Astrodynamics Vol. 2, No. 1, 2537, 2018 https://doi.org/10.

1007/s42064-017-0009-2

Optimization of observing sequence based on nominal trajectories of

symmetric observing configuration

Hongwei Yang , Gao Tang , and Fanghua Jiang (B)

1 1,2 1

1. School of Aerospace Engineering, Tsinghua University, Beijing 100084, China

2. Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA

ABSTRACT KEYWORDS
This paper presents the crucial method for obtaining our teams results in the 8th Global branch and bound
Trajectory Optimization Competition (GTOC8). Because the positions and velocities of optimization
spacecraft cannot be completely determined by one observation on one radio source, the symmetric observing
branch and bound method for sequence optimization of multi-asteroid exploration cannot configuration
be directly applied here. To overcome this difficulty, an optimization method for searching GTOC
the observing sequence based on nominal low-thrust trajectories of the symmetric observing low thrust
configuration is proposed. With the symmetric observing configuration, the normal vector of
the triangle plane formed by the three spacecraft rotates in the ecliptic plane periodically and
approximately points to the radio sources which are close to the ecliptic plane. All possible
observing opportunities are selected and ranked according to the nominal trajectories
designed by the symmetric observing configuration. First, the branch and bound method is Research Article
employed to find the optimal sequence of the radio source with thrice observations. Second, Received: 16 December 2016
this method is also used to find the optimal sequence of the left radio sources. The nominal Accepted: 20 June 2017
trajectories are then corrected for accurate observations. The performance index of our 2017 Tsinghua University
result is 128,286,317.0 km which ranks the second place in GTOC8. Press

a symmetric observing configuration for the sequence

1 Introduction optimization.
In the 8th Global Trajectory Optimization Competition The reason that we used the symmetric observing
(GTOC8), an innovative problem, which is related to configuration and proposed the observing sequence
interplanetary trajectory design, was released by the optimization based on nominal trajectories is because
Jet Propulsion Laboratory. The theme for this new of the particularity and complexity of GTOC8
problem is high-resolution mapping of radio sources compared with the previous GTOCs. For the most
in the universe using space-based very-long-baseline of the previous GTOCs, the problems concern
interferometry (VLBI) [1]. During the competition, low-thrust trajectory optimization for asteroid
we proposed an effective design method for this new missions, such as asteroid defense, asteroid sample
problem. Using this method, all computations were return, multiple asteroid rendezvous, and flyby,
finished efficiently on our personal computers. The etc. (http://sophia.estec.esa.int/gtoc portal/). Extensive
authors team, Tsinghua University, ranks the second papers have been published in the fields of low-
place in GTOC8 [1]. Our team achieved that all observed thrust interplanetary trajectory optimization. The
radio sources are near the ecliptic plane. These observed optimization methods can be classified into two
radio sources are preferable targets according to the categories: indirect methods [28] and direct methods
performance index of GTOC8. The goal of this paper is [9]. Gao and Kluever [10] also proposed a hybrid
to introduce the optimization method of the observing method which combines these two kinds of methods.
sequence as well as the nominal trajectories of As for improving the solution efficiency, there are two

