Extended state observer–based finite-time controller design for coupled spacecraft formation with actuator saturation

Abstract

This article investigates the consensus tracking control problem of the leader–follower spacecraft formation system in the presence of model uncertainties, external disturbances, and actuator saturation, where the relative motion of the leader and the follower need to track a desired time-varying trajectory given in advance. First, the six-degree-of-freedom relative-coupled translational and rotational dynamics models are derived using the exponential coordinates on the Lie group SE(3). Then, a fast terminal sliding mode control law is proposed to guarantee the tracking control objective come true in finite time robust against all the aforementioned drawbacks. As a stepping stone, an extended state observer is designed to estimate and compensate the total composed disturbances of the system, and it is proved that the estimate errors can converge to a really small neighborhood of the origin in finite time. Based on the observer information, a less-conservative modified controller is furthermore developed to eliminate the chattering caused by the signum function. The stability of the closed-loop system is shown by theoretical analysis. Finally, the validity of the proposed schemes is illustrated through numerical simulations.

Introduction

In recent years, spacecraft formation flying (SFF) as a key technology for realizing many important space missions, such as spacecraft rendezvous and docking, monitoring of the Earth and its surrounding atmosphere, deep space imaging and exploration, space-based interferometry, stereo-imaging, and terrain mapping, has received wide attention.^1–4 Since the functionality of a large complex spacecraft can be achieved by the SFF system which consists of multiple simpler distributed ones, the system robustness, control accuracy, flexibility in configuration, and mission cost have all been improved.^5–7 In Scharf et al., ⁸ a survey on the field of SFF was given by dividing the previous synchronization control schemes into five architectures with leader–follower (LF) included. As one of the basic schemes, the developed control schemes on LF have been applied in many aerospace missions. Traditionally, the translational and rotational motion for spacecraft are controlled independently without considering their coupling, which leads to poor control results. In order to meet the requirements of aerospace missions with high control accuracy, taking spacecraft rendezvous and docking for example, the follower is required to finish large angular maneuver and complicated translational maneuver with respect to the leader simultaneously, while high control accuracy needs to be satisfied, a six-degree-of-freedom (6-DOF) model taking the relative translational motion, rotational motion, and their coupling into consideration is required.

The 6-DOF relative-coupled dynamics for SFF system described by the exponential coordinates on the Lie group SE(3), which is the set of positions and orientations of the spacecraft moving in three-dimensional (3D) Euclidean space, as a useful mathematical tool has received increasing attention as in Lee and colleagues^7,9 and Nazari et al. ¹⁰ However, structured and unstructured uncertainties haven’t been considered in these papers, which can’t be ignored due to external environment disturbances or fuel consuming or components/payloads releasing. That is to say, the controllers designed in these papers have poor robustness. Meanwhile, the controllers designed in Lee and colleagues^7,9 could only guarantee asymptotical convergence of the system states, which means that the control objectives will realize when time tends to infinity, and the consensus problem was solved by a local stability controller in Nazari et al. ¹⁰

Terminal sliding mode control (TSMC) proposed in Yu et al. ¹¹ for rigid robotic manipulators could provide finite-time stability in both the reaching phase and the sliding phase, which has been widely studied and explored in the spacecraft control field due to its ability to deal with the external disturbances and model uncertainties, such as finite-time attitude tracking control problem for rigid spacecraft solved in Jin and Sun, ¹² finite-time SFF orbit reconfiguration control in Hui and Li, ¹³ and a terminal sliding mode (TSM)-based observer developed in Xiao et al. ¹⁴ to reconstruct all the states of the attitude system in the presence of actuator failures in finite time. However, TSM has two disadvantages of slower convergence than the sliding mode control (SMC) when the system states far away from the equilibrium and the singularity problem. In Liang et al., ¹⁵ the above two problems were solved properly; a fast terminal sliding mode (FTSM) without singularity was proposed to guarantee robust decentralized attitude control for spacecraft formation. But, the designed controller had a chattering issue since a positive definite constant matrix with a signum function was used to deal with the total system disturbances.

When taking actual system into consideration, due to physical limitations on actuator, controller output saturation can’t be ignored since it can reduce the performance of the closed-loop system. Dealing with this phenomenon, the nonlinearity of the system model will be more complicated leading the chattering issue worse. Recently, combining disturbance observer (DO) with TSMC has shown good control performance and could robust against complicated disturbances, while the chattering problem has been solved well in Wei and Guo ¹⁶ and Chen et al. ¹⁷ Among the DO methods, extended state observer (ESO) is an effective way; by considering the system composed disturbances as an extended state of the system, all the states including the extended one can be estimated by the ESO in finite time with high accuracy. ¹⁸ In Zhai and Xia, ¹⁹ the control objective was achieved in finite time by introducing a switching-technique-based error observer into the design framework, and further an adaptive finite-time controller with an anti-windup compensator and the error observer was developed to achieve finite-time tracking in Zhai and Xia. ²⁰ High-order sliding mode (HOSM) technique moves the discontinuous action to the higher derivatives of the system, and compared with other chattering elimination methods, it can not only eliminate the chattering but also increase the accuracy such as observers with finite-time convergence and high accuracy developed in Van et al. ²¹ and Zou ²² to estimate system states. Meanwhile, an alternative solution to eliminate the couplings in the developed 6-DOF model is output-feedback control design, which has been widely applied such as Wei et al.^23,24

Inspired by these facts, this article attempts to design a control scheme on the LF system robust against external disturbances, model uncertainties, and actuator saturation, such that the follower states can converge to the predefined states with respect to the leader in finite time suitable for missions with high accuracy and quick maneuver tracking requirements. The main contributions of this article are as follows:

1. With consideration of model uncertainties, external disturbances, and actuator saturation, a 6-DOF relative-coupled translational and rotational dynamics models are derived using the exponential coordinates on the Lie group SE(3).

2. A robust controller with a fast terminal sliding manifold constructed by the exponential coordinates and velocity tracking errors is proposed to guarantee the finite-time stability of the closed-loop system.

3. An ESO motived by HOSM is designed to estimate the system composed disturbances, based on which a modified controller with less conservation is constructed to eliminate the chattering and enhance control accuracy.

The remainder of this article is organized as follows: in section “Spacecraft modeling and preliminaries,” the mathematical model of a single rigid spacecraft, the relative-coupled dynamics, and some preliminaries which will be used in controller design part are presented; then, the main purpose of this article is given in section “Problem statement.” In section “Controller design and stability analysis,” the main results of this article are given, in which the designed controllers, the ESO, and their stability theoretical analysis are finished. In section “Simulation results,” numerical simulations are presented to prove the effectiveness of the proposed control schemes, followed by conclusions in section “Conclusion.”

