#### Abstract

This paper studies the problem of distributed coordinated attitude tracking control with a dynamic leader for spacecraft formation systems in the presence of parameter uncertainties and external disturbances. Under the constraint that the time-varying attitude trajectory of the leader is only available for a subset of the followers, a fixed-time distributed estimator for each follower is proposed. Moreover, a fixed-time distributed attitude tracking control law combined with a fixed-time distributed estimator and distributed adaptive control algorithm is proposed. This control law is dedicated to reducing the effects of system uncertainty and external bounded disturbances, and the settling time of this algorithm is independent of the initial state of the system. The fixed-time stability of the spacecraft formation system is proven with the Lyapunov-based approach, and numerical simulations clearly verify the effectiveness of the proposed control algorithm.

#### 1. Introduction

With the development of science and technology, the scientific tasks undertaken by space technology have spread to many fields, including meteorology, geography, oceanography, communications, navigation, and space exploration. Moreover, scientific missions in the field of aerospace are becoming more complex and varied, and the traditional model of a single spacecraft working alone is no longer sufficient to satisfy the requirements of practical engineering. For this situation, the concept of spacecraft formation flying (SFF) was proposed. SFF carries several advantages in accomplishing space missions and creates new opportunities and applications for such missions [1]. As the basis for ensuring the realization of formation missions, coordinated attitude control, which can generally be categorized into centralized and distributed control, guarantees that the attitudes of spacecraft synchronize to a common value. Compared with traditional centralized coordination methods, distributed cooperative attitude control, which has the advantages of high efficiency, strong robustness, and high reliability, has received considerable attention.

Distributed cooperative attitude control can be divided into two cases: with a leader spacecraft and without a leader spacecraft. First, the coordinated attitude control of leaderless spacecraft formation systems is studied in [2, 3], where the attitude of each spacecraft synchronizes to a common value, which is an unknown quantity that is not set in advance. Because the desired trajectory of each spacecraft is identical, predefined, and time varying, Wu et al. [4] proposed a decentralized adaptive sliding-mode control algorithm that maintains synchronization of spacecraft attitudes in formation. Considering the existence of communication delay in spacecraft formation systems, Abdessameud et al. [5] presented an attitude synchronization scheme that solves this problem. Second, the case when there is a leader spacecraft in a spacecraft formation system is discussed. Coordinated attitude tracking control was investigated in [6], and the control algorithm guaranteed that the attitude of a follower spacecraft was asymptotically synchronized to the specified attitude of the leader spacecraft. In [7], a robust adaptive sliding-mode control strategy supported by a neural network with radial basis functions was developed, and attitude tracking was achieved with this adaptive framework. However, all these methods in the mentioned literature only guarantee the asymptotic synchronization of a spacecraft’s attitude. In [8, 9], a control law that can make a system achieve asymptotic stability was designed for the multiagent dynamic cooperative control problem with nonvanishing disturbances and time-varying networks, respectively.

Compared to asymptotic synchronization, finite-time synchronization implies faster application efficiency for SFF. The finite-time attitude synchronization problem for SFF is investigated in [10], and the control algorithm can ensure that the attitudes of spacecraft in the formation are synchronized in a finite time with zero final angular velocity. To improve the robustness of the system, a distributed attitude coordination control scheme using the terminal sliding mode (TSM) was proposed for a group of spacecrafts in the presence of external disturbances in [11], and the proposed control algorithm ensures that the system achieves finite-time stability. In [12–14], the finite-time algorithm was also applied, and the system obtained a faster convergence rate. The literature on finite-time cooperative attitude control algorithms for SFF is fruitful, but the settling time of finite-time control algorithms depends on the initial conditions of the system. The concept of fixed-time convergence, which ensures that the settling time is independent of the initial conditions, was proposed in [15]; thus, the settling times of fixed-time algorithms only depend on the design parameters. Based on fixed-time convergence theory, attitude control algorithms that can provide fixed-time convergence have been successively presented. Ning et al. [16] investigated the fixed-time coordinated tracking problem of nonlinear multiagent systems, and the settling time was directly estimated. A disturbance observer was employed in [17], and the observer combined with the fixed-time control algorithms effectively reduces the influence of external interference for a single spacecraft.

