# Quaternion-Based Attitude Control System Design of Single and Cooperative Spacecrafts: Boundedness of Solution Approach

Authors: Harry Septanto; Riyanto T. Bambang; Arief Syaichu-Rohman; Ridanto Eko Poetro; Adrianto Ravi Ibrahim
Journal: Modelling and Simulation in Engineering
 (2014)

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## Abstract

It is well known that single equilibrium orientation point in matrix rotation is represented by two equilibrium points in quaternion. This fact would imply nonefficient control effort as well as problem in guaranteeing stability of the two equilibrium points in quaternion. This paper presents a solution to design quaternion-based spacecraft attitude control system whose saturation element is in its control law such that those problems are overcome. The proposed feature of methodology is the consideration on boundedness of solution in the control system design even in the presence of unknown external disturbance. The same methodology is also used to design cooperative spacecrafts attitude control system. Through the proposed method, the most relaxed information-state topology requirement is obtained, that is, the directed graph that contains a directed spanning tree. Some numerical simulations demonstrate effectiveness of the proposed feature of methodology.


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## Body

## 1. Introduction

Over the past decades, spacecraft developments have been done to bring wide-range missions, including earth observation and communication. Attitude control system plays an important role in these missions. Many research efforts on spacecraft attitude control design have been reported, that is, linear matrix inequality- (LMI-) based robust mixedH2/H∞ attitude control system design with linearization of dynamics and kinematics approach [1]; LMI-based nonlinear continuous attitude control design, [2]; proportional-derivative+ (PD+) type output feedback attitude control system with uniformly practically asymptotically stability guarantee for its equilibrium points [3]; attitude control system with discontinuous control law applying inverse cotangent function [4]; and hybrid attitude control system with property of robustness to measurement noise [5], to name a few.From an attitude determination calculation, orientation of spacecraft is obtained as a rotation matrix that belongs to special orthogonal order-3 space,SO(3). However, some parameterizations are usually employed in a design of attitude control that is also employed in [1–5]—all of them employ quaternion parameterization, except [1], which uses Euler angle parameterization.Because quaternion can represent spacecraft attitude globally, it is useful for spacecraft whose missions require doing reorientation maneuver over a large rotation angle. In contrast, Euler angle parameterization cannot represent spacecraft attitude globally due to its singularity property. Note that quaternion—also called Euler parameters—is the only parameterization consisting of four parameters and can represent attitude or orientation globally [6]. Nevertheless, its representation is not unique. There are two-antipodal values that correspond to a single physical orientation or a single orientation in matrix rotation representation. Therefore, single equilibrium orientation point in matrix rotation representation is represented by two equilibrium points in quaternion. This fact would imply nonefficient control effort called unwinding phenomenon when only one equilibrium point is regarded in quaternion-based attitude control design [7]. Besides, difficulty in guaranteeing stability appeared when the two equilibrium points in quaternion are regarded. There are many research results giving solution to this problem. They include discontinuous control approach [4], optimal control and finite time stability approach [8], backstepping-based attitude control design [9], hybrid mechanism attitude control [5, 10], and continuous nonlinear attitude control employing augmented dynamic [11], to name a few.In addition to above problem, numerous spacecraft formation flying missions have been conceived and some of them have been launched, for example, interferometric synthetic aperture radar spacecrafts [12] and rendezvous technology experiment [13]. Cooperative control is a main issue for spacecraft formation control. In [14], Ren proposed proportional-derivative (PD) type attitude controller with several consensus algorithms for attitude formation. Hybrid attitude controller with connected and acyclic exchange of information-state topology for attitude formation has been reported in [15]. In [16], PD type controller with connected (undirected) exchange of information-state topology for attitude formation has been designed through Input to state stability approach where cooperative controller term is regarded as external input of each spacecraft.In this paper, attitude controller appearing in [17] is utilized. Main contributions of this paper are attitude control system designed through boundedness of solution approach, that is, ultimately bounded solution and input to state stability for single attitude control case and boundedness of solution for cooperative attitude control case. Unlike the attitude control system employing PD controller that considers only one equilibrium point in quaternion representation, the attitude controller used in this paper considers two equilibrium points in quaternion representation. Less strict information-state topology requirement for attitude formation is also obtained, that is, topology that consists of a directed spanning tree. Preliminary result of this research has been presented in [18].Overall, this paper is organized as follows. Section2 discusses the system model. This section consists of notation, kinematics, and dynamics model of spacecraft and information-state exchange framework of the cooperative spacecrafts used in the rest of the paper. Section 3 describes controllers design and simulations. Finally, some concluding remarks are stated in Section 4.


## 2. Modeling

In this paper, a spacecraft is considered a rigid body. In the following descriptions, subscripts or/and superscriptsb, l, and d denote spacecraft’s fixed body frame, inertial reference frame, and spacecraft’s desired frame, respectively. For brevity, the spacecraft’s fixed body frame, the inertial reference frame, and the spacecraft’s desired frame may be written asbody frame,inertial frame, anddesired frame, respectively. Here, set of n-dimension real column matrices, set of n×m real matrices, and set of positive integers are denoted by Rn, Rn×m, and Z>0, respectively, where n,m∈Z>0. Given y∈Rn and Y∈Rn×n, ∥y∥ is 2-norm of column matrix y; λmin⁡(Y) and λmax⁡(Y) are minimum and maximum Eigen value of Y, respectively; yT and YT are transpose matrix of y and Y, respectively; Y>0 means that Y is a positive definite matrix.Following [19],vectors are defined as follows:
(1)r→=rbTFb,
where Fb is avectrix associated to the body frame, that is, a column matrix with three unit vectors l^1, l^2, and l^3—Fl=[l^1l^2l^3]T; rb∈R3 is a column matrix whose three components of r→ are expressed or decomposed into the body frame; rbμ is skew-symmetric matrix of rb.


### 2.1. Spacecraft Kinematics and Dynamics Model

Consider the rotation matrixRbd to transform a vector expressed in Fb to be expressed in Fd that satisfies (2):
(2)Rbd=Fb·FdT⟺FdT=FbTRbd.
Taking the time derivative of (2) w.r.t. (with respect to) Fd, 0=FbT(ωbdbμRbd+R˙bd) is satisfied. Moreover, it implies (3), that is, the SO(3) rotation matrix based kinematics equation of body frame w.r.t. desired frame:
(3)R˙bd=-ωbdbμRbd.
The corresponding quaternion-based kinematics equation of body frame w.r.t. desired frame is given by
(4)q˙bd=[η˙bdε˙bdT]T,
where
(5)η˙bd=-12εbdTωbdb,ε˙bd=12(εbdμ+ηbdI)ωbdb.
Since quaternion is member ofunit sphere order-3, S3, qbd=[ηbdεbdT]T satisfies ∥qbd∥2=1, where ηbd=cos⁡(ϕbd/2)∈R and εbd=a^bdsin(ϕbd/2)∈R3; a^bd is the corresponding Euler axis and ϕbd is the corresponding Euler angle. Note that, here, this kinematics also represents attitude error.Dynamic of the rigid body rotation system decomposed inFb is given by the Euler equation [19] (6):
(6)Jω˙blb=-ωblbμJωblb+τ+d,
where positive definite matrix J∈R3×3 is a spacecraft moment of inertia about its center of mass located in the origin of Fb (kg m2); ωblb∈R3 is angular velocity of Fb w.r.t. Fl decomposed in Fb, (rad/s); τ∈R3 is the total external control torque about its center of mass located in the origin of Fb (Nm); and d∈R3 is the external disturbance torque or total environmental effects, for example, gravity-gradient effects, solar radiation, magnetic field torques, and air-drag.


