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Introduction to the theory for optimal design of satellite constellations
for earth periodic coverage
Yury N. Razoumny
Abstract
Design of satellite constellations is associated with continuous or periodical coverage of an Earth region of interest depending on the concrete task of the space mission. Satellite constellations for continuous and periodical coverage are principally differed by orbit parameters as well as by methods of constellation design. In this study route theory for design of constellations for circular low Earth orbits providing periodical coverage with minimum revisit time criteria is described. The route theory includes: 1) consideration of so called route orbital pattern as mathematical abstraction of arbitrary constellation; 2) analytical solution for calculation problem of distribution of revisit time values on Earth surface for one satellite and multi-satellite route pattern; 3) formulating several regularities for revisit time as a function of satellites positions in constellation; 4) development of method for optimal design of constellation under given criterion and requirements. Using of route orbital patterns is practically not connected with restriction for type of constellation. It is also shown that optimal constellations calculated using the route theory are in general not worse and in many cases much better (in respect of criteria mentioned above) comparing with the analogs found using other well-known methods for periodical coverage.
1. History of the problem
From the very beginning of the space age the Earth artificial satellites serve the mankind. Moreover, the satellite importance for the human being was emphasized when group of satellites functioning together to solve common task, had been created. These groups are commonly called satellite constellations. Efficiency of such groups functioning depends on how correct these satellites orbits are chosen, and how the satellites relative orbital movement is organized. This problem is discussed below.
The way to solve the problem to optimize the satellite constellations was outlined in principle in the 60th - 70th, when the importance of study of so called function of satellite coverage (continuous or periodic) of the Earth was recognized [1-11]. This function allows interpreting the
purposes of big number of satellite systems achieved by means of different on-board equipment, in unified abstract aspect. Namely, this aspect took the form of establishing (continuous or discontinuous, respectively) necessary geometric conditions of relative positioning of satellites and Earth surface objects observed.
Important theoretical generalization of ballistic design methods for satellite constellations for continuous and periodic coverage known by the late 70th, is presented in [11]. The most valuable result was obtained, independently and almost at once, by Walker [5, 6, 9, 10] and Mozhaev [7, 8]. This result yielded in the development and comprehensive grounding of so called symmetrical constellations for continuous coverage problem. Each symmetrical constellation corresponds to a certain code of symmetry describing relative satellite disposition in the network. In Russian literature the symmetrical constellations are called "kinematically regular networks", in English - "Walker constellations". Later the symmetrical constellation properties for the continuous coverage problem were subjected to close inspection of both these constellations inventors -Walker and Mozhaev, and other authors (see, for example, [12-15]).
Some other methods to optimize the continuous coverage satellite constellations were introduced later on. These are the "streets of coverage" method [16, 17], original Draim's method to analyze constellations using elliptical orbits with minimum needed number of satellites [18], and others. However, such particular methods involve the essential narrowing the classes of constellation in use, and are not universal and simple like symmetrical constellation method. Even nowadays most of investigators of this problem refer first to symmetrical constellations at designing the continuous coverage satellite system.
Simplicity of continuous coverage system design on the base of symmetrical constellations did not allow to fully recognize the complexity of periodic coverage problem, although it was emphasized by some investigators both many years ago [15], and not long ago [26]. Mozhaev, in particular, points to that the main criterion function in the periodic coverage problem (revisit time) is not differentiable (as in the continuous coverage problem), but "even discontinuous and unable to be set by formulas" [15]. It can be spelled out that mathematics did not offer any ready method able to be directly applied to optimize satellite constellations for periodic coverage. Moreover, the word "optimum" was conditional in the periodic coverage problems down to recent times. Usually, this word referred to rational (from different points of view) alternatives of satellite constellations chosen in a priori predetermined narrow classes of orbital structures.
In order to provide equal precession of satellite orbits the Earth surface coverage satellite constellations are usually organized on circular orbits of equal radius and inclination. In this context it can be pointed to some practically validated narrowing of the optimization range of satellite
constellations for periodic coverage (as compared with those for continuous coverage) that is defined by applying for periodic coverage relatively low circular orbits up to about 5,000 km (this maximum altitude can vary depending on available width of the satellite coverage swaths). However, the said circumstance does not simplify, if not complicates, the problem under consideration.
Due to the complexity of the periodic coverage problem the first attempts to solve it were reduced to a simple application of symmetrical constellations method. The latter approved itself at solving the continuous coverage problem, despite that seldom some other particular methods were used [15, 19-26]. As a result, symmetrical constellations have been widely propagated in practice of satellite constellation design not only for continuous but also for periodic Earth coverage, lacking something better.
However, soon afterwards the Soviet (Russian) authors have revealed the versions of satellite constellation for periodic coverage with "poor" symmetry. These constellations badly fit in symmetrical constellation theory but stably have the better or the same periodicity values comparing with symmetrical constellations [27-37]. Unfortunately, judging by the publications, these results were overlooked and not assessed by the world scientific society till nowadays. "The literature on low-altitude satellite constellations that provide non-continuous coverage of a small part of the globe is sparse", authors of [23] pointed out. Really, there were many Russian-language papers devoted to the subject of periodic coverage of the Earth surface areas. Obviously, the reason was that far from all Russian-language papers dealing with the synthesis of periodic coverage orbital structures were translated into English.
As a matter of fact, so called secure constellations were revealed and substantiated in the mentioned papers [27-37]. Secure constellation is designed by means of N satellites (or N identical frontal satellite groups) disposition along common trace(s) on the Earth surface with time lag equal to the 1/N of maximum revisit time provided for a certain Earth region by one of N satellites (by one of N frontal satellite groups) of the constellations. Secure constellation provides maximum revisit time which is equal to the said time lag in the movement of satellites (frontal satellite groups) along the trace(s).
Frontal satellite group consists of satellites whose sub-satellite points at any moment have the same latitude and are shifted relative one another by longitude only. It should be noted that the "secure constellations" term was introduced by the author of present paper. In referenced papers of other Russian authors the said constellations either have other appellations or do not have any.
Steadily good features of secure constellation at the periodic coverage allowed some researchers to throw out a suggestion that these constellations are absolutely optimal in solving the periodic coverage problem: Mozhaev [15, p. 259], Saulsky [29, p. 112].
However, the author had queried this statement in [38], and later he obtained the final negative conclusion about optimality of secure constellations [39, 40]. It was proved by revealing some effective satellite constellations that did not belong to any known class. Revealed constellations were testified to surpass secure constellations in minimum revisit time criterion in a series of different cases.
Therefore, the situation was contradictory: comparative features of different satellite constellation classes essentially depend on the choice of common comparison conditions, and do not allow to prefer one to another in a general case. Author supposes that his route theory for orbital design of periodic coverage satellite constellations, serves to resolve this situation. Fundamentals of the said theory are stated in [41] and are subject of author's lecture courses in MSTU named after Bauman.
For clear reasons only some basic principles of the theory and restricted amount of numerical results are demonstrated in present paper. Only a small part of the theory statements is represented in the paper. Statements represented are given with no proof.
2. Route theory: brief description
Ballistic parameters of the constellations on circular orbits of equal altitude and inclination are specified by vector
^ = § J, Sj = (H, i, Qj, u] ), j = 1(1) N, (1)
where H - orbit altitude; i - orbit inclination; Q j - longitude of the orbit ascending node of j-th satellite; Uj - latitude argument of j-th satellite of the constellation in the initial point of time; N -number of satellites in the constellation.
Q u .
