Cascade-Based Controlled Attitude Synchronization and Tracking of Spacecraft in Leader-Follower Formation

Abstract

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, [1–4].

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 [7–10], 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 [13–15], underactuated marine vessels [16, 17], and rigid bodies [18–20].

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.

Figure 1 

Cascade interconnection of two dynamical systems.

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 𝐱̇ the time derivative of a vector 𝐱, that is, 𝐱̇=𝑑𝐱/𝑑𝑡; moreover, 𝐱̈=𝑑²𝐱/𝑑𝑡², and ‖⋅‖ denotes the ℓ² norm of vectors and induced ℓ² norm of matrices. Coordinate reference frames are denoted by ℱ^(⋅), where the superscript denotes the frame in question. Moreover, we denote a rotation matrix between frame 𝑎 and frame 𝑏 by 𝐑^𝑏𝑎∈SO(3), and the angular velocity of frame 𝑎 relative to frame 𝑏, referenced in frame 𝑐, is denoted by 𝝎^{𝑐𝑏,𝑎}∈ℝ³. We denote by 𝐱(𝑡;𝑡₀,𝐱(𝑡₀)) the solutions of the differential equation 𝐱̇=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 ℱ^𝑖 and has its origin in the center of the Earth. The axes are denoted by 𝑥_𝑖,𝑦_𝑖, and 𝑧_𝑖, where the 𝑧_𝑖-axis is directed along the axis of rotation of the Earth toward the celestial North Pole, the 𝑥_𝑖-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 𝑦_𝑖-axis completes the right-handed orthonormal frame.

Spacecraft Orbit Reference Frame
The orbit frame is denoted by ℱ^𝑠𝑜, where the sub-/superscript 𝑠=𝑙,𝑓 denotes the leader and follower spacecraft, respectively, such that we throughout the paper denote, for example, ℱ^𝑙𝑜 and ℱ^𝑓𝑜 as ℱ^𝑠𝑜, which has its origin located in the center of the mass of the spacecraft. The 𝐞_𝑟-axis in the frame coincides with the vector 𝐫=[𝑟_𝑥,𝑟_𝑦,𝑟_𝑧]^⊤ from the center of the Earth to the spacecraft, and the 𝐞_ℎ-axis is parallel to the orbital angular momentum vector, pointing in the orbit normal direction. The 𝐞_𝜃-axis completes the right-handed orthonormal frame. The basis vectors of the frame can be defined as 𝐞_𝑟=𝐫𝑟,𝐞^𝜃=𝐞_ℎ×𝐞_𝑟,𝐞_ℎ=𝐡ℎ,(1) where 𝐡=𝐫×𝐫̇ is the angular momentum vector of the orbit, ℎ=|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 ℱ^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 𝐑∈SO(3) fulfillingSO(3)=𝐑∈ℝ^3×3∶𝐑^⊤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={𝐱∈ℝ⁴∶𝐱^⊤𝐱=1}, where 𝜂∈ℝ is the scalar part and 𝝐∈ℝ³ 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×=⎡⎢⎢⎣0−𝜖^𝑧𝜖_𝑦𝜖_𝑧0−𝜖_𝑥−𝜖_𝑦𝜖_𝑥0⎤⎥⎥⎦,(4) where 𝝐=[𝜖_𝑥,𝜖_𝑦,𝜖_𝑧]^⊤. 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]𝐪₁⊗𝐪₂=𝜂¹𝜂₂−𝝐^⊤1𝝐₂𝜂₁𝝐₂+𝜂₂𝝐₁+𝐒𝝐₁𝝐₂.(5)

2.3. Kinematics and Dynamics

The time derivative of (3) can be written as [33]𝐑̇^𝑎𝑏=𝐒𝝎^{𝑎𝑎,𝑏}𝐑^𝑎𝑏=𝐑^𝑎𝑏𝐒𝝎^{𝑏𝑎,𝑏},(6) and the kinematic differential equations can be expressed as [33]𝐪̇_𝑠=𝐓𝐪_𝑠𝝎^{𝑠𝑏𝑖,𝑠𝑏},(7) where𝐓𝐪_𝑠=12𝜂_𝑠𝐈+−𝝐𝐒^𝑇𝑠𝝐_𝑠∈ℝ^4×3.(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]𝐉_𝑠𝝎̇^{𝑠𝑏𝑖,𝑠𝑏}=−𝐒𝝎^{𝑠𝑏𝑖,𝑠𝑏}𝐉^𝑠𝝎^{𝑠𝑏𝑖,𝑠𝑏}+𝝉^𝑠𝑠𝑏,(9) where 𝝉^𝑠𝑠𝑏∈ℝ³ is the total torque working on the body frame and 𝐉_𝑠=diag{𝐽_𝑠𝑥,𝐽_𝑠𝑦,𝐽_𝑠𝑧}∈ℝ^3×3 is the moment of inertia. The torque working on the body is expressed as 𝝉^𝑠𝑠𝑏=𝝉^{𝑠𝑏𝑠𝑎}+𝝉^{𝑠𝑏𝑠𝑑}, where 𝝉^{𝑠𝑏𝑠𝑑} is the disturbance torque and 𝝉^{𝑠𝑏𝑠𝑎} 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 applying𝝎^{𝑠𝑏𝑠,𝑠𝑏}=𝝎^{𝑠𝑏𝑖,𝑠𝑏}−𝐑^𝑖𝑠𝑏𝝎^{𝑖𝑖,𝑠},(10) where 𝝎^{𝑖𝑖,𝑠}=𝐒(𝐫^𝑖𝑠)𝐯^𝑖𝑠/𝐫^𝑠𝑖⊤𝐫^𝑖𝑠, and 𝐫^𝑖𝑠 and 𝐯^𝑖𝑠 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 asΣ₁∶𝑥Σ̇₁=𝑓₁𝑡,𝑥₁+𝑔(𝑡,𝑥)𝑥₂,(11)₂∶𝑥̇₂=𝑓₂𝑡,𝑥₂,(12) where 𝑥₁∈ℝ^𝑛, 𝑥₂∈ℝ^𝑚, 𝑥=[𝑥^⊤1,𝑥^⊤2]^⊤, 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 dynamics𝑥̇₁=𝑓₁𝑡,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 𝑉∶ℝ_≥0×ℝ^𝑛→ℝ_≥0 is positive definite, radially unbounded, 𝑉̇(𝑡,𝑥₁)≤0 and ‖𝜕𝑉𝜕𝑥₁‖‖‖‖‖‖𝑥𝜕₁𝑉‖‖≤𝑐₁𝑉𝑡,𝑥₁∀‖‖𝑥₁‖‖≥𝛿^,(14)𝜕𝑥₁‖‖‖≤𝑐₂∀‖‖𝑥₁‖‖≤𝛿^.(15)

