CN103984238A - Satellite formation configuration distribution type control method based on virtual spring damping network - Google Patents

Abstract

The invention discloses a satellite formation configuration distribution type control method based on a virtual spring damping network. The method comprises the following steps: (1) assigning an expected configuration; (2) assigning a virtual spring damping network connection topological structure; (3) selecting the coefficient of a virtual spring damper; (4) sensing the relative motion by satellites through relative motion measurement and communication; (5) respectively calculating the self control accelerations of the satellites; and (6) respectively implementing the control accelerations of the satellites. The self-organization establishment and high-precision keeping of expected relative motion states of the satellites can be realized by adopting the method; the good adaptability to the number increase of the satellites can be achieved; the fuel consumption can be saved; the control method is anti-interference; the stability is theoretically guaranteed, the configuration control precision requirement can be met by adjusting parameters of the virtual spring damper.

Description

Satellites formation configuration distributed control method based on virtual spring damping network

Technical field

The invention belongs to spacecraft technology field, relate to a kind of satellites formation control method, relate in particular to a kind of satellites formation configuration distributed control method based on virtual spring damping network.

Background technology

Satellites formation system utilizes closely flight together of mutual disjunct satellite in many structures, complete space tasks by colony's collaborative work, compared with finishing the work with single satellite, can bring the many advantages such as performance boost, reliability increase, adaptability enhancing, and can realize single satellite impossible mission.And satellites formation task often requires inter-satellite to keep accurate geometrical configuration.Along with the fast development of microsatellite technology, the demand of extensive satellites formation task is more and more stronger, due to scale increase with perception, communicate by letter, the restriction of computing power, central controlled enforcement is very difficult, therefore, how to design effective control architecture, become the key problem that realizes the distributed control of satellites formation configuration.

At present, known satellites formation configuration distributed control method is divided into four classes by architecture, comprises that (English full name is the abbreviation of " Leader-Following " to LF, and Chinese translation is " leader-follow method ".), (Chinese translation is " circulating method " to Cyclic.), (English full name is the abbreviation of " Virtual Structure " to VS, and Chinese translation is " virtual architecture method ".), (Chinese translation is " behavior formula method " to Behavioral.)。In above-mentioned LF method, the position of tagger's Satellite Tracking leader satellite and towards, the controller of each satellite connects with hierarchical manner, its shortcoming is the explicit feedback from tagger to leader not, very easily disperses once occur disturbing; In circulating method, each inter-satellite is non-level control structure, possesses feedback mechanism, but is only applicable to ring topology structure, and the scope of application is restricted; In VS method, form into columns and be counted as rigid structure, the desired motion of defining virtual structure is also translated into the desired motion of satellite, and derive the tracking control of satellite, and shortcoming is the work of must not making its presence or power felt with mass motion; In behavior formula method, be the multiple expected behavior of satellite predefine, control control weighted stacking corresponding to each behavior, weak point is to be difficult to carry out theoretical analysis and to be difficult to guarantee control performance.In a word, existing four class methods are all short of the adaptive faculty to the distributed control task of extensive satellites formation configuration.

Summary of the invention

The technical problem to be solved in the present invention is, for the defect of known technology, to provide a kind of satellites formation configuration distributed control method based on virtual spring damping network.The control performance of its distributed control architecture should possess sufficient theoretical guarantee.

Technical scheme of the present invention is, a kind of satellites formation configuration distributed control method based on virtual spring damping network providing, that the satellite in forming into columns is coupled together to formation network with virtual spring-damper, what produce according to all virtual spring dampers that are directly connected with satellite applies control to satellite with joint efforts, satellite only needs the relative motion state of neighbours' satellite in sensing network, by and neighbours' satellite between reciprocation just can make satellites formation realize expect configuration and keep expect configuration.Through Lyapunov, (Chinese translation is " Liapunov " to this satellites formation configuration.) stability analysis, carry out the control of gravitational field Satellite Formation Configuration with the above-mentioned satellites formation configuration distributed control method based on virtual spring damping network, satellites formation expects that the stability of configuration can be guaranteed, and the control accuracy of the expectation configuration of this satellites formation also can meet by the parameter of adjusting virtual spring damper.Specifically, the method comprises the steps:

(1), specify and expect configuration.According to the configuration requirement of entrusting satellites formation, use conventional method, specify the expectation relative motion law of entrusting satellites formation, this rule comprises specifying entrusts the expectation relative position of satellites formation and specifies the expectation relative velocity of entrusting satellites formation.Be n if entrust satellites formation all satellites, i (i=2,3 ..., n) individual satellite with respect to the expectation relative position of the 1st satellite is provide i (i=2,3 ..., n) individual satellite with respect to the expectation relative velocity of the 1st satellite is expect relative position vector with expectation relative velocity vector specify the expectation configuration of satellites formation, in this two formula, P ^dfor expecting relative position vector, for expecting relative velocity vector, subscript T represents vectorial transposition;

(2), specify virtual spring damping network to connect topological structure.Obtain and expect relative position vector sum expectation relative velocity vector according to step (1), adopt non-directed graph G=(V, E) to represent that virtual spring damping network connects topological structure, in formula, G is non-directed graph, and V is vertex set, V={v ₁, v ₂..., v _n, v _i(i=1,2 ..., n) be i the summit of G, corresponding with i satellite in satellites formation; E is limit collection, E={e ₁, e ₂..., e _m, m is the number on limit in E, e _k(k=1,2 ..., m) be the k article of limit of G, e _kwith two vertex representation be e _k=(v _i, v _j), wherein v _iand v _jtwo different summits in V.The limit e of non-directed graph G _k=(v _i, v _j) specify its vertex v _iand vertex v _jbetween annexation, specify the connection topological structure of setting up a virtual spring-damper connection between i satellite and j satellite, specifying all virtual spring dampers between all satellites.The non-directed graph of any one connection, Ru Shu, ring, chain, all can be used as virtual spring damping network and connect topological structure.Freely select a connected undirected graph of being convenient to implement and connect topological structure as virtual spring damping network.Satellite in described trust satellites formation is coupled together to formation network with virtual spring-damper, specify the annexation of the virtual spring damper between each satellite in this trust satellites formation.Then, determined the neighborhood of each satellite by virtual spring damping network connection topological structure non-directed graph G=(V, E), the neighborhood of i satellite is N _i={ j| (v _i, v _j) ∈ E}, in formula, N _ibe the neighborhood of i satellite, and by N _ibe stored in the internal memory of i satellite, i=1,2 ..., n;

(3), selected virtual spring damper coefficient.According to configuration control steady-state error and the stability requirement of entrusting satellites formation to propose, it is the selected virtual spring damper coefficient of the virtual spring damper between each satellite in the trust satellites formation of the specified annexation of step (2).If the configuration hold error of entrusting satellites formation to propose is that configuration control steady-state error can not be greater than ε, | P-P ^d| _∞≤ ε, in formula || _∞represent the maximal value of vectorial each component absolute value, the actual relative position vector that P is satellites formation; p in formula _ibe the actual relative position of i+1 satellite with respect to the 1st satellite; p _i=r _i+1-r ₁(i=1,2 ..., n-1), r in formula _i+1be the physical location of i+1 satellite, r ₁it is the physical location of the 1st satellite.

Use as lower inequality and choose and meet the requirement of configuration hold error and can ensure the elasticity coefficient k that configuration control closed-loop system is stable _s:

$k_s\≥\frac{1}{\mathrm{\ϵ}}\underset{t\∈T}{\max }{\left|\right\{{\left[S^{T}D\left(G\right)D{\left(G\right)}^{T}S{\left(S^{T}S\right)}^{-1}\right]}^{-1}\⊗{\▿}^{2}Q\left(r_0\left(t\right)\right)\left\}{P}^d\right|}_{\∞}$