Components H, i of vector (1) define the orbit types, as well as components j and j specify orbital or phase structure of the constellation:
On = Oj J Oj = (Qj, uj ), j = 1(1)N. (2)
Traditional approaches to solve the problems of periodic coverage are based on the elaboration of optimization methods with simplifying assumptions reduced generally to a priori
preset of the orbital structure narrow classes ON in the form of symmetrical or other types of constellations. The considered route theory is based on the refuse of satellite constellation optimization within the framework of narrow classes of orbital structures, and on the revealing of
the regularities for general set of ballistic and system features of periodic coverage satellite constellations. The system features are understood here as revisit time T, and swath width n that is formed on the Earth spherical surface by momentary ranges of the satellites coverage. The revisit time is understood as maximum revisit time provided with a priori preset probability level, and the momentary ranges of the satellites coverage can have different form depending on the type of available on-board facilities for observation.
Classical problem of periodic coverage for area R being observed on the Earth surface
consists in search for the minimum of function t(On ) and value On, where this minimum is reached. The problem is formulated as follows:
Problem A. Given: N> '' n •
To find: O*N = arg min {r(ON) == t(On / N, R,H ,i,n)} (3)
ON
and Vn = *(ON)- (4)
Key point of the route theory (giving the latter its name) is so called route constellation used as a mathematical abstraction for approximation of arbitrary satellite constellation. The satellites in route constellation move on geosynchronous orbits along one or several closed traces (or routes) on the Earth surface with repetition period
Ttr = m ■ Tdr = n ■ Tef , (5)
T T f
where dr - draconic orbital period of satellites; ef - efficient astronomical day or simply day
(time span between two subsequent passes of a fixed equator point over ascending node of the
satellite orbit); m, n - integer coprime numbers characterizing the number of circuits and number of
efficient astronomical days within the repetition period of the satellite traces.
Value m/n is a repetition factor of a geosynchronous orbit. Repetition factor m/n and
inclination i unambiguously define the orbit altitude H of route constellation satellites.
Orbital structure of the route constellation is properly characterized by longitudes of the
traces start on the equator (trace longitudes) Lj and time positions of satellites on their routes (on-
trace positions) , that are in one-to-one conformity with traditional parameters ^j , U of the
orbital structure. Values Lj and vary within the following intervals:
Lj e[0, AL), tj e[0, Ttr) (6)
where
AL = 2x/ m (7)
- distance between neighboring points of intersection with equator of the one satellite trace ascending parts within the repetition period Ttr of this trace (internodal distance).
Internodal distance (7) is smaller by a factor of integer n than the interval between ascending
nodes of the satellite on two subsequent circuits (inter-circuit distance) ^ic defined as
ALic = n -AL. (8)
Application of the route constellation is connected with substitution of vectors (1) and (2) by
vectors (9) and (10), respectively:
^ = ^<2.n+2> = § j, Sj = [m/n,t,Lj ,tj \j = 1(1)N; (9)
ON = On<2-n> = Oj j Oj ={Lj ,tj\j = 1(1) N. (10)
This substitution, i.e., the assumptions defining the route constellation do not restrict in practice the domain of possible orbital dispositions both in altitude H, inclination i of the satellite orbits, and in relative position of the satellites on these orbits. Let's show that.
H J
Really, it is not difficult to note that the infinite discrete multitude " g' of geosynchronous orbit altitudes corresponding to various repetition factors m/n and fixed inclination i, corresponds to
{H J H J
continuous multitude " 8' of all possible values H as the domain of rational numbers
(fractions) and real numbers on any closed interval of axis of altitude H. At the same time, it is known that for any real number within the closed interval a rational number can be found being
arbitrarily close to it. So, for any fixed values H = H°'i = io, a repetition factor m/n of
i = i H
geosynchronous orbit with inclination 0 and altitude H being arbitrarily close to 0, can be
found. Hence, the assumptions defining the route constellations do not really restrict the domain of
possible satellite orbits. Taking into account that it is permitted in the route constellations that
satellites move non-synchronously along arbitrarily located traces, one can conclude that the phase
structure restriction in the route system is also absent.
Taking (10) into consideration, the classical problem of periodic coverage is formulated in
terms of the route theory as follows:
Problem B. Given: N,R,m,n,i, n .
Tofind: °N= argminr(°N) /AT (11)
Ow [=t(On /N, R,m,n,i,n) ^
and Tmin = *(On). (12)
The proposed method to solve Problem B is based on the fact that in any problem of multidimensional optimization a duality of the target function Ti°N) and its definition range °N
takes place. In other words, the problem with complex target function can be reformulated in such a way that target function will be easier but the "complexities" will be transferred to the optimization
area. A reverse transformation is possible, too. The area °N dimensions define the a priori
*
information about location of optimum N : the larger is N the more is a priori information of
*
ON location, but the less difficult is perhaps the search of this point due to simplifying of function <°n ) type.
In our case, "simplifying" of function Ti°N ) is reached owing to the description of phase structure °N by vector (10) instead of traditional vector (2). That allows to essentially reduce a
(°n) t l
N on parameters
priori uncertainty of function r'°N)on parameters Lj. In fact, it is seen from (6) and (7) that
range of L] definition is not large and is 22.5...45 degrees for one-day (n = 1) low orbits about 200...5,000 km, respectively (in this case m varies from 16 to 8). With increasing of the trace
repetition period (increasing of n) the range of Lj definition decreases rapidly down to some degrees and degree fractions.
Disadvantage is that a priori uncertainty of parameters rises sufficiently with increasing of
n. In fact, as it can be seen from (6), for one-day orbits the range of definition of parameters is about 24 hours, and with growing repetition period of the trace this range increases n-fold while expanding to enormous dimensions at large values of n.
At the same time, a large uncertainty of parameters is with usury compensated by that at the description of orbital structure with expression (10) the route theory reveals a series of
regularities of revisit time variation r(°N ) as a function of parameters . In its turn, it allows to
create specific theoretical and program-algorithmic means for optimizing function Ti°N) on
parameters j .
Due to limited volume of the paper the theoretical statements concerning optimization of optional constellations are not represented. Present paper shows the route theory possibilities on the example of new efficient class of satellite constellations revealed and substantiated within the
framework of the route theory. The author calls these satellite constellations as regular constellations.
Regular constellation is N-satellite constellation obtained from one satellite as a basic block by means of its successive (N-l)-fold repetition simultaneously with longitude shift
by ^tr k'k ~ and time shift ^
It • k,k = 7(7) N -1
along the satellite trace (Fig. 1).
Satellite routes
Fig. 1: Relative disposition of the satellites in regular constellation (N=4).
^tr and At used in this definition are called, respectively, longitudinal and time shifts of
the regular constellation. Reasoning from physical meaning of these values, their variation ranges
are determined in the following way:
ALtr e [0, AL) At e [0, Ttr ) (13) and an orbital structure of regular constellations is defined according to formulas: Lj = E[ALtr -(j -1)1
tj = E[At -(j -1)1 (14)
J = 1(1) N,
where function E recalculates the argument values for ranges (6) of admissible variation of Lj and t j magnitudes.
The satellites of regular constellation are shifted along their traces by constant relative value
At, therefore the determination of satellites positions t], j=2(1)N ~ on the traces is reduced to the determination of At value providing function
ÁAt )
minimum. Taking into account that the definition range of time parameter At is large (n efficient astronomical days), this problem solution seems to be highly difficult.
Function property allowing cardinal simplifying of this problem, consists in the
following. For specific area R and a fixed longitudinal shift Atr of N-satellite traces in regular constellation revisit time function
r(At )
versus time shift At meets Lipshitz condition within
interval [0'Ttr ] with constant (N - 1).
It is known that function f (x) satisfies Lipshitz condition in the range [a'b], if there exists such constant Q>0, that f ~f ~ Q ' \u ~ vl for any u'v e [a'b]- Constant Q is called Lipshitz constant. In our case J ( '
and Q=N-1. This statement allows to minimize
the
r(At )
function using known mathematical methods for Lipshitz fuctions.
