Adaptive Fixed-Time 6-DOF Coordinated Control of Multiple Spacecraft Formation Flying with Input Quantization

*is paper investigates the fixed-time coordinated control problem of six-degree-of-freedom (6-DOF) dynamic model for multiple spacecraft formation flying (SFF) with input quantization, where the communication topology is assumed directed. Firstly, a new multispacecraft nonsingular fixed-time terminal sliding mode vector is derived by using neighborhood state information. Secondly, a hysteretic quantizer is utilized to quantify control force and torque. Utilizing such a quantizer not only can reduce the required communication rate but also can eliminate the control chattering phenomenon induced by the logarithmic quantizer. *irdly, a 6-DOF fixed-time coordinated control strategy with adaptive tuning laws is proposed, such that the practical fixed-time stability of the controlled system is ensured in the presence of both upper bounds of unknown external disturbances. It theoretically proves that the relative tracking errors of attitude and position can converge into the regions in a fixed time. Finally, a numerical example is exploited to show the usefulness of the theoretical results.


Introduction
e recent decades have seen an ever increasing research interest in the coordinated control problem of spacecraft formation flying (SFF) due to its successful applications in the space industry such as atmosphere monitoring of the Earth, deep space exploration, and spacecraft on-orbit maintenance [1][2][3][4][5]. As is well known, the coordinated control of attitude and orbit are two equally important technologies. It is essential to achieve the desired attitude and position simultaneously for SFF mission [6,7]. Owning to the dynamical coupling between orbit motion and attitude motion, these two motions can be considered as a whole six-degree-of-freedom (6-DOF) motion. Recently, the 6-DOF coordinated control of SFF has attracted considerable research attention [8,9]. However, the control strategies proposed in the aforementioned literature can only guarantee asymptotic stability of the controlled systems [10][11][12][13].
For coordinated control problems of SFF, fast convergence performance is an important requirement [14][15][16][17][18]. In contrast to asymptotic stabilization controllers, the finitetime stabilization controllers can provide a faster response and better disturbance-rejection ability [19][20][21][22]. erefore, the finite-time controllers have been developed in spacecraft formation control [23][24][25]. Even though the finite-time control methods can ensure the controlled systems finitetime stabilization, the convergence time relies on the information of initial system states, which gives rise to difficulties in practical applications [26]. To cope with this constraint, the fixed-time stable concept was applied to study finite-time controller design, in that the convergence time is upper bounded regardless of initial system states [27][28][29]. To date, fixed-time control strategies have been used for various control systems [30,31] but less attention has been paid to fixed-time 6-DOF coordinated control problem for SFF, especially for external disturbances with unknown upper bounds. Another significant issue in multiple SFF task is that the interspacecraft communication links are not always bidirectional, such as in the unidirectional spacecraft laser communication system. However, in some of the existing results, the coordinated control issue is investigated based on an assumption that the communication topology is undirected.
On another research frontier, networked control systems (NCSs) as an active field of research have been applied successfully in various modern complicated engineering processes [13,[32][33][34][35] such as unmanned vehicles, nuclear power stations, and aerospace engineering systems. In modern low-cost plug-and-play small spacecraft formation systems, the functional components connected by wireless networked media [36][37][38][39]. It is quite common that when the signal is transmitted between the control and actuator module via wireless networks, the SFF systems unavoidably suffer from quantization errors caused by quantization behavior which will degrade the control performance or even lead to instability [40][41][42][43]. us, it is needed to propose new controllers for SFF where the signal quantization is taken into consideration. Although some research attention has been centered on the quantized control problem of SFF, there is still no result available that considers 6-DOF coordinated control of multiple spacecraft formation in the presence of quantized input control signal. e complexity of the multiple spacecraft formation coordinated control task makes the quantized fixed-time coordinated control a serious challenge.
In this paper, we are motivated to deal with the problem of fixed-time 6-DOF adaptive coordinated control for multiple SFF with input quantization under directed communication topology. e main contributions of this paper are highlighted as follows: (1) the communication topology among follower spacecraft is described by a directed graph, which will bring more challenges than the case that the communication topology among follower spacecraft is described by an undirected graph. (2) A novel multispacecraft nonsingular FTTSM based on a 6-DOF dynamic model is designed, on which each spacecraft converges to its desired states while keeping synchronization with other formation spacecraft. (3) A fixed-time adaptive coordinate control strategy is derived to compensate for the effects of hysteretic quantizer and external disturbances on the control performance and guarantee the practical fixed-time stability of the controlled system. e rest of this paper is organized as follows: in section 2, the modelling and preliminaries are presented. In section 3, a multispacecraft nonsingular fixed-time terminal sliding mode vector is designed. In section 4, a fixed-time adaptive coordinated control scheme is proposed. An illustrative example and a conclusion are given in Sections 5 and 6, respectively.

