We propose controllers for leader-follower attitude synchronization of spacecraft formations in the presence of disturbances, that is, the leader spacecraft is controlled to follow a given reference, while a follower spacecraft is controlled to synchronize its motion with the leader's. In the ideal disturbance-free scenario, we show that synchronization takes place asymptotically. Moreover, we show the property of uniform practical asymptotic stability which implies that the synchronization is robust to bounded disturbances.

1. Introduction

In recent years, formation flying has become an increasingly popular subject of study. This is a new method of performing space operations, by replacing large and complex spacecraft with an array of simpler microspacecraft, bringing out new possibilities and opportunities of cost reduction, redundancy, and improved resolution aspects of onboard payload. One of the main challenges is the requirement of synchronization between spacecraft; robust and reliable control of relative position and attitude is necessary to make the spacecraft cooperate to gain the possible advantages made feasible by spacecraft formations. For fully autonomous spacecraft formations, both path- and attitude-planning must be performed online which introduces challenges like collision avoidance and restrictions on pointing instruments towards required targets, with the lowest possible fuel expenditure.

Synchronization of dynamical systems was first studied by Christian Huygens in the XVIIth century. In recent years, the problem has obtained increasing interest in various research areas due to its impact in technology development and challenges it imposes; see, for example, [14].

Model-based controlled synchronization consists in using the physics laws and control theory in order to induce synchronization in dynamical systems. Successful instances include synchronization of robot manipulators [5, 6], leader-follower spacecraft formations [710], and ship replenishment operations [11, 12]. Another form of synchronization is consensus, in which a group of systems coordinate their motion without any subsystem having a higher hierarchy with respect to the others. An instance of consensus is cooperative control in which a group of systems is controlled in a way that they collaborate in order to achieve a task as a team of agents. Examples may be found in the areas of autonomous vehicles [1315], underactuated marine vessels [16, 17], and rigid bodies [1820].

In this paper, we address the simultaneous control problems of attitude tracking and leader-follower synchronization. That is, we propose a tracking controller for the leader spacecraft which makes it follow a prescribed reference. Independently, we construct a synchronization control law for the follower spacecraft which makes it track the attitude of the leader, thereby synchronizing in the classical master-slave configuration.

Our controllers are reminiscent of classical tracking controllers for robot manipulators passivity-based control which exploits the system's physical properties; see [21]. For tracking control, see the passivity-based PD+ of [22] and the wrongly called “sliding-mode” controller of [23] which may rather be casted in the passivity-based framework.

Although insightful, these popular control approaches for robot manipulators may not be directly applied in spacecraft tracking control and synchronization. The first obstacle is the specificity of spacecraft nonlinear models, expressed in quaternion coordinates. We revise the model in the following section.

Besides the difficulties imposed by the modeling of spacecraft, simultaneous tracking control and master-slave synchronization implicitly suggest controlling the leader spacecraft towards a reference independently of the slave system dynamics. Correspondingly, the synchronization controller inevitably couples the follower spacecraft to the dynamics of the leader. However, the synchronization controller is demanded to achieve the task regardless of the master dynamics as well as the reference that system intends to track. The ability to control two coupled systems separately is called separation principle and is known not to hold in general for nonlinear systems (see e.g. [24]). This is where cascades theory enters in play.

Cascaded systems theory consists in analyzing complex systems by dividing them in subsystems simpler to control and to analyze (see [25] and references within). It must be emphasized that such representation is purely schematic, for the purpose of analysis only. Generally speaking, the stability analysis problem consists in finding conditions for two systems as in Figure 1 so that, considering that both subsystems separately are stable, they conserve that property when interconnected.

In the context of the present paper, the block on the left corresponds to the leader system in closed loop with a tracking controller, while the block on the right consists in the follower spacecraft in closed loop with the synchronization controller. The blocks are interconnected via the tracking errors of the leader system. Hence, in the ideal case, when the leader spacecraft is perfectly controlled, the systems are decoupled.

The topic of cascaded systems have received a great deal of attention and has successfully been applied to a wide number of applications. In [26], a cascaded adaptive control scheme for marine vehicles including actuator dynamics was introduced, while [27] solved the problem of synchronization of two pendula through use of cascades. The authors of [28] studied the problem of global stabilizability of feed-forward systems by a systematic recursive design procedure for autonomous systems, while time-varying systems were considered in [29] for stabilization of robust control, while [30] established sufficient conditions for uniform global asymptotical stability (UGAS) of cascaded nonlinear time-varying systems. The aspect of practical and semiglobal stability for nonlinear time-varying systems in cascade was pursued in [31, 32]. A stability analysis of spacecraft formations including both leader and follower using relative coordinates is presented in [10], where the controller-observer scheme is proven input-to-state stable, and backstepping was applied in [17] for leader-follower formation control of multiple underactuated autonomous underwater vehicles (AUVs). For the control problems at hand, we show that the closed-loop system has the property of uniform asymptotic stability. Significantly, uniform asymptotic stability guarantees robustness with respect to bounded disturbances. In this regard, we extend our result to the case where bounded perturbations affect the system (atmospheric drag, gravity gradient, etc.). In this scenario, we guarantee uniform practical asymptotic stability. This pertains to the case when there exists a steady-state tracking and synchronization error which can be arbitrarily diminished via an appropriate tuning of the control parameters.

The contribution of this paper is application of the framework for stability analysis of cascaded systems of rigid bodies in leader-follower formation and synchronizvation of PD+ and sliding surface control laws adapted for quaternion space. The equilibrium points of the PD+ controller in closed loop with the rigid body dynamics are proven uniformly asymptotically stable (UAS) when disturbances are considered known, while a sliding surface controller is utilized to prove uniform practical asymptotical stability (UPAS) when disturbances are considered unknown but bounded. Simulation results of a leader-follower spacecraft formation are presented to show the performance of our proposed control laws.

The rest of the paper is organized as follows: in Section 2, we recall the modeling of rigid bodies in quaternion coordinates and present the main theoretical tools of cascaded systems theory, which are instrumental to our control design. Controller design is presented in Section 3 for known disturbances while for unknown but upper-bounded disturbances in Section 4. Simulation results are presented in Section 5, and everything is wrapped up with conclusions in Section 6.

