CN118182875A

CN118182875A - Synchronous unfolding design method and system for sailboard of flat-plate satellite

Info

Publication number: CN118182875A
Application number: CN202410606121.4A
Authority: CN
Inventors: 郭金生; 熊意盛; 王宏旭; 吴凡; 奚瑞辰; 刘天喜
Original assignee: Harbin Institute of Technology
Current assignee: Harbin Institute of Technology
Priority date: 2024-05-16
Filing date: 2024-05-16
Publication date: 2024-06-14

Abstract

The embodiment of the disclosure discloses a synchronous unfolding design method and a synchronous unfolding design system for a sailboard of a flat plate satellite, and relates to the technical field of satellite structures; the synchronous unfolding design method comprises the following steps: in the unfolding process of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard, a first corresponding relation between the respective first unfolding angles of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard and the rigidity coefficients of the corresponding torsion springs is obtained; and when the first unfolding angle is a second unfolding angle when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are unfolded synchronously, acquiring rigidity coefficients of torsion springs corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard based on the first corresponding relation. The synchronous unfolding design method provided by the embodiment of the disclosure can realize synchronous unfolding of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard.

Description

Synchronous unfolding design method and system for sailboard of flat-plate satellite

Technical Field

The embodiment of the disclosure relates to the technical field of satellite structures, in particular to a synchronous unfolding design method and system for a sailboard of a flat-plate satellite.

Background

Flat satellites typically have large solar panels (hereinafter simply referred to as "sailboards") that are typically stowed during the launch phase, given the space constraints in the vehicle and the large overloads that need to be sustained during the launch, until the flat satellite is separated from the vehicle, the sailboards can be unlocked for deployment.

In general, in order to ensure that each sub-windsurfing board in the windsurfing boards can move along a predetermined track and be unfolded synchronously, a synchronous unfolding mechanism is usually required to be added on each windsurfing board to ensure synchronous unfolding of each sub-windsurfing board. However, the deployment control mechanism used at present has a large mass, which is not only unfavorable for the lightweight design of the sailboard, but also reduces the mass of the payload on the flat satellite.

Disclosure of Invention

In view of the foregoing, it is desirable for embodiments of the present disclosure to provide a synchronous deployment design method and system for a windsurfing board of a flat-plate satellite; the light-weight design of the sailboards can be realized while the synchronous unfolding of all the sub sailboards in the sailboards is ensured.

The technical scheme of the embodiment of the disclosure is realized as follows:

In a first aspect, an embodiment of the present disclosure provides a synchronous unfolding design method for a sailboard of a flat-plate satellite, where the sailboard includes a first sub-sailboard, a second sub-sailboard, and a third sub-sailboard, and a main body of the flat-plate satellite, the first sub-sailboard, the second sub-sailboard, and the third sub-sailboard are sequentially connected through a hinge assembly with torsion springs, the synchronous unfolding design method includes:

in the unfolding process of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard, a first corresponding relation between the respective first unfolding angles of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard and the corresponding rigidity coefficients of the torsion springs is obtained;

When the first unfolding angle is a second unfolding angle of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard during synchronous unfolding, respectively acquiring rigidity coefficients of torsion springs corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard based on the first corresponding relation;

The stiffness coefficients of the torsion springs corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard, which are respectively obtained, can enable the first sub-sailboard, the second sub-sailboard and the third sub-sailboard to be unfolded synchronously.

In a second aspect, embodiments of the present disclosure provide a synchronous deployment design system for a panel satellite including a main body of the panel satellite, a first sub-panel, a second sub-panel, and a third sub-panel, the synchronous deployment design system comprising: a first acquisition unit and a second acquisition unit; wherein,

The first obtaining part is configured to obtain a first corresponding relation between respective first unfolding angles of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard and rigidity coefficients of corresponding torsion springs in the unfolding process of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard;

the second obtaining part is configured to obtain stiffness coefficients of torsion springs corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard respectively based on the first corresponding relationship when the first unfolding angle is a second unfolding angle when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are unfolded synchronously;

The embodiment of the disclosure provides a synchronous unfolding design method and system for a sailboard of a flat-plate satellite; the main body, the first sub-sailboard, the second sub-sailboard and the third sub-sailboard of the flat plate satellite are sequentially connected through the hinge assembly with the torsion springs, so that the quality of the sailboard is reduced, and the light-weight design of the sailboard is realized. In addition, in the unfolding process of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard, a first corresponding relation between the respective first unfolding angles of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard and the rigidity coefficients of the corresponding torsion springs is obtained; and when the first unfolding angles are the second unfolding angles of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard during synchronous unfolding, the rigidity coefficients of the torsion springs corresponding to the sub-sailboards are obtained based on the first corresponding relation, so that the first sub-sailboard, the second sub-sailboard and the third sub-sailboard can be unfolded synchronously.

Drawings

FIG. 1 is a schematic diagram of the related art wherein the sub-windsurfing boards are connected by a rope linkage;

FIG. 2 is a schematic diagram of a rope linkage;

FIG. 3 is a schematic structural view of a hinge assembly provided in an embodiment of the present disclosure;

Fig. 4 is a schematic flow chart of a synchronous unfolding design method for a sailboard of a flat-plate satellite according to an embodiment of the disclosure;

FIG. 5 is a simplified schematic diagram of a sailboard without a synchronous deployment mechanism provided by an embodiment of the present disclosure;

FIG. 6 is a simplified schematic diagram of a sailboard with a synchronous deployment mechanism provided in an embodiment of the present disclosure;

FIG. 7 is a schematic representation of the constraint of a flat satellite on a first sub-windsurfing board provided by an embodiment of the present disclosure;

FIG. 8 is a schematic representation of the constraint of a first sub-windsurfing board on a second sub-windsurfing board provided by an embodiment of the present disclosure;

FIG. 9 is a schematic representation of the constraint of a second sub-windsurfing board on a third sub-windsurfing board provided in an embodiment of the present disclosure;

FIG. 10 is a schematic view of a collision lock between a first sub-windsurfing board and a flat satellite according to an embodiment of the present disclosure;

FIG. 11 is a schematic view of a bump lock of a second sub-windsurfing board and a first sub-windsurfing board provided according to an embodiment of the present disclosure;

FIG. 12 is a schematic view of a bump lock of a third sub-windsurfing board and a second sub-windsurfing board provided according to an embodiment of the present disclosure;

FIG. 13 is a schematic diagram of a mechanical analysis of a first sub-windsurfing board provided according to an embodiment of the present disclosure;

FIG. 14 is a schematic diagram of a mechanical analysis of a second sub-windsurfing board provided according to an embodiment of the present disclosure;

FIG. 15 is a schematic diagram of a mechanical analysis of a third sub-windsurfing board provided according to an embodiment of the present disclosure;

FIG. 16 is a schematic view of the motion constraint of a third sub-windsurfing board provided by an embodiment of the present disclosure;

FIG. 17 is a schematic view of the motion constraint of a second sub-windsurfing board provided by an embodiment of the present disclosure;

FIG. 18 is a schematic diagram of a mechanical analysis of a flat plate satellite provided by an embodiment of the present disclosure;

FIG. 19 is a schematic diagram of a synchronous deployment design system for a windsurfing board of a flat-plate satellite according to an embodiment of the present disclosure;

FIG. 20 is a graph of the angular variation of the synchronous deployment of the sub-windsurfing boards provided by embodiments of the present disclosure;

FIG. 21 is a graph of angular velocity variation for synchronous deployment of various sub-windsurfing boards provided by embodiments of the present disclosure;

FIG. 22 is a graph of the stiffness coefficient variation of the torsion springs corresponding to each of the sub-windsurfing boards provided by embodiments of the present disclosure;

FIG. 23 is a schematic illustration of a first sub-windsurfing board not deployed in place, a second sub-windsurfing board and a third sub-windsurfing board deployed in place provided by an embodiment of the present disclosure;

FIG. 24 is a graph of the variation of the spread angle of each of the sub-windsurfing boards of the case shown in FIG. 23;

FIG. 25 is a schematic view of a first and third sub-windsurfing boards deployed in place and a second sub-windsurfing board not deployed in place provided in an embodiment of the present disclosure;

FIG. 26 is a graph of the variation in the spread angle of each of the sub-windsurfing boards of the case shown in FIG. 25;

FIG. 27 is a schematic illustration of a first and second sub-windsurfing boards deployed in place and a third sub-windsurfing board not deployed in place provided by an embodiment of the present disclosure;

FIG. 28 is a graph of the variation in the spread angle of each of the sub-windsurfing boards of the case shown in FIG. 27;

FIG. 29 is a graph of simulation results of the synchronous deployment of a single stage hinge assembly provided by an embodiment of the present disclosure;

FIG. 30 is a graph of synchronous expansion simulation results corresponding to a first-stage stiffness coefficient provided by an embodiment of the present disclosure;

FIG. 31 is a graph of simulation results of a synchronous deployment with the addition of a second stage stiffness coefficient provided by an embodiment of the present disclosure;

fig. 32 is a diagram of simulation results of synchronous deviation angles of the second and third sub-windsurfing boards respectively relative to the first sub-windsurfing board during the unfolding process according to the embodiment of the present disclosure.