Spacecraft modeling and preliminaries

In this article, two Earth-orbiting spacecrafts known as the LF spacecraft formation is considered, both of which are regarded as a rigid body. The follower spacecraft is tasked to keep a specified consensus state consisting of synchronized attitude and predetermined position configuration with respect to the leader one, whose trajectory in a constrained orbital motion is computed offline. In this section, based on the kinematics and dynamics of a single rigid spacecraft, the relative-coupled dynamics of the leader and the follower based on Lie group SE(3) are derived. Meanwhile, some preliminaries are given to finish the theoretical analysis in this article.

Mathematical model of a single rigid spacecraft

Before developing the models, the following coordinate frames will be introduced first as shown in Figure 1. The Earth-centered inertial frame $\{I\}$ with its origin at the center of the Earth, x axis points to the vernal, z axis parallel to the rotation axis of the Earth, and y axis is given by the right-hand rule. The coordinates are denoted by $(x_I,y_I,z_I)$ ; the body-fixed frame $\{B\}$ with its origin on the mass center of the spacecraft and axes coincide with the principal axes of inertia, and the coordinates denoted by $(x_b,y_b,z_b)$ ; the orbital-reference frame $\{O\}$ , which is a right-handed orthogonal system with its origin located in the mass center of the spacecraft, x axis points to the opposite direction of the center of the Earth, y axis is in the flight direction, and z axis is found using the right-hand rule, and the coordinates denoted by $(x_o,y_o,z_o)$ . We use $f$ to represent the follower and $l$ to represent the leader.

Figure 1.

Definition of the coordinate reference frames.

Let $\mathit{C}\in \mathrm{SO}(3)$ be the rotation matrix from $\{B\}$ to $\{I\}$ , and $\mathit{R}\in {\mathbb{R}}^3$ be the spacecraft position expressed in $\{I\}$ , where SO(3) is the Lie group of rotation matrices with $\mathit{C}$ included, and ${\mathbb{R}}^3$ is the 3D real Euclidean space of positions of the spacecraft. Then, the kinematics of a rigid spacecraft is given as follows

$$\{\begin{array}{c}\stackrel{·}{\mathit{C}}=\mathit{CS}(\mathbf{\omega }\mathbf{)}\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\mathit{R}}=\mathit{Cv}}\end{array}$$

(1)

where $\mathit{v}$ and $\mathbf{\omega }$ are translational and angular velocity expressed in $\{B\}$ , and the vector cross product operator $\mathit{S}(·)$ is defined by

$$\mathit{S}(·)=\left[\begin{array}{ccc}0\hfill & -{\mathit{\omega }}_3\hfill & {\mathit{\omega }}_2\hfill \\ \multicolumn{1}{c}{{\mathit{\omega }}_3}& 0\hfill & -{\mathit{\omega }}_1\hfill \\ \multicolumn{1}{c}{-{\mathit{\omega }}_2}& {\mathit{\omega }}_1\hfill & 0\hfill \end{array}\right]$$

(2)

The SE(3) is a Lie group expressed by the semidirect product $\mathrm{SE}(3)={\mathbb{R}}^3⋉\mathrm{SO}(3)$ , and an element $\mathbf{\chi }$ of SE(3) with the form

$$\mathbf{\chi }=\left[\begin{array}{cc}\mathit{C}\hfill & \mathit{R}\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]\in \mathrm{SE}(3)$$

(3)

can be used to represent the orientation and position of a spacecraft in a compact form. ¹⁰ Using this representation, the kinematics equation can be expressed as

$$\stackrel{·}{\chi }=\mathbf{\chi }(\mathit{V}{)}^{\mathit{v}}$$

(4)

where $\mathit{V}=[\begin{array}{cc}{\mathbf{\omega }}^{\mathrm{T}}\hfill & {\mathit{v}}^{\mathrm{T}}\hfill \end{array}{]}^{\mathrm{T}}$ , and the vector space isomorphism ${\mathbb{R}}^6\to \varsigma e(3)$ is defined as

$${\mathit{V}}^{\mathit{v}}=\left[\begin{array}{cc}\mathit{S}(\mathbf{\omega }\mathbf{)}\hfill & \mathit{v}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]\in \varsigma \mathrm{e}(3)$$

(5)

$\varsigma e(3)$ is the Lie algebra (tangent space at the identity element) associated with Lie group SE(3).

Let the mass of the spacecraft be $m$ and its moment of inertia matrix be $\mathit{J}$ . Then, the dynamics of a rigid spacecraft in $\{B\}$ are given the same as ²⁵

$$\{\begin{array}{c}\mathit{J}\stackrel{·}{\mathit{\omega }}+\mathit{S}(\mathbf{\omega }\mathbf{)}\mathit{J}\mathbf{\omega }\mathbf{=}{\mathit{M}}_{\mathbf{g}}\mathbf{+}{\mathbf{\tau }}_{\mathbf{c}}\mathbf{+}{\mathbf{\tau }}_{\mathbf{d}}\hfill \\ m\stackrel{·}{\mathit{v}}+m\mathit{S}(\mathbf{\omega }\mathbf{)}\mathit{v}\mathbf{=}{\mathit{f}}_{\mathbf{g}}\mathbf{+}{\mathit{u}}_{\mathbf{c}}\mathbf{+}{\mathit{u}}_{\mathbf{d}}\hfill \end{array}$$

(6)

where ${\mathit{u}}_c$ and ${\mathbf{\tau }}_c$ are the control force and torque, ${\mathit{u}}_d$ and ${\mathbf{\tau }}_{\mathbf{d}}$ are the external disturbance force and torque, and ${\mathit{f}}_g$ and ${\mathit{M}}_g$ are gravity force and torque on the spacecraft, respectively.