In addition to the convergence rate of a system, parameter uncertainties and external disturbances that affect the control performance and even undermine the stability of the system are another problem of coordinated attitude tracking control of spacecraft formations. In the presence of parametric uncertainties, a distributed adaptive control algorithm combined with distributed sliding-mode estimators was proposed in [18], and this approach can effectively reduce the effect of parametric uncertainties. In [19], a distributed adaptive control algorithm for reducing the influence of external interference combined with a distributed fixed-time sliding-mode control algorithm, which proved that the attitude error converges to the regions containing the origin in a fixed time, was proposed. Considering both parameter uncertainty and bounded external disturbances, a distributed adaptive control algorithm was proposed in [20]; however, this method can only guarantee the asymptotic stability of SFF. It is noted that the literature is unable to guarantee fixed-time stability for an SFF system with a dynamic leader in the presence of system uncertainties and external disturbances.

This study investigates the problem of distributed fixed-time attitude coordination tracking control with a dynamic leader for SFF in the presence of parameter uncertainties and bounded external disturbances. The objective is that the tracking attitude error of a group of follower spacecraft modeled by Euler–Lagrange equations converges to the regions containing the origin in a fixed time. The novel fixed-time controller designed in this study combined with fixed-time observers and adaptive algorithms that handle external disturbances and parameter uncertainties can effectively deal with the uncertainty of external disturbance and system inertia parameters. Since the convergence time of a fixed-time algorithm is independent of the initial system state, a fixed-time algorithm has wider application scenarios in practical engineering. Compared with existing works, the main contributions of this study are given as follows.(1)In this study, only some follower spacecrafts can obtain the leader’s attitude information. A fixed-time distributed estimator is employed to estimate the time-varying attitude trajectory of the leader for each spacecraft by using its neighbors’ attitude information. The assumption that the leader’s information is available for all follower spacecraft in [21, 22] is unnecessary. Compared with the traditional estimator in [20], the fixed-time estimator, which is independent of the initial conditions, provides faster application efficiency. The fixed-time estimator also provides a basis for the design of subsequent fixed-time attitude tracking control algorithms.(2)Distributed adaptive control algorithms for both parameter uncertainties and bounded external disturbances, which are more general in the actual working environment of spacecraft, are designed. Compared with the algorithms that model a system only for the case of uncertainties or external disturbances in [18, 19, 23], the proposed algorithms in this study are more adapted to the actual situation. In particular, compared with [19, 23], the controller designed in this study can effectively deal with the inertia uncertainty of spacecraft.(3)Combined with a fixed-time distributed estimator and distributed adaptive control algorithms, a fixed-time distributed attitude tracking control law is proposed. Compared with those in [11, 24], the distributed coordinated attitude tracking control algorithms, which are independent of the initial conditions designed in this study, guarantee that the attitude tracking error converges to the regions containing the origin in a fixed time. Combined with the fixed-time stability theory, the settling time of the system only depends on the controller parameters, and by choosing the appropriate parameters, the settling time can be designed to satisfy the requirements of practical applications.

Notations: represents the Kronecker product. Given a vector , we use to denote the Euclidean norm and represents the sum norm of the vector. For a vector , , , and . For , define . and represent the minimum and maximum eigenvalues of matrix , respectively.

#### 2. Background and Problem Statement

##### 2.1. Spacecraft Attitude Dynamics

Consider a multiple rigid spacecraft system consisting of *n* followers, labeled spacecraft 1 to *n*, and one leader, labeled spacecraft 0. The attitude of the *i*th follower is described by modified Rodriguez parameters (MRPs) aswhere is the MRPs of the *i*th spacecraft denoting the attitude orientation of the body-fixed frame with respect to the inertial frame. is the angular velocity of the *i*th rigid body with respect to the inertial frame, and is expressed as

The matrix is provided with the following properties:

The dynamics of the *i*th spacecraft in the formation is described bywhere , , and are the inertia matrix, control torque, and external disturbance torque of the *i*th spacecraft, respectively.

The *n* followers are represented with Euler–Lagrange equations by combining (1) and (4) aswhere , , , and . According to (3a) and (3b), is the symmetric positive definite inertia matrix. Throughout the subsequent analysis, the following fundamental properties of system (5) are given in [25]

*Property 1 (parameter boundedness). *for any *i*th spacecraft, there exist positive constants , , and , where and .

*Property 2. *(linearity in the dynamic parameters). for any and , where is the constant parameters vector associated with the *i*th spacecraft and is a known regression matrix.

*Assumption 1. *The time-varying state quantities of the leader spacecraft , , and are bounded.We assume and , where .

*Assumption 2. *The external disturbance torque has an unknown upper bound; we assume , where and .

##### 2.2. Graph Theory

To represent the topology of the information flow among spacecraft in the formation, graph theory is briefly introduced. An undirected weighted graph can be denoted as , where is a finite nonempty set of nodes and is a set of unordered pairs of nodes. An edge denotes that the *j*th spacecraft can obtain the information from the *i*th spacecraft and vice versa. The adjacency matrix of graph is defined such that adjacency elements satisfy if , and otherwise. The Laplacian matrix associated with is defined as and , where . The leader adjacency matrix is defined as , where if the *i*th spacecraft can obtain the information from the leader, and otherwise. Here, . Obviously, and are symmetric matrices.

