Robust distributed control of spacecraft formation flying with adaptive network topology

Author’s Accepted Manuscript Robust Distributed Control of Spacecraft Formation Flying with Adaptive Network Topology Behrouz Shasti, Aria Alasty, Nima Assadian www.elsevier.com/locate/actaastro

PII: DOI: Reference:

S0094-5765(16)31138-9 http://dx.doi.org/10.1016/j.actaastro.2017.03.001 AA6226

To appear in: Acta Astronautica Received date: 31 October 2016 Accepted date: 2 March 2017 Cite this article as: Behrouz Shasti, Aria Alasty and Nima Assadian, Robust Distributed Control of Spacecraft Formation Flying with Adaptive Network Topology, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2017.03.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Robust Distributed Control of Spacecraft Formation Flying with Adaptive Network Topology Behrouz Shastia*, Aria Alastya, Nima Assadianb a

Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran

b

[email protected] (B. Shasti), [email protected] (A. Alasty), [email protected] (N. Assadian) *Corresponding author. Office phone: +98 (21) 6616 5504, Fax: +98 (21) 6600 0021.

Abstract In this study, the distributed six degree-of-freedom (6-DOF) coordinated control of spacecraft formation flying in low earth orbit (LEO) has been investigated. For this purpose, an accurate coupled translational and attitude relative dynamics model of the spacecraft with respect to the reference orbit (virtual leader) is presented by considering the most effective perturbation acceleration forces on LEO satellites, i.e. the second zonal harmonic and the atmospheric drag. Subsequently, the 6-DOF coordinated control of spacecraft in formation is studied. During the mission, the spacecraft communicate with each other through a switching network topology in which the weights of its graph Laplacian matrix change adaptively based on a distance-based connectivity function between neighboring agents. Because some of the dynamical system parameters such as spacecraft masses and moments of inertia may vary with time, an adaptive law is developed to estimate the parameter values during the mission. Furthermore, for the case that there is no knowledge of the unknown and time-varying parameters of the system, a robust controller has been developed. It is proved that the stability of the closed-loop system coupled with adaptation in network topology structure and optimality and robustness in control is guaranteed by the robust contraction analysis as an incremental stability method for multiple synchronized systems. The simulation results show the effectiveness of each control method in the presence of uncertainties and parameter variations. The adaptive and robust controllers show

1

their superiority in reducing the state error integral as well as decreasing the control effort and settling time. Keywords: Spacecraft formation flying, Adaptive network topology, Contraction analysis, Robust control law, Adaptive control law

I.

Introduction

Formation control of multiple homogeneous or heterogeneous agents working together to achieve a common goal has been studied for several years. However, Spacecraft Formation Flying (SFF) is still one of the most challenging problems in the field of multi-agent network systems. This is due to its complexity in dynamical modeling as well as miscellaneous control objectives which must be taken into account depending on the mission requirements. In the past, the dynamical equations representing the relative translational motions of two satellites in proximity of each other used to be modeled decoupled from attitude dynamics [1-3]. However, in the recent studies the relative attitude dynamics of neighboring satellites are generally modeled considering the coupling between translational and attitude dynamics which makes it a challenging problem in the field of formation control [4-10]. Considering these couplings, in this study the high-fidelity 6-DOF relative modeling of nearby spacecraft in LEO has been utilized. In the LEO orbits, the effects of gravitational zonal harmonics as well as the atmospheric drag gravitational harmonics with
 as the dominant term, atmospheric drag, solar radiation pressure, are reflected on the translational and attitude dynamics. Due to the external disturbances such as

and etc. acting on all satellites in formation, some researches focus on the robust control of SFF [11-14]. Similarly, to overcome the parameter uncertainties in the SFF modeling, some adaptive control methods have been utilized previously [15-19]. An important approach in the context of group motion is the network topology. Usually a network topology shows how state information of an agent is considered in the control law of another agent. This network topology may be fixed or switching during the control process. The switching may be discrete or continuous. The discrete switching law changes the network

2

topology in specific times during the motion , while the continuous switching method shapes the topology constantly based on an adaptive law as we developed in this paper . Fixed or switching, at any time instant a specific network topology is confronted in which each agent connected to the others by a weighted graph Laplacian whose connectivity determines the strength of the communication. In the fixed topology, the graph representing relationship among agents in the network does not vary and the connecting weights are constant. While, in the switching topology, the weights may vary due to the change in the topology itself (discrete switching) or based on an adaptive law governs the network at each time instant (continuous switching). The latter has been investigated for the first time by Chung et al. who updated the adaptation law based on distance between neighboring spacecraft. In this adaptation law, lower the distance between two spacecraft, higher the value of weighting coupling gain is. Considering directed graph of topology makes the problem of network formation more real and also more complex. In a directed graph (digraph), weights of influence of adjacent states on each other are selected differently. These weights may be constant or varying with time considering the problem under study. However, in a digraph network the effect of two spacecraft on each other is not similar. In this study, by defining different distance-based connectivity function for each agent, dissimilar connecting weights and then directed graph network is achieved. The additional concern in the concept of spacecraft formation flying is the control. There are different kind of control objectives [4, 17, 26-29]. Decentralized control has been utilized to maintain the spacecraft tightly near each other in desired locations with their attitude coordinated. The schematics of control demand is shown in Fig. 1. As it can be seen, the satellites are desired to revolve in relative orbits around the chief spacecraft. The chief spacecraft is supposed to orbit the Earth in a reference orbit without any control on its motion. For this purpose, they are divided into some groups. Each group contains a number of satellites with a leader or leaderless which are designated to be located in specific orbits around the reference and all of them try to coordinate their attitudes toward the desired goals. Generally, the control signal in the decentralized scheme is a combination of two terms: 1) The term for tracking the desired trajectory/attitude, and 2) the diffusive coupling gains for preserving the formation shape . It is shown that the time delay in transforming data between two spacecraft in vicinity can affect both the control objectives as well as stability analysis. Combination of our precise 6-DOF modeling 3

as well as distance-based digraph of network topology and distributed control necessitate us to utilize an executive analysis to show the stability of the closed-loop system. So an input-to-state stability (ISS) based on contraction analysis is developed to show the effectiveness of our method in this paper.

z

Followers

Leader

φ

y x Fig. 1. Multiple spacecraft on concentric elliptical relative orbits in 3D space with phase angle differences.

Organization of this paper is as follow: Section II expresses some preliminaries and definitions used here. In Section III, a high-fidelity modeling of the reference orbit as well as the relative translational and attitude dynamics of each spacecraft w.r.t the reference orbit has been described. Section IV is devoted to the control process and its elements. Stability analysis is shown in section V and then the simulation results are depicted in section VI. Finally, the concluding remark are summarized in section VII.

4

II.

Preliminaries and Definitions

A. Euler-Lagrange Systems Many mechanical systems can be expressed in Euler-Lagrange (EL) form as,  + ,   +  + ,   = 

In this equation  =  , … ,   ∈ ℜ

is the generalized coordinates,  ∈ ℜ× is (1)

symmetric positive definite matrix of general inertia which is bounded i.e. ∃ as lower bound and  as upper bound such that:  ≤ ‖‖ ≤  , ∀. ,   ∈ ℜ× is the Coriolis and

centrifugal forces matrix and  ∈ ℜ is the gravitational force vector (GFV) which is

bounded as 0 ≤ !"∈ ℜ# ‖ ‖ ≤ $ . Lastly, the perturbative forces and control input of the system are ,   and  ∈ ℜ , respectively.

In EL systems such as many robotics and also spacecraft architecture, matrices  and are such

that  − 2 ,   is a skew-symmetric matrix. However, in the systems which do not have

this property, it is achievable by multiplying the above matrices from left or right with a matrix. This skew-symmetric matrix plays an important role in stability proof.