Assumption 2. There exist two continuous functions 𝜉₁, 𝜉₂∶ℝ_≥0→ℝ_≥0 such that 𝑔(𝑡,𝑥) satisfies ‖𝑔(𝑡,𝑥)‖≤𝜉₁‖‖𝑥₂‖‖+𝜉₂‖‖𝑥₂‖‖‖‖𝑥₁‖‖.(16)

Assumption 3. There exists a class 𝒦 function 𝛼(⋅) such that, for all 𝑡₀≥0, the trajectories of the system (12) satisfy ^∞𝑡₀‖‖𝑥₂𝑡;𝑡₀,𝑥₂𝑡₀‖‖𝑑𝑡≤𝛼‖‖𝑥₂𝑡₀‖‖.(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 𝝉^{𝑙𝑏𝑙𝑑} and 𝝉^{𝑓𝑏𝑓𝑑} 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 𝐪_𝑙(𝑡), 𝝎^{𝑙𝑏𝑖,𝑙𝑏}(𝑡), and 𝝎̇^{𝑙𝑏𝑖,𝑙𝑏}(𝑡) converge towards the generated references specified as 𝐪_𝑑(𝑡), 𝝎^{𝑙𝑏𝑖,𝑑}(𝑡), and 𝝎̇^{𝑙𝑏𝑖,𝑑}(𝑡), satisfying the kinematic equation𝐪̇_𝑑=𝐓𝐪_𝑑𝝎^{𝑙𝑏𝑖,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 𝐪_𝑓(𝑡) and 𝝎^{𝑓𝑏𝑖,𝑓𝑏}(𝑡) converge towards 𝐪_𝑙(𝑡) and 𝝎^{𝑓𝑏𝑖,𝑙𝑏}(𝑡). The error quaternion 𝐪̃_𝑠=[𝜂̃_𝑠,𝝐̃^⊤𝑠]^⊤ is found by applying the quaternion product𝐪̃_𝑠=𝐪_𝑠⊗𝐪_𝑑=𝜂^𝑠𝜂_𝑑+𝝐^⊤𝑠𝝐_𝑑𝜂_𝑑𝝐_𝑠−𝜂_𝑠𝝐_𝑑−𝐒𝝐_𝑠𝝐_𝑑,(19) where the sub-/superscript 𝑠=𝑙,𝑓 denotes the leader and follower spacecraft, respectively, and the error kinematics can be expressed as [39]𝐪̇̃_𝑠=12𝐓𝐪̃^𝑠𝐞_𝑠𝜔,(20) where 𝐞_𝑠𝜔=𝝎^{𝑠𝑏𝑖,𝑠𝑏}−𝝎^{𝑠𝑏𝑖,𝑑} is the angular velocity error. Due to the redundancy in the quaternion representation, 𝐪̃_𝑠 and −𝐪̃_𝑠 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 ℝ^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, 𝐪̃_𝑠+=[1,𝟎^⊤]^⊤ is chosen if 𝜂̃_𝑠(𝑡₀)≥0 and 𝐪̃_𝑠−=[−1,𝟎^⊤]^⊤ is chosen if 𝜂̃_𝑠(𝑡₀)<0. For the positive equilibrium point, an attitude error function is chosen as 𝐞_𝑠𝑞+=[1−𝜂̃_𝑠,𝝐̃^⊤𝑠]^⊤ (see [9]), while, for the negative equilibrium point, the attitude error function is chosen as 𝐞_𝑠𝑞−=[1+𝜂̃_𝑠,𝝐̃^⊤𝑠]^⊤. In accordance with general kinematic relations𝐞̇_𝑠𝑞±=𝐓_𝑠𝑒𝐞_𝑠𝑞±𝐞_𝑠𝜔,(21) where𝐓_𝑠𝑒𝐞_𝑠𝑞±=12𝜂̃_𝑠𝐈+±𝝐𝐒̃^⊤𝑠𝝐̃_𝑠.