### 2.2. Cooperative Attitude Framework

Index of a spacecraft in a group composed byn-spacecrafts is denoted by subscript (and sometimes both subscript and superscript) (i) and its neighbor is denoted by subscript (and sometimes both subscript and superscript) (j), where 1≤i, j≤n∈Z>0. For example, Fb(i) denotes body frame of spacecraft i.The dynamic of spacecrafti in a group is represented by
(7)J(i)ω˙b(i)lb(i)=-ωb(i)lb(i)μJ(i)ωb(i)lb(i)+τ(i)+d(i).
Quaternion-based kinematics equation that represents orientation of body frame of spacecraft i, Fb(i), w.r.t. body frame of spacecraft j, Fb(j), is
(8)q˙b(i)b(j)=[η˙b(i)b(j)ε˙b(i)b(j)T]T,
where
(9)η˙b(i)b(j)=-12εb(i)b(j)Tωb(i)b(j)b(i),(10)ε˙b(i)b(j)=12(εb(i)b(j)μ+ηb(i)b(j)I)ωb(i)b(j)b(i).
Information-state exchange between spacecraft in a group composed by n-spacecraft is modeled by a directed graph topology
(11)Gn≡(Vn,En,An),
where Vn={v(i)} is the node set; En⊆Vn×Vn is the edges set; and An=[a(i)(j)]∈Rn×n is the adjacency matrix of the graph Gn, where 1≤i, j≤n∈Z>0.The entry of adjacency matrixa(i)(j)=1 if spacecraft i receives information-state from spacecraft j, where v(j) is the parent node and v(i) is child node and (v(i)v(j))∈En. If there is no information-state exchange from spacecraft j to spacecraft i, then a(i)(j)=0. Self-edge is not allowed, that is, a(i)(i)=0. In a directed graph, a(i)(j)≠a(j)(i). If the graph Gn has at least one node with a directed path to all other nodes then Gn is called to have a directed spanning tree.


## 2.1. Spacecraft Kinematics and Dynamics Model

Consider the rotation matrixRbd to transform a vector expressed in Fb to be expressed in Fd that satisfies (2):
(2)Rbd=Fb·FdT⟺FdT=FbTRbd.
Taking the time derivative of (2) w.r.t. (with respect to) Fd, 0=FbT(ωbdbμRbd+R˙bd) is satisfied. Moreover, it implies (3), that is, the SO(3) rotation matrix based kinematics equation of body frame w.r.t. desired frame:
(3)R˙bd=-ωbdbμRbd.
The corresponding quaternion-based kinematics equation of body frame w.r.t. desired frame is given by
(4)q˙bd=[η˙bdε˙bdT]T,
where
(5)η˙bd=-12εbdTωbdb,ε˙bd=12(εbdμ+ηbdI)ωbdb.
Since quaternion is member ofunit sphere order-3, S3, qbd=[ηbdεbdT]T satisfies ∥qbd∥2=1, where ηbd=cos⁡(ϕbd/2)∈R and εbd=a^bdsin(ϕbd/2)∈R3; a^bd is the corresponding Euler axis and ϕbd is the corresponding Euler angle. Note that, here, this kinematics also represents attitude error.Dynamic of the rigid body rotation system decomposed inFb is given by the Euler equation [19] (6):
(6)Jω˙blb=-ωblbμJωblb+τ+d,
where positive definite matrix J∈R3×3 is a spacecraft moment of inertia about its center of mass located in the origin of Fb (kg m2); ωblb∈R3 is angular velocity of Fb w.r.t. Fl decomposed in Fb, (rad/s); τ∈R3 is the total external control torque about its center of mass located in the origin of Fb (Nm); and d∈R3 is the external disturbance torque or total environmental effects, for example, gravity-gradient effects, solar radiation, magnetic field torques, and air-drag.


## 2.2. Cooperative Attitude Framework

Index of a spacecraft in a group composed byn-spacecrafts is denoted by subscript (and sometimes both subscript and superscript) (i) and its neighbor is denoted by subscript (and sometimes both subscript and superscript) (j), where 1≤i, j≤n∈Z>0. For example, Fb(i) denotes body frame of spacecraft i.The dynamic of spacecrafti in a group is represented by
(7)J(i)ω˙b(i)lb(i)=-ωb(i)lb(i)μJ(i)ωb(i)lb(i)+τ(i)+d(i).
Quaternion-based kinematics equation that represents orientation of body frame of spacecraft i, Fb(i), w.r.t. body frame of spacecraft j, Fb(j), is
(8)q˙b(i)b(j)=[η˙b(i)b(j)ε˙b(i)b(j)T]T,
where
(9)η˙b(i)b(j)=-12εb(i)b(j)Tωb(i)b(j)b(i),(10)ε˙b(i)b(j)=12(εb(i)b(j)μ+ηb(i)b(j)I)ωb(i)b(j)b(i).
Information-state exchange between spacecraft in a group composed by n-spacecraft is modeled by a directed graph topology
(11)Gn≡(Vn,En,An),
where Vn={v(i)} is the node set; En⊆Vn×Vn is the edges set; and An=[a(i)(j)]∈Rn×n is the adjacency matrix of the graph Gn, where 1≤i, j≤n∈Z>0.The entry of adjacency matrixa(i)(j)=1 if spacecraft i receives information-state from spacecraft j, where v(j) is the parent node and v(i) is child node and (v(i)v(j))∈En. If there is no information-state exchange from spacecraft j to spacecraft i, then a(i)(j)=0. Self-edge is not allowed, that is, a(i)(i)=0. In a directed graph, a(i)(j)≠a(j)(i). If the graph Gn has at least one node with a directed path to all other nodes then Gn is called to have a directed spanning tree.


## 3. Control Systems Design and Simulations

Equations (12) and (13) [17] are the feedback term for (6) and (7), respectively, where k∈R and L∈R3×3 are tuning parameters. The function Φεbd is a column matrix of saturation function that, defined element-wise, follows the scalar saturation function ϕx (14), with the saturation limit 0<ϕ-<1/3∈R:
(12)τ=-k(ηbdεbd+Φεbd-εbd)-Lωblb=-k(ηbdεbd-Ψεbd)-Lωblb,(13)τ(i)=-∑j=1na(i)(j)(k(ηb(i)b(j)εb(i)b(j)+Φεb(i)d(j)-εb(i)d(j))=-∑j=1na(i)(j)w+Lωb(i)lb(i)lk(ηb(i)b(j)εb(i)b(j)+Φεb(i)d(j)-εb(i)d(j)))=-∑j=1na(i)(j)(k(ηb(i)b(j)εb(i)b(j)-Ψεb(i)d(j))+Lωb(i)lb(i)l),(14)ϕx={ϕ-,x>ϕ-x,-ϕ-≤x≤ϕ--ϕ-,x<-ϕ-.