3. Numerical results: comparing of secure and regular constellations
So as to estimate the possibilities of regular constellations it seems to be expedient to compare them with secure systems considered up to the recent time as most preferable for periodic coverage. Equal comparison conditions can be provided by analyzing one-trace alternatives of
regular and secure constellations (ALtr ~ 0 ).
It is easy to see that in the one-trace disposition case secure constellations are a small subset
of wide class of regular constellations at At ~ Tmax ^ where Tmax - one satellite revisit time for
the Earth surface specific region. As it was mentioned before secure constellation provides revisit t =At
time N (hereinafter for convenience number N of satellites in the constellation is added as an
index to the notation of revisit time N for N-satellite system).
Fig. 2: Periodicities of coverage of Earth latitude belt ... 70 ° for secure ( rmax / N ) and optimum regular
*
( Tn ) constellations versus (mn for the satellite swath n=2790 km.
*
T
According to [27-37] it can be concluded that the revisit time N for optimum regular
*
constellations is always reached on the secure constellations set, and it is Tn = = Tmax . Numerical analysis presented below yields that it is really true but in some cases only. Furthermore,
application of optimum alternatives of regular constellations in comparison with secure
*
t / N — t
constellations can provide exceptionally large benefit, i.e. value max N.
Example to consider are constellations of N {2'3'4'5} satellites. The said constellations
are formed on geosynchronous orbits with repetition factor m/n= 29/2 and inclination i = 96° (altitude H= 731 km).
Figs. 2-4 show periodicity (revisit time) magnitudes for coverage of Earth latitude belt
(70° ) t /N T*
(min ' ' for secure (Tmax ) and optimum regular ( Tn ) constellations under said restrictions,
versus lower limit Pmin of latitude belt for the following values of the satellite swath n=2790, 2500, 2000 km.
Fig. 3: Periodicities of coverage of Earth latitude belt ... 70 ° for secure ( Tmx / N ) and optimum regular
*
( Tn ) constellations versus for the satellite swath n=2500 km.
*
Value of revisit time benefit max N using optimum regular constellations for various
values of ^mn in the said figures is defined by the value of shaded range of the line being normal
to the abscissa axis at the ^mn point.
Figs. 2-4 show that due to using of optimum regular constellations under the above restrictions the revisit time benefit value can reach 5 hours for two-satellite constellations, and 1 hour - for three-, four-, and five-satellite constellations comparing with secure constellations.
Fig. 4: Periodicities of coverage of Earth latitude belt ç^n •••
70 0 for secure ( rmax / N ) and optimum regular
*
( Tn ) constellations versus ç^n for the satellite swath n=2000 km.
The mentioned revisit time benefit values are not maximum possible. At other initial data, they can reach even bigger magnitudes. So, for example, the author has revealed some alternatives when, other things being equal, two-satellite optimum regular system provides lower revisit time value than the four-satellite secure constellation.
4. Analysis of limiting possibilities of regular satellite systems
The analysis presented above was aimed to reveal the fact of actual preferability of regular constellations as compared with secure constellations. In order to provide equal conditions for comparing the possibilities of two classes, an assumption was used that the satellites move along the same trace ("one-route" motion). As for secure constellations, the increase of the number of routes cannot be used to reduce coverage periodicity, but for regular constellations this factor is an additional source for enhancing the orbital structure quality. Regular constellations are in this case difficult to be analyzed due to the lack of analogs.
Therewith, the evaluation of limiting possibilities of regular constellations has another, more important, aspect. As it appears from the regular constellation's definition, their extreme modifications cannot pretend on being absolutely optimal because all regular systems meet only the
condition of local optimality. With this, the question arises how close are such modifications to absolute optimum.
t t t
So, it is expediently to compare three values of periodicity: G R A corresponding to the best orbital structures, within the following areas:
1) area G of secure constellations (tg );
2) area R of regular constellations (tr );
3) area A of arbitrary constellations (t) .
In the first case, optimization is provided by the method described in [28, 31, 37], in the second and third cases the methods presented in the framework of the route theory are used [41].
It is clear that G c R c A- Thus, in general case Tg'Tr'ta can not to coincide. At the same time, particular cases when G and R, and R and A or even all three values coincide cannot be excluded. All these situations are really available for tg'Tr'Ta at various initial data:
TG -TR -TA- (g)
Therewith, in order to determine significance of regular constellations in the practice of ballistic design of the systems of periodical coverage it is desirable to evaluate maximal distinctions
between the values of T< and Tr ' Tr and TA .
This evaluation was not so detailed as the research of the regular constellation extreme modifications presented above owing to high labour input needed to find optimal orbital structure. As a result, a series of 500 versions of initial data corresponding with one-day and two-day orbits at N=3 satellites in the system, was analyzed (version with N=2 was not considered because an extremal two-satellite regular system is always absolutely optimal one). Present research led to the following conclusion.
It proved to be that the advantage of extremal versions of regular orbital structures over the guaranteed ones is characterized as follows:
1:3,6 — tR:tG ^ 1:1' (9)
and the analogous advantage of absolutely optimal orbital structures over the regular ones is characterized as follows:
1:1,2 — ^G : ta ^ 1:1. (10)
It is necessary to add that the distinctions in T< and Tr values took place in the great majorities of cases. It would not be correct to give some magnitude because the coincidence of
these two values can be to a great extent foreseen a priori1). Lack of coincidences of Tr and T A occurred very seldom (in about 5% of cases), and sometimes they could be eliminated by the accuracy rise of the extremal system calculation.
5. Parameters characterizing periodic coverage geometry and their functional
interrelation
Revisit time calculated under the assumption that the satellite swath value is fixed is a traditional criterion of the satellite constellation quality for periodic coverage. But in many important applications the value of revisit time could be a priori given on the basis of specific requirements to the satellite constellation. There is a chance in this case to reduce additionally the observations cost in terms of minimized satellite swath. This problem is discussed below.
Let's introduce the parameters characterizing the geometry of the Earth surface coverage by the satellite swath at periodical coverage, and obtain the interrelations of these parameters.
It will be assumed that the observation region is the Earth latitudinal belt R bounded by
R — —
parallels V in latitudes ç = Çmin and ç = çmax-R = URç, Ç^Çmin'Çmax J
ç, ç e
ç (15)
With allowance for (1), the restriction is applied on the satellite orbit inclination i : i0 + P>çm, i0 = min(i, 7 - i ),
Çm = Tn^^m^ çmax |). (16)
Here P is the angular width of the satellite swath corresponding to the linear swath width n = 2reP , re is the Earth mean radius.
Let's consider the coverage of fixed parallel R(p in latitude V. In case of V < i(0 every satellite crosses twice within one circuit the parallel on latitude V : on the circuit ascending part in
A- D A- D ■
point J and on its descending part in point J (Fig. 5,a). Points J and J will be called
1 In fact, it is not difficult to explain theoretically the coincidences of TG and Tr : they have a
tendency to occur in situations being trivial enough, when the satellite swath width corresponds to small repetition factors of the area under observation by a single satellite within the period of the trace repetition, and they always occur at a single coverage. In the mentioned cases, when the coverages of single satellites are not great in number and simple in structure, the optimum has to be reached on the multitude of guaranteed systems.
T
ascending and descending latitudinal nodes. Over a period of TR the satellite will cross a fixed parallel R in m ascending nodes A and in m descending nodes ' Either of the two systems of points A A D ^ j _ 1 ' 2 ' 3 ' "'' m is uniform, and distance between neighbouring
points in these systems (of points) is ^ ~ 2 n^ m (internodal distance - the distance between neighbouring points of intersection with equator of the one satellite trace ascending parts within the
T
repetition period TR of this trace).