6-DOF Dynamic Model.
e 6-DOF dynamic model of spacecraft formation is represented as follows [8]: where superscript i stands for the ith follower spacecraft; ρ i � [x i y i z i ] T represents the relative position vector from the ith follower spacecraft to the leader spacecraft; ω i ∈ R 3 denotes the angular velocity; q i ∈ R 4 is the quaternion defined as q i � q 0i q vi T , where n 0 ∈ R represents angular velocity of the virtual leader spacecraft; q 0i is the scalar part and q vi is the vector part; m fi ∈ R denotes the mass; J fi ∈ R 3×3 is the inertia matrix; u fi ∈ R 3 represents the control force; u it ∈ R 3 is the control torque; F di ∈ R 3 is disturbance force; and 2 Complexity disturbance torque and T iGT ∈ R 3 is the gravity gradient torque. e notation ı × for the vector ı � ı 1 ı 2 ı 3 T represents the skew-symmetric matrix as follows: It is worth to mention that the attitude and orbit are mutually coupled by T iGT ∈ R 3 , which is given as where R fi ∈ R 3 is the position unit vector. Since T iGT is much smaller compared with control torque, T iGT is always treated as a disturbance. We define the following error states: where q ei is error quaternion defined as q ei � q 0ei q T vei T � q ei ⊗ q di ; R(q ei ) is the rotation matrix from the ith follower spacecraft's reference frame to its bodyfixed frame; and ρ di , ρ . di , ω di , and q di are the desired position, desired velocity, desired angular velocity, and desired attitude, respectively. en the 6-DOF relative error dynamic model of SFF can be expressed by

Graph eory.
It is supposed that the information flow among n follower spacecraft is described by a directed graph G � (V, χ, A), where V � V 1 , V 2 , . . . , V n represents the set of nodes, χ ⊆ V × V represents the set of edges, and (V i , V j ) ∈ χ represents if and only if node V i can receive the information of node V j . In spacecraft 6-DOD coordinated control application, (V i , V j ) ∈ χ represents only the jth spacecraft can obtain the ith spacecraft's states information. A � [a ij ] ∈ R n×n denotes the weighted adjacency matrix of the graph G with entries where a ij is the nonnegative element of A, which denotes communication quality between the ith spacecraft and jth spacecraft. It is noticeable that self-edges are not allowed, meaning that a ii � 0. e in-degree matrix of the graph G is D with entries where e Laplacian matrix L ∈ R n×n of the graph G is [44] L � D − A.

Hysteretic Quantizer.
To eliminate the control chattering phenomenon induced by logarithmic quantizer, a hysteretic quantizer is used to quantify control torque and force in this paper, which is similar to [45]. It can be expressed by . e map of the hysteresis quantizer q(u(t)) for u > 0 is illustrated in Figure 1.

Remark 1.
e parameter ρ can be termed as a measure of quantization density. From the definition of ρ, we can see that the smaller parameter ρ is, the coarser the hysteretic quantizer becomes [45]. erefore, the design of parameter ρ Complexity 3 should be based on a criterion to guarantee the control performance with a smaller quantization density. In addition, compared with a traditional logarithmic quantizer, the hysteretic quantizer (10) can avoid oscillations by adding additional quantization levels.

Preliminaries.
For deriving the 6-DOF fixed-time coordinated controller, the lemmas are made as follows.