In formula: the Kronecker product of representing matrix; Q is terrestrial gravitation potential function, r ₀(t) be the motion of the reference orbit that is deployed in of known satellites formation; T is the time; T is the time interval of configuration control effect; S=[-1 _n-1i _n-1] ^t, 1 _n-1for the vector that n-1 capable 1 is listed as and all elements is 1, I _n-1for the unit matrix of n-1 dimension; D (G) is the incidence matrix of virtual spring damping network connection topological structure non-directed graph G, the matrix of the capable m row of n-1, d in formula _ikfor the element of the capable k row of the i of D (G).By being every limit e in G _k=(v _i, v _j) assigned direction, specify e _kvertex v _iand vertex v _jin two summits, any one summit is head, and another summit is tail, determines that by following formula described virtual spring damping network connects the element d of the capable k row of i of the incidence matrix D (G) of topological structure non-directed graph G _ikvalue (i=1,2 ..., n-1; K=1,2 ..., m):

In formula: d _ikfor the element of the capable k row of the i of D (G); v _ifor i the summit of non-directed graph G; e _kfor k article of limit in non-directed graph G.

Configuration control closed-loop system stability is to ratio of damping k _drequirement be k _d>0, accordingly selected k _d;

(4), each passing of satelline relative motion is measured and the perception relative motion state of communicating by letter.The virtual spring damping network specified according to step (2) connects topological structure, use conventional method, measure and communicate by letter by relative motion and obtain that to entrust each satellite and its neighbours' satellite in satellites formation be the relative motion state between the satellite in neighborhood.Directly its neighbours' satellite is carried out to relative motion measurement by each satellite, obtain the relative motion state of neighbours' satellite to described each satellite; Or by neighbours' satellite, described each satellite is carried out to relative motion measurement, more indirectly obtain the relative motion state of neighbours' satellite to described each satellite by described each satellite and communicating by letter of neighbours' satellite;

(5), each satellite calculates respectively the control acceleration of self.Use the selected virtual spring damper coefficient of step (3), according to step (4) relative motion state between each satellite that obtains and the neighbours' satellite being connected by virtual spring damper, adopt the conventional mode of implementing respectively the control of virtual spring damping network along three change in coordinate axis direction, obtain the Linear Control rule based on relative motion state, calculate and control acceleration according to gained Linear Control rule.

First calculate respectively successively by following two formulas the satellites formation being defined by all limits of virtual spring damping network connection topological structure non-directed graph G and expect relative position vector expect relative velocity vector with satellites formation

$P_G^d=[D{\left(G\right)}^{T}S{\left(S^{T}S\right)}^{-1}\⊗I_3]P^d$

${\stackrel{\·}{P}}_G^d=[D{\left(G\right)}^{T}S{\left(S^{T}S\right)}^{-1}\⊗I_3]{\stackrel{\·}{P}}^d$

I in this two formula ₃be 3 dimension unit matrixs; for the p article of limit e of topological structure non-directed graph G _pdefined expectation relative position (p=1,2 ..., m); for the p article of limit e of topological structure non-directed graph G _pdefined expectation relative velocity (p=1,2 ..., m).If k article of limit e in topological structure non-directed graph G _ktwo summits be respectively vertex v _iand vertex v _j, i.e. e _k=(v _i, v _j), calculate respectively expectation relative position by following two formulas with expectation relative velocity

In this two formula: be expectation relative position to i satellite of j neighbours' satellite of i satellite (i=1,2 ..., n; J ∈ N _i); e _kfor k article of limit in non-directed graph G, k=1,2 ..., m, e _kdirection according to the appointment in step (3); for the k article of limit e of non-directed graph G _kdefined expectation relative position, k=1,2 ..., m; v _ifor i the summit of non-directed graph G, i=1,2 ..., n; be expectation relative velocity to i satellite of j neighbours' satellite of i satellite (i=1,2 ..., n; J ∈ N _i); for the k article of limit e of non-directed graph G _kdefined expectation relative velocity, k=1,2 ..., m.