Lemma 1 (see [26]). *If the graph is an undirected connected graph, then all eigenvalues of the matrix have an ascending order, where .*

Lemma 2 (see [27]). *If the graph is an undirected connected graph, then the matrix is a symmetric matrix, and all eigenvalues of the matrix are positive.*

##### 2.3. Mathematic Background

Consider the following system:where and is a nonlinear function that can be discontinuous. For system (6), the following definitions and lemmas are given.

*Definition 1 (see [15]). *The equilibrium of system (6) is fixed-time stable if it is finite-time stable, and the settling time is uniformly bounded for any initial state, that is, such that , where .

Lemma 3 (see [15]). *If there exists a continuous radially unbounded function such that*(1)*(2)**For any solution of (6) satisfies the inequality for some , then the origin is globally fixed-time stable for system (6) and the following estimate holds:**where .*

Lemma 4 (see [28]). *Let . Then,*

Lemma 5 (see [29]). *For any and any symmetric positive definite matrix ,*

##### 2.4. Problem Statement

Consider an SFF system consisting of *n* followers, which are described by (5), and a leader spacecraft. The leader’s attitude information is only available for a some of the follower spacecraft under an undirected communication graph . The attitude tracking error and its derivative for each follower spacecraft is defined as

This study aims to design a distributed attitude control algorithm for each follower spacecraft in the presence of parameter uncertainties and external disturbances such that all spacecraft attitudes achieve synchronization, which means that and converge to the regions containing the origin in a fixed time.

#### 3. Distributed Fixed-Time Coordinated Attitude Tracking Control

##### 3.1. Fixed-Time Distributed Estimator Design

In this section, since only some of the following spacecraft can obtain the attitude information of the leader, to achieve attitude synchronization, state estimators dedicated to obtaining accurate estimates of and for each follower based on fixed-time convergence theory are designed. The estimators are designed aswhere is the estimate of and is the estimate of , , , , , , and .

The errors of the estimated states are defined as

Theorem 1. *Considering the spacecraft formation attitude system (5) under an undirected connected communication graph . and converge to zero in a fixed time with a settling time by estimators (12) and (13) under Assumption 1.*

*Proof. *Define . Based on Lemma 2, the Lyapunov function is described asTaking the time derivative of the Lyapunov function , we haveDefine . Based on Lemma 4, we obtainAccording to Lemma 3, converges to zero in a fixed time with a settling time , which is described asThe proof process for the two estimators is similar. This completes the proof of Theorem 1.

*Remark 1. *According to Lemma 2, the matrix is a positive definite symmetric matrix; then, there is certainly an orthogonal matrix such thatThen, we can obtainwhere and is the eigenvalue of matrix , and based on the property of Kronecker product, and .

##### 3.2. Fixed-Time Distributed Coordinated Adaptive Attitude Tracking Control Algorithm Design

In this section, the fixed-time distributed coordinated adaptive attitude tracking control algorithm is designed based on the fixed-time distributed estimator designed in the previous section.

The attitude state errors between the *i*th and *j*th spacecraft are defined by

The lumped attitude state errors are defined as

Here, , , , and . Based on Lemma 2, is a symmetric positive definite inertia matrix.

An auxiliary variable is constructed as

It was proven that the states are bounded in (18); moreover, , , and are bounded from (13) and (27).

Construct another auxiliary variable as

By Property 2, we obtain

And is bounded for bounded states and .

Combined with (5), the following is obtained:

The coordinated attitude tracking control law for the *i*th follower spacecraft is designed aswhere , , , and . and are the estimates of and , respectively.

*Remark 2. *It is proven in Theorem 1 that and converge to zero in a fixed time, which means and in a fixed time; thus, after , is equivalent toAfter , is equivalent to

Theorem 2. *Considering the spacecraft formation attitude system (5) under an undirected connected communication graph , control law (31) combined with estimators (12) and (13) guarantee the attitude errors and converge to the regions and in the fixed time , respectively.*

*Proof. *First, we prove the boundedness of the system-related variables and define and . , , , , and are column stack vectors of , , , , and , respectively. , , and are block diagonal matrices of , , and , respectively. Substituting (31), (32), and (33) into (30), the system can be expressed in the following form:The Lyapunov function candidate is denoted asTaking the time derivative of the Lyapunov function , we haveThis completes the proof of the boundedness of , , and .

