Distributed and cooperative quaternion-based attitude synchronization and tracking control for a network of heterogeneous spacecraft formation flying mission

Author's Accepted Manuscript

Distributed and cooperative quaternion-based attitude synchronization and tracking control for a network of heterogeneous spacecraft formation flying mission Alireza Mehrabian, Khashayar Khorasani

www.elsevier.com/locate/jfranklin

PII: DOI: Reference:

S0016-0032(15)00158-1 http://dx.doi.org/10.1016/j.jfranklin.2015.04.007 FI2314

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

10 October 2014 5 April 2015 12 April 2015

Cite this article as: Alireza Mehrabian, Khashayar Khorasani, Distributed and cooperative quaternion-based attitude synchronization and tracking control for a network of heterogeneous spacecraft formation flying mission, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2015.04.007 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.

1

Distributed and Cooperative Quaternion-Based Attitude Synchronization and Tracking Control for a Network of Heterogeneous Spacecraft Formation Flying Mission Alireza Mehrabian and Khashayar Khorasani

Abstract Distributed control strategies for the attitude synchronization and set-point tracking control of multiple heterogeneous spacecraft (SC) in a formation flying mission are proposed in this work. The first scheme requires feedback and exchange of angular velocity measurements among the SC in the formation. However, the second scheme does not require measurement and exchange of angular velocities (or their estimates) among the SC in the formation. We have employed unit-quaternion, which is a singularity free attitude representation, to describe the SC attitude so that large attitude maneuvers can be executed. We have also developed two constrained control schemes for attitude synchronization and set-point tracking control for (i) a single SC with and without angular velocity feedback, and (ii) SC formation flying with and without angular velocity feedback. A number of simulation case studies are provided to demonstrate the advantages and benefits of our proposed algorithms as compared to the available results in the literature.

I. I NTRODUCTION Spacecraft (SC) formation flying is a new technology which plays an important role in the future space missions such as NASA’s Terrestrial Planet Finder (TPF) Mission and Space Telescope assembly [1], [2], the European Space Agency’s (ESA) similar mission, called Darwin [3], among many others. Several algorithms have been proposed for the attitude and/or position synchronization and control of multiple SC in deep space and in low orbit [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. In this paper, we consider attitude synchronization and set-point tracking control, therefore, we review recent literature on this topic below. Position synchronization problem for spacecraft formation flying missions is not considered in this paper and is left as a topic of future research. The single SC attitude control and fault-tolerant attitude control with/without using angular velocity measurement is also studied in the literature [18], [19], [20], [21], [22]. Reference [5] is one of the earliest papers on coordinated attitude control of SC. This paper investigates the use of The authors are with the Department of Electrical and Computer Engineering, Concordia University, Montreal, Quebec, H3G-1M8, Canada. Contact email: [email protected]. This publication was made possible by NPRP Grant No. 5-045-2-017 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.

April 25, 2015

DRAFT

2

one-leader, multiple-leader, and barycenter coordination strategies. The one-leader coordination strategy requires that one SC serves as the reference SC, the leader, for the rest of the SC, the followers, in the formation. The followers then track the leader, possibly with a constant offset. The multiple-leader approach involves splitting the formation into two or more groups and assigning one or more fleet leaders. In this case, the fleet leaders act as the reference SC for the group leaders, which in turn, act as the reference SC for the group followers. This approach results in a hierarchical communication topology. The most interesting coordination strategy discussed is the barycenter strategy. In this strategy, the j-th SC uses the position information of the neighboring SC to determine the barycenter of their locations. The barycenter is then used as the desired location of the j-th SC. In a subsequent paper [6] the authors use the same type of formulation to develop one-leader based coordinated control laws for position and attitude control of a SC formation. The interesting addition of this paper is the application of the one leader coordinated control strategy to the problem of Michelson stellar interferometry. The authors in [10] developed a distributed controller for the SC formation attitude control problem that they term the coupled dynamics controller. The coupled dynamics controller uses a ring communication topology, where each SC knows the state of two other SC in the formation. The desired state and the state of the two other SC are used to determine the appropriate control torque. A convergence proof is provided; however the proof does not ensure global convergence of the formation attitude. It requires that the SC begin with no angular rate and that the initial formation error is below a certain limit. In [12] the authors developed a passivity-based controller for the SC formation attitude control problem. The passivity-based controller uses only attitude information to determine control actions, thus alleviating the need for angular rate measurements. The authors also analytically determine the domain of attraction for the passivity-based controller and the coupled dynamics controller. Later in [4] a more general architecture for SC formation attitude control is introduced by the same authors. The architecture is designed to subsume the leader-follower, behaviorbased, and virtual-structure coordination strategies. The authors claim that the architecture is “amenable to analysis via control theoretic methods.” A brief descriptive list of some formation control problems that can be analyzed using the architecture is given. The authors demonstrate the usefulness of the architecture by applying it to the practical problem of Michelson stellar interferometry. In [23] the authors investigate a centralized implementation of virtual structure coordination strategy using the general architecture. The primary contribution of the paper is the addition of formation feedback to the SC formation. The authors prove the virtual structure control law guarantees the stability and convergence of the system. A fundamentally different approach is proposed in [8] for dealing with the SC formation control problems. In this paper, each SC in the formation uses its current desired state and state information communicated by the other SC to determine a quasi-desired state using the reference projection. The quasi-desired state is then used by the SCs attitude controller to determine the appropriate control action. Different types of coordination are possible using the appropriate reference projection. In the paper, a reference projection is developed for the leader-follower, generalized leader-follower, and the virtual desired attitude coordination strategies. The leaderfollower reference projection for the leader is the desired state of the formation, and the current state of the leader is the reference projection for the follower SC. The generalized leader-follower April 25, 2015

DRAFT

3

strategy differs in that the reference projection for the followers is a compromise between the desired state and the current state of the leader. The only truly decentralized coordination strategy is the virtual desired attitude strategy, where the reference projection for each SC is a compromise between the desired state and the average state of the SC in the formation. In a later paper [16], the authors discuss applying the idea of reference projections to tracking control and in [17] the authors investigate the idea further and present simulation results. The authors in [24] introduce a distributed algorithm for SC formation attitude control. The authors consider unit-quaternion in their analysis, which enables large attitude maneuvers. However, the authors consider fixed communication network and full state measurement in their analysis. Communication delays are considered in [25], which extends the results in [24]. Modified Rodriguez Parameters (MRP) are also used in the controller design for SC formation flying in [26]. Communication time-delays are considered in [27]. However, when using MRP SC full attitude rotation maneuvers cannot be executed. The SC attitude synchronization and coordination control without requiring angular velocity feedback is very useful specially when this information is not available due to sensor failure. This problem is considered recently in the literature. Specifically, authors in [15] developed a velocity free attitude tracking algorithm for SC by using leader-follower approach. Consequently, failure of the leader SC will result in failure of the mission. More recently, [14] developed an attitude synchronization and set-point tracking algorithm for multiple SC formation by using MRP for attitude representation. The advantage of these two studies ([15] and [14]) is that they do not require sharing the estimate of the angular velocity among the SC in the formation, and this considerably reduces the communication load in the formation. Another study on velocity-free attitude control of SC formation is reported in [13]. In this paper, the authors use unit-quaternion to describe the SC attitude and extend the results reported in [20] for velocity control of a single SC to SC formation flying. However, in this algorithm it is required that an estimate of the angular velocity is shared among the SC in the formation. Therefore, the algorithm does not reduce the communication among the SC in the formation. However, it is interesting to note that in this algorithm boundedness of the control effort is guaranteed. More recent work related to the formation flight control of spacecraft has also appeared in [28], [29], and [30]. In view of the above works, the main contributions of the present work are stated as follows: 1) Two quaternion-based attitude synchronization and set-point tracking control schemes for networked spacecraft (SC) that are tasked to a formation flying mission are developed. In our first proposed control laws the desired attitude coordinates are only provided to a subset of the SC in the formation designated as the formation leaders. This essentially increases the flexibility in the design of the formation structure by improving the robustness of the formation to actuator, sensor, and component faults. Our second proposed control laws do not require exchange of actual spacecraft angular velocities among the spacecraft in the network. 2) We develop two schemes for the constrained control of a single SC with and without angular velocity feedback. These control schemes are then extended and generalized for the attitude synchronization and set-point tracking of SC formation flying with and without angular velocity feedback. Moreover, in simulations provided our proposed constrained attitude controller for single SC (both with and without velocity measurements) is compared with April 25, 2015

DRAFT

4

the controller that is proposed in [13] and [20], and it is shown in simulations that our proposed controller has a better performance. We also demonstrate and illustrate the superior performance of our proposed velocity-free SC formation flying attitude controllers in several simulation case studies. II. N ETWORKED S PACECRAFT R EPRESENTATION AND P ROBLEM S TATEMENT Below, we provide necessary background information on the Euler-Lagrange (EL) systems, the spacecraft attitude dynamics, and unit quaternions representation that will be used throughout the remainder of this paper. A. Definition of a Saturation Function Definition 1. A saturation function denoted by Sat(x) : R → R is a strictly increasing odd function with the following properties ∀x ∈ R, namely (i) Sat(0) = 0, (ii) |Sat(x)| ≤ 1, (iii) Sat(−x) = −Sat(x), and (iv) ∂ Sat(x) ∂ x ≥ 0. The following lemma will be used subsequently in this work. Lemma II.1. The saturation function defined above has the following property, namely, Z x1 0

1 Sat(x)dx ≥ Sat(x1 )x1 ≥ 0, ∀x, x1 ∈ R 2

(1)

Proof. Proof can be found in [31]. B. Euler-Lagrange (EL) Systems In this work, we consider m > 1 Euler-Lagrange (EL) systems, where the j-th EL system is governed by the following nonlinear dynamic equation, namely, D j (q j )q¨ j + C j (q j , q˙ j )q˙ j + g j (q j ) + F¯ j = u j + d(t)