Then, be calculated as follows out the control acceleration u of i satellite according to above-mentioned gained Linear Control rule _i:

$u_i={-k}_s\underset{j\∈N_i}{\mathrm{\Σ}}(r_{\mathrm{ji}}^d-r_{\mathrm{ji}})-k_d\underset{j\∈N_i}{\mathrm{\Σ}}({\stackrel{\·}{r}}_{\mathrm{ji}}^d-{\stackrel{\·}{r}}_{\mathrm{ji}}),(i=\mathrm{1,2},...,n)$

R in formula _jibe j neighbours' satellite of i satellite relative position to i satellite, it is j neighbours' satellite of i satellite relative velocity to i satellite;

(6), each satellite implements respectively to control acceleration.I satellite uses its actuator implementation step (5) to calculate the control acceleration u of gained _i, i=1,2 ..., n, and at a control cycle, a for example fixed value within the scope of 0.1～1s, keeps this control acceleration in the time.When this control cycle finishes, if configuration control task not yet completes, repeating step (4)～(6); If configuration control task completed already, step finishes.

The invention has the beneficial effects as follows: proposed the satellites formation configuration distributed control method based on virtual spring damping network, can realize satellites formation and expect that the self-organization foundation of relative motion state and high precision keep; Only rely on the intersatellite mutual perception of neighbours that connects in topological structure and communicate by letter, the feature of this Local Interaction makes the method be easy to distributed enforcement, and the increase of satellites formation Satellite quantity is had to good adaptability; Only need the intersatellite relative motion status information of neighbours and do not need the absolute orbit of knowing each satellite to move; Control has the character of internal force to satellites formation, only relative motion is played to control action, does not change orbital motion thereby the consumption of energy fuel saving of satellites formation barycenter; Thereby the amphicheirality that virtual spring damper connects provides Internal feedback, mechanism can be anti-interference; Stability obtains theoretical guarantee, and configuration control accuracy requirement can meet by the parameter of adjusting virtual spring damper.

Embodiment

Embodiment 1;

(1), specify and expect configuration.According to the configuration requirement of entrusting satellites formation, use conventional method, specify the expectation relative motion law of entrusting satellites formation, this rule comprises specifying entrusts the expectation relative position of satellites formation and specifies the expectation relative velocity of entrusting satellites formation.Be n if entrust satellites formation all satellites, i (i=2,3 ..., n) individual satellite with respect to the expectation relative position of the 1st satellite is provide i (i=2,3 ..., n) individual satellite with respect to the expectation relative velocity of the 1st satellite is expect relative position vector with expectation relative velocity vector specify the expectation configuration of satellites formation, in this two formula, P ^dfor expecting relative position vector, for expecting relative velocity vector, subscript T represents vectorial transposition;

(2), specify virtual spring damping network to connect topological structure.Obtain and expect relative position vector sum expectation relative velocity vector according to step (1), adopt non-directed graph G=(V, E) to represent that virtual spring damping network connects topological structure, in formula, G is non-directed graph, and V is vertex set, V={v ₁, v ₂..., v _n, v _i(i=1,2 ..., n) be i the summit of G, corresponding with i satellite in satellites formation; E is limit collection, E={e ₁, e ₂..., e _m, m is the number on limit in E, e _k(k=1,2 ..., m) be the k article of limit of G, e _kwith two vertex representation be e _k=(v _i, v _j), wherein v _iand v _jtwo different summits in V.The limit e of non-directed graph G _k=(v _i, v _j) specify its vertex v _iand vertex v _jbetween annexation, specify the connection topological structure of setting up a virtual spring-damper connection between i satellite and j satellite, specifying all virtual spring dampers between all satellites.The non-directed graph tree of any one connection, ring, chain, all can be used as virtual spring damping network and connect topological structure.Freely select a connected undirected graph tree of being convenient to implement and connect topological structure as virtual spring damping network.Satellite in described trust satellites formation is coupled together to formation network with virtual spring-damper, specify the annexation of the virtual spring damper between each satellite in this trust satellites formation.Then, determined the neighborhood of each satellite by virtual spring damping network connection topological structure non-directed graph G=(V, E), the neighborhood of i satellite is N _i={ j| (v _i, v _j) ∈ E}, in formula, N _ibe the neighborhood of i satellite, and by N _ibe stored in the internal memory of i satellite, i=1,2 ..., n;