(22) The attitude error function is chosen a priori and kept throughout the maneuver even if 𝜂̃_𝑠(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=[𝐞^⊤𝑓𝑞,𝐞^⊤𝑓𝜔]^⊤ and 𝐱₂=[𝐞^⊤𝑙𝑞,𝐞^⊤𝑙𝜔]^⊤. The control objective is to stabilize the equilibrium point (𝐞_𝑓𝑞(𝑡),𝐞_𝑓𝜔(𝑡),𝐞_𝑙𝑞(𝑡),𝐞_𝑙𝜔(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 𝐪_𝑙, angular velocity 𝝎^{𝑙𝑏𝑖,𝑙𝑏}, and inertia matrix 𝐉_𝑙, and, temporarily, we assume to know external perturbations. We choose the equilibrium such that 𝐞_𝑙𝑞±=[1∓𝜂̃_𝑙,𝝐̃^⊤𝑙]^⊤ is either the positive or negative equilibrium point, which does not change during the maneuver. By pure conventionalism, consider the positive equilibrium, that is, 𝐞_𝑙𝑞=𝐞_𝑙𝑞+. We define desired attitude 𝐪_𝑑(𝑡), desired angular velocity 𝝎^{𝑙𝑏𝑖,𝑑}(𝑡), and desired angular acceleration 𝝎̇^{𝑙𝑏𝑖,𝑑}(𝑡) which are all bounded functions. The control law is expressed as𝝉^{𝑙𝑏𝑙𝑎}=𝐉_𝑙𝝎̇^{𝑙𝑏𝑖,𝑑}−𝐒𝐉_𝑙𝝎^{𝑙𝑏𝑖,𝑙𝑏}𝝎^{𝑙𝑏𝑖,𝑑}−𝑘_𝑙𝑞𝐓^⊤𝑙𝑒𝐞_𝑙𝑞−𝑘_𝑙𝜔𝐞_𝑙𝜔−𝝉^{𝑙𝑏𝑙𝑑},(23) where 𝑘_𝑙𝑞>0 and 𝑘_𝑙𝜔>0 are feedback gain scalars. By insertion of (23) into (9), the system is on form (12), and, by applying the property 𝐒(𝝎^{𝑙𝑏𝑖,𝑙𝑏})𝐉_𝑙𝝎^{𝑙𝑏𝑖,𝑙𝑏}=−𝐒(𝐉_𝑙𝝎^{𝑙𝑏𝑖,𝑙𝑏})𝝎^{𝑙𝑏𝑖,𝑙𝑏}, we obtain the closed-loop dynamics𝐉_𝑙𝐞̇_𝑙𝜔+𝑘_𝑙𝜔𝐈−𝐒𝐉_𝑙𝝎^{𝑙𝑏𝑖,𝑙𝑏}𝐞^𝑙𝜔+𝑘_𝑙𝑞𝐓^⊤𝑙𝑒𝐞_𝑙𝑞=𝟎.(24) A radial unbounded, positive definite Lyapunov function candidate is defined as𝑉_𝑙=12𝐞^⊤𝑙𝜔𝐉_𝑙𝐞_𝑙𝜔+12𝐞^⊤𝑙𝑞𝑘_𝑙𝑞𝐞_𝑙𝑞>0∀𝐞_𝑙𝜔≠𝟎,𝐞_𝑙𝑞≠𝟎.(25) Indeed, we have 12min𝑗^𝑙𝑚,𝑘_𝑙𝑞‖𝐱₂‖²≤𝑉_𝑙≤12max𝑗^𝑙𝑀,𝑘_𝑙𝑞‖‖𝐱₂‖‖²,(26) where 𝑗_𝑙𝑚≤‖𝐽_𝑙‖≤𝑗_𝑙𝑀. By differentiation of (25) and inserting (24) and (21), we obtain𝑉_𝑙̇=𝐞^⊤_l𝑞𝑘_𝑙𝑞𝐓_𝑙𝑒𝐞_𝑙𝜔+𝐞^⊤𝑙𝜔𝐒𝐉^𝑙𝝎^{𝑙𝑏𝑖,𝑙𝑏}−𝑘^𝑙𝜔𝐈𝐞_𝑙𝜔−𝐞^⊤𝑙𝜔𝑘_𝑙𝑞𝐓^⊤𝑙𝑒𝐞_𝑙𝑞,(27) where the first part of the second term in (27) is zero because 𝐒(𝐉_𝑙𝝎^{𝑙𝑏𝑖,𝑙𝑏}) is a skew-symmetric matrix, which leads to𝑉_𝑙̇=−𝐞^⊤𝑙𝜔𝑘_𝑙𝜔𝐞_𝑙𝜔≤0(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 𝒱_𝑙(𝑡)=𝑉_𝑙(𝐞_𝑙𝑞(𝑡),𝐞_𝑙𝜔(𝑡)) and 𝐱₂(𝑡)=[𝐞_𝑙𝑞(𝑡)^⊤,𝐞_𝑙𝜔(𝑡)^⊤]^⊤. Then, from (28), we obtain by integrating on both sides^𝑡𝑡₀−𝒱𝒱_𝑙̇_𝑙(𝑠)𝑑𝑠=−^𝑡𝑡₀𝐞^⊤𝑙𝜔(𝑠)𝑘_𝑙𝜔𝐞_𝑙𝜔(𝑠)𝑑𝑠,(29)(𝑡)+𝒱_𝑙𝑡₀=𝑘_𝑙𝜔^𝑡𝑡₀‖‖𝐞_𝑙𝜔(𝑠)‖‖²𝑑𝑠.(30) Since −𝒱_𝑙(𝑡)≤0, we can write𝑘_𝑙𝜔^𝑡𝑡₀‖‖𝐞_𝑙𝜔‖‖²𝑑𝑠≤=𝒱1_𝑙𝑡₀2𝐞^⊤𝑙𝜔𝑡₀𝐉_𝑙≤1𝐞_𝑙𝜔𝑡₀+𝐞^⊤𝑙𝑞𝑡₀𝑘_𝑙𝑞𝐞_𝑙𝑞𝑡₀2max𝑗^𝑙𝑀,𝑘_𝑙𝑞‖‖𝐱₂𝑡₀‖‖²,(31) and thus^𝑡𝑡₀‖‖𝐞_𝑙𝜔(𝑠)‖‖²𝑑𝑠≤𝑐₃‖‖𝐱₂𝑡₀‖‖²,(32) where 𝑐₃=max{𝑗_𝑙𝑀,𝑘_𝑙𝑞}/2𝑘_𝑙𝜔.

