Shape control of spacecraft formation using a virtual spring-damper mesh

Chen Qifeng,Meng Yunhe,Xing Jinjun

aSchool of Aeronautics and Astronautics,Central South University,Changsha 410083,China

bCollege of Aerospace Science and Engineering,National University of Defense Technology,Changsha 410073,China

Shape control of spacecraft formation using a virtual spring-damper mesh

Chen Qifenga,*,Meng Yunheb,Xing Jianjuna

aSchool of Aeronautics and Astronautics,Central South University,Changsha 410083,China

bCollege of Aerospace Science and Engineering,National University of Defense Technology,Changsha 410073,China

This paper derives a distance-based formation control method to maintain the desired formation shape for spacecraft in a gravitational potential field.The method is an analogy of a virtual spring-damper mesh.Spacecraft are connected virtually by spring-damper pairs.Convergence analysis is performed using the energy method.Approximate expressions for the distance errors and control accelerations at steady state are derived by using algebraic graph representations and results of graph rigidity.Analytical results indicate that if the underlying graph of the mesh is rigid,the convergence to a static shape is assured,and higher formation control precision can be achieved by increasing the elastic coefficient without increasing the control accelerations.A numerical example of spacecraft formation in low Earth orbit confirms the theoretical analysis and shows that the desired formation shape can be well achieved using the presented method,whereas the orientation of the formation can be kept pointing to the center of the Earth by the gravity gradient.The method is decentralized,and uses only relative measurement information.Constructing a distributed virtual structure in space can be the general application area.The proposed method can serve as an active shape control law for the spacecraft formations using propellantless internal forces.

1.Introduction

The goal offormation control is the coordination of a team of agents to satisfy a particular shape or relative state.The formation shape is a rotational invariant,whereas the relative state specifies the orientation of the formation in addition to the geometric shape.Accordingly,shape-based and relative statebased coordination strategies are noted in the literature.1Formation shapes are usually achieved by controlling inter-agent distances;thus,shape-based strategies are also called‘distance-based”,and relative state-based coordination strategies are also called ‘position-based”.2Because the available information is limited,the design and analysis of distancebased formation control are more involved.To answer the question which set of distances should be controlled to maintain the formation shape,the concept of graph rigidity and persistence was investigated.Researchers also developed methods to maintain the rigidity of the formation when merging,splitting,or losing agents.3,4Potential function-based control laws2,5,6are natural selections for maintaining inter-agent distances.Jacobi shape coordinates,which are invariant under the translation and rotation of the formation,were also explored to design the distributed control law for achieving shape consensus.7However,most research in the literature of spacecraft formation control is relative state-based,and the research on shape control of spacecraft formation is rare.

Maintaining formation shape is fundamental for constructing virtual structures in space.In addition to the use of conventional chemical thrusters,the use of innovational devices that generate internal and propellantless forces,such as Coulomb,8electromagnetic9,10and photonic laser thrusters and tethers,11has been proposed.Compared to the relative large amount offuel required for spacecraft formation using conventional thrusters,the property of zero fuel consumption makes these internal-forces based spacecraft formation very attractive.The invariant shape of relative equilibrium in spacecraft formation flying actuated by such internal forces becomes a research topic of particular interest.8,9Rather than solving the static relative equilibrium,Natarajan and Schaub12reported the first research on the active control of a spacecraft formation using internal forces.A charge feedback law was introduced to stabilize the relative distance between the spacecraft to a constant value for an electrostatic Coulomb formation.Internal forces change neither the motion of the mass center nor the total angular momentum of the system.Thus the active shape control of the formation is the main concern using internal forces.In Ref.12,only a two-craft system was considered,and the problem of active control for a general formation shape using internal forces has not been addressed in the literature.