List of reference numerals:

1: a flat plate satellite; 10: a body of a flat plate satellite; 20: a windsurfing board; 201: a first sub-windsurfing board; 202: a second sub-windsurfing board; 203: a third sub-windsurfing board; 2: a rope linkage mechanism; 21: a guide ring; 22: a soft wire rope; 3: a hinge assembly; 31 torsion springs; 32: a first connection portion; 33: a second connecting portion; 34: a pivot; 35: and a limiting piece.

Detailed Description

The technical solutions in the embodiments of the present disclosure will be clearly and completely described below with reference to the drawings in the embodiments of the present disclosure.

Referring to fig. 1, a schematic structural diagram of a flat plate satellite 1 is schematically shown. As shown in fig. 1, the flat satellite 1 includes a main body 10 of the flat satellite and a windsurfing board 20. The windsurfing board 20 comprises a first windsurfing board 201, a second windsurfing board 202 and a third windsurfing board 203. In the related art, in order to allow the first, second and third sub-windsurfing boards 201, 202 and 203 to be unfolded synchronously, the main body 10, the first, second and third sub-windsurfing boards 201, 202 and 203 of the flat plate satellite 1 are connected in sequence by the rope linkage 2 shown in fig. 2 and the hinge assembly 3 shown in fig. 3. The rope linkage mechanism 2 is used as a synchronous unfolding mechanism for controlling each sub-sailboard to synchronously move in the unfolding process of each sub-sailboard. The hinge assembly 3 is used to drive each sub-windsurfing board to be unfolded towards the target position. That is, in the related art, the corresponding rope linkage 2 and the hinge assembly 3 are required to cooperate in order to ensure that each sub-windsurfing board can be deployed simultaneously. In some examples, as shown in fig. 2, the rope linkage 2 is composed of a guide ring 21, and a wire rope 22 guided and supported by the guide ring 21. The rope guide ring 21 is fixedly connected with the corresponding sub-sailboard. In some examples, as shown in fig. 3, the hinge assembly 3 includes a torsion spring 31, a first connection portion 32, a second connection portion 33, and a pivot 34. The first and second connection portions 32, 33 are rotatable about a pivot 34. In another example, torsion spring 31 is mounted on pivot 34.

The synchronous unfolding means that in the unfolding process of each sub-sailboard, the unfolding angles corresponding to the sub-sailboards at any same time point are the same.

It should be noted that, as shown in fig. 1 and 2, the first sub-windsurfing board 201 connected to the main body 10 (shown as a hatched rectangular area in the figure) of the flat satellite 1 is generally referred to as an inner windsurfing board, the third sub-windsurfing board 203 located at the outermost side is generally referred to as an outer windsurfing board, and the second sub-windsurfing board 202 located between the first sub-windsurfing board 201 and the third sub-windsurfing board 203 is generally referred to as an intermediate windsurfing board.

In some examples, the technical solutions of the embodiments of the present disclosure can also be applied to other satellites, such as cylindrical satellites, etc.

In some examples, the flat plate satellite 1 may include a payload or the like in addition to the body 10 and the windsurfing board 20.

As can be seen from fig. 1 and 2, the first sub-windsurfing board 201 and the main body 10 of the flat satellite, the first sub-windsurfing board 201 and the second sub-windsurfing board 202, and the third sub-windsurfing board 203 and the second sub-windsurfing board 202 all need to be provided with rope linkage mechanisms 2 to ensure that the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203 can be unfolded synchronously. These rope linkages 2 increase the mass of the windsurfing board 20 and thus the overall system, not only increasing the launch cost, but also limiting the flexibility of the overall system and limiting the mass of the payload.

Based on this, for the flat satellite 1 shown in fig. 1 and including 3 sub-sailboards, the embodiment of the disclosure is expected to provide a technical solution that can achieve both light weight and synchronous deployment of each sub-sailboard. The inventors have noted that synchronous deployment of the sub-windsurfing boards can also be achieved for the flat plate satellite 1 shown in fig. 1 by means of the hinge assembly 3 alone without the installation of the rope linkage 2. Thus, in the disclosed embodiment, the main body 10, the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203 of the flat plate satellite 1 are connected in sequence by the hinge assembly 3 with torsion springs 31 as shown in fig. 3. It should be noted that, in the embodiment of the present disclosure, the hinge assembly 3 corresponding to the first sub-windsurfing board 201 is used for connecting the first sub-windsurfing board 201 with the main body 10 of the flat satellite, the hinge assembly 3 corresponding to the second sub-windsurfing board 202 is used for connecting the first sub-windsurfing board 201 with the second sub-windsurfing board 202, and the hinge assembly 3 corresponding to the third sub-windsurfing board 203 is used for connecting the third sub-windsurfing board 203 with the second sub-windsurfing board 202.

The inventors have also noted that when each sub-windsurfing board is in a collapsed state, the torsion springs 31 in each hinge assembly are compressed and generate a driving moment. When each sub-windsurfing board is unfolded, the driving moment drives the corresponding sub-windsurfing board to be unfolded. After each sub-windsurfing board is unfolded in place, because the torsion spring 31 in each hinge assembly still keeps a certain compression amount, the pretightening force corresponding to the compression amount can lock each corresponding sub-windsurfing board. Of course, in some examples, the hinge assembly 3 further includes a limiting member 35, and after each sub-windsurfing board is unfolded in place, the pretightening force generated by the torsion spring 31 in the corresponding hinge assembly can press the corresponding sub-windsurfing board against the limiting member 35 to achieve locking of the corresponding sub-windsurfing board.

In the embodiment of the present disclosure, the rope linkage mechanism 2 is not provided in the windsurfing board 20 to realize the synchronous unfolding of each sub windsurfing board, but the hinge assembly 3 is adopted to realize the synchronous unfolding of each sub windsurfing board, so that the quality of the windsurfing board 20 is reduced, and the light-weight design of the windsurfing board 20 is realized.

Based on the above, referring to fig. 4, there is shown a synchronous unfolding design method for a sailboard of a flat-plate satellite according to an embodiment of the present disclosure, which includes the following steps.

In step S401, in the process of unfolding the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board, a first correspondence between the respective first unfolding angles of the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board and the stiffness coefficients of the corresponding torsion springs is obtained.

In some examples, a simplified model is built for each sub-windsurfing board provided by embodiments of the present disclosure solely by windsurfing boards 20 connected by a hinge assembly 3, as shown in particular in fig. 5. It should be noted that, since the windsurfing board 20 shown in fig. 5 does not have a synchronous unfolding mechanism, the first unfolding angle, the first unfolding angular speed and the first unfolding angular acceleration of each sub windsurfing board are not equal in the unfolding process, and further three degrees of freedom are provided in the simplified model shown in fig. 5, and three generalized coordinate parameters (α, θ, β) are used to establish the configuration coordinates of the windsurfing board 20. It should be noted that, the origin of the coordinate system in fig. 5 is located at the axis of the hinge assembly 3 corresponding to the first sub-windsurfing board 201, the x-axis is perpendicular to the plane where the windsurfing board 20 is in the folded state, and points to the outside of the main body 10 of the flat satellite, and the y-axis is perpendicular to the plane where the windsurfing board 20 is unfolded.

Specifically, the first deployment angle of the first sub-windsurfing board 201 represents the angle of the first sub-windsurfing board 201 with respect to the y-axis negative direction of the inertial frame, i.e. the angle α in fig. 5; the first deployment angle of the second sub-windsurfing board 202 represents the angle of the second sub-windsurfing board 202 with respect to the positive y-axis direction of the inertial frame, i.e. the angle θ in fig. 5; the first deployment angle of the third sub-windsurfing board 203 represents the angle of the third sub-windsurfing board 203 with respect to the negative y-axis direction of the inertial frame, i.e. the angle β in fig. 5.

In step S402, when the first expansion angle is the second expansion angle when the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board are synchronously expanded, respectively obtaining rigidity coefficients of torsion springs corresponding to the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board based on the first corresponding relation;

The rigidity coefficients of the torsion springs corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard can enable the first sub-sailboard, the second sub-sailboard and the third sub-sailboard to be unfolded synchronously.

The second deployment angle refers to the deployment angle of each of the sub-windsurfing boards when a synchronous deployment mechanism, such as a rope linkage mechanism 2, is provided in the flat satellite 1 to drive the first sub-windsurfing board 201, the second sub-windsurfing board 202, and the third sub-windsurfing board 203 to be deployed synchronously. It will be appreciated that when the first, second and third sub-windsurfing boards 201, 202 and 203 are deployed simultaneously, their respective second deployment angles are the same, as are the second deployment angular speeds and the second deployment angular accelerations.

Therefore, when the first correspondence between the respective first deployment angles of the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203 and the stiffness coefficients of the corresponding torsion springs is known, the stiffness coefficients of the torsion springs in the hinge assemblies corresponding to the respective sub-windsurfing boards can be obtained when the first deployment angles of the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203 are set to be the second deployment angles when the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203 are deployed synchronously. Since the corresponding sub-windsurfing boards are driven to be unfolded by the torsion springs 31 in the hinge assemblies 3 in the embodiment of the present disclosure, the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203 can be synchronously unfolded based on the obtained rigidity coefficient of the torsion springs 31 in the hinge assemblies 3 corresponding to the sub-windsurfing boards.

