missions, a minimum satellites (nodes) Synodic Flower Constellation or Relative Flower Constellation can be proposed to constitute the Space Network Architecture (SNA) for communications between Earth and the exploration planet. The SNA would consist of multi-hop links, a Relative Flower Constellation or a Synodic Flower Constellation of spacecrafts connecting the Earth with the planet to be explored. It is obvious that the stability of the orbits and the frequency/delta velocity required to maintain the orbits in the presence of perturbations and typical injection errors should be studied and used to establish relative advantages of alternate configurations. Minimum cost solutions, which are significantly more compact and efficient, can be identified among those fully complying with the network requirements and constraints, and analyzed. The effects of the orbital geometry on the network topology and the resulting effects of path delay and handover on network traffic (due to the great distances involved) must also be taken into consideration. Actually, a wide variety of requirements and constraints must be taken into consideration while proposing the solution scenarios. These are:
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(a) Service continuity: if anyone of the nodes becomes inoperative (either, permanently or momentarily), then the communications are still guaranteed. (b) Power efficiency: minimize inter-node angles variations (to narrow the antennae FOV), minimize inter-node distances (to limit communication power), etc, (c) Time efficiency: minimize the overall distance (to limit communication times), and (d) Fuel efficiency: seek to minimize the orbit maintenance requirements by optimizing amongst feasible orbit configurations. It is to be mentioned that, for this kind of application (communications), also of interest is another different conceptual configurations: using Halo orbits and Libration points. In fact, the L1 and L2 Halo nodes are always visible to each other and to the Earth at all times. L1 and L2 offer certain obvious advantages, however orbit stability will dictate station-keeping requirements. Alternatively, for the Earth-Moon system, it is known that positions near L4 and L5 are more stable than L1 and L2 and, therefore, a trade study should be carried out to establish the most attractive configuration considering both orbit maintenance requirements and communication system metrics. 2. Global Navigation Architecture for the Solar System. A new Global Navigation system for the Earth using Flower Constellation was already presented [5]. The proposed Global Navigation Flower Constellation (GNFC) was compared in terms of Geometric Dilution Of Precision (GDOP) and Attitude Dilution Of Precision (ADOP) with the US-GPS, the GLONASS, and the GalileoSat constellations by selecting a set of observation locations. As extension for that development, the Relative Flower Constellations or the Synodic Flower Constellations can be used to design a global navigation constellation for interplanetary missions. The proposed system could be made of two constellations. The first could provide navigation information for spacecrafts flying at distances less than the Jupiter orbit radius, while a second constellation can cover distances less than the Neptune orbit radius. Also for this application, the design parameters (which are mostly integer variables) can be sub-optimally estimated using Genetic Algorithms. Acknowledgements The first author wants to dedicate this paper to his friend Dr. Martin W. Lo, with the wish of a prompt rescue of all his energies!
References [1] Draim, J. E., “Satellite Constellations,” 55th International Astronautical Congress, 2004, IAC04-A.5.01, The Breakwell Memorial Lecture. [2] Mortari, D., Wilkins, M. P., and Bruccoleri, C., “The Flower Constellations,” The Journal of the Astronautical Sciences, Vol. 52, No. 1 and 2, January–June 2004, pp. 107–127, Special Issue: The John L. Junkins Astrodynamics Symposium. [3] Wilkins, M. P., The Flower Constellations – Theory, Design Process, and Applications, PhD Thesis, Texas A&M University, October 2004. [4] Wilkins, M. P., Bruccoleri, C., and Mortari, D., “Constellation Design Using Flower Constellations,” Paper AAS 04-208 of the 2004 Space Flight Mechanics Meeting Conference, Maui, Hawaii, February 9–13, 2004.
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[5] Park, K. J., Wilkins, M. P., Bruccoleri, C., and Mortari, D., “Uniformly Distributed Flower Constellation Design Study for Global Positioning System,” Paper AAS 04-297 of the 2004 Space Flight Mechanics Meeting Conference, Maui, Hawaii, February 9–13, 2004. [6] Bruccoleri, C., Wilkins, M. P., and Mortari, D., “On Sun-Synchronous Orbits and Associated Constellations,” Paper of the 6-th Dynamics and Control of Systems and Structures in Space Conference, Riomaggiore, Italy, July 18–22, 2004. [7] Wilkins, M. P. and Mortari, D., “Constellation Design via Projection of an Arbitrary Shape onto a Flower Constellation Surface,” Paper of the 2004 AIAA/AAS Astrodynamics Specialist Conference, Providence, Rhode Island, August 16–19, 2004. [8] Park, K., Ruggieri, M., and Mortari, D., “Comparisons Between GalileoSat and Global Navigation Flower Constellations,” Paper of the 2005 IEEE Aerospace Conference, Big Sky, Montana, March 5–12 2005. [9] Bruccoleri, C. and Mortari, D., “The Flower Constellation Visualization and Analysis Tool,” Paper of the 2005 IEEE Aerospace Conference, Big Sky, Montana, March 5–12 2005.
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