The interactions between the individual agents affect the behavior of a networked multi-agent system.The scalability for implementation requires local interactions that lead to the desired global behavior.Several distributed interaction approaches from multi-agent system theory have been investigated for spacecraftformation control.Mesbahiand Hadaegh13investigated the leader-following(LF)architecture and derived control laws for LF by means of linear matrix inequalities.Ramirez-Riberos et al.14developed cyclic pursuit control laws for spacecraft formations and performed experiments using a test bed operating on the International Space Station.Zhang and Gurfil15designed a distributed controller to maintain cluster flight by utilizing a cyclic control algorithm based on a mean orbital elements feedback.Izzo and Pettazzi16exploited a behavior-based approach to achieve an autonomous and distributed control by designing ‘gather”,‘avoid”and ‘dock” behavior.Nag and Summerer17addressed the problem of distributed evasive maneuvers of the formation by using behavior based method and artificial potential functions.Ren and Beard18proposed a decentralized virtual structure approach in which a local copy of a coordination vector is instantiated at each spacecraft and synchronized by communication with neighbors.Ahn and Kim19integrated an adaptive sliding mode controller to the virtual structure framework to perform the synchronized formation maneuvers for pointing to a desired target.Ren20proposed control laws for spacecraft formation-keeping and attitude alignment under a general directed information-exchange topology with arbitrary feedback between neighboring spacecraft.Zhang and Song21proposed decentralized formation controllersbycombining consensus algorithms with behavior-based control.Some of these formation control approaches13,15,18,20assume that every spacecraft knows its own absolute position or velocity,or equivalently the orbital elements.However,for spacecraft formation missions,accurate absolute position measurements are often not available,whereas relative measurements can reach a much higher accuracy.The absolute position of a spacecraft is usually not subject to as stringent constraints as the relative position or the formation shape.Thus,it is reasonable to control relative motion using only relative measurements.Other formation control approaches14,16,17,19,21use only relative state information.But these approaches usually adopt simplified relative dynamics of spacecraft motion.

This paper investigates a distance-based strategy using a virtual spring damper mesh(VSDM)to control the formation shape of spacecraft swarms in a gravitational field.The formation control strategy is a physical analogy of a system offree point masses connected by spring-dampers.Because of the local interaction of the mesh,the presented method is decentralized.No information about the absolute positions or velocities of spacecraft is required.In a previous study,22the authors proposed a relative state-based VSDM formation control method.A linear relative state feedback is used to achieve the desired relative state between spacecraft.In this study,the distance-based control law is nonlinear.The goal of the formation control is to maintain the desired formation shape,and the rotation of the formation is free or controlled by other means to satisfy the mission requirement.Only the information of the distance and the rate of distance change between spacecraft are needed.By using a distance-based control,every spacecraft can operate in its local frame,whereas the method in Ref.22requires all spacecraft to operate in a common reference frame,which may cause problems for implementation.Moreover,the VSDM generates only virtual internal‘forces”between spacecraft which can be implemented using actual internal forces such as electrostatic Coulomb forces,electromagnetic forces,light pressure,and tether tension.Thus,the VSDM method can serve as an active shape control method for spacecraft formations using novel propellantless internal forces.A different virtual spring mesh algorithm was explored for the deployment of mobile sensors in Ref.23,where virtual dampers are separately used to decrease the absolute velocity of each agent to a stationary state,whereas in this study one spring and one damper are combined as a connection unit for relative distance control.Palmerini24investigated the use of a virtual spring mesh for satellite constellation station keeping,but no dampers were adopted.

The remaining of the paper is organized as follows.Section 2 provides the preliminaries of the theory of algebraic graphs and graph rigidity.The distance-based VSDM method for spacecraft formation control is formulated in Section 3.In Section 4,the convergence and steady-state performance are investigated.In Section 5,spacecraft formation in low Earth orbit for Earthpointing is simulated.Section 6 concludes the paper.