Moreover, there is a linearity in the parameters such that for all vectors ' and ( ∈ )  , and

* ∈ )+:

' + ,  ( +  + ,   = ,,  , ', (*

where , is the regressor matrix of order - × " and * is the unknown parameters vector with " (2)

elements. This regressor matrix contains the state and control variables but separates the parameters. The role of regressor matrix is to substitute the part of the system from which the of inertia as the parameters of the system * = .
 ,
 ,
// ,
/ ,
/ ,
 01 , the multiplication of parameters can be extracted linearly. For instance, in spacecraft attitude dynamics with elements

inertia

matrix

(2)

by

an

arbitrary

vector

' = .3 , 3 , 3/ 01 can

4'* (i.e. 2' = 4'*) using the following regressor matrix 3 4' = 5 0 0

0 3 0

0 0 0 3/ 3/ 3 5

3/ 0 3

3 3 6 0

be

replaced

(3)

by

B. Communication Network Topology based on Graph Theories In spacecraft formation flying similar to other multi-agent network systems, a communication network topology must be considered to show how information exchange among agents ability to show relationship among agents with an understandable architecture . Each graph 7 has

(spacecraft). For this purpose, graph theory has been utilized in this study due to its superior common elements i.e. a node set 8 =  1, . . . , -  representing number of agents in communication, an edge set ℰ ⊆ 8 × 8 which shows the existence or non-existence of

information exchange among pairs of agents and finally a weighted adjacency matrix Λ = >?@A B ∈ ℜ× whose values express the strength of communication between members. In a

network topology, - spacecraft as nodes of the graph communicate with each other through edge

sets. Usually, a positive weight is devoted to each edge as ?@A showing the effect of the states of C DE spacecraft on the control effort of F DE spacecraft. In this way, a weighted adjacency matrix Λ is defined elements of which represent the degree of relatedness between adjacent spacecraft.

Whether the node F has the same effect on node C as the effect of node C on the node F or not, the bidirectional or directional network topology may form. It is obvious that each two nodes without any relationship at a time instance would result to a zero weight, i.e. there is no edge

between them. Also, to each Λ a Laplacian matrix ℒ = >H@A B ∈ ℜ× is devoted whose diagonal element of F DE row is built by summation of all other weighting elements in the same row, i.e.

H@@ = ∑@J,@ K A ?@A and the off-diagonal terms are set to H@A = −?@A (where F ≠ C). One of the M , an - × 1 column vector of ones. Another property is the condition for the connectivity of important properties of this matrix is that it has a zero eigenvalue associated with the eigenvector

graph 7. This graph is connected if all other eigenvalues (except the zero eigenvalue) of its

Laplacian matrix would be positive. This latter property plays a significant role on the stability of closed-loop multi-agent systems.

III.

Relative Dynamics Governing Equations

In this study, the 6-DOF relative dynamics of spacecraft is modeled with respect to the reference orbit where the chief spacecraft is orbiting. For this purpose, at first, the translational dynamics of each spacecraft relative to this orbit are stated. Thus, it necessitates that the reference orbit to

6

be simulated simultaneously with the translational motion of each spacecraft. The relative rotational dynamics are then derived coupled with the translational dynamics.

A. Relative Translational Dynamics Similar to previous studies , the governing orbital motion of a leader or a virtual leader moving in the reference orbit is derived. For the sake of accuracy, a high-fidelity dynamics of the reference orbits in LEO has been considered. It is well-known that the main disturbances in LEO are the Earth oblateness (second zonal harmonics) and the atmospheric drag. The coupling effect originates from both gravitational and drag forces which have terms in translational as well as attitude dynamic equations. expressed in the Local Horizontal-Local Vertical (LVLH) frame (non-inertialNO, PO, Q̂  frame

The differential equations describing the motion of chief spacecraft in the reference orbit

attached to the chief spacecraft as in Fig. 2.) are as follows : S = TU

VTU X ℎ Z[\ = −  + / − ] 1 − 3 sin F sin b − c ‖de ‖ TU VW S S S Z[\ sin F sin 2b Vℎ = − − c ‖de ‖ ℎ − fg S  cos F / VW S 2Z[\ cos F sin b c‖de ‖ fg S  sin 2b VΩ = − − VW ℎS / 2ℎ

Z[\ sin 2F sin 2b c‖de ‖ fg S  sin F cos  b VF = − − VW 2ℎS / ℎ

Vb ℎ 2Z[\ cos  F sin b c‖de ‖ fg S  cos F sin 2b = + + VW S  ℎS / 2ℎ

7

(4)

n ƒ

‚

Ω

Deputy

„A ν

θ

ω

„

‡ O

† O

… O

Relative orbit

Chief

Periapsis

i

2‚

Nodal Line (NL)

n) and LVLH (NO, PO, Q̂ ) frames. Fig. 2. ECI (kl,
l, m

It is assumed that there is no active control on the chief spacecraft in the reference orbit. These S, TU , ℎ, Ω, F, b as shown in Fig. 2. The first three coordinated parameters are the position of equations are expressed as a combination of Cartesian states and orbital elements chief spacecraft S, its radial velocity TU , and angular momentum ℎ, respectively. The latter three

orbital parameters are the right ascension of the ascending node Ω, inclination F, and argument of

latitude b = f + o of the leader or the virtual leader, in which the f and o are the argument of perigee and true anomaly, respectively. The last three elements along with 3 as the semi-major

axis, p as the eccentricity, and f as the argument of perigee of the leader satellite orbit build the

classical orbital elements set as æ = .3, p, f, Ω, F, b0. There are also other parameters including

X = 398600.4418 .m/ /  0 : gravitational constant, )g : Radius of the Earth, Z[\ = 
 X)g = /

2.633 × 10w .mx /  0 : coefficient of
 , de : velocity of the chief spacecraft relative to the

atmosphere and fg : angular velocity of the Earth. Also c as drag constant is defined as c =

z c |  y{ 

with cy , }, , and | are the drag coefficient, cross sectional area of the chief spacecraft, 8

its mass and the air density, respectively. In this study, an exponential density model for the atmosphere is adopted as follows:

| = |w p .Eˆ ‰E0/Š

where |w is the atmosphere density at the reference altitude, ℎw and ℎ are the reference and actual (5)

altitude and ‹ is the scaled height. The values of |w , ℎw , and ‹ in different altitude ranges can be

found in.

The gravitational potential function of the reference orbit in presence of
 perturbation can be

written as

X Z[ 1 Œ = − − /\  − sin Ž S S 3

(6)

where Ž is the geocentric latitude of the chief orbit defined as sin Ž = sin b sin F = S /S, in

which ‘ is the third component of chief spacecraft position vector in ECI frame (Fig. 1) with

components in LVLH frame as:

n = sin F sin b … ’ + sin F cos b † ’ + cos F ‡O ƒ

(7)

„ = −“Œ + 'y”e•

(8)

Expressing dynamic equations in presence of second zonal harmonics and the atmospheric drag would result in

In this equation the drag acceleration is define as 'y”e• = −c‖–e ‖ –e where –e as said earlier is the relative velocity vector of the leader in reference orbit to the rotating atmosphere expressed in LVLH frame

–e = —˜™š→œœŠ „ ˜™š − žg × „˜™š 

—˜™š→œœŠ is a 3-1-3 rotation matrix transformation from ECI to the LVLH frame defined as —˜™š→œœŠ

In above matrix,

U

Ÿ  Ÿ¡ −   Ÿ@ ¡ = 5−   Ÿ¡ − Ÿ  Ÿ@ ¡ @ ¡

Ÿ  ¡ +   Ÿ@ Ÿ¡ −   ¡ + Ÿ  Ÿ@ Ÿ¡ − @ Ÿ¡

(9)

  @

Ÿ  @ 6 Ÿ@

and ŸU are standing for the sin N and cos N, respectively.

(10)

Left hand side of Eq.(8) needs angular velocity components of LVLH frame which can be stated in terms of orbital elements as

9

fU = VF ⁄VW Ÿ  + Ω   @ ¢f¤ = −VF ⁄VW   + Ω Ÿ  @ = 0 f = b + Ω Ÿ@ = ℎ/S 

(11)

n and b ‡O in which the projection of the first term (in nodal line § , Ω ƒ of rotation are VF ⁄VW¥¦ In this equation, the second component of the frame’s angular velocity is zero, because sources

’ direction. It should be noted that direction. Moreover, the third term has no component in the † direction as it can be seen in Fig. 2) is canceled by the second term projection on the tangential

fU and f are the steering and orbital rate of the LVLH frame, respectively.

