Synodic and Relative Flower Constellations with Applications to Planetary Explorations

Transcription

1 AAS 5-5 Synodic and Relative Flower Constellations with Applications to Planetary Explorations Daniele Mortari, Ossama Abdelkhalik, and Christian Bruccoleri Texas A&M University, College Station, Texas 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 55 th International Astronautical Congress specified that the field of constellation design is still in its infancy, and areas for improvement are almost without limits []. This statement was particularly true when the theory of the Flower Constellations [] 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 5-5 of the 5th AAS/AIAA Space Flight Mechanics Meeting, January 3 7, 5, Copper Mountain, Colorado

2 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 (N p ), the number of sidereal days to repeat the ground track (N d ), the number of satellites (N s ), and three integers to rule the phasing (F n, F d, and F h ), 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 (h p ). Following the very first presentation [], some studies and research have been done to understand and to design Flower Constellations [3,, 5,, 7]. The Flower Constellations present beautiful and interesting dynamical features (see 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 (FC- VAT) 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 n and n be the mean motions of two distinct planets orbiting the Sun (as the Earth and Mars), then T syn = π n n 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 π. During the synodic period, the two planets will have the exceeding mean anomaly α = n T syn n n n where x indicates the greatest integer lower than x. π = n T syn n n n () π () Disclosure of FCVAT Software, by Daniele Mortari, Matthew P. Wilkins, and Christian Bruccoleri, signed with the Technology Licensing Office, TAMU 339, 77 Texas Ave, College Station, TX in October 5, 3

3 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto ,33,77 3,589 59,8 9,588 Table : Planets Orbital Periods (days) Using the values for the planets orbital periods given in Table, the α values associated with any Planet pair are given in Table. Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Table : α Associated with Planets Pairs (degrees) Let us to introduce a set of reference frames that rotate with the angular velocities defined as ω = α + π K n K d T syn ( n = n K ) n n n = ( n n K d n = n K ) n n n n n K d (3) where K n and K d can be any integer numbers (even negative), and K d. 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 K n =, 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 K d synodic periods during which the reference frame has performed K n complete rotations (π) plus the exceeding angle K d 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 where F n and F d can be any positive integer numbers. Ω k+ = Ω k + α F n F d () 3

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. (). An orbit is compatible with respect to a reference frame rotating at the angular velocity ω if its orbital period T satisfies the relationship T r = N p T = N d T ω = N d π ω where T r is the repetition period, and N p and N d 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 n = π T = N p ω = N { ( p n n K ) } n n n = N d N d n n K d = N { ( p n n K ) } () n n n N d n n K d Once the orbit period has been quantified, then the orbits apsidal lines are distributed accordingly with Eq. () and the satellites are allocated into the admissible positions defined by the theory of the Flower Constellations []. 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. (5) SFC with Earth and Mars; Time =. days SFC with Earth and Mars; Time = days Figure : First Alignment Figure : 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.

5 SFC with Earth and Mars; Time = days SFC with Earth and Mars; Time = days Figure 3: Second Alignment Figure : Second Anti-Alignment SFC with Earth and Mars; Time = days SFC with Earth and Mars; Time = days Figure 5: Third Alignment Figure : Third Anti-Alignment Figures 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., 3, 5, and 7) and anti-alignments (Figs.,,, 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: K n =, K d =, N s = 3, F n = 3, F d =, N p = 3, N d =, and the eccentricity was chosen to be e =.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

6 SFC with Earth and Mars; Time = days SFC with Earth and Mars; Time = days 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 T and T 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 N p T = N d T (7) N p T = N d T (8) where N p, N p, N d, and N d 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 T = N p N d (9) T N p N d 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 = T T N d N d N p N p () 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 T r = N p T = N d T () T r = N p T = N d T () respectively. This suggests us that the double synchronization can be introduced with the same logic, by finding two integers, K and K, to satisfy T r = K T r = K T r (3)

