# US 11,897,635 B2 · Method for maintaining walker constellation formation and terminal device

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- Number: US 11,897,635 B2
- Document ID: US11897635B2
- Status: Granted
- Kind: B2
- Application number: 17/458,537
- Filed: 2021-08-26
- Published: 2022-03-03
- Granted: 2024-02-13
- Earliest claimed: 2020-08-27
- Assignees: Space Engineering University
- Applicants: Space Engineering University
- Inventors: Min Hu; Jiuyang Li; Xuyu Wang; Yongjing Ruan; Feifei Li; Xianjun Pan; Tianyu Sun; Ge Zhang
- Web page: https://patentgrep.com/patent/US11897635B2

## CPC

- B64G 1/1085: {Swarms and constellations}
- B64G 1/242: {Orbits and trajectories}
- G05B 17/02: electric
- B64G 1/247: {Advanced control concepts for autonomous, robotic spacecraft, e.g. by using artificial intelligence, neural networks or autonomous agents}
- B64G: COSMONAUTICS; VEHICLES OR EQUIPMENT THEREFOR
- G05B: CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS (systems for controlling or regulating non-electric variables G05D; systems for regulating electric or magnetic variables G05F; control devices or systems insofar as characterised by mechanical features only G05G)

## Abstract

Provided are a method for maintaining Walker constellation formation and a terminal device. The method comprises: determining a first offset amount of each satellite within a simulation time period according to parameters of a Walker constellation; performing first offset on each satellite according to the first offset amount to obtain a Walker constellation after the first offset; determining a second offset amount of each satellite within the simulation time period according to parameters of the Walker constellation after the first offset; and superimposing the first offset amount and the second offset amount, and performing second offset on each satellite so as to maintain the formation of the Walker constellation.

## Claims

### Claim 1

A method for maintaining Walker constellation formation, comprising:
determining, according to parameters of a Walker constellation, a first offset amount of each satellite within a simulation time period;
performing, according to the first offset amount, first offset on each satellite to obtain a Walker constellation after the first offset;
determining, according to parameters of the Walker constellation after the first offset, a second offset amount of each satellite within the simulation time period; and
superimposing the first offset amount and the second offset amount, and performing second offset on each satellite according to a superimposing result so as to maintain the formation of the Walker constellation.

### Claim 2 (depends on 1)

The method for maintaining Walker constellation formation according to claim 1, wherein the step of determining the first offset amount of each satellite within the simulation time period according to the Walker constellation parameters further comprises:
determining a first relative drift amount of each satellite at each simulation moment according to the Walker constellation parameter; and
determining the first offset amount of each satellite within the simulation time period according to the first relative drift amount.

### Claim 3 (depends on 2)

The method for maintaining Walker constellation formation according to claim 2, wherein the parameters of the Walker constellation comprise a nominal semi-major axis α, an eccentricity e, an orbit inclination i, the total number N of satellites, the number P of orbital planes and a phase factor F of the Walker constellation.

### Claim 4 (depends on 2)

The method for maintaining Walker constellation formation according to claim 2, wherein the step of determining the first drift amount of each satellite at each simulation moment according to the parameters of the Walker constellation further comprises:
determining the first relative drift amount of each satellite at each simulation moment according to a first formula, which is written as:
{
ΔΩ
i
=
(
Ω
i
-
Ω
i
*
)
-
∑
i
=
1
N
(
Ω
i
-
Ω
i
*
)
N
Δ
⁢
λ
i
=
(
λ
i
-
λ
i
*
)
-
∑
i
=
1
N
/
P
(
λ
i
-
λ
i
*
)
(
N
/
P
)
,
where ΔΩi is a relative drift amount of a right ascension of ascending node of an ith satellite, Δλj is a relative drift amount of an along-track angle of the ith satellite, Ωi is a right ascension of ascending node of the ith satellite under a perturbative force, λi is an along-track angle of the ith satellite under a perturbative force, Ωi* is a right ascension of ascending node of a nominal orbit of the ith satellite, λi* is an along-track angle of the nominal orbit of the ith satellite, N is the total number of satellites, and P is the number of orbit planes.

### Claim 5 (depends on 4)

The method for maintaining Walker constellation formation according to claim 4, wherein the step of determining the first offset amount of each satellite within the simulation time period according to the first relative drift amount further comprises:
determining a relational expression of the first relative drift amount varying over time within the simulation time period;
performing quadratic polynomial fitting on the relational expression to obtain a first-order variation rate and a variation acceleration of relative drift of each satellite within the simulation time period; and
determining the first offset amount of each satellite according to the first-order variation rate and the variation acceleration.

