Low-Orbit Satellite Orbit Prediction Algorithm Based on Near-Polar

Circular Orbit

Delu Wei

, Chao Zhao

, Hangzai Luo

, Sheng Zhong

and Zhengwen Cao

1,*

School of Information Science and Technology, Northwest University, Xi’an, China

Xi’an Institute of Microelectronics Technology, Xi’an, China

Keywords: Low Orbit Satellite, Constellation Network, Orbit Prediction.

Abstract: Satellite orbit prediction is a significant research problem in visualizing low-orbit giant constellation networks.

However, due to the limited resources of the onboard network equipment, the existing satellite orbit prediction

methods are challenging to balance the accuracy and rapidity of the prediction. These traditional forecasting

methods tend to construct perturbation models and obtain accurate orbital dynamics equations for calculation.

Due to the complicated establishment of the perturbation model and the tedious calculation process, the

prediction accuracy of the low-order analytical solution is relatively low, and the calculation efficiency of the

high-order analytical solution is not high, which is not suitable for the orbit prediction of large-scale long-

period low earth orbit (LEO) satellites. This paper proposes an orbit prediction algorithm for LEO satellites

based on near-polar circular orbits. By simplifying the satellite motion model and using the least squares

method to fit the data errors, we finally obtain the position information of the LEO satellite constellation

network. Experimental results show that the method can perform orbit forecasting of large-scale LEO satellite

constellation networks while ensuring accuracy and rapidity compared with satellite tool kit (STK) software.

1 INTRODUCTION

In developing low-orbit giant constellation

networking visualization systems, satellite orbit

prediction technology is one of the key research

directions (Ren et al., 2019; Deng et al., 2021). The

prediction speed and accuracy play a vital role in the

design and optimization of inter-satellite link

visualization, which also directly reflects the

authenticity and reliability of the system. Currently,

most the satellite orbit prediction is based on the

classical mechanics model, which defines the

perturbation model by analyzing the satellites’ forces,

and then establishes the satellite orbit dynamics

equations and performs the orbit calculation through

the numerical integration method. For LEO satellites,

the main regenerative forces affecting the satellite

orbit calculation are the second-order band harmonic

terms of the earth’s non-spherical gravitational field,

atmospheric drag, other harmonic terms of the earth’s

non-spherical gravitational field, and the gravitational

force of the Sun and Moon. Since the accuracy of the

perturbation model is low, it can cause errors in the

traditional calculation methods. The other parameters

introduced by the perturbation model can lead to the

low prediction accuracy of the low-order analytical

solution and the cumbersome process of the high-

order analytical solution. (Wang et al., 2018).

This paper proposes a near-polar circular orbit-

based orbit prediction algorithm for LEO satellites.

The satellite motion model is simplified to a uniform

circular motion model with only the force provided by

gravity, and the computational errors caused by other

perturbations are fitted to the time-dependent primary

term by the least squares method and added to the

simplified satellite orbital dynamics equation to form

a new satellite orbital dynamics equation for orbit

calculation. By comparing the data obtained from the

high-precision satellite orbit prediction model in STK

software with a large amount of data obtained through

several experiments, the accuracy and feasibility of

the algorithm are verified. The fast and accurate

prediction of satellite orbit positions in the LEO giant

constellation network visualization system shown in

Fig. 1 is achieved with a limited computer system.

482

Wei, D., Zhao, C., Luo, H., Zhong, S. and Cao, Z.

Low-Orbit Satellite Orbit Prediction Algorithm Based on Near-Polar Circular Orbit.

DOI: 10.5220/0011953400003612

In Proceedings of the 3rd International Symposium on Automation, Information and Computing (ISAIC 2022), pages 482-487

ISBN: 978-989-758-622-4; ISSN: 2975-9463

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

Figure 1: Visualization system diagram of constellation

networking.

2 SATELLITE CONSTELLATION

MOTION MODEL

2.1 The Law of Motion of the Satellite

Taking the near-polar circular orbit as an example, the

motion law of the satellite in an ideal state can be

simplified to a two-body problem (Ge et al., 2020).

According to the law of gravity, the linear and angular

velocities of the satellite orbiting at a given orbital

altitude

are calculated as:

(1)

(2)

here

is the universal gravitational constant in

/( )mkgs⋅

is the total mass of the earth in

is the mean radius of the earth in

2.2 Position Description of the Satellite

The position description of a satellite in a LEO

constellation network includes how to represent the

position of a point in the orbit and then consider how

to represent a point in the constellation network.

There are two main methods for describing the

position of a point in orbit: namely, phase

representation, and latitude and longitude

representation (Rabjerg et al., 2020).