Now let 𝒲_𝑙(𝑡)=𝑊_𝑙(𝐞_𝑙𝑞(𝑡),𝐞_𝑙𝜔(𝑡)) such that𝒲_𝑙(𝑡)=𝐞_𝑙𝑞(𝑡)^⊤𝐓_𝑙𝑒(𝑡)𝑘_𝑙𝑞𝐉_𝑙𝐞_𝑙𝜔(𝑡),(33) and, by differentiation, we obtain𝒲̇_𝑙(𝑡)=+𝐞̇^⊤𝑙𝑞𝐞(_⊤𝑙𝑞𝑡)𝐓_𝑙𝑒(𝑡)𝑘_𝑙𝑞𝐉_𝑙𝐞_𝑙𝜔(𝑡)(𝑡)𝐓̇_𝑙𝑒(𝑡)𝑘_𝑙𝑞𝐉_𝑙𝐞_𝑙𝜔(𝑡)+𝐞^⊤𝑙𝑞(𝑡)𝐓_𝑙𝑒(𝑡)𝑘_𝑙𝑞𝐉_𝑙𝐞̇_𝑙𝜔(𝑡).(34) By inserting 𝐓̇^⊤𝑙𝑒(𝑡)𝐞_𝑙𝑞(𝑡)=𝐆_𝑙(𝑡)𝐞_𝑙𝜔(𝑡), where𝐆_𝑙(𝑡)=12𝜂̃^𝑙(𝑡)𝐈+𝐒𝝐̃_𝑙(𝑡)−14𝐈,(35) and (21) and (24) into (34), we obtain𝒲̇_𝑙=−𝐞^⊤𝑙𝜔𝐞(_⊤𝑙𝑞𝑡)𝐓^⊤𝑙𝑒(𝑡)𝐓_𝑙𝑒(𝑡)𝑘_𝑙𝑞𝐉_𝑙𝐞_𝑙𝜔(𝑡)+𝐞^⊤𝑙𝜔𝐆^⊤𝑙(𝑡)𝑘_𝑙𝑞𝐉_𝑙𝐞_𝑙𝜔(𝑡)−𝐞_⊤𝑙𝑞(𝑡)𝐓_𝑙𝑒(𝑡)𝑘_𝑙𝑞−𝐒𝐉^𝑙𝝎^{𝑙𝑏𝑖,𝑙𝑏}(𝑡)+𝑘_𝑙𝜔𝐈𝐞_𝑙𝜔(𝑡)≤−𝑐₄(𝑡)𝐓_𝑙𝑒(𝑡)𝑘^2𝑙𝑞𝐓^⊤𝑙𝑒(𝑡)𝐞_𝑙𝑞(𝑡)‖‖𝐞_𝑙𝑞‖‖²+𝑐₅‖‖𝐞_𝑙𝜔‖‖²+‖‖𝐞_𝑙𝑞‖‖𝑐₆‖‖𝐞_𝑙𝜔‖‖,(36) where 𝑐₄=𝑘^2𝑙𝑞/4, 𝑐₅=𝑘_𝑙𝑞𝑗_𝑙𝑀‖𝐆^⊤𝑙(𝑡)‖, 𝑐₆=𝑘^𝑙𝑞2‖‖𝐒𝐉_𝑙𝝎^{𝑙𝑏𝑖,𝑙𝑏}(𝑡)‖‖+𝑘_𝑙𝜔,(37) and ‖𝐆^⊤𝑙(𝑡)‖≤3/4. The last term of (36) can be rewritten as‖‖𝐞_𝑙𝑞‖‖𝑐₆‖‖𝐞_𝑙𝜔‖‖≤𝜅‖‖𝐞_𝑙𝜔(𝑡)‖‖²+𝑐²⁶𝜅‖‖𝐞^𝑙𝑞(𝑡)‖‖²,(38) and by choosing 𝜅≫1 such that 𝑐₄>2𝑐²⁶/𝜅, we obtain𝒲̇_𝑙(𝑡)≤−𝑐⁴2‖‖𝐞_𝑙𝑞(𝑡)‖‖²+𝑐₅+𝜅‖‖𝐞_𝑙𝜔(𝑡)‖‖².(39) By applying the same line of arguments as in (29)–(31) and inequality (32), (39) may be expressed as𝒲_𝑙𝑡₀+𝑐₅+𝜅𝑐₃‖‖𝐱₂𝑡₀‖‖²≥𝑐⁴2^𝑡𝑡₀‖‖𝐞_𝑙𝑞(𝑠)‖‖²𝑑𝑠.(40) By inserting the upper bound‖𝒲_𝑙𝑡₀‖≤12max𝑘^𝑙𝑞≤max𝑘_𝑙𝑞,𝑗_𝑙𝑀‖‖𝐞_𝑙𝑞𝑡₀‖‖‖‖𝐞_𝑙𝜔𝑡₀‖‖≤max𝑘_𝑙𝑞,𝑗_𝑙𝑀‖‖𝐞_𝑙𝑞𝑡₀‖‖²+‖‖𝐞_𝑙𝜔𝑡₀‖‖²,𝑗_𝑙𝑀‖‖𝐱₂𝑡₀‖‖²(41) into (40), the expression may be written as^𝑡𝑡₀‖‖𝐞_𝑙𝑞(𝑠)‖‖²𝑑𝑠≤𝑐₇‖‖𝐱₂𝑡₀‖‖²,(42) where 𝑐₇=2(1/2max{𝑘_𝑙𝑞,𝑗_𝑙𝑀}+(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, 𝐉_𝑓, and that all perturbations are known and accounted for. In the following, it is assumed that the equilibrium point is chosen such that 