### Claim 6 (depends on 5)

The method for maintaining Walker constellation formation according to claim 5, wherein the first offset amount comprises an offset amount of a nominal semi-major axis, an offset amount of an eccentricity and an offset amount of an orbit inclination.

### Claim 7 (depends on 5)

The method for maintaining Walker constellation formation according to claim 5, wherein the step of determining the relational expression of the first relative drift amount varying over time within the simulation time period further comprises:
the relational expression of the first relative drift amount varying over time within the simulation time period is a second formula, which is written as:
{
Δ
⁢
Ω
⁡
(
t
)
=
Δ
⁢
Ω
0
+
k
Ω
′
⁢
t
+
k
1
Ω
⁢
t
+
0
.
5
⁢
k
2
Ω
⁢
t
2
Δ
⁢
λ
⁡
(
t
)
=
Δλ
0
+
k
λ
′
⁢
t
+
k
1
λ
⁢
t
+
0
.
5
⁢
k
2
λ
⁢
t
2
,
where ΔΩ(t) represents a right ascension of ascending node as a function of time, Δλ(t) represents an along-track angle as a function of time, ΔΩ0 is an initial value of relative drift of the right ascension of ascending node, Δλ0 is an initial value of relative drift of the along-track angle, kΩ′ is a first-order variation rate generated by an initial offset amount of the right ascension of ascending node, kλ′ is a first-order variation rate generated by an initial offset amount of the along-track angle, k1Ω is a first-order variation rate of the relative drift of the right ascension of ascending node, k1λ is a first-order variation rate of the relative drift of the along-track angle, k2Ω is a variation acceleration of the relative drift of the right ascension of ascending node, and k2λ is a variation acceleration of the relative drift of the along-track angle.

### Claim 8 (depends on 7)

The method for maintaining Walker constellation formation according to claim 7, wherein the relational expression satisfies a third formula, which is written as:
{
Δ
⁢
Ω
⁡
(
t
0
=
0
)
=
Δ
⁢
Ω
⁡
(
t
e
⁢
n
⁢
d
)
Δ
⁢
λ
⁡
(
t
0
=
0
)
=
Δ
⁢
λ
⁡
(
t
e
⁢
n
⁢
d
)
,
where t0 is a simulation start time, and tend is a simulation end time.

### Claim 9 (depends on 8)

The method for maintaining Walker constellation formation according to claim 8, wherein the step of performing quadratic polynomial fitting on the relational expression to obtain the first-order variation rate and the variation acceleration of relative drift of each satellite within the simulation time period further comprises:
performing quadratic polynomial fitting on the relational expression to obtain a first-order variation rate k1Ω of the relative drift of the right ascension of ascending node, a variation acceleration k2Ω of the relative drift of the right ascension of ascending node, a first-order variation rate k1λ of the relative drift of the along-track angle and a variation acceleration k2λ of the relative drift of the along-track angle; and
determining, according to a fourth formula, a first-order variation rate kΩ′ generated by the initial offset amount of the right ascension of ascending node and a first-order variation rate kλ′ generated by the initial offset amount of the along-track angle, wherein the fourth formula is written as:
{
k
Ω
′
=
-
k
1
Ω
⁢
t
e
⁢
n
⁢
d
+
0
.
5
⁢
k
2
Ω
⁢
t
e
⁢
n
⁢
d
2
t
e
⁢
n
⁢
d
k
λ
′
=
k
1
λ
⁢
t
e
⁢
n
⁢
d
+
0
.
5
⁢
k
2
λ
⁢
t
e
⁢
n
⁢
d
2
t
e
⁢
n
⁢
d
.