Considering the Earth’s oblateness and the coupled relationship between translational and rotational motion, ${\mathit{f}}_g$ and ${\mathit{M}}_g$ expressed in $\{B\}$ are

$${\mathit{f}}_g=\mathit{G}+{\mathit{f}}_{\mathrm{J}2}$$

(7)

$$\mathit{G}=-\left(\frac{m\mu }{{\Vert \mathit{R}\Vert }^3}\right){\mathit{R}}_b-3\left(\frac{\mu }{{\Vert \mathit{R}\Vert }^5}\right){\mathit{J}}_1{\mathit{R}}_b+\frac{15}{2}\left(\frac{\mu ({\mathit{R}}_b^{\mathrm{T}}\mathit{J}{\mathit{R}}_b)}{{\Vert \mathit{R}\Vert }^7}\right){\mathit{R}}_b$$

(8)

$${\mathit{f}}_{\mathrm{J}2}=m{\mathit{C}}^{\mathrm{T}}\left[\begin{array}{c}-\frac{3{\mathit{J}}_2\mu {\mathit{R}}_e^2{\mathit{R}}_x}{2{\Vert \mathit{R}\Vert }^5}(1-\frac{5{\mathit{R}}_z^2}{{\Vert \mathit{R}\Vert }^2})\hfill \\ \multicolumn{1}{c}{-\frac{3{\mathit{J}}_2\mu {\mathit{R}}_e^2{\mathit{R}}_y}{2{\Vert \mathit{R}\Vert }^5}(1-\frac{5{\mathit{R}}_z^2}{{\Vert \mathit{R}\Vert }^2})}\\ \multicolumn{1}{c}{-\frac{3{\mathit{J}}_2\mu {\mathit{R}}_e^2{\mathit{R}}_z}{2{\Vert \mathit{R}\Vert }^5}(3-\frac{5{\mathit{R}}_z^2}{{\Vert \mathit{R}\Vert }^2})}\end{array}\right]$$

(9)

$${\mathit{M}}_g=3\left(\frac{\mu }{{\Vert \mathit{R}\Vert }^5}\right)(\mathit{S}({\mathit{R}}_b)\mathit{J}{\mathit{R}}_b)$$

(10)

where $\mu =398,600.47\mathrm{k}{\mathrm{m}}^3/{\mathrm{s}}^2$ is the gravitational constant of the Earth, ${\mathit{J}}_2=1.08263\times {10}^{-3}$ , ${\mathit{R}}_e=6378.14\mathrm{km}$ is the equatorial radius of the Earth. ${\mathit{J}}_1=0.5\mathrm{tr}(\mathit{J})\mathit{I}+\mathit{J}$ , $\mathrm{tr}(·)$ is used to calculate the trace of a matrix, $\Vert ·\Vert \mathrm{stands}$ for the Euclidean norm of a vector. ${\mathit{R}}_b={\mathit{C}}^{\mathrm{T}}\mathit{R}$ and ${\mathit{R}}_x,{\mathit{R}}_y,\mathrm{and}{\mathit{R}}_z$ are R ’s components.

The dynamic equations can be expressed in a compact form as

$$\mathit{\Xi }\stackrel{·}{\mathit{V}}={\mathrm{ad}}_{\mathit{V}}^{*}\mathit{\Xi }\mathit{V}+{\mathit{\Gamma }}_g+{\mathit{\Gamma }}_c+{\mathit{\Gamma }}_d$$

(11)

where $\mathit{\Xi }=\mathrm{diag}(\mathit{J},m\mathit{I})$ , ${\mathit{\Gamma }}_g=[{\mathit{M}}_g^{\mathrm{T}},{\mathit{f}}_g^{\mathrm{T}}{]}^{\mathrm{T}}$ , ${\mathit{\Gamma }}_c=[{\mathbf{\tau }}_c^{\mathrm{T}},{\mathit{u}}_c^{\mathrm{T}}{]}^{\mathrm{T}}$ , and ${\mathit{\Gamma }}_d=[{\mathbf{\tau }}_d^{\mathrm{T}},{\mathit{u}}_d^{\mathrm{T}}{]}^{\mathrm{T}}$ . The co-adjoint operator ${\mathrm{ad}}_{\mathit{V}}^{*}$ for $\mathit{V}$ is defined as

$${\mathrm{ad}}_{\mathit{V}}^{*}={\mathrm{ad}}_{\mathit{V}}^{\mathrm{T}}=\left[\begin{array}{cc}-\mathit{S}(\mathbf{\omega }\mathbf{)}\hfill & -\mathit{S}(\mathit{v})\hfill \\ \multicolumn{1}{c}{0}& -\mathit{S}(\mathbf{\omega }\mathbf{)}\hfill \end{array}\right]$$

(12)

and the adjoint operator $\mathrm{a}{\mathrm{d}}_{\mathit{V}}$ is given by

$$\mathrm{a}{\mathrm{d}}_{\mathit{V}}=\left[\begin{array}{cc}\mathit{S}(\mathbf{\omega }\mathbf{)}\hfill & 0\hfill \\ \multicolumn{1}{c}{\mathit{S}(\mathit{v})}& \mathit{S}(\mathbf{\omega }\mathbf{)}\hfill \end{array}\right]$$

(13)

Equations (4) and (11) are the kinematics and dynamics of a single rigid spacecraft on SE(3), respectively.

Relative-coupled dynamics

Next, based on the mathematical model of a single rigid spacecraft, the relative-coupled dynamics between the leader and follower spacecraft will be given.

Let ${\chi }_f$ and ${\chi }_l$ be the actual configuration of the follower and the leader, respectively. Then, the configuration between the follower and the leader is

$${\chi }_{f/l}={\chi }_l^{-1}{\chi }_f$$

(14)

If the desired configuration of the follower with respect to the leader is

$${\chi }_d=\left[\begin{array}{cc}{\mathit{C}}_d& {\mathit{R}}_d\\ 0& 1\end{array}\right]$$

(15)

the configuration tracking error would be

$${\chi }_1={\left({\chi }_d\right)}^{-1}{\chi }_{f/l}$$

(16)

Then, the configuration tracking error expressed by exponential coordinates

$$\tilde{\mathbf{\eta }}=\left[\begin{array}{c}\tilde{\mathit{\Phi }}\hfill \\ \multicolumn{1}{c}{\tilde{\mathit{\phi }}}\end{array}\right]\in {\mathbb{R}}^6$$

(17)

can be calculated by ${\chi }_1$

$$\tilde{\eta }={\left[{\log }_{\mathrm{SE}(3)}{\chi }_1\right]}^{-1}$$

(18)

where $(·{)}^{-1}$ is the inverse mapping of $(·{)}^v$ , and $\log (·{)}_{\mathrm{SE}(3)}$ is the logarithm map, that is, the inverse of the exponential map. If we define ${\chi }_1$ as $\left[\begin{array}{cc}{\mathit{C}}_1\hfill & {\mathit{R}}_1\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]$ , $\mathrm{lo}{\mathrm{g}}_{\mathrm{SE}(3)}$ can be expressed as

$${\log }_{\mathrm{SE}(3)}{\chi }_1=\left[\begin{array}{cc}S(\tilde{\Phi })& \tilde{\phi }\\ 0& 0\end{array}\right]$$

(19)

where $\tilde{\mathit{\Phi }}$ and $\tilde{\mathit{\phi }}$ are the tracking errors expressed by exponential coordinates corresponding to attitude and position, respectively, with

$$\mathit{S}(\tilde{\mathit{\Phi }})=\{\begin{array}{cc}0\hfill & \theta =0\hfill \\ \multicolumn{1}{c}{\frac{\theta }{2\sin \theta }({\mathit{C}}_1-{\mathit{C}}_1^{\mathrm{T}})}& \theta \in (-\pi ,\pi ),\theta \ne 0\hfill \end{array}$$