7 where T r is the repetition time of the three rotating frames. Unfortunately, the identity (3) yields no additional information. In fact, by substituting Eqs. () and () into Eq. (3) we obtain K N p T = K N d T = K N p T = K N d T () By equating the first and third terms, and the second and fourth terms, we obtain K K = N p N p = N d T N d T (5) 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 K K = N d T N p T = N p T N d T () from which we derive again Eq. (). This proves that by looking at this problem as done by Eq. () through Eq. (), we add no information. Using the theory of the Flower Constellations, an orbit satisfying Eq. (7) allows us to build a relative trajectory with N p petals with respect to the reference frame rotating with an angular velocity equal to the n = π mean motion. The key parameter to distribute the satellites such T that we obtain this effect is ruled by the N p ratio. Similarly, an orbit satisfying Eq. (8) allows N d us to build a relative trajectory with N p petals with respect to the reference frame rotating with an angular velocity equal to the n = π mean motion if we distribute the satellites such using the T ration N p. N d Approximative Solutions for Earth-Planet Systems In general, when T and T 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 be found by finding the N p, N p, N d, and N d integers such that N p N d approximates T. N p N d T 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 which implies the error T /T N/D. T T N D = N p N d N p N d (7) 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 T () = N d N p T, T () = N d N p T, and T (3) = which, in theory, should be equals. Therefore, the relative percentage error T T N d N d N p N p (8) δ = T (3) T () T () (9) 7

8 # N p N d N p N d N D N D 3 N D N D Table 3: Combinations for N and D primes quantifies how good the quartet of integers is to be an approximated solution. Note that, since represents the geometrical mean of T () and T (), its value will always be between T () and T (). Moreover, note that min(δ) necessarily implies min( T /T N/D ). From a mathematical point of view the Relative Flower Constellation approximated solution can be mathematically re-casted as follows: Given T and T, find the positive integers N p, N p, N d, and N d, minimizing N d T N d T N p N p. In general, N and D, are not primes and, therefore, they must be decomposed into a product of prime factors P k. Therefore, in general, we have T (3) N = N p N d = k P n(k) k and D = N p N d = k P d(k) k () and, there will be a set of integer quartets associated with every possible two subset prime products N p = k P n(k) k and N d = k P n(k) k () and N p = k P d (k) k and N d = k P d (k) k () For instance, if N = 5 = 5, then we have P =, n() =, P = 5, and n() = ; and the associated possibilities for N p and N d are given in Table. Using an easy computer program it is possible to identify all the possible combinations of two integers (e.g. N p and N d ) whose product is an assigned integer (e.g. N). N p N d Table : Example for N = 5 Let us apply the theory to the interesting example where T and T are the Earth and Mars orbital periods, respectively. In this case, for instance, the T /T ratio has the numerical value T T = N p N d N p N d (3) Table 5 lists the results obtained to design an Earth-Mars Relative Flower Constellation. For the Earth-Jupiter Relative Flower Constellation, Table provides the values of the Relative Flower Constellation design parameters. Once the N p, N p, N d, and N d 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

9 N Factors D Factors N p /N d N p /N d δ 59 [59] [3, 37] / 59/.3 [, 3, 7] 79 [79] 79/ 3/.9 7 [7] [, 3, 7] 7/ 7/ [5 ] 7 [7] / 5/ [, 5 ] 9 [, 7] / 5/ [5 ] 7 [7] /5 5/ [3, 7] 9 [ 5, 3] 9/7 3/.33 7 [7] 3 [ 5 ] 8/7 /.33 3 [, 7] [ ] 8/7 /8.33 [, 3] 9 [7 ] 9/3 / [5, 7] [, 3, ] /5 7/.75 8 [ 3 ] 5 [3, 5] 5/ /3.385 [ ] 3 [, 3, 5] 5/ 8/ [3 ] 7 [7] /9 / [3 ] 7 [7] 7/3 3/.99 5 [3, 5] 8 [, 7] 7/5 3/ [7] 3 [3] / 7/3.85 [, 7] 7 [3 3 ] 3/ 7/9.89 Table 5: Design Parameters for a Earth-Mars Relative Flower Constellation (T /T =.5359) 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 where the integer parameter L can be set a priori. Ω = N pn d N p N d N d N d π L () As example, Figures 9 through provide the dynamics of a 3-spacecraft Relative Flower Constellation for the Earth-Jupiter systems that use N d =, N p =, N d =, and N p = 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. Earth Jupiter RFC with Nd/Np=/, and Nd/Np=/3; Time =. days Earth Jupiter RFC with Nd/Np=/, and Nd/Np=/3; Time = 75.7 days x 9 Figure 9: EJ-RFC t = days Figure : EJ-RFC t = 75 days 9