B jiangfh@tsinghua.edu.cn
26 H. Yang, G. Tang, F. Jiang

levels of techniques The first level is about global First, the symmetric observing configuration and its
searching. One of the most popular approaches is nominal trajectories are further analyzed. Second, the
to approximate low-thrust trajectories by impulse algorithms for optimization of the observing sequence
trajectories and then search the optimal asteroid are given in detail.
sequence through the branch and bound method [5]. The rest of the paper is organized as follows. In
Because impulse trajectories are not equivalent to Section 2, the problem of GTOC8 is briefly introduced.
low-thrust trajectories, fast methods for evaluating Section 3 presents and analyzes the symmetric observing
low-thrust trajectories are presented as well [6, 7]. configuration. In Section 4, the designed nominal
Additionally, there are studies about determining trajectories of the symmetric observing configuration is
optimal gravity assist sequences [8,11]. The second level introduced and analyzed. Section 5 presents the branch
is about low-thrust trajectory optimization. Amount and bound method for searching the optimal observing
sequence. Section 6 concludes the paper.
of powerful techniques have been proposed, such as
smoothing techniques [2, 4], parallel computation [3],
a shape-based trajectory technique [9], etc. However, 2 Background
two new features appear in the problem of GTOC8
compared with the previous GTOCs, leading the In this section, the problem of GTOC8 [1] is briefly
problem to be more complex. Three spacecraft need introduced. Three low-thrust propelled spacecraft are
to cooperate for observations. More importantly, the launched to formulate a series of triangles to observe
radio sources. The equation of motion of each spacecraft
directions of the radio sources cannot be used to
is
determine the position and/or velocity of any of the
r Tmax u
spacecraft. Although the problem of GTOC7 is also r + 3 = (1)
r m
about a multiple spacecraft mission [12], searching of
target sequences of each spacecraft can still be regarded where r is the position vector of the spacecraft in the
independently. Besides, the position and velocity of each Earth Mean Ecliptic and Equinox of J2000 frame,
spacecraft are equal to that of the target asteroid at each is the gravitational constant of the Earth, Tmax is the
maximal thrust magnitude, and m is the mass of the
rendezvous time.
spacecraft. The maximum magnitude of the thrust is
A symmetric observing configuration is found by the
0.1 N. The control variable consists of the unit vector of
authors during GTOC8. With this configuration, the
thrust direction and the engine thrust ratio u [0, 1].
observing direction rotates periodically in the ecliptic
The mass variation is computed by
plane, which is preferable for repetitive observations of
the radios close to the ecliptic plane. This configuration Tmax u
m = (2)
is also taken as an efficient tool by some other Isp g0
teams such as the team of State Key Laboratory where Isp is the specific impulse, and g0 is the standard
of Astronautic Dynamics and the team of Chinese acceleration of gravity at the sea level.
Academy of Sciences [13, 14]. These teams [13, 14] both The three spacecraft are launched from a 400-km
got excellent results. One of main differences between altitude circular orbit around the Earth. The whole
our method and their method is that the optimization mission is no more than 3 years.
The range of the spacecraft relative to the Earth must
of the observing sequence. In this paper, we take the
obey the following condition:
idea of using the branch and bound method to search
the optimal observing sequence. Different from the 6578.14 km 6 r 6 1,000,000.0 km (3)
application of the branch and bound method in sequence An observation is recorded when the normal direction
searching of the multiple asteroid rendezvous mission of the observing triangle is aligned with the direction
[5], the searching in the current paper is based on the of a radio source. The direction error must be less
pre-designed nominal trajectories so that no lambert than 0.1 deg. The time intervals between two adjacent
problems need to be solved. observations must be no less than 15 days.
Our method for GTOC8 has been briefly introduced The performance index J to be maximized is [1]:
in our previous conference paper [15]. Compared with X
P h 0.2 + cos2


J= (4)
that paper [15], this paper makes two contributions.
Optimization of observing sequence based on nominal trajectories of symmetric observing configuration 27

where P is the weighting factor, h is the smallest of the is shown in Fig. 2. As for this configuration, the following
three altitudes of the observing triangle, and is the four conditions are satisfied: (1) two of the three
declination of the observed radio sources. An altitude spacecraft, denoted as S1 and S2 , orbit symmetrically
of a triangle is the perpendicular distance from a vertex about the ecliptic plane; (2) the other one of the three
to the opposite side (or its extension) [1]. The observing spacecraft, denoted as S3 , stays in the ecliptic plane;
altitude h must be larger than 10,000 km for effective (3) the orbits are all circular with the same radius; (4)
observations. The weighting factor P takes on the values the arguments of latitude of S1 and S2 are the same (the
of 1, 3, 6, and 0 for repeat observations according to the ascending nodes are opposite) while the difference in the
rules. The most attractive case is to observe the same
radio source for three times and the three observing
altitudes satisfy h2 /h1 > 3 and h3 /h2 > 3.