’ and then differentiate it twice in that Writing the chief position in the LVLH frame as „ = S … frame, the following equations are yield:

’ + Sf † ’ „ = S …

(12)

ℎS  − 2SS ℎ S]

(14)

’ + 2S f + Sf   † ’ + SfU f ‡O „ = S − Sf  …

The derivative of f = ” \ with respect to time is E

f  =

(13)

Finally, substituting f and f  and letting S = TU , by using Eqs. (6)-(14), the Eq.(8) can be rewritten after simplifications as ℎ Vℎ/VW fU ℎ ’+ ’+ ¨TU − / © … † ‡O S S S ª«««««««««¬«««««««««­ „

Z[ @   Z[ @   X Z[ ’+ \ ] † ’ + \ ] ‡O© − c‖–e ‖ ª««««««««««««¬««««««««««««­ = − ¨®  + ]\ ¯1 − 3 @   °± … —˜™š→œœŠ .TU Sf − Sfg Ÿ@ Sfg @ Ÿ  01 S S S S ª««««««««««««««««¬««««««««««««««««­ –³ ª«««««««««««««««¬«««««««««««««««­ “²

(15)

'´µ³¶

Equating both sides of Eq.(15) leads to three equations for T , ℎ and fU . Combining these

equations with angular velocities of Eq.(11) results in six differential equations of chief orbit as in Eq.(4). Similarly, with second zonal harmonic perturbation and the atmospheric drag considered in the modeling the relative translation of each spacecraft (known as deputy) moving with respect to the chief spacecraft orbit, the following equations in LVLH frame can be found:

10

NA = 2PA f − NA ¯·A − f ° + PA ¸ − CfU f − ¯¹A − ¹° sin F sin b − S¯·A − · ° − cA ºdeA º¯NA − PA f °

− ¯cA ºdeA º − c‖de ‖°TU + !,A + »gUD,A

PA = −2NA f + 2QA fU − NA ¸ − PA ¯·A − f − fU ° + QA ¸U

− ¯¹A − ¹° sin F cos b − cA ºdeA º¯PA + NA f − QA fU °

(16)

ℎ − ¯cA ºdeA º − c‖de ‖°  − fg S cos F + !,A + »gUD,A S

QA = −2PA fU − NA fU f − PA ¸U − QA ¯·A − fU ° − ¯¹A − ¹° cos F

− cA ºdeA º¯QA + PA fU ° − ¯cA ºdeA º − c‖de ‖°fg S cos b sin F + !/,A + VgUD/,A

In these equations, the subscript C stands for the deputy spacecraft and terms without this subscript show the corresponding value for the reference orbit of the chief spacecraft. The symbols used in these equations are : ¹ =

¼½\ ¾¿À @ ¾¿À   ”Á

and ¹A =

¼½\ ”Âà ”ÂÄ

effect on satellites moving toward the pole in the meridian and · = ” Æ + and ·A = ” Æ + Å Â

¼½\ ”ÂÄ

−

\ x¼½\ ”ÂÃ

”ÂÇ

Å

which reflect the
 ¼½\ ”Ä

−

x¼½\ ¾¿À\ @ ¾¿À\   ”Ä



as the results of two-body gravitation as well as
 in the radial

 n= direction toward the Earth center. Also SA = ȯS + NA ° + PA + QA and SAÉ = „A . Ê

¯S + NA ° sin F sin b + PA sin F cos b + QA cos F are the deputy spacecraft distance to the Earth

center and its projection on the polar direction, respectively.

It should be noted that  ¸U , ¸¤ , ¸  in Eq.(16) are the LVLH frame’s angular accelerations

obtained directly by differentiating angular velocities  fU , f¤ , f . Additionally, cA and deA are defined for the deputy spacecraft similarly to c and de for the chief spacecraft.

Finally, to achieve the results of Eq.(16), the same form of Eq.(8) for the deputy spacecraft is utilized with the following change in the potential energy ŒA = −

X Z[\ 1 −  − sin ŽA  SA SA/ 3 11

(17)

with sin ŽA = SAÉ /SA , which its components are defined as for the chief spacecraft. Eq.(16) can be written in Euler-Lagrange form as

A,D” W A,D” + A,D” W A,D” + A,D” ËA,D” , æWÌ + A,D” ËA,D” ,  A,D” , æWÌ = A,D” + ÍgUD ,D”

where

A,D” = A / , A,D”

0 = 2A 5f 0

−f 0 fU

NA ¯·A − f ° − PA ¸ + QA fU f + ¯¹A − ¹°

A,D” = A Î

A,D”

@  

0 −fU 6 0

+ S¯·A − · °

NA ¸ + PA ¯·A − f − fU ° − QA ¸U + ¯¹A − ¹° @ Ÿ  NA fU f + PA ¸U +

QA ¯·A

−

fU °

+ ¯¹A − ¹°Ÿ@

Ï

(18)

(19)

−cA ºdeA º¯NA − PA f ° − ¯cA ºdeA º − c‖de ‖°TU Ò Õ ℎ Ñ Ô = A Ñ−cA ºdeA º¯PA + NA f − QA fU ° − ¯cA ºdeA º − c‖de ‖° − fg S Ÿ@ Ô S Ñ Ô ‖°f −c ºd º¯Q + P f ° − ¯c ºd º − c‖d S Ÿ Ð Ó A eA A A U A eA e g   @ A,D” is the translational control force on the C DE spacecraft. The external disturbances ÍgUDÂ,D”

consist of aerodynamics torques as well as gravity gradient as a function of orbital motion of spacecraft, which are modeled through standard formulations.

B. Relative Attitude Dynamics

Modified Rodrigues parameters (MRPs), „Ö× = Ø = ÙU , Ù¤ , ِ , are utilized as the generalized

coordinates to represent rotational kinematics of spacecraft flying in formation. Analogous to the translational equations, MRPs are stated relative to the leader satellite LVLH frame. These parameters are preferred to the Euler angles and also to the quaternions due to their nonsingularity in a large domain of space i.e. .−2Ú, 2Ú0 as well as no need to check normality as it

must be done in case of quaternions. In this way, attitude kinematics would be

12

Ø = ÊØž„Ö×

(20)

1 1 − Ø1 Ø ÊØ = ہ/ ¨ © + ØØ1 + ÜØÝ 2 2

(21)

in which the kinematic matrix is

In this relation, subscript ′SßW′ is added to the relative angular velocity of each spacecraft with previous section. Also matrix Ü.  stands for the skew-symmetric matrix as

respect to the reference orbit to distinguish it from the angular velocity of LVLH frame in the 0 ÜØ = à ِ −Ù¤

−ِ 0 ÙU

Ù¤ −ÙU á 0

(22)

Relative attitude dynamics of C DE spacecraft can also be stated in Euler-Lagrange form as A,”âD ¯A,”âD ° A,”âD + A,”âD ¯A,”âD ,  A,”âD ° A,”âD

‰1 = A,”âD + ÍgUDÂ,”âD + Ê A ܯãAä °åæây¤Â ª««««¬««««­ DE”çèDg” âéé‰êgDg”

(23)

where Ø, Ø , Ø  are substituted by A,”âD ,  A,”âD ,  A,”âD  to harmonize with the equations of

relative motion. The last term shows that the point of effect of the thrusters are off-centered from the center of mass with a distance ãAä . Other parameters are defined as follows: A,”âD ¯A,eD ° = ÊA‰1 2èê ÊA‰

A,”âD ¯A,”âD ,  A,”âD ° = −ÊA‰1 2èê ÊA‰ ÊA + Ü Ë2èꠞA,”âD ̏ ÊA‰ A,”âD = ÊA‰1 ′A,”âD

ÍgUD ,”âD = ÊA‰1 Í′gUD ,”âD

(24)

As it can be seen, both sides of the above equations are multiplied by ÊA‰1 to the left side of all

terms, even real external torques and disturbances i.e. ′A,”âD and Í′gUD ,”âD , to compensate for the skew-symmetry properties mentioned in section II-A. Moreover, ÊA can be easily evaluated

considering relation ÊA ÊA‰ = /. 2èê is the inertia matrix of the C DE spacecraft.