(2)

where j ∈ {1, . . . , m}, q j = {q1, j , . . . , qk, j } ∈ Rk is the generalized coordinates vector, D j (q j ) ∈ Rk×k is symmetric positive definite general inertia matrix, C j (q j , q˙ j ) ∈ Rk×k is the matrix of Coriolis and centrifugal forces, g j (q j ) is the gravitational force vector (GFV), u j ∈ Rk is the vector of control forces/torques, and F¯ j is the damping function. Furthermore, d(t) represents the external time-varying disturbances on the EL system. The dynamic model (2) has the following properties [31], [32], [33], namely, P1: The general inertia matrix is bounded, specifically, ∃k j , k j such that: k j Ik < D j (q j ) < k j Ik , ∀q j , where Ik is an k × k identity matrix. P2: GFV is assumed to be upper bounded, that is, 0 ≤ supq j ∈Rk {|gi, j (q j )|} ≤ gi, j , ∀i ∈ {1, . . . , k}, where gi, j (q j ) denotes the elements of g j (q j ), ˙ j (q j ) − 2C j (q j , q˙ j ) is skew-symmetric matrix. and P3: D C. Spacecraft Attitude Dynamics Sets of coordinates that completely describe the orientation (attitude) of a rigid body relative to some reference coordinate frame are known as the attitude coordinates. There are many different

April 25, 2015

DRAFT

5

ways to describe the attitude of a rigid body in the 3D space. The four fundamental facts on the rigid body attitude coordinates are listed below [34]: 1) A minimum of three coordinates is required to describe the relative angular displacement between two reference frames. 2) Any minimal set of three attitude coordinates will contain at least one geometrical orientation where the coordinates are singular, at least two coordinates are undefined or not unique. 3) At or near such a geometric singularity, the corresponding kinematic differential equations are also singular. 4) The geometric singularities and associated numerical difficulties can be avoided altogether through a regularization. Redundant sets of four or more coordinates exist which are universally determined and contain no geometric singularities. We review the two most commonly used attitude representations below. An interested reader can refer to [34] for more details. 1) Euler angles: The most commonly used set of attitude parameters is the Euler angles. Aircraft and spacecraft orientations are commonly described through the Euler angles roll, pitch and yaw (θ , φ , ψ). The popularity of Euler angles stems from the fact that the relative attitude is easy to visualize. The equations of motion for the j-th spacecraft attitude dynamics are given by [34], [35], ¯ j q˙j =Rω J j ω˙ j = − S(ω j )J j ω j + ρ j

(3a) (3b)

where q j = [θ j , φ j , ψ j ]T is the vector of the Euler angles, ω j = [ω j1 , ω j2 , ω j3 ]T is the vector of spacecraft angular velocities in the body frame, ρ j = [ρ j1 , ρ j2 , ρ j3 ]T is the vector of external torque inputs in the body frame, J j = JTj ∈ R3×3 is the spacecraft positive definite moment of ¯ is defined by, inertia matrix, and R   cθ sφ sθ cφ sθ ¯ = 1  0 cφ cθ −sφ cθ  R cθ 0 sφ cφ where cθ stands for cos(θ ), sθ stands for sin(θ ), sφ stands for sin(φ ), and cφ stands for cos(φ ). In addition, S(x) is the skew-symmetric matrix operator that is given by,   0 −x3 x2 0 −x1  S(x) =  x3 −x2 x1 0 ˙¯ −1 q˙ + R ¯ −1 q˙j , which implies that ω˙ j = R ¯ −1 q¨j . Conse¿From equation (3a), we have ω j = R j ˙¯ −1 q˙ + R ¯ −T J j R ¯ −1 q¨j + R ¯ −T J j R ¯ −T S(R ¯ −1 q˙j )J j R ¯ −1 q˙j = quently, one can re-write equation (3b) as R j −T ¯ ρ j . Therefore, the 3-degrees of freedom (DOF) attitude dynamics of a spacecraft can be R

April 25, 2015

DRAFT

6

written in the form of equation (2) with g(q j ) =

∂ F¯ (q˙j ) ∂ q˙j

= 0, and where we specifically have,

¯ −T J j R ¯ −1 D j (q j ) =R ˙¯ −1 + R ¯ −T S(R ¯ −1 q˙j )J j R ¯ −1 ¯ −T J j R C j (q j , q˙j ) =R ¯ −T ρ j u j =R

(4a) (4b) (4c)

When we replace the above terms in equation (2), it follows that all the terms are left multiplied ¯ −T . However, one should not cancel out this common term, as it would then result in having by R a non-symmetric D j (q j ). Remark II.1. Note that the Euler angle kinematic differential equations encounter a singularity at θ = ±90 degrees for three successive rotations about the 3rd, 2nd and 1st body axis (labeled (3-2-1) set for short). The spacecraft inertia matrix with a vector a j = [a j1 , a j2 , a j3 ]T can be written as [36], J j a j = O(a j )Θ j   a j1 0 0 0 a j3 a j2 where O(a j ) =  0 a j2 0 a j3 0 a j1  and 0 0 a j3 a j2 a j1 0  T Θ j = J11, j J22, j J33, j J23, j J13, j J12, j

(5)

Consequently, one obtains ˙¯ −1 b + R ¯ −T J j R ¯ −1 a j + R ¯ −T J j R ¯ −1 b j ¯ −T S(R ¯ −1 q˙j )J j R D j (q j )a j + C j (q j , q˙j )b j =R j

(6)

,Y j (q j , q˙j , a j , b j )Θ j h i ˙ −T −1 −1 −1 −1 ¯ ¯ ¯ ¯ ¯ where Y j (q j , q˙j , a j , b j ) = R O(R a j ) + O(R b j ) + S(R q˙j )O(R b j ) . The Euler angles provide a compact, three parameter attitude description whose coordinates are easy to visualize. One main drawback of these angles is that a rigid body or reference frame is never further than a 90 degree rotation away from a singular orientation. Therefore, their use in describing large, and especially, arbitrary rotations is limited. One can then use the unit quaternion to overcome this difficulty as described next. 2) Unit quaternion (Euler parameters): Unit quaternion (also known as Euler parameters) is another popular set of attitude coordinates. It provides a redundant, nonsingular attitude description and is well-suited to describe arbitrary, large rotations [34]. The unit quaternion for a spacecraft is defined as:    ϕ  q¯ j e j sin( 2j ) ~q j = = (7) ϕj qˆ j,4 cos( 2 ) where e j = [e j1 , e j2 , e j3 ]T ∈ R3×1 is the Euler axis, ϕ j is the Euler angle, q¯ j is the vector part, and qˆ j,4 is the scalar part of the quaternion, satisfying the constraint qˆ2j,4 + q¯ Tj q¯ j = 1 April 25, 2015

(8) DRAFT

7

¯ q j) ∈ Therefore, qˆ j,4 = ±1 correspond to the same orientation in SO(3). The matrix denoted by R(~ SO(3) represents the rotation from the inertial frame F I to the body frame of the spacecraft, F B . The rotation matrix is related to the quaternion through [35]: ¯ q j ) = (qˆ2j,4 − q¯ Tj q¯ j )I3 + 2¯q j q¯ Tj − 2qˆ j,4 q¯ × R(~ j

(9)

where In is an n × n identity matrix. In general, +~q j and −~q j both represent the same rotation matrix. The equations of motion for the attitude dynamics and kinematics of a spacecraft are then given by [35]: J j ω˙ j + C j (¯q j , ω j )ω j =u j 1¯ q˙¯ j − E(~ q j )ω j =0 2 1 q˙ˆ j,4 + q¯ Tj ω j =0 2

(10)

where uj = [u j1 , u j2 , u j3 ]T ∈ R3×1 is the control input vector and ωj = [ω j1 , ω j2 , ω j3 ]T ∈ R3×1 is the angular velocity of the spacecraft with respect to the inertial frame. In addition, the matrix ¯ q j ) is given by E(~ ¯ q j ) = qˆ j,4 I3×3 + q¯ ×j (11) E(~ When thrusters are used C j (¯q j , ω j ) can be chosen as C j (¯q j , ω j ) = ω × j J j , and when momentum actuators (e.g. reaction wheels and control moment gyros) are chosen C j (¯q j , ω j ) will be a function of the spacecraft rotation matrix and the angular momentum of actuators and the spacecraft in the body frame [37], [38]. In both cases, however, C j (¯q j , ω j ) is a skew symmetric matrix [38]. Remark II.2. Note that the parameters of the dynamic equation (10) satisfy the Properties P1-P3 that are provided in Section II-B. With reference to Remark II.2 and by assuming that the spacecraft inertia matrix is expressed in the principal axis, one concludes that J j is a diagonal matrix. III. N ETWORK OF E ULER -L AGRANGE (EL) S PACECRAFT S YSTEMS AND E RROR R EPRESENTATION A. Information Graph Structure and Neighboring Set In this paper, the information exchanges among the m EL systems is represented by a graph. We provide here some basic terminologies and definitions from graph theory in order to facilitate understanding of the subsequent analyses and developments. An interested reader can refer to [39], [40] for more details. A directed-graph (digraph) G consists of a node set V = {1, . . . , m}, an edge set E ⊆ V × V , and a weighted adjacency matrix Λ = [λ jn ] ∈ Rm×m . The m SC in the network are considered as nodes of a digraph. The communication links among the EL systems are considered as the digraph edge set, where self-connection is not allowed.

April 25, 2015

DRAFT

8

Two vertices j and n are called adjacent if at least one edge exists between them i.e. ( j, n) ∈ E , which is also denoted by j ↓ n. If j ↓ n then node j is the parent of node n and node n is the child of node j. The weighted adjacency matrix Λ = [λ jn ] is defined such that λ jn is a positive weight if n ↓ j, while λ jn = 0, otherwise. The indegree and outdegree of node j are given by di ( j) = ∑n λn j and do ( j) = ∑n λ jn respectively. Associated with Λ we introduce a matrix known as the Laplacian matrix L = [l jn ] ∈ Rm×m such that l j j = ∑m n=1,n6= j λ jn and l jn = −λ jn , where k 6= j. This implies that L is a zero row sum matrix. A path of length l p in a digraph is a sequence ( j0 , . . . , jl ) of l p distinct vertices such that for every i ∈ {0, . . . , l p − 1}, ( ji , ji+1 ) is an edge. A digraph is strongly connected if for any pair of distinct vertices j and n, there is a directed path from j to n. Furthermore, if the digraph is strongly connected, L has a simple eigenvalue ¯ m , where 1m is an m × 1 column vector of ones and 0 with an associated right eigenvector of k1 k¯ is a positive number, i.e. L 1m = 0. All the other eigenvalues of L are positive if and only if the digraph G is strongly connected [40]. For a given node j, the set of SC from which it can receive information is called a neighboring set N j , that is ∀ j = 1, . . . , n : N j = {n|(n, j) ∈ E }. In addition, the number of neighbors of the j-th node is denoted by N j (which is also known as the cardinality of the j-th node). Also, the graph size |E | is the number of edges. An undirected-graph is a digraph with an additional property, i.e. j ↓ n ⇔ n ↓ j. This implies that λ jn = λn j and di ( j) = do ( j) for all j, n ∈ V . An undirected-graph is connected if and only if it is strongly connected. Let us denote (L ⊗ I p )x as a column stack vector of all ∑mj=1 λn j (xn − x j ), n ∈ V with x = [xT1 , . . . , xTm ]T ∈ R p.m×1 . It can be shown that xT (L ⊗ I p )x =

