CN114021352A - Modeling and control method of rope net towing system considering failed satellite flexible sailboard - Google Patents

Modeling and control method of rope net towing system considering failed satellite flexible sailboard Download PDF

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CN114021352A
CN114021352A CN202111311510.7A CN202111311510A CN114021352A CN 114021352 A CN114021352 A CN 114021352A CN 202111311510 A CN202111311510 A CN 202111311510A CN 114021352 A CN114021352 A CN 114021352A
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祁瑞
张扬
姚傅祯
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Beijing Institute of Technology BIT
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Abstract

The invention discloses an optimization modeling and control method of a rope net towing system considering failure satellite flexible sailboards, and belongs to the technical field of aerospace. The invention reasonably simplifies the winding and wrapping of the rope net on the failure satellite into a plurality of sub-ropes attached to the failure satellite, and the attachment points of the sub-ropes can be freely arranged on the central rigid body or the flexible sailboard of the failure satellite, thereby not only avoiding the problem of complex coupling caused by the contact collision of the rope net and the sailboard, but also reflecting the vibration condition of the sailboard in the dragging process. A Keynen equation is adopted to establish a dynamic model, and higher efficiency is achieved during resolving. The present invention also embodies the quality and flexibility of the tether by inserting mass beads in the tether. According to the method, the dynamic model giving consideration to both the resolving efficiency and the model precision is established, the model can approximate the conditions of capturing and wrapping of various rope nets, and analysis of vibration of the flexible sailboard and swinging conditions of the main ropes in the dragging process is facilitated, so that the dragging system is stably controlled.

Description

Rope net towing system modeling and control method considering failure satellite flexible sailboard
Technical Field
The invention relates to a dynamic modeling and control method of a failure satellite rope net towing system, in particular to a dynamic modeling method considering flexible sailboards on a failure satellite and considering system model precision and operation efficiency, which is suitable for a towing and off-orbit clearing process of large space fragments such as a failure satellite with sailboards and belongs to the technical field of aerospace.
Background
With the continuous reduction of the satellite launching cost, more and more satellites are launched into the space, and the satellite launching system provides services for daily life or scientific research work of people. SpaceX corporation further plans to move thousands of minisatellites into space to form "chains," and has completed the launch of over 600 "chain satellites" since 3 months 2021. It is expected that the huge launching volume will generate more failed satellites caused by faults or losses to stay in the orbit, which not only occupies limited space resources and blocks the launching window, but also causes the current in-orbit spacecraft to have to waste more fuel on evasive maneuvers to influence the normal operation of the spacecraft. In addition, if a failed satellite collides with uncontrolled space debris, the resulting secondary debris also tends to induce the "Kessler effect" with irreversible consequences to the space environment. Therefore, under the development trend that a large number of satellites are about to be flooded into space, the speed of the space debris represented by the failed satellites in the orbit is far from enough by only depending on the natural destruction speed, and how to safely and efficiently actively remove the space debris is a problem to be researched and solved urgently.
The currently proposed active clearing method for space debris can be divided into three categories of pushing derailment, resistance increasing derailment and capturing derailment (Liu Huawei, Liu Yong Jian, Tan Chun forest, Liu Yong Qiang, Liu Yong Jian, space debris removal key technical analysis and suggestion [ J ] spacecraft engineering, 2017,26(02): 105-. The catching means comprises a fish fork, a mechanical arm, a rope net and the like. Although the use of netting is less technically sophisticated than robotic arms, the use of netting is a more promising means of clearance for space debris such as dead satellites. Because the posture of the failed satellite is often unstable, the unfolded flexible solar sailboard is reserved, and secondary fragment pollution is easily caused by collision in the capturing process by directly using a mechanical arm and other rigid capturing devices. The rope net is lighter in weight, stronger in compatibility with the shapes of the captured fragments, and farther in capture range, and the flexible rope enables the rope net to directly capture and drag targets with unstable postures, so that rope net dragging is more advantageous.
For the towing phase, building an accurate and efficient kinetic model is the basis for subsequent further studies. If the dynamic model of the towing system is not accurately established, and the deviation between the numerical simulation and the actual situation is large, the follow-up researches such as a control scheme and the like designed on the basis are feared to be difficult to be applied to the actual situation. A dynamic model of a rope net towing failure satellite with sailboards mainly relates to two difficulties, namely how to embody the flexibility of the sailboards in the towing process and how to model a rope net structure on the premise of ensuring the operation efficiency. If a research method in a capture stage is adopted, a fine model of the rope net is directly used for analyzing the dragging process, and the large-area contact collision between the flexible rope net and the satellite sailboard causes the low operation efficiency of the model, thereby being not beneficial to the development of research work. Therefore, a simplified dynamic model of the system needs to be established on the premise of ensuring the accuracy of the model.
Disclosure of Invention
Aiming at the rope net dragging and clearing process of an invalid spacecraft with a flexible sailboard in space and solving the problem that the resolving efficiency and the model precision are difficult to balance caused by complex collision contact between the rope net and the sailboard, the invention aims to provide a rope net dragging system optimization modeling and control method considering the invalid satellite flexible sailboard. Meanwhile, the method adopts a Kane equation to establish a dynamic model, and has higher efficiency in resolving. In addition, the method also embodies the quality and flexibility of the tether by inserting the mass bead points into the tether, and is more close to the characteristics of an actual rope net dragging system. According to the method, a dynamic model which gives consideration to calculation efficiency and model precision is reasonably optimized and established, the model can approximate various rope net capturing and wrapping conditions, analysis of vibration of the flexible sailboard and swinging conditions of the main ropes in the dragging process is facilitated, and therefore the control method is designed according to the obtained dynamic characteristics to enable the dragging system to be stably controlled. The invention can provide a flexible and efficient simulation analysis model for the actual rope net dragging work, and is beneficial to the development of space debris clearing work.
The purpose of the invention is realized by the following technical scheme:
the invention discloses an optimization modeling method of a rope net towing system considering a failure satellite flexible sailboard, which comprises the following steps:
the method comprises the following steps: the method comprises the steps of simplifying a failure satellite with a sailboard and a rope net towing system, establishing a related coordinate system related to a simplified towing model, reasonably simplifying winding and wrapping of the failure satellite by a rope net into a plurality of sub ropes attached to the failure satellite, wherein the attachment points of the sub ropes can be freely arranged on a central rigid body or a flexible sailboard of the failure satellite, so that the problem of complex coupling caused by contact collision of the rope net and the sailboard can be avoided, and the vibration condition of the sailboard in the towing process can be reflected.
The central platform of the tug boat and the failure spacecraft is simplified into a rigid body, the sailboard is simplified into a flexible board with neglected thickness, and the rope net is simplified into a plurality of sub-ropes connected to the failure spacecraft and the sailboard.
In the dynamic modeling of the towing system, the following coordinate systems are required: the equatorial inertial frame of the Earth is denoted as fe(Oexeyeze) Origin of coordinate system OeAt the center of the earth, xeAxis pointing to spring equinox, zeThe axis pointing normal to the equatorial plane to the north pole, yeThe axis satisfies the right hand rule in the equatorial plane. Track coordinate system of center of mass of tug is recorded as fo(Ob1xoyozo) Origin Ob1Taken at the center of mass of the tug, zoThe axis pointing to the centre of the earth, xoThe axis being perpendicular to z in the plane of the trackoAxis, yoThe axis is perpendicular to the centroid orbital plane and is opposite to the angular momentum vector, meeting the right hand rule.
Based on the direction of the track system, a coordinate system of each part of the system is also defined, including a tug body system fb1(Ob1xb1yb1zb1) Failure satellite body system fb2(Ob2xb2yb2zb2) And a body coordinate system f of two sailboardsa1(Oa1xa1ya1za1) And fa2(Oa2xa2ya2za2) The origin of the body coordinate system is taken at each mass center, and the directions of the three axes are consistent with the directions of respective inertia main shafts.
Step two: and establishing a conversion matrix between coordinate systems through the attitude quaternion, and deducing an attitude kinematic equation of the tug and the failed satellite.
By quaternions Qb1And Qb2Indicating tug and failed satellite relative foIs specifically written as
Figure BDA0003341732350000031
Figure BDA0003341732350000032
Then using quaternion to obtain foTo fb1And fb2Is converted into a matrix
Figure BDA0003341732350000033
Figure BDA0003341732350000034
In the formula: i is3Is a third order identity matrix, the superscript "-" defines an operation to convert a 3 × 1 vector into a 3 × 3 matrix, with βb2As an example, βb2＝[βb21,βb22,βb23]Then, then
Figure BDA0003341732350000035
Is expressed as
Figure BDA0003341732350000036
Sailboard body system fa1And fa2From fb2Obtained by simple rotation. System of inertia feAnd a track system foA conversion matrix C betweenoeAnd CeoThe number of tracks is determined. On the basis of the transformation matrix, the transformation relation between any two coordinate systems in the rope net dragging system is obtained by combining the properties of the transformation matrix.