13

Finally, by combining relative translational and rotational dynamics into one equation, the following Euler-Lagrange formed equations are yield:

 + ,   +  + ,   = å + ÍgUD

(25)

 is the generalized coordinate including the translational as well as the rotational states in the

ë ë , ,”âD , … , ëA,D” , ëA,”âD , … , ë,D” , ë,”âD B , where - is the number of spacecraft in form of >,D” ë

formation. The other parameters are defined as  = ì

,   = ì

,D” ,D”  í/×/ î ⊕ … í/×/ ,”âD ,”âD  A,D” ¯A,D” ° ⊕ ® í/×/

í/×/

A,”âD ¯A,”âD °

± ⊕ …

,D” ¯,D” ° í/×/ ⊕ ® ± í/×/ ,”âD ¯,”âD °

,D” ,D” ,  ,D”  í/×/ î ⊕ … í/×/ ,”âD ,”âD ,  ,”âD  A,D” ¯A,D” ,  A,D” ° ⊕ ® í/×/

í/×/

A,”âD ¯A,”âD ,  A,”âD °

(26)

± ⊕ …

,D” ¯,D” ,  ,D” ° í/×/ ⊕ ® ± í/×/ ,”âD ¯,”âD ,  ,”âD °

where the direct sum operator is utilized to constitute block diagonal matrices from square matrices as ð ⊕ ñ = VF3òð, ñ. and

ë ë  = > ,D” , í×/ , … , A,D” , í×/ , … , ë,D” , í×/ B

ë

ë ë

,   = > ,D” , í×/ , … , A,D” , í×/ , … , ë,D” , í×/ B

å=®

/×/ ʉ1 ܯã,ä °

í/×/ /×/ ± ⊕ … ⊕ ® ‰1 /×/ Ê ܯã,ä °

í/×/ ± /×/

ë ë ë ë  = >,D” , ,”âD , … , A,D” , A,”âD , … , ë,D” , ë,”âD B

14

ë

ë

(27)

IV.

Control Design Procedure

In this paper, the behavioral control of the formation is included using two different terms: 1) the single-platform control term which keep each spacecraft tracking a predefined time varying desired position and attitude, and 2) the group-formation control term which control the position and attitude of spacecraft relative to each other in the formation. For every mission to be accomplished in the field of SFF, these two behaviors should be considered at the same time. In order to control the relative states of spacecraft, the position and attitude of each one are controlled with respect to the reference orbit in which the chief spacecraft is orbiting the Earth. To do so, equations of relative translation and attitude are written with respect to the chief orbit.

A. Adaptive Synchronization Control Law The adaptive synchronization control law is utilized throughout this paper and its structure is introduced in this section. This controller consists of a feedforward term compensating the modeling parameters along with the single-platform and group-formation control terms as the elements of the behavioral control as follows [48-50]: nA æ ∗A + n A  ∗A + nA ¯A∗ ,  ∗A , æ° − A =  −

÷ A ¯ô, |A °øA ª««¬««­

è@•ög‰+öeDéâ”{ êâD”âö Dg”{

•”âç+‰éâ”{eD@â êâD”âö Dg”{

Zõ A ôA

(28)

n and n, n are the estimated matrices of their counterparts in (25) for the cases that the where  n as the estimating parameters vector, regressor matrix can be obtained as and by choosing ù

parameters are unknown or time-varying. Based on the final property mentioned in section II-A

where

nA æ ∗A + nA n A  ∗A + nA ¯∗A ,  ∗A , æ° ≜ ûA ¯∗A ,  ∗A ,  ∗A , æ°ù 

ü ‰1 ý4¯Ê ü ‰  ∗A ° + 4ËÊ ü ‰  ∗A Ì + Ü¯Ê ü ‰  ∗A °4¯Ê ü ‰  ∗A °þ ûA ¯A∗ ,  ∗A ,  ∗A , æ° = Ê

(29)

(30)

In the above equation, the matrices 4.  and Ü.  are defined in Eqs. (3) and (22), respectively. ü matrix can be stated as: Also the Ê

 ü = ì/×/ í/×/ î ⊕ … ⊕ ì /×/ Ê í/×/ í/×/ Ê

í/×/ /×/ í/×/ ÊA î ⊕ … ⊕ ìí/×/ Ê î 15

(31)

Substituting Eq. (29) in (28) yields

nA − ZA ôA − ÷A ¯ô, |A °øA A = ûA ¯∗A ,  ∗A ,  ∗A , æ°ù

(32)

In these equations, ôA =  A −  ∗A is the sliding manifold where ∗A is the time-varying reference

trajectory whose derivative is defined as  ∗A =  y,A − Λ ËA − y,A WÌ. Also, y,A W is the desired trajectory for each spacecraft which can be the same for all agents or not, depending on

the mission requirements to be established by their collaboration. The first three terms in the control law formula (28) are feedforward control which are directly the functions of the dynamic states of the system. The last two terms are included to achieve the goals of control. The first of which, ZA ôA , is the single-platform control term in which each spacecraft uses the information of

its own desired trajectory and its derivatives (if available) to track a desired trajectory. The final

term, ÷A ¯ô, |A °øA , plays the role of formation keeping. For this purpose, each agent receives the information (containing state and/or it derivative) of all adjacent spacecraft whose connection is established by a predefined graph of network topology. This information are gathered into an

- × Z matrix ÷A and multiplied by a Z × 1 vector øA as control gains. It is assumed that the numbers

of

connection

is

now

fixed

at

Z.

By

definition,

matrix

÷A ∶= > |A ô ⋯ |A,A‰ ôA‰ ôA |A,A ôA ⋯ |A¼ ô¼ B and vector øA ∶= > ŸA ⋯ ŸAA ⋯ ŸA¼ B. It

should be noted that the elements of vector øA are containing both the tracking control and

diffusive coupling gains for the C DE spacecraft resembling tracking and formation control parts,

respectively. It should be also noted that ŸAA in this vector does the same task that ZA in the single-

platform control term does. However, it is required to exist in øA to avoid Laplacian matrix from

singularity and therefore instability of the closed-loop system. Moreover, |A@ is the distance-

based connectivity function that mentioned for the first time in and then utilized by in the subject of SFF. Based on their idea, in a graph of network topology, the neighboring in agent territory is defined based on the distance between adjacent agents. It works in such a way that the less distance between two agents leads to more effectiveness in graph of network topology and vice-versa. This simple idea works well in this study, because the spacecraft are always maneuvering and neighboring zone varies with time. Therefore, defining a fixed network topology with no earlier knowledge about the neighboring agents, as it is usually used in group of robot motions, would not result in systematic reconfiguration and fuel consumption 16

optimization. Thus, is this study the structure of the network topology is changing regularly based on the arrangement of the spacecraft in the network. This distance-based connectivity is F VA@ ≤ V¿ ¿  \ ‰” \ Ì  Ëy |A@ V = ¢  1+p F VA@ > V¿ ¿  0

defines as

1

(33)

In the above connectivity functions, VA@ stands for the distance between agents as VA@ ≔

º„A − „@ º where „A is defined as the distance of the C DE spacecraft to the Earth center as already been introduced. If the distance is less than a certain value V¿ ¿ , the connectivity |A@ appears

and changes from 1 to 0 in this boundary. Out of this zone, the connectivity is vanished. Two

parameters  and Sê as the inclination and critical radius play an important role in this function.

As Fig. 3 shows, the parameter  determines how fast the connectivity function would change from 1 for spacecraft in close vicinity of each other to 0 in far distances and the shifting happens

ji

ji

in critical radius, Sê .

Fig. 3. Role of  (part a) and Sê (part b) in distance-based connectivity function

17

As it is said earlier, there are several control objectives in subject of formation flying. It is obvious that the optimality is generally desirable in control algorithm design. Furthermore, due to the source of uncertainties like external disturbances or actuator/sensor faults, a robust controller should be considered. In addition, to overcome the parameter uncertainties and variations, an adaptive controller is developed. For each type of control, a standard structure is developed and utilized in this paper as follows.