𝐞_𝑓𝑞±=[1∓𝜂̃_𝑓,𝝐̃^⊤𝑓]^⊤ is either the positive or negative equilibrium point and does not change during the maneuver. The control law is expressed as𝝉^{𝑓𝑏𝑓𝑎}=𝐉_𝑓𝝎̇^{𝑓𝑏𝑖,𝑑}−𝐒𝐉_𝑓𝝎^{𝑓𝑏𝑖,𝑓𝑏}𝝎^{𝑓𝑏𝑖,𝑑}−𝑘_𝑓𝑞𝐓^⊤𝑓𝑒𝐞_𝑓𝑞−𝐓^⊤𝑙𝑒𝐞_𝑙𝑞−𝑘^𝑓𝜔𝐞_𝑓𝜔−𝐞_𝑙𝜔−𝝉^{𝑓𝑏𝑓𝑑},(43) where 𝑘_𝑓𝑞>0 and 𝑘_𝑓𝜔>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 dynamics𝐞̇_𝑓𝜔=𝐉^𝑓−1−𝑘^𝑓𝜔−𝐒𝐉_𝑓𝝎^{𝑓𝑏𝑖,𝑓𝑏}𝐞^𝑓𝜔=𝑓𝐱−₁𝑘_𝑓𝑞𝐓^⊤𝑓𝑒𝐞_𝑓𝑞+𝑘_𝑓𝑞𝐓^⊤𝑙𝑒𝐞_𝑙𝑞+𝑘_𝑓𝜔𝐞_𝑙𝜔+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 is𝑔(𝐱)=𝐉^𝑓−1𝑘𝟎_𝑓𝑞𝐓^⊤𝑙𝑒𝐉_𝑓−1𝟎𝑘_𝑓𝜔.(45) In what follows we show that Assumptions 1–3 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 𝑉_𝑓, we obtain ‖‖𝐞^⊤𝑓𝜔𝐉_𝑓+𝐞^⊤𝑓𝑞𝑘_𝑓𝑞‖‖‖‖𝐱₁‖‖≤=𝑐𝑐₁₁𝑉_𝑓𝐱₁2𝐞^⊤𝑓𝜔𝐉_𝑓𝐞_𝑓𝜔+𝐞^⊤𝑓𝑞𝑘_𝑓𝑞𝐞_𝑓𝑞,(46) and by applying the triangular inequality on the left side of (46) and squaring, we obtain ‖‖𝐞^⊤𝑓𝜔𝐉_𝑓‖‖+‖‖𝐞^⊤𝑓𝑞𝑘_𝑓𝑞‖‖²‖‖𝐱₁‖‖²≤𝑐²¹4𝐞^⊤𝑓𝜔𝐉_𝑓𝐞_𝑓𝜔+𝐞^⊤𝑓𝑞𝑘_𝑓𝑞𝐞_𝑓𝑞².(47) On the left side of (47), we apply ‖𝑥‖²+‖𝑦‖²+2‖𝑥‖‖𝑦‖≤2(‖𝑥‖²+‖𝑦‖²) and 𝑗_𝑓𝑚≤‖𝐉_𝑓‖≤𝑗_𝑓𝑀 such that ‖‖𝐞^⊤𝑓𝜔𝐉_𝑓‖‖+‖‖𝐞^⊤𝑓𝑞𝑘_𝑓𝑞‖‖²≤2𝜑𝐞^⊤𝑓𝜔𝐉_𝑓𝐞_𝑓𝜔+𝐞^⊤𝑓𝑞𝑘_𝑓𝑞𝐞_𝑓𝑞,(48) where 𝜑=max{𝑗_𝑓𝑀,𝑘_𝑓𝑞}. By insertion of (48) in (47), we obtain 2𝜑𝐞^⊤𝑓𝜔𝐞_𝑓𝜔+𝐞^⊤𝑓𝑞𝐞_𝑓𝑞≤𝑐²¹𝜌4𝐞^⊤𝑓𝜔𝐞_𝑓𝜔+𝐞^⊤𝑓𝑞𝐞_𝑓𝑞,(49) where 𝜌=min{𝑗_𝑓𝑚,𝑘_𝑓𝑞}, and thus, we have to choose 𝑐₁≥8𝜑𝜌(50) to fulfill (14). For (15), we have that ‖‖𝐞^⊤𝑓𝜔𝐉_𝑓+𝐞^⊤𝑓𝑞𝑘_𝑓𝑞‖‖≤𝑐₂,(51) and by using the triangular inequality and squaring and applying (48) we obtain 2𝜑²‖‖𝐱₁‖‖²≤𝑐²².(52) Since ‖𝐱₁‖≤1, we have to choose 𝑐₂≥√2𝜑(53) to fulfill (15), and thus Assumption 1 is fulfilled.