### 3.1. Problem Statements of the Control Systems Design

Next subsection would discuss the proposed attitude control design for single spacecraft case as well as cooperative spacecraft case. By utilizing Lyapunov stability theory,k∈R and L∈R3×3 have to be found such that the solution of single spacecraft attitude control system composed by (6) and (12) is ultimately bounded if d=0 and is input to state stable if d≠0. Similarly, a cooperative spacecraft attitude control has to be designed by determining k∈R and L∈R3×3 and information-state exchange topology such that the solution of each spacecraft in a group is ultimately bounded if d=0 and is input to state stable if d≠0. The authors suggest the readers should refer to [20] for definition of ultimately bounded solution and input to state stability.


### 3.2. Single Spacecraft Case

It is well known that Lyapunov stability theory requires the existence and uniqueness of a solution, for a given initial condition, for all future time. The following proposition states that the system composed by (6) and (12) satisfies existence and uniqueness solution requirement because the system is locally Lipschitz [19].Proposition 1.
Consider the system (6). Suppose d=0. The spacecraft attitude control system composed by (4)–(6) and (12) is locally Lipschitz in (qbd=[ηbdεbd]T,ωblb)∈S3×R3.Proof.
For brevity, letωblpb=ωp, dωblpb/dt=ω˙blpb, εbdp=εp, and ηbdp=ηp, where p={1,2}. Then, consider the following:
(15)∥η˙1-η˙2ε˙1-ε˙2Jω˙1-Jω˙2∥=∥η˙1-η˙2∥2+∥ε˙1-ε˙2∥2+∥Jω˙1-Jω˙2∥2⟺∥η˙1-η˙2ε˙1-ε˙2Jω˙1-Jω˙2∥2=∥η˙1-η˙2∥2+∥ε˙1-ε˙2∥2+∥Jω˙1-Jω˙2∥2.
Since (5) are continuously differentiable, then the following inequalities can be obtained
(16)∥η˙1-η˙2∥≤la∥ε1-ε2ω1-ω2∥≤la∥η1-η2ε1-ε2ω1-ω2∥∥ε˙1-ε˙2∥≤lb∥η1-η2ε1-ε2ω1-ω2∥,
where la,lb∈R>0.
Now consider∥Jω˙1-Jω˙2∥ as follows:
(17)∥Jω˙1-Jω˙2∥=∥-ω1μJω1-k(η1ε1+Φε1-ε1)-Lω1w+ω2μJω2+k(η2ε2+Φε2-ε2)+Lω2∥,(18)∥Jω˙1-Jω˙2∥≤∥-ω1μJω1+ω2μJω2∥+∥-k(η1ε1-ε1)+k(η2ε2-ε2)∥+∥-kΦε1+kΦε2∥+∥-Lω1+Lω2∥.
Since fa(ωp)=-ωpμJωp and fb(ηp,εp)=-k(ηpεp-εp), where p={1,2}, are continuously differentiable functions, then the following inequalities can be obtained:
(19)∥fa(ω1)-fa(ω2)∥≤lc∥ω1-ω2∥≤lc∥η1-η2ε1-ε2ω1-ω2∥∥fb(ω1)-fb(ω2)∥≤ld∥η1-η2ε1-ε2∥≤ld∥η1-η2ε1-ε2ω1-ω2∥,
where lc,ld∈R>0.
Now, recall (18) and (19), since the term with saturation function satisfies ∥-kΦε1+kΦε2∥ ≤ k∥ε1-ε2∥ ≤ k∥η1-η2ε1-ε2ω1-ω2∥, where k>0, then the following inequalities can be obtained:
(20)∥Jω˙1-Jω˙2∥≤lc∥η1-η2ε1-ε2ω1-ω2∥+ld∥η1-η2ε1-ε2ω1-ω2∥+k∥η1-η2ε1-ε2ω1-ω2∥+λmax⁡(L)∥η1-η2ε1-ε2ω1-ω2∥≤(lc+ld+k+λmax⁡(L))∥η1-η2ε1-ε2ω1-ω2∥,
where symmetric matrix L>0.
Recall (15), (16), and (20), then the following inequalities can be obtained:
(21)∥η˙1-η˙2ε˙1-ε˙2Jω˙1-Jω˙2∥2≤(la2+lb2+(lc+ld+k+λmax⁡(L))2)∥η1-η2ε1-ε2ω1-ω2∥2⟺∥η˙1-η˙2ε˙1-ε˙2Jω˙1-Jω˙2∥≤la2+lb2+(lc+ld+k+λmax⁡(L))2∥η1-η2ε1-ε2ω1-ω2∥,
where la,lb,lc,ld,k>0 and symmetric matrix L>0.
The last inequality in (21) states that the system composed by (4)–(6) and (12) is locally Lipschitz in (qbd=[ηbdεbd]T,ωblb)∈S3×R3. It implies that the system has a unique solution for all future time.Proposition 2.
Consider the system (6). If d is continuous and bounded for all t≥0, then the attitude control system composed by (4)–(6) and (12) satisfies local Lipschitz in (qbd=[ηbdεbd]T,ωblb,d)∈S3×R3×R3.Proof.
Follow idea of the proof of Proposition1.Theorem 3.
Consider the system (6) and suppose there is no external disturbance, that is, d=0. If scalar k is positive, matrix L and moment of inertia J are symmetric and positive definite, then solution of the quaternion-based attitude control system composed by (4)–(6) and (12) is ultimately bounded.Proof.
ConsiderS, a set that consists of all εbd, (22), and positive definite function (23):
(22)S={εbd∈R3:-1≤ηbd≤1∈R,ηbd2+εbdTεbd=1}(23)V=kεbdTεbd+12ωblbTJωblb=[εbdωblb]TP[εbdωblb],
where
(24)P=[kI0012J],I=[100010001].
To make sure that P is a positive definite matrix, that is, P>0, hence k>0∈R and symmetric matrix J>0∈R3×3. If J has three different eigenvalues, then V will be bounded below by α1 and bounded above by α2, where α1 and α2 are class K∞ functions given below:
(25)α1=λmin⁡(P)∥εbdωblb∥2≤V≤α2=λmax⁡(P)∥εbdωblb∥2.
Now, take time derivative of (23). Since the desired angular velocity ωdlb is zero, ωbdb=ωblb. In addition, note that εbdTεbdμ=ωblbTωblbμ=0 (see Fact 3.10.1 in [21]). Following these, (26) is satisfied:
(26)V˙=kωblbTΨεbd-ωblbTLωblbV is a nondecreasing function since V˙ is not always negative. Fortunately, the unit quaternion is bounded by the unit sphere order-3 property (27):
(27)∥qbd∥2=ηbd2+εbdTεbd=1.
Therefore, the solution εbd is bounded above by 1 as shown below:
(28)∥εbd(t)∥≤1,t≥0.
As a direct consequence of (28), Ψεbd(t) is bounded above by κ1, that is,
(29)∥Ψεbd∥≤κ1=max⁡⁡{3·(13-ϕ-)2,(1-ϕ-)2}<1.