For the technical solution shown in fig. 4, the main body 10, the first sub-sailboard 201, the second sub-sailboard 202 and the third sub-sailboard 203 of the flat satellite are sequentially connected through a hinge assembly with torsion springs, so that the quality of the sailboard 20 is reduced, and the light-weight design of the sailboard 20 is realized. In addition, in the unfolding process of the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203, a first corresponding relation between the first unfolding angles of the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203 and the rigidity coefficients of torsion springs in the corresponding hinge assemblies is obtained; and when the first unfolding angles are the second unfolding angles when the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203 are unfolded synchronously, based on the first corresponding relation, the rigidity coefficient of the torsion spring in the hinge assembly corresponding to each sub-windsurfing board is obtained, so that the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203 are unfolded synchronously.

For the technical solution shown in fig. 4, in some possible embodiments, before acquiring the first correspondence, the synchronous expansion design method further includes:

Based on a set synchronous unfolding model, in the synchronous unfolding process of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard, respectively obtaining the inertia force and the inertia moment of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard according to the length, the mass and the inertia radius around the mass center of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard;

Based on the darebel principle, determining a second correspondence between the inertial force and the moment of inertia and a second deployment angle;

and acquiring a second unfolding angle based on the inertia force, the inertia moment and the second corresponding relation.

In some examples, the synchronous deployment model set forth above builds a simplified model for the windsurfing board 20 with rope linkages 2 shown in fig. 1, as shown in particular in fig. 6. It will be appreciated that in the model shown in fig. 6, the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203 can be deployed synchronously under the drive of the corresponding rope linkage 2. The second deployment angle refers to the angle between the sub-sails relative to the y-axis of the inertial frame, i.e. the angle in FIG. 6Is included in the above-mentioned range. As shown in fig. 6, in the embodiment of the present disclosure, the length of the first sub-windsurfing board 201 is set to/>The center of mass eccentricity of the first sub-windsurfing board 201 isThe second sub-windsurfing board 202 has a length/>The length of the third sub-windsurfing board is/>The distance of the end a in the first sub-windsurfing board 201 from the centre of mass of the first sub-windsurfing board 201 is/>The distance of the end B in the first sub-windsurfing board 201 from the centre of mass of the first sub-windsurfing board 201 is/>. The distance of end B in the second sub-windsurfing board 202 from the centre of mass of the second sub-windsurfing board 202 is/>The distance of the end C in the second sub-windsurfing board 202 from the centre of mass of the second sub-windsurfing board 202 is/>. The distance of the end C in the third sub-windsurfing board 203 from the centre of mass of the third sub-windsurfing board 203 is/>The distance of the end point D in the third sub-windsurfing board 203 from the centre of mass of the third sub-windsurfing board 203 is/>。

It should be noted that, in fig. 5 and fig. 6, the centroids of the second and third sub-windsurfing boards 202 and 203 are exemplarily shown to be located on the x-axis, but in the implementation process, the ordinate of the centroids of the second and third sub-windsurfing boards 202 and 203 still needs to be calculated by the calculation method of the foregoing technical solution.

In some examples, the complementary angle to the second deployment angle is based on the length of the first sub-windsurfing board 201, the centroid eccentricity, and the complementary angle to the second deployment angleThe centroid coordinates of the first sub-windsurfing board 201 can be calculated. Specifically, the calculation can be performed according to formula (1):

（1）

Where x ₀₁ represents the abscissa of the centroid of the first sub-windsurfing board 201 and y ₀₁ represents the ordinate of the centroid of the first sub-windsurfing board 201.

In the formula (1),。

In other examples, the centroid coordinates of the second and third sub-windsurfing boards 202 and 203 can be calculated from the lengths of the second and third sub-windsurfing boards 202 and 203, respectively, and the complementary angles of the second deployment angle. Specifically, the calculation can be performed according to formula (2):

（2）

Where, when i=2, the centroid coordinates of the second sub-windsurfing board 202 shown in the BC segment are calculated according to the formula (2), and when i=3, the centroid coordinates of the third sub-windsurfing board 203 shown in the CD segment are calculated according to the formula (2).

In the formula (2),。

It will be appreciated that the technical solution of the present disclosure is explained in detail based on the fact that the second and third sub-windsurfing boards 202 and 203 have the same length. However, in some examples, the technical solution according to the present disclosure can also perform synchronous unfolding design for the case that the lengths of the first sub-windsurfing board 201, the second sub-windsurfing board 202, and the third sub-windsurfing board 203 are different.

In the disclosed embodiments, the inertial force and moment of inertia of each sub-windsurfing board are related to the quadratic differentiation of the corresponding centroid coordinates with respect to time. Thus, once differentiating the time t based on the formula (1) and the formula (2), respectively, can be obtained:

（3）

（4）

Then, the time t is differentiated again based on the equation (3) and the equation (4), respectively, to obtain:

（5）

Therefore, based on the secondary differentiation result of the centroid coordinates of each sub-sailboard with respect to the time t, the inertia force and the inertia moment of each sub-sailboard can be obtained through the darebel principle.

（6）

Wherein F _x1 represents the inertial force of the first sub-windsurfing board in the x-direction; f _y1 represents the inertial force of the first sub-windsurfing board in the y direction; m ₁ represents the moment of inertia of the first sub-windsurfing board; Representing the radius of inertia of the first sub-windsurfing board about its centroid; when i=2,/> Representing the inertial radius of the second sub-windsurfing board around the centroid, F _xi representing the inertial force of the second sub-windsurfing board in the x direction, F _yi representing the inertial force of the second sub-windsurfing board in the y direction, and M _i representing the inertial moment of the second sub-windsurfing board; when i=3,/>The inertia radius of the third sub-windsurfing board around the centroid is shown, F _xi is the inertia force of the third sub-windsurfing board in the x direction, F _yi is the inertia force of the third sub-windsurfing board in the y direction, and M _i is the inertia moment of the third sub-windsurfing board.

For the simplified model shown in fig. 6, based on the darebel principle, it is possible to obtain:

（7）

Wherein, The driving moment of the torsion spring in the hinge component corresponding to the first sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are synchronously unfolded is represented; /(I)Representing the resistance moment of the hinge component corresponding to the first sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are unfolded synchronously; /(I)Representing a virtual displacement; /(I)Representing the virtual displacement of the centroid of the ith sub-windsurfing board in the x direction; /(I)Representing the virtual displacement of the centroid of the ith sub-windsurfing board in the y direction; when i=2,/>Representing the driving moment of the torsion spring in the hinge component corresponding to the second sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are synchronously unfolded,Representing the resistance moment of the hinge component corresponding to the second sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are unfolded synchronously; when i=3,/>Representing the driving moment of a torsion spring in a hinge assembly corresponding to the third sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are synchronously unfolded,/>And the moment of resistance of the hinge assembly corresponding to the third sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are unfolded synchronously is represented.

In the model shown in fig. 6, the virtual displacements corresponding to the first sub-windsurfing board 201, the second sub-windsurfing board 202 and the third sub-windsurfing board 203 are equal.

As can be appreciated, equation (7) is used to characterize a second correspondence between inertial force and moment of inertia and a second deployment angle.

In formula (7):

Wherein i=2, 3 (8)

Substitution of formula (8) into (7) yields:

（9）

Wherein,

（10）

Wherein,The rigidity coefficient of a torsion spring in a hinge component corresponding to the first sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are synchronously unfolded is represented; /(I)The resistance coefficient of the hinge component corresponding to the first sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are synchronously unfolded is represented; when i=2,/>The rigidity coefficient of the torsion spring in the hinge component corresponding to the second sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are synchronously unfolded is represented; /(I)The resistance coefficient of the hinge component corresponding to the second sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are synchronously unfolded is represented; when i=3,/>The rigidity coefficient of the torsion spring in the hinge component corresponding to the third sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are synchronously unfolded is represented; /(I)And the resistance coefficient of the hinge component corresponding to the third sub-sailboard when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are unfolded synchronously is represented.

It will be appreciated that, in the implementation process, when each parameter in the formula (10) corresponding to each sub-windsurfing board is known for the model shown in fig. 6, the complementary angle Φ about the second deployment angle when each sub-windsurfing board is deployed synchronously can be obtained through the formula (9), and thus the second deployment angle can be obtained.