Lemma 1. e hysteretic quantizer q(u(t)) is decomposed into two parts as
where D(u) and Q(t) satisfy e proof of Lemma 1 is similar to eorem 1 in [45].
Lemma 2 (see [44]). If G is a directed graph with N nodes, then all the eigenvalues of the weighted Laplace matrix L have nonnegative real part.
Lemma 3 (see [46]). For any matrix M ∈ R m×m , N ∈ R n×n , X ∈ R m×m , and Y ∈ R n×n , then the following equalities hold: Lemma 4 (see [47]). For any x, y ∈ R, if ] ∈ R + and ] > 1, then Lemma 5 (see [48]). If Lemma 6 (see [47]). Consider the nonlinear system given by Suppose that there exists a Lyapunov function V(x) that satisfies the following condition: where α, β, p, g ∈ R + , pk < 1, gk > 1, and 0 < υ < ∞. en, the origin of system (17) is practical fixed-time stable and the residual set of the solution satisfies where θ is a scalar satisfying 0 < θ ≤ 1. e setting time is bounded by

Multispacecraft Nonsingular Fixed-Time Terminal Sliding Mode
In this section, a multispacecraft nonsingular fixed-time terminal sliding mode (FTTSM) vector is proposed to realize the orbit and attitude coordinated control for SFF. e following assumptions are presented regarding the 6-DOF dynamic model.

Assumption 1.
e total disturbance τ i is assumed to be bounded due to the fact that magnetic forces, J 2 perturbations, gravitation, and solar radiation pressure are bounded.

Assumption 2.
e desired angular velocity ω di and its time derivative ω . di are assumed to be bounded. e desired trajectory ρ di and its time derivative ρ . di are assumed to be bounded.

Remark 2.
It is supposed that the communication topology of SFF is undirected; we can obtain a ij � a ji , which simplifies the design and analysis of the controller. However, for the directed communication topology, a ij � a ji does not hold. erefore, compared with bidirectional communication topology, the coordinated control of formation spacecraft under directed communication topology is more challenging.
Remark 3. Note that the multispacecraft FTTSM (21) can be simplified as a modified terminal sliding mode (TSM) designed in [47] if k 1 � 1 and p 2 � 0; moreover, (21) coincides with the modified fast TSM designed in [49] for k 1 � 1. It is worth mentioning that when e 1ik converges to the region |e 1ik | ≤ ε, the multispacecraft FTTSM is converted to the general sliding mode for s i,1 ≠ 0. us, the singularity problem of (21) can be effectively avoided. Moreover, by the choice of l 1 and l 2 , the continuity of α i,k and its first-order time derivative is guaranteed.
By Kronecker product, the sliding mode function (21) can be described by where L is the weighted Laplace matrix, which is determined by directed topology,

Design of Fixed-Time Adaptive Coordinated Control Scheme
In this section, a fixed-time adaptive 6-DOF coordinated control scheme is presented for multiple SFF with input quantization and external disturbances.
To design the control scheme, (6) can be derived as Complexity 5 Under Assumption 1, it can be seen that (27) where c i are nonnegative constant numbers. e fixed-time adaptive controller is designed as where 0 < c1 < 1, c2 > 1, I i > 0, κ i > 0, and ϱ i > 0 are the controller parameters.
Theorem 1. Consider the 6-DOF control system (6) with the fixed-time coordinated control law (28). If the parameter uncertainty and external disturbance satisfy Assumptions 1-2, then the sliding mode vector s i will converge into Proof. We construct the following Lyapunov function candidate: with where c i � c i − c i . By the Kronecker product, the controller (28) can be rewritten as It follows from (34) that × α sgn(S) c1 + β sgn(S) c2 + cS , × α sgn(S) c1 + β sgn(S) c2 + cS , Considering (24) and (25), it can be shown that 6 Complexity
From (49), we have Assume that there exists a compact set Π i satisfying where Δ i is an unknown constant.
On the other hand, from (67), we can obtain that system state e 2i will converge to the region in a fixed time. Furthermore, we can conclude that us, e 1i and e 2i , i � 1, 2, 3, will converge to the regions Δ e1i and Δ e2i in a fixed time, respectively. Remark 4. Based on the multispacecraft FTTSM results, the property of graph theory and adaptive technique, a fixedtime 6-DOF coordinated control strategy is designed under directed communication topology. Subsequently, the system tracking error states can be guaranteed to converge their desired trajectories in a fixed time even with external disturbances and quantized control input. Note that this small region is determined by the controller parameters α i > 0, β i > 0, and κ i > 0. us, this small region is adjustable and can be reduced as needed.
Remark 5. By employing the adaptive method, the precise information of the external disturbance and parameters uncertain is not needed for the controller (28) design. Moreover, it is no required to make an additional assumption about interspacecraft communication topology in the designed fixed-time coordinated controller. Hence, the designed controller is suitable for any communication topology. Even if there is no communication link between the formation spacecraft, this proposed controller can still guarantee the practical fixed-time stability of each formation spacecraft.