2. Mathematical Background

In the following, we denote by ̇𝐱 the time derivative of a vector 𝐱, that is, ̇𝐱=𝑑𝐱/𝑑𝑡; moreover, ̈𝐱=𝑑2𝐱/𝑑𝑡2, and denotes the 2 norm of vectors and induced 2 norm of matrices. Coordinate reference frames are denoted by (), where the superscript denotes the frame in question. Moreover, we denote a rotation matrix between frame 𝑎 and frame 𝑏 by 𝐑𝑏𝑎SO(3), and the angular velocity of frame 𝑎 relative to frame 𝑏, referenced in frame 𝑐, is denoted by 𝝎𝑐𝑏,𝑎3. We denote by 𝐱(𝑡;𝑡0,𝐱(𝑡0)) the solutions of the differential equation ̇𝐱=𝑓(𝑡,𝐱) with initial conditions (𝑡0,𝐱(𝑡0)). When the context is sufficiently explicit, we may omit the arguments of a function, vector or matrix.

2.1. Cartesian Coordinate Frames

The coordinate reference frames used throughout the paper are defined as follows.

Earth-Centered Inertial Frame
The Earth-centered inertial (ECI) frame is denoted by 𝑖 and has its origin in the center of the Earth. The axes are denoted by 𝑥𝑖,𝑦𝑖, and 𝑧𝑖, where the 𝑧𝑖-axis is directed along the axis of rotation of the Earth toward the celestial North Pole, the 𝑥𝑖-axis is pointing in the direction of the vernal equinox, 𝚼, which is the vector pointing from the center of the sun toward the center of the Earth during the vernal equinox, and finally the 𝑦𝑖-axis completes the right-handed orthonormal frame.

Spacecraft Orbit Reference Frame
The orbit frame is denoted by 𝑠𝑜, where the sub-/superscript 𝑠=𝑙,𝑓 denotes the leader and follower spacecraft, respectively, such that we throughout the paper denote, for example, 𝑙𝑜 and 𝑓𝑜 as 𝑠𝑜, which has its origin located in the center of the mass of the spacecraft. The 𝐞𝑟-axis in the frame coincides with the vector 𝐫=[𝑟𝑥,𝑟𝑦,𝑟𝑧] from the center of the Earth to the spacecraft, and the 𝐞-axis is parallel to the orbital angular momentum vector, pointing in the orbit normal direction. The 𝐞𝜃-axis completes the right-handed orthonormal frame. The basis vectors of the frame can be defined as 𝐞𝑟=𝐫𝑟,𝐞𝜃=𝐞×𝐞𝑟,𝐞=𝐡,(1) where ̇𝐫𝐡=𝐫× is the angular momentum vector of the orbit, =|𝐡| and 𝑟=|𝐫|. This frame is also known as the local vertical/local horizontal (LVLH) frame.

Spacecraft Body Reference Frame
The body frame of the spacecraft is denoted by 𝑠𝑏 and is located at the center of the mass of the spacecraft, and its basis vectors are aligned with the principle axis of inertia.

2.2. Quaternions

The attitude of a rigid body is often represented by a rotation matrix 𝐑SO(3) fulfillingSO(3)=𝐑3×3𝐑,𝐑=𝐈,det𝐑=1(2) which is the special orthogonal group of order three. Quaternions are often used to parameterize members of SO(3) where the unit quaternion is defined as 𝐪=[𝜂,𝝐]S3={𝐱4𝐱𝐱=1}, where 𝜂 is the scalar part and 𝝐3 is the vector part. The rotation matrix may be described by [33]𝐑=𝐈+2𝜂𝐒(𝝐)+2𝐒2(𝝐),(3) where the matrix 𝐒() is the cross product operator defined as𝐒(𝝐)=𝝐×=0𝜖𝑧𝜖𝑦𝜖𝑧0𝜖𝑥𝜖𝑦𝜖𝑥0,(4) where 𝝐=[𝜖𝑥,𝜖𝑦,𝜖𝑧]. The inverse rotation can be performed by using the inverse conjugate of 𝐪 as 𝐪=[𝜂,𝝐]. The set S3 forms a group with quaternion multiplication, which is distributive and associative, but not commutative, and the quaternion product of two arbitrary quaternions 𝐪1 and 𝐪2 is defined as [33]𝐪1𝐪2=𝜂1𝜂2𝝐1𝝐2𝜂1𝝐2+𝜂2𝝐1𝝐+𝐒1𝝐2.(5)

2.3. Kinematics and Dynamics

The time derivative of (3) can be written as [33]̇𝐑𝑎𝑏𝝎=𝐒𝑎𝑎,𝑏𝐑𝑎𝑏=𝐑𝑎𝑏𝐒𝝎𝑏𝑎,𝑏,(6) and the kinematic differential equations can be expressed as [33]̇𝐪𝑠𝐪=𝐓𝑠𝝎𝑠𝑏𝑖,𝑠𝑏,(7) where𝐓𝐪𝑠=12𝝐𝑇𝑠𝜂𝑠𝝐𝐈+𝐒𝑠4×3.(8) The dynamical model of a rigid body can be described by a differential equation for angular velocity and is deduced from Euler’s moment equation. This equation describes the relationship between applied torque and angular momentum on a rigid body as [34]𝐉𝑠̇𝝎𝑠𝑏𝑖,𝑠𝑏𝝎=𝐒𝑠𝑏𝑖,𝑠𝑏𝐉𝑠𝝎𝑠𝑏𝑖,𝑠𝑏+𝝉𝑠𝑠𝑏,(9) where 𝝉𝑠𝑠𝑏3 is the total torque working on the body frame and 𝐉𝑠=diag{𝐽𝑠𝑥,𝐽𝑠𝑦,𝐽𝑠𝑧}3×3 is the moment of inertia. The torque working on the body is expressed as 𝝉𝑠𝑠𝑏=𝝉𝑠𝑏𝑠𝑎+𝝉𝑠𝑏𝑠𝑑, where 𝝉𝑠𝑏𝑠𝑑 is the disturbance torque and 𝝉𝑠𝑏𝑠𝑎 is the actuator (control) torque. It might be desirable to express the rotation between the body frame and the orbit frame which can be done by applying𝝎𝑠𝑏𝑠,𝑠𝑏=𝝎𝑠𝑏𝑖,𝑠𝑏𝐑𝑖𝑠𝑏𝝎𝑖𝑖,𝑠,(10) where 𝝎𝑖𝑖,𝑠=𝐒(𝐫𝑖𝑠)𝐯𝑖𝑠/𝐫𝑠𝑖𝐫𝑖𝑠, and 𝐫𝑖𝑠 and 𝐯𝑖𝑠 are the spacecraft radius and velocity vector, respectively, represented in the inertial frame.