2.Preliminaries

2.1.Graphical description of relative motions in a formation

The relative states between the agents in a formation can be associated with the incidence matrix of a directed graph(or digraph),G= （V，E）,whereis the vertex set corresponding to n agents,the edge set, I={1，2，...，n} the vertex index set,J={1，2，...，m}the edge index set,and m the number of edges in the graph.If （vi，vj）∈ E,then viand vjare adjacent,i.e.,they are neighbors.Any edge ek= （vi，vj）∈ E of the digraph is an ordered pair,where k∈J is the edge index.The starting vertex viis defined to be the tail of the edge,and the ending vertex vjis the head.The incidence matrix of the digraph G is defined as1

The columns of the incidence matrix represent the edges of the digraph.The size of the matrix D（G）is n×m,where m is the number of edges in G.The relative position vector corresponding to the kth directed edge ek= （vi，vj）of the digraph G can be defined as

where PG=[pT1，pT2，...，pTm]T.

2.2.Formation rigidity and rigid graph

A formation is said to be rigid when the distance between every pair of agents remains constant along any trajectory on which the lengths of all edges of its underlying graph G are kept if xed.3For distance-based formation control,the concept of rigidity is important because it reveals the feasibility of maintaining the formation shape by only maintaining the desired length of edges in the graph G of the connecting topology.This subsection borrows several de finitions and conclusions that were previously summarized.3

To examine the rigidity of a given formation,the key is to study the trajectories on which the lengths of all edges of the connecting graph G= （V，E）are constant.Along such a trajectory,for every ek= （vi，vj）∈ E,the Euclidean distance between the pairs of agents,‖pk‖ = ‖rj-ri‖,is constant.Therefore,the following can be written for these trajectories:

The relation of Eq.(4)can be represented in matrix form as

whereℜ（R）is a specially structured m×3n matrix called the rigidity matrix.The rigidity matrix can be represented by the relative position vectors and the incidence matrix of the edges in G as

In practice,actual agent groups cannot be expected to move exactly in a rigid formation because of sensing,modeling,and actuation errors.Thus,another type of rigidity,called ‘generic rigidity”in which the topology will be robust for maintaining formations under small perturbations,is more useful for our purposes.

A formation is generically rigid when the formation is rigid for almost all choices of R in Θnd,where Θ is the set of real numbers and d the dimension.A formation with at least three agents(n≥3)in three-dimensional space is generically rigid if and only if the generic rank ofℜequals to 3n-6,where the generic rank ofℜis defined as the largest value of the rank ofℜ（R）as R ranges over all values in Θ3n.The generic rigidity is a property of only the underlying graph G;thus,such graphs are denoted as generically rigid graphs,or rigid graphs.1A formation is said to be strongly generically rigid when the formation is generically rigid and when the rank ofℜ（R）equals to the generic rank ofℜ.The set of R that satisfies the condition that the rank of ℜ（R）equals to the generic rank ofℜ is a dense open subset of Θ3n.Hence,a strongly generically rigid formation is rigid and remains rigid under small perturbations.The following theorem provides a justification for regarding a formation with a connecting topology of generically rigid graphs as strongly generically rigid.

Theorem 1.3For a formation in a three-dimensional space with at least 3 agents,the following are equivalent:(1)the formation’s underlying graph G= （V，E）is generically rigid;(2)for some R,Rank （ℜ（R））=3n-6;(3)for almost all R,the formation is strongly generically rigid.

Every formation with a complete connecting graph G is rigid.The converse,however,is not generally true.A graph is minimally rigid when the graph is rigid but does not remain rigid after the removal of a single edge.Minimally rigid graphs have 3n-6 edges in three-dimensional space.The maintained edges of the connecting graph may not be all independent to allow a formation to be rigid.Minimally rigid formations are also maximally independent formations,for which ‘independent”indicates that the maintenance edge set resides in independent rows in the rigidity matrix.Thus,the maximum rank of the rigidity matrixℜ（R）is 3n-6.