Proof of Assumption 2. Since ‖𝐓_𝑙𝑒‖=1/2, (45) obviously fulfills the growth rate criteria of (16), such as ‖𝑔(𝐱)‖≤12𝐽1^2𝑥+1𝐽^2𝑦+1𝐽^2𝑧𝑘^2𝑓𝑞+4𝑘^2𝑓𝜔^1/2=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 (𝐞_𝑙𝑞,𝐞_𝑙𝜔,𝐞_𝑓𝑞,𝐞_𝑓𝜔)=(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=[𝑘_𝑓𝑞,𝑘_𝑓𝜔], 𝜃₂=[𝑘_𝑙𝑞,𝑘_𝑙𝜔]∈Θ₁=Θ₂=ℝ^2>0. The uniform asymptotic stability in Section 3.2 is a result of the assumption that 𝝉^{𝑙𝑏𝑙𝑑} is known and accounted for in the control law. We relax this assumption and assume that 𝝉^{𝑙𝑏𝑙𝑑} is unknown, but bounded, and thus there exists a 𝛽_𝑙>0 such that ‖𝝉^{𝑙𝑏𝑙𝑑}‖≤𝛽_𝑙. Note that many of the disturbances for Earth-orbiting spacecraft can be reasonably well modeled as 𝐟̂^sb_sd, which can be added to the overall analysis such that 𝐟̃^sb_sd=𝐟^𝑠𝑏sd−𝐟̂^𝑠𝑏sd. This strategy can reduce the upper bound such that ‖𝐟𝑠^𝑏sd̃(𝑡)‖<𝛽𝑠̃<𝛽_𝑠, based on the quality of the disturbance modeling. We apply the control law𝝉^{𝑙𝑏𝑙𝑎}=𝐉_𝑙𝝎̇𝝎^{𝑙𝑏𝑖,𝑟}−𝐒𝐉_𝑙𝝎^{𝑙𝑏𝑖,𝑙𝑏}𝝎^{𝑙𝑏𝑖,𝑟}−𝑘_𝑙𝑞𝐓^⊤𝑙𝑒𝐞_𝑙𝑞−𝑘_𝑙𝜔𝐬_𝑙,(55)_{𝑙𝑏𝑖,𝑟}𝐬=𝝎^{𝑙𝑏𝑖,𝑑}−𝚪_𝑙𝐓^⊤𝑙𝑒𝐞_𝑙𝑞𝐞_𝑙𝑞,(56)_𝑙=𝝎^{𝑙𝑏𝑖,𝑙𝑏}−𝝎^{𝑙𝑏𝑖,𝑟}=𝐞_𝑙𝜔+𝚪_𝑙𝐓^⊤𝑙𝑒𝐞_𝑞𝑙,(57) where 𝑘_𝑙𝑞>0, 𝑘_𝑙𝜔>0 and Γ_𝑙=Γ^⊤𝑙>0 are feedback gains, and, by inserting (55) into (9), we obtain the closed-loop dynamics𝐬_𝑙̇=𝐉^𝑙−1𝐒𝐉^𝑙𝝎^{𝑙𝑏𝑖,𝑙𝑏}𝐬^𝑙−𝑘_𝑙𝑞𝐓^⊤𝑙𝑒𝐞_𝑙𝑞−𝑘_𝑙𝜔𝐬_𝑙.(58) A radial unbounded, positive definite Lyapunov function candidate is defined as𝑉_𝑙=12𝐬^⊤𝑙𝐉_𝑙𝐬_𝑙+𝐞^⊤𝑙𝑞𝑘_𝑙𝑞𝐞_𝑙𝑞>0∀𝐬^𝑙≠𝟎,𝐞_𝑙𝑞≠𝟎,(59) and, by differentiation of (59) and insertion of (58), we obtain𝑉̇_𝑙=−𝐬^⊤𝑙𝑘_𝑙𝜔𝐬_𝑙−𝐞^⊤𝑙𝑞𝐓_𝑙𝑒𝚪𝑘_𝑙𝑞𝐓^⊤𝑙𝑒𝐞_𝑙𝑞+𝐬^⊤𝑙𝝉^{𝑙𝑏𝑙𝑑}=−𝝌^⊤2𝐐_𝑙𝝌₂+𝐬^⊤𝑙𝝉^{𝑙𝑏𝑙𝑑}≤−𝑞_𝑙𝑚‖‖𝝌₂‖‖²+𝛽_𝑙‖‖𝝌₂‖‖,(60) where 𝝌₂=[𝐞^⊤𝑙𝑞,𝐬^⊤𝑙]^⊤, 𝐐_𝑙=diag{𝐓_𝑙𝑒𝑘_𝑙𝑞Γ_𝑙𝐓^⊤𝑙𝑒,𝑘_𝑙𝜔𝐈}, and 𝑞_𝑙𝑚>0 is the smallest eigenvalue of 𝐐_𝑙. Accordingly, 𝑉̇_𝑙<0 when ‖𝝌₂‖>𝛽_𝑙/𝑞_𝑙𝑚=𝛿₂, and, as 𝛽_𝑙 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 ‖𝝉^{𝑓𝑏𝑓𝑑}‖≤𝛽_𝑓, and we apply a synchronizing controller based on the Slotine and Li structure such as𝝉^{𝑙𝑏𝑓𝑎}=−𝐉_𝑓𝑘𝝎̇^{𝑓𝑏𝑖,𝑟}_𝑓𝜔−𝐒𝐉_𝑓𝝎^{𝑓𝑏𝑖,𝑓𝑏}𝝎^{𝑓𝑏𝑖,𝑟}−𝑘_𝑓𝑞𝐓^⊤𝑒𝑓𝐞_𝑓𝑞−𝐓^⊤𝑙𝑒𝐞_𝑙𝑞𝐬_𝑓−𝐬_𝑙,(61) where 𝝎^{𝑓𝑏𝑖,𝑟} and 𝐬_𝑓 are defined similar to (56)-(57). By inserting (61) into (9), we obtain𝐬_𝑓̇=𝑓𝝌₁+𝑔̃(𝝌)𝝌₂,(62) where 𝝌₁=[𝐞^⊤𝑓𝑞,𝐬^⊤𝑓]^⊤ and 𝝌=[𝝌^⊤1,𝝌^⊤2]^⊤, and 𝑓(𝝌₁) and 𝑔̃(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 𝑔̃(𝝌)𝝌₂ is uniformly bounded both in time and in 𝜃₂ and vanishes with 𝝌₂; thus, for any 𝜃₁∈Θ₁, we choose 𝐺_𝜃1(‖𝝌‖)=12𝐽1^2𝑥+1𝐽^2𝑦+1𝐽^2𝑧𝑘^2𝑓𝑞Ψ_𝜃1+4𝑘^2𝑓𝜔^1/2,‖‖𝝌₂‖‖=‖𝝌₂‖,(63) thus, 𝐺_𝜃1(‖𝝌‖) is constant and Ψ_𝜃1(‖𝝌₂‖) is of class 𝒦_∞, and the assumption is fulfilled for all 𝜃₂∈Θ₂ and all 𝝌∈𝒮³×ℝ³×𝒮³×ℝ³.