In order to directly use coordinates under different coordinate systems for operation, a space vector is represented by a vector array. Taking inertia as an example, feUnit vector i of three coordinate axese、jeAnd keForm a vector array of the inertial system
Figure BDA0003341732350000037
. And performing dot product operation among the vector arrays of different coordinate systems to obtain a matrix, namely a conversion matrix among the coordinate systems.
First, a corresponding attitude kinematics equation is established for the failed satellite. Definition fb2Relative to foAngular velocity vector of (1) is at fb2The component array below is omegarb2The following expression holds
ωrb2＝[ωrb2x,ωrb2y,ωrb2z]T＝ω2-ωo (4)
In the formula: omegaoIs foRelative to feAngular velocity vector of (1) is at fb2Component of lower, ω2Is fb2Relative to feAngular velocity vector of (1). Thus, the attitude kinematics equation for a failed satellite is written as
Figure BDA0003341732350000038
Similarly, the attitude kinematics equation of the tug is written as
Figure BDA0003341732350000039
Step three: and selecting the generalized velocity corresponding to the translational motion of the towing system, the rotational motion of the towing system and the vibration of the sailboard to calculate the corresponding deviation velocity and the generalized inertia force.
In order to determine the generalized speed of the rope net towing system, any infinitesimal or point position is selected from each part of the system, and the speed of the infinitesimal or point position is recorded as the speed of the infinitesimal or point position in sequence
Figure BDA0003341732350000041
Are respectively
Figure BDA0003341732350000042
In the formula: dm1Is a rigid body of tugboat with arbitrary micro-elements dm2Is any infinitesimal, dm, of fragment central bodya1And dma2Is any infinitesimal element on the two sailboards. Each speed expression is respectively
Figure BDA0003341732350000043
Figure BDA0003341732350000044
Figure BDA0003341732350000045
Figure BDA0003341732350000046
Figure BDA0003341732350000047
Figure BDA0003341732350000048
Wherein R is1、R2、RBAnd REkRespectively showing the mass center of the tug, the mass center of the failure satellite, the rope bifurcation point and the mass bead points at feAn array of lower position components;
Figure BDA0003341732350000049
and
Figure BDA00033417323500000410
respectively representing the position component arrays of the micro elements with any mass under respective systems; omega1And ω2Respectively fb1And fb2Relative to feThe component arrays of angular velocity vectors of (a) under respective systems; q. q.sa1And q isa2The first three-order modal coordinate of the sailboard is fa1And fa2An array of lower components.
Figure BDA00033417323500000411
And
Figure BDA00033417323500000412
as the third order mode at the infinitesimal position at fa1And fa2A 3 x 3 order matrix formed by the lower component arrays; infinitesimal dma1Is elastically displaced by
Figure BDA00033417323500000413
The sailboard on the other side is the same; cb2a1And Cb2a2Is a coordinate transformation matrix.
Finding out common velocity vector from each velocity vector expression to obtain a group of generalized velocity describing system motion condition
Figure BDA00033417323500000418
Is specifically shown as
Figure BDA00033417323500000414
Thus, multiple sets of yaw rates at arbitrary positions in various parts of the system are determined
Figure BDA00033417323500000415
Is composed of
Figure BDA00033417323500000416
And further calculates the generalized inertia force of each order of the towing system as
Figure BDA00033417323500000417
Step four: and solving various external forces borne by the towing system, and solving the generalized main force borne by the system by combining the stress condition and the deflection speed at the position of the force action point.
Neglecting the rest disturbing force of the space, the system mainly relates to three forces of gravity, engine thrust and tether tension in the dragging process. The specific design of the propulsion scheme is not the focus of dynamic research, and only the thrust vector is at foThe component ofpIn numerical simulation, the direction is set to be a constant value and passes through the center of mass of the tug. Mass point m at any position alphaαIs recorded as
Figure BDA0003341732350000051
Then the general expression of gravity is
Figure BDA0003341732350000052
In the formula: mu-3.986X 105km3/s2Is the constant of gravity of the earth, RαIs that the mass point is at feA lower position vector.
Bringing the position and mass of tugs, satellites, sailboards, bifurcation points and ball points into (17), i.e.Calculate feGravity component array of tow boat, failed satellite, sailboard, bifurcation point and mass bead point, marked as
Figure BDA0003341732350000053
And
Figure BDA0003341732350000054
the main rope mass is uniformly distributed on each bead point, each sub rope mass is concentrated at the branch point, the rope section between the mass points adopts a no-mass spring damping model, and feLower arbitrary rope segment vector labThe calculation formula of the tether tension is described as an example. labRepresenting the segment of the tether line pointing from point a to point b, the tension of the tether line at point a being
Figure BDA0003341732350000055
Wherein
Figure BDA0003341732350000056
Is the initial moment of the rope segment labThe length of (2) can be regarded as the original length of the rope section. λ and η are the elastic coefficient and damping coefficient of the tether, respectively, as a function of the cross-sectional area S, Young' S modulus E, damping ratio ζ, linear density ρ, and original length of the strand
Figure BDA0003341732350000057
The following relationship is satisfied:
Figure BDA0003341732350000058
the corresponding tension can be calculated from the individual rope segment vectors by means of the equation (18).
The generalized dominant force in the kahn equation is defined as the dot product of the deflection velocity at the location of the force effect and the force. By using
Figure BDA0003341732350000059
Representing the ith order yaw rate of the element at position alpha,
Figure BDA00033417323500000510
representing the force acting at that location, the ith order generalized primary force for that location is represented as
Figure BDA00033417323500000511
It is not difficult to find from the formula (19) that the thrust force F is calculatedpGravity FgAnd tether tension FtOn the basis, different deviation speeds are selected and substituted to obtain generalized main power corresponding to different motions of the towing system. And then, solving generalized main power of each order in sequence.
Generalized velocity upsilon1Corresponding to the translational motion of the tug, the 1 st order deviation speed at the center of mass of the tug is considered
Figure BDA00033417323500000512
Obtain a generalized main power of
Figure BDA00033417323500000513
Generalized velocity upsilon2、υ3And upsilon(7+k)Corresponding to the translational motion of the failure satellite, the bifurcation point and the bead point, the corresponding generalized main force is shown as formulas (21) to (23).
Figure BDA00033417323500000514
Figure BDA0003341732350000061
Figure BDA0003341732350000062
The calculation results of the generalized main forces corresponding to the translational motion are all in feThe vector array is consistent with the calculation result of the generalized inertia force。
Tugboats and satellite failures also have rotational motion, at which point the generalized yaw rate at the point of force action actually has the physical significance of the moment arm. To avoid calculating the integral of each infinitesimal gravity, the gravity gradient moment of the tug and the satellite central body is directly calculated, including
Figure BDA0003341732350000063
Figure BDA0003341732350000064
Wherein is zob1、zob2Is that
Figure BDA0003341732350000065
Unit vector z ofoAt fb1And fb2Projection of (5), Jb1And Jb2Respectively, are the inertia matrices of the moment of inertia of the tug and chip central body under the respective systems.
Thus, the generalized velocity v4And upsilon5The corresponding generalized principal forces are respectively expressed as
Figure BDA0003341732350000066
Figure BDA0003341732350000067
Generalized velocity upsilon6And upsilon7Corresponding to the vibration of the flexible sailboard, the external force acts on the flexible structure to cause the flexible structure to generate modal force, so the generalized main force in the vibration equation also needs to be added with the structural rigidity term and the structural damping term of the flexible sailboard, and the sum of the two terms is recorded as FqaCorresponding to two sailboards are respectively
Fqa1＝-(Ka1qa1+Ca1υ6) (28)
Fqa2＝-(Ka2qa2+Ca2υ7) (29)
Wherein KaIs the modal stiffness matrix of the sailboard, CaIs its modal damping matrix, the expressions are respectively
Figure BDA0003341732350000068
Figure BDA0003341732350000069
Wherein sigmaa,h＝2πνa,h(h＝1,2,3)，νa,hShowing the h-th order natural frequency, ζ, of the windsurfing boardaRepresenting the modal damping ratio of the windsurfing board. The generalized main force corresponding to the vibration of the windsurfing board can then be expressed as
Figure BDA0003341732350000071
Figure BDA0003341732350000072
So far, expressions of all generalized main forces have been obtained. The general main force in the Kane method summarizes and unifies external forces with different action effects such as concentration force, distribution force, moment and the like into the same expression form, and is beneficial to synchronous calculation of a computer.
Step five: and substituting the solved generalized inertia force and the generalized main force into a Kane equation to obtain the rope net dragging system optimization dynamic model considering the failure satellite flexible sailboard.