(20)

$$\{\begin{array}{c}\tilde{\mathit{\phi }}={\mathit{M}}^{-1}(\tilde{\mathit{\Phi }}){\mathit{R}}_1\hfill \\ \mathit{M}(\tilde{\mathit{\Phi }})=\mathit{I}+\frac{1-\cos \theta }{{\theta }^2}\mathit{S}(\tilde{\mathit{\Phi }})+\frac{\theta -\sin \theta }{{\theta }^3}{\mathit{S}}^2(\tilde{\mathit{\Phi }})\hfill \end{array}$$

(21)

where $\theta =\arccos (0.5(\mathrm{tr}({\mathit{C}}_1)-1))$ is the norm of $\tilde{\mathit{\Phi }}$ and corresponds to the principal rotation angle.

The relative velocity of the follower with respect to the leader expressed in the body frame of the follower is

$$\tilde{\mathit{V}}=\mathit{V}-{\mathrm{Ad}}_{\chi f/l}^{-1}{\mathit{V}}_l$$

(22)

where the adjoint action map defined for $\mathbf{\chi }$ is

$$\mathrm{A}{\mathrm{d}}_{\mathbf{\chi }}=\left[\begin{array}{cc}\mathit{C}\hfill & 0\hfill \\ \multicolumn{1}{c}{\mathit{S}(\mathit{R})\mathit{C}}& \mathit{C}\hfill \end{array}\right]\in {\mathbb{R}}^{6\times 6}$$

(23)

Then, the kinematics in exponential coordinates as given in Brás et al. ²⁶ is

$$\stackrel{·}{\stackrel{~}{\mathbf{\eta }}}=\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathit{V}}$$

(24)

where $\mathit{G}(\tilde{\mathbf{\eta }})$ is given by

$$\mathit{G}(\tilde{\mathbf{\eta }})=\left[\begin{array}{cc}\mathit{A}(\tilde{\mathit{\Phi }})\hfill & 0\hfill \\ \multicolumn{1}{c}{\mathit{T}(\tilde{\mathit{\Phi }},\tilde{\mathit{\phi }})}& \mathit{A}(\tilde{\mathit{\Phi }})\hfill \end{array}\right]$$

(25)

where

$$\mathit{A}(\tilde{\mathit{\Phi }})=\mathit{I}+\frac{1}{2}\mathit{S}(\tilde{\mathit{\Phi }})+(\frac{1}{{\theta }^2}-\frac{1+\cos \theta }{2\theta \sin \theta }){\mathit{S}}^2(\tilde{\mathit{\Phi }})$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathit{T}(\tilde{\mathit{\Phi }},\tilde{\mathit{\phi }})=\frac{1}{2}\mathit{S}(\mathit{M}(\tilde{\mathit{\Phi }})\tilde{\mathit{\phi }})\mathit{A}(\tilde{\mathit{\Phi }})}\\ \multicolumn{1}{c}{+(\frac{1}{{\theta }^2}-\frac{1+\cos \theta }{2\theta \sin \theta })(\tilde{\mathit{\Phi }}{\tilde{\mathit{\phi }}}^{\mathrm{T}}+{\tilde{\mathit{\Phi }}}^{\mathrm{T}}\tilde{\mathit{\phi }}\mathit{A}(\tilde{\mathit{\Phi }}))}\\ \multicolumn{1}{c}{-\frac{(1+\cos \theta )(\theta -\sin \theta )}{2\theta \mathrm{si}{\mathrm{n}}^2\theta }\mathit{M}(\tilde{\mathit{\Phi }})\tilde{\mathit{\phi }}{\tilde{\mathit{\Phi }}}^{\mathrm{T}}}\\ \multicolumn{1}{c}{+(\frac{(1+\cos \theta )(\theta +\sin \theta )}{2{\theta }^3\mathrm{si}{\mathrm{n}}^2\theta }-\frac{2}{{\theta }^4}){\tilde{\mathit{\Phi }}}^{\mathrm{T}}\tilde{\mathit{\phi }}\tilde{\mathit{\Phi }}{\tilde{\mathit{\Phi }}}^{\mathrm{T}}}\end{array}$$

According to equation (22) and $d({\mathrm{Ad}}_{\chi f/l}^{-1}(t))/dt=-{\mathrm{ad}}_{\tilde{V}}{\mathrm{Ad}}_{\chi f/l}^{-1}(t)$ as proved in Lee, ⁷ the relative acceleration of the follower will be given by

$$\stackrel{.}{\stackrel{~}{\mathit{V}}}=\stackrel{.}{\mathit{V}}+{\mathrm{ad}}_{\tilde{\mathit{V}}}{\mathrm{Ad}}_{{\chi }_{f/l}}^{-1}{\mathit{V}}_l-{\mathrm{Ad}}_{{\chi }_{f/l}}^{-1}{\stackrel{.}{\mathit{V}}}_l$$

(26)

Substituting equation (11) into equation (26) produces

$$\Xi \stackrel{.}{\stackrel{~}{V}}={\mathrm{ad}}_V^{\ast }\Xi V+{\Gamma }_g+{\Gamma }_c+{\Gamma }_d+\Xi \left({\mathrm{ad}}_{\tilde{V}}{\mathrm{Ad}}_{{\chi }_{f/l}}^{-1}{V}_l-{\mathrm{Ad}}_{{\chi }_{f/l}}^{-1}{\stackrel{.}{V}}_l\right)$$

(27)

Then, the relative-coupled dynamics between the follower and the leader can be modeled by equations (24) and (27).