In addition, for all positive scalar k and positive definite symmetric matrix L, if V˙>0, then the solution ωblb(t) is bounded, that is,
(30)∥ωblb∥≤kλmin⁡(L)κ1.
Note that L has to be a symmetric and positive definite matrix to make sure that all eigenvalue of L are positive so that, along with k>0, (30) is satisfied.
From (28)–(30), V is bounded as follows:
(31)V≤k+12λmax⁡(J)(k·κ1λmin⁡(L))2=κ2.
Consider (26); using property (29), the following inequalities are obtained:
(32)V˙≤k∥Ψεbd∥∥ωblb∥-λmin⁡(L)∥ωblb∥2≤kκ3∥ωblb∥-λmin⁡(L)∥ωblb∥2≤kκ3∥εbdωblb∥-λmin⁡(L)∥εbdωblb∥2+λmin⁡(L)∥εbd∥2≤kκ3∥εbdωblb∥-λmin⁡(L)∥εbdωblb∥2+λmin⁡(L)=V˙1,
where ∥Ψεbd∥≤κ3≤κ1 and V˙1 is given as follows:
(33)V˙1=-(1-θ)λmin⁡(L)∥εbdωblb∥2-θλmin⁡(L)∥εbdωblb∥2+kκ3∥εbdωblb∥+λmin⁡(L),0<θ<1.
Therefore, inequality (34) is fulfilled for every ∥εbdωblb∥ satisfying (35):
(34)V˙≤-(1-θ)λmin⁡(L)∥εbdωblb∥2,(35)∥εbdωblb∥≥kκ3+(kκ3)2+4θλmin⁡2(L)2θλmin⁡(L)=ρ.
In according to Theorem 4.18 in [20], solution of the system is ultimately bounded with the ultimate bound as follows:
(36)∥εbdωblb∥≤α1-1(α2(ρ)).Remark 4.
Note thatκ3 in (35) clearly satisfies ∥Ψεbd∥≤κ3≤κ1. This fact implies that ρ in (35) is possible to be smaller when εbd is getting smaller. Since ρ is getting smaller, then the ultimate bound is smaller. Thus, if lim⁡t→∞⁡ρ=0, then it will be equivalent to asymptotic stability of the set consisting of two equilibrium points {(qbd=[ηbdεbd]T,ωblb)∈S3×R3:εbd=0,ωblb=0}.The solid line in Figure1 depicts the response of Euler angle attitude control system designed based on Theorem 3 using parameters in Table 1. The response is also compared by Euler Angle response of the attitude control system (4)–(6) using a well-known PD-like control law, τ=-kεbd-Lωbdb—depicted by dash-dot line.Table 1
Parameter simulation for single spacecraft case.
Parameters
Value
J, [22]
[
1.49
0.054
0.0442
0.054
1.51
0
0.0442
0
1.56
]
q
d
l
[
1
0
0
0
]
T
q
b
l
(
0
)
[
-
1
/
3
0
1
/
3
1
/
3
]
T
ω
b
l
b
(
0
); ωdlb(t), t≥0
[
0
0
0
]
T
k
1
L
[
1
0
0
0
1
0
0
0
1
]
ϕ
-
0.57Figure 1
Euler angle response,ϕbd; Theorem 3 (solid line) and PD-like control law (dash-dot line).As seen from Figure1, the attitude control system employing PD-like control law exhibits unwinding phenomenon because it is designed by considering only one equilibrium point in quaternion parameterization. Figure 2 shows more clearly that unlike the attitude control system employing PD-like control law, the designed control system converges to the closer equilibrium point, [-1000]. Figure 3 shows the opposite direction of rotation between the two control systems. This situation implies the efficiency of energy consumption as shown in Figure 4. In addition, the responses depicted in Figures 2 and 3 show that the situation explained in Remark 4 is promising.Attitude responses,qbd=[ηbdεbd]T=[ηbdεbd1εbd2εbd3]T; Theorem 3 (solid line) and PD-like control law (dash-dot line).
(a)
(b)
(c)
(d)Angular velocity; Theorem3 (solid line) and PD-like control law (dash-dot line).
(a)
(b)
(c)Figure 4
Energy consumption; Theorem3 (solid line) and PD-like control law (dash-dot line).Now, suppose that the external disturbance of system (6) is continuous and bounded, that is,
(37)∥d(t)∥≤c,d∈R3,c∈R,t≥0.Theorem 5.
If scalark is positive and matrix L is symmetric and positive definite, quaternion-based attitude control system composed by (4)–(6) and (12) is input to state stable.Proof.
Using the same energy-like function (23), its time derivative is given as follows:
(38)V˙=kωblbTΨεbd-ωblbTLωblb+ωblbTd.
Note that, according to (37), the disturbance is bounded by ∥d∥≤c. Following the way of proof of Theorem 3, for 0<θ<1, (40) is satisfied:
(39)V˙≤-(1-θ)λmin⁡(L)∥εbdωblb∥,
where κ3 satisfies ∥Ψεbd∥≤κ3≤κ1, κ1 is the same as (29) and every ∥εbdωblb∥ satisfies (40):
(40)∥εbdωblb∥≥(kκ3+c)+(kκ3+c)2+4θλmin⁡2(L)2θλmin⁡(L)=ρISS(c).
According to Theorem 4.19 in [20], the system is input to state stable with ultimate bound of the system’s solution being as follows:
(41)∥εbdωblb∥≤α1-1(α2(ρISS(c))).To verify Theorem5, a simulation is run using the same parameters in Table 1 and a disturbance function,
(42)d(t)=[3·10-2sin⁡(10πt)5·10-2sin⁡(10πt)2·10-2sin⁡(10πt-π2)],
as shown in Figure 5. The boundedness of solution properties of the designed attitude control system in the presence of disturbance is confirmed by Figures 6 and 7. Figure 9 shows the control action of the designed control system in order to have robustness property in the presence of disturbance as shown in inset of Figure 6. In addition, Figure 8 is presented as comparison control action without the presence of disturbance.External disturbance;d(t)=[d1(t)d2(t)d3(t)]T.
(a)
(b)
(c)Figure 6
Euler angle response,ϕbd; Theorem 5.Attitude responses; Theorem5.
(a)
(b)
(c)
(d)Control torque,τ=[τ1τ2τ3]T; Theorem 3.
(a)
(b)
(c)
(d)
(e)
(f)Control torque; Theorem5.
(a)
(b)
(c)
(d)
(e)
(f)