### Claim 10 (depends on 5)

The method for maintaining Walker constellation formation according to claim 5, wherein the step of determining the first offset amount of each satellite according to the first-order variation rate and the variation acceleration further comprises:
determining the first offset amount of each satellite according to a fifth formula, which is written as:
{
k
1
Ω
=
∂
Ω
˙
∂
a
⁢
Δ
⁢
a
0
+
∂
Ω
˙
∂
e
⁢
Δ
⁢
e
0
+
∂
Ω
˙
∂
i
⁢
Δ
⁢
i
0
k
1
λ
=
∂
λ
˙
∂
a
⁢
Δ
⁢
a
0
+
∂
λ
˙
∂
e
⁢
Δ
⁢
e
0
+
∂
λ
˙
∂
i
⁢
Δ
⁢
i
0
k
2
Ω
=
∂
Ω
˙
∂
a
⁢
Δ
⁢
a
˙
+
∂
Ω
˙
∂
e
⁢
Δ
⁢
e
.
+
∂
Ω
˙
∂
i
⁢
Δ
⁢
i
˙
k
2
λ
=
∂
λ
˙
∂
a
⁢
Δ
⁢
a
˙
+
∂
λ
˙
∂
e
⁢
Δ
⁢
e
˙
+
∂
λ
˙
∂
i
⁢
Δ
⁢
i
˙
,
where Δα0 is an offset amount of the nominal semi-major axis, Δe0 is an offset amount of an eccentricity, Δi0 is an offset amount of an orbit inclination,
∂
Ω
˙
∂
a
,
∂
Ω
˙
∂
e
,
∂
Ω
˙
∂
i
are partial derivatives of a variation rate of a right ascension of ascending node influenced by a perturbative force with respect to the semi-major axis, the eccentricity and the orbit inclination, respectively,
∂
λ
˙
∂
a
,
∂
λ
˙
∂
e
,
∂
λ
˙
∂
i
are partial derivatives of a variation rate of an along-track angle influenced by the perturbative force with respect to the semi-major axis, the eccentricity and the orbit inclination, respectively, and Δ{dot over (a)}, Δė, Δ{dot over (i)} are respective variation rates, which are determined by the perturbative forces applied to the respective satellite.

### Claim 11 (depends on 1)

The method for maintaining Walker constellation formation according to claim 1, wherein the step of determining the second offset amount of each satellite within the simulation time period according to the parameters of the Walker constellation after the first offset further comprises:
determining a second relative drift amount of each satellite at each simulation moment according to the parameters of the Walker constellation after the first offset; and determining the second offset amount of each satellite within the simulation time period according to the respective second relative drift amount.

### Claim 12 (depends on 11)

The method for maintaining Walker constellation formation according to claim 11, wherein the step of determining the second offset amount of each satellite within the simulation time period according to the respective second relative drift amount further comprises:
determining a relational expression of the second relative drift amount varying over time within the simulation time period;
performing quadratic polynomial fitting on the relational expression to obtain a first-order variation rate and a variation acceleration of relative drift of each satellite within the simulation time period; and
determining the second offset amount of each satellite according to the first-order variation rate and the variation acceleration.

### Claim 13 (depends on 12)

The method for maintaining Walker constellation formation according to claim 12, wherein the step of determining the relational expression of the second relative drift amount varying over time within the simulation time period further comprises:
the relational expression of the second relative drift amount varying over time within the simulation time period is a sixth formula, which is written as:
{
Δ
⁢
Ω
⁡
(
t
)
=
Δ
⁢
Ω
0
+
k
∼
1
Ω
⁢
t
+
0
.
5
⁢
k
∼
2
Ω
⁢
t
2
+
O
⁡
(
t
2
)
Δ
⁢
λ
⁡
(
t
)
=
Δλ
0
+
k
∼
1
λ
⁢
t
+
0
.
5
⁢
k
∼
2
λ
⁢
t
2
+
O
⁡
(
t
2
)
,
where ΔΩ(t) represents a right ascension of ascending node as a function of time, Δλ(t) represents an along-track angle as a function of time, ΔΩ0 is an initial value of relative drift of the right ascension of ascending node, Δλ0 is an initial value of relative drift of the along-track angle, {tilde over (k)}1Ω and {tilde over (k)}2Ω are a residual term of a first-order variation rate and a residual term of a variation acceleration of the relative drift of the right ascension of ascending node after the first offset, respectively, {tilde over (k)}1λ and {tilde over (k)}2λ are a residual term of a first-order variation rate and a residual term of a variation acceleration of the relative drift of the along-track angle after the first offset, respectively, and O(t2) is a higher-order term after the first offset.

### Claim 14

A terminal device, comprising a non-transitory memory, a processor and a computer program which is stored on the memory and can be run on the processor, wherein the computer program, when executed by the processor, implements a method for maintaining Walker constellation formation, comprising steps of: determining, according to parameters of a Walker constellation, a first offset amount of each satellite within a simulation time period; performing first offset on each satellite according to the first offset amount to obtain a Walker constellation after the first offset; determining, according to parameters of the Walker constellation after the first offset, a second offset amount of each satellite within the simulation time period; and superimposing the first offset amount and the second offset amount, and performing second offset on each satellite according to a superimposing result so as to maintain the formation of the Walker constellation.