CN109975838B - Space-earth equivalent experimental method of spin closed loop tethered formation system - Google Patents
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Abstract
The invention discloses a space-earth equivalent experimental method of a spin closed loop rope system formation system, which comprises the following steps: 1. according to Newton's second law, under the non-inertial reference system, establishing a spin closed loop tether formation system running in an orbit plane, and obtaining a kinetic equation of the closed loop tether formation system; 2. establishing a ground equivalent experimental system and obtaining a kinetic equation of the ground equivalent experimental system; 3. providing parameter conditions for converting a dynamic equation of a closed loop rope system formation system and a dynamic equation of a ground equivalent experiment system into the same dimensionless dynamic equation so as to ensure consistency of a space-earth system; 4. and converting the ground equivalent experimental system into a static balance system by utilizing the darebel principle. The method can verify the spin stability of the space tethered formation system through a ground experiment method, is simple and easy to operate, and has higher reliability than a computer numerical simulation result.
Description
Technical Field
The invention belongs to the technical field of spacecraft flight, and particularly relates to a space-earth equivalent experimental method of a spin closed loop tethered formation system.
Background
In order to maintain a stable spatial configuration, in-orbit tethered formation systems typically require the use of spin-induced centrifugal forces to maintain a stable spatial attitude, which have been extensively studied by researchers. Obviously, the on-orbit experiment is the most reliable method for verifying various theories, but has huge cost; therefore, people turn their eyes to the ground equivalent experiment. Numerous studies have also been conducted by students on ground verification experiments of tethered formation systems. For example, nakaya et al developed a ground test system consisting of a satellite simulator, a tether reel, and a control mechanism that successfully achieved the ground deployment process of a three-body tether formation system. In order to make the space structure detection plan of the U.S. space navigation bureau of sub-millimeter level more technically feasible, chung et al build a tethered satellite experiment platform, can complete the experiment of synchronous position maintenance, contact and repositioning of tethered satellites, and has obtained a great deal of experimental data. Wen Hao and the like develop a set of air floatation experiment platform which can simulate the position and the attitude control of a tethered formation system in a track plane. Bindra et al also developed a set of tethered formation test systems based on the ground environment, which completed a comparison of the dynamic behavior of the ground test system and the on-track system by scaling the parameters of the on-track system, but the ground test was difficult to implement accurate control strategies due to complex perturbation factors. Olivieri et al specifically designed a tether release mechanism for use in tether deployment experiments in a ground tether formation system.
Previous studies have shown that the ground experiments of the multi-body tethered formation system are mainly focused on aspects of tethered deployment, pose control, mechanism design and the like, and the spin stability analysis of the tethered formation system is mainly focused on the numerical method level. Because the mass center eccentricity problem of the rigid body of the satellite simulator exists, no suitable method for effective experimental verification exists at present.
Disclosure of Invention
The invention aims to: aiming at the problems in the prior art, a space-earth equivalent experimental method of a spin closed loop rope formation system is provided, corresponding parameter conditions are given through dimensionless transformation, and the dynamic equations of the ground equivalent system and a space rope formation system are transformed into the same dimensionless dynamic equation, so that the equivalence of the ground experimental system and the space system is ensured; and then the spinning system is converted into a static balance system by utilizing the Darby principle, so that experimental uncertainty caused by factors such as rigid body eccentricity of a satellite simulator is avoided.
The technical scheme is as follows: in order to solve the technical problems, the invention provides a space-earth equivalent experimental method of a spin closed loop rope system, which is characterized by comprising the following steps:
(1) According to Newton's second law, under the non-inertial reference system, establishing a spin closed loop tether formation system running in an orbit plane, and obtaining a kinetic equation of the closed loop tether formation system;
(2) Establishing a ground equivalent experimental system and obtaining a kinetic equation of the ground equivalent experimental system;
(3) The dynamic equation of the closed loop rope system formation system obtained in the step (1) and the dynamic equation of the ground equivalent experimental system obtained in the step (2) are converted into the parameter condition of the same dimensionless dynamic equation so as to ensure the consistency of the heaven-earth system;
(4) And converting the ground equivalent experimental system into a static balance system by utilizing the darebel principle.