2.4. Cascaded Systems Theory

A typical nonlinear cascaded time-varying system on closed-loop dynamical form is expressed asΣ1̇𝑥1=𝑓1𝑡,𝑥1+𝑔(𝑡,𝑥)𝑥2Σ,(11)2̇𝑥2=𝑓2𝑡,𝑥2,(12) where 𝑥1𝑛, 𝑥2𝑚, 𝑥=[𝑥1,𝑥2], and the functions 𝑓1(,), 𝑓2(,), and 𝑔(,) are continuous in their arguments, locally Lipschitz in 𝑥, and uniform in 𝑡, and 𝑓1(,) is continuously differentiable in both arguments. Note that (12) is decoupled from (11), hence, it will be called the driving system, and its state enters as an input to the upper system with state 𝑥1 through the interconnection term 𝑔(𝑡,𝑥)𝑥2.

In the context of this paper, the dynamicṡ𝑥1=𝑓1𝑡,𝑥1(13) represents the synchronization error dynamics of the leader-follower configuration, assuming that perfect tracking is achieved for the leader system, that is, the tracking error for the latter is represented by 𝑥2, and its closed-loop dynamics under tracking control will be represented by (12).

We will evoke [25, Theorem 1] to prove uniform asymptotic stability of the equilibrium point of a closed-loop system on the form (11)-(12) under the following assumptions.

Assumption 1. There exist constants 𝑐1,𝑐2,𝛿>0 and a Lyapunov function 𝑉(𝑡,𝑥1) for (13) such that 𝑉0×𝑛0 is positive definite, radially unbounded, ̇𝑉(𝑡,𝑥1)0 and 𝜕𝑉𝜕𝑥1𝑥1𝑐1𝑉𝑡,𝑥1𝑥1𝛿,(14)𝜕𝑉𝜕𝑥1𝑐2𝑥1𝛿.(15)

Assumption 2. There exist two continuous functions 𝜉1, 𝜉200 such that 𝑔(𝑡,𝑥) satisfies 𝑔(𝑡,𝑥)𝜉1𝑥2+𝜉2𝑥2𝑥1.(16)

Assumption 3. There exists a class 𝒦 function 𝛼() such that, for all 𝑡00, the trajectories of the system (12) satisfy 𝑡0𝑥2𝑡;𝑡0,𝑥2𝑡0𝑥𝑑𝑡𝛼2𝑡0.(17) The Theorem cited above may be extended to the case when the subsystems possess the weaker property of practical asymptotic stability. This pertains to the situation in which the errors do not converge to zero but to a bounded region that may be made arbitrarily small; see [35]. A related popular concept, for instance, in control of mechanical systems is that of ultimate boundedness. However, practical asymptotic stability is stronger than ultimate boundedness since the later is only a notion of convergence and does not imply stability in the sense of Lyapunov. In other words, the fact that the errors converge to a bounded region does not imply that they remain always arbitrarily close to it.

3. Uniform Asymptotic Stabilization

We are ready to present the first results on tracking and synchronization cascade-based control. The control strategy relies on using models for two single spacecraft coupled through synchronized control, and stability properties are proved using cascade theory for known disturbances, that is, we assume that the disturbances 𝝉𝑙𝑏𝑙𝑑 and 𝝉𝑓𝑏𝑓𝑑 can be modeled correctly (see, e.g., [34, 36, 37]).

3.1. Problem Formulation

The control problem is to design two controllers; the first one makes the states 𝐪𝑙(𝑡), 𝝎𝑙𝑏𝑖,𝑙𝑏(𝑡), and ̇𝝎𝑙𝑏𝑖,𝑙𝑏(𝑡) converge towards the generated references specified as 𝐪𝑑(𝑡), 𝝎𝑙𝑏𝑖,𝑑(𝑡), and ̇𝝎𝑙𝑏𝑖,𝑑(𝑡), satisfying the kinematic equatioṅ𝐪𝑑𝐪=𝐓𝑑𝝎𝑙𝑏𝑖,𝑑,(18) and acts as a solution to the dynamical model presented in (9). It should be noted that we apply (10) and its derivative to the generated reference rather than the dynamical equation to keep a simple control law structure compared to [38]. The second controller is designed as a synchronizing controller to make the states 𝐪𝑓(𝑡) and 𝝎𝑓𝑏𝑖,𝑓𝑏(𝑡) converge towards 𝐪𝑙(𝑡) and 𝝎𝑓𝑏𝑖,𝑙𝑏(𝑡). The error quaternion ̃𝐪𝑠=[̃𝜂𝑠,̃𝝐𝑠] is found by applying the quaternion product̃𝐪𝑠=𝐪𝑠𝐪𝑑=𝜂𝑠𝜂𝑑+𝝐𝑠𝝐𝑑𝜂𝑑𝝐𝑠𝜂𝑠𝝐𝑑𝝐𝐒𝑠𝝐𝑑,(19) where the sub-/superscript 𝑠=𝑙,𝑓 denotes the leader and follower spacecraft, respectively, and the error kinematics can be expressed as [39]̇̃𝐪𝑠=12𝐓̃𝐪𝑠𝐞𝑠𝜔,(20) where 𝐞𝑠𝜔=𝝎𝑠𝑏𝑖,𝑠𝑏𝝎𝑠𝑏𝑖,𝑑 is the angular velocity error. Due to the redundancy in the quaternion representation, ̃𝐪𝑠 and ̃𝐪𝑠 represent the same physical attitude, but, mathematically, it differs by a 2𝜋 rotation about an arbitrary axis. Therefore, we are not able to achieve a global representation since the term global refers to the whole state space 𝑛 according to [35]. Since both equilibrium points represent the same physical representation, the equilibrium point which requires the shortest rotation is usually chosen, thus minimizing the path length; hence, ̃𝐪𝑠+=[1,𝟎] is chosen if ̃𝜂𝑠(𝑡0)0 and ̃𝐪𝑠=[1,𝟎] is chosen if ̃𝜂𝑠(𝑡0)<0. For the positive equilibrium point, an attitude error function is chosen as 𝐞𝑠𝑞+=[1̃𝜂𝑠,̃𝝐𝑠] (see [9]), while, for the negative equilibrium point, the attitude error function is chosen as 𝐞𝑠𝑞=[1+̃𝜂𝑠,̃𝝐𝑠]. In accordance with general kinematic relationṡ𝐞𝑠𝑞±=𝐓𝑠𝑒𝐞𝑠𝑞±𝐞𝑠𝜔,(21) where𝐓𝑠𝑒𝐞𝑠𝑞±=12±̃𝝐𝑠̃𝜂𝑠̃𝝐𝐈+𝐒𝑠.(22) The attitude error function is chosen a priori and kept throughout the maneuver even if ̃𝜂𝑠(𝑡) should happen to switch sign for some 𝑡. The control problem is presented as a cascaded system on the form (11)-(12), where the states are defined as 𝐱1=[𝐞𝑓𝑞,𝐞𝑓𝜔] and 𝐱2=[𝐞𝑙𝑞,𝐞𝑙𝜔]. The control objective is to stabilize the equilibrium point (𝐞𝑓𝑞(𝑡),𝐞𝑓𝜔(𝑡),𝐞𝑙𝑞(𝑡),𝐞𝑙𝜔(𝑡))=(𝟎,𝟎,𝟎,𝟎) as 𝑡 for all initial values.