Union A UD in general case yields an irregular set of points characterized by angular
§ §
shifts R and L (right and left shifts of the systems of points A and D - see Fig. 5,a). In
§ §
substance, shifts R and L are minimal angles to turn round a system of points A towards
§ §
increasing ( R) and decreasing ( L) longitudes of the nodes till it coincides with a system D of
points. Obviously, ^R
Fig.5: Relative displacement on the Earth parallel of the ascending and descending latitudinal nodes (view from the pole)
To evaluate shifts ^ R and ^ L, it is convenient to consider angle of turn ^^ (turn of the systems of points A and D ) that is the difference of longitudes of latitudinal nodes Ak and Dk belonging to any k th-circuit of one satellite. It follows from Fig. 5,b that ^^ = ^BM ^'MN' where is the difference of longitudes of points B and M of the orbit crossing with parallel
R
V on non-rotating Earth; ^MN - rotational displacement (angle of rotation) of the Earth during the time period of the satellite orbital flight from point B to point M . From the analysis of
spherical triangle ABC (Fig. 5,b) the following expression can be obtained for angle of turn ^^ :
A\y = n
^ _ ^
v
m
2 ■
arcsin
tgP tgi
n
— arcsin m
i ■ \ v sini j
(17)
a
C d
Suppose that G is a selection function of the residual of division, the expressions for R
and ^ will be: dR = G{A¥lAL), = ^
d d
As it is seen from the expressions given above, shifts R' L are functions of m, n, 1, 9. Alteration of these parameters depending on latitude ^ is of oscillatory nature which is related to
specific values of m, n, 1. Dependencies dR and dL are shown for example in Fig. 6
for
m = 1 n = 16 1 = 65o
(orbit height
H = 226
km).
8r, d, degrees 20
10
0
20
40 9, degrees
Fig. 6: Dependencies between right and left shifts and the parallel latitude (m=16, n= 1, i=65 )
R B
Let n be a satellite swath on the rotating Earth surface realized for angular width P of
the satellite swath. Let's call a length aR (aL) of the circuit part R(p covered by Rn from its axial line, i.e. from the satellite trace, to its right (left) border, as right-side (left-side) lock-on of
R Rd
swath n on parallel $. Locks-on in ascending and descending latitudinal nodes will be distinguished by identifying them in case of need, by upper and lower index marks "A" and "D ",
respectively:
• aR' aL' aR ' aL ■
R R
It can be shown that owing to the symmetry of parallel ^ coverage by swath n, the
equalities
AD AD
aR =aR =aL' aL =aL = aR are right. Sum a = aR +aL will be called lock-
RR on of swath n on parallel 9.
The locks-on a, aR, aL are functions of the swath width n, latitude 9 of covered
parallel R<p' as well as inclination 1 and repetition factor m /n of the satellite geosynchronous
orbit. These locks-on are monotonous functions of the swath width n at fixed parameters m, n , i, (
a = a(n), (Xr = (XR (n), aL = aL (n) (1g)
Geometric characteristics considered above are referred to the parallel coverage by a single satellite swath. As for the system of several satellites, total coverage of the parallel by their swaths is formed by superposable coverages provided by separate satellites. Relative position of the
R
coverages of different satellites on any parallel ( is characterized unambiguously by specified
longitudes LJ'^ 11N of the system satellites traces [41].
By analogy with the Earth continuous observation, a notion of the Earth surface coverage
repetition factor l in the problem of periodical coverage is introduced. Magnitude l will be used to characterize the coverage of point r or region R of the Earth (in particular case, of the parallel or latitudinal belt) created by the swaths of one or several (all) satellites of the system. It will be
1 T
understood that l -fold coverage of point r is provided, if during the time interval TR between the
satellites (satellite) traces repetition this point l times finds itself in the instantaneous zones of the
/ T
satellites (satellite) coverage, and l -fold coverage of region R is provided, if during the time TR
not less than l -fold coverage of every point r s R is provided. If point r never comes upon the
T
instantaneous coverage zones of the satellites (satellite) during the whole period TR, it should be said that the coverage uniformity of r is not provided (or there is no coverage uniformity of r). If there is even one such point among points r s R, it will be said that the coverage uniformity of R is not provided (there is no coverage uniformity of region R).
It should be noted that according to the definition of l -fold coverage, the minimal swath
width providing that coverage for latitudinal belt R is connected to the values ni of
R s R
this parameter for separate parallels ( by the formula:
n [r] = maxnj R4 R^s R ^
So, in order to solve the problem of the latitudinal belt multiple coverage it is enough to do it R s R
for one parallel ( . This fact is used from now on. At that, for simplicity sake, a parameter
' is used instead of ni [R( ].
6. Problem of satellite design for periodic coverage with minimal satellite swath: general solution and some known particular cases
With allowance for introduced designations, a general solution of the problem of multiple
(l -fold) coverage of the parallel by a single satellite swath for odd (l = 2k ~1) and even (l = 2k) cases (k ~ 1 2> ...) is as follows [33].
n = n [r ] =
= max < mm
a
R
(k - l)-AL + SR
R
2
min
a
(k - l)-AL + SL 2
a
/k-AL + dL ^ 2
aR
rk -AL + SR A 2
(20)
nz =n I [R.]=a"1(k-AL),
l = 2k .
(21)
Functions being inverse to functions (18) are designated here as
a
-i R <
a
-1
l , a
-i
Expressions (20) and (21) allow to define multiple swaths (i.e. minimal swaths providing coverage of prescribed repetition factor l) for any given factors m/n, values of geosynchronous orbits inclinations 1 and latitude 9.
It follows from the analysis of expression (21) for even coverage that a required swath width
n R
l is in that case maximal for parallels located in the lesser (in absolute value) latitude ^. Actually, a linear size of the arc of constant angular value AL decreasing with increasing 9 on
each of analysed parallels, is of determining influence on ^l, l = .
The known method widely used in the engineering practice - to design the satellite systems on the basis of assigning the Earth surface non-miss observation swath equal in magnitude to intercircuit distance (the distance between ascending nodes of the satellite orbit on the equator (latitude) on two subsequent circuits) in a minimal latitude of the observation region, can serve as
an application example of the abovementioned property of relation ^1at l _ 2,4,6,....
nv=ALc = n ■AL. (22)
characterizes the swath width measured not perpendicular to the satellite orbit plane (as it was thought before, see [41]) but along the equator (the parallel in a minimal observation
l
l
l
latitude). It is easy to see that swath chosen in such a way is the swath of even 2n -fold coverage
1/ ^ because internodal distance is here always equal to ' n part of inter-circuit distance IC . In
a particular case of round-the-clock (n = 1) orbit it will be a 2-fold (l = 2) swath (ALic _AL)
Another way to choose a minimum needed swath is the method [44] including the swath
representation reasoning from repetition factor m /n of the used geosynchronous orbit of the satellite in accordance with expression
U(=ALIc/ = AL = (23) ( /n m
Formula (23) is a generalization of (22), and it fully defines the 2-fold swath width in compliance with the terminology introduced above in the paper.
It can be shown that the observation periodicity T of the Earth regions with 2-fold swath is
within the limits ts(TTr/2, TTr) So, if the 2-fold swaths are used, the chosen swath can turn out to be unduly high, and periodicity - needlessly low, when it is enough to solve the problem with T = T
periodicity TR The right way in this situation is the use of swaths with the odd repetition factor l and first of all single swaths, i.e. the least swaths providing non-miss observation of the
t — T
region with periodicity TR
n (()
Application of formulas (22) and (23) is based on that the function l decreases
monotonously with growing ( for any even repetition factor l. Width n( of the swath can be thus prescribed in minimal latitude of the observation region (owing to the symmetry of the Earth coverage in the Southern and Northern hemispheres it is here and after supposed that the latitude
changes within the interval 0 — (< 90 ). Relationship ni (() for the odd repetition factor l has qualitatively another character, with sudden changes, because in this case a relative disposition of
the parallel ascending and descending nodes has additional influence upon n1 value. Physical essence of such an influence is illustrated in Fig. 3 by means of a single-swath example. The
R S
parallel ( coverage diagrams are given in Fig. 3 for some ranges of right shift R variation.