Further, in the step (1), according to newton's second law, under a non-inertial reference system, a spin closed loop formation system operating in an orbit plane is established, and a specific step of obtaining a kinetic equation of the closed loop formation system is as follows:
build a spin closed loop tethered formation system operating in an orbital plane consisting of n satellites M 1 、M 2 、…、M n The elastic ropes are sequentially connected through n elastic ropes to form a closed loop; wherein the mass of n satellites is m respectively 1 、m 2 、…、m n The original lengths of the n tethers are L 0 The rigidity is EA, and the pressure can not be borne by the EA only by the tensile force; the closed loop system is set to perform spin rotation around the mass center o of the system at the angular velocity omega, and the system always performs ground-winding movement on a fixed circumferential track at the angular velocity omega;
establishing a non-inertial reference system o-xy taking a system centroid o as an origin, wherein an x-axis points in a direction opposite to the movement of the centroid o, and a y-axis points to the system centroid from the earth centroid; the kinetic equation of the closed loop tether formation system is therefore expressed as:
wherein subscripts j, k represent two satellites adjacent to satellite i;
representing the first derivative of time t,
representing the second derivative, μ, of time t E Represents the gravitational constant, r oE Representing the distance between the centroid of the earth and the origin o of the reference frame, wherein the variable sign delta ij(k) Is defined as
Wherein r is i ＝(x i ,y i ) T Representing the position vector of satellite i in the non-inertial frame of reference, subscript ij (k) represents ij or ik.
Further, the specific steps of establishing a ground equivalent experimental system and obtaining a kinetic equation of the ground equivalent experimental system are as follows:
constructing a ground equivalent experiment system aiming at the space tethered formation system established in the step (1); sequentially connecting n satellite simulators on a smooth experiment platform by using n flexible experiment tethers to form a closed loop; the satellite simulator is suspended on an experimental platform through an air floatation method, and a ground equivalent spin closed loop rope system formation system is built; the n satellite simulators are respectively marked as a satellite simulator 1, a satellite simulator 2, a satellite simulator … and a satellite simulator n; then, a ground reference system o is constructed g -x g y g Wherein the origin of coordinates o g Centroid, x located in formation system g Axes and y g The axis is parallel to the experimental platform;
let m be gi And L g The mass of the satellite simulator i and the experimental tether length omega are respectively g And omega g Respectively the equivalent angular velocity of the ground and the spin angular velocity t g For the ground experiment time, a kinetic equation of a ground experiment system is established
Wherein the method comprises the steps of
Represents the experimental time t to the ground g First derivative is calculated>
Represents the experimental time t to the ground g Second derivative, mu Eg Is equivalent to the gravitational constant, r g Is the origin o of the ground reference system g Equivalent gravity from groundThe distance of the field, the tether tension is expressed as
Wherein, the subscript x or y of the pulling force P represents the pulling force in the x direction or the y direction, and the subscript ij or ik represents the pulling force of the satellite simulator j or the satellite simulator k at two adjacent ends to the satellite simulator i;
in the formula (3), the tension term is directly provided by the self tension of the tether in the experiment, and the other items are obtained by equivalent air injection force.
Further, in the step (3), the parameter condition that the dynamic equation of the closed loop rope system formation system in the step (1) and the dynamic equation of the ground equivalent experiment system in the step (2) are transformed to the same dimensionless dynamic equation is obtained, so as to ensure the consistency of the heaven and earth systems, and the specific steps are as follows:
introduction of dimensionless time τ=Ω·t and reference length l r ＝(μ E /Ω 2 ) 1/3 The following dimensionless transformation is carried out on the space rope formation system equation (1)
Obtaining a dimensionless kinetic equation of the system
Wherein "'" means to obtain a first derivative of the dimensionless time τ and "" means to obtain a second derivative of the dimensionless time τ,
and->
As dimensionless coordinate variables, there are the following parameters
The non-dimensional length, the non-dimensional tether stiffness, the non-dimensional tether length and the non-dimensional winding angular velocity are respectively; wherein the subscript ij (k) represents ij or ik;
then, the following dimensionless transformation is carried out on the dynamics equation (3) of the ground experiment system
At this time, the following parameter conditions should be satisfied
r g ＝r oE L g /(μ E /Ω 2 ) 1/3 (11)
The dynamics equation (3) of the ground experiment system can be converted into the dimensionless dynamics equation (6) which is the same as the space on-orbit system, namely the equivalent space tethered formation system of the ground experiment system can be completely used.