The kahn method uses generalized velocity to describe the dynamic behavior of the system. For the selected ith generalized velocity, calculating corresponding generalized inertia force
Figure BDA0003341732350000073
And generalized main power
Figure BDA0003341732350000074
Then a set of Kane equations describing the corresponding motion is obtained as
Figure BDA0003341732350000075
Therefore, the derivation process of the dynamic equation becomes the solving process of the generalized inertia force and the generalized main force. The calculation of the two forces only relates to vector point multiplication, so that the dynamic modeling of the rope net towing system by adopting the Kane method has the advantages of uniform equation form, easy programming realization and high operation efficiency.
Because the equation forms are unified, a group of Kane equations corresponding to each order of generalized velocity can be written into a matrix form, and an optimized dynamic model of the rope net towing system considering the failure satellite flexible sailboard is obtained, wherein the optimized dynamic model includes
Figure BDA0003341732350000076
Wherein the expressions of the terms are specifically
Figure BDA0003341732350000077
Figure BDA0003341732350000078
Figure BDA0003341732350000081
For simplicity, a part of expressions are abbreviated as letter forms, specifically:
Figure BDA0003341732350000082
Figure BDA0003341732350000083
Figure BDA0003341732350000084
Figure BDA0003341732350000085
Sak＝∫(rak+uak)dmak (43)
Figure BDA0003341732350000086
Figure BDA0003341732350000087
Sba1＝∫rba1dma1+Lb2a1Sa1 (46)
Ta1＝∫Na1dma1 (47)
Figure BDA0003341732350000088
Figure BDA0003341732350000091
Figure BDA0003341732350000092
Figure BDA0003341732350000093
the invention discloses an optimization control method of a rope net dragging system considering a failure satellite flexible sailboard, which comprises the following steps of: and D, according to the rope net dragging system optimization dynamic model considering the failure satellite flexible sailboard obtained in the step five, analyzing the evolution conditions of parameters such as the attitude change of the failure satellite, the tension change of the main rope, the swinging condition of the main rope, the vibration condition of the sailboard and the like in the dragging process, applying control force to the tug based on the system characteristics, and realizing the stable control of the dragging system under the condition of ensuring the precision and the operation efficiency so as to realize the complete process of dragging the failure satellite with the sailboard by using the rope net.
Has the advantages that:
1. the rope net towing system optimization modeling and control method considering the failure satellite flexible sailboard reasonably simplifies the precise rope net wound on the failure satellite into a plurality of sub-ropes attached to the failure satellite, avoids the complex coupling problem caused by the contact collision of the rope net and the sailboard, improves the resolving efficiency, and can simulate different rope net wrapping conditions through the change of the sub-rope connecting points, thereby improving the universality of the model.
2. The rope net towing system optimization modeling and control method considering the failure satellite flexible sailboard considers the flexible sailboard on the failure satellite and the connection condition of the rope net and the flexible sailboard, can analyze the vibration of the sailboard in the towing process compared with a common rope net towing system, and is beneficial to avoiding secondary fragment pollution caused by sailboard breakage.
3. According to the rope net dragging system optimization modeling and control method considering the failure satellite flexible sailboard, the plurality of mass bead points are added into the main rope, the swinging condition in the main rope dragging process can be reflected, the extra moment caused by swinging can be calculated, and the rope net dragging system optimization modeling and control method has higher precision compared with a common rope net dragging model.
4. The rope net towing system optimization modeling and control method considering the failure satellite flexible sailboard disclosed by the invention has the advantages that the Kenn equation is used for deducing the dynamic model, the translation motion, the rotation motion and the sailboard vibration of the system are unified into the form of the Kenn equation, the computer programming operation is facilitated, and therefore, the method has higher resolving efficiency compared with the traditional modeling method.
5. The rope net dragging system optimization modeling and control method considering the failure satellite flexible sailboard has higher precision than that of the modeling of a common dragging system under the condition of ensuring the model resolving efficiency, can simulate the actual rope net dragging situation more truly, lays a foundation for subsequent related research, and is beneficial to the development of actual space debris clearing work.
Drawings
FIG. 1 is a rope net towing system optimization modeling and control method considering a failure satellite flexible sailboard, disclosed by the invention;
FIG. 2 is a schematic view of a rope net towing system according to the present invention;
FIG. 3 is a schematic diagram of different analysis conditions in the example of the present invention, in which FIG. 3a) is a case of sub-rope connection when a bifurcation point of a small mesh rope network is farther, FIG. 3b) is a case of sub-rope connection when a bifurcation point of a small mesh rope network is closer, and FIG. 3c) is a case of sub-rope connection when a bifurcation point of a large mesh rope network is closer;
FIG. 4 is a schematic diagram of attitude changes of a failed satellite under different conditions in an example of the present invention, wherein FIG. 4a) is a case where the failed satellite has an initial angular velocity in a pitch direction, and wherein FIG. 4b) is a case where the failed satellite has an initial angular velocity in a yaw direction;
FIG. 5 is a schematic diagram of the tension of the main rope under different working conditions in the example of the present invention, in which FIG. 5a) is the case where the failed satellite has the initial angular velocity in the pitch direction, and FIG. 5b) is the case where the failed satellite has the initial angular velocity in the yaw direction;
FIG. 6 is a schematic diagram of the main line sway under different operating conditions in an example of the present invention, where FIG. 6a) is a case where the failed satellite has an initial angular velocity in the pitch direction, and where FIG. 6b) is a case where the failed satellite has an initial angular velocity in the yaw direction;
FIG. 7 is a schematic diagram of the magnitude of the extra moment generated by the main rope swinging under different working conditions in the example of the present invention, wherein FIG. 7a) is the case that the failed satellite has the initial angular velocity in the pitch direction, and wherein FIG. 7b) is the case that the failed satellite has the initial angular velocity in the yaw direction;
fig. 8 is a schematic diagram of the vibration of the sailboard under different conditions in the example of the invention, wherein fig. 8a) is the case where the failed satellite has the initial angular velocity in the pitch direction, and fig. 8b) is the case where the failed satellite has the initial angular velocity in the yaw direction.
Detailed Description
For a better understanding of the objects and advantages of the present invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings and examples.
In order to verify the feasibility of the method, a tugboat with the mass of 2500kg runs on a geosynchronous orbit, the semi-major axis of the orbit is 42164km, the eccentricity is 0.008, the inclination angle of the orbit is 7 degrees, the ascension angle of a rising intersection is 50 degrees, the amplitude angle of an approach point is 230 degrees, and the true approach point angle is 170 degrees. The distance between the failure satellite with the mass of 1800kg and the tug is 80m, the failure satellite is provided with a flexible sailboard and has self-rotation, the edge length of a central rigid body is 4m, and the length and the width of a single sailboard are 6.7m and 2.4 m. Three sub-rope connection modes are set to simulate different rope net wrapping and winding modes, as shown in the attached figure 2. The case where the failed satellite at the initial time had a rotational speed of 5 °/s in each of the yaw and pitch directions was analyzed.
Figure BDA0003341732350000101
Figure BDA0003341732350000111
As shown in fig. 1, the method for optimally modeling and controlling a rope net towing system considering a failed satellite flexible windsurfing board disclosed in this embodiment includes the following steps:
the method comprises the following steps: the method comprises the steps of simplifying a failure satellite with a sailboard and a rope net towing system, establishing a related coordinate system related to a simplified towing model, reasonably simplifying winding and wrapping of the failure satellite by a rope net into a plurality of sub ropes attached to the failure satellite, wherein the attachment points of the sub ropes can be freely arranged on a central rigid body or a flexible sailboard of the failure satellite, so that the problem of complex coupling caused by contact collision of the rope net and the sailboard can be avoided, and the vibration condition of the sailboard in the towing process can be reflected.
The central platform of the tug boat and the failure spacecraft is simplified into a rigid body, the sailboard is simplified into a flexible board with neglected thickness, and the rope net is simplified into a plurality of sub-ropes connected to the failure spacecraft and the sailboard.
In the dynamic modeling of the towing system, the following coordinate systems are required: the equatorial inertial frame of the Earth is denoted as fe(Oexeyeze) Origin of coordinate system OeAt the center of the earth, xeAxis pointing to spring equinox, zeThe axis pointing normal to the equatorial plane to the north pole, yeThe axis satisfies the right hand rule in the equatorial plane. Track coordinate system of center of mass of tug is recorded as fo(Ob1xoyozo) Origin Ob1Taken at the center of mass of the tug, zoThe axis pointing to the centre of the earth, xoThe axis being perpendicular to z in the plane of the trackoAxis, yoThe axis is perpendicular to the centroid orbital plane and is opposite to the angular momentum vector, meeting the right hand rule.