In the phase representation, the orbit ascending

intersection point is specified as the 0 phase point. As

the satellite moves from south to north, the phase

varies between

(0, 2 )

. It increases linearly along the

direction of satellite motion, with the points on the

orbit corresponding to the phase values. In order to

extend the phase description method to the satellite

constellation, it is sufficient to add the orbit number

again, denoted by

(,)

In the latitude-longitude representation, at any

moment, the subsatellite points of the satellite

uniquely map a latitude-longitude coordinate. The

satellite's orbit and the earth’s rotation will cause the

mapping relationship to change with time. The

latitude and longitude description method can be

directly extended to the constellation.

In order to illustrate the calculation process of the

longitude and latitude coordinates of the satellite

subsatellite, a typical satellite orbit model is

introduced as shown in Fig. 2. OA represents the

rotation axis of the earth, the orbital inclination can be

expressed as

, and the current phase angle of

satellite M is

. Under the premise that the current

phase of the satellite is known, the latitude and

longitude of the satellite can be obtained according to

the geometric relationship in the Fig. 2 as follows:

sin (sin sin )

lat

θϕ

−

(3)

lng

tan (tan cos )i

θϕ

−

(4)

Figure 2: Satellite orbit model.

2.3 Location Distribution Strategy of

Satellites

Consider the problem of uniform coverage of the

satellite constellation network, the location

distribution of satellites can be carried out according

to the following strategy. Assume that a satellite

constellation network contains M orbits and N

Low-Orbit Satellite Orbit Prediction Algorithm Based on Near-Polar Circular Orbit

483

satellites on each satellite orbit, and the total number

of satellites in the constellation is S. The first orbit is

selected as the reference orbit, and the first satellite on

the first orbit is the reference satellite. The longitude

of the ascending node of the reference orbit is 0°, and

the initial phase of the reference satellite is 0 phase

point. The satellite orbit starts from the ascending

intersection point of the reference orbit and is evenly

distributed from west to east. The longitude

difference between adjacent orbits (

) is

;

the satellites in the same orbit start from the 0 phase

point and are evenly distributed along the direction of

the increasing phase. The phase difference between

adjacent satellites in the same orbit (

) is

2/N

From the above, for any satellite in the satellite

constellation network

(denoting the n-th

satellite in the

m-th orbit, where

1, 2, ,mM= 

1, 2, ,nN= 

) the latitude and longitude of the initial

state can be expressed as:

( ) sin (sin sin )

lat

θϕ

−

sin (sin(( 1) )sin )ji

−

=−

(5)

lng 0 0

( ) tan (tan cos )

θθ ϕ

−

( 1) tan (tan cos )

ϕϕ

−

=−Δ +

(6)

Since the satellite constellation network is in

constant motion, the positions of the satellites in the

constellation are also changing. The satellite orbit

forecast is to calculate the position information of the

satellite at

based on the initial position of the

satellite and then to obtain the position information of

all the satellites in the entire constellation network at

t based on the distribution law of the satellites in the

constellation network. Considering the time factor,

the calculation formula of latitude and longitude is

rewritten as

( ) sin (sin ( ) sin )

lat

tti

θϕ

−

(7)

lng

() tan (tan ()cos)

jst

tti

θθ ϕ

−

(8)

3 POSITION DATA ERROR

CORRECTION METHOD OF

SATELLITE

When using the satellite orbit motion model and

method composed of formula (1) to formula (8),

which is called the MNC model algorithm, the

obtained satellite position data has a significant error

compared with the data obtained by STK high-

precision orbit prediction model (MSGP4). The error

increases with the increase in prediction time.

The MNC calculation model simplifies the

satellite motion problem into a two-body problem.

This model does not consider the perturbation caused

by the second harmonic term of the earth’s aspheric

gravitational field, atmospheric resistance, and other

disturbances. It results in a significant error in the

calculation results.

Because of the difficulty of establishing an

accurate perturbation model and the complexity of

calculating the acceleration caused by each regression

force on satellite motion. The more orders of spherical

harmonic coefficients are considered in the

integration, the more time consumed. It is not

conducive to large-scale orbit position prediction and

visualization of LEO satellites with limited system

resources.

Therefore, in the subsequent, the least-squares

method is used for fitting. We fit the data errors

caused by various disturbances into polynomials

related to the running time and add them to the

original latitude and longitude calculation motion

model, which is called the MAC model. It solves the

problem that the data obtained by the MNC model and

the data obtained by the MSGP4 model have a

significant error. Under the condition of real-time and

accuracy, the position prediction and visualization of

large-scale LEO satellite constellation networks are

realized.

4 EXPERIMENTAL RESULTS

The target satellites in our experiments are low orbit

satellites in a near-polar circular orbit. A constellation

network of 6 orbits with 9 satellites in each orbit is

used as an example, and the satellites are distributed

according to the strategy described in subsection 2.3,

with an orbit inclination of 90°, and the experiments

are conducted under the settings of orbital altitudes of

500 km, 800 km, and 1000 km, respectively.