B. Distributed Optimal Formation Control Indeed, the main part of the control containing single-platform and group-formation control terms are developed in an optimal manner üA + A@ = A =  â+

•”âç+‰éâ”{eD@â êâD”âö Dg”{

1 − ¯1 − ¸A °mA ôA ª««««¬««««­ 2

 m A,@  ¸ A −  ô 2 A A@ @∈Â

è@•ög‰+öeDéâ”{ êâD”âö Dg”{

(34)

üA as the single-platform control term and A@ as the group-formation control As it can be seen, 

term have the similar structure as introduced in (28). The first term utilizes the desired trajectory

of its own and the second term is dependent on the state of the neighbors whose connection is determined by a predefined network topology . In this equation, the parameter 0 ≤ ¸A ≤ 1 plays

an important role in distributing appropriate weights to the two specific parts of the control

efforts. The smaller the value of ¸A , the less emphasis is put on the state synchronization over the single-platform control requirement and vice-versa. Also mA and mA,@ are the control parameters

which are computed such that the following performance index (PI) is minimized:  1 1 üA1 )A  üA +  ôA@1 A@ ôA@ á ℐA =  à ôA1 A ôA +  2 4 w @∈Â

(35)

where )A is a symmetric positive definite matrix and A and A@ are symmetric positive semidefinite matrices representing the control and state weighting factors in PI, respectively. With

this control in hand, not only the single-platform and group-formation control goals as two main goals are achieved, but also a Linear Quadratic Regulator (LQR) type of optimality is satisfied by the selection of proper control gains of Eq. (34). These set of control gains can guarantee that

18

all the spacecraft in a neighborhood would synchronize their states and follow the desired position and attitude by satisfying the Hamilton-Jacobi-Bellman (HJB) equation [20, 52]: −

A ¯W, ôA ° üA ) = min ΦA (W, ôA ,  ü  W

(36)

üA ) and A (W, ôA ) are the Hamiltonian and the value function, respectively. This where ΦA (W, ôA , 

üA for minimizing the HJB equation should be satisfied to find a smooth control input 

performance index of (35). For this purpose, the Hamiltonian is defined as: üA ° = ΦA ¯W, ôA , 

A ¯W, ôA ° 1 üA1 )A  üA + ôA1 A ôA + ôA 2

1 1 +  ¯ôA − ô@ ° A@ ¯ôA − ô@ ° 4

(37)

@∈Â

and the solution of the following nonlinear Riccati equation for A ¯ôA ° satisfies the corresponding HJB equation:

A ¯ôA ° + A ¯ôA °A ¯ôA ° + 1A ¯ôA °A A ¯ôA ° + A +  A@ @∈Â

(38)

− A ¯ôA °A ¯ôA °)A‰ × A1 ¯ôA °A ¯ôA ° = 0

where A , A@ , and )A are defined as in (35) and the other parameters determine the control input

function. It is shown in [20] that by introducing the value function as A ¯W, ôA ° =  ôA1 A (ôA )ôA , 

the following distributed control law for the C DE system is the solution of the HJB equation:

where

A@

A ¯W, ôA ° 1 Aâ+ = − )A‰ A1 ¯ôA ° ® ±+  2  ôA

is chosen such that A A ∑@∈Â

A@

@∈Â

A@ ô@

(39)

= 1/2 ∑@∈Â A@ , is an optimal controller in the

sense that it minimizes the PI in (35). Finally, by selecting the appropriate matrices for A , A@ , and )A in terms of the control gains mA and mA@ , the establishment of (34) from (39) is fulfilled

[20].

19

C. Distributed Adaptive Formation Control of Uncertain Spacecraft Systems In this subsection, it is assumed that the parameters of system are constant but there is no a priori knowledge on these parameters. Thus, a feedforward controller coupled with an adaption law is augmented to the optimal control effort in the previous subsection to compensate for the effects of the parametric uncertainties in spacecraft system as

üA A = A + ,A ¯A ,  A , ôA , ôA °ù â+

üA = −!"A ,A1 ¯A ,  A , ôA , ôA °ôA ù

(40)

üA denotes an estimate of the uncertain constant parameters of the C DE In these equations, ù

spacecraft system. The matrix !"A is a diagonal positive definite matrix in the parameter adaptation law whose elements determine the degree of stability of the closed-loop system. This is further elaborated in the next section.

D. Distributed Robust Synchronization Control of Uncertain Spacecraft Systems In developing the adaptive control law, no a priori knowledge of the constant parameters of the system was available. However, it is supposed in this subsection that certain pre-knowledge of

nA ) does exist. Nevertheless, the exact values of these the nominal values of the parameters (ù parameters (ùA ) are not known precisely and the system’s parameters could be time-varying as well. To find a robust control in these circumferences, usually the parameters are assumed to be #A º = ºùA − ù nA º ≤ ?A where ?A > 0 is a known constant. bounded as ºù

In this manner, the control law is composed of the optimal control structure augmented with the following robust control law to compensate for the effects of parametric uncertainties nA + $A ° A = Aâ+ + ,A ¯A ,  A , ôA , ôA °¯ù

−?A ò-¯,A ôA ° F º,A ôA º > %A $A = ¢ ?A − ,A ôA F º,A ôA º ≤ %A %A

(41)

where %A is a small positive constant whose values determine the depth of the sliding surface.

This parameter has a critical role in region of stability of the closed-loop system. ?A is also a positive constant which determine the speed of convergence to the sliding surface. The selection 20

of these two parameters is problem-dependent and should be selected by the designer in each case.

V.

Stability Analysis

In this section, the stability conditions of the proposed control law are investigated. Considering the optimal control introduced in Part IV.A and rewriting the control effort in (28), the following control law is utilized

1 nA æ) ∗A + n A  ∗A + nA ¯A∗ ,  ∗A , æ° − ZA ôA − ¯1 − ¸A °mA ôA A =  ª««««« 2«¬««««««­

è@•ög‰+öeDéâ”{ êâD”âö Dg”{è

¸A mA,@ −  ô − ÷A ¯ô, |A °øA 2 A A@ @∈ ª««««««««¬««««««««­ Â

(42)

•”âç+‰éâ”{eD@â êâD”âö Dg”{è

Combining the single-platform control terms into one expression and also considering only two adjacent neighbors of each spacecraft in the first group-formation control term, the control law can be written in the following form: nA æ) ∗A + n A  ∗A + nA ¯A∗ ,  ∗A , æ° − Z ôA + A =  − ÷A ¯ô, |A °øA

Z (ô + ôA,A ) 4 A,A‰

(43)

where Z and Z are substituted as tracking and diffusive coupling gains for simplicity. It is required to introduce two adaptation laws that make the system stable globally. One of these laws is assigned to adapt the diffusive coupling gains of control input as øA = &A ÷A1 ôA − &A Üê øA

(44)

It is worth mentioning that the above adaptation law benefits from preserving the configuration

of the formation during the transition. In the above equation, &A and Üê are the diagonal positive definite and semi-definite matrices for the C DE spacecraft, respectively. The elements of Z × Z

matrix Üê is positive constants if the elements of vector øA are limited to a maximum allowable value of 'ŸA@ '{eU ; otherwise, they are set to zero. It should be noted that the tracking elements of 21

øA i.e., ŸAA , are not appearing in the above equation and they are selected such that the Laplacian matrix is connected as

¼Â

ŸAA =  |A@ 'ŸA@ '

(45)

@J,@KA

Similarly, ZA is the numbers of spacecraft in neighbor of the C DE spacecraft whose states are affecting the behavior of this spacecraft. Eq.(44) is interpreted as a smoothing coupling On/Off law . The first part of equation (44) determines the rate of change in the value of diffusive coupling gain. However, whenever the neighborhoods of a specific spacecraft is getting empty of another spacecraft e.g. the F DE spacecraft, the second term lets the corresponding coupling gain (ŸA@ ) exponentially drop to zero by an adaptation law as ŸA@ = −ÙA@

ê A@ ŸA@ .

The next adaptation law is defined to deal with the parameter uncertainties as n A = −!A ,A1 ôA − !A Ü n ù n ù Â A

(46)

Again !A and ܠn are the diagonal positive definite and semi-definite matrices of order " for the

C DE spacecraft, respectively. It should be noted that the selection of the elements of these matrices is dependent on the problem and also on the extreme values of the estimating parameters. Similarly, the " × " matrix Ü n is composed of the positive or zero elements when the elements

nA are limited by 'b‚A@ ' of vector ù or not. {eU

Theorem 1: The proposed control law in Eq. (43) coupled with two adaptation laws in Eqs.(44) and (46) globally exponentially synchronizes the states of spacecraft formation flying system and exponentially converge them to an error ball around desired state in confront of the external disturbances and parameters uncertainties. Proof: The proof of Theorem 1 is provided in Appendix A.

