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

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 systemwhose saturation element is in its control law such that those problems are overcome.The proposed feature ofmethodology 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.


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 mixed  2 / ∞ 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][2][3][4][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

Modeling
In this paper, a spacecraft is considered a rigid body.In the following descriptions, subscripts or/and superscripts , , and  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 as body frame, inertial frame, and desired frame, respectively.Here, set of -dimension real column matrices, set of  ×  real matrices, and set of positive integers are denoted by R  , R × , and  >0 , respectively, where ,  ∈  >0 .Given  ∈ R  and  ∈ R × , ‖‖ is 2-norm of column matrix ;  min () and  max () are minimum and maximum Eigen value of , respectively;   and   are transpose matrix of  and , respectively;  > 0 means that  is a positive definite matrix.
Following [19], vectors are defined as follows: where F  is a vectrix associated to the body frame, that is, a column matrix with three unit vectors l1 , l2 , and l3 -F  = [ l1 l2 l3 ]

𝑇
;   ∈ R 3 is a column matrix whose three components of ⃗  are expressed or decomposed into the body frame;    is skew-symmetric matrix of   .

Spacecraft Kinematics and Dynamics
Model.Consider the rotation matrix   to transform a vector expressed in F  to be expressed in F  that satisfies (2): Taking the time derivative of ( The corresponding quaternion-based kinematics equation of body frame w.r.t.desired frame is given by where Since quaternion is member of unit sphere order-3, where positive definite matrix  ∈ R 3×3 is a spacecraft moment of inertia about its center of mass located in the origin of F  (kg m 2 );    ∈ R 3 is angular velocity of F  w.r.t.F  decomposed in F  , (rad/s);  ∈ R 3 is the total external control torque about its center of mass located in the origin of F  (Nm); and  ∈ R 3 is the external disturbance torque or total environmental effects, for example, gravity-gradient effects, solar radiation, magnetic field torques, and air-drag.
Modelling and Simulation in Engineering 3 2.2.Cooperative Attitude Framework.Index of a spacecraft in a group composed by -spacecrafts is denoted by subscript (and sometimes both subscript and superscript) () and its neighbor is denoted by subscript (and sometimes both subscript and superscript) (), where 1 ≤ ,  ≤  ∈  >0 .For example, F () denotes body frame of spacecraft .
The dynamic of spacecraft  in a group is represented by Quaternion-based kinematics equation that represents orientation of body frame of spacecraft , F () , w.r.t.body frame of spacecraft , F () , is where Information-state exchange between spacecraft in a group composed by -spacecraft is modeled by a directed graph topology where V  = {V () } is the node set; E  ⊆ V  × V  is the edges set; and A  = [ ()() ] ∈ R × is the adjacency matrix of the graph G  , where 1 ≤ ,  ≤  ∈  >0 .The entry of adjacency matrix  ()() = 1 if spacecraft  receives information-state from spacecraft , where V () is the parent node and V () is child node and (V () V () ) ∈ E  .If there is no information-state exchange from spacecraft  to spacecraft , then  ()() = 0. Self-edge is not allowed, that is,  ()() = 0.In a directed graph,  ()() ̸ =  ()() .If the graph G  has at least one node with a directed path to all other nodes then G  is called to have a directed spanning tree.

Control Systems Design and Simulations
Equations ( 12) and ( 13) [17] are the feedback term for ( 6) and (7), respectively, where  ∈ R and  ∈ R 3×3 are tuning parameters.The function Φ   is a column matrix of saturation function that, defined element-wise, follows the scalar saturation function   (14), with the saturation limit 0 <  < √1/3 ∈ R: 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,  ∈ R and  ∈ R 3×3 have to be found such that the solution of single spacecraft attitude control system composed by ( 6) and ( 12) is ultimately bounded if  = 0 and is input to state stable if  ̸ = 0. Similarly, a cooperative spacecraft attitude control has to be designed by determining  ∈ R and  ∈ R 3×3 and information-state exchange topology such that the solution of each spacecraft in a group is ultimately bounded if  = 0 and is input to state stable if  ̸ = 0.The authors suggest the readers should refer to [20] for definition of ultimately bounded solution and input to state stability.