3.Formulation of spacecraft formation control method

3.1.Model setup for spacecraft motion

A swarm of spacecraft in a gravitational field is considered.Each spacecraft is considered as a point with unit mass.The translational motion of the ith spacecraft is described as

where Q is the potential function of the gravitational field and uithe control acceleration input to the ith spacecraft.For the spacecraft formation application in low Earth orbit,this model can account for the zonal gravitational harmonics of the perturbing potential of the oblateness of the Earth.For example,when the main perturbation(the J2term from the oblateness of the Earth)is included,the gravitational potential function is as follows:25

where r=[x，y，z]Tis the position vector in the Earth-centered inertial(ECI)reference frame;REis the mean equatorial radius of the Earth;μ is the gravitational parameter of the Earth;and μ =3.986004405×1014m3·s-2,RE=6378137 m and J2=1.082626675×10-3.

3.2.Distance-based VSDM control law

The VSDM method is an analogy of a system offree point masses connected by massless spring-dampers(Fig.1).The natural lengths of the springs are set to the desired distances between the point masses.Because of the elastic and damping forces,the system is expected to eventually reach equilibrium,i.e.,the desired shape of geometric distribution of these point masses.In implementation,no actual spring or damper is used,but the control forces that drive the spacecraft are generated based on virtual spring-damper pairs ‘connected” to the spacecraft.

The virtual spring-damper mesh used for formation control can be represented by a graph G,in which the vertices of G correspond to the spacecraft,and the edges correspond to the spring-damper connections.Because each connection is double-sided in the VSDM method,G is an undirected graph.By endowing every edge of G with an arbitrary orientation,a corresponding directed graph of G can be specified.With a slight abuse of notations,the derived directed graph is still denoted as G in this paper.

The distance-based VSDM control law is the direct analogy of the physics illustrated in Fig.1.The elastic force is proportional to the difference between the relative distance of the pair of agents and the natural length of the spring,and the damping force is proportional to the rate of change of their relative distance.For the kth edge of the connecting graph,ek= （vtk，vhk）,where vtkis the tail vertex of ekand vhkthe head(tk，hk∈ I),and the control acceleration generated by the corresponding spring-damper is

where ks＞0 is the elastic coefficient;kd＞0 is the damping coefficient;is the distance between the pair of agents;ek= （rhk-rtk）/lkis the unit vector from the tail agent to the head agent;l˙kis the rate of change of the distance lk;and ldkis the desired constant distance between the two agents.The term ldkcan also be regarded as the natural length of the spring corresponding to the kth edge.The direction of akis in accordance with the predefined orientation of the kth edge.Note that˙lkcan be represented using the relative velocities between the pair of agents as

Fig.1 Point masses connected by a spring-damper mesh.

where vk=˙rhk-˙rtkis the relative velocity defined by the kth edge,and this term is equivalent to

The overall control acceleration input for the ith agent can be described using the edges in the connecting graph G:

where dikis the element of the incidence matrix of G defined in Eq.(1).Then,Eq.(12)can be reformulated as

where UG=[aT1，aT2，...，aTm]Tis the vector form of the generated control accelerations by all virtual spring-dampers in the mesh.Note that[di1，di2，...，dim]is the ith row of D（G）,and the control input to the swarm of spacecraft in vector form can be represented using the incidence matrix of the connecting topology:

where U=[uT1，uT2，...，uTn]T.The orientations of the edges of G should be consistent,i.e.,the identical digraph derived from G must be used when we specify the edge control acceleration UG,the incidence matrix D（G）,and the relative state PG.

The distributed nature of the virtual spring-damper mesh provides scalability to accommodate large fleets.Only internal forces are generated by the control law,leaving the translational and rotational motion of the formation as a whole unaffected.The required information for the formation control only includes the distance and the rate of distance change,which can be represented in a local coordinate frame of each spacecraft.Without using a common reference frame,distributed implementation is direct.No centralized computation,communication,or control is required.Spacecraft are required to exert continuously variable control accelerations,which may cause difficulties for implementation using conventional thrusters.