3.2. Control of Leader Spacecraft

For control of the leader spacecraft attitude, we incorporate a model-dependent control law as in [40]. It is assumed that we have available information of its attitude 𝐪𝑙, angular velocity 𝝎𝑙𝑏𝑖,𝑙𝑏, and inertia matrix 𝐉𝑙, and, temporarily, we assume to know external perturbations. We choose the equilibrium such that 𝐞𝑙𝑞±=[1̃𝜂𝑙,̃𝝐𝑙] is either the positive or negative equilibrium point, which does not change during the maneuver. By pure conventionalism, consider the positive equilibrium, that is, 𝐞𝑙𝑞=𝐞𝑙𝑞+. We define desired attitude 𝐪𝑑(𝑡), desired angular velocity 𝝎𝑙𝑏𝑖,𝑑(𝑡), and desired angular acceleration ̇𝝎𝑙𝑏𝑖,𝑑(𝑡) which are all bounded functions. The control law is expressed as𝝉𝑙𝑏𝑙𝑎=𝐉𝑙̇𝝎𝑙𝑏𝑖,𝑑𝐉𝐒𝑙𝝎𝑙𝑏𝑖,𝑙𝑏𝝎𝑙𝑏𝑖,𝑑𝑘𝑙𝑞𝐓𝑙𝑒𝐞𝑙𝑞𝑘𝑙𝜔𝐞𝑙𝜔𝝉𝑙𝑏𝑙𝑑,(23) where 𝑘𝑙𝑞>0 and 𝑘𝑙𝜔>0 are feedback gain scalars. By insertion of (23) into (9), the system is on form (12), and, by applying the property 𝐒(𝝎𝑙𝑏𝑖,𝑙𝑏)𝐉𝑙𝝎𝑙𝑏𝑖,𝑙𝑏=𝐒(𝐉𝑙𝝎𝑙𝑏𝑖,𝑙𝑏)𝝎𝑙𝑏𝑖,𝑙𝑏, we obtain the closed-loop dynamics𝐉𝑙̇𝐞𝑙𝜔+𝑘𝑙𝜔𝐉𝐈𝐒𝑙𝝎𝑙𝑏𝑖,𝑙𝑏𝐞𝑙𝜔+𝑘𝑙𝑞𝐓𝑙𝑒𝐞𝑙𝑞=𝟎.(24) A radial unbounded, positive definite Lyapunov function candidate is defined as𝑉𝑙=12𝐞𝑙𝜔𝐉𝑙𝐞𝑙𝜔+12𝐞𝑙𝑞𝑘𝑙𝑞𝐞𝑙𝑞>0𝐞𝑙𝜔𝟎,𝐞𝑙𝑞𝟎.(25) Indeed, we have 12𝑗min𝑙𝑚,𝑘𝑙𝑞𝐱22𝑉𝑙12𝑗max𝑙𝑀,𝑘𝑙𝑞𝐱22,(26) where 𝑗𝑙𝑚𝐽𝑙𝑗𝑙𝑀. By differentiation of (25) and inserting (24) and (21), we obtaiṅ𝑉𝑙=𝐞l𝑞𝑘𝑙𝑞𝐓𝑙𝑒𝐞𝑙𝜔+𝐞𝑙𝜔𝐒𝐉𝑙𝝎𝑙𝑏𝑖,𝑙𝑏𝑘𝑙𝜔𝐈𝐞𝑙𝜔𝐞𝑙𝜔𝑘𝑙𝑞𝐓𝑙𝑒𝐞𝑙𝑞,(27) where the first part of the second term in (27) is zero because 𝐒(𝐉𝑙𝝎𝑙𝑏𝑖,𝑙𝑏) is a skew-symmetric matrix, which leads tȯ𝑉𝑙=𝐞𝑙𝜔𝑘𝑙𝜔𝐞𝑙𝜔0(28) so the origin of the system is uniformly stable and the trajectories are bounded. The rest of the proof consists in showing that the position errors and the velocity tracking errors are square integrable. Then it suffices to invoke [41, Lemma 3].

Let 𝒱𝑙(𝑡)=𝑉𝑙(𝐞𝑙𝑞(𝑡),𝐞𝑙𝜔(𝑡)) and 𝐱2(𝑡)=[𝐞𝑙𝑞(𝑡),𝐞𝑙𝜔(𝑡)]. Then, from (28), we obtain by integrating on both sides𝑡𝑡0̇𝒱𝑙(𝑠)𝑑𝑠=𝑡𝑡0𝐞𝑙𝜔(𝑠)𝑘𝑙𝜔𝐞𝑙𝜔(𝑠)𝑑𝑠,(29)𝒱𝑙(𝑡)+𝒱𝑙𝑡0=𝑘𝑙𝜔𝑡𝑡0𝐞𝑙𝜔(𝑠)2𝑑𝑠.(30) Since 𝒱𝑙(𝑡)0, we can write𝑘𝑙𝜔𝑡𝑡0𝐞𝑙𝜔2𝑑𝑠𝒱𝑙𝑡0=12𝐞𝑙𝜔𝑡0𝐉𝑙𝐞𝑙𝜔𝑡0+𝐞𝑙𝑞𝑡0𝑘𝑙𝑞𝐞𝑙𝑞𝑡012𝑗max𝑙𝑀,𝑘𝑙𝑞𝐱2𝑡02,(31) and thus𝑡𝑡0𝐞𝑙𝜔(𝑠)2𝑑𝑠𝑐3𝐱2𝑡02,(32) where 𝑐3=max{𝑗𝑙𝑀,𝑘𝑙𝑞}/2𝑘𝑙𝜔.