1 m m

xn − x j 2 . λ ∑ ∑ n j 2 n=1 j=1 B. Synchronization Error Let us denote the position (attitude) synchronization error between the j-th and the n-th EL system as, ~q jn (t) =~q j (t) −~qn (t) −~q[jn , j ∈ V , n ∈ N j (12) where ~q[jn is a positive constant vector 1 added to allow non-zero distances/angles among the EL systems in the steady-state. The velocity (rate) synchronization error, which is the time derivative of ~q jn (t), is given by ~q˙ jn (t) = ~q˙ j (t) −~q˙ n (t), j ∈ V , n ∈ N j . One can decompose the term ~q[jn as follows ~q[jn =~q[j −~q[n , j, n ∈ V , j 6= n, where ~q[j and ~q[n are constant positive numbers that are provided to the j-th and the n-th EL systems by the command and control center to avoid the EL systems collision. Furthermore, let us denote the desired constant position (attitude) for the networked EL system to be ~q? . Definition III.1. The EL systems that receive the desired position/attitude, ~q? , are defined as the leaders. The other EL systems, which do not have access to this desired position/attitude are denoted as the followers. We label, without loss of generality, the EL systems 1 to ‘l’ as the leaders and the EL systems “l + 1” to “m” as the followers.  1A

vector x ∈ Rn is positive if and only if xi > 0, ∀i.

April 25, 2015

DRAFT

9

Let us now define the error vectors for the leaders, q˜ j (t), according to: q˜ j (t) =~q j (t) −~q[j −~q? , j ∈ {1, . . . , l}

(13)

and the error vectors for the followers according to q˜¯ j (t) =~q j (t) −~q[j , j ∈ {l + 1, . . . , m}

(14)

where we also define q˙˜ jn = q˙˜ j − q˙˜ n and q˜ jn = q˜ j − q˜ n . Remark III.1. It should be noted that the constant term ~q[jn is introduced to effectively avoid the EL agents from colliding at the steady-state. However, during the transient phase of the mission one needs to consider an obstacle avoidance penalty function and add an extra term to the controllers in order to avoid collision among the agents. This aspect is, however, beyond the scope of this paper. An interested reader can refer to [41] and [42] for more details. The following two lemmas will be used subsequently in this paper. Lemma III.1. Consider a symmetric matrix Λ = ΛT ∈ Rk×k . Let us denote λi j as the i j-th element of this matrix. The following equality then holds: k k 1 k k ∑ ∑ λi j (yi − y j ) χ(xi − x j ) = ∑ ∑ λi j yi χ(xi − x j ) 2 i=1 j=1 i=1 j=1

(15)

where χ(x) is any odd function. Proof: The left hand side of equation (15) can be written as: 1 k k ∑ ∑ λi j (yi − y j ) χ(xi − x j ) 2 i=1 j=1 =

1 k k 1 k k λ y χ(x − x ) − i j i i j ∑∑ ∑ ∑ λi j y j χ(xi − x j ) 2 i=1 2 i=1 j=1 j=1

=

1 k k 1 λi j yi χ(xi − x j ) − ∑ ∑ 2 i=1 j=1 2

k

k

∑ ∑ λ ji yi χ(x j − xi)

j=1 i=1

By noting λi j = λ ji and χ(xi − x j ) = −χ(x j − xi ) one obtains (15). This completes the proof of the lemma.  Lemma III.2. Consider the following algebraic equations that correspond to a strongly connected network of m spacecraft systems Λ pj χ(q˜ j ) +

∑

Λ pjn χ(q jn ) = 0, j, n ∈ {1, . . . , m}, j 6= n

(16)

n∈N j

where q˜ j ∈ Rk , χ(x) is a monotonically increasing odd function, and Λ pjn = Λnp j are positive definite matrices. Furthermore, assume that Λ pj is a positive definite diagonal matrix for only 0 < l ≤ m number of equations (corresponding to “l” leaders and explicitly defined in Definition April 25, 2015

DRAFT

10

III.1) and is zero, otherwise. If we have ∑lj=1 Λ pj χ(q˜ j ) = 0, then the only solution to equation (16) is q˜ j = 0, ∀ j ∈ {1, . . . , m}. Proof: We prove this lemma by contradiction. First note that equation (16) implies that if for the j-th algebraic equation we have Λ pj = 0 (corresponding to m − l ≥ 0 “followers”), then p p p ∑n∈N j Λ jn χ(q jn ) = 0. Therefore, equation (16) essentially reduces to Λ j χ(˜q j )+ ∑n∈N j Λ jn χ(q jn ) = 0, j, n ∈ {1, . . . , l}, j 6= n. Now, let us assume that the claim does not hold, i.e. q˜ j 6= 0, ∀ j ∈ {1, . . . , l}. This in view of ∑lj=1 Λ pj χ(˜q j ) = 0, implies that there exists at least one SC (let’s say p q j ) = −Λlp χ(˜ql ) ≡ the l-th SC, without loss of any generality) for which we have: ∑l−1 j=1 Λ j χ(˜ Λlp χ(−˜ql ), which implies that the sign of the l-th SC error is opposite to that of the others in the network. Without loss of generality, let us assume q˜ l = −ε j q˜ j , j = 1, . . . , l − 1, where ε j > 0, and that q˜ j > 0, j = 1, . . . , l − 1. Thus, from equation (16) we have: Λlp χ(˜ql ) + p p p χ(˜ql − q˜ l−1 ) = 0, which can be re-written as: χ(˜ql − q˜ 2 ) + · · · + Λl,l−1 χ(˜ql − q˜ 1 ) + Λl,2 Λl,1 p p p −Λl χ(ε1 q˜ 1 ) − Λl,1 χ[(ε1 + 1)˜q1 ] − · · · − Λl,l−1 χ[(εl−1 + 1)˜ql−1 ] = 0. The statement above does not hold when q˜ j 6= 0, ∀ j ∈ {1, . . . , l}, which is a contradiction. Therefore, the only solution to the problem is to have q˜ j = 0, j = 1, . . . , l. Consequently, from equation (16) we have ∑n∈N j Λ pjn χ(q jn ) = 0, ∀ j, n ∈ {1, . . . , m}, j 6= n, which by the strong connectivity of the communication graph, and the fact that χ(x) is a monotonically increasing odd function implies that q jn = 0, ∀ j, n ∈ {1, . . . , m}, j 6= n. Therefore, in view of equation (12), one obtains q˜ j = 0, j = l + 1, . . . , m. Hence, we have: q˜ j = 0, ∀ j ∈ {1, . . . , m}. This completes the proof of the lemma.  The following lemma, which is known as the Barbalat’s lemma will be used subsequently in our work. The proof of this lemma can be found on page 323 of [43]. LemmaR III.3. Let α¯ : R → R be a uniformly continuous function on [0, ∞). Suppose that ¯ ¯ limt→∞ 0t α(τ)dτ exists and is finite. Then, α(t) → 0 as t → ∞. C. Integral Input-to-State Stability In this subsection, we generalize the standard definition (Definition 2.1 in [44]) of the integral input-to-state stability (iISS) to general networked nonlinear systems. The iISS is defined such that it reflects the qualitative property of small overshoot when disturbances have finite energy. Also iISS is a strictly stronger property than asymptotic stability in absence of external disturbances [44]. Definition III.2. Consider a network of m multiple heterogeneous nonlinear systems where the dynamics of the j-th agent can be expressed by x˙j = f j (x j ) + g¯ j (x j )w j

(17)

¯ . For a given input w and any ξ ∈ Rn¯ , there is a where x j ∈ Rn¯ , w j ∈ Rl , and g¯ j (x j ) ∈ Rn×l j j unique maximal solution of the initial value problem x˙j = f j (x j ) + g¯ j (x j )w j , x j (0) = ξ j . This solution is defined on some maximal open interval, and it is denoted by x j (., ξ j , w j ). The networked nonlinear system, where the dynamics of the j-th agent is given by (17), is ¯ β¯ , and γ¯ denote class K∞ , iISS if there exist functions α¯ ∈ K∞ , β¯ ∈ K L , and γ¯ ∈ K (α,

April 25, 2015

DRAFT

11

K L , K functions, respectively. For details on the definitions of these classes refer to [43]), such that, for all ξ j ∈ Rn¯ and all w j , the solution x j (t, ξ j , w j ) is defined for all t ≥ 0, and



¯ x j (t, ξ j , w j ) + x jn (t, ξ jn , w jn ) ) ≤ α( Z t (18)





¯



¯ w j (s) + w jn (s) )ds β ( ξ j + ξ jn ,t) + γ( 0

where ξ jn = ξ j − ξn and w jn (s) = w j (s) − wn (s). Lemma III.4. [44] The j-th nonlinear system (17) is iISS if and only if there exists an output function y j = h j (x j ) (continuous and with h j (0) = 0) which provides zero-detectability (w j ≡ 0 and y j ≡ 0 ⇒ x j (t) → 0) and dissipativity in the following sense: there exists a storage function W j and α¯ j ∈ K∞ , γ¯ j ∈ K , such that:



W˙j ≤ α¯ j ( w j ) − γ¯ j ( y j ) holds for all (x j , w j ). This storage function is denoted as the iISS-Lyapunov function. Lemma III.5. [45] Let us assume that the j-th nonlinear system (17) is iISS, then we have, Z +∞

α¯ j ( w j (ξ ) )dξ < +∞ ⇒ lim inf x j = 0 0

t→+∞

where α¯ j ∈ K∞ . This system is denoted as bounded energy weakly converging state. In addition, if the j-th nonlinear system (17) is iISS we have Z +∞

α¯ j ( w j (ξ ) )dξ < +∞ ⇒ lim inf x j < +∞ 0

t→+∞

where α¯ j ∈ K∞ . This system is denoted as bounded energy frequently bounded state. D. Spacecraft Attitude Error Dynamics For a SC in formation we define two error measures. These measures are the station-keeping and the formation-keeping attitude state errors. The station-keeping error is defined as the attitude state error of an individual SC with respect to its absolute desired attitude state. The stationkeeping error, δ~q j , is defined as: δ~q j = Q(~q?j −1 )~q j (19) where ~q?j is the desired attitude of the SC formation, the inverse of the quaternion is defined as ~q−1 = [−¯qT qˆ4 ]T , and the matrix Q(~q) is defined as:   ¯ q) q¯ E(~ Q(~q) = −¯qT qˆ4 ¯ q) = qˆ4 I3×3 + q¯ × . One can decompose the station-keeping error into a vector and a where E(~ scalar part, namely, δ~q j = [δ q¯ Tj , δ qˆ j,4 ]T . The station-keeping angular velocity error, δ ω j , is defined as: δωj = ωj −Ωj