3.3.Closed-loop system with linearized dynamics

Note that for spacecraft formation flying problems,usually all spacecraftmove close to a reference orbitr0（t）,i.e.,ri-r0（i=1，2，...，n）is small relative to r0;therefore,the nonlinear dynamics of Eq.(7)can be linearized as

Then,the vector form of the approximated closed-loop dynamics with the virtual spring-damper mesh control is

Substitute Eq.(16)into the twice differentiated Eq.(3)with respect to time,and the full relative motion equation is obtained corresponding to all edges of the connecting graph G for the approximate closed-loop system:

4.Convergence and steady state analysis

4.1.Convergence analysis

Theorem 2.For a spacecraft swarm with the dynamics of Eq.(7)under the distance-based VSDM formation control law of Eqs.(9)and(14),the distances between the spacecraft pairs corresponding to all edges of the connecting topology G,i.e.,all lk(k=1，2，...，m),become constant values as t→ ∞.Furthermore,if the connecting topology of G is rigid,the distances between all spacecraft approach constant values as t→∞.

Proof.Consider the energy function

where Q0is a negative constant satisfying Q（ri）-Q0＞ 0 for the trajectories of ri（t）（i=1，2，...，n）.For the spacecraft formation flying problems considered in this paper,all spacecraft are close to a natural reference orbit.The gravity potentials Q（ri）are bounded during spacecraft motion.Thus,finding such a Q0is always possible.Then,substituting Eqs.(7)and(12)into the time derivative of the energy function yields the following:

By substituting Eqs.(10)and(11)into the above form,the following can be written:

Therefore,according to LaSalle’s invariance principle,26every˙lkapproacheszeroast→∞,andalllk(k=1，2，...，m)approach constant values as t→ ∞.Then,graph rigidity of G indicates that the lengths between every pair of agents are constant.Theorem 2 is thus proven.□

According to Theorem 2,the steady-state motion of spacecraft formation under distance-based VSDM control can be regarded as the motion of a ‘rigid body” as t→ ∞.The translational motion is equivalent to the orbital motion of the mass center in the gravitational field,and the rotational motion of the formation is only driven by the gravity gradient because the composition of the forces and torques generated by the virtual spring-damper mesh on the formation is zero.

4.2.Steady state errors

At steady state,the errors of the relative configuration between spacecraft with respect to the desired relative configuration can be represented by the set of distance errors corresponding to the edges of the connecting graph.Two sets of linear equations of the edge distance errors at steady state are obtained.One is derived from the linearized closed-loop dynamics of Eq.(17).The other is derived from the geometric constraints of graph rigidity when small edge distance errors are assumed.The edge distance errors are solved from the two sets of equations,and steady-state error is found to be inversely proportional to the elastic coefficient ks,when ksis sufficiently large.

4.2.1.Error equations from closed-loop dynamics

We suppose that the instantaneous angular rate of the steadystate formation(‘rigid body”)is ε（t）,the instantaneous angular velocity is ω（t）,and the instantaneous transition matrix from the ECI reference frame to a body frame of the ‘rigid body” is M（t）.Because the relative position vector between a pair of spacecraft corresponding to the kth edge,pk=lkek,is fixed in the formation rigid body,the second-order time derivative can be derived:

Additionally,the following can be obtained:

where ωk，⊥= ω - ωekek.

We construct a local coordinate frame for each edge with the origin at the position of the tail agent and the x axis toward the head agent;the y and z axes can be arbitrarily set provided that they complete the definition of a dextral coordinate framework.We suppose that the transition matrix from the body frame to the local coordinate frame associated with the kth edge is Mk(k=1，2，...，m).The transition matrices Mkare constant because all local coordinate frames are fixed in the body frame.Then,the closed-loop dynamics of Eq.(17)can be expressed using coordinates in the set of local frames as follows:

where MG=diag（M1，M2，...，Mm）and the leading superscript ‘lb” denotes the local edge coordinates of the vector.