For the technical solution shown in fig. 4, in some possible embodiments, during the unfolding process of the first, second and third sub-windsurfing boards, respectively, a first correspondence relationship between the first unfolding angles of the first, second and third sub-windsurfing boards and the stiffness coefficients of the corresponding torsion springs is obtained, including:

in the unfolding process of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard, the kinetic energy equation of the sailboard is determined as follows:

Wherein, ，/>，，/>，/>，/>; M ₁ represents the mass of the first sub-windsurfing board; m ₂ represents the mass of the second sub-windsurfing board; m ₃ represents the mass of the third sub-windsurfing board; alpha represents a first unfolding angle of the first sub-windsurfing board; /(I)A first spread angular velocity representing a first sub-windsurfing board; θ represents the first deployment angle of the second sub-windsurfing board; /(I)A first spread angular velocity representing a second sub-windsurfing board; beta represents the first unfolding angle of the third sub-windsurfing board; /(I)A first spread angular velocity representing a third sub-windsurfing board; /(I)Representing the length of the first sub-windsurfing board; /(I)Representing the length of the second sub-windsurfing board; /(I)Representing the length of the third sub-windsurfing board; /(I)Representing a distance between an end of the second sub-windsurfing board connected to the first sub-windsurfing board and a center of mass of the second sub-windsurfing board; /(I)Representing a distance between an end of the third sub-windsurfing board connected to the second sub-windsurfing board and a center of mass of the third sub-windsurfing board;

based on the driving moment of the torsion springs corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard, the potential energy equation of the sailboard is obtained as follows:

Wherein, Representing the driving moment of a torsion spring in the hinge assembly corresponding to the first sub-sailboard; /(I)Representing the rigidity coefficient of a torsion spring in the hinge assembly corresponding to the first sub-sailboard; /(I)Representing the driving moment of the torsion spring in the hinge assembly corresponding to the second sub-sailboard; /(I)Representing the rigidity coefficient of the torsion spring in the hinge component corresponding to the second sub-sailboard; /(I)Representing the driving moment of the torsion spring in the hinge assembly corresponding to the third sub-sailboard; /(I)Representing the rigidity coefficient of the torsion spring in the hinge component corresponding to the third sub-sailboard;

Based on the kinetic energy equation and the potential energy equation, a first corresponding relation is obtained by using the second Lagrangian equation as follows:

Wherein, A first angular acceleration representative of a first sub-windsurfing board; /(I)A first angular acceleration representing a second sub-windsurfing board; /(I)Representing the first angular acceleration of the third sub-windsurfing board.

In some examples, as shown in fig. 5, the centroids of each sub-windsurfing board are set to be at their respective geometric centers, so the centroid coordinates of the three sub-windsurfing boards can be expressed in generalized coordinates as:

（11）

（12）

Wherein, An abscissa representing the centroid of the first sub-windsurfing board 201; /(I)An ordinate representing the centre of mass of the first sub-windsurfing board 201; /(I)An abscissa representing the center of mass of the second sub-windsurfing board 202; /(I)An ordinate representing the center of mass of the second sub-windsurfing board 202; /(I)An abscissa representing the center of mass of the third sub-windsurfing board 203; /(I)Representing the ordinate of the centre of mass of the third sub-windsurfing board 203.

In some examples, the generalized speed of the centroids of the individual sub-windsurfing boards is derived first order from the above equation:

（13）

（14）

the kinetic energy equation for the windsurfing board is thus obtained as follows:

（15）

In some examples, the driving moment generated by the torsion spring of the hinge assembly during the process of driving the corresponding sub-windsurfing board to unwind may be expressed in mathematical form as:

Wherein, Representing the driving moment of the torsion spring; /(I)Indicating the remaining preload of the torsion spring when the windsurfing board 20 is extended; /(I)Representing the stiffness coefficient of the torsion spring; λ represents the corresponding angle of the hinge assembly in the folded state, and λ=2Φ.

The potential function V _q of the torsion spring can be expressed mathematically as:

The potential energy equation of the simplified model shown in fig. 5 can be expressed as the following potential function:

。

and solving the second Lagrangian equation to obtain a dynamic differential equation set of the simplified model shown in FIG. 5.

Substituting the expression of the kinetic energy equation Q and the potential energy equation V obtained by calculation into a Lagrangian equation of a second class:

Wherein, ; Q _k represents the generalized coordinates (α, θ, β) described above.

Further, the kinetic equation shown below is obtained:

（16）

In a specific implementation process, the above-mentioned dynamic equation (16) is used to characterize a first correspondence between the first deployment angles of the first, second and third sub-windsurfing boards and the stiffness coefficients of the torsion springs in the corresponding hinge assemblies.

For the technical solution shown in fig. 4, in some possible embodiments, when the first deployment angle is the second deployment angle when the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board are deployed synchronously, based on the first correspondence, the stiffness coefficients of torsion springs corresponding to the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board are obtained respectively, including:

When the first unfolding angle is the second unfolding angle, respectively acquiring the rigidity coefficients of torsion springs corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard according to the complementary angle of the second unfolding angle and the first corresponding relation, wherein the rigidity coefficients are as follows:

（17）

Wherein, The complementary angles of the second unfolding angles when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are synchronously unfolded are represented; /(I)Representing a second angular spread velocity when the first, second, and third sub-windsurfing boards are synchronously spread; /(I)Representing a second angular spread acceleration of the first, second, and third sub-windsurfing boards when synchronously spread; and

；

Representing the moment of inertia of the first sub-windsurfing board about its centre of mass; /(I)Representing the moment of inertia of the second sub-windsurfing board about its centre of mass; /(I)Representing the moment of inertia of the third sub-windsurfing board about its centre of mass; m ₁ represents the mass of the first sub-windsurfing board; m ₂ represents the mass of the second sub-windsurfing board; m ₃ represents the mass of the third sub-windsurfing board; /(I)Representing the length of the first sub-windsurfing board; /(I)Representing the length of the second sub-windsurfing board; /(I)Representing a distance between an end of the second sub-windsurfing board connected to the third sub-windsurfing board and a center of mass of the second sub-windsurfing board; /(I)Representing a distance between an end of the second sub-windsurfing board connected to the first sub-windsurfing board and a center of mass of the second sub-windsurfing board; /(I)Representing the distance between the end of the third sub-windsurfing board that is connected to the second sub-windsurfing board and the centre of mass of the third sub-windsurfing board.

It can be understood that, when the synchronous unfolding of the sub-sailboards in the model shown in fig. 5 is achieved, the first unfolding angles of the sub-sailboards are the same, and the first unfolding angles of the sub-sailboards are the second unfolding angles. Therefore, the first unfolding angles alpha, theta and beta of the sub-sailboards in the formula (16) are all set to be equal to the second unfolding angles, so that the rigidity coefficients of the torsion springs in the hinge assemblies corresponding to the sub-sailboards can be obtained when the sub-sailboards are unfolded synchronously.

Specifically, the relation equation between the stiffness coefficient of the torsion spring in the hinge assembly corresponding to each sub-sailboard and each parameter is as follows:

。

Based on the above, when the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are connected and unfolded through the hinge assemblies in sequence, the first sub-sailboard, the second sub-sailboard and the third sub-sailboard can be unfolded synchronously by setting the rigidity coefficient of the torsion spring in the hinge assembly corresponding to each sub-sailboard to be the numerical value calculated in the formula (17).

For the solution shown in fig. 4, in some possible embodiments, the synchronous unfolding design method further includes:

According to the unfolding sequence of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard, a first constraint counter moment mathematical model among the first sub-sailboard, the second sub-sailboard and the third sub-sailboard is built;

Establishing a second constraint counter moment mathematical model among the first sub-sailboard, the second sub-sailboard and the third sub-sailboard according to the first unfolding angles of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard;

And correcting the rigidity coefficients of torsion springs corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard in the unfolding process of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard based on the first constraint counter-torque mathematical model and the second constraint counter-torque mathematical model.

It should be noted that, the stiffness coefficients of the torsion springs corresponding to the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board can be corrected in an auxiliary manner by using the first constraint counter-torque mathematical model and the second constraint counter-torque mathematical model, so as to ensure the continuity of the model shown in fig. 5 at the starting and ending critical points of the expansion of the windsurfing board 20.

For the above embodiments, in some examples, building a first constrained counter moment mathematical model between the first, second, and third sub-windsurfing boards according to an order of deployment of the first, second, and third sub-windsurfing boards comprises:

When the second sub-sailboard is unfolded before the first sub-sailboard, the first constraint counter moment mathematical model of the main body of the first sub-sailboard subjected to the flat plate satellite is as follows:

Wherein G ₁ represents a first constraint counter moment of the first sub-windsurfing board subject to the body of the flat-plate satellite; alpha represents a first unfolding angle of the first sub-windsurfing board; Representing an extrusion elastic coefficient between the first sub-windsurfing board and the main body of the flat satellite;

when the first sub-sailboard is unfolded before the second sub-sailboard, the first constraint counter-moment mathematical model of the second sub-sailboard subjected to the first sub-sailboard is as follows:

Wherein G ₂ represents a first constraint counter moment to which the second sub-windsurfing board is subjected to the first sub-windsurfing board; a first spread angular velocity representing a first sub-windsurfing board; /(I) A first spread angular velocity representing a second sub-windsurfing board; /(I)Representing the extrusion elasticity coefficient between the first sub-sailboard and the second sub-sailboard;

when the second sub-windsurfing board is unfolded before the third sub-windsurfing board, the first constraint counter moment mathematical model of the third sub-windsurfing board subjected to the second sub-windsurfing board is as follows:

Wherein G ₃ represents a first constraint counter moment to which the third sub-windsurfing board is subjected to the second sub-windsurfing board; A first spread angular velocity representing a third sub-windsurfing board; /(I) Representing the coefficient of extrusion elasticity between the second and third sub-windsurfing boards.