VI.

Simulation Studies

In order to validate and compare the results of the controllers developed in the previous sections, some numerical simulations have been done. In these simulations, 8 spacecraft are initially 22

distributed near each other. Then, they are commanded to divide into two groups; the first 3 spacecraft move to a relative elliptical orbit around the chief spacecraft with semi-minor axis of

0.56 m and semi-major axis of 1.12 m, and the next 5 spacecraft move to another relative orbit with semi-minor axis of 1.12 m and semi-major axis of 2.24 m in the same plane.

Furthermore, they are arranged to control their attitude toward a desired direction as follows: b = 0.3 sin(0.002ÚW)

Ž = 0.2 sin(0.004ÚW + Ú/6)

(47)

)=0

These three command Euler angles with the rotation order of 3-2-1 are in radians and are transformed to desired MRPs for the purpose of control. This can be used for a surveillance space mission in which the spacecraft are designed to track a common attitude while they are divided into two groups in specific orbit to increase the domain of surveillance. The spacecraft would reach the goal through synchronizing their attitude along with revolving in their predetermined orbits. For the simulation purposes, the initial conditions for the reference orbit are required. The initial semi-major axis, eccentricity, inclination, the right ascension of the ascending node, argument of periapsis, and the true anomaly are chosen as 3 = 6978.137 Z, p = 0, F = Ú/4, Ω = Ú/6, f = Ú/18, and o = 0, respectively.

For two periodic relative orbits, initial radial distances to the reference orbit are a normal

distribution (+, Ù  ) with mean value + = .Nw , Pw , Qw 0 = .0.5,0.5,0.250m and variance

Ù  = 0.25 m in all directions. Also N w , P w and Qw as the initial relative velocities of both orbits relative to the reference orbit are chosen to satisfy three important requirement in the

context of swarm motion. First of all, to reduce the drift rate of the swarm, the period-matching (energy-matching) condition has been adopted. Then, to avoid collision among spacecraft especially in initial motion, the condition making passive relative orbits (PROs) to be concentric is used. Finally, the condition to avoid growth in the amplitude of the cross-track motion has been utilized. These conditions have been combined into the Hill-Clohessy-Wiltshire (HCW) initial conditions to result in

23

1 N w = f Pw , P w = −2f Nw 3-V Qw = −f Qw tan bw 2

(48)

where f is introduced in Eq. (11) and bw is the initial argument of latitude. It is worth

mentioning that these conditions are rigorously valid for the linearized circular Keplerian orbits, but they are good approximations for slightly eccentric and perturbed LEO orbits. The initial angular positions are chosen randomly between 0 and 0.5 (S3V) and the initial angular velocities are selected randomly in the interval ±0.05 (S3V/ pŸ) for conducting the

simulations. The physical parameters are the nominal mass  = 90 Zò, the moments of inertia

matrix
èê = .30 5 4; 5 15 4; 4 4 200, the effective cross-sectional area of each spacecraft

} = 1  with a sphere shape with drag coefficient c0 = 2.0. The distance to the center of the mass of each spacecraft for the thruster off-center is considered as ãAä = .0.1; 0.1; 0.10.

The performances of three proposed controllers have been evaluated. The common term in all three controllers is the formation-keeping term built from the distance-based connectivity function. Three parameters of this function are chosen carefully as  = .2; 1.50, Sê = .1.5; 20 and

Vö@{@D = .3; 40 for the first and second desired orbits, respectively.

A. Distributed Optimal Formation Control The translational tracking control gains are chosen as mA = 3 and mA@ = 1 for all spacecraft going to the first or second orbits. The values for rotational tracking control are opted as mA = 62.5 and

mA@ = 25 for both orbits. These results are obtained to minimize PI of Eq.(35) via the following weighting matrices: )A = VF3ò.k/ , 1.x k/ 0, A = VF3ò(.0.125,0.125,0.125,62.5,62.5,62.5]), 

and ∑@∈Â A@ = VF3ò([0.5,0.5,0.5,25,25,25]) for the C DE spacecraft. The weighting matrices are selected in a way to diminish the oscillation of controlled motion. Moreover, the parameter

¸A is opted as 0.5 to put equal weights both on the tracking and formation-keeping control costs.

For each spacecraft C, only two adjacent neighbors i.e., F ∈ {1,2}, are considered to interchange

state data between and play a role in the control command of Eq. (34). Because there is the other controller with the same goal working parallel with this control to satisfy the mission requirements and it is the distance-based connectivity function controller. The key role of 24

controller here is to guarantee the connectivity in the graph of network topology to be established and therefore the close-loop system would be stable. For the connectivity based controller, the

initial values of all diffusive coupling gain ŸA@ are set to zeros. Furthermore, the diagonal elements of matrix &A in adaptation law of coupling gain are selected as ÙAA = 0.03 and (Ù@@ )A = 0.01 (for F = 1, 2, ⋯ , 8 and F ≠ C).

By applying this type of controller to the system of 8 spacecraft initially placed near each other, as it can be seen in Fig. 4, they would gradually synchronize their error distance and then go to the desired orbits. These results can also be seen in Fig. 5 which reflects the synchronization plus tracking of different spacecraft attitudes relative to the desired trajectories. The optimal control forces and torques acting on each spacecraft are shown in Figs. 6 and 7, respectively.

To show the necessity of the distance-based connectivity function, i.e., |A@ , the time variations of

this function for the 4DE spacecraft is shown in Fig. 8. Also in this figure, the tracking and

diffusive coupling gain changes due to the distance-based connectivity effect have been

displayed. As this figure depicts, the numbers of spacecraft in the neighborhood of this spacecraft may change several times even in one maneuver. At the start of the maneuvers, almost However, after initial transition phase, only the spacecraft 1 and 8 keep connected and other all spacecraft have connectivity due to the deployment of all spacecraft in vicinity of each other.

spacecraft loss the interaction and therefore the state data transmission with the 4DE spacecraft.

Anyways, after the W = 1000 s the connectivity with the 8DE spacecraft is weakened due to

increase in its distance with 4DE spacecraft. At the same time, the 5DE spacecraft goes into the boundary domain and preserve the numbers of neighborhood. This shows the effectiveness of the

distance-based connectivity function as a well-organized method for synchronization of spacecraft system. Because the elements of this system persistently change their position and the need for such a continuous varying network topology seems crucial. It should be noted that if

there is no connectivity method, all ŸA@ coefficients may have values regardless of the relative

distances between spacecraft, which is neither possible nor economical due to the extensive communication load. On the other hand, with connectivity function in hand, the unnecessary long-distance communications can be eliminated.

25

Crosstrack [km] Fig. 4. Initial and desired positions and the transfer trajectories of the spacecraft using optimal control method.

26

Fig. 5. Desired (solid line) and controlled attitude angles of the spacecraft using optimal control method.

27

Force Efforts of Optimal Control (in KN)

0.1

u 1tr [KN]

0.05 0 -0.05 -0.1 0

20

40

60

80

100

120

140

160

180

200 SC1

0.1

SC2

u 2tr [KN]

0.05

SC3 SC4

0

SC5 SC6

-0.05

SC7

-0.1 0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

0.1

u 3tr [KN]

0.05 0 -0.05 -0.1

t(s)

Fig. 6. The control force effort of the spacecraft using optimal control method.

28

SC8

u 1rot [N.m] u 2rot [N.m] u 3rot [N.m]

Fig. 7. The control torque effort of the spacecraft using optimal control method.

29

Variation of the connectivity for the 4 th spacecraft

connectivity

4k

1

0.5

0 0

500

1000

1500

tracking and diffusive coupling gains SC

0.1

c4k

SC

Gains

SC

4k

SC

0.05

SC SC SC SC

0 0

500

1000

1 2 3 4 5 6 7 8

1500

Fig. 8. Variation of the connectivity and diffusive coupling gains weighted by the connectivity for the 4DE spacecraft (F = 4)

B. Distributed Adaptive Formation Control of Uncertain Spacecraft Systems In this case, while there is no knowledge on the physical parameters of system in hand, it is assumed that these parameters are constant. Therefore, the same nominal values are considered

for the masses and moments of inertia of the spacecraft as mentioned before. Almost 10% error in physical masses is considered as  = [80; 75; 90; 95; 100; 80; 75; 85] Zò and up to 50%

error in principal moments of inertia as in Table 1.