From the definition of the local frames,As t→ ∞ atsteady state,From Eq.(19),the first coordinate inThus,extracting the equations along the x axis of the local frames in Eq.(20)yields

Eq.(21)is the relation of edge distances derived from the closed-loop formation dynamics at steady state.

Proposition 1.Let ℜ（R）denote the rigidity matrix of the formation,where R is the position vector in the inertial frame.Then,the following is obtained:

Proof.Let ℜ（R）ℜ（R）T=[ϑij]i=1，2，...，mj=1，2，...，m,then,according to Eq.(6),the following can be written:

Using coordinate transformations,pk=M（t）Mklbpkyields the following:

Thus,Eq.(23)is obtained.□

According to the theory of graph rigidity,the formation is rigid if and only if Rank（ℜ）=3n-6= κ.Additionally,m≥κ must hold for the formation to be rigid.From Eq.(23),Rank（Γ）=Rank（ℜℜT）=Rank（ℜ）= κ.

Because Γ is real and symmetric,it can be diagonalized by an orthogonal matrix Υ satisfying the following:

where σi（i=1，2，...，κ）is the nonzero real eigenvalues of Γ.

Let~LG=LG-LdGdenote the error of the edge distances.By left multiplication of the matrix Υ,Eq.(21)becomes the following:

where only the first set of equations,i.e.,Eq.(27),is relevant to the formation distance errors,~LG.Eq.(27)can be reformed in the relationship between~LGand LdGas

Note that[Iκ×κ，0κ×（m-κ）]Υ are the first κ rows of the matrix Υ,which are linearly independent.Thus,Eq.(29)provides κ linear independent equations for solving~LG.However,m unknown parameters are present in~LG.If m＞κ,then another m-κ equations are needed to solve~LG.

4.2.2.Error equations from constraints of graph rigidity

According to the theory of graph rigidity,κ=Rank（ℜ）is equal to the minimal number of independent edges in the graph.If m ＞ κ,then the distance setmust not all be independent,and m-κ geometric constraints must be exerted on the distance set{l1，l2，...，lm}.Therefore,LGcan be divided into two parts,i.e.,the part corresponding to the κ independent edges,Lκ,and the part corresponding to the remaining m-κ edges,LC.It is assumed that the edges are permuted with a proper order such thatSimilarly,LdGcan be divided into two parts,i.e.,the desired edge distances corresponding to the κ independent edges,and the desired edge distances corresponding to the remaining m-κ edges,We suppose that the vector form of the m-κ geometric constraints is

Because the geometric constraints come from the property of the rigid graph in the sense of strong generic rigidity,it is invariant to small perturbations of specific edge distances.Thus,LdGalso satisfies the geometric constraints of Eq.(30)when the steady-state errors~LGare small.Eq.(30)can then be linearized at LdGas

where~Lκ=Lκ-Ldκare the errors of the independent edge distances,and~LC=LC-LdC.Eq.(31)can be solved as follows:

4.2.3.Solution of steady-state error

Because the κ independent error equations of Eq.(29)are derived from the closed-loop dynamics of the relative motion,whereas the m-κ independent error equations of Eq.(31)originate from the properties of the rigid graph G,the two sets of equations must be mutually independent.Therefore,these equations can be used together to solve the distance errors in the independent edge set:

The instantaneous angular velocity ω（t）of the formation is assumed to be bounded.Thus,Λ-Ω is bounded.If ksis sufficiently large,then the term[Iκ×κ，0κ×（m-κ）]Υ（Λ - Ω）on the left side of Eq.(33)can be dropped,and an approximate solution for~Lκcan be obtained:

Fig.2 Hexagon pyramid formation and connecting topology.

Then,the full set of edge distance errors of G can be written as

From Eqs.(34)and(35),the error of the edge distances is inversely proportional to ks.Therefore,the steady-state error can be reduced by using a larger elastic coefficient.