Referring specifically to fig. 7-9, there is shown the restraint of the first, second and third sub-windsurfing boards 201, 202 and 203 during deployment.

In some examples, as shown in fig. 7, when the second sub-windsurfing board 202 is spread before the first sub-windsurfing board 201, since the first sub-windsurfing board 201 is constrained by the body of the flat satellite (not shown in fig. 7), the first sub-windsurfing board 201 cannot be spread towards the body of the flat satellite, i.e. the first spread angle of the first sub-windsurfing board 201 cannot be less than 0 degrees, and thus the first sub-windsurfing board 201 is constrained against the constraint reaction force of the body of the flat satellite (shown by the dotted arrow in fig. 7) to prevent the first sub-windsurfing board 201 from being spread towards the body of the flat satellite.

In this case, the constrained counter moment of the first sub-windsurfing board 201 by the body of the flat satellite may be expressed as:

。

In other examples, as shown in fig. 8, when the first sub-windsurfing board 201 is spread before the second sub-windsurfing board 202, since the second sub-windsurfing board 202 is constrained by the first sub-windsurfing board 201, the second sub-windsurfing board 202 cannot be spread in the opposite direction to avoid that the second sub-windsurfing board 202 passes through the first sub-windsurfing board 201, i.e. the first spread angle of the opposite direction of the second sub-windsurfing board 202 cannot be larger than the first spread angle of the first sub-windsurfing board 201, and thus the second sub-windsurfing board 202 is constrained by the constraint counterforce of the first sub-windsurfing board 201 (shown by the dot-and-dash arrow in fig. 8) to prevent the second sub-windsurfing board 202 from spreading in the opposite direction to pass through the first sub-windsurfing board 201.

In this case, the constrained counter moment of the second sub-windsurfing board 202 to the first sub-windsurfing board 201 may be expressed as:

。

in still other examples, as shown in fig. 9, when the second sub-windsurfing board 202 is spread before the third sub-windsurfing board 203, since the third sub-windsurfing board 203 is constrained by the second sub-windsurfing board 202, the third sub-windsurfing board 203 cannot be spread in the opposite direction to avoid that the third sub-windsurfing board 203 passes through the second sub-windsurfing board 202, i.e. the first spread angle of the third sub-windsurfing board 203 in the opposite direction cannot be larger than the first spread angle of the second sub-windsurfing board 202, and thus the third sub-windsurfing board 203 is constrained by the constraint reaction force (shown by the dot-and-dash arrow in fig. 9) of the second sub-windsurfing board 202 to prevent the third sub-windsurfing board 203 from spreading in the opposite direction through the second sub-windsurfing board 202.

In this case, the constrained counter moment of the third sub-windsurfing board 203 by the second sub-windsurfing board 202 may be expressed as:

。

For the above embodiments, in some examples, building a second constrained counter moment mathematical model between the first, second, and third sub-windsurfing boards from the first deployment angles of the first, second, and third sub-windsurfing boards comprises:

when the first unfolding angle of the first sub-sailboard is larger than 90 degrees, the first sub-sailboard is subjected to the second constraint counter moment mathematical model of the limiting piece in the corresponding hinge assembly, and the second constraint counter moment mathematical model is as follows:

Wherein S ₁ represents a second constraint counter moment of the first sub-windsurfing board subjected to the limiting member; a first spread angular velocity representing a first sub-windsurfing board; /(I) Representing a collision elasticity coefficient between the first sub-windsurfing board and the main body of the flat satellite;

When the sum of the first unfolding angle of the first sub-sailboard and the first unfolding angle of the second sub-sailboard is larger than 180 degrees, the second constraint counter moment mathematical model of the second sub-sailboard subjected to the first sub-sailboard is as follows:

Wherein S ₂ represents a second constraint counter moment to which the second sub-windsurfing board is subjected to the first sub-windsurfing board; a first spread angular velocity representing a second sub-windsurfing board; /(I) Representing a bump spring rate between the second sub-windsurfing board and the first sub-windsurfing board;

When the sum of the first unfolding angle of the third sub-sailboard and the first unfolding angle of the second sub-sailboard is larger than 180 degrees, the second constraint counter moment mathematical model of the third sub-sailboard subjected to the second sub-sailboard is as follows:

Wherein S ₃ represents a second constraint counter moment to which the third sub-windsurfing board is subjected to the second sub-windsurfing board; A first spread angular velocity representing a third sub-windsurfing board; /(I) Representing the coefficient of bump elasticity between the third and second sub-windsurfing boards.

In some examples, as shown in fig. 10, the first deployment angle of the first sub-windsurfing board 201 cannot exceed 90 degrees due to the constraint of the stop 35 in the corresponding hinge assembly 3. Therefore, when the first unfolding angle of the first sub-windsurfing board 201 is 90 degrees, the first sub-windsurfing board 201 collides with the limiting piece 35 in the corresponding hinge assembly if being unfolded, so that the first sub-windsurfing board 201 is constrained against moment (indicated by the dotted arrow in fig. 10) of the limiting piece 35, so as to prevent the first unfolding angle of the first sub-windsurfing board 201 from being too large, and further, the first sub-windsurfing board 201 is locked to the position with the first unfolding angle of 90 degrees, that is, the position of the first sub-windsurfing board 201 shown in fig. 10.

In this case, the constrained counter moment of the first sub-windsurfing board by the stop in the corresponding hinge assembly can be expressed as:

。

In other examples, as shown in fig. 11, since the respective sub-windsurfing boards are restrained by corresponding limiting members, the second sub-windsurfing board 202 cannot be unfolded again when being unfolded parallel to the first sub-windsurfing board 201, that is, the sum of the first unfolding angle of the first sub-windsurfing board 201 and the first unfolding angle of the second sub-windsurfing board 202 cannot exceed 180 degrees, so as to avoid collision with the first sub-windsurfing board 201 when the second sub-windsurfing board 202 is unfolded continuously. Based on this, the second sub windsurfing board 202 will be constrained against the first sub windsurfing board 201 (shown by the dotted arrow in fig. 11). But due to the interaction between the first and second sub-windsurfing boards 201, 202, the first sub-windsurfing board 201 will also be constrained against the second sub-windsurfing board 202 (indicated by the two-dot-dash arrow in fig. 11), thereby preventing the second sub-windsurfing board 202 from continuing to spread out, eventually causing the second sub-windsurfing board 202 to be locked into a position parallel to the first sub-windsurfing board 201 and co-moving with the first sub-windsurfing board 201.

。

In still other examples, as shown in fig. 12, in the implementation process, when the third sub-windsurfing board 203 is unfolded to be parallel to the second sub-windsurfing board 202, the unfolding cannot be continued, that is, the sum of the first unfolding angle of the third sub-windsurfing board 203 and the first unfolding angle of the second sub-windsurfing board 202 cannot exceed 180 degrees, so as to avoid collision with the second sub-windsurfing board 202 when the third sub-windsurfing board 203 is unfolded continuously. Based on this, the third sub-windsurfing board 203 will be constrained against the second sub-windsurfing board 202 (shown by the dotted arrow in fig. 12). But due to the interaction between the third sub-windsurfing board 203 and the second sub-windsurfing board 202, the second sub-windsurfing board 202 will also be subjected to a restraining counter moment (indicated by the two-dot chain arrow in fig. 12) of the third sub-windsurfing board 203, thereby preventing the third sub-windsurfing board 203 from being spread further, eventually causing the third sub-windsurfing board 203 to be locked into a position parallel to the second sub-windsurfing board 202 and to co-move with the second sub-windsurfing board 202.

。

Respectively carrying out mechanical analysis on the first sub-sailboard, the second sub-sailboard and the third sub-sailboard to obtain dynamic equations respectively corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard;

based on the dynamic equation, the supporting force of the hinge assembly corresponding to the first sub-sailboard is obtained as follows:

Wherein, ；/>；/>; M ₁ represents the mass of the first sub-windsurfing board; m ₂ represents the mass of the second sub-windsurfing board; m ₃ represents the mass of the third sub-windsurfing board; /(I)Representing the length of the first sub-windsurfing board; /(I)Representing the length of the second sub-windsurfing board; /(I)Representing a distance between an end of the first sub-windsurfing board connected to the body of the flat satellite and a center of mass of the first sub-windsurfing board; /(I)Representing a distance between an end of the second sub-windsurfing board connected to the third sub-windsurfing board and a center of mass of the second sub-windsurfing board; /(I)Representing a distance between an end of the third sub-windsurfing board connected to the second sub-windsurfing board and a center of mass of the third sub-windsurfing board; /(I)Representing the component force of the supporting force of the hinge component corresponding to the first sub-sailboard in the x direction; /(I)Representing the component force of the supporting force of the hinge component corresponding to the first sub-sailboard in the y direction;

the supporting force of the hinge component corresponding to the first sub-sailboard is used for representing the disturbance performance of the main body of the flat plate satellite when the first sub-sailboard is unfolded.

Specifically, as shown in fig. 13, the first sub-windsurfing board 201 is subjected to mechanical analysis. The first sub-sailboard is respectively subjected to the torque of the torsion spring at the end part A and the end part BAnd/>And is acted upon by the supporting forces of the body of the flat satellite at end a and by the forces of the second sub-windsurfing board 202 at end B.