30

Table 1. The moments of inertia of the 8 spacecraft (in mò.  ).

1

2MM 30

233 15

244 19

2M3 5

2M4 4

234

2

28

14

17

5

4

4

3

32

19

16

5

4

4

4

35

22

20

5

4

4

5

36

20

22

5

4

4

6

30

17

17

5

4

4

7

26

17

15

5

4

4

8

30

18

16

5

4

4

Spacecraft

4

The gain matrix in the parameter adaptation law is composed of a diagonal 7 × 7 matrix as: !"A = VF3ò(5p2,1p5 1 ). Only one mass and six moments of inertia build the parameter

distribution of the gain matrix. Considering above assumptions, Figs. 9 and 10 show the position and attitude synchronization along with tracking to the desired trajectories, respectively. Note that the results are simulated with the same initial condition as the previous scenario for comparison. As it can be seen in Fig. 10 compared to the Fig. 5, the settling time of synchronization has been increased relative to the previous subsection results. It is because the time to estimate the values of the physical parameters has been added to the time of synchronization in the present case. With the same reasoning, the control commands increase with respect to the earlier optimal control where the parameters were known. Fig. 11 is showing the results of torque efforts in adaptive control method. The results of the force efforts are not reported due to its similarity to the previous case.

31

Crosstrack [km] Fig. 9. Initial and desired positions and the transfer trajectories of the spacecraft using adaptive control method.

32

Fig. 10. Desired (solid line) and controlled attitude angles of the spacecraft using adaptive control method.

33

Torque Efforts of Adaptive Control (in N.m)

0.1 0.05 0 -0.05 -0.1 0

10

20

30

40

50

60 SC

0.1

1

SC 2

0.05

SC

3

SC 4

0

SC 5 SC 6

-0.05

SC 7

-0.1 0

10

20

30

40

50

60

0

10

20

30

40

50

60

SC 8

0.1 0.05 0 -0.05 -0.1

t(s)

Fig. 11. The control torque effort of the spacecraft using adaptive control method.

It should be noted that if the adaptation law is not used in this case (only using the optimal control law), the errors would be much higher with the change in physical parameters of the system with respect to nominal. The control efforts (integral of the norm of the control forces and torques with weighting factor of 1) would also grow considerably and the settling time (the time that error in states reach to the 99% of the final value and stays there) can increase moderately. As Table 2 shows, the increases in errors and rates of error of rotational state are

13.3% and 38.5%, respectively. Also the control effort has been improved up to 27.3% and the settling time can be saved to 16.0% for the case the adaptation law is applied. The same results

can be explored if the translational state is selected for comparison in two cases. Another point is 34

that these results have been obtained for 100 cases with random initial conditions and then integration is selected to be [40,140], because the phase of synchronization has been passed and

averaged in order to eliminate the sensitivity to different initial conditions. The time interval of

only the tracking error and control demand is studied.

Table 2 : Comparison of control demand and state error integrals with/without applying adaptation law averaged over 100 random initial conditions

control demand/state error 8

M7í

8

M7í



5JM ×J7í



5JM ×J7í 8

'5,„Ö× − 6,5 '6× ' 5,„Ö× −  6,5 '6×



M7í

5JM ×J7í

'5,„Ö× '6×

settling time (in worst case) [s]

with adaptation law

without adaptation law

336.7

381.4

1.3

1.8

2.2

2.8

76.8

89.1

C. Distributed Robust Synchronization Control of Uncertain Spacecraft Systems

The parameter ?A which reflects the maximum limit of change in parameters value respect to

their nominal magnitude is set to 20. The control parameter %A is also selected to be 1 which cover the sliding surface both in translation and attitude. The simulation results with the designed

controller parameters leads to the Figs. 12 and 13 that show the attitude synchronization and tracking would be achieved by more control effort than the two previous cases. Again, these results have been evaluated with similar initial conditions for comparison.

35

Fig. 12. Desired (solid line) and controlled attitude angles of the spacecraft using robust control method.

36

Torque Efforts of Robust Control in [N.m]

0.4

u 1rot

0.2

0

-0.2 0

10

20

30

40

50

60 SC

0.2

SC

u 2rot

0.1

SC SC

0

SC SC

-0.1

1 2 3 4 5 6

SC 7 SC

-0.2 0

10

20

0

10

20

30

40

50

60

30

40

50

60

8

u 3rot

0.4 0.2 0 -0.2

t(s)

Fig. 13. The control torque effort of the spacecraft using robust control method.

Table 3 summarizes the behavior of the controllers with and without the robust control term while the parameters of systems are changing. In both cases, the simulations have been repeated one time for the small uncertainties in parameters and another for the large uncertainties as ten times larger than the previous case. As it can be seen in Table 3, large uncertainties do not make over 100 simulations with random initial conditions. It is due to the nature of Eq.(41) in which

significant changes in results of the robust controller especially when the results are averaged

the control gain would grow as long as the uncertainties are growing. But for the case that the

robust controller has been eliminated and only the main controller in Eq. (43) is applied, the 10.6% and 27.2% with respect to the case of using robust control, for small and large

errors and control efforts will increase noticeably. The errors in rotational state are increased

uncertainties, respectively. In the same way, the rates of state errors are growing 45.5% and 37

72.7% in both cases, respectively. Similarly, the control effort for small uncertainties is 5.4%

more than the controlled case, while the large uncertainties would need 15.4% more control effort compared to the small uncertainties case. Finally, the settling time using the robust

controller is reached 15.8% and 21.2% sooner for the small and large uncertainties cases, respectively.

Table 3: Comparison of control demand and state error integrals with/without applying robust controller for small/large uncertainties averaged over 100 random initial conditions robust control (large

Without robust control

Without robust control

uncertainties)

(small uncertainties)

(large uncertainties)

5JM ×J7í

322.6

356.7

410.3

5JM ×J7í

1.1

1.6

1.9

3.7

3.9

4.5

83.4

96.6

101.1

control demand/state error 8

M7í

8

M7í

  8

'5,„Ö× − 6,5 '6× ' 5,„Ö× −  6,5 '6×



M7í

'5,„Ö× '6×

5JM ×J7í

settling time(in worst case) [s]

VII.

Concluding Remarks

In this study, a framework for synchronization and tracking control of arbitrary numbers of spacecraft located in elliptical orbits is established, considering both the translational and rotational dynamics (EL approach). Due to the constant need of spacecraft to translational and attitude maneuvers for the sake of the mission, they are frequently moving from the neighborhood of each other. Therefore, it necessitates a topology structure based on the formation of agents in network which is built based on the relative positions of agents in each time instant. For this purpose, the idea of distance-based connectivity function is utilized. According to this idea, the network connecting agents in a group is established by the relative distance between agents on that topology. The distance-based connectivity function also 38

determines the weight of connectivity between agents based on the degrees of neighborhood. As a result, the graph of topology is not fixed and is changing with time based on an adaptive formula to satisfy the need of a large network moving arbitrarily in a reconfiguration maneuver. This adaptation law is utilized in optimal, adaptive, and robust control methods. It is shown that the optimal control works well without presence of the parameter uncertainties. However, using the adaptation law of the adaptive and robust control, the performance of the formation control is improved in tracking error as well as control effort in the presence of even large uncertainties. The proposed methods are very beneficial in very large networks, because only a few adjacent spacecraft can contribute in control of each spacecraft. This can help reducing the computational time of control and in turn using the proposed method in real-time applications. Using the adaptation law coupled with the optimal and robust terms help the controller to deal with the parameter uncertainties and/or external disturbances. These characteristics of the proposed SFF controller are very useful in the long period missions in which the variation of spacecraft mass and inertia is inevitable and some of the systems parameters (e.g. thrust force and thruster offcenter) cannot be specified exactly.

