CN114021352A - Rope net towing system modeling and control method considering failure satellite flexible sailboard - Google Patents
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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
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
Then using quaternion to obtain foTo fb1And fb2Is converted into a matrix
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
Is expressed as
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
. 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
Similarly, the attitude kinematics equation of the tug is written as
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
Are respectively
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
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;
and
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.
And
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
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
Is specifically shown as
Thus, multiple sets of yaw rates at arbitrary positions in various parts of the system are determined
Is composed of
And further calculates the generalized inertia force of each order of the towing system as
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
Then the general expression of gravity is
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
And
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
Wherein
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
The following relationship is satisfied:
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
Representing the ith order yaw rate of the element at position alpha,
representing the force acting at that location, the ith order generalized primary force for that location is represented as
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
Obtain a generalized main power of
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).
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
Wherein is zob1、zob2Is that
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
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
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
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
And generalized main power
Then a set of Kane equations describing the corresponding motion is obtained as
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
Wherein the expressions of the terms are specifically
For simplicity, a part of expressions are abbreviated as letter forms, specifically:
Sak＝∫(rak+uak)dmak (43)
Sba1＝∫rba1dma1+Lb2a1Sa1 (46)
Ta1＝∫Na1dma1 (47)
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.
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
Then using quaternion to obtain foTo fb1And fb2Is converted into a matrix
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
Is expressed as
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
. 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
Similarly, the attitude kinematics equation of the tug is written as
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
Are respectively
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
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.
And
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
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
Is particularly shown as
Thus, multiple sets of yaw rates at arbitrary positions in various parts of the system are determined
Is composed of
And further calculates the generalized inertia force of each order of the towing system as
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
The general expression for gravity is
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
And
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
Wherein
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
The following relationship is satisfied:
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
Representing the ith order yaw rate of the element at position alpha,
representing the force acting at that location, the ith order generalized primary force for that location is represented as
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
Obtain a generalized main power of
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).
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
Wherein is zob1、zob2Is that
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
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
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
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
And generalized main power
Then a set of Kane equations describing the corresponding motion is obtained as
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
Wherein the expressions of the terms are specifically
For simplicity, a part of expressions are abbreviated as letter forms, specifically:
Sak＝∫(rak+uak)dmak (43)
Sba1＝∫rba1dma1+Lb2a1Sa1 (46)
Ta1＝∫Na1dma1 (47)
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. A rope net dragging system optimization modeling method considering failure satellite flexible sailboards is characterized by comprising the following steps: comprises the following steps of (a) carrying out,
the method comprises the following steps: simplifying a failed satellite with a sailboard and a rope net towing system, and establishing a related coordinate system related to a simplified towing model; the winding and wrapping of the failure satellite by the rope net are reasonably simplified into a plurality of sub-ropes attached to the failure satellite, and 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 dragging process can be reflected;
step two: establishing a conversion matrix between coordinate systems through the attitude quaternion, and deducing an attitude kinematic equation of the tug and the failure satellite;
step three: selecting generalized rates corresponding to the translation motion of the towing system, the rotation motion of the towing system and the vibration of the sailboard, and solving corresponding deviation speed and generalized inertia force;
step four: solving various external forces borne by the towing system, and solving generalized main force borne by the system by combining the stress condition and the deflection speed at the position of the force action point;
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.
2. The optimized modeling method for the rope net towing system considering the failed satellite flexible sailboard as claimed in claim 1, is characterized in that: the first implementation method comprises the following steps of,
simplifying a tugboat and a center platform of the failed spacecraft into rigid bodies, simplifying a sailboard into a flexible board with neglected thickness, and simplifying a rope net into a plurality of sub-ropes connected to the failed 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 meets 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 plane of the centroid orbit and is opposite to the angular momentum vector, so that the right-hand rule is met;
based on railThe direction of the track system, and the body coordinate system of each part of the system, including the 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.
