IAA-AAS-DyCoSS1-12-02
CONSTELLATION DESIGN FOR RESPONSIVE REVISITING
Ming Xu* and C.R. McInnes†
Responsive communications or Earth observation missions typically require frequent revisit to any point of the Earth’s surface and employ a constellation with
two satellites to improve coverage performance. As a new approach to constellation design, a novel constellation is presented by fixing the interval between
successive revisits at all latitudes. Analytic and numerical simulations are implemented to illustrate that such a revisiting constellation exists in a large region
parameterized by the relative values of right ascension of ascending node and
argument of latitude. A global investigation is presented to address the practical
time-resolution of the constellation and its topological shape, the existence region, and the relationship to the semi-major axis and inclination.
INTRODUCTION
In recent years, there has been a growing interest in responsive space missions, including the
both civil and security applications. Some typical missions, such as TacSat-1, 2, 3 and 4 were implemented to build a deployable Operationally Responsive Space (ORS) System1. Moreover, the
constellation architecture employed is used to significantly improve the coverage performance.
Generally, responsive communications or Earth observation missions typically require frequent revisits to any point on the Earth’s surface, and usually adopt LEO orbits with altitudes of
800 to 1500 km for long mission life (low air drag) and acceptable launch cost2. It is expected to
decrease the revisiting time for specified points, which is referred as the time-resolution (TR) in
the constellation design methodology. Wertz proposed five potential responsive orbits to improve
the coverage performance for responsive missions3. Both Cobra and Magic orbits are highly elliptical and at the critical inclination, and so provide mean time-resolution of 24 and 12 hrs respectively; low altitude circular Sun synchronous orbits can provide a time-resolution of 6 hrs; LEO
fast access orbits provide 5 minutes of coverage once or twice per day; LEO repeat coverage orbits adopt resonant orbits to provide a mean time-resolution of 9 hrs.
For the purpose of responsive revisit, the orbits are expected to provide good performance on
revisit intervals to some point on the Earth’s surface. For an orbit at an altitude of 1100 km with
an inclination of 63.43°, the revisit intervals vary from 9.5 to 12 hrs as the station’s latitude increases, shown in Figure 1. Therefore, a constellation with two satellites is preferred to improve
coverage performance. The constellation parameters can be identified by the pair (/u), where
 and u are respectively the relative values of right ascension of ascending node and argument
*
Assistant Professor, Dept. of Aerospace Engineering, School of Astronautics, Beihang University, 37 Rd. of Xueyuan,
Dist. Haidian, Beijing, China.
† Professor, Advanced Space Concepts Laboratory, University of Strathclyde, James Weir Building, Glasgow G1 1XJ,
UK.
1
of latitude. Walker presented a general constellation, the ‘Walker-’ pattern to deploy different
satellites, which only involve circular orbits and is identified in a T/P/F form, where T is the total
number of satellites, P is the number of orbital planes and F defines the relative phases of the satellites 4 . Kantsiper, et al, addressed the design of constellations employing the ORS satellites
Tacsat-1, 2, 3 and 4, with the assistance of Walker-like patterns5.
12.5
Time-Resolution [h]
12
11.5
11
10.5
10
9.5
9
0
10
20
30
40
50
60
70
Latitude [ 0]
Figure 1. Time Resolutions of a Single Satellite at Different Latitudes.
For the purposes of optimization, the Walker- constellation cannot achieve optimal results to
improve the time-resolution to any point on the Earth’s surface. So optimization methods are employed to decrease the revisit intervals; for example, genetic algorithms have potential application
in designing the constellation6. However, genetic algorithms are good at searching for the minimum TR on a specified latitude, but weak in the further goal of holding TRs on different the latitudes.
As an extension of traditional constellations, an interesting constellation can be found by fixing the interval between successive revisits on all latitudes, as presented in this paper. Analytic
and numerical simulations are implemented to illustrate that the revisiting constellation exists in a
large region parameterized by the relative values of right ascension of ascending node and argument of latitude. A global investigation is presented for the practical time-resolution constellation
on its topological shape, the existence region, and the relationship with the common orbital parameters as semi-major axis and inclination.
For the constellation the satellites could arrive at symmetrical regions on the northern or
southern hemisphere; and so it is then demonstrated from symmetry that the time-resolution for
any latitude on the northern hemisphere is equivalent to the one on the southern hemisphere. So
only the TR on the northern hemisphere is referred in the paper.
OPTIMAL PRACTICAL TIME-RESOLUTION ON SPECIFIED LATITUDE
Definition of Practical Time-Resolution
The coverage performance for revisiting includes two different variables: one is the maximum
interval between target revisits, and the other is the frequency with which the maximum interval
reoccurs. For quasi-periodic coverage, some revisits are quite frequent (generally, less than 10
2
days per revisit), but others are infrequent (usually, more than 30 days per revisit). Only the interval between the frequent revisits is significant for responsive communications, which is referred
to as the practical time-resolution (PTR), even through the interval between infrequent revisits
has achieved the true maximum.
For the common orbital parameters with an altitude of 1000 km and the inclination of 63.43°,
the time resolution of the constellation pair (108°/240°) is quite different from the pair
(105°/240°), demonstrated in Figure 2. Both can achieve the maximum time-resolution of 6h;
however, the frequency of the former is about 4.18 days, but the frequency of the latter is up to
58.82 days. So the two pairs are considered to have different practical time-resolutions of 6 and 4
hrs, respectively.
Figure 2. Comparison of TRs on the Pairs of (108°/240°) and (105°/240°) on Specified Latitude of 0°
Based on the definition of PTR, it is practicable to check PTR by the following method: for
the threshold value of revisit frequency f (days per revisit), the revisiting schedule during Nf
(where N>1) days, including the starting time and duration for every revisit, is achieved from the
integration of orbital dynamics associated with the J2 or J4 perturbation7; and then the frequency
of maximum interval is checked whether it is less or more then the threshold value.
Following the method above, the ergodic representation of PTR in space of ‘-u’ for the
specified latitude of 25° is demonstrated in Figure 3, where the pair (108°/240°) will achieve the
optimal PTR of 3.8167 hrs. However, the pair has a poor PTR of 5.6 hrs on latitude from 0° to
+13° (shown in Fig.4), which is not acceptable for global observation missions. There is a ladderlike change of PTR in the low latitudes near 13°, which is not expected from further constellation
design.
3
PTR [h]
7
98
350 11
10
65
5
6
300
98
11
10
5
4
6
11
25010
5
Optimal Constellation
9
4
98
0
u [ ]
10
7
10
200
7
9
45
54
8
6
10
8
150
109
7
7
5
7
5
6
10
5
6
8
100
10
11
50
5
10
9
7
65
5
10
8
5
0
9
10
0
20
8
40
7
60
4
5
80
100
120
140
160
180
 [0]
Figure 3. Ergodic Representation of PTR in Space of ‘-u’ on Specified Latitude of 25°.
6
PTR [h]
5.5
5
4.5
4
3.5
0
10
20
30
40
50
60
70
0
Latitude [ ]
Figure 4. PTRs on Different Latitudes for the Pair of (108°/240°).
Analytic Solution to the Ladder-like Change on PTR
Without loss of generality, the geographic latitude is taken as   0,70 0  , descending and ascending nodes on this latitude are marked as ‘/’ and ‘×’, respectively. So the distribution of the
descending and ascending nodes of the two satellites in the constellation is shown in Figure 5.
4
C
    