3 Analysis of the symmetric observing

configuration
3.1 Reason of using the symmetric
observing configuration
According to Eq. (4), radio sources near the ecliptic
plane are preferable because a low declination greatly
contributes to the performance index. Therefore, an
observing configuration with its observing direction
rotating in the ecliptic plane is considered in the (a) All given radio sources
preliminary design. In GTOC8, 420 radio sources are
given by the organizer. Because the time interval
between two adjacent observations should be no less
than 15 days, no more than 16 different radio sources,
if each one is observed thrice, can be observed in two
years. Actually, 37 radio sources, whose declinations
are from 3 to 3 deg, can be selected from all 420 given
radio sources, as shown in Fig. 1. Therefore, the radio
sources near the ecliptic plane are enough for designing
a competitive observing mission. Because the number
of radio sources is reduced to 37, the complexity of
computations is greatly reduced.
Additionally, it contributes greatly to the
(b) The selected radio sources
performance index if the weighting factor P takes
Fig. 1 Selection of the radio sources.
the maximum value. But, it requires that a radio source
is observed thrice. In order to repeatedly observe many
radio sources, it is beneficial to rotate the observing
direction periodically.
Therefore, we propose to use the symmetric observing
configuration which enables the observing direction
rotating periodically in the ecliptic plane. The geometry
of the desired configuration and its properties will be
given below.

3.2 Geometry of the symmetric observing

configuration
The geometry of the symmetric observing configuration Fig. 2 Geometry of the symmetric observing configurations.
28 H. Yang, G. Tang, F. Jiang

arguments of latitude of S1 and S3 is . then = + 2. The history of with for different

Denote the inclination and the argument of latitude i is shown in Fig. 3. According to this figure, has
of S1 as i and , respectively. Then, the position vectors periodicity for each inclination. It also can be found
of the three spacecraft are derived as follows: that the curve of to becomes straight when the
r1 = r[ cos sin cos i sin sin i ] (5) inclination decreases. Nevertheless, the effect of i on
r2 = r[ cos sin cos i sin sin i ] (6) is small according to the shapes of the curves.
r3 = r[ cos sin 0 ] (7)
The normal direction of the observing triangle is 4 Nominal trajectories of the symmetric
(r1 r2 ) (r1 r3 ) observing configuration
n= (8)
k(r1 r2 ) (r1 r3 )k
where In this paper, we only concern the low-thrust
trajectories after the Moon gravity assists because there
(r1 r2 ) (r1 r3 ) =
is no effective observation in our design result before
r2 [ 2sin2 sini(cosi+1) 4sinsinicos 0 ] (9) gravity assists. Before the Moon gravity assists, we
use the analytical control law which maximizes the
According to Eqs. (8) and (9), the normal direction
variation of the apogee to design the multiple-revolution
of the observing triangle keeps parallel to the ecliptic
trajectories. In this way, the orbits of spacecraft
plane, which is also shown in Fig. 2.
can be raised to the Moon rapidly. At the end stage,
Moreover, the observing altitude is low if S1 and S2
the trajectory optimization is conducted to obtain the
are close to the ecliptic plane but it becomes higher
if S1 and S2 move away from the ecliptic plane. To desired final state before the Moon gravity assists. More
obtain the maximum h, the following conditions must details for designing the low-thrust trajectory before
be satisfied: (1) the inclination of S1 is 60 deg; (2) the the Moon gravity assists can be found in Ref. [15].
three spacecraft consist of an equilateral triangle; (3) Additionally, we use the homotopy method [4] with
the radius of each spacecrafts orbit is 1106 km. Once the logarithmic homotopy [2] for low-thrust trajectory
these conditions are satisfied, h is equal to 1.5106 km. optimization. And the equation of motion is described
The configuration with i = 60 deg is chosen as the final by the equinoctial elements [10]. More details on the
configuration in our strategy. low-thrust trajectory optimization can also be found in
Ref. [15].
3.3 Properties of the symmetric observing The state variables of the three spacecraft are
configuration shown in Table 1. After the Moon gravity assists,
Because n is always parallel to the ecliptic plane, the the orbits of all spacecraft are ellipses with high
first property of the symmetric observing configuration eccentricities. Besides, the orbital planes of S1 and
is that the observing direction can be regarded as
staying in the ecliptic plane.
The second property is about the periodicity of the
observing direction. According to Eqs. (8) and (9), the
right ascension of the observing direction is
1 = arctan 2 4sinsinicos, 2sin2 sini(cosi+1)

(10)
or
2 = arctan 2 4sinsinicos, 2sin2 sini(cosi+1)