April 25, 2015

(20)

DRAFT

12

¯ q j )ω ? , and ω ? is the absolute desired angular velocity vector expressed in the where Ω j = R(δ~ absolute desired reference frame such that ω˙ ? ≡ 0. We make the following assumption explicit. Assumption III.1. Let there exist positive constant c1 such that supt≥0 {ω ? } < c1 . The first derivative of the station-keeping angular velocity error is obtained as [19]: δ ω˙ j = ω˙ j + ω × j Ωj

(21)

We can now express the governing equations for the attitude error δ~q j and the angular velocity error δ ω j as follows: 1 δ~q˙ j = E(δ~q j )δ ω j 2 × J j δ ω˙ j = u j − ω × j J j ω j + J j ω j Ω j + d(t)

(22) (23)

where d(t) denotes the external environmental disturbances and noise and the matrix E(~q) is given by     ¯ q) qˆ4 I3×3 + q¯ × E(~ E(~q) = ≡ (24) −¯qT −¯qT Formation-keeping error, for the j-th SC is the attitude state error of the j-th SC with respect to the other SC in the formation. The relative attitude error between the j-th and the n-th SC is defined as: −1 ~q jn = Q(~q−1 (25) n )~q j ≡ Q(δ~qn )δ~q j The relative angular velocity vector of the j-th SC with respect to the n-th SC, ω jn , can be ¯ q jn ) ∈ SO(3) as follows written in terms of δ ω j and δ ωn , and the rotation matrix denoted by R(~ ¯ q jn )δ ωn ω jn = δ ω j − R(~

(26)

Consequently, the dynamics of the relative attitude error, ~q jn is obtained as 1 ~q˙ jn = E(~q jn )ω jn (27) 2 The following equations corresponding to the relative states of the j-th and the n-th SC will be used subsequently [11], namely ¯ q jn ) = R ¯ T (~qn j ) and q¯ n j = −¯q jn = −R(~ ¯ qn j )¯q jn R(~

(28)

We define two objectives in this work. Our first objective is to design a distributed and a cooperative controller for each SC which commands the actuators in order guarantee coordinated SC attitude and angular velocity alignment, i.e. ~q j →~qn (or equivalently, ~q jn → 0) and ω jn → 0. This objective is designated as the formation-keeping. Our second objective is to ensure that the designed distributed and cooperative controllers guarantee that each SC attitude converges to the commanded attitude, i.e. δ~q j → 0 and δ ω j → 0. This objective is designated as the station-keeping. Note that in the development of the control laws it is assumed that the final angular velocity for the spacecraft network is zero. We impose two constraints in the design, which are (1) there should be no information exchange regarding the angular velocities of the April 25, 2015

DRAFT

13

SC in the formation, and (2) there exist actuator constraints on the maximum control efforts, max i.e. ur | j (t) ≤ u¯max r | j , where r = 1, 2, 3, j ∈ V and u¯r | j is a known positive scalar. IV. S INGLE S PACECRAFT ATTITUDE S ET-P OINT T RACKING C ONTROL We first make the following definition regarding the actuators constraint explicit in this section. Definition IV.1. The bound on the j-th SC control effort is denoted by umax j , i.e. ku(t)k ≤ max u , ∀t ≥ 0. A. Single Spacecraft Attitude Tracking with Bounded Input We are now in the position to provide our first result. Lemma IV.1. Let the dynamics and kinematics of a single SC be governed by equations (22) and (23) where the subscript j can be removed for simplicity. Consider the following control law, namely, u = −∆[δ q¯ + β (t)Sat(δ ω)] + Ω× J Ω (29) where 0 < β ≤ β (t) ≤ β , and ∆ = αI ∈ R3×3 is a positive definite diagonal matrix (α a positive scalar) and Sat(x) = [sat(x1 ), sat(x2 ), sat(x3 )]T . Under the control law (29) the closed-loop system is iISS where the input is the disturbance and the output is the SC angular velocity and quaternion. Furthermore, the control effort is bounded for all t ≥ 0 and for all initial conditions, i.e. √ ||u(t)|| ≤ umax provided that the controller gains, α and β (t) are selected such that 0 < α(1 + 3β ) + c21 ||J|| ≤ umax , where umax is given in Definition IV.1. Proof. The SC dynamics (23) under the control law (29) can be re-written as: J δ ω˙ = −∆[δ q¯ + β (t)Sat(δ ω)] + Ω× J Ω − ω × J ω + Jω × Ω + d(t)

(30)

Consider the following radially unbounded positive definite decrescent iISS-Lyapunov function candidate for the system, 1 W = δ ω T ∆−1 J δ ω + δ q¯ T δ q¯ + (1 − δ qˆ4 )2 2

(31)

Note that ∆−1 is a positive definite diagonal matrix. The time derivative of the iISS-Lyapunov function candidate is given by, d W = δ ω T ∆−1 J δ ω˙ + 2δ q˙¯ T δ q¯ − 2δ q˙ˆ4 (1 − δ qˆ4 ) dt Using equations (22), (23), and (30) one obtains,   d −1 T Ω× J Ω W =δ ω −δ q¯ − β (t)Sat(δ ω) + ∆ dt   × × −1 − ω J ω + Jω Ω + ∆ d(t) + δ ω T δ q¯

April 25, 2015

(32)

DRAFT

14

Noting the fact that

δ ωT

  × × × Ω J Ω − ω J ω + Jω Ω = 0 (for proof refer to the Appendix at

the end of this paper), we obtain the following equation, namely, W˙ = − β (t)δ ω T Sat(δ ω) + δ ω T ∆−1 d(t) ≤ −k1 kδ ωk2 + k2 kd(t)k2

(33)

for some k1 , k2 > 0. Note that the above inequality always hold for sufficiently large diagonal entries in the matrix ∆ and a positive constant k2 such that δ ω T ∆−1 d(t) ≤ kδ ω T kk∆−1 kkd(t)k ≤ k2 ||d(t)||2 → kδ ω T kk∆−1 kkd(t)k − k2 kd(t)k2 = kd(t)k(k∆−1 kkδ ω T k − k2 kd(t)k) < 0. One can show that δ ω ≡ 0 and d(t) ≡ 0 implies that δ q¯ ≡ 0. This implies that the closed-loop system is weakly zero-detectable and dissipative. By invoking Lemma III.4 one can conclude that the system is iISS, where the input is the disturbance and the output is the SC angular velocity and quaternion. Now we show that the control effort is bounded under the control law (29). First note that ¯ ||R(δ~q)|| ≤ 1 for all δ~ √q, therefore ||Ω|| ≤ c1 . Consequently, under Assumption III.1 one can show that ||u(t)|| ≤ α(1 + 3β ) + c21 ||J|| ≤ umax , ∀t ≥ 0, where umax is given in Definition IV.1. Corollary IV.1. Let the dynamics and kinematics of a single SC be governed by equations (22) and (23) where the subscript j is removed for simplicity. Let the control law be provided as in R +∞ equation (29). Then the closed-loop system is asymptotically stable when 0 d(ξ )dξ < +∞, i.e. δ q¯ → 0 and δ ω → 0 as t → ∞. Proof. The proof follows from Lemmas III.5 and IV.1. V. S PACECRAFT ATTITUDE S ET-P OINT T RACKING WITH B OUNDED I NPUT WITHOUT V ELOCITY F EEDBACK In this section, we extend the results of Lemma IV.1 to the case when the angular velocity is not available for feedback. This scheme will be extended to the synchronized attitude control of SC formation in the next section. The following lemma is presented first to guarantee asymptotic stability of a single SC without requiring angular velocity feedback. Lemma V.1. Consider the SC error kinematics governed by equation (22) and the SC error dynamics governed by equation (23) where the subscript j is removed for simplicity. Let the control input, u, be computed according to the dynamic controller that utilizes no angular velocity measurement, namely  T  ¯  u = − ∆[δ q¯ + β E¯ (δ~q)Sat (−γz + γδ q)] (34) + Ω× J Ω   z˙ = −γz + γδ q¯ where ∆ = αI ∈ R3×3 is a positive definite diagonal matrix (α a positive scalar) and β , γ > 0. Under the control law (34) the closed-loop system is iISS where the input is the disturbance and the output is the SC angular velocity, the SC quaternion and z. Furthermore, under Assumption III.1 the control effort is bounded for all t ≥ 0 and for all initial conditions, i.e. ||u(t)|| ≤ umax

April 25, 2015

DRAFT

15

√ provided that the controller gains α and β are selected such that 0 < α(1+ 3β )+c21 ||J|| ≤ umax , where umax is given in Definition IV.1. Proof. Consider the following radially unbounded iISS-Lyapunov function candidate: 1 1 1 W¯ = δ ω T ∆−1 Jδ ω + δ q¯ T δ q¯ + (1 − δ qˆ4 )2 2 2 2 3 Z −γzk +γδ qˆk β + ∑ Sat(x)dx γ k=1 0

(35)

Note that by using Lemma IV.1 one can show that W¯ is a positive definite function. The time derivative of this function along the trajectories of the closed-loop system is given by β W˙¯ =δ ω T ∆−1 Jδ ω˙ + δ q¯ T δ q˙¯ − (1 − δ qˆ j,4 )δ q˙ˆ j,4 + γ

3

∑ z¨k Sat(˙zk ) k=1

¯ + ∆−1 d(t)] =δ ω [−δ q¯ − β E¯ T (δ~q)Sat (−γz + γδ q) T

+ δ ω T δ q¯ +

β T z¨ Sat(˙z) γ

¯ = − β δ ω T E¯ T (δ~q)Sat (−γz + γδ q) β ˙¯ T Sat(˙z) + δ ω T ∆−1 d(t) (−γ z˙ + γδ q) γ = − β δ ω T E¯ T (δ~q)Sat(˙z) − β z˙ T Sat(˙z) +