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 𝑉̇_𝑓<0 when ‖𝝌₁‖>𝛽_𝑓/𝑞_𝑓𝑚=𝛿₁, where 𝑞_𝑓𝑚>0 is the smallest eigenvalue of 𝐐_𝑓=diag{𝐓_𝑓𝑒𝑘_𝑓𝑞Γ_𝑓𝐓^⊤𝑓𝑒,𝑘_𝑓𝜔𝐈}, which is defined similar to 𝐐_𝑙 in (60). An increase of 𝛽_𝑓 can as well be counteracted by increasing the controller gains; thus, given any Δ₁>𝛿₁>0, there exists a parameter 𝜃^⋆1(𝛿₁,Δ₁)∈Θ₁. We choose 𝛼_𝛿1,Δ1(‖𝝌₁‖)=1/2min{𝑗_𝑓𝑚,𝑘_𝑓𝑞}‖𝝌₁‖² and 𝛼_𝛿1,Δ1(‖𝝌₁‖)=1/2max{𝑗_𝑓𝑀,𝑘_𝑓𝑞}‖𝝌₁‖², and thus the first part of the assumption is fulfilled, and the second part is fulfilled for 𝛼_𝛿1,Δ1(‖𝝌₁‖)=min{𝑘_𝑓𝑞/4,𝑘_𝑓𝜔}‖𝝌₁‖². The last inequality also holds similar to (51)–(53) with 𝑐_𝛿1,Δ1(‖𝝌₁‖)=√2𝜎‖𝝌₁‖^1/2, and thus Assumption 3.5 holds for all 𝝌₁∈ℋ(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 𝑉_𝑙𝑓(𝑡,𝑥)=12𝐬^⊤𝑙𝐉_𝑙𝐬_𝑙+𝐞^⊤𝑙𝑞𝑘_𝑙𝑞𝐞_𝑙𝑞+𝐬^⊤𝑓𝐉_𝑓𝐬_𝑓+𝐞^⊤𝑓𝑞𝑘_𝑓𝑞𝐞_𝑓𝑞,(64) which is lower and upper bounded by 𝛼_𝑙𝑓=12min𝑗^𝑙𝑚,𝑗_𝑓𝑚,𝑘_𝑙𝑞,𝑘_𝑓𝑞‖𝝌‖²,(65)𝛼_𝑙𝑓=12max𝑗^𝑙𝑀,𝑗_𝑓𝑀,𝑘_𝑙𝑞,𝑘_𝑓𝑞‖𝝌‖²,(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 𝛼_𝑙𝑓(𝛿₁)<𝛼_𝑙𝑓(Δ₁). As mentioned in Section 4.1, given any 𝛽_𝑙, we can achieve any 𝛿₂ by increasing the gains 𝜃₂, and thus there exits a parameter 𝜃^⋆2∈𝒟_𝑓2(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 𝛾Δ₁=⎷min𝑗_𝑙𝑚,𝑗_𝑓𝑚,𝑘_𝑙𝑞,𝑘_𝑓𝑞Δ²¹max𝑗_𝑙𝑀,𝑗_𝑓𝑀,𝑘_𝑙𝑞,𝑘_𝑓𝑞.(67) We have that 𝛼^𝛿−1_1,Δ1∘𝛼_𝛿1,Δ1𝛿₁=⎷max𝑗_𝑓𝑀,𝑘_𝑓𝑞𝛿²¹min𝑗_𝑓𝑚,𝑘_𝑓𝑞,(68) then, for any Δ_⋆>𝛿_⋆>0, there exist parameters 𝛿₁,𝛿₂,Δ₁, and Δ₂ such that min⎧⎪⎨⎪⎩Δ₁,Δ₂,⎷min𝑗_𝑙𝑚,𝑗_𝑓𝑚,𝑘_𝑙𝑞,𝑘_𝑓𝑞Δ²¹max𝑗_𝑙𝑀,𝑗_𝑓𝑀,𝑘_𝑙𝑞,𝑘_𝑓𝑞⎫⎪⎬⎪⎭≥Δ^⋆(69) since Δ₁ and Δ₂ can be chosen arbitrarily large and the constants 𝑗_𝑠𝑚, 𝑗_𝑠𝑀, 𝑘_𝑠𝑞, 𝑘_𝑠𝜔>0, and max⎧⎪⎨⎪⎩𝛿₂,⎷max𝑗_𝑓𝑀,𝑘_𝑓𝑞𝛿²¹min𝑗_𝑓𝑚,𝑘_𝑓𝑞⎫⎪⎬⎪⎭≤𝛿^⋆(70) is fulfilled since, by decreasing 𝛿₁,𝑘_𝑓𝑞 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 