Further, the specific steps of converting the ground equivalent experimental system into the static balance system by using the darebel principle in the step (4) are as follows:
inertial force
Introducing the static equilibrium equation into a ground equivalent system dynamics equation (3) to obtain the following static equilibrium equation
Wherein the inertial force component is written as
The position vector of the satellite simulator in the ground reference system is written as r gi ＝(x gi ,y gi ) T The method comprises the steps of carrying out a first treatment on the surface of the Since the ground closed loop tethered formation experiment system is at a constant angular velocity omega g Spin movement is performed so that the inertial force is expressed as +.>
During the experiment, it can be determined that the spin motion of the system is unstable as long as one satellite simulator is moving or the tether remains relaxed.
Compared with the prior art, the invention has the advantages that:
the method can verify the spin stability of the space tethered formation system through a ground experiment method, is simple and easy to operate, and has higher reliability than a computer numerical simulation result.
Drawings
FIG. 1 is a schematic diagram of a spin closed loop tether formation system operating in an orbital plane in an exemplary embodiment;
FIG. 2 is a schematic diagram of a ground equivalent experimental system in an embodiment;
FIG. 3 is an equivalent schematic diagram of centrifugal force compensation in an embodiment;
FIG. 4 is a schematic diagram of a first experimental set of trajectories of a satellite simulator in an embodiment;
FIG. 5 is a plot of inter-simulator distance as a function of v for a first set of experiments in an exemplary embodiment g A graph of the variation of (2);
FIG. 6 is a plot of the angle between tethers as a function of v for a first set of experiments in accordance with an exemplary embodiment g A graph of the variation of (2);
FIG. 7 is a schematic diagram of a motion trajectory of a satellite 1 for a first set of experiments in an embodiment;
FIG. 8 is a schematic diagram of a motion trajectory of the satellite 2 for a first set of experiments in an embodiment;
FIG. 9 is a schematic diagram of a motion trajectory of the satellite 3 for a first set of experiments in an embodiment;
FIG. 10 is a diagram illustrating the variation of inter-satellite distance with v for a first set of experiments in accordance with an exemplary embodiment;
FIG. 11 is a schematic diagram of a second set of experimental satellite simulator trajectories in an embodiment;
FIG. 12 is an enlarged view of the trajectory of the satellite simulator 1 of the second set of experiments in an embodiment;
FIG. 13 is a plot of inter-simulator distance as a function of v for a second set of experiments in an exemplary embodiment g A graph of the variation of (2);
fig. 14 is a plot of angle between tethers versus v for a second set of experiments in accordance with an exemplary embodiment g A graph of the variation of (2);
FIG. 15 is a schematic diagram of the trajectories of three satellites in a second set of experiments in an embodiment;
fig. 16 is a diagram illustrating the variation of the inter-satellite distance with v for the second set of experiments in accordance with the present invention.
Detailed Description
The invention is further elucidated below in connection with the drawings and the detailed description. The described embodiments of the invention are only some, but not all, embodiments of the invention. Based on the embodiments of the present invention, other embodiments that may be obtained by those of ordinary skill in the art without making any inventive effort are within the scope of the present invention.
As shown in FIG. 1, the present invention contemplates a spin closed loop tethered formation system that runs in the orbital plane. The system consists of three satellites M 1 、M 2 And M 3 Is formed and sequentially connected through three elastic ropes to form a closed loop. The mass of the three satellites is m respectively 1 、m 2 And m 3 The original lengths of the three tethers are all L 0 The rigidity is EA, and the pressure can not be born only by the tensile force. The closed loop system is set to spin around the system centroid o at angular velocity omegaAnd always moves around a fixed circumferential track at an angular velocity Ω.
A non-inertial reference frame o-xy is established with the system centroid o as the origin, as shown in fig. 1, where the x-axis points in a direction opposite to the centroid o motion and the y-axis points from the earth centroid to the system centroid.