Based on the above, the kinetic equation for the first sub-windsurfing board 201 may be derived as follows:

Wherein, Representing the moment of inertia of the first sub-windsurfing board 201 about the end a.

As shown in fig. 14, a mechanical analysis is performed on the second sub-windsurfing board 202. The second sub-windsurfing board 202 is subjected to torsion of the torsion spring at the end B, the end C, respectivelyAnd/>And is acted upon by the supporting force of the first sub-windsurfing board 201 at end B and by the acting force of the third sub-windsurfing board 203 at end C.

Based on the above description, the corresponding kinetic equation for the second sub-windsurfing board 202 can be obtained as follows:

Wherein, Representing the moment of inertia of the second sub-windsurfing board about its centre of mass.

As shown in fig. 15, the third sub-windsurfing board 203 is subjected to a mechanical analysis. The third sub-windsurfing board 203 is subjected to the torque of the torsion spring at end CIs supported by the second sub-windsurfing board 202 at the end C.

Based on the above description, the kinetic equation corresponding to the third sub-windsurfing board 203 may be obtained as follows:

Wherein, Representing the moment of inertia of the third sub-windsurfing board about its centre of mass.

In addition, during the deployment of the windsurfing boards, the connection of the first sub windsurfing board 201 with the body of the flat satellite at end a creates a constraint, the connection of the first sub windsurfing board 201 with the second sub windsurfing board 202 at end B creates a constraint, and the connection of the second sub windsurfing board 202 with the third sub windsurfing board 203 at end C creates a constraint.

The following set of constraint equations can be derived from the constraint relationships shown in fig. 16 and 17:

and then can obtain the support force of the hinge component that the first sub-sailboard corresponds:

Wherein, ；/>；/>。

It should be noted that, the supporting force of the hinge assembly corresponding to the first sub-windsurfing board refers to the force applied to the end portion of the first sub-windsurfing board 201 connected to the main body of the flat satellite, that is, the force at the end portion a in fig. 5, which is also referred to as "root hinge supporting force".

under the condition that the supporting force of the hinge component corresponding to the first sub-sailboard is obtained, carrying out mechanical analysis on the main body of the flat satellite to obtain a dynamic equation corresponding to the main body of the flat satellite;

And carrying out disturbance analysis on the main body of the flat satellite based on a dynamic equation corresponding to the main body of the flat satellite.

In a specific implementation process, as shown in fig. 18, the sub-sailboards connected to two sides of the main body of the flat satellite are coupled to the main body of the flat satellite and then unfolded, and the forces and moments generated by the hinge assemblies of the sub-sailboards located at two sides of the main body of the flat satellite on the main body of the flat satellite are analyzed. In some examples, the body of the flat satellite is subjected to forces and moments in the x 'and y' directions by the hinge assemblies of the two side sub-windsurfing boards. If the sub-sailboards on two sides are not unfolded synchronously, the forces on two sides of the main body of the flat plate satellite are different at the same moment, so that the main body of the flat plate satellite generates accelerations along the x 'direction and the y' direction respectively, and simultaneously, angular acceleration is generated, and the posture of the main body of the flat plate satellite is influenced. In the implementation process, the two side sub-sailboards are symmetrically arranged about the mass center of the main body of the flat plate satellite, and the forces in the x 'direction pass through the mass center, so that the forces in the x' direction of the hinge assemblies of the two side sub-sailboards do not generate moment which can cause the disturbance of the main body posture of the flat plate satellite.

Thus, the following kinetic equation can be derived based on the simplified model shown in fig. 18:

wherein m represents the mass of the body of the flat plate satellite; j represents the moment of inertia of the body of the flat satellite about its centroid; representing the distance between the sub sailboards at two sides and the main body of the flat plate satellite respectively; /(I) The main body of the flat plate satellite starts to rotate in the positive direction of the y' axis; /(I)Representing a component force of a supporting force of a hinge assembly corresponding to a first sub-sailboard positioned on the right side of a main body of the flat-plate satellite in the x' direction; /(I)Representing a component force of a supporting force of a hinge assembly corresponding to a first sub-sailboard positioned on the right side of a main body of the flat-plate satellite in the y' direction; /(I)Representing a component force of a supporting force of a hinge assembly corresponding to a first sub-sailboard positioned at the left side of a main body of the flat-plate satellite in an x' direction; /(I)Representing a component force of a supporting force of a hinge assembly corresponding to a first sub-sailboard positioned at the left side of a main body of the flat-plate satellite in a y' direction; /(I)Representing torque generated by a first sub-windsurfing board located on the right side of a body of the flat satellite to the body of the flat satellite; /(I)Representing the torque produced by a first sub-windsurfing board located to the left of the body of the flat satellite to the body of the flat satellite.

It should be noted that, the origin of the coordinate system in fig. 18 is located at the centroid position of the body of the flat satellite, the x 'axis is perpendicular to the plane where the sailboard 20 is in the folded state, and points to the outside of the body 10 of the flat satellite, and the y' axis is perpendicular to the plane where the sailboard 20 is unfolded.

In addition, in the implementation, the windsurfing boards 20 are symmetrically installed on both sides of the main body 10 of the flat satellite.

Finally, referring to fig. 19, embodiments of the present disclosure also provide a synchronous deployment design system 190 for a windsurfing board of a flat-plate satellite, the synchronous deployment design system 190 comprising: a first acquisition unit 1901 and a second acquisition unit 1902; wherein,

The first obtaining portion 1901 is configured to obtain a first corresponding relationship between a first expansion angle of each of the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board and a stiffness coefficient of a corresponding torsion spring respectively in a process of expanding the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board;

The second obtaining portion 1902 is configured to obtain, based on the first correspondence, stiffness coefficients of torsion springs corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard when the first deployment angle is a second deployment angle at which the first sub-sailboard, the second sub-sailboard and the third sub-sailboard are deployed synchronously;

The technical solutions provided by the embodiments of the present disclosure are described in detail below through simulation analysis examples.

Referring to table 1, specific parameters of the windsurfing board 20 are shown.

TABLE 1

The angular variation and angular velocity variation curves of the windsurfing board 20 with rope linkage 2 shown in fig. 6 are obtained through simulation analysis, and particularly, refer to fig. 20 and 21. The second expansion angle change and the second expansion angular velocity change curves of the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board shown in fig. 20 and 21 are led into the formula (17) described in the foregoing technical scheme, and each parameter in table 1 is input, so that the stiffness coefficient change curve of the torsion spring corresponding to each sub-windsurfing board in the model shown in fig. 5 can be obtained, and specifically, as shown in fig. 22. The rigidity coefficient change curves of the torsion springs corresponding to the sub-sailboards shown in fig. 22 can be fitted to obtain the approximate rigidity coefficients of the torsion springs corresponding to the sub-sailboards, if each torsion spring is only fitted with a single-value rigidity coefficient, the torsion spring is a single-stage hinge assembly, and if the rigidity coefficient change curves of the torsion springs corresponding to the sub-sailboards shown in fig. 22 are fitted in a segmented mode, each torsion spring is fitted with a multi-value combined rigidity coefficient, the torsion spring is a multi-stage hinge assembly.

The sailboard unfolding process connected with the single-stage hinge assembly is simulated, and the simulation is specifically as follows:

In some examples, if the driving moment of the torsion spring in the hinge assembly corresponding to the first sub-windsurfing board is smaller, and the driving moment of the torsion spring in the hinge assembly corresponding to the second sub-windsurfing board and the third sub-windsurfing board is relatively larger, the situation shown in fig. 23 may occur, that is, the second sub-windsurfing board and the third sub-windsurfing board are both unfolded in place, but the first sub-windsurfing board is not yet unfolded in place. In this case, the second and third sub-windsurfing boards are locked and together with the first sub-windsurfing board rotate together under the drive of the torsion spring in the corresponding hinge assembly of the first sub-windsurfing board (indicated by the dashed arrow in fig. 23). As shown in fig. 24, the second sub-windsurfing board is deployed in place for around 5.2 seconds, followed by the third sub-windsurfing board being deployed in place for around 5.6 seconds. The second and third sub-windsurfing boards are then rotated to the target position with the first sub-windsurfing board at the same first angular velocity. At this time, it is necessary to increase the driving moment of the torsion spring in the hinge assembly corresponding to the first sub-windsurfing board, and adjust the driving moment of the torsion spring in the hinge assembly corresponding to the second sub-windsurfing board and the third sub-windsurfing board respectively, so that the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board are unfolded synchronously.

In the collision sequence, the second sub-sailboard collides with the first sub-sailboard, so that the first angular velocity of the second sub-sailboard and the first sub-sailboard is reduced, then the third sub-sailboard collides with the second sub-sailboard with the reduced first angular velocity, at this time, the first angular velocity of the second sub-sailboard is reversely increased, the first angular velocity of the third sub-sailboard is reduced, finally the first sub-sailboard, the second sub-sailboard and the third sub-sailboard rotate to the target position at the same first angular velocity, and finally the first sub-sailboard collides with the main body of the flat satellite and is locked.