(11)
Because the difference between 1 and 2 keeps
, only the variation of 1 is analyzed herein. Both
directions will be considered in searching the optimal
sequence. Besides, the considered range for is from 0 Fig. 3 Periodicity of the observing direction with different
to 2 in the rest of this paper. If is less than zero, inclinations.
Optimization of observing sequence based on nominal trajectories of symmetric observing configuration 29

Table 1 State variables of spacecraft immediately after the Moon gravity assists

Spacecraft Epoch (MJD) x (km) y (km) z (km) vx (km) vy (km) vz (km) Mass (kg)
1 59077.839 52941.717 372202.105 0 0.672 0.900 0.028 1994.145
2 59077.839 52941.717 372202.105 0 0.672 0.900 0.028 1995.612
3 59090.705 54833.246 385500.332 0 0.876 0.653 0 1994.060

S2 are very close to the ecliptic plane. To achieve The nominal trajectories in each stage are presented
the final observing configuration as well as adjust the and analyzed below.
size of the configuration for obtaining desired observing Nominal trajectories in stage 1
altitudes, both circularization and adjustments of The designed nominal trajectories for stage 1 are
orbital inclination are essential. shown in Fig. 5. According to the figure, the semimajor
The designed nominal trajectories in our methods axis of each spacecrafts orbit increases and all orbits
belong to four stages. In the first stage, the orbits of are circularized during this stage.
the three spacecraft are circularized and adjusted to The history of the observing altitude and direction
construct the symmetric observing configuration. The for stage 1 are shown in Fig. 6. The reason that there
target radius is set to 0.99106 km. In the second stage, are two color lines in Fig. 6(b) is because has two
the inclinations of S1 and S2 are changed to 13 deg and solutions. This explanation also holds for the figures
the time spent on this stage is 2 orbital periods. In order in other stages. According to the figure, the observing
to achieve observations with high performance, a radio altitude and direction are not periodic in this stage.
source needs to be observed three times and the three This is because the symmetric observing configuration
observing altitudes should satisfy hmax /hmid > 3 and has not been constructed before this stage.
hmid /hmin > 3 [1]. The h for each i takes its maximum Nominal trajectories in stage 2
when = . The variation of h with respect to i when The designed nominal trajectories for stage 2 are
= is shown in Fig. 4. The inclination of 13 deg is shown in Fig. 7. According to the figure, both the
chosen to make the maximum observing height at the inclinations of S1 and S2 change during this stage and
last coast stage as about 1/3 of the largest observing then stay at the desired values.
height. The length of this period is approximate to The history of the observing altitude and direction for
those of the first stage and the last coast stage. The stage 2 are shown in Fig. 8. The first wave in Fig. 8(a)
reason that this ratio is not exactly 1/3 is to make the is not the same as other three waves because the
period of the third stage as integral multiples of the half inclination adjustment is conducted. After that, both
orbital period. In the third stage, the inclinations of S1 the observing altitude and direction becomes periodic.
and S2 are changed to 60 deg. In the fourth stage, all And the range of the observing direction is from 0 to
spacecraft coast until the mission ends. 2.

Fig. 5 Nominal trajectories in stage 1.

Fig. 4 Variation of h with respect to i when = .
30 H. Yang, G. Tang, F. Jiang

(a) (a)

(b) (b)
Fig. 6 History of the observing altitude and direction in stage 1. Fig. 8 History of the observing altitude and direction in stage 2.

Fig. 7 Nominal trajectories in stage 2. Fig. 9 Nominal trajectories in stage 3.

Nominal trajectories in stage 3 The history of the observing altitude and direction
The designed nominal trajectories for stage 3 are for stage 3 are shown in Fig. 10. According to the
shown in Fig. 9. According to the figure, both the figure, the peaks of the wave increases. This is because of
inclinations of S1 and S2 change continuously during this the inclination adjustment. But, the observing direction
stage. is almost periodic. And the range of the observing
Optimization of observing sequence based on nominal trajectories of symmetric observing configuration 31

The history of the observing altitude and direction for

stage 4 are shown in Fig. 12. According to the figure,
both the observing altitude and direction are periodic.
And the range of the observing direction is from 0 to
2. Compared Fig. 12 with Fig. 8, we see that the
shapes of the waves of the observing altitude are similar
and the histories of the observing direction are similar
as well. Because of this phenomenon, we predict that
if an observing sequence is found in stage 4 then the
same observing sequence can be found in stage 2 with
the same order it appears. Therefore, we search the
observing sequence in stage 2 the same as that in stage
(a)
4. It greatly reduces the computation effort when the
branch bound method is applied.