*
Critical points r of a single swath, i.e. the mating points of the swaths providing the coverage continuity, are shown, too. Every range of R alteration corresponds to single (l = 1) coverage characterized by angular swath width '
All designations in Fig. 3 are related to a particular case of expression (20) - swath width
= n1 (2Re ) of single (l = 1) coverage in angular measurement: fa = max [min (, fa(4)), min (fa(2), fa(3)), (24)
where
fa^a^R/ 2),
M2)=a_ (dj 2),
fa(3)=XR 1 ((AL + Sr )/ 2), fa4) = a__ ((AL + SL )/ 2).
This implies that in general case function n is not monotonous at single (l =1) coverage (and also at any other coverage with the odd repetition factor l , as can be shown), and can
have several local extremes. Local maxima n(() are thus reached in latitudes where a pair of
*
r
ascending and a pair of descending latitudinal nodes mate simultaneously in critical points (that
is case "c" in Fig. 3, and right shift is here Sr = Sr ). Local minima are reached in latitudes where
*
there are not one (as in all other cases) but two "equivalent" critical points r of mating ascending
tt
and descending nodes (that is case "e", R = R ) on each interval of length . It is interesting
S = Sf a = a
to note that shift R R at L R corresponds to full coincidence of ascending and
tt
S _S
descending nodes, and shift R = R corresponds to a uniform distribution of all latitudinal nodes along the parallel. Diagrams in Fig. 7 are represented for a_ — Xr , and possesses one of
three values , . It can be shown that at Xr — a_ the picture of the parallel coverage
R(3) B(4)
will be analogous, and instead of ^ will "work" ^ in formula (24).
ac
Bp+i\
a
B
Hq+1 Bp+1
AL
H,
Bp-i
B
1 C
p-2
H
H
q-2
Bp-1
B
1 L
Hq+ 1
I_|
H
: l
p-2
H
B
p+1
B
Bp-i
B
H.
q+ 1
. SD=d'c
R R
H
p-2
H„
B
p+1
B
Bp-1
B
I I
H.
I I
p-2
q+ 1
B
G
p+1
Bp r*_r* Bp-1
B
H,
q+1
1« dR=SR Hq
p-2
B
p+1
B
rfe *
Bp-1
B
p-2
e
H„
q+2 q+1 q q-1
Fig. 7: Typical diagrams for single coverage of the parallel depending on relative disposition of the systems of ascending and descending nodes:
a) Sr = 0 (Sr = AL), fa = fa(3) (fa = fa(1)); b) 0 < Sr < dR, fa = p(3);
?(3) _ a(2). A^x ^ R - /?(2).
c) Sr = Sr , fa = fa(3) = fa(2) ; e) Sr = SR, Pi = P(2) = P(1)
d) Sr < Sr < Sr , fa = fa(2) ;
It follows from above that the width of the parallel non-miss observation can be reduced by means of modification of relative position of ascending and descending latitudinal nodes. It is easy to make sure that in such a way the swath width of non-miss observation of large latitudinal belts
can be decreased. Really, it can be obtained from the analysis of the plots ^1in Fig. 8 that maximal swath for the odd (l = 2k ~1 )-fold coverage can be less than analogous swath of even (l = 2k)-fold repetition for some latitudinal belts (k = 1,2,3,...). For example, such latitude
• + 10°...65% m = 15, n = 1, 1 = 65o q in interval is for (Fig. 8,b).
r
a
r
p
b
-1
r
c
r
d
g 40-
30-
20
10-
40
^ 30
C a
20
10
1=4 a
J=2 3?\/
$
B y 1=4 b
1=3^
1=2
1=1 A
10
20
30 40
50 60
deg
Fig. 8. Plots of the swath width Ui for /-fold coverage of the parallel against its latitude ty for
different values l = {l,2,3,4\ and parameters of the satellite orbit: a - m=15, n=1, i=650 b - m=16, n=1, i=650
Several special methods to design the satellite constellations with minimum swath width for non-miss coverage of the Earth latitudinal belts, are considered below. All these methods are based on the consideration of relative dispositions of the ascending and descending nodes on the parallels.
7. Method of satellite design for latitude belt coverage with minimal swath on basis of optimization of orbit altitude and inclination [45]
Alteration of the ascending and descending nodes relative position will be considered over p e [0, (Pj ], <Pj = min( i, 7i — i)
of latitudes covered by the swath of a single satellite
the range
moving in the orbit with inclination 1. For that, the values of right shift dR will be calculated as a function of latitude 9 on the equator 9 = 0)
and on the maximal latitude 9 From (3) at 9 = 0, angle of turn is Al = ~ n)/n\ With allowance for that, right shift dR is as follows:
0
0
Ay _ n(m -n ) _ m-n
AL
m
2n
m
2
a m-n Ay =--AL,
2
dR = Ay -AL ■ E
Ay AL
m-n 2
AL ■
■ AL AL ■ E
/ m-n^
2
m-n
2
E
v 2 y Y
v
m-n 2
y
(25)
It results from (25) that if difference (m n) is an even number (m-n = 2k,k = 12'3' •••), than the systems of latitudinal nodes A and D on the equator
coincide:
dR = AL ■
"2k J 2k A
--E —
2 V 2 y
(26)
= AL ■[k - E (k)] = 0.
If difference (m-n) is an odd number (m-n = 2k-1,k = 12>3>---ascending and descending latitudinal nodes are located on the equator uniformly:
), than all
dR = AL ■
2k +1 ( 2k +1
E
2
2
== AL ■
k + -1-E 2
k +1
. 2 y
= AL 2
— (27)
At 9 = 9i = 1 ^2, angle of turn Aiy = 0, wherefrom at any m, n the right shift Sr = 0. similarly, it will be obtained at 9 = 9i >n^2 that dR = AL' So, in a
maximal latitude 9 = 9i the ascending and descending latitudinal nodes always coincide.
The obtained conditions define the right shift values dR 9) on the latitude range
9 £ l0,9i ] borders. The right shift dR alteration as well as the left shift dL = A dL alteration
within this range are in general case of oscillatory character depending on specific values m, n, 1 (see Fig. 6). The non-miss coverage swath of any parallel will be determined by maximal distance
\ R> LJ between the latitudinal nodes. That is why the most economical coverage of the
equator is realized at moving of the satellite on geosynchronous orbits with the odd difference (m — n) providing a uniform distribution of the nodes on the equator. In this case, for a non-miss coverage of the equator
n^=AL/2 = n/2. (28)
In order to provide swath (28) non-miss coverage of not only the equator but also any other
parallel R( ( e 0 ] it is necessary vector v of the subsatellite point displacement rate in any equatorial node be in the meridian plane. Displacement rate v of the subsatellite point relative to
the Earth is a resulting vector of absolute velocity Va due to the satellite motion along the orbit and
v *
of transportation velocity E' caused by the Earth rotation (Fig. 9). Inclination l conditioning the
velocity v direction being strictly at a tangent to the meridian is defined by the expression
I * = arccos {vE / vA ), (29)
where Va, Ve are the values of respective vectors Va' Ve' defined as
vA =
2xR
E
T
vE =
2xR
E
T
, (30)
DR T EF
It will be obtained after substitution of (16) into (15) and transformation:
i = arccos\n/m
(n/m).