Sat2
Sat1

C

    

Figure 5. The Distribution of Descending and Ascending Nodes of the Constellation.
The interval angle between the adjacent ascending or descending nodes is


 T
  E  

(1)
which is constant for any adjacent nodes on this latitude; while the angle between descending
node and its adjacent ascending node is
    
(2)
where
    2u 
T
 , u  arcsin  sin   .
 E  
2
 sin i 


Because all the nodes cannot arrive at the latitude at the same time, the relative right ascension of
ascending node  is introduced as
   
u
  .
2
(3)
So the distribution of descending and ascending nodes of the constellation can labeled as
S       
      
which will maintain their interval angles with the same rotating rate of
(4)

 to achieve
 E  
T


the global coverage.
So the time-resolution can be defined as    min max S  , where the function max is em,u
ployed to assign the constellation equally, and the function min is used to optimize the minimum
time-resolution.
More discussions of the components of S are implemented as follows:
The first component   has the relationship with the number of orbital revolution N and the
full angle of field-of-view  with a nadir-pointing conical sensor as
5
        N  
.

        N  1  
(5)
So the time-resolution for revisiting   is 1  N  1  T  1   2 , where  1  
start time for the coverage on ,  2  


 T is the
          N  1
 T is the end time, and

T is the duration for the total coverage on  just dependent on the common orbital parameters
a and i.
The second component      has the relationship with the number of orbital revolutions N and the coverage angle  to the Earth, as
            N  
.

            N  1  
(6)
So the time-resolution for the section is 2  N  1  T   1   2 , where 1  

1
 T  T is the

2
               N  1
 T is the
start time for the coverage on the section,  2  

end time, and T is the nodal period just dependent on the common orbital parameters a and i.
The third component      has the relationship with the number of orbital revolutions
N and the coverage angle  to the Earth, as
            N  
.