However, considering actual system, the model uncertainties can’t be ignored. Let the actual matrix with moment of inertia matrix and mass be ${\mathit{\Xi }}_1=\mathit{\Xi }+\mathbf{\Delta }\mathit{\Xi }$ , where $\mathbf{\Delta }\mathit{\Xi }$ is the uncertainty part. $(\mathit{\Xi }+\mathbf{\Delta }\mathit{\Xi }{\mathbf{)}}^{-1}$ can be expressed as ${\mathit{\Xi }}^{-1}+\mathbf{\Delta }\tilde{\mathit{\Xi }}$ , and $\mathbf{\Delta }\tilde{\mathit{\Xi }}$ is also an uncertainty part. Then, equation (27) can be modified as

$$\stackrel{·}{\stackrel{~}{\mathit{V}}}=\mathit{H}+\mathbf{\Delta }\tilde{\mathit{d}}\mathbf{+}{\mathit{\Xi }}^{-1}{\mathit{\Gamma }}_{\mathbf{c}}\mathbf{+}{\mathit{\Xi }}^{-1}{\mathit{\Gamma }}_{\mathbf{g}}$$

(28)

where $H={\Xi }^{-1}{\mathrm{ad}}_V^{\ast }\Xi V+{\mathrm{ad}}_{\tilde{V}}{\mathrm{Ad}}_{{\chi }_{f/l}}^{-1}{V}_l-{\mathrm{Ad}}_{{\chi }_{f/l}}^{-1}{\stackrel{.}{V}}_l$ , $\Delta \tilde{\mathit{d}}=\Delta \tilde{\mathit{\Xi }}({\mathrm{ad}}_{\mathit{V}}^{*}{\mathit{\Xi }}_1\mathit{V}+{\mathit{\Gamma }}_c+{\mathit{\Gamma }}_g)+{\mathit{\Xi }}^{-1}{\mathrm{ad}}_{\mathit{V}}^{*}\Delta \mathit{\Xi }\mathbf{V}+{\mathit{\Xi }}_1^{-1}({\mathit{\Gamma }}_d+{\mathit{\Gamma }}_G-{\Gamma }_g)$ is the composed disturbance of the system, and ${\mathit{\Gamma }}_G$ is computed by substituting ${\mathit{\Xi }}_1$ into equations (7)–(10).

Remark 1

Employing exponential coordinates on the Lie group SE(3), the relative-coupled dynamics between the leader and follower is formed by equations (24) and (28) in a united framework. In equation (24), the existence of $\mathit{G}(\tilde{\mathbf{\eta }})$ guarantees the decouple of the nonlinear dynamics in equation (1), and in equation (28), the dynamical couplings is involved in $\mathit{H}$ , which makes the couplings being handled using feedback technique be possible. The trajectory of the leader is computed offline in an ideal environment without external disturbances. When the relative configuration approaches the desired one, the configuration tracking error $\tilde{\mathbf{\eta }}$ goes to zero, and the desired formation is achieved. And for equation (20), the logarithm map $\mathrm{SE}(3)\to \varsigma e(3)$ is bijective when $\theta \in (-\pi ,\pi )$ , which means that a control scheme designed on the relative-coupled dynamics is almost global in its convergence over the state space.

Preliminaries

For the simplicity of expression, throughout this article, the following notations are defined as follows: for a real number $\gamma$ and a given vector $\mathit{x}=[{\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_m{]}^{\mathrm{T}}\in {\mathbb{R}}^m$ , where $(\mathit{x}{)}_k$ represents the kth component of $\mathit{x}$

$$\mathrm{si}{g}^{\gamma }(\mathit{x})={[{\left|{\mathit{x}}_1\right|}^{\gamma }\mathrm{sgn}({\mathit{x}}_1),\dots ,{\left|{\mathit{x}}_m\right|}^{\gamma }\mathrm{sgn}({\mathit{x}}_m)]}^{\mathrm{T}}$$

(29)

where $\mathrm{sgn}(·)$ is the signum function. $\left|{\mathit{x}}_k\right|$ denotes the absolute value of ${\mathit{x}}_k$ , and $\left|\mathit{x}\right|\in {\mathbb{R}}^{m\times 1}$ is a vector composed by $\left|{\mathit{x}}_k\right|$ . Meanwhile, $\Vert \mathit{x}\Vert$ denotes the Euclidean norm or its induced norm. ${\lambda }_{\max }(·)$ , and ${\lambda }_{\min }(·)$ denote the maximum and minimum eigenvalues of a matrix, respectively.

In order to facilitate the stability analysis of the controller, the following assumptions and lemmas are presented.

Assumption 1

The composed disturbance of the spacecraft $\Delta \tilde{\mathit{d}}$ , including force and torque caused by fuel consuming or components/payloads releasing, gravitational perturbation and torque, radiation force and torque, aerodynamic force and torque, and other environmental or non-environmental forces and torques, are assumed to be differentiable and bounded. Thus, it is reasonable to assume that

$$\{\begin{array}{c}\left|\Delta {\tilde{\mathit{d}}}_i\right|\le {\mathbf{\delta }}_{1i},i=1,2,\dots ,6\hfill \\ \multicolumn{1}{c}{\left|\Delta {\stackrel{·}{\stackrel{~}{\mathit{d}}}}_i\right|\le {\mathbf{\delta }}_{2i},i=1,2,\dots ,6}\end{array}$$

(30)

where ${\mathbf{\delta }}_1\mathrm{and}{\mathbf{\delta }}_2\in {\mathbb{R}}^{6\times 1}$ are positive constant vectors.

Lemma 1 ¹¹

For a continuous positive definite function $V(\mathit{x}):{\mathbb{R}}^n\to \mathbb{R}$ , if the following differential inequality holds with ${\rho }_1>0,{\rho }_2>0,\mathrm{and}0<\rho <1$

$$\stackrel{·}{V}(\mathit{x})\le -{\rho }_{1}V(\mathit{x})-{\rho }_2{V}^{\rho }(\mathit{x}),\vee t>t_0$$

(31)

then, $V(\mathit{x})$ can reach the equilibrium in finite time, and the convergence time satisfies

$$t_f\le t_0+\frac{1}{{\rho }_1(1-\rho )}\ln \frac{{\rho }_1{V}^{1-\rho }(x(t_0))+{\rho }_2}{{\rho }_2}$$

(32)

where $\mathit{x}(t_0)$ is the initial value of $x$ .

Lemma 2 ¹²

For a continuous positive definite function $V(\mathit{x}):{\mathbb{R}}^n\to \mathbb{R}$ , which satisfies the following differential inequality

$$\stackrel{·}{V}(\mathit{x})+\vartheta V^{\varpi }(\mathit{x})\le 0,(\mathit{x}\in \mathbf{\Omega },\mathit{x}\ne \mathit{0})$$

(33)

where $\vartheta >0$ and $\varpi \in (0,1)$ , and $\mathbf{\Omega }\in {\mathbb{R}}^n$ is a neighborhood of the equilibrium, then $V(\mathit{x})$ can reach the equilibrium in finite time, and the convergence time satisfies

$$t=\frac{V^{1-\varpi }(\mathit{x}(t_0))}{\vartheta (1-\varpi )}$$

(34)

Lemma 3

The eigenvalues of matrix $\mathit{G}(\tilde{\mathbf{\eta }})$ in equation (24) are all positive.