Under this non-inertial reference frame o-xy we can write the dynamics equation of the above closed loop tethered formation system
In the method, in the process of the invention,
representing the first derivative of time t,/>
Representing the second derivative of time t, the three satellites are rotated by indices i, j and k, i.e., j=2, k=3 when i=1; when i=2, j=3, k=1; when i=3, j=1, k=2. Mu (mu) E Represents the gravitational constant, r oE Representing the distance between the centroid of the earth and the origin o of the reference system, the variable symbol delta ij(k) Is defined as
Wherein r is i ＝(x i ,y i ) T Representing the position vector of satellite i in the non-inertial frame of reference, subscript ij (k) represents ij or ik.
The dimensionless time τ=Ω·t and the reference length l are now introduced r ＝(μ E /Ω 2 ) 1/3 The following dimensionless transformation is carried out on the space rope formation system equation (1)
Dimensionless kinetic equation of readily available system
Wherein "'" means to obtain a first derivative of the dimensionless time τ and "" means to obtain a second derivative of the dimensionless time τ,
and->
As dimensionless coordinate variables, there are the following parameters
The non-dimensional length, the non-dimensional tether stiffness, the non-dimensional tether length and the non-dimensional angular velocity are respectively. Wherein the subscript ij (k) represents ij or ik.
Now, a ground equivalent experimental system is constructed for the above space system, as shown in fig. 2. Three satellite simulators were sequentially connected with three flexible test tethers on a smooth test platform to form a closed loop. The simulator is suspended on an experimental platform through an air floatation method, so that a ground equivalent spin closed loop tether formation system is created. Meanwhile, for convenience of discussion of the equivalent experiment system, three satellite simulators are respectively denoted as a satellite simulator 1, a satellite simulator 2 and a satellite simulator 3. Then, a ground reference system o is reconstructed g -x g y g Wherein the origin of coordinates o g Centroid, x located in formation system g Axes and y g The axis was parallel to the experimental platform.
Let m be gi And L g The mass of the satellite simulator i and the experimental tether length omega are respectively g And omega g Respectively the equivalent angular velocity of the ground and the spin angular velocity t g For ground experimentsBetween them, the dynamics equation of the ground experiment system is easy to be established
Wherein,,
represents the experimental time t to the ground g First derivative is calculated>
Represents the experimental time t to the ground g Second derivative, mu Eg Is equivalent to the gravitational constant, r g Is the origin o of the ground reference system g The distance from the ground equivalent gravitational field, the tether tension can be expressed as
Here, the subscript x or y of the pulling force P indicates the pulling force in the x direction or the y direction, and the subscript ij or ik indicates the pulling force of the satellite simulator j or the satellite simulator k at the two adjacent ends on the satellite simulator i
Note that in equation (6), the tension term may be directly provided by the tether tension itself in the experiment, and the rest of the tension term is obtained by air injection force.
The following dimensionless transformation is performed on the dynamics equation (6) of the ground experiment system
When the following parameter conditions are satisfied
r g ＝r oE L g /(μ E /Ω 2 ) 1/3 (11)
The dynamics equation (6) of the ground experiment system can be converted into the same dimensionless dynamics equation (4), namely the equivalent space tethered formation system of the ground experiment system can be completely used.
Finally, the ground dynamic spin system is converted into a static balance system by utilizing the darebel principle. As shown in FIG. 3, an inertial force generated by the spin motion of the system is applied to each of the three simulators, and the original spin system is converted into a dynamically balanced non-spin system. Inertial force
Introducing into a ground equivalent system dynamics equation (6) to obtain the following static equilibrium equation
Wherein the inertial force component is written as
The position vector of the satellite simulator in the ground reference system is written as r gi ＝(x gi ,y gi ) T . Since the ground closed loop tethered formation experiment system is at a constant angular velocity omega g Performs a spin motion, so the inertial force is expressed as
Wherein d g12 、d g23 And d g31 Representing the distance, θ, between simulators g1 、θ g2 And theta g3 Indicating the angle between the experimental tethers.
During the experiment, it was determined that the spin motion of the system was unstable as long as the satellite simulator moved or the tether remained relaxed.
First set of experiments: according to the following experimental parameters, carrying out ground simulation experiment, selecting three experiment tethers with the length of 1m, and setting the equivalent track angular velocity as omega g =0.1 rad/s, taking the satellite simulator mass as m g 8.5kg (the simulator average mass is taken in the experiment since the simulator will generate gas mass consumption due to jet force and the simulator mass variation ranges from 8.75kg to 8.25 kg), the equivalent centrifugal force of the spin motion is applied to the three simulators according to equation (10).