(31)
Fig. 9. To the evaluation of inclination (31)
Relationship (31) for the range of altitude 200...4000 km is represented in Fig. 10. It is supposed that the plot is not an uninterrupted line but consists of discrete points corresponding to
rational numbers m / n on the abscissa axis.
a
i, deg
84-
82
r
8 10 12 14 K=m/n
Fig. 10. Dependence of inclination (31) on the geosynchronous orbit repetition factor
Thus, the satellite moving along geosynchronous orbit with repetition factor
m / n
odd
difference m — n, and having inclination (31) provides the most efficient coverage of the Earth surface - it is enough to have for that the swath of width (28). The physical essence of the resulting coverage efficiency can be illustrated in Fig. 11 similar to Fig. 8 and differing by that the inclination defined by formula (31) is used in the figure. It can be seen from these two figures comparison that
spasmodic graphs n represented in Fig. 8 for l = 1 and l = 65 , become straight in Fig. 11 at inclination (31). When difference of the orbit parameters m and n is an odd number, then maximal
value ( = 0) for repetition factor l = 1 becomes sufficiently less than the analogous value for repetition factor l = 2. It should be noted that the same conclusion would hold with respect to
maximal value of ^l ( = 0)
case
l = 3 and l = 4
', and also at the comparison of repetition factors of higher order - the
is analysed in detail below.
1 40-
g 40-5£
0 10 20 30 40 50 60
9, deg
Fig. 11. Plots of the swath width Ui for l -fold coverage of the parallel against its latitude ty for
different values l = {l,2,3,4\ and parameters of the satellite orbit: a - m=15, n=1; i=arccos(1/15)~86,20 b - m=16, n=1; i=arccos(1/16)~86,40
To compare the efficiency of this method of the orbit formation with traditional method
0 70°
described by expression (23), the observation of the latitude belt "' by means of a single satellite moving on geosynchronous orbit with repetition factor m /n =16/1 (with the odd difference m-n =16-1), will be considered. Taking after traditional approach, inclination 1 = 70°
can be proposed for the region observation, and the swath width is chosen in accordance with (23) for a double swath. The method under consideration supposes the use of inclination
1 — arccos(16/1)~ 86,4 , swath corresponds to the single one. The dependencies between
Ulmax = maxUl (9),9£\0,9i ]
9 and inclination 1 for traditional (l = 2) and proposed (l = !)
methods allowing to choose the swath U are presented in Fig. 12. It is seen that in the first case
U is 2410 km (point A,1 = 70 ), and in the second case - 1250 km (point B, 1 = 86 ,4). So, the proposed method permits to reduce the necessary swath of the satellite by a factor of 2410/1250 «1,9
A 20-
B 10-
l=2
/
l=1
30 50 70 90 110 130 i, deg
Fig. 12: Relationships between I] max value and inclination for traditional (l = 2) and proposed (l = 1) methods to choose the swath width
Observation by one satellite connected with large time breaks was considered above. The placement of a big number of satellites on one circuit under the same conditions allows to apply directly the proposed method for designing the satellite systems for observation with small periodicities.
8. Method of satellite design for low-cost multiple coverage using optimal altitude
and inclination [46]
The coverage repetition increase is one of the ways to rise the observation periodicity. So, it is important to find the methods to minimise the swaths for multiple coverages. Let's appeal with this end in view to Fig. 9 showing the arrangement of ascending and descending latitudinal nodes on the equator at the odd difference m — n of parameters m, n of the geosynchronous orbit. It is seen in this figure that one latitudinal node (ascending or descending) occurs in the satellite swath
of width n/ m within the period TR of the trace repetition, two nodes (ascending and descending)
occur in the swath of width 2n / m, three nodes - in the swath of width 3n / m, etc. The number of the nodes falling into the swath of nodes corresponds actually to the number of observation sessions
of the arbitrary equator point during time TR' i.e. to the repetition factor l of the equator coverage. The choice of inclination according to (31) provides the repetition factor value being not less than that for the equator.
l n/m
3n/m
2■ n/m
%/m
Equator
A!
B„
H„
B
p-1
H
q-1
B.
p-2
Fig. 13: Relative position of ascending and descending equatorial nodes at an odd difference (m — n) of parameters m , n of the satellite geosynchronous orbit
Thus, the conclusion can be made that in order to obtain the l -fold observation of the Earth surface by the satellite during the period of its trace repetition it is enough to provide under prescribed conditions the swath width of
U (n=nV. (32)
9 /m v '
The particular case of expression (32) at l =1 results in (14). At even repetition factors
I = 2k, k = l,2,... expression (32) is reduced to (23). The cases of odd repetition factor numbers
I = 2k +1k = 1,2,... give new wayS t0 provide the Earth coverage. Let's analyse them.
Suppose the need to provide a single satellite observation of the Earth latitudinal belt
°...(55 with periodicity not more than 24 hours and, if possible, lesser width of the satellite swath. Have also the restrictions for altitude H and inclination 1 of the satellite orbit defined by the
following ranges of these parameters alterations often used in the practice: H = 600...800 km,
1 = 65 ...90 . Fig. 14 illustrates the dependencies Ulmax(1 ^ similar to those in Fig. 8 but calculated for various repetition factors l of the coverage and for different parameters m / n, corresponding to the (n = 1)-, (n = 2)- and (n = 3)- orbits from the altitude range 500...900 km.
with altitudes 500.. .900 km during one - (a - n=1), two- (b - n=2) or three (c - n=3) efficient astronomical days
The relationships between altitudes and inclinations of the analysed orbits are shown in
Fig. 11.
H, km 900-
800-
700-
600-
50 70 90 i, deg Fig. 15. Graph of altitudes for geosynchronous orbits within the altitude range 500...900 km of one-, two- and three astronomical days, versus inclination
It is clear that points A, B, C in Fig. 14 are related to the realisation of the method
presented in section 7. Only one of these point meets at that the requirements for periodicity
A T ^ 1 D T ^ 2 /"*
(periodicity in point A « astronomical day, in point B - « astronomical days, in point C
t ^ 3
- ~ astronomical days). However, this point proved to be out off the acceptable range of altitude, namely 600...800 km. Only the realisations of a single-satellite observation corresponding
to parts DE, D E ' , F G ' in Fig. 14 meets fully all the demands in altitude and periodicity. Each of these parts corresponds to analogous (designations without primed symbols) parts DE, FG, HJ in Fig. 15.
D ' E' E ' (T ' ht
Parts ' G ' HJ in Fig. 14 conform to the methods of the swath width choice considered above. The lowest swath width under the prescribed restricting conditions is for these « 2500
methods ~ km. As it can be seen the gain in the swath size due to the use of the method
, (2500-2000) -100%/2500 = 20%. T, , ,
under consideration is v J The example of the realisation
method being proposed can be easily modified in the context of other initial data including multi-satellite systems.
9. Method of multi-satellite network design for low-cost multiple coverage using
optimal altitude and inclination [47]
In use of the orbits with optimal altitude and inclination in accordance with section 7, a
possibility appears to lower the needed swath for latitudinal belt 0...ty1 still more while applying additional N -1 satellites in the orbital group. For instance, if the traces of all N satellites within
internodal interval AL (in particular case, within inter-circuit interval A1C = n A), the next generalisation can be obtained as N -route satellite system. It can be shown that in this case the choice of the repletion factor will be realised under condition N(m n) =2k 1
(k =1, 2,3,...), and the swath will be defined by expression U(p ^^(mN), generalising formula (14). In the event, such placement of the satellite traces does not always give the most economical advantage. The more efficient method for the satellite traces disposition in order of the swath minimisation will be examined below.