Illustrative Example
To validate the proposed coordinated controller, we give an illustrative example in this section. e communicate topology of three follower spacecraft is described in Figure 2, in which "Sat i(i � 1, 2, 3)" denotes the ith formation spacecraft. e leader spacecraft is specified to a circular orbit with a radius of 6878 km and orbit angel velocity is n 0 � 1.11 × 10 − 3 rad/s. e weighted Laplace matrix L is designed as (77) In order to form a triangle of three follower spacecraft, the desired relative position and velocity are specified as   Complexity z i � 0.06cos(0.3t) 0.01sin(0.1t) 0.06cos(0.2t) T Nm, e maximum of input control force and torques are limited to 5 N and 1 Nm, respectively. e following performance indexes are defined to describe the synchronization accuracy and tracking accuracy of the proposed control law (28): where ρ i � ρ i − ρ di , ρ j � ρ j − ρ j d , ρ i,j � ρ i − ρ j , and q i,jve is the vector part of q i,je � q i,j0e q T i,jve T � q ei ⊗ q ej . From the definition of ϖ1, ϖ2, ϖ3, and ϖ4, it can be shown that smaller ϖ1 and ϖ3 can guarantee the better formation and attitude tracking performance, respectively; smaller ϖ2 and ϖ4 can ensure the better formation and attitude synchronization performance, respectively. e control scheme (28), adaptive updating law (29), and hysteretic quantizer (12) parameters are chosen as c1 � 0.9, c2 � 1.1, a 12 � 0.5, a 23 � 0.5, a 31 � 0.5, α i � 0.1, β i � 1, p 1 � 0.4, p 2 � 1.5, k 1 � 2, c i � 0.1, b i � 0.04, κ i � 0.000002, I i � 0.5, ϱ i � 0.000002, ∀i � 1, 2, 3, δ � 0.25, and u min � 0.00001. e simulation results of the controller (28) are shown in Figures 3 to 9, where Figure 3 depicts the relative attitude error and Figure 4 plots the relative angular velocity error. It can be observed that the relative attitude errors converge to near zero within 50 s, which has a fast convergence rate. e relative position error and relative velocity error are shown in Figures 5 and 6, respectively. It can be seen that the relative position errors converge to near zero about 85 s, which has a fast convergence rate. e quantized control force and torque are shown in Figures 7 and 8, respectively. e curves of the performance indexes ϖ1 to ϖ4 are depicted in Figure 9. As seen from the simulation results, the proposed fixed-time control strategy provides a good performance of tracking and synchronization.

Conclusion
In this paper, the fixed-time 6-DOF coordinated control problem has been studied for multiple spacecraft formation with input quantization under directed communication graph. A fixed-time adaptive coordinated control strategy is designed by using multispacecraft FTTSM vector such that, in the presence of the upper bounds of unknown external disturbances, the controlled system is practical fixed-time stable and, at the same time, the tracking errors converge to their desired trajectories in a fixed time. Compared with the existing finite-time stabilization controllers, the designed adaptive fixed-time coordinated controller in this paper is more suitable for practical engineering application due to its convergent time regardless of initial system states. An illustrative example is given to illustrate the performance of the presented fixed-time coordinated controller. It was shown that the presented controller not only can ensure each spacecraft's convergence to its desired states but also can provide the desired synchronization and tacking performance. Future study will focus on the extension of the presented controller under time-varying communication topology and communication time delay.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.