### 3.3. Cooperative Spacecraft Case

Bounded solution of cooperative spacecraft attitude control system involves attitude errors between spacecrafts, that is,∥εb(i)b(j)ωb(i)lb(i)∥. In this subsection it will be shown that through the boundedness of solution approach, cooperative attitude system could be designed.Theorem 6.
Consider the cooperative spacecraft attitude control system composed by (7)–(10) and (13). Suppose scalar k is positive and matrix L and moment of inertia J(i) are symmetric and positive definite, for 1≤i≤n∈Ζ>0. Suppose external disturbance d(i) is continuous and bounded function, ∥d(i)∥≤c, c∈R, for 1≤i≤n∈Ζ>0. If at least there is a directed spanning tree in the topology representing information-state exchange between spacecraft i and spacecraft j in a group composed by n-spacecrafts, then solution of the cooperative system is bounded.Proof.
Consider the energy-like function (43):
(43)V=k∑i,jna(i)(j)εb(i)b(j)Tεb(i)b(j)+∑in12ωb(i)lb(i)TJ(i)ωb(i)lb(i).
Then, its time derivatives can be written as follows:
(44)V˙=k∑i,jna(i)(j)εb(i)b(j)Tηb(i)b(j)ωb(i)b(j)b(i)+∑inωb(i)lb(i)Tk∑jna(i)(j)(-ηb(i)b(j)εb(i)b(j)+Ψεb(i)b(j)-Lωb(i)lb(i))+∑inωb(i)lb(i)Td(i).
Since ωb(i)b(j)b(i)=ωb(i)lb(i)-Rb(i)b(j)ωb(j)lb(j)=ωb(i)lb(i)-ωb(j)lb(i), then (45) is satisfied.                                    (45)V˙=-k∑i,jna(i)(j)εb(i)b(j)Tηb(i)b(j)ωb(j)lb(i)+∑inωb(i)lb(i)Tk∑jna(i)(j)(Ψεb(i)b(j)-Lωb(i)lb(i))+∑inωb(i)lb(i)Td(i).
Regarding property of spacecraft orientation and angular velocity, there is inequality as follows:
(46)V˙≤k∑i,jna(i)(j)∥ηb(i)b(j)εb(i)b(j)∥·∥ωb(j)lb(i)∥+k∑i,jna(i)(j)ωb(i)lb(i)T(Ψεb(i)b(j)-Lωb(i)lb(i))+∑inωb(i)lb(i)Td(i).
Noting that ∥ηb(i)b(j)εb(i)b(j)∥≤1/2, ∥d(i)∥≤c and the bound of ∥Ψεb(i)b(j)∥ then the following inequality can be obtained:
(47)V˙≤k12∑i,jna(i)(j)∥ωb(j)lb(i)∥+kκ3∑i,jna(i)(j)∥ωb(i)lb(i)∥-k∑i,jna(i)(j)λmin⁡(L)∥ωb(i)lb(i)∥2+c∑in∥ωb(i)lb(i)∥,
where ∥Ψεb(i)b(j)∥≤κ3≤κ1=max⁡⁡{3·(1/3-α)2,(1-α)2}<1. And
(48)V˙≤V-˙=k12∑i,jna(i)(j)∥ωb(j)lb(i)εb(i)b(j)∥+kκ3∑i,jna(i)(j)∥ωb(i)lb(i)εb(i)b(j)∥-k∑i,jna(i)(j)λmin⁡(L)∥ωb(i)lb(i)εb(i)b(j)∥2+k∑i,jna(i)(j)λmin⁡(L)∥εb(i)b(j)∥2+c∑in∥ωb(i)lb(i)∥.
Therefore, there are k>0 and symmetric matrix L>0 such that V-˙ implies that ∥ωb(i)lb(i)εb(i)b(j)∥ is bounded for 1≤i, j≤n and t≥0.
Now, suppose∥ωb(i)lb(i)εb(i)b(j)∥ is not bounded. Without loss of generality, taking k=λmin⁡(L)=c=1, the following inequality is obtained:
(49)V-˙≤V^˙=12∑i,jna(i)(j)∥ωb(j)lb(i)εb(i)b(j)∥+∑i,jna(i)(j)∥ωb(i)lb(i)εb(i)b(j)∥-∑i,jna(i)(j)∥ωb(i)lb(i)εb(i)b(j)∥2+∑i,jna(i)(j)∥εb(i)b(j)∥2+∑in∥ωb(i)lb(i)∥.
Then, there is a sufficiently large ∥ωb(i)lb(i)εb(i)b(j)∥, where, inherently, also sufficiently large ∥ωb(j)lb(i)εb(i)b(j)∥=∥Rb(i)b(j)ωb(j)lb(j)εb(i)b(j)∥ and ∥ωb(i)lb(i)∥, such that V^˙>0, and the increasing ∥ωb(i)lb(i)εb(i)b(j)∥ are resulting V^˙>0.
Let the information-state exchange topology consist of one directed spanning tree and∥ωb(i)lb(i)εb(i)b(j)∥=σ≠∞; therefore, the following inequality is obtained:
(50)V^˙<V~˙=12(n-1)σ+(n-1)σ-(n-1)σ2+(n-1)+nσ=-(n-1)σ2+12(5n-3)σ+(n-1).
This contradicts that, for σ>((1/2)(5n-3)+((1/2)(5n-3))2+4(n-1))/2(n-1), V~˙<0. Note that it implies that V˙<0. Therefore, by contradiction, it is shown that ∥ωb(i)lb(i)εb(i)b(j)∥ is bounded.Remark 7.
Through the proof by contradiction described above, solution of the cooperative system∥ωb(i)lb(i)εb(i)b(j)∥ is bounded for all k>0 and symmetric matrix L>0 where that only requires a directed spanning tree in information-state exchange topology.Remark 8.
Note that here the condition of information-state exchange topology is less-conservative condition, for example, directed spanning tree. In contrast, for example, [15] requires acyclic topology and [16] requires connected (undirected) topology.Simulation of cooperative attitude control case is run usingϕ-, k, and L as shown in Table 1 and the same disturbance function as used in the previous subsection. The simulation used three spacecrafts with different moment of inertia and three different initial attitudes as shown in Table 2. Meanwhile, initial angular velocities of all spacecrafts are zero. A cyclic directed graph representing information-state exchange is applied in the simulation (Figure 10). Figure 11 shows that each spacecraft converges to absolute attitude about 95°. It implies, as shown in Figure 12, that the relative attitudes between spacecrafts are regulated.Table 2
Parameter simulation for cooperative spacecrafts case.
Parameters
Value
J
(
1
)
[
1.49
0.054
0.442
0.054
1.51
0
0.0442
0
1.56
]
J
(
2
)
[
1.49
0
0.0442
0
1.51
0.054
0.0442
0.054
1.56
]
J
(
3
)
[
1.49
0
0.054
0
1.56
0.442
0.054
0.442
1.51
]
q
(
1
)
l
(
0
)
[
-
1
/
3
0
1
/
3
1
/
3
]
T
q
(
2
)
l
(
0
)
[
0
1
/
3
1
/
3
1
/
3
]
T
q
(
3
)
l
(
0
)
[
1
/
3
0
1
/
3
-
1
/
3
]
TFigure 10
Information-state exchange topology.Absolute Euler angles; Theorem6.
(a)
(b)
(c)
(d)
(e)
(f)Relative Euler angle between spacecrafts; Theorem6.
(a)
(b)
(c)


## 3.1. Problem Statements of the Control Systems Design

Next subsection would discuss the proposed attitude control design for single spacecraft case as well as cooperative spacecraft case. By utilizing Lyapunov stability theory,k∈R and L∈R3×3 have to be found such that the solution of single spacecraft attitude control system composed by (6) and (12) is ultimately bounded if d=0 and is input to state stable if d≠0. Similarly, a cooperative spacecraft attitude control has to be designed by determining k∈R and L∈R3×3 and information-state exchange topology such that the solution of each spacecraft in a group is ultimately bounded if d=0 and is input to state stable if d≠0. The authors suggest the readers should refer to [20] for definition of ultimately bounded solution and input to state stability.