In other examples, if the driving moment of the torsion spring in the hinge assembly corresponding to the second sub-windsurfing board is relatively small, the situation shown in fig. 25 may occur, that is, the first sub-windsurfing board and the third windsurfing board are both unfolded in place, but the second sub-windsurfing board is not yet unfolded in place. In this case, the first sub-windsurfing board is locked with the main body of the flat satellite, and the third sub-windsurfing board is locked with the second windsurfing board and rotates together with the second sub-windsurfing board under the driving of the torsion spring of the hinge assembly corresponding to the second sub-windsurfing board (indicated by the dotted arrow in fig. 25). As shown in fig. 26, the first sub-windsurfing board is spread in place at about 5.4 seconds, then the third sub-windsurfing board is spread in place at about 5.6 seconds, and then the first sub-windsurfing board and the third sub-windsurfing board are rotated to the target position at the same first angular velocity with the second sub-windsurfing board. At this time, it is necessary to increase the driving moment of the torsion spring in the hinge assembly corresponding to the second sub-windsurfing board, and correspondingly reduce the driving moment of the torsion spring in the hinge assembly corresponding to the third sub-windsurfing board, so that the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board are synchronously unfolded.

In the collision sequence, firstly, the first sub-sailboard collides with the main body of the flat plate satellite so as to lock the first sub-sailboard, then the third sub-sailboard collides with the second sub-sailboard, the first angular velocity is reduced after the third sub-sailboard collides with the second sub-sailboard, finally the third sub-sailboard and the second sub-sailboard rotate to the target position at the same first angular velocity, and the second sub-sailboard collides with the first sub-sailboard, and because the first angular velocity of the second sub-sailboard is smaller at the moment, the first sub-sailboard cannot be caused to be too greatly deviated.

In still other examples, if the driving torque of the torsion spring in the corresponding hinge assembly of the third sub-windsurfing board is relatively small, the situation shown in fig. 27 may occur, that is, both the first sub-windsurfing board and the second sub-windsurfing board have been unfolded in place, but the third sub-windsurfing board has not yet been unfolded in place. In this case the first sub-windsurfing board is locked to the main body of the flat satellite, the second sub-windsurfing board is locked to the first sub-windsurfing board and the third sub-windsurfing board continues to spread out (indicated by the dashed arrow in fig. 27). As shown in fig. 28, the second sub-windsurfing board is spread out in place around 5.2 seconds, and then the first sub-windsurfing board is spread out in place around 6.2 seconds. In this case the third sub-windsurfing board is also unfolded into position, but the second sub-windsurfing board is bumped out of the locked position by the third sub-windsurfing board, and the third sub-windsurfing board is then rotated with the second sub-windsurfing board to the target position. At this time, it is necessary to increase the driving moment of the torsion spring in the hinge assembly corresponding to the second sub-windsurfing board, and correspondingly reduce the driving moment of the torsion spring in the hinge assembly corresponding to the third sub-windsurfing board, so that the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board are synchronously unfolded.

In the collision sequence, the second sub-windsurfing board collides with the first sub-windsurfing board first, and the second sub-windsurfing board is locked and transferred to the target position along with the first sub-windsurfing board. The first sub-windsurfing board then collides with the body of the flat satellite to complete the lock. And finally, the third sub-sailboard is unfolded to be in place and collides with the second sub-sailboard. However, the third sub-sailboard has a larger first angular velocity, and after colliding with the second sub-sailboard, the second sub-sailboard deviates from the original locking position, so that a larger deviation is formed, and finally the third sub-sailboard and the second sub-sailboard take 3 seconds to rotate again to the target position.

Based on the above description, the synchronous unfolding of each sub-sailboard can be basically realized by adjusting the parameters of the hinge assembly corresponding to each sub-sailboard. However, in order to avoid the collision when the sub-windsurfing boards are unfolded in place from causing the position deviation, the driving moment of the torsion springs in the corresponding hinge assemblies needs to be finely adjusted to ensure the collision sequence.

The preliminary simulation results obtained after optimizing the driving torque of the torsion spring in the hinge assembly are shown in fig. 29. As can be seen from fig. 29, the first, second and third sub-windsurfing boards are simultaneously unfolded to 90 degrees at 6.2 seconds, i.e. the first, second and third sub-windsurfing boards are simultaneously unfolded in place. Specifically, during the first two seconds, the second sub-sailboard is deployed first, the first sub-sailboard remains stationary at 0 degrees due to the constraints of the body of the flat satellite, and the third sub-sailboard is deployed gradually following the second sub-sailboard due to the constraints of the second sub-sailboard. After two seconds, the first sub-windsurfing board also begins to spread, and the third sub-windsurfing board still clings to the second sub-windsurfing board, and then spreads reversely. After three seconds, the third sub-windsurfing board begins to spread forward. The first, second and third sub-windsurfing boards are all unfolded in the forward direction at this time. The second sub-sailboard is unfolded at a constant speed, the first sub-sailboard is unfolded at a constant speed, and the third sub-sailboard is unfolded reversely and then is unfolded forward in an accelerating mode. The final first, second and third sub-windsurfing boards are deployed simultaneously to 90 degrees at 6.2 seconds.

It should be noted that, the reverse unfolding of the sub-sailboard means that the sub-sailboard rotates in a direction away from the target position, and the forward unfolding means that the sub-sailboard rotates in a direction close to the target position.

The sailboard unfolding process connected with the secondary hinge assembly is simulated, and the simulation is specifically as follows:

The simulated optimization of the secondary hinge assembly is performed on the basis of the simulation of the single-stage hinge assembly. Firstly, fitting a first-stage rigidity coefficient according to a rigidity coefficient change curve of a torsion spring corresponding to each sub-sailboard shown in fig. 22 so that each sub-sailboard can be synchronously unfolded in an initial unfolding process, then, when each sub-sailboard is in an asynchronous unfolding condition, fitting a second-stage rigidity coefficient according to a rigidity coefficient change curve of a torsion spring corresponding to each sub-sailboard shown in fig. 22 so that each sub-sailboard can be synchronously unfolded in a subsequent unfolding process, and then, adjusting the first-stage rigidity coefficient and the second-stage rigidity coefficient so as to further optimize the synchronous unfolding process of each sub-sailboard.

In some examples, the first order stiffness coefficients are first fitted such that each sub-windsurfing board achieves synchronous deployment during initial deployment, and a deployment graph for each sub-windsurfing board is shown in fig. 30. As can be seen from fig. 30, the sub-windsurfing boards can be deployed synchronously at substantially every time point before 2.5 seconds. After 2.5 seconds, the second sub-windsurfing board is spread out with hysteresis, while the first sub-windsurfing board remains spread out in synchronization with the third sub-windsurfing board. After 3.5 seconds, the third sub-windsurfing board is deployed ahead of the first windsurfing board. It can be seen from fig. 30 that a larger second order stiffness factor is required when the corresponding hinge assembly of the second sub-windsurfing board reaches 0.8rad with the radian of the first order stiffness factor deployment to avoid the second sub-windsurfing board from lagging deployment. And when the radian of the hinge component corresponding to the third sub-sailboard unfolded by the first-stage rigidity coefficient reaches 2.2rad, the second-stage rigidity coefficient needs to be smaller, so that the advanced unfolding of the third sub-sailboard is avoided. FIG. 31 is a graph showing the development of each sub-windsurfing board with the addition of a second level stiffness factor.

As can be seen from fig. 31, compared to the synchronous unfolding process of the single-stage hinge assembly, the synchronous unfolding process of the two-stage hinge assembly not only can ensure that the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board reach the target position at the same time, but also can ensure that the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board maintain better synchronism in the unfolding process.

Fig. 32 shows the synchronous offset angle of the second and third sub-windsurfing boards, respectively, with respect to the first sub-windsurfing board during deployment. As can be seen from fig. 32, the maximum synchronization deviation angle of the second sub-windsurfing board in the spreading process is about 0.14rad compared with the first sub-windsurfing board, the maximum synchronization deviation angle of the third sub-windsurfing board in the spreading process is about 0.17rad compared with the first sub-windsurfing board, and the shift amounts of the second sub-windsurfing board and the third sub-windsurfing board are smaller relative to the maximum synchronization deviation angle of the first sub-windsurfing board in the spreading process, which indicates that the synchronous spreading of the first sub-windsurfing board, the second sub-windsurfing board and the third sub-windsurfing board is realized by adopting the secondary hinge assembly.

It should be noted that: the technical schemes described in the embodiments of the present disclosure may be arbitrarily combined without any conflict.

The foregoing is merely specific embodiments of the disclosure, but the protection scope of the disclosure is not limited thereto, and any person skilled in the art can easily think about changes or substitutions within the technical scope of the disclosure, and it is intended to cover the scope of the disclosure. Therefore, the protection scope of the present disclosure shall be subject to the protection scope of the claims.