ISAIC 2022 - International Symposium on Automation, Information and Computing

484

The experimental environment was configured

with a processor AMD Ryzen 7 5800H with Radeon

Graphics 3.20 GHz, 16 GB of RAM, and an operating

system of 64-bit Windows 10 version 19044.1706.

4.1 Accuracy of Orbit Prediction

Algorithms for LEO Satellites in

near Polar Circular Orbit

Based on the given orbits, the satellite positions are

distributed according to the strategy proposed in the

previous section, and a satellite constellation network

is established. The initial latitude and longitude of all

satellites in the constellation network are obtained by

traversing the number of satellite orbits, the number

of satellites in orbit, and the latitude and longitude

position of any satellite at a given observation

moment is obtained by the latitude and longitude

calculation algorithm (MNC).

Without loss of generality, we selected any two

satellites among 54 stars with an orbital altitude of

500 km and observation time of 7 days for

position

data calculation and then compared them with the

position data calculated by the identical numbered

satellites in the same scale constellation network of

STK 11.6 software under the MSGP4 model, and the

error of the satellite latitude data obtained from the

MNC model experiment and the latitude data

obtained from the MSGP4 model in STK. The

distribution of error with time is shown in Figure 3.

(a) Error of S11 (b) Error of S66

Figure 3: Error distribution plots.

Since our experiment is for a constellation of

satellites in a near-polar circular orbit, it can be seen

in the longitude calculation equation that when the

orbital inclination

i is 90, the longitude variation

depends only on the mean angular velocity of the

earth’s rotation

. The value of the mean angular

velocity of the earth selected in the experiment is

consistent with the value set in the MSGP4 of STK

software. For the calculation of longitude, the errors

in the data obtained from the two models are minimal

and negligible (

−

). Therefore, only the errors in the

calculation of latitude are considered in this paper.

Figure 3 shows that the error in the latitude

direction is approximately linear for time. The

positive and negative error is that the latitude takes a

range of (-90°, 90°), and there are negative values.

Observing a large amount of data, it is found that the

error is positive when the satellite moves from south

to north and negative when the satellite moves from

north to south. Therefore, we set

as the direction

factor and find a primary polynomial by least squares

fitting

yatb=+

to fit the error curve to a time-

dependent primary term and add it to the latitude

calculation expression as an error correction term to

improve the accuracy of satellite position data

forecasting. The correction terms at three different

altitudes are shown in Table 1.

Table 1 Expressions of correction terms at different orbital

heights.

Track

height

(km)

Correction

function

(fitting

degree)

RMSE

500

0.005094 0.00671yt=+

0.9996 0.294

800

0.004378 0.005281yt=+

0.9997 0.212

1000

0.003973 0.00312yt=+

0.9997 0.204

The correction term is added to the satellite

latitude calculation model to form a new calculation

model, which can be expressed as:

() ()

wt lat

tdyt

θθ

(9)

here

is the orientation factor.

is -1 when the

satellite is moving from south to north,

is 1 when

the satellite is moving from north to south, and

()yt

represents the expression of the correction term at the

corresponding orbital altitude.

We refer to the satellite orbit prediction model

after adding the correction term as the MAC model.

By experimenting with the MAC model, the latitude

data of the satellite are calculated at three different

orbital altitudes. The latitude data of a satellite at three

different orbital altitudes are randomly selected and

compared with the latitude data of the corresponding

satellite in the MSGP4 model of STK. The error of

Low-Orbit Satellite Orbit Prediction Algorithm Based on Near-Polar Circular Orbit

485

the data obtained from the MAC model compared

with the MSGP4 model of STK and the error of the

data obtained from the MNC model compared with

the MSGP4 model of STK are shown in Fig. 4. From

the figure, it can be seen that the data calculated by

the MAC model is closer to the data calculated by the

MSGP4 model in STK.

We experimentally tested the accuracy of the

satellite orbit prediction models for satellite

constellation networks with orbital altitudes of 500

km operating for 7, 15, and 30 days, and calculated

the correlation degree of the data obtained from the

MAC model proposed in this paper and the GSRPS

model in STK, calculated the MSE (mean square error)

of the data obtained from the two models, and

compared them with the MNC model. The

experimental results are shown in Table 2.

(a)Error of S22 on MNC (b)Error of S22 on MAC

(c)Error of S16 on MNC (d)Error of S16 on MAC

(c)Error of S35 on MNC (d)Error of S35 on MAC

Figure 4: Error comparison graph. (a) (b) with h = 500 km,

Table 2 Comparison of the accuracy of satellite orbit

prediction models before and after adding correction terms.