Based on the direction of the track system, a coordinate system of each part of the system is also defined, including a tug body system fb1(Ob1xb1yb1zb1) Failure satellite body system fb2(Ob2xb2yb2zb2) And a body coordinate system f of two sailboardsa1(Oa1xa1ya1za1) And fa2(Oa2xa2ya2za2) The origin of the body coordinate system is taken at each mass center, and the directions of the three axes are consistent with the directions of respective inertia main shafts.
Step two: and establishing a conversion matrix between coordinate systems through the attitude quaternion, and deducing an attitude kinematic equation of the tug and the failed satellite.
By quaternions Qb1And Qb2Indicating tug and failed satellite relative foIs specifically written as
Figure BDA0003341732350000112
Figure BDA0003341732350000113
Then using quaternion to obtain foTo fb1And fb2Is converted into a matrix
Figure BDA0003341732350000114
Figure BDA0003341732350000115
In the formula: i is3Is a third order identity matrix, the superscript "-" defines an operation to convert a 3 × 1 vector into a 3 × 3 matrix, with βb2As an example, βb2＝[βb21,βb22,βb23]Then, then
Figure BDA0003341732350000116
Is expressed as
Figure BDA0003341732350000121
Sailboard body system fa1And fa2From fb2Obtained by simple rotation. System of inertia feAnd a track system foA conversion matrix C betweenoeAnd CeoThe number of tracks is determined. On the basis of the transformation matrix, the transformation relation between any two coordinate systems in the rope net dragging system is obtained by combining the properties of the transformation matrix.
In order to directly use coordinates under different coordinate systems for operation, a space vector is represented by a vector array. Taking inertia as an example, feThree coordinates ofAxial unit vector ie、jeAnd keForm a vector array of the inertial system
Figure BDA0003341732350000122
. And performing dot product operation among the vector arrays of different coordinate systems to obtain a matrix, namely a conversion matrix among the coordinate systems.
First, a corresponding attitude kinematics equation is established for the failed satellite. Definition fb2Relative to foAngular velocity vector of (1) is at fb2The component array below is omegarb2The following expression holds
ωrb2＝[ωrb2x,ωrb2y,ωrb2z]T＝ω2-ωo (4)
In the formula: omegaoIs foRelative to feAngular velocity vector of (1) is at fb2Component of lower, ω2Is fb2Relative to feAngular velocity vector of (1). Thus, the attitude kinematics equation for a failed satellite is written as
Figure BDA0003341732350000123
Similarly, the attitude kinematics equation of the tug is written as
Figure BDA0003341732350000124
Step three: and selecting the generalized velocity corresponding to the translational motion of the towing system, the rotational motion of the towing system and the vibration of the sailboard to calculate the corresponding deviation velocity and the generalized inertia force.
In order to determine the generalized speed of the rope net towing system, any infinitesimal or point position is selected from each part of the system, and the speed of the infinitesimal or point position is recorded as the speed of the infinitesimal or point position in sequence
Figure BDA0003341732350000125
Are respectively
Figure BDA0003341732350000126
In the formula: dm1Is a rigid body of tugboat with arbitrary micro-elements dm2Is any infinitesimal, dm, of fragment central bodya1And dma2Is any infinitesimal element on the two sailboards. Each speed expression is respectively
Figure BDA0003341732350000131
Figure BDA0003341732350000132
Figure BDA0003341732350000133
Figure BDA0003341732350000134
Figure BDA0003341732350000135
Figure BDA0003341732350000136
Wherein R is1、R2、RBAnd REkRespectively showing the mass center of the tug, the mass center of the failure satellite, the rope bifurcation point and the mass bead points at feAn array of lower position components; r isdm1、rdm2、rdma1And rdma2Respectively representing the position component arrays of the micro elements with any mass under respective systems; omega1And ω2Respectively fb1And fb2Relative to feThe component arrays of angular velocity vectors of (a) under respective systems; q. q.sa1And q isa2In the first three-order mode of sailboardThe coordinate is at fa1And fa2An array of lower components.
Figure BDA0003341732350000137
And
Figure BDA0003341732350000138
as the third order mode at the infinitesimal position at fa1And fa2A 3 x 3 order matrix formed by the lower component arrays; infinitesimal dma1Is elastically displaced by
Figure BDA0003341732350000139
The sailboard on the other side is treated in the same way; cb2a1And Cb2a2Is a coordinate transformation matrix.
Finding out common velocity vector from each velocity vector expression to obtain a group of generalized velocity describing system motion condition
Figure BDA00033417323500001310
Is particularly shown as
Figure BDA00033417323500001311
Thus, multiple sets of yaw rates at arbitrary positions in various parts of the system are determined
Figure BDA00033417323500001312
Is composed of
Figure BDA00033417323500001313
And further calculates the generalized inertia force of each order of the towing system as
Figure BDA00033417323500001314
Step four: and solving various external forces borne by the towing system, and solving the generalized main force borne by the system by combining the stress condition and the deflection speed at the position of the force action point.
Neglecting the rest disturbing force of the space, the system mainly relates to three forces of gravity, engine thrust and tether tension in the dragging process. The specific design of the propulsion scheme is not the focus of dynamic research, and only the thrust vector is at foThe component ofpIn numerical simulation, the direction is set to be a constant value and passes through the center of mass of the tug. Mass point m at any position alphaαIs recorded as
Figure BDA00033417323500001315
The general expression for gravity is
Figure BDA00033417323500001316
In the formula: mu-3.986X 105km3/s2Is the constant of gravity of the earth, RαIs that the mass point is at feA lower position vector.
Bringing the position and mass of the tug, satellite, sailboard, bifurcation point and ball point into (17), i.e. calculating feGravity component array of tow boat, failed satellite, sailboard, bifurcation point and mass bead point, marked as
Figure BDA0003341732350000141
And
Figure BDA0003341732350000142
the main rope mass is uniformly distributed on each bead point, each sub rope mass is concentrated at the branch point, the rope section between the mass points adopts a no-mass spring damping model, and feLower arbitrary rope segment vector labThe calculation formula of the tether tension is described as an example. labRepresenting the segment of the tether line pointing from point a to point b, the tension of the tether line at point a being
Figure BDA0003341732350000143
Wherein
Figure BDA0003341732350000144
Is the initial moment of the rope segment labThe length of (2) can be regarded as the original length of the rope section. λ and η are the elastic coefficient and damping coefficient of the tether, respectively, as a function of the cross-sectional area S, Young' S modulus E, damping ratio ζ, linear density ρ, and original length of the strand
Figure BDA0003341732350000145
The following relationship is satisfied:
Figure BDA0003341732350000146
the corresponding tension can be calculated from the individual rope segment vectors by means of the equation (18).
The generalized dominant force in the kahn equation is defined as the dot product of the deflection velocity at the location of the force effect and the force. By using
Figure BDA0003341732350000147
Representing the ith order yaw rate of the element at position alpha,
Figure BDA0003341732350000148
representing the force acting at that location, the ith order generalized primary force for that location is represented as
Figure BDA0003341732350000149
It is not difficult to find from the formula (19) that the thrust force F is calculatedpGravity FgAnd tether tension FtOn the basis, different deviation speeds are selected and substituted to obtain generalized main power corresponding to different motions of the towing system. And then, solving generalized main power of each order in sequence.
Generalized velocity upsilon1Corresponding to the translational motion of the tug, the 1 st order deviation speed at the center of mass of the tug is considered
Figure BDA00033417323500001410
Obtain a generalized main power of
Figure BDA00033417323500001411
Generalized velocity upsilon2、υ3And upsilon(7+k)Corresponding to the translational motion of the failure satellite, the bifurcation point and the bead point, the corresponding generalized main force is shown as formulas (21) to (23).
Figure BDA00033417323500001412
Figure BDA00033417323500001413
Figure BDA00033417323500001414
The calculation results of the generalized main forces corresponding to the translational motion are all in feThe vector array below is consistent with the calculation result of the generalized inertia force.
Tugboats and satellite failures also have rotational motion, at which point the generalized yaw rate at the point of force action actually has the physical significance of the moment arm. To avoid calculating the integral of each infinitesimal gravity, the gravity gradient moment of the tug and the satellite central body is directly calculated, including
Figure BDA0003341732350000151
Figure BDA0003341732350000152
Wherein is zob1、zob2Is that
Figure BDA0003341732350000153
Unit vector z ofoAt fb1And fb2Projection of (5), Jb1And Jb2Respectively, are the inertia matrices of the moment of inertia of the tug and chip central body under the respective systems.