Now let 𝒲𝑙(𝑡)=𝑊𝑙(𝐞𝑙𝑞(𝑡),𝐞𝑙𝜔(𝑡)) such that𝒲𝑙(𝑡)=𝐞𝑙𝑞(𝑡)𝐓𝑙𝑒(𝑡)𝑘𝑙𝑞𝐉𝑙𝐞𝑙𝜔(𝑡),(33) and, by differentiation, we obtaiṅ𝒲𝑙̇𝐞(𝑡)=𝑙𝑞(𝑡)𝐓𝑙𝑒(𝑡)𝑘𝑙𝑞𝐉𝑙𝐞𝑙𝜔(𝑡)+𝐞𝑙𝑞̇𝐓(𝑡)𝑙𝑒(𝑡)𝑘𝑙𝑞𝐉𝑙𝐞𝑙𝜔(𝑡)+𝐞𝑙𝑞(𝑡)𝐓𝑙𝑒(𝑡)𝑘𝑙𝑞𝐉𝑙̇𝐞𝑙𝜔(𝑡).(34) By inserting ̇𝐓𝑙𝑒(𝑡)𝐞𝑙𝑞(𝑡)=𝐆𝑙(𝑡)𝐞𝑙𝜔(𝑡), where𝐆𝑙1(𝑡)=2̃𝜂𝑙̃𝝐(𝑡)𝐈+𝐒𝑙1(𝑡)4𝐈,(35) and (21) and (24) into (34), we obtaiṅ𝒲𝑙=𝐞𝑙𝜔(𝑡)𝐓𝑙𝑒(𝑡)𝐓𝑙𝑒(𝑡)𝑘𝑙𝑞𝐉𝑙𝐞𝑙𝜔(𝑡)+𝐞𝑙𝜔𝐆𝑙(𝑡)𝑘𝑙𝑞𝐉𝑙𝐞𝑙𝜔(𝑡)𝐞𝑙𝑞(𝑡)𝐓𝑙𝑒(𝑡)𝑘𝑙𝑞𝐉𝐒𝑙𝝎𝑙𝑏𝑖,𝑙𝑏(𝑡)+𝑘𝑙𝜔𝐈𝐞𝑙𝜔(𝑡)𝐞𝑙𝑞(𝑡)𝐓𝑙𝑒(𝑡)𝑘2𝑙𝑞𝐓𝑙𝑒(𝑡)𝐞𝑙𝑞(𝑡)𝑐4𝐞𝑙𝑞2+𝑐5𝐞𝑙𝜔2+𝐞𝑙𝑞𝑐6𝐞𝑙𝜔,(36) where 𝑐4=𝑘2𝑙𝑞/4, 𝑐5=𝑘𝑙𝑞𝑗𝑙𝑀𝐆𝑙(𝑡), 𝑐6=𝑘𝑙𝑞2𝐒𝐉𝑙𝝎𝑙𝑏𝑖,𝑙𝑏(𝑡)+𝑘𝑙𝜔,(37) and 𝐆𝑙(𝑡)3/4. The last term of (36) can be rewritten as𝐞𝑙𝑞𝑐6𝐞𝑙𝜔𝐞𝜅𝑙𝜔(𝑡)2+𝑐26𝜅𝐞𝑙𝑞(𝑡)2,(38) and by choosing 𝜅1 such that 𝑐4>2𝑐26/𝜅, we obtaiṅ𝒲𝑙𝑐(𝑡)42𝐞𝑙𝑞(𝑡)2+𝑐5𝐞+𝜅𝑙𝜔(𝑡)2.(39) By applying the same line of arguments as in (29)–(31) and inequality (32), (39) may be expressed as𝒲𝑙𝑡0+𝑐5𝑐+𝜅3𝐱2𝑡02𝑐42𝑡𝑡0𝐞𝑙𝑞(𝑠)2𝑑𝑠.(40) By inserting the upper bound𝒲𝑙𝑡012𝑘max𝑙𝑞,𝑗𝑙𝑀𝐞𝑙𝑞𝑡0𝐞𝑙𝜔𝑡0𝑘max𝑙𝑞,𝑗𝑙𝑀𝐞𝑙𝑞𝑡02+𝐞𝑙𝜔𝑡02𝑘max𝑙𝑞,𝑗𝑙𝑀𝐱2𝑡02(41) into (40), the expression may be written as𝑡𝑡0𝐞𝑙𝑞(𝑠)2𝑑𝑠𝑐7𝐱2𝑡02,(42) where 𝑐7=2(1/2max{𝑘𝑙𝑞,𝑗𝑙𝑀}+(𝑐5+𝜅)𝑐3)/𝑐4.

We conclude from Lemma  3 of [41] that the origin is uniformly exponentially stable.

3.3. Control of Follower Spacecraft

For control of the follower spacecraft attitude, we incorporate a similar control law as in Section 3.2 but add terms for synchronization. For the control law, it is assumed that we have available information of the attitude and angular velocity for both spacecraft and inertia matrix, 𝐉𝑓, and that all perturbations are known and accounted for. In the following, it is assumed that the equilibrium point is chosen such that 𝐞𝑓𝑞±=[1̃𝜂𝑓,̃𝝐𝑓] is either the positive or negative equilibrium point and does not change during the maneuver. The control law is expressed as𝝉𝑓𝑏𝑓𝑎=𝐉𝑓̇𝝎𝑓𝑏𝑖,𝑑𝐉𝐒𝑓𝝎𝑓𝑏𝑖,𝑓𝑏𝝎𝑓𝑏𝑖,𝑑𝑘𝑓𝑞𝐓𝑓𝑒𝐞𝑓𝑞𝐓𝑙𝑒𝐞𝑙𝑞𝑘𝑓𝜔𝐞𝑓𝜔𝐞𝑙𝜔𝝉𝑓𝑏𝑓𝑑,(43) where 𝑘𝑓𝑞>0 and 𝑘𝑓𝜔>0 are feedback gain scalars and the last two terms are for synchronization. By insertion of (43) into (9), we obtain the closed-loop rotational dynamicṡ𝐞𝑓𝜔=𝐉𝑓1𝑘𝑓𝜔𝐉𝐒𝑓𝝎𝑓𝑏𝑖,𝑓𝑏𝐞𝑓𝜔𝑘𝑓𝑞𝐓𝑓𝑒𝐞𝑓𝑞+𝑘𝑓𝑞𝐓𝑙𝑒𝐞𝑙𝑞+𝑘𝑓𝜔𝐞𝑙𝜔𝐱=𝑓1+𝑔(𝐱)𝐱2,(44) where 𝑓(𝐱1) is similar to the closed-loop system derived in Section 3.2 hence the proof of uniform asymptotic stability follows along similar lines.