4.3.Steady-state control accelerations

At steady state,the control acceleration corresponding to the k-th edge of the connecting graph is

From Eqs.(38)and(35),the approximate control acceleration for the spacecraft swarm at steady state will not be affected by the elastic coefficient ks.Thus,the distance-based VSDM control method has a desirable property that the control effort will not increase when a larger elastic coefficient ksis used to improve the precision of the formation control.

Fig.3 Distance errors of edge distances.

Fig.4 Pointing errors offormation orientation with respect to direction of center of the Earth.

Fig.5 Control accelerations of spacecraft.

5.Numerical simulations

A potential application of Earth pointing formation in low Earth orbit was simulated.The idea is to use the distancebased VSDM control to maintain the formation shape,whereas the orientation of the formation in inertial space is stabilized by the gravity gradient and kept pointing to the center of the Earth.The principle is identical to the gravity gradient stabilization of spacecraft in low earth orbit,which will not be explained here.A hexagon pyramid formation with 7 spacecraft was assumed to be the desired formation shape.The hexagon pyramid formation serves as a virtual aperture for observation,reconnaissance,and other types of missions.The direction from the center of the hexagon to the spacecraft at the peak of the pyramid was the orientation of the virtual aperture.The side length of the hexagon was 100 m.The distance from the center of the hexagon to the spacecraft at the peak of the pyramid was 500 m.The hexagon pyramid formation and the graph structure used for the connecting topology for the distance-based VSDM control are shown in Fig.2.Circles stand for the spacecraft,dashed lines stand for the VSDM connection between spacecraft,and the arrow shows the direction of the formation orientation.

In the simulation,the spacecraft were near a circular reference orbit at an altitude of 500 km.At the initial time,the inclination of the reference orbit was 30°,the right ascension of the ascending node was 60°,and the argument of the perigee and the mean anomaly were both 0°.The simulations were performed by numerical integration.Orbital dynamics with a J2perturbing potential of Eq.(11)were used.The MATLAB built-in function ode45 was used for the numerical integrations.The relative tolerance and the absolute tolerance were both set to 10-10.

Fig.6 Relative motion of spacecraft in hexagon with respect to spacecraft at peak of pyramid.

Three simulation cases with different initial states were presented to illustrate the performance.In the first case,the spacecraft were initially at the precise pyramid formation pointing at the Earth.In the second case,the spacecraft have position errors and velocity errors from the precise initial state of the first case.The initial position errors for each spacecraft were randomly generated with a uniform distribution over[-20,20]m for all three coordinates.Additionally,the initial velocity errors for each spacecraft were randomly generated with a uniform distribution in[-50no,50no]m/s for all three coordinates,where no=1.1068×10-3rad/s is the orbital angular velocity of the reference orbit.In the first and second simulation cases,the elastic coefficient was ks=0.01,and the damping coefficient was kd=0.05.In the third simulation case,all settings were identical to the second case except that ks=0.1 and kd=0.5.The simulation time for all three cases was 15000 s.

Fig.3 shows the distance errors of the edges in the connecting graph.In each sub-graph,a different curve represents the distance error of a different edge in the hexagon pyramid topology illustrated in Fig.2.Fig.4 shows the pointing error of the formation orientation with respect to the direction of the center of the Earth.In all three cases,the steady-state errors of the edge distances are small,and the formation continuously points to the Earth with a coarse precision.Larger initial errors result in larger pointing errors.Fig.5 shows the control acceleration inputs of spacecraft.In each sub-graph,a different curve represents a control acceleration input of a different spacecraft.The steady-state control acceleration inputs are on the order of 10-3for all three cases.Fig.6 shows the relative motion trajectory of the spacecraft with respect to the spacecraft at the peak of the pyramid.In each sub-graph,a different curve represents a different relative motion.The initial topology of the hexagon pyramid is illustrated using triangles and dotted lines.The final topology of the hexagon pyramid is illustrated using circles and lines.The comparison of the results of the third case and the second case in Figs.3 and 5 confirms the theoretical analysis in Section 4 that,at steady state,the error of the edge distances is approximately inversely proportional to the value of the elastic coefficient,and the control acceleration inputs are not affected by the value of the elastic coefficient.The comparison of the results of the third case and the second case in Fig.4 also shows that the elastic coefficient does not affect the pointing accuracy.