Thus, the generalized velocity v4And upsilon5The corresponding generalized principal forces are respectively expressed as
Figure BDA0003341732350000154
Figure BDA0003341732350000155
Generalized velocity upsilon6And upsilon7Corresponding to the vibration of the flexible sailboard, the external force acts on the flexible structure to cause the flexible structure to generate modal force, so the generalized main force in the vibration equation also needs to be added with the structural rigidity term and the structural damping term of the flexible sailboard, and the sum of the two terms is recorded as FqaCorresponding to two sailboards are respectively
Fqa1＝-(Ka1qa1+Ca1υ6) (28)
Fqa2＝-(Ka2qa2+Ca2υ7) (29)
Wherein KaIs the modal stiffness matrix of the sailboard, CaIs its modal damping matrix, the expressions are respectively
Figure BDA0003341732350000156
Figure BDA0003341732350000157
Wherein sigmaa,h＝2πνa,h(h＝1,2,3)，νa,hShowing the h-th order natural frequency, ζ, of the windsurfing boardaRepresenting the modal damping ratio of the windsurfing board. The generalized main force corresponding to the vibration of the windsurfing board can then be expressed as
Figure BDA0003341732350000158
Figure BDA0003341732350000159
So far, expressions of all generalized main forces have been obtained. The general main force in the Kane method summarizes and unifies external forces with different action effects such as concentration force, distribution force, moment and the like into the same expression form, and is beneficial to synchronous calculation of a computer.
Step five: and substituting the solved generalized inertia force and the generalized main force into a Kane equation to obtain the rope net dragging system optimization dynamic model considering the failure satellite flexible sailboard.
The kahn method uses generalized velocity to describe the dynamic behavior of the system. For the selected ith generalized velocity, calculating corresponding generalized inertia force
Figure BDA0003341732350000161
And generalized main power
Figure BDA0003341732350000162
Then a set of Kane equations describing the corresponding motion is obtained as
Figure BDA0003341732350000163
Therefore, the derivation process of the dynamic equation becomes the solving process of the generalized inertia force and the generalized main force. The calculation of the two forces only relates to vector point multiplication, so that the dynamic modeling of the rope net towing system by adopting the Kane method has the advantages of uniform equation form, easy programming realization and high operation efficiency.
Because the equation forms are unified, a group of Kane equations corresponding to each order of generalized velocity can be written into a matrix form, and an optimized dynamic model of the rope net towing system considering the failure satellite flexible sailboard is obtained, wherein the optimized dynamic model includes
Figure BDA0003341732350000164
Wherein the expressions of the terms are specifically
Figure BDA0003341732350000165
Figure BDA0003341732350000166
Figure BDA0003341732350000171
For simplicity, a part of expressions are abbreviated as letter forms, specifically:
Figure BDA0003341732350000172
Figure BDA0003341732350000173
Figure BDA0003341732350000174
Figure BDA0003341732350000175
Sak＝∫(rak+uak)dmak (43)
Figure BDA0003341732350000176
Figure BDA0003341732350000177
Sba1＝∫rba1dma1+Lb2a1Sa1 (46)
Ta1＝∫Na1dma1 (47)
Figure BDA0003341732350000178
Figure BDA0003341732350000181
Figure BDA0003341732350000182
Figure BDA0003341732350000183
the invention discloses an optimization control method of a rope net dragging system considering a failure satellite flexible sailboard, which comprises the following steps of: and D, according to the rope net dragging system optimization dynamic model considering the failure satellite flexible sailboard obtained in the step five, analyzing the evolution conditions of parameters such as the attitude change of the failure satellite, the tension change of the main rope, the swinging condition of the main rope, the vibration condition of the sailboard and the like in the dragging process, applying control force to the tug based on the system characteristics, and realizing the stable control of the dragging system under the condition of ensuring the precision and the operation efficiency so as to realize the complete process of dragging the failure satellite with the sailboard by using the rope net.
And analyzing the change condition of each parameter by combining the drawing. Under the initial condition of different angular velocities, the attitude change of a failed satellite wrapped by three ways in a towed state within 1000 seconds is shown in fig. 4. Simulation results verify that the rope net structure has certain rotation damping characteristics, so that the failure satellite cannot rotate in a single direction but swings back and forth within a certain angle range. When the failed satellite rotates in the pitching direction at the initial moment, the amplitude of the rotation angle of the failed satellite in the 1 st rope net wrapping mode is about twice larger than that of the failed satellite in other modes at the initial towing stage, and then the rotation angle gradually converges. When the failed satellite rotates in the yaw direction, the attitude change conditions of the satellite under different wrapping modes are almost the same.
If the part of the rope net and the failed satellite which are wound together is regarded as a whole, the tension of the sub-ropes is changed into internal force, the connection mode of the sub-ropes does not influence the whole movement of the system, and the main rope is the key influencing the dragging process. The change situation of the tension of the main rope given in fig. 5 illustrates this point, and the change situation of the tension of the main rope is not significantly different in different sub-rope connection situations, and the main rope and the sub-rope alternately become larger and smaller with similar amplitude and period. Through the change situation of the tension, the situation and the characteristics of the rope net dragging process can be found: under the combined action of tugboat thrust and main rope tension, the tugboat and the failed satellite alternately move away from proximity, so that the system advances like a "yo-yo ball" until the failed satellite is towed to the grave orbit.
To analyze the swing of the main rope in the case of the 2 nd wrapping mode, the positions of the bead points in each second during towing are observed by looking at the line AB from the point a on the tug, and the distribution diagram is shown in fig. 6. It is clear from the figure that the amplitude of the oscillation of the ball point in the middle is always the largest at different initial rotation speeds of the satellite, i.e. the oscillation in the middle of the tether is the most obvious. Compared with a single spring damping model without adding the ball points, the movement of the ball points influences the stress condition of the bifurcation point, so that the tug and the failed satellite can be subjected to extra moment caused by the swinging of the main rope. Fig. 7 shows the variation of the magnitude of this part of the additional moment in the case of 3 bead points. It can be seen from the figure that the extra moment suffered by the failed satellite is far larger than that of the tug, which shows that the change of the tension of the subline has relatively obvious influence on the attitude of the failed satellite after the swing of the main line disturbs the motion state of the bifurcation point. In the early stage of towing, the change of the magnitude of the extra moment received by the failed satellite with different spin directions is similar, and the maximum value is about 25N m. But as the towing progresses, the additional moment experienced by the failed satellite turning in the yaw direction is significantly greater, which is consistent with the hunting case of fig. 6. FIG. 8 shows the vibration of the sailboard, and simulation results show that the amplitude of the vibration of the sailboard in the first wrapping mode is significantly larger regardless of the spin in any direction. And no matter which kind of parcel mode, the rotation of inefficacy satellite in the yaw direction all can let the windsurfing board vibration condition more complicated.