5 Optimization of the observing

sequence
In the previous section, we present the nominal

(b)
Fig. 10 History of the observing altitude and direction in stage 3.

direction is from 0 to 2.
Nominal trajectories in stage 4
The designed nominal trajectories for stage 4 are
shown in Fig. 11. According to the figure, all spacecraft
keep coasting during this stage. (a)

(b)
Fig. 11 Nominal trajectories in stage 4. Fig. 12 History of the observing altitude and direction in stage 4.
32 H. Yang, G. Tang, F. Jiang

trajectories. With these nominal trajectories, we are shown in Fig. 13, where the stars denote the
propose an algorithm to select all possible observing observing opportunities and the dashed lines represent
opportunities. After that, a global optimization method, the boundaries of the four stages.
the branch bound method, is employed for optimization As shown in the figure, the widths of stage 1,
of the observing sequence. stage 2, and stage 4 are almost the same. Moreover,
the maximum altitudes of these three stages satisfy
5.1 Selecting possible observing hmax /hmid > 3 and hmid /hmin > 3. We will search the
opportunities optimal sequence of the thrice observed radio sources in
The following algorithm is used for selecting possible these stages. Because the thrice observed radio sources
observing opportunities: dominate the performance index, we conducted this
Step 1 Initialize the discrete time points {t1 ,. . . , search first. After the optimal thrice observing sequence
tN } between ts and tf with the time interval of one day. has been determined, we conducted optimization of the
Step 2 Compute h and at the discrete time sequence with twice and once observed radio sources.
points.
5.2 Optimization of the sequence of the
Step 3 Find all observing possibilities with the
thrice observed radio sources
selected radio sources between tk and tk+1 (k = 1,. . . ,
N 1) based on . Use linear interpolations to obtain The branch and bound method is used for the
observing time points and h at these time points. optimization of the sequence with thrice observed radio
Step 4 Sort all possible observations according to sources. Because the observations with P = 6 dominate
observing time. the performance index, we conducted the searching
The time interval for the discrete time points is set in stage 4 first. The searching procedure is shown in
to 1 h. The positions of the three spacecraft at these Fig. 14. In this figure, PN k represents the number of
time points are calculated by a cubic spline method. the radio sources.
Because the declinations of the selected radio sources As illustrated in Fig. 14, we take last 30 points of the
are small, a radio source is regarded as observed if its observing opportunities in Fig. 13 as the starting point
right ascension is the same as that of the observing in the searching procedure. In this way, the observing
direction. The declination error will be corrected after opportunities near the last wave crest are all considered.
the sequence has been optimized. The observing time Besides, the opportunities with high observing altitude
and the observing altitude between two adjacent discrete are preferred. The points whose observing altitude h
time points are computed by the linear interpolation. is less than 7.0105 km are discarded, as indicated by
With the algorithm mentioned above, we select the dashed boxes in the figure. Then, we increase the
all the possible observing opportunities as well as sequence step by step. As for the point PN k , the points
the corresponding observing altitudes. The results from PN k1 to PN k50 are considered and the point

Fig. 13 History of the observing altitude and all possible observing opportunities.
Optimization of observing sequence based on nominal trajectories of symmetric observing configuration 33