3. The optimized modeling method for the rope net towing system considering the failed satellite flexible sailboard as claimed in claim 2, characterized in that: the second step is realized by the method that,
by quaternions Qb1And Qb2Indicating tug and failed satellite relative foIs specifically written as
Then using quaternion to obtain foTo fb1And fb2Is converted into a matrix
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
Is expressed as
Sailboard body system fa1And fa2From fb2Obtained by simple rotation; system of inertia feAnd a track system foA conversion matrix C betweenoeAnd CeoCalculating according to the number of the tracks; 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 facilitate the direct use of 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
Carrying out dot product operation among the vector arrays of different coordinate systems, wherein the obtained matrix is a conversion matrix among the coordinate systems;
firstly, establishing a corresponding attitude kinematics equation aiming at a failure 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 feThe angular velocity vector of (a); thus, the attitude kinematics equation for a failed satellite is written as
Similarly, the attitude kinematics equation of the tug is written as
4. The optimized modeling method for the rope net towing system considering the failed satellite flexible sailboard as claimed in claim 3, characterized in that: the third step is to realize the method as follows,
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
Are respectively
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
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;
and
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 fa2A component array of down;
and
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
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
Is particularly shown as
Thus, multiple sets of yaw rates at arbitrary positions in various parts of the system are determined
Is composed of
And further calculates the generalized inertia force of each order of the towing system as
5. The optimized modeling method for the rope net towing system considering the failed satellite flexible sailboard as claimed in claim 4, wherein: the implementation method of the fourth step is that,
neglecting other disturbing forces in the space, wherein 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 the numerical simulation, a constant value is set and the direction passes through the center of mass of the tug; mass point m at any position alphaαIs recorded as
The general expression for gravity is
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
And
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 illustrated 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
Wherein
Is the initial moment of the rope segment labThe length of the rope 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
The following relationship is satisfied:
the corresponding tension can be calculated according to each rope segment vector through the formula (18);
the generalized main force in the Kane equation is defined as the dot product of the deflection speed of the force action position and the force; by using
Representing the ith order yaw rate of the element at position alpha,
representing the force acting at that location, the ith order generalized primary force for that location is represented as
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; 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
Obtain a generalized main power of
Generalized velocity upsilon2、υ3And upsilon(7+k)Corresponding failure satelliteThe corresponding generalized main force is shown as formulas (21) to (23) through the translation motion of the bifurcation point and the bead point;
the calculation results of the generalized main forces corresponding to the translational motion are all in feThe lower vector array is consistent with the calculation result of the generalized inertia force;
the tugboat and the failure satellite also have rotary motion, and the generalized deflection speed at the force action point actually has the physical significance of a force 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
Wherein is zob1、zob2Is that
Unit vector z ofoAt fb1And fb2Projection of (5), Jb1And Jb2Inertia matrixes of the rotational inertia of the tugboat and the debris central body under respective systems are respectively provided;
thus, the generalized velocity v4And upsilon5The corresponding generalized principal forces are respectively expressed as
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
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
So far, expressions of all generalized main forces are 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.
6. The optimized modeling method for the rope net towing system considering the failed satellite flexible sailboard as claimed in claim 5, wherein: the fifth step is to realize that the method is that,
the Keynen method uses a generalized velocity to describe the dynamic behavior of the system; for the selected ith generalized velocity, calculating corresponding generalized inertia force
And generalized main power
Then a set of Kane equations describing the corresponding motion is obtained as
Therefore, the derivation process of the dynamic equation is changed into 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 a 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
Wherein the expressions of the terms are specifically
For simplicity, a part of expressions are abbreviated as letter forms, specifically:
Sak＝∫(rak+uak)dmak (43)
Sba1＝∫rba1dma1+Lb2a1Sa1 (46)
Ta1＝∫Na1dma1 (47)
7. the method for optimally controlling the rope net towing system considering the failed satellite flexible sailboard comprises the steps one to five of the method for optimally modeling the rope net towing system considering the failed satellite flexible sailboard according to the claims 1, 2, 3, 4, 5 or 6, and is characterized in that: and step six, optimizing a dynamic model according to the rope net dragging system considering the failure satellite flexible sailboard obtained in the step five, analyzing the evolution conditions of parameters such as the attitude change, the main rope tension change, the main rope swinging condition, the sailboard vibration condition and the like of the failure satellite in the dragging process, applying control force to the tugboat based on the system characteristics, namely realizing that the dragging system is stably controlled under the condition of ensuring the precision and the operation efficiency, and further realizing the complete process of dragging the failure satellite with the sailboard by using the rope net.
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