The calculation results of the examples prove that the rope net towing system optimization modeling and control method considering the failure satellite flexible sailboard, disclosed by the embodiment, can analyze the dynamic characteristics of the towing system under higher operation precision and operation efficiency, and can simulate different rope net winding and wrapping conditions, so that control is carried out according to the characteristics of different wrapping modes, and the effective rope net towing process of the failure satellite with the sailboard is realized.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (7)

1.一种考虑失效卫星柔性帆板的绳网拖曳系统优化建模方法，其特征在于：包括如下步骤，1. a rope net towing system optimization modeling method considering a failure satellite flexible windsurfing board is characterized in that: comprise the steps, 步骤一：对带帆板的失效卫星和绳网拖曳系统进行简化，建立简化拖曳模型所涉及的相关坐标系；将绳网对失效卫星的缠绕包裹合理地简化为若干附着在失效卫星上的子绳，子绳附着点能够自由地设置在失效卫星的中心刚体或是柔性帆板上，既能够避免绳网和帆板接触碰撞带来的复杂耦合问题，又能够体现出拖曳过程中帆板的振动情况；Step 1: Simplify the failed satellite with windsurfing board and the rope net towing system, and establish the relevant coordinate system involved in the simplified towing model; reasonably simplify the winding and wrapping of the failed satellite by the rope net into a number of children attached to the failed satellite. The attachment point of the rope and the sub-rope can be freely set on the central rigid body of the failed satellite or on the flexible sailboard, which can not only avoid the complex coupling problem caused by the contact and collision between the rope net and the sailboard, but also reflect the stability of the sailboard during the towing process. vibration; 步骤二：通过姿态四元数建立坐标系之间的转换矩阵，推导拖船和失效卫星的姿态运动学方程；Step 2: Establish the transformation matrix between coordinate systems through attitude quaternions, and derive the attitude kinematic equations of the tugboat and the failed satellite; 步骤三：选取对应拖曳系统平动运动、拖曳系统转动运动和帆板振动的广义速率，求出相应的偏速度和广义惯性力；Step 3: Select the generalized rate corresponding to the translational motion of the drag system, the rotational motion of the drag system and the vibration of the windsurfing board, and obtain the corresponding deflection velocity and generalized inertial force; 步骤四：求解拖曳系统所受各种外力，并结合受力情况和力作用点位置处的偏速度求出系统所受到的广义主动力；Step 4: Solve the various external forces on the drag system, and obtain the generalized active force on the system based on the force conditions and the deflection velocity at the point where the force is applied; 步骤五：将所求出的广义惯性力和广义主动力代入凯恩方程，得到考虑失效卫星柔性帆板的绳网拖曳系统优化动力学模型。Step 5: Substitute the obtained generalized inertial force and generalized active force into the Kane equation, and obtain the optimal dynamic model of the rope-net towing system considering the failed satellite flexible sailboard. 2.如权利要求1所述的一种考虑失效卫星柔性帆板的绳网拖曳系统优化建模方法，其特征在于：步骤一实现方法为，2. a kind of rope net towing system optimization modeling method considering a failed satellite flexible sailboard as claimed in claim 1, is characterized in that: step 1 realization method is, 将拖船和失效航天器中心平台简化为刚体，帆板简化为忽略厚度的柔性板，绳网简化为若干连接在失效航天器和帆板上的子绳；The center platform of the tugboat and the failed spacecraft is simplified as a rigid body, the windsurfing board is simplified as a flexible board whose thickness is ignored, and the rope net is simplified as a number of strands connected to the failed spacecraft and the windsurfing board; 对拖曳系统进行动力学建模的过程中，需要用到以下坐标系：地心赤道惯性坐标系记为fe(Oexeyeze)，坐标系原点Oe位于地球中心,xe轴指向春分点,ze轴垂直于赤道平面指向北极,ye轴在赤道平面内满足右手定则；拖船质心轨道坐标系记为fo(Ob1xoyozo)，原点Ob1取在拖船质心，zo轴指向地球中心，xo轴在轨道平面内垂直于zo轴，yo轴垂直于质心轨道平面与角动量矢量反向，满足右手定则；In the process of dynamic modeling of the drag system, the following coordinate systems need to be used: the earth-centered equatorial inertial coordinate system is denoted as f e (O e x e y e z e ), the origin of the coordinate system O e is located at the center of the earth, x The e -axis points to the vernal equinox, the z- e -axis points to the north pole perpendicular to the equatorial plane, and the y- e -axis satisfies the right-hand rule in the equatorial plane; the orbital coordinate system of the tugboat barycenter is denoted as f o (O b1 x o y o z o ), the origin O b1 Take it at the center of mass of the tugboat, the z o axis points to the center of the earth, the x o axis is perpendicular to the z o axis in the orbital plane, and the y o axis is perpendicular to the orbital plane of the center of mass and the angular momentum vector is opposite, which satisfies the right-hand rule; 基于轨道系的方向，还要定义系统各部分本体坐标系，包括拖船本体系fb1(Ob1xb1yb1zb1)、失效卫星本体系fb2(Ob2xb2yb2zb2)以及两块帆板的本体坐标系fa1(Oa1xa1ya1za1)和fa2(Oa2xa2ya2za2)，本体坐标系的原点均取在各质心，三轴方向与各自惯性主轴方向一致。Based on the orientation of the orbital system, the body coordinate system of each part of the system is also defined, including the tugboat body system f b1 (O b1 x b1 y b1 z b1 ), the failed satellite body system f b2 (O b2 x b2 y b2 z b2 ) and The body coordinate systems f a1 (O a1 x a1 y a1 z a1 ) and f a2 (O a2 x a2 y a2 z a2 ) of the two windsurfing boards, the origin of the body coordinate system is taken at each center of mass, and the three-axis direction is related to each The directions of the inertial axes are the same. 3.如权利要求2所述的一种考虑失效卫星柔性帆板的绳网拖曳系统优化建模方法，其特征在于：步骤二实现方法为，3. a kind of rope net towing system optimization modeling method considering a failed satellite flexible sailboard as claimed in claim 2, is characterized in that: step 2 realization method is, 用四元数Qb1和Qb2表示拖船和失效卫星相对fo的姿态，具体写为
Figure FDA0003341732340000011
Figure FDA0003341732340000012
于是利用四元数得到从fo到fb1和fb2的转换矩阵The quaternions Q b1 and Q b2 are used to represent the relative attitudes of the tugboat and the failed satellite relative to f o , which are specifically written as
Figure FDA0003341732340000011
Figure FDA0003341732340000012
Then use the quaternion to get the conversion matrix from f o to f b1 and f b2
Figure FDA0003341732340000013
Figure FDA0003341732340000013
Figure FDA0003341732340000021
Figure FDA0003341732340000021
 式中：I3是三阶单位矩阵，上标“～”定义了一种将3×1矢量转换为3×3矩阵的运算，以βb2为例，βb2＝[βb21,βb22,βb23]，则
Figure FDA0003341732340000022
的表达式为In the formula: I 3 is the third-order unit matrix, and the superscript “～” defines an operation to convert a 3×1 vector into a 3×3 matrix. Taking β b2 as an example, β b2 =[β b21 ,β b22 , β b23 ], then
Figure FDA0003341732340000022
The expression is
Figure FDA0003341732340000023
Figure FDA0003341732340000023
 帆板本体系fa1和fa2由fb2通过简单旋转得到；惯性系fe和轨道系fo之间的转换矩阵Coe和Ceo通过轨道根数求出；在以上转换矩阵的基础上，结合转换矩阵之间的性质，便得到绳网拖曳系统中任意两个坐标系之间的转换关系；The main system f a1 and f a2 of the windsurfing board are obtained by simple rotation of f b2 ; the transformation matrices C oe and C eo between the inertial system f e and the orbital system f o are obtained by the orbital roots; on the basis of the above transformation matrix , combined with the properties of the transformation matrices, the transformation relationship between any two coordinate systems in the rope-net towing system can be obtained; 为方便直接使用不同坐标系下的坐标进行运算，用“矢阵”来表示空间矢量；以惯性系为例，fe的三个坐标轴单位矢量ie、je和ke组成惯性系矢阵
Figure FDA0003341732340000024
不同坐标系矢阵之间进行点积运算，所得到的矩阵即为坐标系之间的转换矩阵；In order to directly use the coordinates in different coordinate systems for operation, the "vector array" is used to represent the space vector; taking the inertial system as an example, the unit vectors i e , j e and k e of the three coordinate axes of f e form the inertial system vector array
Figure FDA0003341732340000024
The dot product operation is performed between vector matrices of different coordinate systems, and the obtained matrix is the transformation matrix between coordinate systems; 首先针对失效卫星建立相应的姿态运动学方程；定义fb2相对于fo的角速度矢量在fb2下的分量列阵为ωrb2，有以下表达式成立Firstly, the corresponding attitude kinematics equation is established for the failed satellite; the component array of the angular velocity vector of f b2 relative to f o under f b2 is defined as ω rb2 , and the following expressions are established ωrb2＝[ωrb2x,ωrb2y,ωrb2z]T＝ω2-ωo (4)ω rb2 =[ω rb2x ,ω rb2y ,ω rb2z ] T =ω 2 -ω o (4) 式中：ωo为fo相对于fe的角速度矢量在fb2下的分量，ω2为fb2相对于fe的角速度矢量；于是，失效卫星的姿态运动学方程写为In the formula: ω o is the component of the angular velocity vector of f o relative to f e under f b2 , and ω 2 is the angular velocity vector of f b2 relative to f e ; therefore, the attitude kinematics equation of the failed satellite is written as