(3), selected virtual spring damper coefficient.According to configuration control steady-state error and the stability requirement of entrusting satellites formation to propose, it is the selected virtual spring damper coefficient of the virtual spring damper between each satellite in the trust satellites formation of the specified annexation of step (2).If the configuration hold error of entrusting satellites formation to propose can not be greater than ε, | P-P ^d| _∞≤ ε, in formula || _∞represent the maximal value of vectorial each component absolute value, the actual relative position vector that P is satellites formation; p in formula _ibe the actual relative position of i+1 satellite with respect to the 1st satellite; p _i=r _i+1-r ₁(i=1,2 ..., n-1), r in formula _i+1be the physical location of i+1 satellite, r ₁it is the physical location of the 1st satellite.

Use as lower inequality and choose and meet the requirement of configuration hold error and can ensure the elasticity coefficient k that configuration control closed-loop system is stable _s:

$k_s\≥\frac{1}{\mathrm{\ϵ}}\underset{t\∈T}{\max }{\left|\right\{{\left[S^{T}D\left(G\right)D{\left(G\right)}^{T}S{\left(S^{T}S\right)}^{-1}\right]}^{-1}\⊗{\▿}^{2}Q\left(r_0\left(t\right)\right)\left\}{P}^d\right|}_{\∞}$

In formula: the Kronecker product of representing matrix; Q is terrestrial gravitation potential function, r ₀(t) be the motion of the reference orbit that is deployed in of known satellites formation; T is the time; T is the time interval of configuration control effect; S=[-1 _n-1i _n-1] ^t, 1 _n-1for the vector that n-1 capable 1 is listed as and all elements is 1, I _n-1for the unit matrix of n-1 dimension; D (G) is the incidence matrix of virtual spring damping network connection topological structure non-directed graph G, the matrix of the capable m row of n-1, d in formula _ikfor the element of the capable k row of the i of D (G).By being every limit e in G _k=(v _i, v _j) appointment e _kvertex v i and vertex v _jin any one summit be head, another summit is tail, determines that by following formula described virtual spring damping network connects the element d of the capable k row of i of the incidence matrix D (G) of topological structure non-directed graph G _ikvalue (i=1,2 ..., n-1; K=1,2 ..., m):

In formula: d _ikfor the element of the capable k row of the i of D (G); v _ifor i the summit of non-directed graph G; e _kfor k article of limit in non-directed graph G.

Configuration control closed-loop system stability is to ratio of damping k _drequirement be k _d>0, accordingly selected k _d;

(4), each passing of satelline relative motion is measured and the perception relative motion state of communicating by letter.The virtual spring damping network specified according to step (2) connects topological structure, use conventional method, measure and communicate by letter by relative motion and obtain that to entrust each satellite and its neighbours' satellite in satellites formation be the relative motion state between the satellite in neighborhood.Directly its neighbours' satellite is carried out to relative motion measurement by each satellite, obtain the relative motion state of neighbours' satellite to described each satellite;

(5), each satellite calculates respectively the control acceleration of self.Use the selected virtual spring damper coefficient of step (3), according to step (4) relative motion state between each satellite that obtains and the neighbours' satellite being connected by virtual spring damper, adopt the conventional mode of implementing respectively the control of virtual spring damping network along three change in coordinate axis direction, obtain the Linear Control rule based on relative motion state, calculate and control acceleration according to gained Linear Control rule.