Let omega g /Ω g The ratio of (2) was-2.6, and the experimental results are shown in FIGS. 4 to 6. The running track of the satellite simulator is shown in fig. 4, which shows that under the action of the external force, all three simulators cannot keep static and stable. The angle v of the length of the tether along with the ground true near point g As shown in fig. 5, it can be seen that the tether gradually becomes slack. FIG. 6 shows the angle θ between tethers gi With v g Is a variation of (2). Thus, it can be concluded that the ground equivalent tethered formation system is at ω g ＝-2.6Ω g Is unstable in the case of (a).
The experimental results are verified through numerical simulation calculation, the spin angular velocity omega= -2.6Ω of the on-orbit system is taken, and the computer simulation results are shown in fig. 7-10. The moving tracks of the satellites 1, 2 and 3 are shown in fig. 7-9, and the change of the distance between the satellites along with the true near point angle v is shown in fig. 10. Obviously, the satellite cannot form a closed motion trajectory and the tether is always in a relaxed state, so the spinning motion is unstable. Thus, the experiments were consistent with the numerical results, which illustrates that the experimental methods proposed by the present invention are feasible.
The second set of calculation examples takes ω g /Ω g = -3, the experimental results are shown in fig. 11-14. From fig. 11 it can be seen that there is little change in the position of the three satellite simulators. Fig. 12 is an enlarged view of the movement locus of the simulator 1, and it can be seen that the positional movement of the simulator 1 occurs within a very small range. FIG. 13 shows the process of the experimentAll three tethers are kept in a tight state. In FIG. 14, the included angle θ between tethers gi Angle v along with true and near point of ground g The variation is not great. Therefore, the experimental result shows that the tethered simulator formation system has a constant angular velocity omega g ＝-3Ω g Is asymptotically stable.
The corresponding numerical simulation results are shown in fig. 15-16. Obviously, the motion of the three satellites forms a closed orbit and the tether is always in a tight state, so the spinning motion of the spatial tether system is asymptotically stable when ω= -3Ω. The ground experiment is consistent with the numerical simulation result.
The two groups of experiments and corresponding numerical comparison examples prove that the ground equivalent experimental method of the spin closed loop rope formation system provided by the invention can verify the stability of the spin motion of the system.
Claims (4)
1. The space-earth equivalent experimental method of the spin closed loop rope formation system is characterized by comprising the following steps of:
(1) According to Newton's second law, under the non-inertial reference system, establishing a spin closed loop tether formation system running in an orbit plane, and obtaining a kinetic equation of the closed loop tether formation system;
(2) Establishing a ground equivalent experimental system and obtaining a kinetic equation of the ground equivalent experimental system;
(3) The dynamic equation of the closed loop rope system formation system obtained in the step (1) and the dynamic equation of the ground equivalent experimental system obtained in the step (2) are converted into the parameter condition of the same dimensionless dynamic equation so as to ensure the consistency of the heaven-earth system;
(4) Converting a ground equivalent experimental system into a static balance system by utilizing the Dallangei principle;
in the step (2), a ground equivalent experimental system is established, and a dynamic equation of the ground equivalent experimental system is obtained by the following specific steps:
constructing a ground equivalent experiment system aiming at the space tethered formation system established in the step (1); simulation of n satellites on a smooth experiment platformSequentially connecting n flexible experiment tethers to form a closed loop; the satellite simulator is suspended on an experimental platform through an air floatation method, and a ground equivalent spin closed loop rope system formation system is built; the n satellite simulators are respectively marked as a satellite simulator 1, a satellite simulator 2, a satellite simulator … and a satellite simulator n; then, a ground reference system o is constructed g -x g y g Wherein the origin of coordinates o g Centroid, x located in formation system g Axes and y g The axis is parallel to the experimental platform;
let m be gi And L g The mass of the satellite simulator i and the experimental tether length omega are respectively g And omega g Respectively the equivalent angular velocity of the ground and the spin angular velocity t g For the ground experiment time, a kinetic equation of a ground experiment system is established
Wherein the method comprises the steps of
Represents the experimental time t to the ground g First derivative is calculated>
Represents the experimental time t to the ground g Second derivative, mu Eg Is equivalent to the gravitational constant, r g Is the origin o of the ground reference system g Distance from the ground equivalent gravitational field, tether pull is expressed as
Wherein, the subscript x or y of the pulling force P represents the pulling force in the x direction or the y direction, and the subscript ij or ik represents the pulling force of the satellite simulator j or the satellite simulator k at two adjacent ends to the satellite simulator i;
in the formula (3), the tension term is directly provided by the self tension of the tether in the experiment, and the other items are obtained by equivalent air injection force.
