#### Abstract

This 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. Thirdly, 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.

#### 1. Introduction

The 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 [1–5]. 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 [10–13].

For coordinated control problems of SFF, fast convergence performance is an important requirement [14–18]. In contrast to asymptotic stabilization controllers, the finite-time stabilization controllers can provide a faster response and better disturbance-rejection ability [19–22]. Therefore, the finite-time controllers have been developed in spacecraft formation control [23–25]. Even though the finite-time control methods can ensure the controlled systems finite-time 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 [27–29]. 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, 32–35] 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 [36–39]. 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 [40–43]. Thus, 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. The 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. The 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.

The 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.

#### 2. Modelling and Preliminaries

##### 2.1. 6-DOF Dynamic Model

The 6-DOF dynamic model of spacecraft formation is represented as follows [8]:wherewhere superscript stands for the th follower spacecraft; represents the relative position vector from the th follower spacecraft to the leader spacecraft; denotes the angular velocity; is the quaternion defined as , where represents angular velocity of the virtual leader spacecraft; is the scalar part and is the vector part; denotes the mass; is the inertia matrix; represents the control force; is the control torque; is disturbance force; and , where is disturbance torque and is the gravity gradient torque. The notation for the vector represents the skew-symmetric matrix as follows:

It is worth to mention that the attitude and orbit are mutually coupled by , which is given aswhere is the position unit vector. Since is much smaller compared with control torque, is always treated as a disturbance.

We define the following error states:where is error quaternion defined as ; is the rotation matrix from the th follower spacecraft’s reference frame to its body-fixed frame; and , , , and are the desired position, desired velocity, desired angular velocity, and desired attitude, respectively.

Then the 6-DOF relative error dynamic model of SFF can be expressed bywhere

##### 2.2. Graph Theory

It is supposed that the information flow among follower spacecraft is described by a directed graph , where represents the set of nodes, represents the set of edges, and represents if and only if node can receive the information of node . In spacecraft 6-DOD coordinated control application, represents only the th spacecraft can obtain the th spacecraft’s states information. denotes the weighted adjacency matrix of the graph with entrieswhere is the nonnegative element of , which denotes communication quality between the th spacecraft and th spacecraft. It is noticeable that self-edges are not allowed, meaning that .

The in-degree matrix of the graph is with entrieswhere

The Laplacian matrix of the graph is [44]

##### 2.3. 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 bywhere with , , and . is in the set . The map of the hysteresis quantizer for is illustrated in Figure 1.

*Remark 1. *The parameter can be termed as a measure of quantization density. From the definition of , we can see that the smaller parameter is, the coarser the hysteretic quantizer becomes [45]. Therefore, the design of parameter 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.

##### 2.4. Preliminaries

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

Lemma 1. *The hysteretic quantizer is decomposed into two parts aswhere and satisfy**The proof of Lemma 1 is similar to Theorem 1 in [45].*

Lemma 2 (see [44]). * If is a directed graph with nodes, then all the eigenvalues of the weighted Laplace matrix have nonnegative real part.*

Lemma 3 (see [46]). *For any matrix , , , and , then the following equalities hold:*(1)*(2)**If the matrices and are invertible, then *(3)*If the eigenvalues of are and the eigenvalues of are , then the eigenvalues of can be expressed by *

Lemma 4 (see [47]). * For any , if and , then*

Lemma 5 (see [48]). * If , and , then*

Lemma 6 (see [47]). * Consider the nonlinear system given by**Suppose that there exists a Lyapunov function that satisfies the following condition:where , , , , , , and . Then, the origin of system (17) is practical fixed-time stable and the residual set of the solution satisfieswhere is a scalar satisfying . The setting time is bounded by*

#### 3. 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. The following assumptions are presented regarding the 6-DOF dynamic model.

*Assumption 1. *The total disturbance is assumed to be bounded due to the fact that magnetic forces, perturbations, gravitation, and solar radiation pressure are bounded.

*Assumption 2. *The desired angular velocity and its time derivative are assumed to be bounded. The desired trajectory and its time derivative are assumed to be bounded.

In order to achieve coordinated control for multiple spacecraft formation, the sliding mode based on (6) is defined aswhere , and , given bywith , andwhere , , , , , , is control scheme gain to realize tracking control, is control scheme gain to realize coordinated control between th and th formation spacecraft, and , , , , are designed parameters, , is small constant, , , and is sign function.

*Remark 2. *It is supposed that the communication topology of SFF is undirected; we can obtain , which simplifies the design and analysis of the controller. However, for the directed communication topology, does not hold. Therefore, 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 and ; moreover, (21) coincides with the modified fast TSM designed in [49] for . It is worth mentioning that when converges to the region , the multispacecraft FTTSM is converted to the general sliding mode for . Thus, the singularity problem of (21) can be effectively avoided. Moreover, by the choice of and , the continuity of and its first-order time derivative is guaranteed.

By Kronecker product, the sliding mode function (21) can be described bywhere is the weighted Laplace matrix, which is determined by directed topology, , .

#### 4. 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 aswith

Under Assumption 1, it can be seen thatwhere are nonnegative constant numbers.

The fixed-time adaptive controller is designed aswhere , , , , and 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 will converge intoin fixed time, where , , , , , , , , , .*

*Proof. * We construct the following Lyapunov function candidate:withwhere .

By the Kronecker product, the controller (28) can be rewritten aswhere , , , .

Since can be equivalently expressed asIt follows from (34) thatConsidering (24) and (25), it can be shown thatwhere , , .

Next, it can be derived thatIn addition, taking the derivative of yieldsIt is noticed from (37) and (38) thatFromand by substituting (40) into (39), we havewhere .

*Case 1. *If , we haveSubstituting (42) into (41) yieldsAccording to Lemma 5, it is easy to prove thatwhere , , .

By applying (44), (43) can be rewritten as

*Case 2. *If , one can obtainSubstituting (46) into (41) yieldsFurthermore, (47) can be rewritten asAccording to Lemma 5, we obtainwhere , , .

From (49), we haveAssume that there exists a compact set satisfyingwhere is an unknown constant.

If , we can obtainIf , we can obtainDenoteThen, from the above analyses, we can further conclude thatWith Lemma 6, system (6) is practical fixed-time stable. Furthermore, will converge into the regionin a fixed-time ; that is, will converge into the region in a fixed-time .

Theorem 2. *When reach the boundary in fixed-time, the tracking error and will converge toin a fixed time, where .*

*Proof. * If , we can obtainFurthermore, one can obtainConstruct the following Lyapunov function candidate:DenoteTaking the derivative of and yieldsSimilarly, taking the derivative of yieldsThen, from (62) and (63), we can obtain that system error states converge into regions , at the same time converges to 1 in fixed-time by using Lemma 6.

*Case 4. *If and , which implies that has converged to the region in a fixed time. Then, from (23), we haveMoreover, it is easy to see thatwhich means that converges to the region in a fixed time.

*Case 5. *If and , we can obtain