First calculate respectively successively by following two formulas the satellites formation being defined by all limits of virtual spring damping network connection topological structure non-directed graph G and expect relative position vector expect relative velocity vector with satellites formation

$P_G^d=[D{\left(G\right)}^{T}S{\left(S^{T}S\right)}^{-1}\⊗I_3]P^d$

${\stackrel{\·}{P}}_G^d=[D{\left(G\right)}^{T}S{\left(S^{T}S\right)}^{-1}\⊗I_3]{\stackrel{\·}{P}}^d$

I in this two formula ₃be 3 dimension unit matrixs; for the p article of limit e of topological structure non-directed graph G _pdefined expectation relative position (p=1,2 ..., m); for the p article of limit e of topological structure non-directed graph G _pdefined expectation relative velocity (p=1,2 ..., m).If k article of limit e in topological structure non-directed graph G _ktwo summits be respectively vertex v _iand vertex v _j, i.e. e _k=(v _i, v _j), calculate respectively expectation relative position by following two formulas with expectation relative velocity

In this two formula: be expectation relative position to i satellite of j neighbours' satellite of i satellite (i=1,2 ..., n; J ∈ N _i); e _kfor k article of limit in non-directed graph G, k=1,2 ..., m, e _kdirection according to the appointment in step (3); for the k article of limit e of non-directed graph G _kdefined expectation relative position, k=1,2 ..., m; v _ifor i the summit of non-directed graph G, i=1,2 ..., n; be expectation relative velocity to i satellite of j neighbours' satellite of i satellite (i=1,2 ..., n; J ∈ N _i); for the k article of limit e of non-directed graph G _kdefined expectation relative velocity, k=1,2 ..., m.

Then, be calculated as follows out the control acceleration u of i satellite according to above-mentioned gained Linear Control rule _i:

$u_i={-k}_s\underset{j\∈N_i}{\mathrm{\Σ}}(r_{\mathrm{ji}}^d-r_{\mathrm{ji}})-k_d\underset{j\∈N_i}{\mathrm{\Σ}}({\stackrel{\·}{r}}_{\mathrm{ji}}^d-{\stackrel{\·}{r}}_{\mathrm{ji}}),(i=\mathrm{1,2},...,n)$

R in formula _jibe j neighbours' satellite of i satellite relative position to i satellite, it is j neighbours' satellite of i satellite relative velocity to i satellite;

(6), each satellite implements respectively to control acceleration.I satellite uses its actuator implementation step (5) to calculate the control acceleration u of gained _i, i=1,2 ..., n, and within a control cycle time, keep this control acceleration.When this control cycle finishes, configuration control task not yet completes, repeating step (4)～(6).

Embodiment 2:

Step (1)～(3) are with embodiment 1;

(4), each passing of satelline relative motion is measured and the perception relative motion state of communicating by letter.The virtual spring damping network specified according to step (2) connects topological structure, use conventional method, measure and communicate by letter by relative motion and obtain that to entrust each satellite and its neighbours' satellite in satellites formation be the relative motion state between the satellite in neighborhood.By neighbours' satellite, described each satellite is carried out to relative motion measurement, more indirectly obtain the relative motion state of neighbours' satellite to described each satellite by described each satellite and communicating by letter of neighbours' satellite;

Step (5) is with embodiment 1;

(6), each satellite implements respectively to control acceleration.I satellite uses its actuator implementation step (5) to calculate the control acceleration u of gained _i, i=1,2 ..., n, and within a control cycle time, keep this control acceleration.When this control cycle finishes, configuration control task completed already, and step finishes.

Claims (1)

1. the satellites formation configuration distributed control method based on virtual spring damping network, the method comprises the steps:

(1), specify and expect configuration, according to the configuration requirement of entrusting satellites formation, use conventional method, specify the expectation relative motion law of entrusting satellites formation, this rule comprises specifying entrusts the expectation relative position of satellites formation and specifies the expectation relative velocity of entrusting satellites formation, and establishing and entrusting all satellites of satellites formation is n, i (i=2,3 ..., n) individual satellite with respect to the expectation relative position of the 1st satellite is provide i (i=2,3 ..., n) individual satellite with respect to the expectation relative velocity of the 1st satellite is expect relative position vector with expectation relative velocity vector specify the expectation configuration of satellites formation, in this two formula, P ^dfor expecting relative position vector, for expecting relative velocity vector, subscript T represents vectorial transposition;