stage 2 as {QN ,. . . ,Q1 }. The searching procedure is

illustrated in Fig. 15. For the step k, the radio source
varies from QN to Q1 . If its number matches Tk , then
the following two conditions are checked: (1) the time
interval between this radio source and any other selected
radio source is no less than 15 days; (2) its observing
altitude is less than 1/3 of that of the same radio source
in stage 4. If these two conditions are satisfied, this
radio source is regarded as feasible and the observing
sequence increases.
After the branch and bound searching, we obtain
50,872 observing sequences. The ten sequences with
the largest performance indexes are shown in Table
3. In this table, J represents the increment to the
performance index because of the observations with
Fig. 14 Searching procedure of the branch and bound algorithm
P = 3. We find that the given performance indexes
for observations with P = 6 (dashed box: h is not satisfied; dashed are all larger than 115106 km. Because the number
line: the time interval is not satisfied). of the obtained observing sequences is large, only those
with J > 114 106 km will be used in the next step.
is chosen if the observing time interval between this
Last, we search the observing sequences in the part of
point and PN k is no less than 15 days. A termination
stage 2 where no radio sources are selected and in stage
condition of increasing sequences is either no new radio
source can be found or a sequences length reaches 15.
With the branch and bound method mentioned above,
we obtain 158,112 observing sequences and all these
sequences will be used for the next searching. According
to the performance index, the top ten of the obtained
observing sequences are found and listed in Table 2. The
lengths of these observing sequences are all 12 and the
performance indexes are all over 100106 km.
Then, we search the observing sequences in stage 2
with P = 3. Because the observing sequence in stage
3 must have the same radio sources, the searching
procedure is not the same as that shown in Fig.
14. Denote a observing sequence found in stage 4 as Fig. 15 Searching procedure of the branch and bound algorithm
{T1 ,. . . ,T12 } and the numbers of the radio sources in for observations with P = 3.

Table 2 Top ten of the obtained observing sequences with P = 6

Rank Observing sequence J (km)
1 206-200-212-223-199-193-207-220-214-224-201-213 101830796.8
2 223-199-193-207-220-214-206-200-212-224-201-213 101830790.5
3 223-199-193-206-200-212-207-220-214-224-201-213 101830785.8
4 207-220-214-206-200-212-223-199-193-224-201-213 101830785.1
5 206-200-212-223-199-193-207-220-420-224-201-213 101830440.2
6 207-220-420-206-200-212-223-199-193-224-201-213 101830429.8
7 223-199-193-206-200-212-207-220-420-224-201-213 101830429.2
8 223-199-193-207-220-420-206-200-212-224-201-213 101830426.8
9 206-218-212-223-199-193-207-220-214-224-201-213 101828333.3
10 223-199-193-207-220-214-206-218-212-224-201-213 101828325.2
34 H. Yang, G. Tang, F. Jiang

Table 3 Top ten of the obtained observing sequences with P = 6 and 3

Rank Observing sequence J (km) J (km)

1 223-199-193-206-200-212-207-220-214-224-201-213 13288434.3 115119220.1
2 223-199-193-206-200-212-207-220-420-224-201-213 13288618.4 115119047.5
3 223-199-193-206-218-212-207-220-214-224-201-213 13288184.9 115116506.1
4 223-199-193-206-218-212-207-220-420-224-201-213 13288369.0 115116333.6
5 223-217-193-206-200-212-207-220-214-224-201-213 13283593.2 115099630.1
6 223-217-193-206-200-212-207-220-420-224-201-213 13283777.3 115099457.6
7 223-217-193-206-218-212-207-220-214-224-201-213 13283343.9 115096916.2
8 223-217-193-206-218-212-207-220-420-224-201-213 13283527.9 115096743.6
9 223-199-193-206-200-212-207-220-214-224-201-195 13281708.2 115050641.6
10 223-199-193-206-200-212-207-220-420-224-201-195 13281892.3 115050469.1

1 with P = 1. The searching procedure is similar to that minor to the performance index, we introduce the
for stage 2. After the branch and bound searching, we approach briefly. First, radio sources that will be
obtain 1,384,288 observing sequences. Then, the best repeatedly observed twice (P = 1, 3) is searched. The
observing sequence is taken as the optimal observing searching procedure is similar to that for thrice
sequence. The result is shown in Table 4. The increment observations, but the searching area for P = 3 are in
to the performance index in this stage is small because stage 3 and stage 1 while searching area for P = 1 is
P = 1 and the observing altitudes are small. in stage 1. Only the radio sources which have not been
The distribution of the optimal observations with selected for thrice observation are considered. And the
three times is shown in Fig. 16. As the prediction stated selected radio sources in this section must satisfy the
in Section 4, the observing sequence in stage 4 has the constraint of time intervals. Second, all other radio
same appearing order as that in stage 2. sources that can be observed once (P = 1) is searched.
The result of the observing sequence with two
5.3 Optimization of the sequence with observations is listed in Table 5 and the result of the
twice and once observed radio sources observing sequence with one observation is listed in
After the optimal sequence with thrice observations Table 6. The increments due to twice observation and
is obtained, the branch and bound method is used once observation is much smaller compared with that of
for optimization of the sequence with twice and once the thrice observation. The total performance index is
observations. Because these observations contribute 126,298,367.8 km which is a little smaller than our final
Table 4 Optimal observing sequence with three times of observations
Observing sequence J (km) J (km)
223-199-193-206-200-212-207-220-214-224-201-213 679587.2 115798807.3