Figure FDA0003341732340000025
Figure FDA0003341732340000025
 同理，拖船的姿态运动学方程写为Similarly, the attitude kinematics equation of the tugboat is written as
Figure FDA0003341732340000026
Figure FDA0003341732340000026
 4.如权利要求3所述的一种考虑失效卫星柔性帆板的绳网拖曳系统优化建模方法，其特征在于：步骤三实现方法为，4. a kind of rope net towing system optimization modeling method considering a failed satellite flexible sailboard as claimed in claim 3, is characterized in that: step 3 realization method is, 为确定绳网拖曳系统的广义速率，从系统各部分选取任意微元或点位置，将它们的速度依次记为
Figure FDA0003341732340000027
分别是In order to determine the generalized velocity of the rope-net towing system, any micro-element or point position is selected from each part of the system, and their velocity is recorded as
Figure FDA0003341732340000027
respectively
Figure FDA0003341732340000031
Figure FDA0003341732340000031
 式中：dm1是拖船刚体任意微元，dm2是碎片中心体任意微元，dma1和dma2是两块帆板上的任意微元；各速度表达式分别为In the formula: dm 1 is an arbitrary micro-element of the rigid body of the tugboat, dm 2 is an arbitrary micro-element of the fragment center body, dm a1 and dm a2 are arbitrary micro-elements on the two sailboards; the speed expressions are respectively:
Figure FDA0003341732340000032
Figure FDA0003341732340000032
Figure FDA0003341732340000033
Figure FDA0003341732340000033
Figure FDA0003341732340000034
Figure FDA0003341732340000034
Figure FDA0003341732340000035
Figure FDA0003341732340000035
Figure FDA0003341732340000036
Figure FDA0003341732340000036
Figure FDA0003341732340000037
Figure FDA0003341732340000037
 其中，R1、R2、RB和REk分别表示拖船质心、失效卫星质心、绳系分叉点以及各质量珠点在fe下的位置分量列阵；
Figure FDA0003341732340000038
和
Figure FDA0003341732340000039
分别表示任意质量微元在各自本体系下的位置分量列阵；ω1和ω2分别fb1和fb2相对于fe的角速度矢量在各自本体系下的分量列阵；qa1和qa2为帆板的前三阶模态坐标在fa1和fa2下的分量列阵；
Figure FDA00033417323400000310
和
Figure FDA00033417323400000311
为微元位置处的三阶模态在fa1和fa2下的分量列阵组成的3×3阶矩阵；微元dma1的弹性位移为
Figure FDA00033417323400000312
另一侧帆板同理；Cb2a1和Cb2a2是坐标转换矩阵；Among them, R 1 , R 2 , R B and R Ek represent the center of mass of the tugboat, the center of mass of the failed satellite, the bifurcation point of the tether and the positional component array of each mass bead point under f e , respectively;
Figure FDA0003341732340000038
and
Figure FDA0003341732340000039
Respectively represent the position component arrays of micro-elements of arbitrary mass under their respective systems; ω 1 and ω 2 are the component arrays of the angular velocity vectors of f b1 and f b2 relative to f e under their respective systems; q a1 and q a2 is the component array of the first three modal coordinates of the windsurfing board under f a1 and f a2 ;
Figure FDA00033417323400000310
and
Figure FDA00033417323400000311
is a 3×3-order matrix composed of the component arrays of the third-order mode at the position of the micro-element under f a1 and f a2 ; the elastic displacement of the micro-element dm a1 is
Figure FDA00033417323400000312
The same is true for the other side of the windsurfing board; C b2a1 and C b2a2 are coordinate transformation matrices; 从各个速度向量表达式中找到共同的速度矢量，便得到一组描述系统运动情况的广义速率
Figure FDA00033417323400000313
具体表示为Find the common velocity vector from each velocity vector expression, and then get a set of generalized velocity describing the motion of the system
Figure FDA00033417323400000313
Specifically expressed as
Figure FDA00033417323400000314
Figure FDA00033417323400000314
 于是求出系统各个部分中任意位置处的多组偏速度
Figure FDA00033417323400000315
为Then, the multiple sets of deflection velocities at any position in each part of the system are obtained
Figure FDA00033417323400000315
for
Figure FDA00033417323400000316
Figure FDA00033417323400000316
 进而求出拖曳系统的各阶广义惯性力为Then, the generalized inertial forces of each order of the drag system are obtained as
Figure FDA00033417323400000317
Figure FDA00033417323400000317
 5.如权利要求4所述的一种考虑失效卫星柔性帆板的绳网拖曳系统优化建模方法，其特征在于：步骤四实现方法为，5. a kind of rope net towing system optimization modeling method considering a failed satellite flexible sailboard as claimed in claim 4, is characterized in that: step 4 realization method is, 忽略空间其余扰动力，拖曳过程中系统主要涉及重力、发动机推力和系绳张力这三种力；推进方案的具体设计不是动力学研究的重点，仅将推力矢量在fo下的分量记为Fp，在数值仿真中设置为常值且方向经过拖船质心；将任意位置α处质量点mα的重力记为
Figure FDA0003341732340000041
则重力的通用表达式为Ignoring the remaining disturbance forces in space, the towing process mainly involves three forces: gravity, engine thrust and tether tension; the specific design of the propulsion scheme is not the focus of dynamic research, only the component of the thrust vector under f o is denoted as F p , set as a constant value in the numerical simulation and the direction passes through the center of mass of the tugboat; the gravity of the mass point m α at any position α is recorded as
Figure FDA0003341732340000041
Then the general expression for gravity is
Figure FDA0003341732340000042
Figure FDA0003341732340000042
 式中：μ＝3.986×105km3/s2是地球引力常数，Rα是该质量点在fe下的位置矢量；In the formula: μ=3.986×10 5 km 3 /s 2 is the earth’s gravitational constant, R α is the position vector of the mass point under f e ; 将拖船、卫星、帆板、分叉点和珠点的位置和质量带入式(17)，即计算出fe下拖船、失效卫星、帆板、分叉点和质量珠点的重力分量列阵，记为
Figure FDA0003341732340000043
和
Figure FDA0003341732340000044
Bring the positions and masses of the tugboat, satellite, windsurfing board, bifurcation point and bead point into equation (17), that is, calculate the gravity component column of the tugboat, failed satellite, windsurfing board, bifurcation point and mass bead point under f e array, denoted as
Figure FDA0003341732340000043
and
Figure FDA0003341732340000044
 将主绳质量均布在各珠点上，各子绳质量集中在分叉点处，质量点之间的绳段采用无质量弹簧阻尼模型，以fe下任意绳段矢量lab为例说明系绳张力的计算式；lab表示从a点指向b点的系绳绳段，a点所受系绳张力为The mass of the main rope is evenly distributed on each bead point, the mass of each sub-rope is concentrated at the bifurcation point, the rope segment between the mass points adopts a massless spring damping model, and the vector l ab of any rope segment under f e is used as an example to illustrate The calculation formula of the tether tension; l ab represents the tether rope segment from point a to point b, and the tether tension at point a is
Figure FDA0003341732340000045
Figure FDA0003341732340000045
 其中
Figure FDA0003341732340000046
是初始时刻绳段lab的长度，可视为绳段的原长；λ和η分别是系绳的弹性系数和阻尼系数，它们与绳的横截面积S、杨氏模量E、阻尼比ζ、质量线密度ρ以及绳段原长
Figure FDA0003341732340000047
满足以下关系：
Figure FDA0003341732340000048
Figure FDA0003341732340000049
通过式(18)能够根据各绳段矢量计算出相应张力；in
Figure FDA0003341732340000046
is the length of the rope segment l ab at the initial moment, which can be regarded as the original length of the rope segment; λ and η are the elastic coefficient and damping coefficient of the tether, respectively, which are related to the cross-sectional area S of the rope, the Young's modulus E, and the damping ratio. ζ, mass linear density ρ and original length of rope
Figure FDA0003341732340000047
Satisfy the following relationship:
Figure FDA0003341732340000048
Figure FDA0003341732340000049
According to formula (18), the corresponding tension can be calculated according to the vector of each rope segment; 凯恩方程中广义主动力定义为力作用位置的处偏速度与力的点积；用
Figure FDA00033417323400000410
表示在位置α处微元的第i阶偏速度，
Figure FDA00033417323400000411
表示作用于该位置的力，则该位置的第i阶广义主动力表示为The generalized active force in the Kane equation is defined as the dot product of the deflection velocity and the force at the position where the force acts;
Figure FDA00033417323400000410
represents the i-th order deflection velocity of the element at position α,
Figure FDA00033417323400000411
represents the force acting on the position, then the i-th order generalized active force at the position is expressed as
Figure FDA00033417323400000412
Figure FDA00033417323400000412
 根据式(19)不难发现，在计算出推力Fp、重力Fg与系绳张力Ft的基础上，选择不同的偏速度代入即可得到拖曳系统不同运动所对应的广义主动力；接下来依次求取各阶广义主动力；According to formula (19), it is not difficult to find that on the basis of calculating the thrust force F p , the gravity F g and the tether tension F t , the generalized main force corresponding to the different motions of the drag system can be obtained by selecting different deflection velocities to substitute; Next, find the generalized active force of each order in turn; 广义速率υ1对应拖船的平动运动，考虑拖船质心处的1阶偏速度