## 3.2. Single Spacecraft Case

It is well known that Lyapunov stability theory requires the existence and uniqueness of a solution, for a given initial condition, for all future time. The following proposition states that the system composed by (6) and (12) satisfies existence and uniqueness solution requirement because the system is locally Lipschitz [19].Proposition 1.
Consider the system (6). Suppose d=0. The spacecraft attitude control system composed by (4)–(6) and (12) is locally Lipschitz in (qbd=[ηbdεbd]T,ωblb)∈S3×R3.Proof.
For brevity, letωblpb=ωp, dωblpb/dt=ω˙blpb, εbdp=εp, and ηbdp=ηp, where p={1,2}. Then, consider the following:
(15)∥η˙1-η˙2ε˙1-ε˙2Jω˙1-Jω˙2∥=∥η˙1-η˙2∥2+∥ε˙1-ε˙2∥2+∥Jω˙1-Jω˙2∥2⟺∥η˙1-η˙2ε˙1-ε˙2Jω˙1-Jω˙2∥2=∥η˙1-η˙2∥2+∥ε˙1-ε˙2∥2+∥Jω˙1-Jω˙2∥2.
Since (5) are continuously differentiable, then the following inequalities can be obtained
(16)∥η˙1-η˙2∥≤la∥ε1-ε2ω1-ω2∥≤la∥η1-η2ε1-ε2ω1-ω2∥∥ε˙1-ε˙2∥≤lb∥η1-η2ε1-ε2ω1-ω2∥,
where la,lb∈R>0.
Now consider∥Jω˙1-Jω˙2∥ as follows:
(17)∥Jω˙1-Jω˙2∥=∥-ω1μJω1-k(η1ε1+Φε1-ε1)-Lω1w+ω2μJω2+k(η2ε2+Φε2-ε2)+Lω2∥,(18)∥Jω˙1-Jω˙2∥≤∥-ω1μJω1+ω2μJω2∥+∥-k(η1ε1-ε1)+k(η2ε2-ε2)∥+∥-kΦε1+kΦε2∥+∥-Lω1+Lω2∥.
Since fa(ωp)=-ωpμJωp and fb(ηp,εp)=-k(ηpεp-εp), where p={1,2}, are continuously differentiable functions, then the following inequalities can be obtained:
(19)∥fa(ω1)-fa(ω2)∥≤lc∥ω1-ω2∥≤lc∥η1-η2ε1-ε2ω1-ω2∥∥fb(ω1)-fb(ω2)∥≤ld∥η1-η2ε1-ε2∥≤ld∥η1-η2ε1-ε2ω1-ω2∥,
where lc,ld∈R>0.
Now, recall (18) and (19), since the term with saturation function satisfies ∥-kΦε1+kΦε2∥ ≤ k∥ε1-ε2∥ ≤ k∥η1-η2ε1-ε2ω1-ω2∥, where k>0, then the following inequalities can be obtained:
(20)∥Jω˙1-Jω˙2∥≤lc∥η1-η2ε1-ε2ω1-ω2∥+ld∥η1-η2ε1-ε2ω1-ω2∥+k∥η1-η2ε1-ε2ω1-ω2∥+λmax⁡(L)∥η1-η2ε1-ε2ω1-ω2∥≤(lc+ld+k+λmax⁡(L))∥η1-η2ε1-ε2ω1-ω2∥,
where symmetric matrix L>0.
Recall (15), (16), and (20), then the following inequalities can be obtained:
(21)∥η˙1-η˙2ε˙1-ε˙2Jω˙1-Jω˙2∥2≤(la2+lb2+(lc+ld+k+λmax⁡(L))2)∥η1-η2ε1-ε2ω1-ω2∥2⟺∥η˙1-η˙2ε˙1-ε˙2Jω˙1-Jω˙2∥≤la2+lb2+(lc+ld+k+λmax⁡(L))2∥η1-η2ε1-ε2ω1-ω2∥,
where la,lb,lc,ld,k>0 and symmetric matrix L>0.
The last inequality in (21) states that the system composed by (4)–(6) and (12) is locally Lipschitz in (qbd=[ηbdεbd]T,ωblb)∈S3×R3. It implies that the system has a unique solution for all future time.Proposition 2.
Consider the system (6). If d is continuous and bounded for all t≥0, then the attitude control system composed by (4)–(6) and (12) satisfies local Lipschitz in (qbd=[ηbdεbd]T,ωblb,d)∈S3×R3×R3.Proof.
Follow idea of the proof of Proposition1.Theorem 3.
Consider the system (6) and suppose there is no external disturbance, that is, d=0. If scalar k is positive, matrix L and moment of inertia J are symmetric and positive definite, then solution of the quaternion-based attitude control system composed by (4)–(6) and (12) is ultimately bounded.Proof.
ConsiderS, a set that consists of all εbd, (22), and positive definite function (23):
(22)S={εbd∈R3:-1≤ηbd≤1∈R,ηbd2+εbdTεbd=1}(23)V=kεbdTεbd+12ωblbTJωblb=[εbdωblb]TP[εbdωblb],
where
(24)P=[kI0012J],I=[100010001].
To make sure that P is a positive definite matrix, that is, P>0, hence k>0∈R and symmetric matrix J>0∈R3×3. If J has three different eigenvalues, then V will be bounded below by α1 and bounded above by α2, where α1 and α2 are class K∞ functions given below:
(25)α1=λmin⁡(P)∥εbdωblb∥2≤V≤α2=λmax⁡(P)∥εbdωblb∥2.
Now, take time derivative of (23). Since the desired angular velocity ωdlb is zero, ωbdb=ωblb. In addition, note that εbdTεbdμ=ωblbTωblbμ=0 (see Fact 3.10.1 in [21]). Following these, (26) is satisfied:
(26)V˙=kωblbTΨεbd-ωblbTLωblbV is a nondecreasing function since V˙ is not always negative. Fortunately, the unit quaternion is bounded by the unit sphere order-3 property (27):
(27)∥qbd∥2=ηbd2+εbdTεbd=1.
Therefore, the solution εbd is bounded above by 1 as shown below:
(28)∥εbd(t)∥≤1,t≥0.
As a direct consequence of (28), Ψεbd(t) is bounded above by κ1, that is,
(29)∥Ψεbd∥≤κ1=max⁡⁡{3·(13-ϕ-)2,(1-ϕ-)2}<1.
In addition, for all positive scalar k and positive definite symmetric matrix L, if V˙>0, then the solution ωblb(t) is bounded, that is,
(30)∥ωblb∥≤kλmin⁡(L)κ1.
Note that L has to be a symmetric and positive definite matrix to make sure that all eigenvalue of L are positive so that, along with k>0, (30) is satisfied.
From (28)–(30), V is bounded as follows:
(31)V≤k+12λmax⁡(J)(k·κ1λmin⁡(L))2=κ2.
Consider (26); using property (29), the following inequalities are obtained:
(32)V˙≤k∥Ψεbd∥∥ωblb∥-λmin⁡(L)∥ωblb∥2≤kκ3∥ωblb∥-λmin⁡(L)∥ωblb∥2≤kκ3∥εbdωblb∥-λmin⁡(L)∥εbdωblb∥2+λmin⁡(L)∥εbd∥2≤kκ3∥εbdωblb∥-λmin⁡(L)∥εbdωblb∥2+λmin⁡(L)=V˙1,
where ∥Ψεbd∥≤κ3≤κ1 and V˙1 is given as follows:
(33)V˙1=-(1-θ)λmin⁡(L)∥εbdωblb∥2-θλmin⁡(L)∥εbdωblb∥2+kκ3∥εbdωblb∥+λmin⁡(L),0<θ<1.
Therefore, inequality (34) is fulfilled for every ∥εbdωblb∥ satisfying (35):
(34)V˙≤-(1-θ)λmin⁡(L)∥εbdωblb∥2,(35)∥εbdωblb∥≥kκ3+(kκ3)2+4θλmin⁡2(L)2θλmin⁡(L)=ρ.
In according to Theorem 4.18 in [20], solution of the system is ultimately bounded with the ultimate bound as follows:
(36)∥εbdωblb∥≤α1-1(α2(ρ)).Remark 4.
Note thatκ3 in (35) clearly satisfies ∥Ψεbd∥≤κ3≤κ1. This fact implies that ρ in (35) is possible to be smaller when εbd is getting smaller. Since ρ is getting smaller, then the ultimate bound is smaller. Thus, if lim⁡t→∞⁡ρ=0, then it will be equivalent to asymptotic stability of the set consisting of two equilibrium points {(qbd=[ηbdεbd]T,ωblb)∈S3×R3:εbd=0,ωblb=0}.The solid line in Figure1 depicts the response of Euler angle attitude control system designed based on Theorem 3 using parameters in Table 1. The response is also compared by Euler Angle response of the attitude control system (4)–(6) using a well-known PD-like control law, τ=-kεbd-Lωbdb—depicted by dash-dot line.Table 1
Parameter simulation for single spacecraft case.
Parameters
Value
J, [22]
[
1.49
0.054
0.0442
0.054
1.51
0
0.0442
0
1.56
]
q
d
l
[
1
0
0
0
]
T
q
b
l
(
0
)
[
-
1
/
3
0
1
/
3
1
/
3
]
T
ω
b
l
b
(
0
); ωdlb(t), t≥0
[
0
0
0
]
T
k
1
L
[
1
0
0
0
1
0
0
0
1
]
ϕ
-
0.57Figure 1
Euler angle response,ϕbd; Theorem 3 (solid line) and PD-like control law (dash-dot line).As seen from Figure1, the attitude control system employing PD-like control law exhibits unwinding phenomenon because it is designed by considering only one equilibrium point in quaternion parameterization. Figure 2 shows more clearly that unlike the attitude control system employing PD-like control law, the designed control system converges to the closer equilibrium point, [-1000]. Figure 3 shows the opposite direction of rotation between the two control systems. This situation implies the efficiency of energy consumption as shown in Figure 4. In addition, the responses depicted in Figures 2 and 3 show that the situation explained in Remark 4 is promising.Attitude responses,qbd=[ηbdεbd]T=[ηbdεbd1εbd2εbd3]T; Theorem 3 (solid line) and PD-like control law (dash-dot line).
(a)
(b)
(c)
(d)Angular velocity; Theorem3 (solid line) and PD-like control law (dash-dot line).
(a)
(b)
(c)Figure 4
Energy consumption; Theorem3 (solid line) and PD-like control law (dash-dot line).Now, suppose that the external disturbance of system (6) is continuous and bounded, that is,
(37)∥d(t)∥≤c,d∈R3,c∈R,t≥0.Theorem 5.
If scalark is positive and matrix L is symmetric and positive definite, quaternion-based attitude control system composed by (4)–(6) and (12) is input to state stable.Proof.
Using the same energy-like function (23), its time derivative is given as follows:
(38)V˙=kωblbTΨεbd-ωblbTLωblb+ωblbTd.
Note that, according to (37), the disturbance is bounded by ∥d∥≤c. Following the way of proof of Theorem 3, for 0<θ<1, (40) is satisfied:
(39)V˙≤-(1-θ)λmin⁡(L)∥εbdωblb∥,
where κ3 satisfies ∥Ψεbd∥≤κ3≤κ1, κ1 is the same as (29) and every ∥εbdωblb∥ satisfies (40):
(40)∥εbdωblb∥≥(kκ3+c)+(kκ3+c)2+4θλmin⁡2(L)2θλmin⁡(L)=ρISS(c).
According to Theorem 4.19 in [20], the system is input to state stable with ultimate bound of the system’s solution being as follows:
(41)∥εbdωblb∥≤α1-1(α2(ρISS(c))).To verify Theorem5, a simulation is run using the same parameters in Table 1 and a disturbance function,
(42)d(t)=[3·10-2sin⁡(10πt)5·10-2sin⁡(10πt)2·10-2sin⁡(10πt-π2)],
as shown in Figure 5. The boundedness of solution properties of the designed attitude control system in the presence of disturbance is confirmed by Figures 6 and 7. Figure 9 shows the control action of the designed control system in order to have robustness property in the presence of disturbance as shown in inset of Figure 6. In addition, Figure 8 is presented as comparison control action without the presence of disturbance.External disturbance;d(t)=[d1(t)d2(t)d3(t)]T.
(a)
(b)
(c)Figure 6
Euler angle response,ϕbd; Theorem 5.Attitude responses; Theorem5.
(a)
(b)
(c)
(d)Control torque,τ=[τ1τ2τ3]T; Theorem 3.
(a)
(b)
(c)
(d)
(e)
(f)Control torque; Theorem5.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.3. Cooperative Spacecraft Case