Proof

In Lee et al., ⁹ $\mathit{G}(\tilde{\mathbf{\eta }})$ satisfies the relation

$$\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathbf{\eta }}=\tilde{\mathbf{\eta }}$$

(35)

Multiplying both sides of equation (35) by ${\tilde{\mathbf{\eta }}}^{\mathrm{T}}$ produces

$${\tilde{\mathbf{\eta }}}^{\mathrm{T}}\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathbf{\eta }}={\tilde{\mathbf{\eta }}}^{\mathrm{T}}(\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathbf{\eta }})={\tilde{\mathbf{\eta }}}^{\mathrm{T}}\tilde{\mathbf{\eta }}$$

(36)

When $\tilde{\mathbf{\eta }}\ne \mathit{0}$ and ${\tilde{\mathbf{\eta }}}^{\mathrm{T}}\tilde{\mathbf{\eta }}>0$ , thus we have $\mathit{G}(\tilde{\mathbf{\eta }})$ is a positive definite matrix, which means that the eigenvalues must be positive; when $\tilde{\mathbf{\eta }}$ is equal to zero, by taking it into equation (25), we have $\mathit{G}(\tilde{\mathbf{\eta }})$ is an identity matrix, then the eigenvalues are all equal to one. The above analysis concludes the proof.

Problem statement

In this research, a LF spacecraft formation is considered. The control objective herein is to design a robust controller against the model uncertainties and external disturbances based on the relative-coupled dynamics equations (24) and (28) by the use of the SMC algorithm, such that the follower trajectory can converge to the reference state with respect to the leader in finite time, and then maintain the certain relative configuration, which can be expressed as $\mathrm{li}{\mathrm{m}}_{t\to T}\tilde{\mathbf{\eta }}=\mathrm{li}{\mathrm{m}}_{t\to T}\tilde{\mathit{V}}=\mathit{0}$ , $T$ is the convergence time. Furthermore, the control objective should be met in the presence of actuator saturation $\left|{\mathbf{\tau }}_{\mathrm{ci}}\right|\le {\tau }_{\max },i=1,2,3;\left|{\mathit{u}}_{\mathrm{ci}}\right|\le u_{\max },i=4,5,6$ , where the positive scalars ${\tau }_{\max }$ and $u_{\max }$ are the upper bound on the allowed control torque and control force components, respectively.

Controller design and stability analysis

In this section, based on the relative-coupled translational and rotational dynamics described by equations (24) and (28), a class of robust terminal SMC laws using the exponential coordinates and relative velocities is proposed to achieve the control objective stated in section “Problem statement.” The proposed control schemes are effective against the composed disturbance of the system, and by employing Lyapunov methods, the convergence of the closed-loop system is proved theoretically.

Finite-time controller synthesis

For the convenience of the following analysis, we put the relative-coupled dynamics equations (24) and (28) together with one equation number

$$\{\begin{array}{c}\stackrel{·}{\stackrel{~}{\mathbf{\eta }}}\mathbf{=}\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathit{V}}\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\stackrel{~}{\mathit{V}}}\mathbf{=}\mathit{H}\mathbf{+}\mathbf{\Delta }\tilde{\mathit{d}}\mathbf{+}{\mathit{\Xi }}^{-1}{\mathit{\Gamma }}_{\mathbf{c}}\mathbf{+}{\mathit{\Xi }}^{-1}{\mathit{\Gamma }}_{\mathbf{g}}}\end{array}$$

(37)

To guarantee the system states converge to the equilibrium in finite time, TSM is first developed. However, its convergence time is longer than that of the conventional SMC when far away from the equilibrium point. To enhance the convergence speed, based on Liang et al., ¹⁵ we first propose a FTSM which is designed with the following form

$$\mathit{S}=\tilde{\mathit{V}}+{\mathit{C}}_1\tilde{\mathbf{\eta }}+{\mathit{C}}_2\mathrm{si}{g}^{\alpha }(\tilde{\mathbf{\eta }})$$

(38)

where ${\mathit{C}}_1\mathrm{and}{\mathit{C}}_2\in {\mathbb{R}}^{6\times 6}$ are positive constant definite diagonal matrices, and $\alpha \in (0.5,1)$ . If the states of the system reach the sliding surface, then we have $\mathit{S}=0$ , which implies that

$$\tilde{\mathit{V}}=-{\mathit{C}}_1\tilde{\mathbf{\eta }}-{\mathit{C}}_2\mathrm{si}{g}^{\alpha }(\tilde{\mathbf{\eta }})$$

(39)

Consider a Lyapunov function candidate as $V_1=(1/2){\tilde{\mathbf{\eta }}}^{\mathrm{T}}\tilde{\mathbf{\eta }}$ , calculating the derivative of $V_1$ with respect to time and substituting equation (39) into the expression yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1={\tilde{\mathbf{\eta }}}^{\mathrm{T}}\stackrel{·}{\stackrel{~}{\mathbf{\eta }}}={\tilde{\mathbf{\eta }}}^{\mathrm{T}}\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathit{V}}}\\ ={\tilde{\mathbf{\eta }}}^{\mathrm{T}}\mathit{G}(\tilde{\mathbf{\eta }})(-{\mathit{C}}_1\tilde{\mathbf{\eta }}-{\mathit{C}}_2\mathrm{si}{g}^{\alpha }(\tilde{\mathbf{\eta }}))\hfill \\ \le -{\lambda }_{\min }(\mathit{G}(\tilde{\mathbf{\eta }}))({\mathit{C}}_1{\tilde{\mathbf{\eta }}}^{\mathrm{T}}\tilde{\mathbf{\eta }}+{\mathit{C}}_2{\tilde{\mathbf{\eta }}}^{\mathrm{T}}\mathrm{si}{g}^{\alpha }(\tilde{\mathbf{\eta }}))\hfill \\ \le -{\overline{\mathit{C}}}_1{V}_1-{\overline{\mathit{C}}}_2{V}_1^{\frac{1+\alpha }{2}}\hfill \end{array}$$

(40)

where ${\overline{\mathit{C}}}_1=-2{\lambda }_{\min }\mathit{G}(\tilde{\mathbf{\eta }}){\mathit{C}}_1$ , and ${\overline{\mathit{C}}}_2=-2^{(1+\alpha )/2}{\lambda }_{\min }\mathit{G}(\tilde{\mathbf{\eta }}){\mathit{C}}_2$ . Since ${\lambda }_{\min }\mathit{G}(\tilde{\mathbf{\eta }})$ is positive using Lemma 3, thus we have ${\overline{\mathit{C}}}_1$ and ${\overline{\mathit{C}}}_2$ are positive definite diagonal matrices. Then, $\tilde{\mathbf{\eta }}\equiv 0$ is achieved in finite time using Lemma 1, and it is easy to obtain that $\tilde{\mathit{V}}$ converges to zero in finite time by substituting $\tilde{\mathbf{\eta }}=0$ into equation (39).