Although the steady-state control accelerations are small,long-term formation shape keeping will consume relatively large amount offuel when conventional chemical thrusters are used.However,if novel internal forces such as electrostatic Coulomb forces,electromagnetic forces,light pressure,tether tension and their combinations can be used to implement the distance-based VSDM control,no fuel consumption will be needed for maintaining the formation shape.During the transient state of establishing the formation shape,larger control accelerations are required,and the conventional chemical thrusters may be adopted.Because the period of the transient state is short,the amount offuel consumption will be acceptable.However,the details for generating the control accelerations are out of the research scope of this paper.

6.Conclusions

(1)This paper investigated a decentralized shape control method for spacecraft formation in a general gravitational potential field.A nonlinear distance-based control law with an analogy of a virtual spring-damper mesh is presented to maintain the shape of the formation,whereas we leave the formation as a whole to rotate freely.A theoretical analysis shows that if the underlying graph of the mesh is rigid,the convergence to a static shape is assured,and higher formation control precision can be achieved by increasing the elastic coefficient without increasing the control accelerations.Other favorable characteristics of the proposed method include relying only on relative measurements and requiring no common reference frame.The simulation study confirms the theoretical analysis and shows a possible application of the distance-based virtual spring-damper mesh control to maintaining the formation shape,whereas leaving the orientation of the formation stabilized by the gravity gradient for Earth pointing missions.

(2)Because the proposed method generates only virtual internal ‘forces” between spacecraft,it naturally provides an active shape control law for the spacecraft formations using propellantless internal forces such as Coulomb formation,electromagnetic formation,and photon tether formation.Distributed virtual structure in space can be constructed using these novel internalforce methods,and the proposed method can be used to maintain the rigid shape of the structure in resisting disturbances.In order to make the proposed method more concrete,further research effort may focus on the integration with specific types of internal forces.Designing and integrating with distributed control laws for the formation rotation as a whole also deserve further study.

Acknowledgments

This study was supported by the National Natural Science Foundation of China(Nos.61273351 and 61673390).

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Chen Qifeng is a professor at School of Aeronautics and Astronautics,Central South University.He received the B.S.and Ph.D.degrees in aerospace science and engineering from National University of Defense Technology in 1997 and 2003 respectively.His area of research includes coordinated control of multiple flight vehicles,spacecraft formation flying and aerospace system analysis.

Meng Yunhe is an associate professor at College of Aerospace Science and Engineering,National University of Defense Technology.He received the B.S.degree in automatic control engineering in 2000,and Ph.D.degree in aeronautical and astronautical science and technology in 2006 from National University of Defense Technology.His area of research includes spacecraft dynamics and control,and spacecraft formation flying.

Xing Jianjun is an associate professor at School of Aeronautics and Astronautics,Central South University.He received the Ph.D.degree from National University of Defense Technology.His current research interests are satellite formation flying,spacecraft orbit design,and aerospace vehicle navigation,guidance and control.

28 July 2015;revised 17 September 2015;accepted 20 June 2016

Available online 21 October 2016

Formation shape control;

Graph rigidity;

Internal forces;

PD control;

Spacecraft formation flying;

Spacecraft guidance and control;

Spring-damper mesh

Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.Tel.:+86 731 88830945.

E-mail addresses:chenqifeng@csu.edu.cn(Q.Chen),myh_world@163.com(Y.Meng),xjj@csu.edu.cn(J.Xing).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2016.09.009

1000-9361Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).