Second, we prove the fixed-time stability of (36). The Lyapunov function is described asTaking the time derivative of the Lyapunov function , based on Lemma 4 and Lemma 5, we haveSince , , and are bounded, there exist a constant such thatThen, we obtainHere, (42) is rewritten as the following two equations:where .

From (43), if , then the fixed-time stability is guaranteed. By Lemma 3, auxiliary variable converges to the region in fixed-time . From (44), if , auxiliary variable converges to the region in fixed time . Then, the auxiliary variable converges to the region:which means , where and , in a fixed time asTherefore, we obtain the following equations after :The Lyapunov function candidate is denoted asTaking the time derivative of the Lyapunov function , we haveSimilar to the analysis of , can be rewritten as the following two equations:From (51), if , then the fixed-time stability is guaranteed. By Lemma 3, the lumped attitude state error converges to the region in fixed time . From (52), if , the lumped attitude state error converges to the region in fixed time . Then, the lumped attitude state error converges to the region:Based on (48), the lumped attitude state error converges to the region:Moreover, the settling time of and isTherefore, after , we obtainThen, the attitude state errors and converge to the regions:in fixed time , respectively. This completes the proof of Theorem 2.

*Remark 3. *The controller proposed in this study contains several parameters, where , , , and affect the convergence accuracy of the system. The smaller the values of , , , and , the higher the accuracy of the estimator and the control performance; however, smaller values cause the system to produce larger chattering. , , , and affect the control system response speed; the larger the values of , , , and are, the higher the obtained system response speed is; however, large values will cause a large overshoot.

As shown in Figure 1, the control system designed in this study includes fixed-time state observers, fixed-time attitude tracking controllers, and adaptive disturbance estimators for the *i*th spacecraft, where represents the spacecraft that is available to provide the *i*th spacecraft with the attitude information.

#### 4. Simulation Results

In this section, to demonstrate the validity of the proposed control algorithms, numerical simulations for a spacecraft formation system are conducted, and the communication topology is described in Figure 2. A system consisting of four follower spacecraft and one leader spacecraft satisfies Assumption 1. Table 1 lists the inertia matrix and initial conditions of the follower spacecraft. To obtain better control performance in the simulation process, after several debugging operations, the parameters of the distributed estimator in (12) and (13) and the auxiliary variable and the control law (31) are set as , , , , , , , , , , , and , respectively. The additional variation inertia matrix for each follower spacecraft is set as N *m*. The additional variation inertia matrix for each follower spacecraft is set as kg .

Figure 3 shows that the attitude tracking error converges to in four seconds in the presence of external disturbing torques and periodic changes in the inertia matrix. The estimation errors and are illustrated in Figures 4 and 5, respectively. and can converge to regions and with a dynamic leader, respectively. As illustrated in Figure 6, the auxiliary variable converges to region . Figure 7 shows the control torques of the follower spacecraft. As the desired attitude is dynamic, the control torques will never be zero, which is illustrated in Figure 7. To reflect the performance of the controller proposed in this study, comparisons are made with the controllers in [19] (the controller for comparison 1) and [23] (the controller for comparison 2). The station-keeping attitude error metric (SKAEM) is defined as . The control performance comparison is shown in Figure 8. The controller proposed in this study can provide a faster convergence rate and a more accurate attitude tracking performance.

#### 5. Conclusion

The distributed coordinated attitude tracking control problem for a spacecraft formation system with a dynamic leader is investigated in the presence of external disturbing torques and parameter uncertainty in this study. Considering that the attitude information of the leader is only available to some of the follower spacecraft, fixed-time distributed estimators that only utilize the neighbors’ information are proposed to estimate the leader’s attitude information. Based on the estimated information, auxiliary variables are designed for the subsequent design. By employing the fixed-time control method, a fixed-time control algorithm combined with an adaptive control algorithm, which is used to cope with external disturbances and parameter uncertainties, is proposed. The algorithm achieves fixed-time convergence of the auxiliary variable, and the convergence time is independent of the initial condition. By employing the Lyapunov approach, the proposed control algorithms demonstrate that the attitude of each follower spacecraft can be synchronized with the attitude of the dynamic leader in a fixed time. The fixed-time control algorithm can be used to reduce the convergence time in the case of tracking a dynamic target, as the convergence time is influenced by the initial conditions, which change at all times due to the dynamic target in conventional control methods and finite-time control algorithms. The effectiveness of the proposed control algorithms is demonstrated by numerical simulations. In future work, based on this research, I will continue to deeply study the fixed-time attitude tracking control of multispacecraft formations and strive to focus on the state estimator that can be used in the case of directed graph communication.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The author declares that there are no conflicts of interest.