The interconnection function is𝐉𝑔(𝐱)=𝑓1𝑘𝑓𝑞𝐓𝑙𝑒𝟎𝟎𝐉𝑓1𝑘𝑓𝜔.(45) In what follows we show that Assumptions 13 hold and, hence, that the origin of the closed-loop system is uniformly asymptotically stable.

Proof of Assumption 1. We start by considering 𝛿=1 (where in this case 𝛿 is the parameter given in Assumption 1). By evaluating (14) on 𝑉𝑓, we obtain 𝐞𝑓𝜔𝐉𝑓+𝐞𝑓𝑞𝑘𝑓𝑞𝐱1𝑐1𝑉𝑓𝐱1=𝑐12𝐞𝑓𝜔𝐉𝑓𝐞𝑓𝜔+𝐞𝑓𝑞𝑘𝑓𝑞𝐞𝑓𝑞,(46) and by applying the triangular inequality on the left side of (46) and squaring, we obtain 𝐞𝑓𝜔𝐉𝑓+𝐞𝑓𝑞𝑘𝑓𝑞2𝐱12𝑐214𝐞𝑓𝜔𝐉𝑓𝐞𝑓𝜔+𝐞𝑓𝑞𝑘𝑓𝑞𝐞𝑓𝑞2.(47) On the left side of (47), we apply 𝑥2+𝑦2+2𝑥𝑦2(𝑥2+𝑦2) and 𝑗𝑓𝑚𝐉𝑓𝑗𝑓𝑀 such that 𝐞𝑓𝜔𝐉𝑓+𝐞𝑓𝑞𝑘𝑓𝑞2𝐞2𝜑𝑓𝜔𝐉𝑓𝐞𝑓𝜔+𝐞𝑓𝑞𝑘𝑓𝑞𝐞𝑓𝑞,(48) where 𝜑=max{𝑗𝑓𝑀,𝑘𝑓𝑞}. By insertion of (48) in (47), we obtain 𝐞2𝜑𝑓𝜔𝐞𝑓𝜔+𝐞𝑓𝑞𝐞𝑓𝑞𝑐21𝜌4𝐞𝑓𝜔𝐞𝑓𝜔+𝐞𝑓𝑞𝐞𝑓𝑞,(49) where 𝜌=min{𝑗𝑓𝑚,𝑘𝑓𝑞}, and thus, we have to choose 𝑐18𝜑𝜌(50) to fulfill (14). For (15), we have that 𝐞𝑓𝜔𝐉𝑓+𝐞𝑓𝑞𝑘𝑓𝑞𝑐2,(51) and by using the triangular inequality and squaring and applying (48) we obtain 2𝜑2𝐱12𝑐22.(52) Since 𝐱11, we have to choose 𝑐22𝜑(53) to fulfill (15), and thus Assumption 1 is fulfilled.

Proof of Assumption 2. Since 𝐓𝑙𝑒=1/2, (45) obviously fulfills the growth rate criteria of (16), such as 1𝑔(𝐱)21𝐽2𝑥+1𝐽2𝑦+1𝐽2𝑧𝑘2𝑓𝑞+4𝑘2𝑓𝜔1/2=𝜉1,(54) where 𝜉1 is constant, and thus Assumption 2 is fulfilled.

Proof of Assumption 3. This assumption holds because 𝐱2(𝑡) converges to zero exponentially fast.
We conclude that the equilibrium point (𝐞𝑙𝑞,𝐞𝑙𝜔,𝐞𝑓𝑞,𝐞𝑓𝜔)=(𝟎,𝟎,𝟎,𝟎) of the cascaded system is UAS.
The proof for the other equilibria follows mutatis mutandis.

4. Robustness to Disturbances

In this section, we develop our results from Section 3 by introducing unknown bounded disturbances. We use a control law reminiscent of the Slotine and Li controller for manipulators; see [23], based on a control structure which has often been shown to be favorable from a stability analysis point of view. In the case of bounded additive nonvanishing disturbances, a steady-state error is unavoidable; hence, only practical asymptotic stability may be obtained. Although the control approach is the same, the technical analysis tools are more sophisticated. We refer the reader to [32].

4.1. Control of Leader Spacecraft

We start by assuming that we have available controller gains according to 𝜃1=[𝑘𝑓𝑞,𝑘𝑓𝜔], 𝜃2=[𝑘𝑙𝑞,𝑘𝑙𝜔]Θ1=Θ2=2>0. The uniform asymptotic stability in Section 3.2 is a result of the assumption that 𝝉𝑙𝑏𝑙𝑑 is known and accounted for in the control law. We relax this assumption and assume that 𝝉𝑙𝑏𝑙𝑑 is unknown, but bounded, and thus there exists a 𝛽𝑙>0 such that 𝝉𝑙𝑏𝑙𝑑𝛽𝑙. Note that many of the disturbances for Earth-orbiting spacecraft can be reasonably well modeled as ̂𝐟sbsd, which can be added to the overall analysis such that ̃𝐟sbsd=𝐟𝑠𝑏sd̂𝐟𝑠𝑏sd. This strategy can reduce the upper bound such that 𝐟̃𝑠𝑏sd(𝑡)<𝛽̃𝑠<𝛽𝑠, based on the quality of the disturbance modeling. We apply the control law𝝉𝑙𝑏𝑙𝑎=𝐉𝑙̇𝝎𝑙𝑏𝑖,𝑟𝐉𝐒𝑙𝝎𝑙𝑏𝑖,𝑙𝑏𝝎𝑙𝑏𝑖,𝑟𝑘𝑙𝑞𝐓𝑙𝑒𝐞𝑙𝑞𝑘𝑙𝜔𝐬𝑙𝝎,(55)𝑙𝑏𝑖,𝑟=𝝎𝑙𝑏𝑖,𝑑𝚪𝑙𝐓𝑙𝑒𝐞𝑙𝑞𝐞𝑙𝑞𝐬,(56)𝑙=𝝎𝑙𝑏𝑖,𝑙𝑏𝝎𝑙𝑏𝑖,𝑟=𝐞𝑙𝜔+𝚪𝑙𝐓𝑙𝑒𝐞𝑞𝑙,(57) where 𝑘𝑙𝑞>0, 𝑘𝑙𝜔>0 and Γ𝑙=Γ𝑙>0 are feedback gains, and, by inserting (55) into (9), we obtain the closed-loop dynamicṡ𝐬𝑙=𝐉𝑙1𝐒𝐉𝑙𝝎𝑙𝑏𝑖,𝑙𝑏𝐬𝑙𝑘𝑙𝑞𝐓𝑙𝑒𝐞𝑙𝑞𝑘𝑙𝜔𝐬𝑙.(58) A radial unbounded, positive definite Lyapunov function candidate is defined as𝑉𝑙=12𝐬𝑙𝐉𝑙𝐬𝑙+𝐞𝑙𝑞𝑘𝑙𝑞𝐞𝑙𝑞>0𝐬𝑙𝟎,𝐞𝑙𝑞𝟎,(59) and, by differentiation of (59) and insertion of (58), we obtaiṅ𝑉𝑙=𝐬𝑙𝑘𝑙𝜔𝐬𝑙𝐞𝑙𝑞𝐓𝑙𝑒𝚪𝑘𝑙𝑞𝐓𝑙𝑒𝐞𝑙𝑞+𝐬𝑙𝝉𝑙𝑏𝑙𝑑=𝝌2𝐐𝑙𝝌2+𝐬𝑙𝝉𝑙𝑏𝑙𝑑𝑞𝑙𝑚𝝌22+𝛽𝑙𝝌2,(60) where 𝝌2=[𝐞𝑙𝑞,𝐬𝑙], 𝐐𝑙=diag{𝐓𝑙𝑒𝑘𝑙𝑞Γ𝑙𝐓𝑙𝑒,𝑘𝑙𝜔𝐈}, and 𝑞𝑙𝑚>0 is the smallest eigenvalue of 𝐐𝑙. Accordingly, ̇𝑉𝑙<0 when 𝝌2>𝛽𝑙/𝑞𝑙𝑚=𝛿2, and, as 𝛽𝑙 increases, it can be counteracted by increasing the controller gains. Hence, the perturbed system is uniformly practically asymptotically stable (UPAS); see [35]. We cannot claim semiglobal results for the same reasons as we cannot claim global results. Nevertheless, we assume that both Δ1 and Δ2 can be chosen arbitrary large to make it easier to follow the line of the proof.