Fig. 16 Distribution of the optimal observations with three times.

Optimization of observing sequence based on nominal trajectories of symmetric observing configuration 35

Table 5 Result of the observing sequence with two observations Table 7 Result of the observing sequences submitted
Observing sequence J (km) J (km) Category Number of points Radio source
204-209-190 4259296.0 120058103.3 223, 199, 193, 206, 200, 212,
Rsp136 12
207, 220, 214, 224, 201, 213
Table 6 Result of the observing sequence with one observation Rsp13 3 204, 209, 190
Rsp11 1 211
Observing sequence J (km) J (km)
Rsp1 6 198, 203, 195, 218, 205, 216
211-198-203-195-218-205-211-216 6240264.5 126298367.8
Rsp136 = radio sources observed exactly thrice with P =1, 3,
and 6.
result. This is because the observations designed by Rsp13 = radio sources observed exactly twice with P =1, 3.
the nominal trajectories are not accurate. After orbit Rsp11 = radio sources observed exactly twice with P = 1 both
times.
corrections for accurate observations, the performance
Rsp1 = radio sources observed exactly once with P = 1.
index will change.
The distribution of the optimal observations with one spacecraft for observing radio sources. With the
and two times is shown in Fig. 17. According to this observing configuration, the normal vector of the
figure, all these observations happen in stage 1 and stage triangle plane formed by the three spacecraft rotates in
3. the ecliptic plane periodically and approximately points
to the radio sources which are close to the ecliptic
5.4 Results with accurate observations
plane. The searching procedures using the branch and
After the observations are approximately designed, bound method based on nominal trajectories for optimal
orbit corrections are conducted to obtain accurate observing sequence is then proposed.
observations. The introduction on the orbit corrections The obtained optimal sequence has 50 observations
can be found in Ref. [15]. The trajectories of the three and 22 unique radio sources. Specifically, 12 radio
spacecraft for accurate observations are shown in Fig. sources observed exactly thrice with P = 1, 3, and
18. 6; 3 radio sources observed exactly twice with P =
The summary of the optimal observing sequence 1, 3; 1 radio source observed exactly twice with
submitted is shown in Table 7. The final performance both P = 1; and 6 radio sources observed exactly
index is J = 128 286 317.0 km. once with P = 1. The performance J of the optimized
low-thrust trajectories for accurate observations is
128,286,317.0 km. These results are exactly the same as
6 Conclusions that we submitted in GTOC8. The second place ranking
In this paper, the symmetric observing configuration is of our performance in GTOC8 implies advantage and
used for designing low-thrust trajectories of three effectiveness of our method.

Fig. 17 Distribution of the optimal observations with one and two times.
 36 H. Yang, G. Tang, F. Jiang