Remark 2

In the proposed FTSM manifold equation (38), the sliding surface is defined by configuration errors described by exponential coordinates $\tilde{\mathbf{\eta }}$ and velocity tracking errors $\tilde{\mathit{V}}$ , which will converge to zero in finite time by designing a suitable controller. However, when $\mathit{S}$ reaches the equilibrium, the existence of $\mathit{G}(\tilde{\mathbf{\eta }})$ makes the finite-time stability proof of $\tilde{\eta }=\tilde{\mathit{V}}\equiv \mathit{0}$ be difficult. Fortunately, by first proposing Lemma 3, the problem has been solved properly.

Theorem 1

Consider the spacecraft system governed by equation (37), and based on the fast terminal sliding surface equation (38), if Assumption 1 $(\left|\mathrm{\Delta }{\tilde{\mathit{d}}}_i\right|\le {\mathbf{\delta }}_{1i},i=1,2,\dots ,6)$ is satisfied and the control scheme is designed as equation (41), then the sliding surface $\mathit{S}$ with its two components $\tilde{\eta }$ and $\tilde{\mathit{V}}$ will converge to zero in finite time

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{\Gamma }}_c=-\mathit{\Xi }({\mathit{C}}_1\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathit{V}}+\alpha {\mathit{C}}_2\mathrm{si}{g}^{\alpha -1}(\tilde{\mathbf{\eta }})\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathit{V}})}\\ -{\mathit{\Gamma }}_g-\mathit{\Xi }\mathit{H}-{\mathit{K}}_1\mathrm{sgn}(\mathit{S})-{\mathit{K}}_2\mathit{S}\hfill \end{array}$$

(41)

where ${\mathit{K}}_1=\mathrm{diag}(k_{11},\dots ,k_{16})$ , and ${\mathit{K}}_2=\mathrm{diag}(k_{21},\dots ,k_{26})$ , they are positive constant definite matrices, and ${\mathit{K}}_1$ satisfies the following inequality

$${\mathit{K}}_{1i}>(\mathit{\Xi }{\mathbf{\delta }}_1{)}_i,i=1,2,\dots ,6$$

(42)

Proof

Consider the following Lyapunov function candidate

$$V_2=\frac{1}{2}{\mathit{S}}^{\mathrm{T}}\mathit{\Xi }\mathit{S}$$

(43)

calculating the derivative of $V_2$ with respect to time and substituting equation (37) into it, yields

$$\begin{array}{c}{\stackrel{.}{V}}_2=S^{\mathrm{T}}\Xi \stackrel{.}{S}\\ =S^{\mathrm{T}}\Xi \left(H+\Delta \tilde{d}+{\Xi }^{-1}{\Gamma }_c+{\Xi }^{-1}{\Gamma }_g+C_{1}G(\tilde{\eta })\tilde{V}+\alpha C_{2}si{g}^{\alpha -1}(\tilde{\eta })G(\tilde{\eta })\tilde{V}\right)\end{array}$$

(44)

Then, substituting control law (41) into equation (44), we have

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2={\mathit{S}}^{\mathrm{T}}[-{\mathit{K}}_1\mathrm{sgn}(\mathit{S})-{\mathit{K}}_2\mathit{S}+\mathit{\Xi }\Delta \tilde{\mathit{d}}]}\\ \le -\sum _{i=1}^6({\mathit{K}}_{1i}\left|{\mathit{S}}_i\right|-{(\mathit{\Xi }{\mathbf{\delta }}_1)}_i\left|{\mathit{S}}_i\right|)-{\mathit{K}}_2{\mathit{S}}^{\mathrm{T}}\mathit{S}\hfill \\ \le -{\vartheta }_1{V}_2^{\frac{1}{2}}-{\vartheta }_2{V}_2\hfill \end{array}$$

(45)

where ${\vartheta }_1=\sqrt{2}{\lambda }_{\min }({\mathit{K}}_1-\mathit{\Xi }{\mathbf{\delta }}_1\mathbf{)}\mathbf{/}{\mathbf{\lambda }}_{\mathbf{max}}\mathbf{(}\mathit{\Xi }\mathbf{)}>0$ , and ${\vartheta }_2=2{\lambda }_{\min }({\mathit{K}}_2)/{\lambda }_{\max }(\mathit{\Xi }\mathbf{)}>0$ . Using Lemma 1, the sliding surface $\mathit{S}=0$ can be reached in finite time. Meanwhile, through the analysis in equation (40), we have the control objective $\tilde{\mathbf{\eta }}=\tilde{\mathit{V}}=0$ can be obtained in finite time, and therefore the follower tracks the reference trajectory with respect to the leader in finite time. The above analysis concludes the proof.

Remark 3

Because of the term $\alpha {\mathit{C}}_2\mathrm{si}{g}^{\alpha -1}(\tilde{\mathbf{\eta }})\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathit{V}}$ in the control law equation (41), when $\tilde{\eta }$ reaches the equilibrium before $\tilde{\mathit{V}}$ , the output of the controller will be infinite. To solve this singularity problem, Zou et al. ²⁷ found a region $\mathbf{\Omega }$ , in which the singularity problem will not occur. But when the sliding mode is in the region, the modified expression is not suitable for the relative-coupled dynamics proposed in this article since the existence of $\mathit{G}(\tilde{\mathbf{\eta }})$ . In this article, no singularity will occur because when $\tilde{\eta }=0$ , the following equation holds

$$\begin{array}{c}\alpha {\mathit{C}}_2\mathrm{si}{g}^{\alpha -1}(\tilde{\mathbf{\eta }})\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathit{V}}\hfill \\ =\alpha {\mathit{C}}_2\mathit{G}(\tilde{\mathbf{\eta }})\mathrm{si}{g}^{\alpha -1}(\tilde{\mathbf{\eta }})(-{\mathit{C}}_1\tilde{\mathbf{\eta }}-{\mathit{C}}_2\mathrm{si}{g}^{\alpha }(\tilde{\mathbf{\eta }}))\hfill \\ =-\alpha {\mathit{C}}_2\mathit{G}(\tilde{\mathbf{\eta }})({\mathit{C}}_1\left|\tilde{\mathbf{\eta }}\right|-{\mathit{C}}_2\mathrm{si}{g}^{2\alpha -1}(\tilde{\mathbf{\eta }}))=\mathit{0}\hfill \end{array}$$

(46)