4.2. Control of Follower Spacecraft

A similar disturbance as in Section 4.1 is introduced which are bounded such that 𝝉𝑓𝑏𝑓𝑑𝛽𝑓, and we apply a synchronizing controller based on the Slotine and Li structure such as𝝉𝑙𝑏𝑓𝑎=𝐉𝑓̇𝝎𝑓𝑏𝑖,𝑟𝐉𝐒𝑓𝝎𝑓𝑏𝑖,𝑓𝑏𝝎𝑓𝑏𝑖,𝑟𝑘𝑓𝑞𝐓𝑒𝑓𝐞𝑓𝑞𝐓𝑙𝑒𝐞𝑙𝑞𝑘𝑓𝜔𝐬𝑓𝐬𝑙,(61) where 𝝎𝑓𝑏𝑖,𝑟 and 𝐬𝑓 are defined similar to (56)-(57). By inserting (61) into (9), we obtaiṅ𝐬𝑓=𝑓𝝌1+̃𝑔(𝝌)𝝌2,(62) where 𝝌1=[𝐞𝑓𝑞,𝐬𝑓] and 𝝌=[𝝌1,𝝌2], and 𝑓(𝝌1) and ̃𝑔(𝝌) are on the same form as (58) and (45), respectively.

The rest of the proof consists in verifying the conditions of [32, Theorem 3.3].

Proof of Assumption 3.2 (see [32]). The function ̃𝑔(𝝌)𝝌2 is uniformly bounded both in time and in 𝜃2 and vanishes with 𝝌2; thus, for any 𝜃1Θ1, we choose 𝐺𝜃11(𝝌)=21𝐽2𝑥+1𝐽2𝑦+1𝐽2𝑧𝑘2𝑓𝑞+4𝑘2𝑓𝜔1/2,Ψ𝜃1𝝌2=𝝌2,(63) thus, 𝐺𝜃1(𝝌) is constant and Ψ𝜃1(𝝌2) is of class 𝒦, and the assumption is fulfilled for all 𝜃2Θ2 and all 𝝌𝒮3×3×𝒮3×3.

Proof of Assumption 3.4 (see [32]). This Assumption was proven in Section 4.1.

Proof of Assumption 3.5 (see [32]). By applying the same reasoning as in Section 4.1, we achieve ̇𝑉𝑓<0 when 𝝌1>𝛽𝑓/𝑞𝑓𝑚=𝛿1, where 𝑞𝑓𝑚>0 is the smallest eigenvalue of 𝐐𝑓=diag{𝐓𝑓𝑒𝑘𝑓𝑞Γ𝑓𝐓𝑓𝑒,𝑘𝑓𝜔𝐈}, which is defined similar to 𝐐𝑙 in (60). An increase of 𝛽𝑓 can as well be counteracted by increasing the controller gains; thus, given any Δ1>𝛿1>0, there exists a parameter 𝜃1(𝛿1,Δ1)Θ1. We choose 𝛼𝛿1,Δ1(𝝌1)=1/2min{𝑗𝑓𝑚,𝑘𝑓𝑞}𝝌12 and 𝛼𝛿1,Δ1(𝝌1)=1/2max{𝑗𝑓𝑀,𝑘𝑓𝑞}𝝌12, and thus the first part of the assumption is fulfilled, and the second part is fulfilled for 𝛼𝛿1,Δ1(𝝌1)=min{𝑘𝑓𝑞/4,𝑘𝑓𝜔}𝝌12. The last inequality also holds similar to (51)–(53) with 𝑐𝛿1,Δ1(𝝌1)=2𝜎𝝌11/2, and thus Assumption 3.5 holds for all 𝝌1(𝛿1,Δ1), where Δ1 can be chosen arbitrary large by assumption.