+ β δ q˙¯ T Sat(˙z) + δ ω T ∆−1 d(t) By noting the fact that δ ω T E¯ T (δ~q) = δ q˙¯ T , we obtain W˙¯ = − β z˙ T Sat(˙z) + δ ω T ∆−1 d(t) ≤ −k1 k˙zk2 + k2 kd(t)k2

(36)

for some k1 , k2 > 0. Note that the above inequality always hold for sufficiently large diagonal entries in the matrix ∆ and a sufficiently large positive constant k2 . One can show from equation ¯ q) is nonsingular (34) that z˙ ≡ 0 implies δ q˙¯ ≡ 0. This from (22) implies that δ ω ≡ 0, since E(δ~ for δ~q 6= 0. These results can be used along with (20), (23) and (34) to demonstrate that δ q¯ ≡ 0, which implies that δ qˆ4 ≡ ±1. Since γ is positive, (34) also implies that z ≡ 0. Consequently, the closed-loop system is weakly zero-detectable and dissipative. By invoking Lemma III.4 one can conclude that the system is iISS, where the input is the disturbance and the output is the SC angular velocity, quaternion and z. Now, we demonstrate that the control strategy given by equation (34) is bounded for all the initial conditions and t ≥ 0. First, observe that equation (8) and the constraint (19) imply that ¯ q)|| ≤ 1. Also note that by definition −1 ≤ Sat(x) ≤ ¯ ≤ 1. It can also be shown that ||E(δ~ ||δ q|| 1, ∀x ∈ R. Therefore, by utilizing these √ inequalities and the upper bounds derived above, from equation (34), we get: ||u|| ≤ α(1 + 3β ) + c21 ||J||. Therefore, under Assumption III.1 by proper selection of the controller gains ∆ and β one can ensure that ||u(t)|| remains less that a maximum

April 25, 2015

DRAFT

16

allowable torque, i.e. ||u(t)|| ≤ umax , ∀ t ≥ 0. Corollary V.1. Let the dynamics and kinematics of a single SC be governed by equations (22) and (23) where the subscript j is removed for simplicity. Let the control law be provided as in R equation (34). Then the closed-loop system is asymptotically stable when 0+∞ d(ζ )dζ < +∞, i.e. z → 0, δ q¯ → 0 and δ ω → 0 as t → ∞. Proof. The proof follows from Lemmas III.5 and V.1. VI. D ISTRIBUTED AND C OOPERATIVE F ORMATION ATTITUDE S YNCHRONIZATION S ET-P OINT T RACKING WITH B OUNDED I NPUT

AND

Our main result in this section is provided in the following theorem. Theorem VI.1. Consider a network of m (m > 1) SC with the dynamics and kinematics as given by equation (10). It is assumed that the desired coordinates are provided to only “l” (l ≤ m) SC in the network that are designated as the network leaders. Let the error dynamics and kinematics of the j-th SC be governed by equations (22) and (23). Consider the following distributed control law for the j-th leader SC, "  # m β jn uleader = − ∆ j δ q¯ j − ∑ λ jn Sat q¯ jn + β jn δ ω j + ω jn + Ω×j J j Ω j (37) j 2 n=1 and the following distributed control law for the j-th follower SC (which does not receive the desired coordinates),   m β jn follower ω jn + Ω×j J j Ω j (38) uj = −∆ j ∑ λ jn Sat q¯ jn + β jn δ ω j + 2 n=1 where ∆ j = α j I ∈ R3×3 is a positive definite diagonal matrix (α j is a positive scalar) and β jn is a positive constant that satisfy the equality β jn = βn j . Furthermore, we assume the communication network is bidirectional and connected, where λ jn is defined as in Section III(A). The application of the control laws (37) and (38) will guarantee that the spacecraft synchronize their states and the set-point tracking error is asymptotically stable, i.e. q¯ jn → 0 and ω jn → 0 as t → ∞, in addition, δ q¯ j → 0 and δ ω j → 0 as t → ∞ for all spacecraft in the network (both leaders and followers). Additionally, the j-th control effort is bounded for all time and for all initial conditions,

max

i.e. u j (t) ≤ u j , provided that the controller gains α j and λ jn for the leader SC are

√ 2 kJ k ≤ umax , and for the follower SC are selected selected such that α j + ∑m 3λ + c

j jn i n=1

1 √

2 max such that ∑m n=1 3λ jn + c1 kJi k ≤ u j . Proof: Consider the following radially unbounded decrescent Lyapunov function candidate

April 25, 2015

DRAFT

17

for the SC formation, m

  m 1 T −1 T 2 U = ∑ β jn δ ω j ∆ j J j δ ω j + ∑ β jn δ q¯ j δ q¯ j + (1 − δ qˆ j,4 ) j=1 2 j=1   m m 1 T 2 + ∑ ∑ β jn λ jn q¯ jn q¯ jn + (1 − qˆ jn,4 ) 2 j=1 n=1

(39)

Note that ∆−1 j is a positive definite diagonal matrix and λ jn ≥ 0, which essentially implies that the above function is positive definite. The time derivative of the Lyapunov function candidate along the trajectories of the closed-loop system (22), (23), (37) and (38) and by neglecting the disturbance d(t) is given by   m 1 m m T T U˙ = ∑ β jn −δ ω j δ q¯ j + δ ω j δ q¯ j + ∑ ∑ β jn λ jn ω Tjn q¯ jn 2 j=1 n=1 j=1   m m β jn T ω jn − ∑ ∑ β jn λ jn δ ω j Sat q¯ jn + β jn δ ω j + 2 j=1 n=1 m T¯ =1 m T ¯ , where by using equations (26) and (28) one can show that ∑mj=1 ∑m jn jn n=1 β jn λ jn δ ω j q 2 ∑ j=1 ∑n=1 β jn λ jn ω jn q and where q¯ jn = Sat(¯q jn ), and which can be re-written as "  #  m m β jn U˙ = − ∑ ∑ β jn λ jn δ ω Tj Sat q¯ jn + β jn δ ω j + ω jn − Sat(¯q jn ) 2 j=1 n=1 # "   β 1 m m jn = − ∑ ∑ β jn λ jn δ ω Tj Sat q¯ jn + β jn δ ω j + ω jn − Sat(¯q jn ) 2 j=1 n=1 2 # "   β jn 1 m m T − ∑ ∑ β jn λ jn δ ω j Sat q¯ jn + β jn δ ω j + ω jn − Sat(¯q jn ) 2 j=1 n=1 2 "  #  β 1 m m jn = − ∑ ∑ β jn λ jn δ ω Tj Sat q¯ jn + β jn δ ω j + ω jn − Sat(¯q jn ) 2 j=1 n=1 2 "  #  β 1 m m jn − ∑ ∑ β jn λ jn ω Tjn Sat q¯ jn + β jn δ ω j + ω jn − Sat(¯q jn ) 4 j=1 n=1 2 #  T "   β jn β jn 1 m m = − ∑ ∑ λ jn β jn δ ω j + ω jn Sat q¯ jn + β jn δ ω j + ω jn − Sat(¯q jn ) 2 j=1 n=1 2 2

Consequently, by noting that χ(x + y) − χ(x) > 0 ⇔ y > 0, and χ(x + y) − χ(x) < 0 ⇔ y < 0 (for χ(x) = Sat(x)), one concludes that U˙ is negative semi-definite, i.e. U˙ ≤ 0 with x , q¯ jn and β

y , β jn δ ω j + 2jn ω jn . β This by invoking Lemma III.3 (where α¯ , U˙ ) implies Rthat β jn δ ω j + 2jn ω jn → 0, therefore ω jn → 0 and δ ω j → 0, j, n ∈ V , j 6= n as t → ∞ when 0∞ d(ζ )dζ < +∞. Consequently, by studying the closed-loop dynamics and invoking Lemma III.2 one can now conclude that since April 25, 2015

DRAFT

18

  β β Sat q¯ jn + β jn δ ω j + 2jn ω jn − Sat(¯q jn ) → 0 and β jn δ ω j + 2jn ω jn → 0, then q¯ jn → 0 as t → ∞. Finally, it follows that δ q¯ j → 0 since given U → 0 and δ ω j → 0, q¯ jn → 0, and all the terms in U are positive and should also → 0 as t → ∞. Note that from the third term in U it follows that qˆ jn,4 → 1, which again confirms that δ q¯ j → 0. Note that the control laws given by equations (37) and (38) are indeed distributed as each SC control strategy is dependent on exchanges of information with its own nearest neighboring set and SC. Now we show

that the control effort is bounded under the control laws (37) and (38).

First note that δ q¯ j

≤ 1 for all times. Consequently, one can show that for all the leader SC one

√

has u j (t) ≤ umax 3λ jn +

provided that the controller gains are selected such that α j + ∑m n=1 j





max

max 2

c1 kJi k ≤ u j . Similarly, one can show that for all the follower SC one has u j (t) ≤ u j

√ 2 kJ k ≤ umax . This provided that the controller gains are selected such that ∑m 3λ + c

jn n=1 j 1 i completes the proof of the theorem.  VII. D ISTRIBUTED AND C OOPERATIVE F ORMATION ATTITUDE S YNCHRONIZATION AND S ET-P OINT T RACKING C ONTROL WITH B OUNDED I NPUT WITHOUT V ELOCITY F EEDBACK Our main result of this section is now provided in the following theorem. Theorem VII.1. Consider a network of m (m > 1) SC with the dynamics and kinematics as given by equation (10). It is assumed that the desired coordinates are provided to only “l” (l ≤ m) SC in the formation that are designated as the leaders. Consider the following distributed control law for the j-th leader SC, "     uleader = − ∆ j δ q¯ j  j     "     m
   − ∑ λ jn Sat q¯ jn + β jn E¯ T (δ~q j ) −γz j + γδ q¯ j  n=1 (40) ##        + E¯ T (~q jn )(−γz jn + γ q¯ jn ) + Ω×j J j Ω j        z˙ = −γz j + γδ q¯ j   j z˙ jn = −γz jn + γ q¯ jn and the following distributed control law for the j-th follower SC (which does not receive the desired attitude coordinates),  "  m 
 follower  ¯ T (δ~q j ) −γz j + γδ q¯ j ¯ u = − ∆ λ Sat q + β E  j jn jn ∑ jn j    n=1    # (41) + E¯ T (~q jn )(−γz jn + γ q¯ jn ) + Ω×j J j Ω j        ¯   z˙ j = −γz j + γδ q j  z˙ jn = −γz jn + γ q¯ jn April 25, 2015