10 N p N d Factors N p N d Factors N p /N d N p /N d (3) δ/ T 7 [7] 83 [83] /7 / [, 3] 7 [7] 7/3 /.88 5 [5] 59 [59] / 5/59.59 [ ] 7 [7] / / [5] [, 3, 5] /5 / [3] 3 [, 3 ] 9/ 3/.798 [, 3] 7 [ 3, 3 ] 9/ 3/8.798 [] [, 3] / / [3] 3 [, 3 ] / 3/.798 [ ] 8 [, 3] 8/ / [7] 8 [, 3, 7] / 7/ [7] 8 [, 3, 7] / 7/7.798 Table : Design Parameters for a Earth-Jupiter Relative Flower Constellation (T /T =.83) Earth Jupiter RFC with Nd/Np=/, and Nd/Np=/3; Time = days Earth Jupiter RFC with Nd/Np=/, and Nd/Np=/3; Time = 5.7 days x x 9 Figure : EJ-RFC t = 357 days Figure : EJ-RFC t = 5 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:. 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

11 Earth Jupiter RFC with Nd/Np=/, and Nd/Np=/3; Time = days Earth Jupiter RFC with Nd/Np=/, and Nd/Np=/3; Time = days x x 9 Figure 3: EJ-RFC t = 7 days Figure : EJ-RFC t = 9 days Earth Jupiter RFC with Nd/Np=/, and Nd/Np=/3; Time = days Earth Jupiter RFC with Nd/Np=/, and Nd/Np=/3; Time = days x x 9 Figure 5: EJ-RFC t = 87 days Figure : EJ-RFC t = 9 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:

12 (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 L and L Halo nodes are always visible to each other and to the Earth at all times. L and L offer certain obvious advantages, however orbit stability will dictate station-keeping requirements. Alternatively, for the Earth-Moon system, it is known that positions near L and L 5 are more stable than L and L and, therefore, a trade study should be carried out to establish the most attractive configuration considering both orbit maintenance requirements and communication system metrics.. 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 [] Draim, J. E., Satellite Constellations, 55th International Astronautical Congress,, IAC- -A.5., The Breakwell Memorial Lecture. [] Mortari, D., Wilkins, M. P., and Bruccoleri, C., The Flower Constellations, The Journal of the Astronautical Sciences, Vol. 5, No. and, January June, pp. 7 7, 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. [] Wilkins, M. P., Bruccoleri, C., and Mortari, D., Constellation Design Using Flower Constellations, Paper AAS -8 of the Space Flight Mechanics Meeting Conference, Maui, Hawaii, February 9 3,.

13 [5] Park, K. J., Wilkins, M. P., Bruccoleri, C., and Mortari, D., Uniformly Distributed Flower Constellation Design Study for Global Positioning System, Paper AAS -97 of the Space Flight Mechanics Meeting Conference, Maui, Hawaii, February 9 3,. [] Bruccoleri, C., Wilkins, M. P., and Mortari, D., On Sun-Synchronous Orbits and Associated Constellations, Paper of the -th Dynamics and Control of Systems and Structures in Space Conference, Riomaggiore, Italy, July 8,. [7] Wilkins, M. P. and Mortari, D., Constellation Design via Projection of an Arbitrary Shape onto a Flower Constellation Surface, Paper of the AIAA/AAS Astrodynamics Specialist Conference, Providence, Rhode Island, August 9,. [8] Park, K., Ruggieri, M., and Mortari, D., Comparisons Between GalileoSat and Global Navigation Flower Constellations, Paper of the 5 IEEE Aerospace Conference, Big Sky, Montana, March 5 5. [9] Bruccoleri, C. and Mortari, D., The Flower Constellation Visualization and Analysis Tool, Paper of the 5 IEEE Aerospace Conference, Big Sky, Montana, March