Satellite

constellation

network

operation

time

MSE

for

MNC

and

STK

MSE

for

MAC

and

STK

Order of

magnitude

improvement

in data

accuracy

7 days 754.41 0.0170 4.438×10

15 days 2810.45 0.0201 1.398×10

30 days 6309.11 0.0315 2.003×10

4.2 Superiority of Orbit Prediction

Algorithms for LEO Satellites in

near Polar Circular Orbit

The superiority of the near-polar circular orbit LEO

satellite forecasting algorithm in terms of execution

time is given in this subsection. The calculation

method proposed in this paper is applied to a network

of LEO satellites with different sizes of a near-polar

circular orbit, the orbital altitude is set to 500 km. The

operation time is set to 7 days, and the step size is 60

s. The position data of all satellites in the whole

constellation are calculated and output as a .txt file,

and the execution time of the MAC model algorithm

is obtained through experiments, and then compared

with the algorithm used by the MSGP4 model in STK

software, the experimental results are shown in Table

Table 3 Comparison of the time consumption of the two

algorithms under different satellite constellation size.

Number

satellite

orbits

Number

satellites

per

orbit

STK

execution

time

(unit: s)

MAC

execution

time

(unit: s)

6 9 269.98 1.005

10 9 701.11 2.190

12 15 975.43 3.309

188 40 4263.61 13.278

With the increase of satellite constellation

network scale, the algorithm execution time gradually

increases. In the calculation of the orbit forecast of the

same scale satellite constellation network, the speed

of the algorithm used in this paper is nearly 300 times

higher than that of STK software, which can calculate

the position data of the large-scale LEO satellite

constellation network more quickly.

ISAIC 2022 - International Symposium on Automation, Information and Computing

486

5 CONCLUSION

This paper proposes a prediction algorithm for LEO

satellite orbits based on near-polar circular orbits.

Based on the prediction above model and algorithm,

the position data of the satellite at a given observation

moment and the time consumed by the algorithm

execution are obtained. In the experimental validation,

the predicted longitude and

latitude position errors are within the acceptable range

compared to the STK high-precision orbit prediction

model. Four orders of magnitude improve the

accuracy compared to before the correction term is

added, and the predicted position errors tend to be

stable for a time. The computation speed of the

algorithm proposed in this paper is nearly 300 times

higher than that of the STK software in the same scale

LEO satellite constellation, which is more suitable for

the long-period orbit prediction of large-scale LEO

satellite constellation networks.

ACKNOWLEDGEMENTS

This work is supported by the National Natural

Science Foundation of China (Grant No. 62071381),

Shaanxi Provincial Key R&D Program General

Project (2022GY-023).

REFERENCES

Deng, R.Q., Di, B.Y., and Song, L.Y.(2021). Ultra-Dense

LEO Satellite Based Formation Flying. IEEE

TRANSACTIONS ON COMMUNICATIONS(5).

Rabjerg, J.W., Leyva-Mayorga, I., and Soret, B. (2020).

Exploiting topology awareness for routing in LEO

satellite constellations.

Liang, J., Nan, X., and Zhang, J. (2011). Constellation

design and performance simulation of LEO satellite

communication system. International Conference on

Applied Informatics and Communication.

Ge, H., Li, B., Ge, M., Nie, L., and Schuh, H. (2020).

Improving low earth orbit (LEO) prediction with

accelerometer data. Remote Sensing, 12(10), 1599.

Zhang, Y., Wu, Y., Liu, A., Xia, X., and Liu, X. (2021).

Deep learning-based channel prediction for LEO

satellite massive MIMO communication system. IEEE

wireless communications letters(10-8).

Ren, H., Chen, X., Guan, B., Wang, Y., and Peng, K.

(2019). Research on Satellite Orbit Prediction Based on

Neural Network Algorithm. the 2019 3rd High

Performance Computing and Cluster Technologies

Conference.

Bl, A., Hga, B., Mg, B., Ln, A., Ys, A., and Hs, B. (2019).

Leo enhanced global navigation satellite system

(LeGNSS) for real-time precise positioning services -

sciencedirect. Advances in Space Research, 63(1), 73-

93.

Li, S., and Wang, H. (2021). Research on navigation and

positioning technology based on opportunity signal of

low orbit satellite. Journal of Physics: Conference

Series, 1952(4), 042134 (13pp).

Wang, D. W., Tang, J. S., Liu, L., Ma, C. W., and Hao, S.

F. (2018). The assessment of the semi-analytical

method in the long-term orbit prediction of earth

satellites. Chinese Astronomy and Astrophysics, 42( 2),

239-266.

Low-Orbit Satellite Orbit Prediction Algorithm Based on Near-Polar Circular Orbit

487