2. The method for space-earth equivalent experiments of a spin closed loop tethered formation system according to claim 1, wherein in the step (1), the spin closed loop tethered formation system operating in an orbit plane is established under a non-inertial reference system according to newton's second law, and the specific steps of obtaining the dynamics equation of the closed loop tethered formation system are as follows:
build a spin closed loop tethered formation system operating in an orbital plane consisting of n satellites M 1 、M 2 、…、M n The elastic ropes are sequentially connected through n elastic ropes to form a closed loop; wherein the mass of n satellites is m respectively 1 、m 2 、…、m n The original lengths of the n tethers are L 0 The rigidity is EA, and the pressure can not be borne by the EA only by the tensile force; the closed loop rope system is set to spin around the mass center o of the system at the angular velocity omega, and the system always moves around the ground on a fixed circumferential track at the angular velocity omega;
establishing a non-inertial reference system o-xy taking a system centroid o as an origin, wherein an x-axis points in a direction opposite to the movement of the centroid o, and a y-axis points to the system centroid from the earth centroid; the kinetic equation of the closed loop tether formation system is therefore expressed as:
wherein subscripts j, k represent two satellites adjacent to satellite i;
representing the first derivative of time t,/>
Representing the second derivative, μ, of time t E Represents the gravitational constant, r oE Representing the distance between the centroid of the earth and the origin o of the reference frame, wherein the variable sign delta ij(k) Is defined as
Wherein r is i ＝(x i ,y i ) T Representing the position vector of satellite i in the non-inertial frame of reference, subscript ij (k) represents ij or ik.
3. The method according to claim 2, wherein the step (3) is performed by transforming the dynamics equation of the closed loop rope formation system in the step (1) and the dynamics equation of the ground equivalent experiment system in the step (2) into the same parameter condition of dimensionless dynamics equation, so as to ensure the consistency of the space-earth system, and the specific steps are as follows:
introduction of dimensionless time τ=Ω·t and reference length l r ＝(μ E /Ω 2 ) 1/3 The following dimensionless transformation is carried out on the space rope formation system equation (1)
Obtaining a dimensionless kinetic equation of the system
Wherein "'" means to take the first derivative of the dimensionless time τ and "" "means to take the second derivative of the dimensionless time τ,
and->
As dimensionless coordinate variables, there are the following parameters
The non-dimensional length, the non-dimensional tether stiffness, the non-dimensional tether length and the non-dimensional winding angular velocity are respectively; wherein the subscript ij (k) represents ij or ik;
then, the following dimensionless transformation is carried out on the dynamics equation (3) of the ground experiment system
At this time, the following parameter conditions should be satisfied
r g ＝r oE L g /(μ E /Ω 2 ) 1/3 (11)
The dynamics equation (3) of the ground experiment system can be converted into the dimensionless dynamics equation (6) which is the same as the space on-orbit system, namely the equivalent space tethered formation system of the ground experiment system can be completely used.
4. The method for the space-earth equivalent experiment of the spin closed loop tethered formation system according to claim 1, wherein the specific steps of converting the ground equivalent experiment system into the static balance system by using the darebel principle in the step (4) are as follows:
inertial force
Introducing the static equilibrium equation into a ground equivalent system dynamics equation (3) to obtain the following static equilibrium equation
Wherein the inertial force component is written as
The position vector of the satellite simulator in the ground reference system is written as r gi ＝(x gi ,y gi ) T The method comprises the steps of carrying out a first treatment on the surface of the Since the ground equivalent experimental system is at a constant angular velocity omega g Spin movement is performed so that the inertial force is expressed as +.>
During the experiment, it can be determined that the spin motion of the system is unstable as long as one satellite simulator is moving or the tether remains relaxed.
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