Claims

1. The utility model provides a synchronous expansion design method for the windsurfing board of flat satellite, characterized in that, the windsurfing board includes first son windsurfing board, second son windsurfing board and third son windsurfing board, and the main part of flat satellite, first son windsurfing board, second son windsurfing board and third son windsurfing board is connected through the hinge subassembly that has the torsional spring in proper order, synchronous expansion design method includes:

2. The synchronous expansion design method according to claim 1, characterized in that before the first correspondence is acquired, the synchronous expansion design method further comprises:

Based on a set synchronous unfolding model, in the synchronous unfolding process of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard, respectively obtaining the inertia force and the inertia moment of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard according to the lengths, the masses and the inertia radiuses around the centroids of the first sub-sailboard, the second sub-sailboard and the third sub-sailboard;

determining a second correspondence between the inertial force and the moment of inertia and the second deployment angle based on darebel's principle;

And acquiring the second unfolding angle based on the inertia force, the inertia moment and the second corresponding relation.

3. The synchronous unfolding design method according to claim 1, wherein the step of obtaining a first correspondence between respective first unfolding angles of the first, second and third sub-windsurfing boards and stiffness coefficients of the corresponding torsion springs in the unfolding process of the first, second and third sub-windsurfing boards comprises:

Wherein, ，/>，，/>，/>，/>; M ₁ represents the mass of the first sub-windsurfing board; m ₂ represents the mass of the second sub-windsurfing board; m ₃ represents the mass of the third sub-windsurfing board; alpha represents a first unfolding angle of the first sub-windsurfing board; /(I)A first spread angular velocity representing the first sub-windsurfing board; θ represents a first deployment angle of the second sub-windsurfing board; /(I)A first spread angular velocity representing the second sub-windsurfing board; beta represents a first unfolding angle of the third sub-windsurfing board; /(I)Representing a deployed first angular velocity of the third sub-windsurfing board; /(I)Representing the length of the first sub-windsurfing board; /(I)Representing the length of the second sub-windsurfing board; /(I)Representing the length of the third sub-windsurfing board; /(I)Representing a distance between an end of the second sub-windsurfing board connected to the first sub-windsurfing board and a center of mass of the second sub-windsurfing board; /(I)Representing a distance between an end of the third sub-windsurfing board connected to the second sub-windsurfing board and a center of mass of the third sub-windsurfing board;

Based on the driving moment of the torsion spring corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard, a potential energy equation of the sailboard is obtained as follows:

Wherein, Representing the driving moment of the corresponding torsion spring when the first sub-sailboard is unfolded; /(I)Representing the rigidity coefficient of the torsion spring corresponding to the unfolding of the first sub-sailboard; /(I)Representing the driving moment of the corresponding torsion spring when the second sub-sailboard is unfolded; /(I)Representing the rigidity coefficient of the torsion spring corresponding to the second sub-sailboard when being unfolded; /(I)Representing the driving moment of the torsion spring corresponding to the unfolding of the third sub-sailboard; /(I)Representing the rigidity coefficient of the torsion spring corresponding to the unfolding of the third sub-sailboard;

based on the kinetic energy equation and the potential energy equation, the first corresponding relation is obtained by using a second Lagrangian equation as follows:

Wherein, A first spread angular acceleration representative of the first sub-windsurfing board; /(I)A first spread angular acceleration representative of the second sub-windsurfing board; /(I)Representing a first spread angular acceleration of the third sub-windsurfing board.

4. The synchronous unfolding design method according to claim 1, wherein when the first unfolding angle is a second unfolding angle when the first, second and third sub-sailboards are unfolded synchronously, respectively acquiring stiffness coefficients of torsion springs corresponding to the first, second and third sub-sailboards based on the first correspondence relation, including:

when the first unfolding angle is the second unfolding angle, respectively acquiring stiffness coefficients of the torsion springs corresponding to the first sub-sailboard, the second sub-sailboard and the third sub-sailboard according to the complementary angle of the second unfolding angle and the first corresponding relation as follows:

Wherein, A complementary angle representing a second deployment angle when the first, second, and third sub-windsurfing boards are deployed in synchronization; /(I)A second angular spread velocity representing when the first, second, and third sub-windsurfing boards are synchronously spread; /(I)Representing a second angular spread acceleration of the first, second, and third sub-windsurfing boards when synchronously spread; and

；

Representing a moment of inertia of the first sub-windsurfing board about its centre of mass; /(I)Representing a moment of inertia of the second sub-windsurfing board about its centre of mass; /(I)Representing a moment of inertia of the third sub-windsurfing board about its centre of mass; m ₁ represents the mass of the first sub-windsurfing board; m ₂ represents the mass of the second sub-windsurfing board; m ₃ represents the mass of the third sub-windsurfing board; /(I)Representing the length of the first sub-windsurfing board; Representing the length of the second sub-windsurfing board; /(I) Representing a distance between an end of the second sub-windsurfing board connected to the third sub-windsurfing board and a center of mass of the second sub-windsurfing board; /(I)Representing a distance between an end of the second sub-windsurfing board connected to the first sub-windsurfing board and a center of mass of the second sub-windsurfing board; /(I)Representing the distance between the end of the third sub-windsurfing board that is connected to the second sub-windsurfing board and the centre of mass of the third sub-windsurfing board.

5. The synchronous deployment design method of claim 1, further comprising:

6. The synchronous spread design method according to claim 5, wherein the establishing a first constrained counter moment mathematical model between the first, second and third sub-windsurfing boards according to the spread sequence of the first, second and third sub-windsurfing boards comprises:

When the second sub-sailboard is unfolded before the first sub-sailboard, the first sub-sailboard is subjected to a first constraint counter moment mathematical model of the main body of the flat plate satellite, wherein the first constraint counter moment mathematical model is as follows:

Wherein G ₁ represents a first constraint counter moment to which the first sub-windsurfing board is subjected to the body of the flat-plate satellite; alpha represents a first unfolding angle of the first sub-windsurfing board; representing an extrusion elastic coefficient between the first sub-windsurfing board and a main body of the flat-plate satellite;

when the first sub-sailboard is unfolded before the second sub-sailboard, the second sub-sailboard is subjected to a first constraint counter-moment mathematical model of the first sub-sailboard, which is as follows:

wherein G ₂ represents a first constrained counter moment to which the second sub-windsurfing board is subjected; A first spread angular velocity representing the first sub-windsurfing board; /(I) A first spread angular velocity representing the second sub-windsurfing board; /(I)Representing an extrusion elasticity coefficient between the first and second sub-windsurfing boards;

When the second sub-windsurfing board is unfolded before the third sub-windsurfing board, the third sub-windsurfing board is subjected to a first constraint counter-moment mathematical model of the second sub-windsurfing board, wherein the first constraint counter-moment mathematical model is as follows:

wherein G ₃ represents a first constrained counter moment to which the third sub-windsurfing board is subjected; a first spread angular velocity representing the third sub-windsurfing board; /(I) Representing the coefficient of extrusion elasticity between the second and third sub-windsurfing boards.

7. The synchronous spread design method according to claim 5, wherein the establishing a second constrained counter moment mathematical model between the first, second and third sub-windsurfing boards according to the first spread angles of the first, second and third sub-windsurfing boards comprises:

When the first unfolding angle of the first sub-sailboard is larger than 90 degrees, the first sub-sailboard is subjected to the corresponding second constraint counter moment mathematical model of the limiting piece in the hinge assembly, wherein the second constraint counter moment mathematical model is as follows:

Wherein S ₁ represents a second constraint counter moment to which the first sub-windsurfing board is subjected by the limiting member; A first spread angular velocity representing the first sub-windsurfing board; /(I) Representing a coefficient of collision elasticity between the first sub-windsurfing board and a body of the flat-plate satellite;

wherein S ₂ represents a second constraint counter moment to which the second sub-windsurfing board is subjected by the first sub-windsurfing board; a first spread angular velocity representing the second sub-windsurfing board; /(I) Representing a coefficient of bump elasticity between the second sub-windsurfing board and the first sub-windsurfing board;

When the sum of the first unfolding angle of the third sub-sailboard and the first unfolding angle of the second sub-sailboard is greater than 180 degrees, the second constraint counter moment mathematical model of the third sub-sailboard subjected to the second sub-sailboard is as follows:

Wherein S ₃ represents a second constraint counter moment to which the third sub-windsurfing board is subjected; a first spread angular velocity representing the third sub-windsurfing board; /(I) Representing the coefficient of bump elasticity between the third and second sub-windsurfing boards.

8. The synchronous deployment design method of any one of claims 1 to 7, further comprising:

Based on the dynamics equation, the supporting force of the hinge assembly corresponding to the first sub-sailboard is obtained as follows:

The supporting force of the hinge component corresponding to the first sub-sailboard is used for representing the disturbance performance of the first sub-sailboard on the main body of the flat plate satellite when the first sub-sailboard is unfolded.

9. The synchronous deployment design method of claim 8, further comprising:

under the condition that the supporting force of the hinge component corresponding to the first sub-sailboard is obtained, carrying out mechanical analysis on the main body of the flat satellite to obtain a dynamic equation of the main body of the flat satellite;

And performing disturbance analysis on the main body of the flat satellite based on a dynamic equation of the main body of the flat satellite.

10. A synchronous deployment design system for a windsurfing board of a flat-plate satellite, wherein the synchronous deployment design system is for a flat-plate satellite comprising a main body of the flat-plate satellite, a first sub windsurfing board, a second sub windsurfing board, and a third sub windsurfing board, the synchronous deployment design system comprising: a first acquisition unit and a second acquisition unit; wherein,