Bounded solution of cooperative spacecraft attitude control system involves attitude errors between spacecrafts, that is,∥εb(i)b(j)ωb(i)lb(i)∥. In this subsection it will be shown that through the boundedness of solution approach, cooperative attitude system could be designed.Theorem 6.
Consider the cooperative spacecraft attitude control system composed by (7)–(10) and (13). Suppose scalar k is positive and matrix L and moment of inertia J(i) are symmetric and positive definite, for 1≤i≤n∈Ζ>0. Suppose external disturbance d(i) is continuous and bounded function, ∥d(i)∥≤c, c∈R, for 1≤i≤n∈Ζ>0. If at least there is a directed spanning tree in the topology representing information-state exchange between spacecraft i and spacecraft j in a group composed by n-spacecrafts, then solution of the cooperative system is bounded.Proof.
Consider the energy-like function (43):
(43)V=k∑i,jna(i)(j)εb(i)b(j)Tεb(i)b(j)+∑in12ωb(i)lb(i)TJ(i)ωb(i)lb(i).
Then, its time derivatives can be written as follows:
(44)V˙=k∑i,jna(i)(j)εb(i)b(j)Tηb(i)b(j)ωb(i)b(j)b(i)+∑inωb(i)lb(i)Tk∑jna(i)(j)(-ηb(i)b(j)εb(i)b(j)+Ψεb(i)b(j)-Lωb(i)lb(i))+∑inωb(i)lb(i)Td(i).
Since ωb(i)b(j)b(i)=ωb(i)lb(i)-Rb(i)b(j)ωb(j)lb(j)=ωb(i)lb(i)-ωb(j)lb(i), then (45) is satisfied.                                    (45)V˙=-k∑i,jna(i)(j)εb(i)b(j)Tηb(i)b(j)ωb(j)lb(i)+∑inωb(i)lb(i)Tk∑jna(i)(j)(Ψεb(i)b(j)-Lωb(i)lb(i))+∑inωb(i)lb(i)Td(i).
Regarding property of spacecraft orientation and angular velocity, there is inequality as follows:
(46)V˙≤k∑i,jna(i)(j)∥ηb(i)b(j)εb(i)b(j)∥·∥ωb(j)lb(i)∥+k∑i,jna(i)(j)ωb(i)lb(i)T(Ψεb(i)b(j)-Lωb(i)lb(i))+∑inωb(i)lb(i)Td(i).
Noting that ∥ηb(i)b(j)εb(i)b(j)∥≤1/2, ∥d(i)∥≤c and the bound of ∥Ψεb(i)b(j)∥ then the following inequality can be obtained:
(47)V˙≤k12∑i,jna(i)(j)∥ωb(j)lb(i)∥+kκ3∑i,jna(i)(j)∥ωb(i)lb(i)∥-k∑i,jna(i)(j)λmin⁡(L)∥ωb(i)lb(i)∥2+c∑in∥ωb(i)lb(i)∥,
where ∥Ψεb(i)b(j)∥≤κ3≤κ1=max⁡⁡{3·(1/3-α)2,(1-α)2}<1. And
(48)V˙≤V-˙=k12∑i,jna(i)(j)∥ωb(j)lb(i)εb(i)b(j)∥+kκ3∑i,jna(i)(j)∥ωb(i)lb(i)εb(i)b(j)∥-k∑i,jna(i)(j)λmin⁡(L)∥ωb(i)lb(i)εb(i)b(j)∥2+k∑i,jna(i)(j)λmin⁡(L)∥εb(i)b(j)∥2+c∑in∥ωb(i)lb(i)∥.
Therefore, there are k>0 and symmetric matrix L>0 such that V-˙ implies that ∥ωb(i)lb(i)εb(i)b(j)∥ is bounded for 1≤i, j≤n and t≥0.
Now, suppose∥ωb(i)lb(i)εb(i)b(j)∥ is not bounded. Without loss of generality, taking k=λmin⁡(L)=c=1, the following inequality is obtained:
(49)V-˙≤V^˙=12∑i,jna(i)(j)∥ωb(j)lb(i)εb(i)b(j)∥+∑i,jna(i)(j)∥ωb(i)lb(i)εb(i)b(j)∥-∑i,jna(i)(j)∥ωb(i)lb(i)εb(i)b(j)∥2+∑i,jna(i)(j)∥εb(i)b(j)∥2+∑in∥ωb(i)lb(i)∥.
Then, there is a sufficiently large ∥ωb(i)lb(i)εb(i)b(j)∥, where, inherently, also sufficiently large ∥ωb(j)lb(i)εb(i)b(j)∥=∥Rb(i)b(j)ωb(j)lb(j)εb(i)b(j)∥ and ∥ωb(i)lb(i)∥, such that V^˙>0, and the increasing ∥ωb(i)lb(i)εb(i)b(j)∥ are resulting V^˙>0.
Let the information-state exchange topology consist of one directed spanning tree and∥ωb(i)lb(i)εb(i)b(j)∥=σ≠∞; therefore, the following inequality is obtained:
(50)V^˙<V~˙=12(n-1)σ+(n-1)σ-(n-1)σ2+(n-1)+nσ=-(n-1)σ2+12(5n-3)σ+(n-1).
This contradicts that, for σ>((1/2)(5n-3)+((1/2)(5n-3))2+4(n-1))/2(n-1), V~˙<0. Note that it implies that V˙<0. Therefore, by contradiction, it is shown that ∥ωb(i)lb(i)εb(i)b(j)∥ is bounded.Remark 7.
Through the proof by contradiction described above, solution of the cooperative system∥ωb(i)lb(i)εb(i)b(j)∥ is bounded for all k>0 and symmetric matrix L>0 where that only requires a directed spanning tree in information-state exchange topology.Remark 8.
Note that here the condition of information-state exchange topology is less-conservative condition, for example, directed spanning tree. In contrast, for example, [15] requires acyclic topology and [16] requires connected (undirected) topology.Simulation of cooperative attitude control case is run usingϕ-, k, and L as shown in Table 1 and the same disturbance function as used in the previous subsection. The simulation used three spacecrafts with different moment of inertia and three different initial attitudes as shown in Table 2. Meanwhile, initial angular velocities of all spacecrafts are zero. A cyclic directed graph representing information-state exchange is applied in the simulation (Figure 10). Figure 11 shows that each spacecraft converges to absolute attitude about 95°. It implies, as shown in Figure 12, that the relative attitudes between spacecrafts are regulated.Table 2
Parameter simulation for cooperative spacecrafts case.
Parameters
Value
J
(
1
)
[
1.49
0.054
0.442
0.054
1.51
0
0.0442
0
1.56
]
J
(
2
)
[
1.49
0
0.0442
0
1.51
0.054
0.0442
0.054
1.56
]
J
(
3
)
[
1.49
0
0.054
0
1.56
0.442
0.054
0.442
1.51
]
q
(
1
)
l
(
0
)
[
-
1
/
3
0
1
/
3
1
/
3
]
T
q
(
2
)
l
(
0
)
[
0
1
/
3
1
/
3
1
/
3
]
T
q
(
3
)
l
(
0
)
[
1
/
3
0
1
/
3
-
1
/
3
]
TFigure 10
Information-state exchange topology.Absolute Euler angles; Theorem6.
(a)
(b)
(c)
(d)
(e)
(f)Relative Euler angle between spacecrafts; Theorem6.
(a)
(b)
(c)


## 4. Concluding Remarks

Dynamics of spacecraft rotational motion based on Euler equation and quaternion-based kinematics describing the spacecraft attitudes have been presented. For cooperative spacecrafts case, information-state exchange is modeled by directed graph. Using these models, design of attitude control through boundedness of solution approach for single and cooperative spacecrafts have been proposed. The control designs are done by regarding two equilibrium points.In single spacecraft attitude control system case, the attitude control system has been designed, for zero disturbances, via ultimately bounded solution and for nonzero disturbances, via input to state stability approach. For zero disturbances, if the ultimate bound is zero att→∞, then it is equivalent to asymptotic stability. In cooperative spacecraft attitude control system case, the cooperative system has been designed via the boundedness of solution approach. If the information-state exchange topology has directed spanning tree, then the cooperative spacecraft attitude system is bounded. In addition, all theorems stated in this paper have been verified in simulations. The transient responses and steady state responses demonstrate effectiveness of the proposed methodology.



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*Source: 652571-2014-05-19.xml*