Traditional methods of the satellites N traces relative placement are reduced to their uniform distribution (with longitudinal spacing ALTR ) within internodal interval AL (in particular case within inter-circuit interval AIC), and to the choice of the satellite swath width ^ty equal to
U9=ALtr =AyN = 2fmN • (33)
Under the conditions of optimal choice of the altitude and inclination according to the above, when each interval between adjacent like nodes on the equator is divided in two equal parts by the node of another type, the placement of another satellite with longitudinal spacing (33) will not lead to the needed swath width reduction because all latitudinal nodes of the second satellite will in this case coincide accurately with those of the first satellite.
It can be shown that the most efficient use of the locks-on of all available N satellites swaths with the aim to their value in need consists in the providing of uniform distribution along the
equator of the set of latitudinal nodes of all N satellites that allows to reduce the observation swath
proportionally to number N. Such uniform displacement is possible to be gained by placing the N satellites traces with longitudinal shift (Fig. 16)
AL
TR
2n
/ N = 2k +1,
mN' k = 1,2,3,... / N = 2k,
mN,
k = 1,2,3,...
H
H
ALr
H3 B2 H1
ALr
(34)
B3 H2
B1 B2 B3 B4 H1 H2 H3 H4 B1
AL
AL
AL
AL
R
R,„
Fig. 16: Optimal relative arrangement of ascending and descending equatorial latitudinal nodes for two (a), three (b), and four satellites
The swath width is obtained by generalization of (32) (35)
I = n ■ l/
1 /mN
Comparison of expressions (35) and (32) attests directly the method efficiency in minimization of the swath width.
10. General method of multi-satellite network design for low-cost coverage [48]
Note that aR (A) h aL (A) characterise the satellite swath width (measured along the parallel) necessary for the parallel arc A covering accordingly on the right and on the left from the
parallel intersection point by the satellite trace. Write symbols Pr (A) h (A). for them. Expression (10) assumes in the new notations the form: P = max {min p (A1), Pl (A 2 )J min [Pl (A3 ),Pr (A 4 )J}, where
A1 =SR/2, A2 =(AL + SL ) /2, A =SL/2, A4 =(AL + Sr ) /2.
(36)
<
2
B
B
B
a
B
B
b
c
Maximal value of (36) obtained for separate parallels in latitudes ^ has to be taken to
provide the coverage of not a single parallel but latitudinal belt •
n1mx = max n1 M 9 e [9min > 9max i
* (37)
*
Maximum (37) can be reached in any latitude 9 = 9 e[9mm9max i One of the
P = p ^[Pmin'Pmcx 1
parameters ^ (A (A A4 calculated for latitude ф will correspond (and
be equal) to the width value of the non-miss coverage swath for the whole latitudinal belt. One of
the parts A' A2' A or A4 (let's denote it X) related to this maximal value is critical - the largest swath is required to cover it.
Thereby, for decreasing the needed swath of the latitudinall belt Pmin'Pmax 1 owing to the
use of not one but several satellites in proportion to their total number N it seems to be expedient to place the satellite traces with longitudinal spacing. Пр=Щк = 2X/N. (38) It is seen from the comparison of (38) with analogous expression (33) and with allowance
for X — AL//'2^ that the method under consideration provides in general case a more effective coverage of the Earth surface. It can be shown that the gain obtained in the swath width depending on requested parameters H, 1 of the orbits (as well as of the observation latitudinal belt) varies from the values corresponding to such an optimal request (see section 5) to zero (there is no gain at X = AL/2).
11. Некоторые закономерности оптимальных решений
Прикладное значение маршрутной теории оптимизации не ограничивается решением классической задачи периодического обзора. Ее более важная роль состоит в том, что она предлагает новую парадигму мышления при анализе взаимовлияния более широкого набора проектно-баллистических характеристик спутниковых систем периодического обзора. Одним из способов такого анализа является рассмотрение так называемых уровенных поверхностей периодичности обзора - поверхностей в пространстве параметров «периодичность обзора - высота орбит - ширина полос обзора спутников». Действительно, на первый взгляд может показаться, что, ограничиваясь анализом геосинхронных орбит, мы
не можем «увидеть», что происходит в пространстве параметров, включающем высоту. Вместе с тем, и здесь предлагаемый подход оказывается крайне полезным. Покажем это.
На рис. 17 показан фрагмент уровенной поверхности периодичности обзора
широтного пояса 0...70 для простейшей системы из одного спутника, движущегося по
орбите с наклонением i = 85 . Можно показать, что любые попытки получить аналогичную
уровенную поверхность для системы из N>1 спутников в рамках лучших известных
(гарантированных) орбитальных структур окончатся неудачей в связи с тем, что уровенные
линии для различных соседних кратностей m/n (сечений H=const) не будут "стыковаться"
между собой. Более того, в этом случае уровенные линии в сколь угодно близких сечениях
H = const И = const 1 и 2 могут столь сильно отличаться, что это вызывает определенные
сомнения в корректности используемых методов оптимизации (в классе гарантированных
орбитальных структур). Именно данное обстоятельство в свое время навело автора на мысль
о существовании более лучших структур чем гарантированные, каковыми и явились
регулярные СС.
На рис. 18 изображена уровенная поверхность, аналогичная показанной на рис. 17, но полученная для оптимальной регулярной СС с численным составом N=2. Сравнение рис. 17 и 18 позволяет заключить, что данные уровенные поверхности схожи по своему общему внешнему виду. Можно показать, что такая повторяемость в основных своих чертах сохраняется и для уровенных поверхностей периодичности обзора с помощью СС любого численного состава N>2. Данное обстоятельство автор считает важным дополнительным аргументом в пользу того, что экстремальные РСС близки по своим характеристикам к абсолютно оптимальным, поскольку только при условии их использования мы имеем физически понятный, естественным образом укладывающийся на шкале высот ряд отдельных уровенных линий периодичности.
Остановимся подробнее на вопросе, какие особенности присущи всем этим поверхностям.
Из рис. 17, 18 видно, что уровенная поверхность периодичности всегда разрывна, причем отдельные ее "куски" почти параллельны плоскости "полоса-высота" и находятся на уровнях некоторых периодичностей приблизительно кратных половине суток для односпутникового наблюдения (см. рис. 17), четверти суток для двухспутникового наблюдения (см. рис. 18) и, как можно показать, 1/2N части суток для N-спутникового наблюдения в общем случае.
В направлении уменьшения периодичности z (увеличения ширины П полос) уровенная поверхность ограничена значениями П, соответствующими нулевому (заданному минимальному) углу места спутника относительно плоскости местного горизонта объектов наблюдения. (Они находятся за пределами рис. 17, 18, справа от них.)
В направлении увеличения периодичности z обзора (уменьшения П) в каждом сечении
m/n=const (39)
уровенная поверхность ограничена некоторым предельным значением величины
T
зависящим от периода повторяемости тр трассы спутника на геосинхронной орбите данной кратности m/n. Однако, в любом хоть сколько-нибудь продолжительном диапазоне высот уровенная поверхность в рассматриваемом направлении становится бесконечной.
Рис.17. Уровенная поверхность периодичности обзора для одиночного спутника в диапазоне высот 405...1061 км при наклонении г=850 и широтном поясе наблюдения 0...700.
Данный факт является следствием обсуждавшейся выше возможности сколь угодно большого повышения дискретности множества геосинхронных орбит, имеющихся в фиксированном диапазоне высот, путем увеличения числа п эффективных суток в периоде
повторяемости трасс спутников. В пределе с увеличением n «нулевой» ширине П полосы соответствует периодичность обзора т ^ На рис. 4 уровенная поверхность представлена до величины т ~ 60 ч, на рис. 5 - т ~ 30 ч. Прерывистыми линиями на обоих рисунках показаны линии равных кратностей m/n, n=1,2,3,4.