(2), specify virtual spring damping network to connect topological structure, obtain and expect relative position vector sum expectation relative velocity vector according to step (1), adopt non-directed graph G=(V, E) represent that virtual spring damping network connects topological structure, in formula, G is non-directed graph, V is vertex set, V={v ₁, v ₂..., v _n, v _i(i=1,2 ..., n) be i the summit of G, corresponding with i satellite in satellites formation, E is limit collection, E={e ₁, e ₂..., e _m, m is the number on limit in E, e _k(k=1,2 ..., m) be the k article of limit of G, e _kwith two vertex representation be e _k=(v _i, v _j), wherein v _iand v _jtwo different summits in V, the limit e of non-directed graph G _k=(v _i, v _j) specify its vertex v _iand vertex v _jbetween annexation, specify and between i satellite and j satellite, set up a virtual spring-damper and connect, specify the connection topological structure of all virtual spring dampers between all satellites, the non-directed graph tree of any one connection, ring, chain, all can be used as virtual spring damping network and connect topological structure, freely select a connected undirected graph of being convenient to implement and connect topological structure as virtual spring damping network, satellite in described trust satellites formation is coupled together to formation network with virtual spring-damper, specify the annexation of the virtual spring damper between each satellite in this trust satellites formation, then, connect topological structure non-directed graph G=(V by virtual spring damping network, E) determine the neighborhood of each satellite, the neighborhood of i satellite is N _i={ j| (v _i, v _j) ∈ E}, in formula, N _ibe the neighborhood of i satellite, and by N _ibe stored in the internal memory of i satellite, i=1,2 ..., n,

(3), selected virtual spring damper coefficient, according to configuration control steady-state error and the stability requirement of entrusting satellites formation to propose, for the selected virtual spring damper coefficient of the virtual spring damper between each satellite in the trust satellites formation of the specified annexation of step (2), if the configuration hold error of entrusting satellites formation to propose can not be greater than ε, | P-P ^d| _∞≤ ε, in formula || _∞represent the maximal value of vectorial each component absolute value, the actual relative position vector that P is satellites formation; p in formula _ibe the actual relative position of i+1 satellite with respect to the 1st satellite; p _i=r _i+1-r ₁(i=1,2 ..., n-1), r in formula _i+1be the physical location of i+1 satellite, r ₁be the physical location of the 1st satellite,

Use as lower inequality and choose and meet the requirement of configuration hold error and can ensure the elasticity coefficient k that configuration control closed-loop system is stable _s:

$k_s\≥\frac{1}{\mathrm{\ϵ}}\underset{t\∈T}{\max }{\left|\right\{{\left[S^{T}D\left(G\right)D{\left(G\right)}^{T}S{\left(S^{T}S\right)}^{-1}\right]}^{-1}\⊗{\▿}^{2}Q\left(r_0\left(t\right)\right)\left\}{P}^d\right|}_{\∞}$

In formula: the Kronecker product of representing matrix; Q is terrestrial gravitation potential function, r ₀(t) be the motion of the reference orbit that is deployed in of known satellites formation; T is the time; T is the time interval of configuration control effect; S=[-1 _n-1i _n-1] ^t, 1 _n-1for the vector that n-1 capable 1 is listed as and all elements is 1, I _n-1for the unit matrix of n-1 dimension; D (G) is the incidence matrix of virtual spring damping network connection topological structure non-directed graph G, the matrix of the capable m row of n-1, d in formula _ikfor the element of the capable k row of the i of D (G), by being every limit e in G _k=(v _i, v _j) appointment e _kvertex v _iand vertex v _jin any one summit be head, another summit is tail, determines that by following formula described virtual spring damping network connects the element d of the capable k row of i of the incidence matrix D (G) of topological structure non-directed graph G _ikvalue (i=1,2 ..., n-1; K=1,2 ..., m):