DRAFT

19

where ∆ j = α j I ∈ R3×3 is a positive definite diagonal matrix (α j is a positive scalar), β jn = βn j > 0 and γ > 0. Furthermore, we assume the communication network is bidirectional and connected, where λ jn is defined as in Section III(A). The application of the control laws (40) and (41) will guarantee that all the spacecraft in the network synchronize their states and set-point tracking error is asymptotically stable, i.e. q¯ jn → 0, z jn → 0 and ω jn → 0 as t → ∞, in addition, δ q¯ j → 0, z j → 0 and δ ω j → 0 as t → ∞ for all spacecraft in the network (both leaders and followers). Furthermore, the control effort is bounded



max

for all time and for all initial conditions, i.e. u j (t) ≤ u j , provided that the controller gains

√ 2 kJ k ≤ umax , and α j and λ jn for the leader SC are selected such that α j + ∑m 3λ + c

n=1 jn 1 i j √

max 2 for the follower SC are selected such that ∑m n=1 3λ jn + c1 kJi k ≤ u j . Proof: Consider the following positive definite radially unbounded Lyapunov function candidate for the SC formation,  l  1 1 1 T −1 T 2 U¯ = ∑ δ ω j ∆ j J j δ ω j + δ q¯ j δ q¯ j + (1 − δ qˆ j,4 ) 2 2 j=1 2 β jn 3 −γzk, j +γδ q¯k, j x dx ∑ 0 γ j=1 k=1  h  i β 3 Z −γzk, jn +γ q¯k, jn m m 1 T p jn 2 + ∑ ∑ λ jn q¯ Λ q¯ + (1 − qˆ jn,4 ) + x dx ∑ 0 4 jn jn jn 2γ k=1 j=1 n=1 m

+∑

Z

(42)

The time derivative of this function along the trajectories of the closed-loop system by neglecting

April 25, 2015

DRAFT

20

the disturbance d(t) can be computed as: l l m β jn U˙¯ = − ∑ δ ω Tj δ q¯ j + ∑ δ ω Tj δ q¯ j + ∑ (−γ z˙ j + γ δ q˙¯ j )T z˙ j γ j=1 j=1 j=1     m m T T T − ∑ ∑ λ jn δ ω j Sat q¯ jn + β jn E¯ (δ~q j )˙z j + E¯ (~q jn )˙z jn j=1 n=1

 β jn 1 T T (−γ z˙ jn + γ q˙¯ jn ) z˙ jn − q¯ jn ω jn + 2 2γ m β jn (−γ z˙ j + γ δ q˙¯ j )T z˙ j =∑ j=1 γ     m m T T T − ∑ ∑ λ jn δ ω j Sat q¯ jn + β jn E¯ (δ~q j )˙z j + E¯ (~q jn )˙z jn j=1 n=1

 β jn 1 T T ˙ (−γ z˙ jn + γ q¯ jn ) z˙ jn − q¯ jn ω jn + 2 2γ m β jn =∑ (−γ z˙ j + γ δ q˙¯ j )T z˙ j γ j=1     m m T T T ¯ ¯ − ∑ ∑ λ jn δ ω j Sat q¯ jn + β jn E (δ~q j )˙z j + E (~q jn )˙z jn j=1 n=1

 m − Sat(¯q jn ) + ∑

m

∑ λ jn

j=1 n=1



β jn (−γ z˙ jn + γ q˙¯ jn )T z˙ jn 2γ



which essentially has the same sign as the following expression,    m β m m jn T ˙ Ξ2 =k1 ∑ (−γ z˙ j + γ δ q¯ j ) z˙ j − k1 ∑ ∑ λ jn β jn δ ω Tj E¯ T (δ~q j )˙z j + E¯ T (~q jn )˙z jn j=1 γ j=1 n=1  β jn − (−γ z˙ jn + γ q˙¯ jn )T z˙ jn 2γ for some positive constant k1 . By noting the facts that ω Tjn E¯ T (~q jn ) = 2q˙¯ Tjn , δ ω Tj E¯ T (δ~q j ) = 2δ q˙¯ Tj and β jn = βn j one can show that     1 n m T ¯T T ˙ ∑ ∑ λ jn δ ω j E (~q jn)˙z jn − 2 q¯ jnz˙ jn = 0 j=1 n=1 and δ ω Tj

  T ¯ E (δ~q j )˙z j − δ q˙¯ Tj z˙ j = 0

Therefore, one can further simplify Ξ2 as follows, m

m

j=1

April 25, 2015

m

β jn T z˙ jn z˙ jn ≤ 0 j=1 n=1 2

Ξ2 = −k1 ∑ β jn z˙ Tj z˙ j − k1 ∑

∑

(43)

DRAFT

21

This by invoking Lemma III.3, essentially implies that z˙ jn → 0 as t → ∞, which by noting strong connectivity of the communication graph implies that z˙ j → 0 as t → ∞. One can show, from equations (40) and (41), that when z˙ jn = z¨ jn ≡ 0 we have q˙¯ jn ≡ 0. This result can be used along with equation (27) to show that ω jn ≡ 0. Furthermore, when z¨ j ≡ 0 and z˙ j ≡ 0, j ∈ V one can use equation (40) to show that δ q˙¯ j ≡ 0. This from equation (22) implies that δ ω j ≡ 0, j ∈ V . Consequently, by studying the closed-loop dynamics and invoking Lemma III.2 one can now conclude that q¯ jn → 0 and δ q¯ j → 0 as t → ∞. Therefore, since γ is positive, equations (40) and (41) also imply that z jn ≡ 0 and z j ≡ 0. Consequently, all the spacecraft in the network synchronize their states and set-point tracking error is asymptotically stable. that

Now

we show that the control effort is bounded under the control laws (40) and (41). Note

δ q¯ j ≤ 1 for all time. Consequently, one can show that for all the leader SC one has u j (t) ≤



√

max

max m 2

u j provided the controller gains are selected such that α j + ∑n=1 3λ jn + c1 kJi k ≤ u j .



Similarly, one can show that for all the follower SC one has u j (t) ≤ umax j provided the

√

max 2 controller gains are selected such that ∑m n=1 3λ jn + c1 kJi k ≤ u j . This now completes the proof of the theorem.  VIII. S IMULATION AND C ASE S TUDIES In this section, first the performance of our proposed constrained controller with and without requiring velocity feedback is studied for a single SC. The obtained simulation results are then compared with the velocity free control scheme that is proposed in [13], [20]. In the second part of this section, the performance of our proposed distributed and cooperative constrained velocityfree SC formation flying control strategies is evaluated in simulations. Note that for simulations it is assumed that the desired angular velocity and its rate are zero, i.e. ω ? = ω˙ ? = 0, ∀t > 0. A. Single SC Attitude Control We demonstrate performance of our constrained attitude controllers for a single SC in this subsection. The initial conditions considered in this part of simulations are δ ω(0) = [−1, 1, 1.5]T and δ q(0) = [0.2, 0.2, 0.2, 0.9381]T . The considered SC inertia matrix is given by   5 0 0 J = 0 5 0 . 0 0 6 Furthermore, the following saturation function is namely,  if  1 α x if Sat(αx) =  −1 if

considered in conducting the simulations, α x>1 −1 < α x < 1 α x < −1

(44)

where the parameter α in the saturation function (44) is set to 5. Note that the parameter α does not play any role in the stability of the closed-loop system. Parameters of the constrained controller (29) are selected as: ∆ = 0.5I3 and β = 1. For the constrained velocity free controller (34) we select the following controller gains, namely, ∆ = 0.5I3 , β = 1, γ = 20 and z(0) = April 25, 2015

DRAFT

22

1

0

1

u [N.m]

0.5

−0.5 −1

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

1

0

2

u [N.m]

0.5

−0.5 −1

1 Tayebi, 2008: β = 2 with no vel. feedback with vel. feedback

0

3

u [N.m]

0.5

−0.5 −1

0

10

20

30

40

50 time [sec]

60

70

80

90

100

Fig. 1. Control efforts for the attitude control of a single SC with α = 5.

[10, 10, 10]T . This selection of gains guarantees that the control effort does not exceed umax = 1 for both constrained controllers. For sake of clarification, we compare the performance of our proposed controller with the velocity free controller that is proposed in [13], [20]. The controller gains for the controller in [13], [20] are selected such that the maximum control effort does not exceed umax = 1. For that controller the parameter β = 2 was obtained by trial and error that yields the best response. The control efforts for the considered three controllers (our two proposed controllers and the one in [13], [20]) are depicted in Fig. 1. The SC angular velocity and attitude errors are depicted in Fig. 2 and Fig. 3, respectively. It follows from Fig. 1 that the constraint on the maximum control effort is satisfied by all the three controllers. However, our proposed constrained controllers (29) and (34) use the control input more efficiently and bring the SC into rest considerably faster when compared to the controller proposed in [13], [20] as verified and demonstrated in Fig. 2 and Fig. 3. B. SC Formation Control without Requiring Velocity Feedback In this subsection, the performance of our proposed cooperative control schemes for distributed synchronization and set-point tracking control of multiple spacecraft with saturation constraints in the network with and without velocity feedback is evaluated and compared to the performance of the velocity-free controller that is proposed in [13], [20]. We consider four spacecraft in the network in the ring topology and assume that the desired spacecraft coordinates are provided to the first spacecraft (that is, the leader SC). The initial conditions for the spacecraft in the network are set to be the same as the initial conditions in [13], [20], i.e. we set δ~q1 (0) = [0, 0, 1, 0]T , April 25, 2015

DRAFT

23

0.6

0.5

0.4

0

δ q2

δ q1

0.2

−0.2

0

−0.4 −0.6 0

20

40

60

80

−0.5

100

1

1

0.5

0.5

δ q4

δ q3

−0.8

0

−0.5

−1

0

20

40

20

40 60 time [sec]

80

−1

100

80

100

0

−0.5

0

60

Tayebi, 2008: β = 2 with no vel. feedback with vel. feedback 0

20

40 60 time [sec]

80

100

Fig. 2. Attitude errors for a single SC with α = 5.

1

1

ω [N.m]

0.5 0 −0.5 −1 −1.5

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

1.5

0.5

2

ω [N.m]

1

0 −0.5

2 Tayebi, 2008: β = 2 with no vel. feedback with vel. feedback

3

ω [N.m]

1.5 1 0.5 0 −0.5

0

10

20

30

40

50 time [sec]

60

70

80

90

100

Fig. 3. Angular velocity errors for a single SC with α = 5.