Appendix A

Proof of Theorem 1: Suppose there are - spacecraft in formation where the closed-loop systems for the C DE spacecraft based on Eqs. (18)- (28) can be written as A ôA + A ôA + Z ôA −

Z #A + ÷A øA = A (ô + ôA,A ) − ,A ù 4 A,A‰

(49)

#A = ù nA − ùA . By combining disturbances and the terms related to the errors in parameters, i.e. ù

It should be noted that the right hand side of this equation can monitor the non-vanishing

above equation with the diffusive coupling control and the parameters adaptation laws in

39

Eqs.(44) and (46), and finally summarizing equations of all spacecraft in one place, it is straightforward to show that the closed-loop formation system can be stated in matrix form as [] à í í

í [& ‰ ] í

ô í [ ] í í á à {ø } á + 5 í í # : í í [! ‰ ] 9ù

.ƒ0 + à−.÷01 .,01

í ô í6 5 ø 6 #< í ;ù

−.,0 ô   ø í á5 6 = 5 í 6 #< .Ü n 0 ;ù í

.÷0 .Üê 0 í

(50)

where matrices .0, . 0, and vector   are defined as in Eqs. (26) and (27). In the same way,

the matrices .& ‰ 0, .! ‰ 0, .÷0, .,0, .Üê 0 and .Ü n 0 can be defined too. For instance, the matrices

.& ‰ 0 and .! ‰ 0 are

.& ‰ 0 = .& ‰ 0 ⨁ … ⨁ .& ‰ 0A ⨁ … ⨁ .& ‰ 0¼

.! ‰ 0 = .! ‰ 0 ⨁ … ⨁ .! ‰ 0A ⨁ … ⨁ .! ‰ 0+

(51)

Also the - × - gain matrices .ƒ0 can be stated as .ƒ0 = 5

Z − Z ⋮ −Z

⋯ −Z −Z Z − Z ⋱ ⋮ 6 ⨂ 1 ⋯ −Z Z

(52)

In each row of this matrix the diagonal element is the tracking gain as Z and two adjacent

elements constitute the diffusive coupling gain as −Z . However, for the first and final

spacecraft, as it can be seen in Eq. (52), the final and first spacecraft are considered as neighbors,

respectively. The following relation exists between ÷, ø, and ô:

.÷(ô, ã)0ø = >øã B ô

in which matrix .ø0 can be stated as

| Ÿ | Ÿ >øã B = Î ⋮ |¼ Ÿ¼

| Ÿ | Ÿ ⋮ |¼ Ÿ¼

⋯ |¼ Ÿ¼ ⋯ |¼ Ÿ¼ ⋱ ⋮ Ï ⨂ 1 ⋯ |¼¼ Ÿ¼¼

40

(53)

(54)

which the maximum number of neighbors is assumed to be Z. It should also be noted that the

above matrix is not necessarily symmetric. In this way, the Laplacian matrix of the closed-loop system is composed of two parts as follows: .¦0 = .ƒ0 + .øã 0

(55)

It is shown in that for a bidirectional graph of network topology with Laplacian matrix .ƒ0, the close-loop system is contracting if the minimum eigen-value of the Laplacian matrix would be positive, i.e., Z − 2Z > 0.

In order to investigate the system global exponential stability, three important lemma from have been reviewed briefly. Readers are referred to this paper for further study and their proofs. Lemma 1: (Contraction Analysis) Suppose there is a smooth nonlinear non-autonomous system

N (W) = (N(W), W) where N(W) ∈ )  . A virtual displacement AN is defined as an infinitesimal displacement at fixed time, and Θ(N, W) is a smooth coordinate transformation of the virtual displacement such that AQ = Θ AN. Then if there exists a positive ? and a uniformly positive

definite metric, C(N, W) = Θ(N, W)D Θ(N, W), such that V V (AQ 1 AQ) = (AN 1 C(N, W)AN) VW VW

 1  = AN ¨C +   C + C © AN ≤ −2?AN 1 C(N, W)AN N N 1

(56)

Then, all system trajectories converge exponentially fast to a single trajectory regardless of the initial conditions (AQ, AN → 0), i.e., contracting at a rate of ?.




Lemma 2: (Contraction & Robustness) Consider a nonlinear non-autonomous system N (W) = (N, W) which is contracting with a contraction rate ?. Let  (W) be a trajectory of the system. If

there exists a perturbed system N (W) = (N, W) + V(N, W) and its trajectory  (W), then the distance )(W) = EF G ‖AQ‖ satisfies ∀ W ≥ Ww F

\

)(W) ≤ p ‰I(D‰Dˆ ) )(Ww ) +

‰g JK(LJLˆ ) I

supU,D ‖ΘV‖,

3-V, 3 W → ∞, )(W) ≤ sup‖Θ(x, t)V(N, W)‖ ? U,D

41

(57)


Lemma 3: (Robust Hierarchical Connection) The hierarchically combined system with a

generalized Jacobian matrix ì

å å

í î is assumed to be subject to a perturbed flow field of å

[6 ; 6 ]. Then, the path length integral )@ (W) = EF G‖AQ@ ‖ , F = 1,2 between the original and F

\

perturbed dynamics can be written as

) + |?{eU (å )|) ≤ ‖Θ V ‖

FG

) + |?{eU (å )|) ≤ ‖Θ V ‖ +  ‖å ‖ ‖AQ@ ‖




(58)

F\

Defining contraction metric as .′()0 = .()0⨁.& ‰M 0⨁.! ‰M 0 and the square virtual length as

Aô Aô Aø Aø d≔à á .′()0 à á # # A;ù< A;ù< 1

(59)

and differentiating (59) respect to the time leads to Ò[ƒ] + >øã B + >øã B Aô Ñ 2 d ≔ −2 à Aø á Ñ í #< Ñ A;ù í Ð 1

1

í

[Üê ] í

í ÕÔ Aô à Aø á + 2Aô1  Aô í ÔÔ A;ù #< [Ü n ]Ó

(60)

It should be noted that the skew-symmetric matrix >()B − 2[ (,  )] is eliminated from the

above equation based on what is mentioned in section II-A as one of the properties of EL systems. Thus, as long as ‖ô‖ ≥ ‖ ‖ and the diffusive coupling gains and the parameters of

systems are outside the boundaries already said in this section, i.e., 'ŸA@ '{eU and 'b‚A@ '{eU , and by

using lemmas 1 and 2, it can be concluded that Eq. (60) is contracting with the negative Jacobian as

42

Aô d ≤ −2 à Aø á #< A;ù

if and only if [ƒ] +

1

Ò[ƒ] + >øã B + >øã B Ñ 2 Ñ í Ñ í Ð S

>øã B>øã B 

1

í

[Üê ] í

í ÕÔ Aô à Aø á < 0 í ÔÔ A;ù #< [Ü n ]Ó

(61)

> 0. Based on ?{@ ([ƒ]) = Z − 2Z and Weyl’s theorem,

?{@ ([ð]) + ?{@ ([ñ]) ≤ ?{@ ([ð] + [ñ]), this implies that 1

Therefore,

>øã B + >øã B ?{@ Û[ƒ] + Ý ≥ Zw , ∃ Zw > 0 2 d ≤ −2 Zw ‖Aô‖ < −2

Zw

?{eU ¯T ()°

d

(62)

(63)

For the diffusive coupling gains and the parameters of systems inside their boundaries ô must be #< − .÷0øº. Eq. (63) would imply that ô converges selected such that ‖ô‖ > supº  + .,0;ù

exponentially to í, in the absence of the disturbances terms (lemma 1) and also converge

exponentially to an error ball around the desired trajectory if the disturbances are present (lemma 2). From the latter case, the incrementally input-to-state stability is concluded too. Finally,

lemma 3 can be utilized to hierarchically result in y → í and  y → í in the absence of disturbances. In this way, the global exponential stability of the closed-loop system has been proved by the contraction analysis.




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Highlights •

A coupled translational and attitude relative dynamics of several spacecraft is used.

•

The Earth second zonal harmonic and aerodynamic drag are modeled.

•

Adaptive switching network topology based on distance-based connectivity is proposed.

•

A distributed optimal control method is suggested based on network topology.

•

Robust and adaptive control methods are developed in the case of uncertainties.

47