            N  1  
(7)
So the time-resolution for the section is 3  N  1  T   1   2 , where  1  

1
 T  T is the

2
               N  1
 T is
start time for the coverage on the section, and  2  

the end time.
The fourth component is the same as the first one.
As a result, the time-resolution of the constellation is   min 1 , 2 , 3 , 4  . The fact that N in
Eqs. (5), (6) and (7) is a positive integer will cause the discontinuous changes on different latitudes and , i.e., the ladder shape in the ergodic representation of PTR in space of ‘-u’ in
Figure 3.
The comparison between the analytic and the numerical solution is implemented in Figure 8
for the discontinuous ladder-like changes on different  for the fixed latitude  =0° and u=0°.
It is indicated that the analytic solution to the time-resolution is accurate enough to predict the
ladder-like changes. So the discrete number of orbital revolutions N is considered to cause the
discontinuous time-resolution, and has the potential application in avoiding the ladder-like changes by fixing the number of orbital revolutions to acquire the practical constellation holding PTRs
on all altitudes.
6
Figure 8. Comparison between Numerical and Analytic Results for Ladder-like Change
PRACTICAL CONSTELLATION HOLDING PTRS ON ALL ALTITUDES
Instead of achieving the minimum value on the specified latitude, the constellation near the
optimal pair is expected to hold continuous PTRs on all latitudes, which is referred as the PTR
constellation in this paper. The ergodic relationships between the PTR and  or u are investigated near the optimal pair (108°/240°) in Figure 9 and 10, respectively. Obviously, u has a significant effect on PTR; however,  has a slight effect on PTR, especially from 97° to 121°. It is
therefore necessary to check the ergodic representation of PTR on the parameters of  and the
latitude , demonstrated in Figure 11.
5
4.8
PTR [h]
4.6
4.4
4.2
4
3.8
0
50
100
150
200
250
300
350
u [0]
Figure 9. The Ergodic Relationship between PTRs and u (=108°, =25°).
7
11
10
9
PTR [h]
8
7
6
5
4
3
0
20
40
60
80
100
120
140
160
180
 [0]
Figure 10. The Ergodic Relationship between PTRs and  (=108°, =25°).
PTR Constellation
70
9.51
8.515
10.505
7.52
6.525
PTR [h]
2.545
3.54
5.53
4.535
10
60
40
0
 [ ]
7.52
6.525
8.515
10.505
9.51
50
7.52
6.525
8.515
10.505
9.51
3.54
5.53
4.535
7.52
8.515
7.52
8.515
10
7.52
5.53
6.525
5.53
0
20
40
60
4.535
5.53
5
4
6.525
7.52
3
4.535
80
6
6.525
6.525
4.535
5.53
0
7
2.545
2.545
3.54
3.54
4.535
5.53
4.535
5.53
5.53
4.535
6.525
8
2.545
3.54
5.53
4.535
30
10.505
9.51
2010.505
9
2.545
9.51
100
120
140
8.515
7.52
160
2
180
 [ ]
0
Figure 11. Ergodic Representation of PTR in Space of ‘-’ with u=240°.
The ergodic representation in the space of ‘-’ has a diffident ladder-like shape with the
one in the space of ‘-u’. An interesting appearance in Figure 11 is that some constellations
holding the PTR for all the latitudes can share the same value of , such as 105°, which is contributed from the same ladder labeled as 4h shared by a line from the latitudes of 0° to 70°. The
existence of the PTR constellation is confirmed by the ergodic representation of the pair
(105°/240°) on different altitudes, shown in Figure 12. The continuous PTR are held by the pair
around 4hrs.
8
4.5
4.4
PTR [h]
4.3
4.2
4.1
4
3.9
3.8
0
10
20
30
40
50
60
70
0
Latitude [ ]
Figure 12. PTRs on Different Latitudes for the Pair (105°/240°).
Based on the existence of the PTR constellation, it is practicable to search the constellation by
the following method: for some sequence selected from key latitudes as 0°, 10°, … , and 70°, the
difference between any point and its adjacent point in this sequence is required to be less than
0.5hr by the PTR constellation. Then the PTR constellation and the unsatisfied constellation are
classified and labeled in the space of ‘-u’ in Figure 13.
Time  Resolution [h]
Time  Resolution [h]
Unsatisfie d Constellat ion
Unsatisfie d Constellat ion
Optimal Constellat ion
Unsatisfie d Constellat ion
Unsatisfie d Constellat ion
Figure 13. PTR Constellations Holding Time-Resolutions on All Latitudes.
However, the optimal pair (108°/240°) does not belong to PTR constellation, which makes the
PTR constellation sub-optimal for time-resolution, but more practical than the optimal pair.
GLOBAL ANALYSIS ON PTR CONSTELLATION
A global investigation is implemented for the PTR constellation on its topological shape, existence region, and the relationship with the common orbital parameters as semi-major axis and
inclination.
9
Numerical simulation is implemented to illustrate that the PTR constellation exists in a quadrangle-shape region surrounded by two parallel lines in the space of (-u). The characteristic
parameters for the PTR constellation are defined as the distance L between the lower and upper
boundaries, and the gradient  of the lower or upper boundary, depicted in Figure 14. For the case
of h=1100km and i=63.43°, the lower and upper boundaries within the reachable region of [88°,
111°][167°, 424°] are formulated as u= mod(13 + 1629°, 360°) and u= mod(13 +
1642°, 360°) respectively, where mod(*, 360°) is the function of modulus after division by 360°.
Therefore, the characteristic measurements for this case are L=13° and =-13.
PTR [h]
140
4.7
lower boundary
upper boundary
4.6
90
4.5
40
L
4.4
0
u [ ]
350
4.3
300
Optimal Constellation
4.2
250
4.1