In formula: d _ikfor the element of the capable k row of the i of D (G); v _ifor i the summit of non-directed graph G; e _kfor k article of limit in non-directed graph G,

Configuration control closed-loop system stability is to ratio of damping k _drequirement be k _d>0, accordingly selected k _d;

(4), each passing of satelline relative motion is measured and the perception relative motion state of communicating by letter, the virtual spring damping network specified according to step (2) connects topological structure, use conventional method, measure and communicate by letter by relative motion and obtain that to entrust each satellite and its neighbours' satellite in satellites formation be the relative motion state between the satellite in neighborhood, directly its neighbours' satellite is carried out to relative motion measurement by each satellite, obtain the relative motion state of neighbours' satellite to described each satellite; Or by neighbours' satellite, described each satellite is carried out to relative motion measurement, more indirectly obtain the relative motion state of neighbours' satellite to described each satellite by described each satellite and communicating by letter of neighbours' satellite;

(5), each satellite calculates respectively the control acceleration of self, use the selected virtual spring damper coefficient of step (3), according to step (4) relative motion state between each satellite that obtains and the neighbours' satellite being connected by virtual spring damper, adopt the conventional mode of implementing respectively the control of virtual spring damping network along three change in coordinate axis direction, obtain the Linear Control rule based on relative motion state, calculate and control acceleration according to gained Linear Control rule

First calculate respectively successively by following two formulas the satellites formation being defined by all limits of virtual spring damping network connection topological structure non-directed graph G and expect relative position vector expect relative velocity vector with satellites formation

$P_G^d=[D{\left(G\right)}^{T}S{\left(S^{T}S\right)}^{-1}\⊗I_3]P^d$

${\stackrel{\·}{P}}_G^d=[D{\left(G\right)}^{T}S{\left(S^{T}S\right)}^{-1}\⊗I_3]{\stackrel{\·}{P}}^d$

I in this two formula ₃be 3 dimension unit matrixs; for the p article of limit e of topological structure non-directed graph G _pdefined expectation relative position (p=1,2 ..., m); for the p article of limit e of topological structure non-directed graph G _pdefined expectation relative velocity (p=1,2 ..., m), if k article of limit e in topological structure non-directed graph G _ktwo summits be respectively vertex v _iand vertex v _j, i.e. e _k=(v _i, v _j), calculate respectively expectation relative position by following two formulas with expectation relative velocity

In this two formula: be expectation relative position to i satellite of j neighbours' satellite of i satellite (i=1,2 ..., n; J ∈ N _i); e _kfor k article of limit in non-directed graph G, k=1,2 ..., m, e _kdirection according to the appointment in step (3); for the k article of limit e of non-directed graph G _kdefined expectation relative position, k=1,2 ..., m; v _ifor i the summit of non-directed graph G, i=1,2 ..., n; be expectation relative velocity to i satellite of j neighbours' satellite of i satellite (i=1,2 ..., n; J ∈ N _i); for the k article of limit e of non-directed graph G _kdefined expectation relative velocity, k=1,2 ..., m,

Then, be calculated as follows out the control acceleration u of i satellite according to above-mentioned gained Linear Control rule _i:

$u_i={-k}_s\underset{j\∈N_i}{\mathrm{\Σ}}(r_{\mathrm{ji}}^d-r_{\mathrm{ji}})-k_d\underset{j\∈N_i}{\mathrm{\Σ}}({\stackrel{\·}{r}}_{\mathrm{ji}}^d-{\stackrel{\·}{r}}_{\mathrm{ji}}),(i=\mathrm{1,2},...,n)$

R in formula _jibe j neighbours' satellite of i satellite relative position to i satellite, it is j neighbours' satellite of i satellite relative velocity to i satellite;

(6), each satellite implements respectively to control acceleration, i satellite uses its actuator implementation step (5) to calculate the control acceleration u of gained _i, i=1,2 ..., n, and within a control cycle time, keep this control acceleration, when this control cycle finishes, if configuration control task not yet completes, repeating step (4)～(6); If configuration control task completed already, step finishes.