𝐞_𝑙𝑞=𝐞_𝑙𝑞− or 𝐞_𝑓𝑞=𝐞_𝑓𝑞−, the other three proofs are performed in a similar way, and we thus conclude that the set of equilibrium points (𝐞_𝑙𝑞±,𝐞_𝑙𝜔,𝐞_𝑓𝑞±,𝐞_𝑓𝜔)=(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 10⁻⁹. The moments of inertia were chosen as 𝐉_𝑙=𝐉_𝑓=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 𝐪_𝑙(𝑡₀)=[−0.3772,−0.4329,0.6645,0.4783]^⊤, 𝐪_𝑓(𝑡₀)=1/√4[1,1,1,1]^⊤, 𝝎_𝑙(𝑡₀)=[0.1−0.30.2]^⊤, and 𝝎_𝑓(𝑡₀)=[0.2−0.30.1]^⊤, controller gains as 𝑘_𝑠𝑞=1,𝑘_𝑠𝜔=2, and Γ_𝑠=𝐈 for (23), (43), (55), and (61), and desired trajectory as 𝝎̇_𝑑=0.1[−10𝑐²⁰cos(8𝑐₀𝑡),48𝑐²⁰sin(16𝑐₀𝑡),−8𝑐²⁰cos(4𝑐₀𝑡)], 𝝎_𝑑 its integrate, and 𝐪_𝑑(𝑡₀)=[1,𝟎]^⊤, where 𝑐₀=𝜋/𝑡_𝑜 and 𝑡_𝑜 denotes the orbital period. For relative attitude (synchronization error) and angular velocity between the leader and follower spacecraft, we define 𝐪̃_𝑠𝑦=𝐪_𝑓⊗𝐪_𝑙 and 𝐞_𝑠𝑦=𝐞_𝑓𝜔−𝐞_𝑙𝜔, 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.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

Attitude and angular velocity error and control torque for leader ((a), (c), (e)) and synchronizing follower ((b), (d), (f)) spacecraft during maneuver without measurement noise and disturbances.

In the second set of simulation results, we introduce measurement noise as 𝜎𝔹^𝑛={𝑥∈ℝ^𝑛∶‖𝑥‖≤𝜎} which was added according to 𝐞_𝑞̃=(𝐞_𝑞+0.05𝔹⁴)/‖𝐞_𝑞+0.05𝔹⁴‖ and 𝐞_𝜔̃=𝐞_𝜔+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.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3 

Attitude and angular velocity error and control torque for leader ((a, (c), (e))) and synchronizing follower ((b, (d), (f))) spacecraft during maneuver including measurement noise and disturbances.

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.