200
4
150
3.9
100
75
80
85
90
95
100
105
110
115
120
 [0]
Figure 14. The Existence Region of PTR Constellations for a=7471.137km and i=63.43°.
The space of ‘-u’ is a topological mapping onto a torus, with the boundaries of the PTR
constellation demonstrated in Figure 15.
Figure 15. The Boundaries of the PTR Constellation on a Torus.
An iterative algorithm to generate the lower and upper boundaries of the PTR constellation is
proposed from the continuation procedure to perform the ergodic representation in the space of
‘a-i’8. The global distribution for the locations of the lower and upper boundaries from the iterative algorithm is interesting. The locations are continuous with the two orbital parameters a and i
because no bifurcation occurs for the boundaries. Thus, the true value of the location for the specified parameters of a and i can work as the successive initial guesses for the updated parameters
of a+a and i+i, with only slight step changes. The initial states for the lower and upper bound-
10
aries can be chosen as [l,  ul0 ] and [u,  uu0 ], where  ul0 =ul-20° and  uu0 =uu+20°. [l,
ul] and [u, uu] are located on the boundaries for the initial parameters of a=7471.137km and
i=63.43°. The updated state is produced by the iteration algorithm as  ul1 = ul0 +1° and
 uu1 = uu0 -1°.
With respect to the parameters a and i, the characteristic measurements  and L can be yielded
from the lower and upper boundaries, shown in Figure 16 and 17, respectively.
L [ 0]
1200
200
1190
180
1180
160
1170
140
h [km]
1160
120
1150
100
1140
80
1130
60
1120
40
1110
20
1100
64
65
66
67
68
69
70
i [0]
Figure 15. The Ergodic Representation of L in the Space of h-i.

1200
0
1190
-2
1180
-4
1170
h [km]
1160
-6
1150
-8
1140
1130
-10
1120
-12
1110
1100
-14
64
65
66
67
68
69
70
i [0]
Figure 16. The Ergodic Representation of  in the Space of h-i.
CONCLUSION
Responsive communications or Earth observation missions typically require frequent revisit to
any point of the Earth’s surface and employ a constellation with two satellites to improve coverage performance. As n extension of traditional constellations, an interesting constellation with can
fix the interval between successive revisits on all latitudes is presented in this paper.
11
The coverage performance for revisiting includes the maximum interval between target revisits and the frequency with which the maximum interval reoccurs. Only the interval between the
frequent revisits is significant for responsive communications, which is referred to as the practical
time-resolution, even through the interval between infrequent revisits has achieved the true maximum.
The analytic solution for the time-resolution is accurate enough to predict the ladder-like
changes. So the discrete number of orbital revolution N is considered to cause the discontinuous
time-resolution, and has the potential application in avoiding the ladder-like changes by fixing the
number of orbital revolution to acquire the practical constellation holding the PTRs on all latitudes. Numerical simulations are implemented to illustrate that the revisiting constellation exists
in a large region parameterized by the relative values of right ascension of ascending node and
argument of latitude.
Furthermore, a global investigation addressed the practical time-resolution constellation on its
topological shape, the existence region, and the relationship with the common orbital parameters
as semi-major axis and inclination.
ACKNOWLEDGMENTS
M. Xu acknowledges the supports of the National Natural Science Foundation of China
(11172020), Aerospace Science and Technology Innovation Foundation of China Aerospace Science Corporation, and Talent Foundation supported by the Fundamental Research Funds for the
Central Universities.
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