April 25, 2015

DRAFT

24

TABLE I T HE TWO SETS OF SELECTED CONTROLLER GAINS

Set #

∆j

λ jn

β jn

1 2

20 2

30 3

0.6 0.6

max max , j = 1, 2, 3

u u

j 1 80 8

60 6

δ~q2 (0) = [1, 0, 0, 0]T , δ~q3 (0) = [0, 1, 0, 0]T , δ~q4 (0) = [0, 0, sin(−π/4), cos(−π/4)]T , δ ω1 (0) = [−0.5, 0.5, −0.45]T , δ ω2 (0) = [0.5, −0.3, 0.1]T , δ ω3 (0) = [0.1, 0.6, −0.1]T , and δ ω4 (0) = [0.4, 0.4, −0.5]T . The moment of inertia matrices are set to J j = Diag(20, 20, 30)kg − m2 , j = {1, . . . , 4}. The desired attitude is also set to ~q?j = [0, 0, 0, −1]T . For conducting simulations two sets of controller gains are considered as provided in Table I. The bounds on the maximum control efforts for the leader and the followers for the controller gain Set #1 is ten times larger than the maximum control efforts of the leader and the followers for the controller gain Set #2. It is important to note that by this selection of the controller gains more emphasis is now placed on the formation-keeping requirement as opposed to the station-keeping requirement, due to the fact that this is more important in networked spacecraft control missions. Additionally, the gain γ is set to 20 for the velocity-free control scheme. The response of the closed-loop networked spacecraft using the controller gains Set #1 with the velocity feedback and exchange among the SC are depicted in Fig. 4. Fig. 5 depicts the response of the closed-loop networked spacecraft using the controller gains Set #1 without the velocity feedback and exchange among the SC. One can observe from the figures (not all the attitude and angular velocity synchronization and set-point tracking signals are shown due to space limitations) that the attitude and angular velocity synchronization and set-point tracking have been achieved and the bounds on the control efforts are ensured and respected. The response of the closed-loop networked spacecraft using the controller gains Set #2 with the velocity feedback and exchange among the agents are depicted in Fig. 6. This figure shows that attitude and angular velocity synchronization and set-point tracking have been achieved and the bounds on the control efforts are ensured and respected. Fig. 7 depicts the response of the closed-loop networked spacecraft using the controller gains Set #2 without the velocity feedback and exchange among the SC. One can observe from the figures (not all the attitude and angular velocity synchronization and set-point tracking signals are shown due to space limitations) that the attitude and angular velocity synchronization and set-point tracking have been achieved and the bounds on the control efforts are ensured and respected. Next, we compare the performance of our proposed constrained distributed controllers with the controller gains in Sets #1 and #2 to study the effects of changing the maximum control efforts on the performance of the networked spacecraft. We execute ten Monte Carlo simulation studies. The initial conditions are selected such that δ qr, j (0) ∈ [0, 0.5] and δ ωr, j (0) ∈ [−0.5, 0.5], where r ∈ {1, 2, 3} and j ∈ {1, . . . , 4}. We consider five performance measures as provided in Table II. Table II summarizes the results by using the velocity-feedback controller and Table III summarizes the results by using the velocity-free controller. One can conclude from results in Table II that by using a higher control effort, the error in the attitude and angular velocity April 25, 2015

DRAFT

1

1

0.5

0.5

q2

q1

25

0

−0.5

−0.5

−1

−1 0

20

40

60

80

100

0

1

1

0.5

0.5

q4

q3

0

0

−0.5

20

40

60

80

100

SC1 (leader) SC2 SC3 SC4

0

−0.5

−1

−1 0

20

40 60 time [sec]

80

100

0

20

40 60 time [sec]

80

100

1

1

0.5

0.5

q2

q1

Fig. 4. The quaternion of the four spacecraft in the network with velocity feedback with the controller gains Set #1. Spacecraft #1 receives the desired attitude.

0

−0.5

−0.5

−1

−1 0

50

100

150

0

1

1

0.5

0.5

q4

q3

0

0

−0.5

50

100

150

SC1 (leader) SC2 SC3 SC4

0

−0.5

−1

−1 0

50

100 time [sec]

150

0

50

100

150

time [sec]

Fig. 5. The quaternion of the four spacecraft in the network without velocity feedback with the controller gains Set #1. Spacecraft #1 receives the desired attitude.

synchronization as well as the error in the angular velocity set-point tracking are reduced. However, the attitude set-point tracking error has increased. This change in the performance comes with a high price, which is an increase in the overall spacecraft control effort by more than 640% for the controller with the gain Set #1. From the results provided in Table III one can conclude that by increasing the bound on the control efforts and by using our distributed velocity-free controllers one can improve the attitude synchronization and set-point tracking performance. However, one can observe degradations in the angular velocity synchronization April 25, 2015

DRAFT

1

1

0.5

0.5

q2

q1

26

0

−0.5

−0.5

−1

−1 0

20

40

60

80

100

0

1

1

0.5

0.5

q4

q3

0

0

−0.5

20

40

60

80

100

SC1 (leader) SC2 SC3 SC4

0

−0.5

−1

−1 0

20

40 60 time [sec]

80

100

0

20

40 60 time [sec]

80

100

1

1

0.5

0.5

q2

q1

Fig. 6. The quaternion of the four spacecraft in the network with velocity feedback with the controller gains Set #1. Spacecraft #1 receives the desired attitude.

0

−0.5

−0.5

−1

−1 0

50

100

150

0

1

1

0.5

0.5

q4

q3

0

0

−0.5

50

100

150

SC1 (leader) SC2 SC3 SC4

0

−0.5

−1

−1 0

50

100 time [sec]

150

0

50

100

150

time [sec]

Fig. 7. The quaternion of the four spacecraft in the network without velocity feedback with the controller gains Set #2. Spacecraft #1 receives the desired attitude.

and set-point tracking performance. This change in the performance comes with a significant (more than 4400%) increase in the overall control effort by the spacecraft in the network. Now, we study the team performance degradation when the velocity feedback is not available for our proposed constrained control algorithms. Table IV summarizes the results. This table shows that the performance of the networked spacecraft considerably degrades when the velocity information is not available for feedback and exchange. Absence of the velocity feedback mainly reduces the angular velocity synchronization and set-point tracking performance of the system, April 25, 2015

DRAFT

27

TABLE II P ERFORMANCE OF OUR PROPOSED CONSTRAINED CONTROLLERS with ANGULAR VELOCITY FEEDBACK

Performance measure

Controller with gain Set #1

Controller with gain Set #2

Performance change of the gain Set #1 with respect to gain Set #2

3000 ∑3i=1 ∑4j=1 0 q2i, j (t)dt R 3000 ∑3i=1 ∑4j=1 0 ωi,2 j (t)dt R 3000 ∑3i=1 ∑4j=2 0 q2j1,i (t)dt R 3000 ∑3i=1 ∑4j=2 0 ω 2j1,i (t)dt R 3000 ∑3i=1 ∑4j=1 0 u2i, j (t)dt

111.93 2.43 6.45 0.6 770.36

90.49 4.82 10.29 3.31 103.43

-23.7% 98.35% 59.52% 455.3% -644.78%

R

TABLE III P ERFORMANCE OF OUR PROPOSED CONSTRAINED CONTROLLERS without ANGULAR VELOCITY FEEDBACK

Performance measure

Controller with gain Set #1

Controller with gain Set #2

Performance change of the gain Set #1 with respect to gain Set #2

3000 ∑3i=1 ∑4j=1 0 q2i, j (t)dt R 3000 ∑3i=1 ∑4j=1 0 ωi,2 j (t)dt R 3000 ∑3i=1 ∑4j=2 0 q2j1,i (t)dt R 3000 ∑3i=1 ∑4j=2 0 ω 2j1,i (t)dt R 3000 ∑3i=1 ∑4j=1 0 u2i, j (t)dt

27.67 161.81 4.315 19.817 25312

165.88 39.27 29.32 16.67 555.49

499.57% -312.09% 579.42% -18.88% -4456.7%

R

and it also results in an increase in the overall control effort of the spacecraft in the network. Finally, we compare the performance of our proposed distributed attitude synchronization controllers with the velocity-free controller proposed in [13], [20]. We consider the network with the same setup and let the maximum control effort for the leader be bounded by 8 N − m and the follower control effort be bounded by 6 N − m. The results of ten Monte Carlo simulation studies are provided in Table V. This table clearly shows that our proposed constrained velocity-free controller outperforms the velocity-free controller proposed in [13], [20] in terms of attitude and angular velocity synchronization and attitude set-point tracking performance with a significant margin. In addition, our proposed velocity-free controller consumes much less energy, 437% less than the controller proposed in [13], [20], which is critically important for networked spacecraft missions as the thruster fuel is limited. It should be noted that the controller proposed in [13], [20] produced lower angular velocity set-point error when compared to our proposed velocityfree controller. However, this should not be seen as a drawback for our proposed algorithm since in networked spacecraft control problems, attitude and velocity synchronization errors are considerably more important than the single spacecraft angular velocity set-point tracking error.

April 25, 2015

DRAFT

28

TABLE IV P ERFORMANCE LOSS IN ABSENCE OF ANGULAR VELOCITY FEEDBACK

Performance measure

Performance change in absence of velocity feedback

3000 ∑3i=1 ∑4j=1 0 q2i, j (t)dt R 3000 ∑3i=1 ∑4j=1 0 ωi,2 j (t)dt R 3000 ∑3i=1 ∑4j=2 0 q2j1,i (t)dt R 3000 ∑3i=1 ∑4j=2 0 ω 2j1,i (t)dt R 3000 ∑3i=1 ∑4j=1 0 u2i, j (t)dt

-83.33% -714.41% -185.04% -404.03% -437.05%

R

TABLE V P ERFORMANCE COMPARISON AND ANALYSIS .