Figure FDA00033417323400000413
得到广义主动力为The generalized rate υ 1 corresponds to the translational motion of the tugboat, considering the first-order deflection velocity at the center of mass of the tugboat
Figure FDA00033417323400000413
The generalized active force is obtained as
Figure FDA00033417323400000414
Figure FDA00033417323400000414
 广义速率υ2、υ3和υ(7+k)对应失效卫星、分叉点和珠点的平动运动，相应的广义主动力如式(21)～式(23)所示；The generalized rates υ 2 , υ 3 and υ (7+k) correspond to the translational motions of the failed satellites, bifurcation points and bead points, and the corresponding generalized active forces are shown in equations (21) to (23);
Figure FDA00033417323400000415
Figure FDA00033417323400000415
Figure FDA00033417323400000416
Figure FDA00033417323400000416
Figure FDA0003341732340000051
Figure FDA0003341732340000051
 平动运动相应的广义主动力计算结果都是在fe下的矢量列阵，与广义惯性力的计算结果一致；The corresponding generalized active force calculation results of translational motion are all vector arrays under f e , which are consistent with the generalized inertial force calculation results; 拖船和失效卫星还具有转动运动，此时力作用点处的广义偏速度实际上具有力臂的物理意义；为了避免计算各微元重力的积分，直接计算拖船和卫星中心体的重力梯度力矩，有The tugboat and the failed satellite also have rotational motion, and the generalized deflection velocity at the point of force application actually has the physical meaning of the force arm. Have
Figure FDA0003341732340000052
Figure FDA0003341732340000052
Figure FDA0003341732340000053
Figure FDA0003341732340000053
 其中，为zob1、zob2是
Figure FDA0003341732340000054
中的单位矢量zo在fb1和fb2中的投影，Jb1和Jb2分别是拖船和碎片中心体的转动惯量在各自本体系下的惯量矩阵；where z ob1 and z ob2 are
Figure FDA0003341732340000054
The projection of the unit vector z o in f b1 and f b2 , J b1 and J b2 are the inertia matrices of the moment of inertia of the tugboat and the debris center body under their respective systems; 于是，广义速率υ4和υ5对应的广义主动力分别表示为Therefore, the generalized active forces corresponding to the generalized rates υ 4 and υ 5 are expressed as
Figure FDA0003341732340000055
Figure FDA0003341732340000055
Figure FDA0003341732340000056
Figure FDA0003341732340000056
 广义速率υ6和υ7对应柔性帆板的振动，而外力作用在柔性结构上会使其产生模态力，因此振动方程中的广义主动力还需加入柔性帆板的结构刚度项与结构阻尼项，将两项之和记为Fqa，对应两块帆板分别为The generalized rates υ 6 and υ 7 correspond to the vibration of the flexible sailboard, and the external force acting on the flexible structure will cause it to generate modal force. Therefore, the generalized active force in the vibration equation needs to be added to the structural stiffness term and structural damping of the flexible sailboard. term, the sum of the two items is recorded as F qa , and the corresponding two windsurfing boards are respectively Fqa1＝-(Ka1qa1+Ca1υ6) (28)F qa1 =-(K a1 q a1 +C a1 υ 6 ) (28) Fqa2＝-(Ka2qa2+Ca2υ7) (29)F qa2 =-(K a2 q a2 +C a2 υ 7 ) (29) 其中Ka是帆板的模态刚度矩阵，Ca是其模态阻尼矩阵，表达式分别为where Ka is the modal stiffness matrix of the windsurfing board, and C a is its modal damping matrix, and the expressions are
Figure FDA0003341732340000057
Figure FDA0003341732340000057
Figure FDA0003341732340000058
Figure FDA0003341732340000058
 其中σa,h＝2πνa,h(h＝1,2,3)，νa,h表示帆板的第h阶自然频率，ζa表示帆板的模态阻尼比；于是帆板振动对应的广义主动力可表示为where σ a,h =2πν a,h (h=1,2,3), ν a,h represents the h-th order natural frequency of the windsurfing board, and ζa represents the modal damping ratio of the windsurfing board; then the windsurfing board vibration corresponds to The generalized active force of can be expressed as
Figure FDA0003341732340000061
Figure FDA0003341732340000061
Figure FDA0003341732340000062
Figure FDA0003341732340000062
 至此，已得到所有广义主动力的表达式；凯恩方法中广义主动力将集中力、分布力、力矩等不同作用效果的外力归纳统一为相同的表达形式，有利于计算机进行同步计算。So far, the expressions of all generalized active forces have been obtained; in the Kane method, the generalized active forces generalize and unify the external forces with different effects, such as concentrated force, distributed force, and moment, into the same expression form, which is beneficial for computers to perform synchronous calculation. 6.如权利要求5所述的一种考虑失效卫星柔性帆板的绳网拖曳系统优化建模方法，其特征在于：步骤五实现方法为，6. a kind of rope net towing system optimization modeling method considering a failed satellite flexible sailboard as claimed in claim 5, it is characterized in that: step 5 realization method is, 凯恩方法运用广义速率来描述系统的动力学行为；对于选定的第i个广义速率，计算出相应的广义惯性力
Figure FDA0003341732340000063
和广义主动力
Figure FDA0003341732340000064
便得到一组描述相应运动的凯恩方程为The Kane method uses generalized rates to describe the dynamic behavior of the system; for the selected ith generalized rate, the corresponding generalized inertial force is calculated
Figure FDA0003341732340000063
and generalized driving force
Figure FDA0003341732340000064
Then a set of Kane equations describing the corresponding motion is obtained as
Figure FDA0003341732340000065
Figure FDA0003341732340000065
 于是，动力学方程的推导过程就变为了广义惯性力和广义主动力的求解过程；所述两种力的计算都只涉及矢量点乘，所以采用凯恩方法对绳网拖曳系统进行动力学建模具有方程形式统一，易于编程实现并且运算效率高的优点；Therefore, the derivation process of the dynamic equation becomes the solution process of the generalized inertial force and the generalized active force; the calculation of the two forces only involves the vector point product, so the Kane method is used to carry out the dynamic construction of the rope-net towing system. The mold has the advantages of unified equation form, easy programming and high computing efficiency; 由于方程形式统一，能够将各阶广义速率所对应的一组凯恩方程写为矩阵形式，即得到考虑失效卫星柔性帆板的绳网拖曳系统优化动力学模型，有Due to the unified equation form, a set of Kane equations corresponding to each order of generalized rates can be written in matrix form, that is, the optimal dynamic model of the rope-net towing system considering the failed satellite flexible sailboard can be obtained.
Figure FDA0003341732340000066
Figure FDA0003341732340000066
 式中各项的表达式具体为The expression of each term in the formula is specifically
Figure FDA0003341732340000067
Figure FDA0003341732340000067
Figure FDA0003341732340000068
Figure FDA0003341732340000068
Figure FDA0003341732340000071
Figure FDA0003341732340000071
 其中为了简便，将部分表达式简写为字母形式，具体为：For simplicity, some expressions are abbreviated as letters, specifically:
Figure FDA0003341732340000072
Figure FDA0003341732340000072
Figure FDA0003341732340000073
Figure FDA0003341732340000073
Figure FDA0003341732340000074
Figure FDA0003341732340000074
Figure FDA0003341732340000075
Figure FDA0003341732340000075
 Sak＝∫(rak+uak)dmak (43)S ak =∫(r ak +u ak )dm ak (43)
Figure FDA0003341732340000076
Figure FDA0003341732340000076
Figure FDA0003341732340000077
Figure FDA0003341732340000077
 Sba1＝∫rba1dma1+Lb2a1Sa1 (46)S ba1 = ∫r ba1 dm a1 +L b2a1 S a1 (46) Ta1＝∫Na1dma1 (47)T a1 =∫N a1 dm a1 (47)
Figure FDA0003341732340000078
Figure FDA0003341732340000078
Figure FDA0003341732340000081
Figure FDA0003341732340000081
Figure FDA0003341732340000082
Figure FDA0003341732340000082
Figure FDA0003341732340000083
Figure FDA0003341732340000083
 7.一种考虑失效卫星柔性帆板的绳网拖曳系统优化控制方法，如权利要求1、2、3、4、5或6所述的一种考虑失效卫星柔性帆板的绳网拖曳系统优化建模方法的步骤一至步骤五，其特征在于：还包括步骤六，根据步骤五得到的考虑失效卫星柔性帆板的绳网拖曳系统优化动力学模型，分析拖曳过程中失效卫星的姿态变化、主绳张力变化、主绳摆动情况、帆板振动情况等参数的演变情况，基于系统特性对拖船施加控制力，即能够在保证精度和运算效率的情况下，实现拖曳系统得到稳定控制，进而实现利用绳网拖曳带帆板失效卫星的完整过程。7. A rope net towing system optimization control method considering a failed satellite flexible windsurfing board, a rope net towing system optimization considering a failed satellite flexible windsurfing board as described in claim 1, 2, 3, 4, 5 or 6 Steps 1 to 5 of the modeling method are characterized in that: it further includes step 6, according to the optimized dynamic model of the rope net towing system considering the flexible sailboard of the failed satellite obtained in step 5, analyzing the attitude change of the failed satellite during the towing process, main The evolution of parameters such as rope tension changes, main rope swings, sailboard vibrations, etc., based on the system characteristics to exert control force on the tugboat, that is, the towing system can be stably controlled under the condition of ensuring accuracy and computing efficiency, and then realize the utilization of The complete process of rope net towing a failed satellite with a windsurfing board.
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