Proof of Assumption 3.6 (see [32]). We define a LFC for the leader and follower spacecraft as 𝑉𝑙𝑓1(𝑡,𝑥)=2𝐬𝑙𝐉𝑙𝐬𝑙+𝐞𝑙𝑞𝑘𝑙𝑞𝐞𝑙𝑞+𝐬𝑓𝐉𝑓𝐬𝑓+𝐞𝑓𝑞𝑘𝑓𝑞𝐞𝑓𝑞,(64) which is lower and upper bounded by 𝛼𝑙𝑓=12𝑗min𝑙𝑚,𝑗𝑓𝑚,𝑘𝑙𝑞,𝑘𝑓𝑞𝝌2,(65)𝛼𝑙𝑓=12𝑗max𝑙𝑀,𝑗𝑓𝑀,𝑘𝑙𝑞,𝑘𝑓𝑞𝝌2,(66) respectively, and it can be shown that the second equation of [32, Proposition 2.13] is fulfilled by differentiation of (64). There exists a positive constant Δ0 such that for any given positive number 𝛿1, Δ1, 𝛿2, Δ2, satisfying Δ1>𝑚𝑎𝑥{𝛿1,Δ0} and Δ2>𝛿2, there exists a 𝛿1 such that 𝛼𝑙𝑓(𝛿1)<𝛼𝑙𝑓(Δ1). As mentioned in Section 4.1, given any 𝛽𝑙, we can achieve any 𝛿2 by increasing the gains 𝜃2, and thus there exits a parameter 𝜃2𝒟𝑓2(𝛿2,Δ2)Θ2, such that by applying the last inequality of [32, Proposition 2.13] using the bounds (65), we see that the first equation of Assumption 3.6 is fulfilled for 𝛾Δ1=𝑗min𝑙𝑚,𝑗𝑓𝑚,𝑘𝑙𝑞,𝑘𝑓𝑞Δ21𝑗max𝑙𝑀,𝑗𝑓𝑀,𝑘𝑙𝑞,𝑘𝑓𝑞.(67) We have that 𝛼𝛿11,Δ1𝛼𝛿1,Δ1𝛿1=𝑗max𝑓𝑀,𝑘𝑓𝑞𝛿21𝑗min𝑓𝑚,𝑘𝑓𝑞,(68) then, for any Δ>𝛿>0, there exist parameters 𝛿1,𝛿2,Δ1, and Δ2 such that Δmin1,Δ2,𝑗min𝑙𝑚,𝑗𝑓𝑚,𝑘𝑙𝑞,𝑘𝑓𝑞Δ21𝑗max𝑙𝑀,𝑗𝑓𝑀,𝑘𝑙𝑞,𝑘𝑓𝑞Δ(69) since Δ1 and Δ2 can be chosen arbitrarily large and the constants 𝑗𝑠𝑚, 𝑗𝑠𝑀, 𝑘𝑠𝑞, 𝑘𝑠𝜔>0, and 𝛿max2,𝑗max𝑓𝑀,𝑘𝑓𝑞𝛿21𝑗min𝑓𝑚,𝑘𝑓𝑞𝛿(70) is fulfilled since, by decreasing 𝛿1,𝑘𝑓𝑞 is increased but only of order one, and thus the two last inequalities of Assumption 3.6 are fulfilled and Assumption 3.6 holds. It can then be concluded that the equilibrium points of the cascaded system are UPAS.
By setting 𝐞𝑙𝑞=𝐞𝑙𝑞 or 𝐞𝑓𝑞=𝐞𝑓𝑞, the other three proofs are performed in a similar way, and we thus conclude that the set of equilibrium points (𝐞𝑙𝑞±,𝐞𝑙𝜔,𝐞𝑓𝑞±,𝐞𝑓𝜔)=(𝟎,𝟎,𝟎,𝟎) are UPAS.

5. Simulations

In the following, simulation results of a leader spacecraft in an elliptic LEO with the follower spacecraft following the same orbit with one-second delay are presented to validate the proposed approach. The simulations were performed in Simulink using a variable sample-time Runge-Kutta ODE45 solver with relative and absolute tolerance of 109. The moments of inertia were chosen as 𝐉𝑙=𝐉𝑓=diag{4.35,4.33,3.664} kgm2, and the spacecraft orbits were chosen with apogee at 750 km, perigee at 600 km, inclination at 71°, and the argument of perigee and the right ascension of the ascending node at 0°. The initial conditions were chosen as 𝐪𝑙(𝑡0)=[0.3772,0.4329,0.6645,0.4783], 𝐪𝑓(𝑡0)=1/4[1,1,1,1], 𝝎𝑙(𝑡0)=[], and 𝝎𝑓(𝑡0)=[], controller gains as 𝑘𝑠𝑞=1,𝑘𝑠𝜔=2, and Γ𝑠=𝐈 for (23), (43), (55), and (61), and desired trajectory as ̇𝝎𝑑=0.1[10𝑐20cos(8𝑐0𝑡),48𝑐20sin(16𝑐0𝑡),8𝑐20cos(4𝑐0𝑡)], 𝝎𝑑 its integrate, and 𝐪𝑑(𝑡0)=[1,𝟎], where 𝑐0=𝜋/𝑡𝑜 and 𝑡𝑜 denotes the orbital period. For relative attitude (synchronization error) and angular velocity between the leader and follower spacecraft, we define ̃𝐪𝑠𝑦=𝐪𝑓𝐪𝑙 and 𝐞𝑠𝑦=𝐞𝑓𝜔𝐞𝑙𝜔, respectively.

In Figure 2, simulation results are presented without disturbances and noise. From Figure 2(a) we see that the leader spacecraft settles at the negative equilibrium, the angular velocity error go towards zero, and the actuator torque is presented in the bottommost plot. On the right-hand side we see that the relative attitude and angular velocity error go towards zero, and thus the follower spacecraft is able to synchronize with the leader spacecraft. The bottommost plot on the right hand depicts the follower actuator torque.

In the second set of simulation results, we introduce measurement noise as 𝜎𝔹𝑛={𝑥𝑛𝑥𝜎} which was added according to ̃𝐞𝑞=(𝐞𝑞+0.05𝔹4)/𝐞𝑞+0.05𝔹4 and ̃𝐞𝜔=𝐞𝜔+0.01𝔹3 for both the leader and follower spacecraft. Also, since we are considering a slightly elliptic LEO, we only considered the disturbance torques which are the major contributors to these kind of orbits, that is, gravity gradient torque and torques caused by atmospheric drag and 𝐽2 effect (cf. [34, 36, 37]). The latter is caused by nonhomogeneous mass distribution of a planet. The torques generated by atmospheric drag and 𝐽2 were induced because of a 10 cm displacement of the center of mass. All disturbances are considered continuous and bounded.

As it can be seen from Figure 3, the control laws derived for unknown disturbances also are able to make the leader track the reference and make the follower synchronize with the leader, similar to the results from the first simulation. One notable difference is that these control laws are in general faster than the results presented in Figure 2, though demanding larger absolute control torque.

6. Conclusions

In this paper, we have presented control laws for leader-follower synchronization of spacecraft, performed stability analysis based on cascade theory, and proven the equilibrium points of the overall system to be uniformly asymptotically stable (UAS) when disturbances were assumed to be known, and uniformly practically asymptotically stable (UPAS) when unknown, but bounded disturbances were included. Simulation results were presented to validate the proposed method for the overall system showing that the follower spacecraft was able to synchronize with the leader spacecraft in a satisfactor way even when disturbances were presented.