Performance measure

Velocity-free controller [13], [20]

Performance change when compared with our velocity-free controller

3000 ∑3i=1 ∑4j=1 0 q2i, j (t)dt R 3000 ∑3i=1 ∑4j=1 0 ωi,2 j (t)dt R 3000 ∑3i=1 ∑4j=2 0 q2j1,i (t)dt R 3000 ∑3i=1 ∑4j=2 0 ω 2j1,i (t)dt R 3000 ∑3i=1 ∑4j=1 0 u2i, j (t)dt

423.55 32.62 123.22 34.00 707.32

+155.33% -20.37% +320.31% +103.97% +27.33%

R

IX. C ONCLUSIONS In this paper, we consider distributed and cooperative controller design and development for the attitude synchronization and set-point tracking control for spacecraft formation flying missions. The control laws are developed subject to two constraints, namely, (1) constraints on the maximum control effort available to each SC in the network, and (2) unavailability of velocity measurements. The closed-loop performance of the control strategies that are proposed in this paper are evaluated extensively through numerical simulations and compared with other similar methods [13] and [20] that are available in the literature. One of the future directions of research is to extend our proposed controllers for the general problem of trajectory tracking and not only set point tracking that is considered in this work. A PPENDIX Our objective is to show that:   × × × ϒ := δ ω Ω j JΩ − ω Jω + Jω Ω = 0 T

April 25, 2015

(45)

DRAFT

29

Let us write the above expression as:
ϒ =(ω − Ω)T Ω× JΩ − ω × Jω + Jω × Ω =ω T Ω× JΩ − ω T ω × Jω + ω T Jω × Ω − ΩT Ω× JΩ + ΩT ω × Jω − ΩT Jω × Ω

(46)

Noting ω × ω = −ω T ω × = 0, one obtains ω T ω × Jω = ΩT Ω× JΩ = 0. Therefore, (45) can be simplified as follows: ϒ =ω T Ω× JΩ + ω T Jω × Ω + ΩT ω × Jω − ΩT Jω × Ω

(47)

By noting (ω × )T = −ω × , the above can be re-written as: ϒ =ω T Ω× JΩ + ω T Jω × Ω − ΩT Jω × ω − ΩT Jω × Ω =ω T Ω× JΩ − ΩT Jω × Ω

(48)

Since ω T Ω× = ω × Ω, we can now conclude that ϒ = 0, which confirms our objective that (45) holds. R EFERENCES [1] S. Unwin and C. Beichman, “Terrestrial planet finder: science and technology overview,” SPIE Astronomical Telescopes and Instrumentation, 2004. [2] B. Mennesson, “Expected science capabilities of the tpf interferometer,” SPIE Astronomical Telescopes and Instrumentation, 2004. [3] L. Kaltenegger, Search for Extra-Terrestrial planets: The DARWIN mission - Target Stars and Array Architectures. Karl Franzens University, Graz, Austria: Ph.D. dissertation, 2004. [4] R. Beard, J. Lawton, and F. Y. Hadaegh, “A coordination architecture for spacecraft formation control,” IEEE Trans. Contr. Syst. Tech., vol. 9, no. 6, pp. 777–789, 2001. [5] P. Wang and F. Hadaegh, “Coordination and control of multiple microspacecraft moving in formation,” J. Astro. Sci., vol. 44, no. 3, pp. 315–355, 1996. [6] P. Wang, F. Hadaegh, and K. Lau, “Synchronized formation rotation and attitude control of multiple free-flying spacecraft,” J. Guid., Contr., and Dynamics, vol. 22, no. 1, pp. 28–35, 1999. [7] V. Kapilal, A. G. Sparks, J. M. Buffington, and D. P. Q. Yan, “Spacecraft formation flying: Dynamics and control,” Amer. Contr. Conf., vol. 6, pp. 4137–4141, 1999. [8] H.-H. Y. W. Kang and A. Sparks, “Coordinated control of relative attitude for satellite formation,” AIAA Guid., Nav., and Cont. Conf., 2001. [9] W. Ren and R. Beard, “Decentralized scheme for spacecraft formation flying via the virtual structure approach,” AIAA J. Guid., Contr., and Dyn., vol. 27, no. 1, pp. 73–82, 2004. [10] R. B. J. Lawton and F. Hadaegh, “Elementary attitude formation maneuvers via leader-following and behavior-based control,” AIAA Guid., Nav., and Cont. Conf., 2000. [11] M. VanDyke and C. D. Hall, “Decentralized coordinated attitude control within a formation of spacecraft,” J. Guid., Contr., and Dyn., vol. 29, no. 5, pp. 1101–1109, 2006. [12] R. B. J.R. Lawton, “Synchronized multiple spacecraft rotations,” Automatica, vol. 38, no. 8, pp. 1359–1364, 2002. [13] A. Abdessameud and A. Tayebi, “Attitude synchronization of a group of spacecraft without velocity measurements,” IEEE Trans. on Auto. Contr., vol. 54, no. 11, pp. 2642–2648, 2009. [14] W. Ren, “Distributed cooperative attitude synchronization and tracking for multiple rigid bodies,” IEEE Trans. on Contr. Syst. Techno., vol. 18, no. 2, pp. 383–392, 2010. [15] R. Kristiansena, A. Loria, A. Chailletc, and P.-J. Nicklasson, “Spacecraft relative rotation tracking without angular velocity measurements,” Automatica, vol. 45, no. 3, pp. 750–756, 2009. [16] W. Kang and H.-H. Yeh, “Co-ordinated attitude control of multi-satellite systems,” Int. J. Robu. & Nonlinear Contr., vol. 12, no. 2–3, pp. 185–205, 2002. [17] W. Kang and A. Sparks, “Coordinated attitude and formation control of multisatellite systems,” AIAA Guid., Nav., and Cont. Conf., 2002.

April 25, 2015

DRAFT

30

[18] F. Lizarralde and J. Wen, “Attitude control without angular velocity measurements: A passivity approach,” IEEE Trans. Auto. Contr., vol. 41, no. 3, pp. 468–472, 1996. [19] M. R. Akella, “Rigid body attitude tracking without velocity feedback,” Syst. and Contr. Lett., vol. 42, pp. 321–326, 2001. [20] A. Tayebi, “Unit quaternion-based output feedback for the attitude tracking problem,” IEEE Trans. Auto. Contr., vol. 53, no. 6, pp. 1516–1520, 2008. [21] Q. H. Bing Xiao and Y. Zhang, “Fault-tolerant attitude control for flexible spacecraft without angular velocity magnitude measurement,” AIAA J. Guid., Contr., and Dyn., vol. 34, no. 5, pp. 1556–1561, 2011. [22] Q. H. Bing Xiao and Y. Zhang, “Fault-tolerant attitude control for spacecraft under loss of actuator effectiveness,” AIAA J. Guid., Contr., and Dyn., vol. 34, no. 3, pp. 927–932, 2011. [23] W. Ren and R. Beard, “Virtual structure based spacecraft formation control with formation feedback,” AIAA Guid., Nav., and Cont. Conf., 2002. [24] M. VanDyke and C. Hall, “Decentralized coordinated attitude control within a formation of spacecraft,” AIAA J. Guid., Contr., and Dyn., vol. 29, no. 5, pp. 1101–1109, 2006. [25] J. Erdonga, J. Xiaoleib, and S. Zhaoweia, “Robust decentralized attitude coordination control of spacecraft formation,” Syst. & Contr. Lett., vol. 57, no. 7, pp. 567–577, 2008. [26] D. V. Dimarogonas, P. Tsiotras, and K. J. Kyriakopoulos, “Leaderfollower cooperative attitude control of multiple rigid bodies,” Syst. & Contr. Lett., vol. 58, no. 6, pp. 429–435, 2009. [27] Z. Meng, W. Ren, and Z. You, “Decentralized cooperative attitude tracking using modified rodriguez parameters,” IEEE Conf. Deci. Contr., pp. 853–858, 2009. [28] K. Zhang and M. A. Demetriou, “Adaptive attitude synchronization control of spacecraft formation with adaptive synchronization gains,” Journal of Guidance, Control, and Dynamics, vol. 37, no. 5, pp. 1644–1651, 2014. [29] S. Weng and D. Yue, “Distributed event-triggered cooperative attitude control of multiple rigid bodies with leaderfollower architecture,” International Journal of Systems Science, p. DOI:10.1080/00207721.2014.891777, 2014. [30] A.-M. Zou and K. D. Kumar, “Quaternion-based distributed output feedback attitude coordination control for spacecraft formation flying,” Journal of Guidance, Control, and Dynamics, vol. 36, no. 2, pp. 548–556, 2013. [31] R. Ortega, P. J. Nicklasson, and H. Sira-Ramirez, Passivity-Based Control of Euler-Lagrange Systems: Mechanical, Electrical, and Electromechanical Applications. New York, NY: Springer-Verlag, 1998. [32] S. A. A. Moosavian and E. Papadopoulos, “Explicit dynamics of space free-flyers with multiple manipulators via spacemaple,” J. of Adv. Robot., vol. 18, no. 2, pp. 223–244, 2004. [33] S. Sagatun and T. Fossen, “Lagrangian formulation of underwater vehicles’ dynamics,” IEEE Int. Conf. Syst., Man, and Cyber., vol. 2, pp. 1029–1034, 1991. [34] H. Schaub and J. L. Junkins, Analytical mechanics of aerospace systems. AIAA, 2002. [35] B. Wie, Space Vehicle Dynamics and Control. Reston, VA: AIAA Education Series, 1998. [36] J. Ahmed, V. T. Coppola, and D. Bernstein, “Adaptive asymptotic tracking of spacecraft attitude motion with inertia matrix identification,” AIAA J. Guid. Navig. and Contr., vol. 21, no. 5, pp. 684–691, 1998. [37] T. Krogstad and J. Gravdahl, “6-dof mutual synchronization of formation flying spacecraft,” IEEE Conf. Deci. Contr., pp. 5706–5711, 2006. [38] S.-J. Chung, U. Ahsun, and J.-J. Slotine, “Application of synchronization to formation flying spacecraft: Lagrangian approach,” AIAA J. of Guid., Contr. and Dyn., vol. 32, no. 2, pp. 512–526, 2009. [39] C. Godsil and G. Royle, Algebraic Graph Theory. New York, NY: Springer-Verlag, 2001. [40] C. Wu, “Agreement and consensus problems in groups of autonomous agents with linear dynamics,” IEEE Int. Symp. Circui. and Syst., pp. 292–295, 2005. [41] H. Tanner, A. Jadbabaie, and G. J. Pappas, “Flocking in fixed and switching networks,” IEEE Trans. on Auto. Contr., vol. 52, no. 5, pp. 863–868, 2007. [42] Y. Kim and M. Mesbahi, “Quadratically constrained attitude control via semidefinite programming,” IEEE Transactions on Automatic Control, vol. 49, pp. 731–735, 2014. [43] H. K. Khalil, Nonlinear Systems. Upper Saddle River, NJ: Prentice Hall, 2002. [44] D. Angeli, E. D. Sontag, and Y. Wang, “A characterization of integral input-to-state stability,” IEEE Transactions on Automatic Control, vol. 45, no. 6, pp. 1082–1096, 2000. [45] E. Sontag, “Input to state stability: Basic concepts and results,” Nonlinear and Optimal Control Theory, vol. 1932, pp. 163– 220, 2008.

April 25, 2015

DRAFT