847.
[9] Petropoulos, A. E., Longuski, J. M. Shape-based
algorithm for the automated design of low-thrust,
gravity assist trajectories. Journal of Spacecraft and
Rockets, 2004, 41(5): 787796.
[10] Gao, Y., Kluever, C. A. Low-thrust interplanetary
orbit transfers using hybrid trajectory optimization
method with multiple shooting. In: Proceedings of the
AIAA/AAS Astrodynamics Specialist Conference and
Exhibit, Guidance, Navigation, and Control and Co-
located Conferences, 2004: AIAA 2004-5088.
[11] Petropoulos, A. E., Longuski, J. M., Bonfiglio, E. P.
Trajectories to Jupiter via gravity assists from Venus,
Earth, and Mars. Journal of Spacecraft and Rockets,
2000, 37(6): 776783.
Fig. 18 Low-thrust trajectories of the three spacecraft for [12] Casalino, L., Colasurdo, G. Problem description
accurate observations. for the 7th Global Trajectory Optimisation
Competition. 2014. Available at http://sophia.estec.
Acknowledgements esa.int/gtoc portal.
[13] Shen, H.-X., Huang, A.-Y., Zhang, Z.-B., Li, H.-N.
This work is supported by the National Natural GTOC8: Results and methods of State Key Laboratory
Science Foundation of China (Grant Nos. 11672146 and of Astronautic Dynamics. In: Proceedings of the
11432001). The authors thank the organizer of GTOC8. AAS/AIAA Space Flight Mechanics Meeting, 2016:
AAS 16-384.
References [14] He, S., Zhu, Z., Gao, Y. GTOC8: Results and
methods of TEAM 8/Chinese Academy of Sciences. In:
[1] Petropoulos, A. E. GTOC8: Problem description and Proceedings of the AAS/AIAA Space Flight Mechanics
summary of the results. In: Proceedings of the 26th Meeting, 2016: AAS 16-552.
AAS/AIAA Space Flight Mechanics Meeting, 2016. [15] Tang, G., Yang, H., Jiang, F., Baoyin, H., Li, J.
[2] Bertrand, R., Epenoy, R. New smoothing techniques GTOC8: Results and methods of TEAM 3Tsinghua
for solving bangbang optimal control problems University. In: Proceedings of the AAS/AIAA Space
numerical results and statistical interpretation. Optimal Flight Mechanics Meeting, 2016: AAS 16-248.
Control Applications and Methods, 2002, 23(4): 171
197.
[3] Olympio, J. T. Optimal control problem for low-thrust
Hongwei Yang received his Ph.D.
multiple asteroid tour missions. Journal of Guidance,
degree in aerospace engineering from
Control, and Dynamics, 2011, 34(6): 17091720.
Tsinghua University, China, in 2017,
[4] Jiang, F., Baoyin, H., Li, J. Practical techniques for
and he was a visiting Ph.D. student of
low-thrust trajectory optimization with homotopic
Rutgers University, USA, in 20152016.
approach. Journal of Guidance, Control, and
Dynamics, 2012, 35(1): 245258. His current research interests include
[5] Jiang, F., Chen, Y., Liu, Y., Baoyin, H., Li, J. GTOC5: dynamics and control near asteroids,
Results from the Tsinghua University. Acta Futura, interplanetary trajectory design, and
2014, 8: 3744. optimization. E-mail: yang.hw.thu@gmail.com.
[6] Casalino, L. Approximate optimization of low-thrust
transfers between low-eccentricity close orbits. Journal Gao Tang received his bachelor
of Guidance, Control, and Dynamics, 2014, 37(3): degree and master degree in aerospace
10031008. engineering from Tsinghua University,
[7] Gatto, G., Casalino, L. Fast evaluation and
China, in 2010 and 2012, respectively.
optimization of low-thrust transfers to multiple
He is currently pursuing his Ph.D.
targets. Journal of Guidance, Control, and Dynamics,
degree in the Department of Mechanical
2015, 38(8): 15251530.
[8] Yang, H., Li, J., Baoyin, H. Low-cost transfer between Engineering and Material Science at
asteroids with distant orbits using multiple gravity Duke University, USA. His current
assists. Advances in Space Research, 2015, 56(5): 837 research interests include dynamics and control in
Optimization of observing sequence based on nominal trajectories of symmetric observing configuration 37

robotics, interplanetary trajectory optimization, and current research interests include astrodynamics, spacecraft
global optimization methods. E-mail: gao.tang@duke.edu. formation flying, and interplanetary trajectory optimization.
E-mail: jiangfh@tsinghua.edu.cn.
Fanghua Jiang received his B.S. degree
in engineering mechanics and Ph.D.
degree in mechanics from Tsinghua
University, China, in 2004 and 2009,
respectively. Since 2009, he has worked
in the School of Aerospace Engineering
at Tsinghua University, China, including
two years of postdoctor, three years of
research assistant, and two years of associate professor up
to now. Currently, he is an AIAA senior member. His