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  = 0.The spacecraft attitude control system composed by (4)-( 6) and ( 12) Proof.For brevity, let     =   ,     / = ω    ,    =   , and    =   , where  = {1, 2}.Then, consider the following: 4 Modelling and Simulation in Engineering Since (5) are continuously differentiable, then the following inequalities can be obtained where ℓ  , ℓ  ∈ R > 0. Now consider ‖ ω 1 −  ω 2 ‖ as follows: Since where  = {1, 2}, are continuously differentiable functions, then the following inequalities can be obtained: where ℓ  , ℓ  ∈ R > 0. Now, recall ( 18) and ( 19), since the term with saturation function satisfies where  > 0, then the following inequalities can be obtained: where symmetric matrix  > 0.
The last inequality in (21) states that the system composed by ( 4)-( 6) and ( 12) is locally Lipschitz in It implies that the system has a unique solution for all future time.Proposition 2. Consider the system (6).If  is continuous and bounded for all  ≥ 0, then the attitude control system composed by (4)-( 6) and (12) Proof.Follow idea of the proof of Proposition 1.
Theorem 3. Consider the system (6) and suppose there is no external disturbance, that is,  = 0.If scalar  is positive, matrix  and moment of inertia  are symmetric and positive definite, then solution of the quaternion-based attitude control system composed by (4)-( 6) and ( 12) is ultimately bounded.
Proof.Consider S, a set that consists of all   , (22), and positive definite function (23): where To make sure that  is a positive definite matrix, that is,  > 0, hence  > 0 ∈ R and symmetric matrix  > 0 ∈ R 3×3 is a nondecreasing function since V is not always negative.Fortunately, the unit quaternion is bounded by the unit sphere order-3 property (27): Therefore, the solution   is bounded above by 1 as shown below: As a direct consequence of (28), Ψ   () is bounded above by  1 , that is, In addition, for all positive scalar  and positive definite symmetric matrix , if V > 0, then the solution    () is bounded, that is, Note that  has to be a symmetric and positive definite matrix to make sure that all eigenvalue of  are positive so that, along with  > 0, (30) is satisfied.From (28)-(30),  is bounded as follows:  ] where ‖Ψ   ‖ ≤  3 ≤  1 and V 1 is given as follows: In according to Theorem 4.18 in [20], solution of the system is ultimately bounded with the ultimate bound as follows: Remark 4. Note that  3 in (35) clearly satisfies ‖Ψ   ‖ ≤  3 ≤  1 .This fact implies that  in (35) is possible to be smaller when   is getting smaller.Since  is getting smaller, then the ultimate bound is smaller.Thus, if lim  → ∞  = 0, then it will be equivalent to asymptotic stability of the set consisting of two equilibrium points The solid line in Figure 1 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,  = −  −   depicted by dash-dot line.
As seen from Figure 1, 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 PDlike control law, the designed control system converges to the closer equilibrium point, [−1 0 0 0]. 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.Now, suppose that the external disturbance of system ( 6) is continuous and bounded, that is, Theorem 5.If scalar  is positive and matrix  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: Note that, according to (37), the disturbance is bounded by ‖‖ ≤ .Following the way of proof of Theorem 3, for 0 <  < 1, (40) is satisfied: (40) 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: To verify Theorem 5, a simulation is run using the same parameters in Table 1 and a disturbance function, 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.
Remark 8.Note that here the condition of informationstate 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 using , , and  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.

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 at  → ∞, 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.

Table 1 :
Parameter simulation for single spacecraft case.

Table 2 :
Parameter simulation for cooperative spacecrafts case.