Уменьшение периодичности т в любом сечении (11) с увеличением ширины П
полосы происходит скачкообразно, т.е. функция ^П здесь претерпевает разрывы первого
рода. В некоторых случаях подобные скачки в периодичности имеют малую величину и, с
целью упрощения очень сложной формы уровенной поверхности, не отражены на рис. 17, 18.
В действительности на указанных уровнях периодичности находятся не одиночные "куски",
и п « » т ■
а группы таких кусков , расположенных через соответствующие отступы по величине
Заметим, кстати, что в тех достаточно нечастых случаях, когда абсолютно оптимальные
структуры превосходят по периодичности регулярные СС, это выражается в дополнительном
дроблении указанной группы "кусков" уровенной поверхности.
Каждый "кусок" реальной уровенной поверхности не строго параллелен плоскости
"полоса-высота", а несколько наклонен к ней под небольшими углами таким образом, что
периодичность обзора возрастает с увеличением высоты H при П=const и остается
постоянной с изменением ширины П при H=const. Такой наклон "кусков" уровенной
поверхности объясняется тем, что, как можно показать, различным точкам любого такого
"куска" соответствуют однотипные по структуре оптимальные последовательности сеансов
наблюдения (потоки наблюдений), которые и определяют наибольший, в том или ином
смысле, возможный перерыв (т.е. периодичность т обзора заданного района). Числовое же
T
выражение этого максимального перерыва зависит от драконического периода dr
обращения спутника таким образом, что для двух различных высот Hl' с
TT тт
драконическими периодами dr1 dr2 значения 1 2 периодичности соотносятся в
т ' т = T T
соответствии с выражением 1 2 dr1 dr2 что и предопределяет указанное изменение периодичности для точек каждого "куска".
Рис.18. Уровенная поверхность периодичности обзора для оптимальной двухспутниковой системы в диапазоне высот 405.1061 км при наклонении г=850 и широтном поясе наблюдения 0...700.
Действительно, в свете сказанного понятно, что каждой точке "куска" уровенной поверхности при N=1 (см. рис. 17) соответствует строго определенный поток наблюдений критической точки наблюдаемого района, поскольку такой точке соответствует и
единственно определенная односпутниковая орбитальная «структура». Вместе с тем, ситуация при N>1 и, в частности, при N=2 (см. рис. 18) выглядит несколько иначе: каждой точке "куска" уровенной поверхности соответствуют однотипные оптимальные потоки наблюдения критической точки, но соответствовать ей, в общем случае, могут разные спутниковые системы. Регулярная СС, используемая в расчетах уровенных поверхностей, по сути является одной из возможных таких систем. С точки зрения практики это нисколько не ухудшает ситуацию, поскольку нас интересует любая из СС с лучшими характеристиками.
Чрезвычайно интересной, на взгляд автора, особенностью уровенной поверхности периодичности обзора заданного района спутниковой системой любого численного состава является повторяемость ее формы в направлении изменения высоты орбит спутников с периодом, соответствующим изменению кратности к=т/п (как действительного числа) на две единицы: например, изменению кратности суточной (п=1) орбиты с 14/1=14 на 16/1=16 или кратности двухсуточной (п=2) орбиты с 27/2=13,5 на 31/2=15,5. Действительно,
представлен на рис. 17, 18. На этих рисунках видно, что при соответствующей компенсации увеличения линейного размера ширины П полосы за счет изменения высоты Н можно обеспечить совпадение сечений каждой уровенной поверхности на границах указанного интервала.
К другим полезным результатам мы можем прийти, если рассмотрим уровенную поверхность в пространстве параметров «периодичность обзора - ширина полос обзора спутников - наклонение орбит» при фиксированной высоте (кратности) орбит. Такая уровенная поверхность периодичности обзора показана на рис. 6. Данная поверхность является хорошей иллюстрацией выявленной в рамках маршрутной теории общей закономерности локализации минимумов потребных полос обзора спутников в поле указанных параметров, состоящей в следующем.
Минимумы ширины 1тах полос нечетного 1-кратного покрытия достигаются при наклонении (31) для нечетной разности т-п параметров т, п геосинхронной орбиты. При этом под 1-кратным покрытием заданного района в задаче периодического обзора понимается такое покрытие, когда в течение периода повторяемости трасс каждая точка наблюдаемого района не менее чем 1 раз попадает в зоны обзора спутника (спутников).
последний указанный диапазон изменения кратности
раз и
о
о см
Рис. 19. Уровенная поверхность периодичности обзора широтного пояса (ö,^), (pi = min (i,K — i) в пространстве параметров, включающем наклонение орбиты спутника (m=29, n=2, N=1).
Указанные минимумы ширины полосы нечетного l-кратного покрытия хорошо видны на приведенной на рис. 19 уровенной поверхности для одного из нечетных чисел m-n=29-2
л \0, Pi 1, ( = min Ь.ж -/}. 0
при наблюдении одиночным спутником широтного пояса L J 1 ' В
данном случае минимумы имеются на "кусках" уровенной поверхности, соответствующих
, 1 с т- 47,8; 23,1;12,4
кратностям 1=1, 3, 5 и уровням периодичности ч.
Одновременно из того же рис. 19 видно, что "куски" подобной уровенной
поверхности, приходящейся на четные кратности 1 покрытия, характеризуются изгибом
несколько другого профиля - в направлении изменения периодичности т ' Интересно то, что получающийся в этом случае минимум по периодичности приходится на то же самое
наклонение i =i ■ Так, на рис. 19 при четных кратностях 1=2, 4, 6 достигаются минимумы
т- 36,0; 12,8; 11,2 Л/Г ,
ч. Можно показать, что в случае, когда кратность m/n геосинхронной
орбиты выбрана из другого условия: m-n=2k, k=1,2,..., указанный изгиб имеют все "куски"
соответствующей уровенной поверхности.
Рассмотренные выше закономерности имеют важное самостоятельное значение для
практики баллистического проектирования спутниковых систем. На их основе могут быть
созданы комплексы автоматизированного проектирования орбитального построения ССПО
по заданным ограничениям, а также разработаны специальные способы построения ССПО,
оптимизирующие те или иные их параметры.
12. Conslusions
Elaboration of the route theory has shown that problems of continuous and periodic coverage have strong qualitative difference (that is dictated by the necessity to take into account the Earth rotation at the periodic coverage). Therefore, it is not suitable to look for optimum alternatives of orbital disposition for these two problems within the framework of equal satellite constellation classes. It is proved in [7, 8] that symmetrical systems satisfy necessary optimality condition for the continuous coverage problem. This means that small variations in the position of one satellite in the constellation cannot decrease the main feature of the constellation as a whole. At the continuous coverage, as it is known, such feature is an angular value of the satellite coverage area on the Earth surface. Likewise, taking into account that the said feature for the periodic coverage system is the revisit time, it is demonstrated that the route systems satisfy the necessary optimality condition for the periodic coverage problem.
Power of the route theory is not at all exhausted by revealing the above regular constellations class being although a particular class of satellite systems. The main worth of the theory is that it gives a general method for optimal design of constellation under given requirements
using, for instance, optimum regular constellations as a good initial approach. This method allows finding a global optimum for the above stated classical problem of periodic coverage. Practical application of the said method would be impossible without some other theoretical results achieved by the route theory - analytical solution for calculating problem of distribution of revisit time values on the Earth surface for one satellite and multi-satellite route pattern, formulating and proving of several regularities for revisit time as a function of satellite positions in route constellations, etc.
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Author
Yury N. Razoumny,
Bauman Moscow State Technical University, Russia. Contacts: yr@viaspace.ru