Remark 4

Due to physical limitations on actuator, special attention should be paid to actuator saturation, then the actual output of the controller acts on the system can be expressed as

$$Sat({\Gamma }_c)=\{\begin{array}{cc}Sat({\tau }_c)=\{\begin{array}{l}{\tau }_{max},if{\Gamma }_{ci}>{\tau }_{max}\hfill \\ {\Gamma }_{ci},if\left|{\Gamma }_{ci}\right|\le {\tau }_{max}\hfill \\ -{\tau }_{max},if{\Gamma }_{ci}<-{\tau }_{max}\hfill \end{array}& i=1,2,3\\ Sat(u_c)=\{\begin{array}{l}{u}_{max},if{\Gamma }_{ci}>u_{max}\hfill \\ {\Gamma }_{ci},if\left|{\Gamma }_{ci}\right|\le u_{max}\hfill \\ -u_{max},if{\Gamma }_{ci}<-u_{max}\hfill \end{array}& i=4,5,6\end{array}$$

(47)

Then, we have $\mathrm{Sat}({\mathit{\Gamma }}_c)={\mathit{\Gamma }}_c+\Delta \mathit{\Gamma }$ , and $\Delta \tilde{d}$ in equation (28) can be modified as $\Delta \tilde{\mathit{d}}=\mathrm{\Delta }\tilde{\Xi }({\mathrm{ad}}_{\mathit{V}}^{*}{\mathit{\Xi }}_1\mathit{V}+\mathrm{Sat}({\mathit{\Gamma }}_c)+{\mathit{\Gamma }}_g)+{\mathit{\Xi }}^{-1}{\mathrm{ad}}_{\mathit{V}}^{*}\Delta \mathit{\Xi }\mathit{V}\mathbf{+}{\mathit{\Xi }}_1^{-1}\mathbf{(}{\mathit{\Gamma }}_{\mathrm{d}}\mathbf{+}{\mathit{\Gamma }}_{\mathbf{G}}-{\mathit{\Gamma }}_{\mathbf{g}}\mathbf{)}\mathbf{+}{\mathit{\Xi }}^{-1}\mathbf{\Delta }\mathit{\Gamma }$ . Meanwhile, Assumption 1 is also satisfied.

Remark 5

To eliminate the chattering phenomenon caused by the signum function in equation (41), the saturation function is employed to replace $\mathrm{sgn}(\mathit{S})$ with

$$\mathrm{sat}({\mathit{S}}_i,ϵ)=\{\begin{array}{cc}\mathrm{sgn}({\mathit{S}}_i),\hfill & \mathrm{if}\left|{\mathit{S}}_i\right|\ge ϵ\hfill \\ \multicolumn{1}{c}{\frac{{\mathit{S}}_i}{ϵ},}& \mathrm{if}\left|{\mathit{S}}_i\right|<ϵ\hfill \end{array}$$

(48)

where $ϵ>0$ is a small constant.

Although by introducing the saturation function, the chattering can be eliminated in a certain degree, in equation (42) we know that only when ${\mathit{K}}_{1i}>(\mathit{\Xi }{\mathbf{\delta }}_1{\mathbf{)}}_{\mathbf{i}}$ , our control objective can be finished in finite time. When taking actual system into consideration, the composed disturbance $\Delta \tilde{\mathit{d}}$ is always unknown and very large, which means that the control parameter ${\mathit{K}}_1$ must be chosen large enough resulting in another chattering, strong conservation of the controller, and the waste of fuel. If we can compensate the composed disturbance in the control law actively, the aforementioned problem will be solved perfectly. Among the disturbance estimation methods, the ESO is an effective way, which will be discussed in the next subsection.

ESO-based controller design

First, a coordinate transformation is defined as ${\mathit{x}}_1=\tilde{\mathit{V}}+\mathit{k}\tilde{\eta }$ for simplicity, where $\mathit{k}$ is a diagonal positive constant matrix. Taking derivative of $\mathit{x}$ with respect to time and substituting equation (37) into it, we have

$${\stackrel{·}{\mathit{x}}}_1=\mathit{A}+\mathit{B}{\mathit{\Gamma }}_c+\Delta \tilde{\mathit{d}}$$

(49)

with $\mathit{A}=\mathit{H}+{\mathit{\Xi }}^{-1}{\mathit{\Gamma }}_g+\mathit{k}\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathit{V}}$ , and $\mathit{B}={\mathit{\Xi }}^{-1}$ . Utilizing the ESO technique, ¹⁸ the composed disturbance $\Delta \tilde{\mathit{d}}$ can be regarded as an extended state variable ${\mathit{x}}_2$ , then equation (49) can be modified as

$$\{\begin{array}{c}{\stackrel{·}{\mathit{x}}}_1=\mathit{A}+\mathit{B}{\mathit{\Gamma }}_c+{\mathit{x}}_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathit{x}}}_2=\mathit{g}(t)}\end{array}$$

(50)

where $\mathit{g}(t)$ is the derivative of $\Delta \tilde{\mathit{d}}$ , which satisfies Assumption 1.

Based on the finite-time observer in Zou, ²² an ESO is designed in the form of

$$\{\begin{array}{c}{\stackrel{·}{\mathit{Z}}}_1=\mathit{A}+\mathit{B}{\mathit{\Gamma }}_c+{\mathit{Z}}_2-{\mathit{k}}_1\mathrm{si}{g}^{\beta }({\mathit{e}}_1)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathit{Z}}}_2=-{\mathit{k}}_2\mathrm{si}{g}^{2\beta -1}({\mathit{e}}_1)}\end{array}$$

(51)

where ${\mathit{Z}}_1\mathrm{and}{\mathit{Z}}_2$ are the outputs of the observer, ${\mathit{e}}_1={\mathit{Z}}_1-{\mathit{x}}_1$ , $0.5<\beta <1$ , ${\mathit{k}}_{1i},{\mathit{k}}_{2i},i=1,2,\dots ,6$ are positive observer gains.

Define the observer errors as $\mathit{e}=[{\mathit{e}}_1^{\mathrm{T}},{\mathit{e}}_2^{\mathrm{T}}{]}^{\mathrm{T}}$ , with ${\mathit{e}}_2={\mathit{Z}}_2-{\mathit{x}}_2$ , and based on equations (50) and (51), the dynamics of the observer errors $\mathit{e}$ can be established as

$$\{\begin{array}{c}{\stackrel{·}{\mathit{e}}}_1=-{\mathit{k}}_1\mathrm{si}{g}^{\beta }({\mathit{e}}_1)+{\mathit{e}}_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathit{e}}}_2=-{\mathit{k}}_2\mathrm{si}{g}^{2\beta -1}({\mathit{e}}_1)-\mathit{g}(t)}\end{array}$$

(52)

Theorem 2

Consider the system described by equation (50) in combination with the ESO equation (51). Give the initial estimation of ${\mathit{Z}}_{10}$ and ${\mathit{Z}}_{20}$ , by choosing appropriate observer gains ${\mathit{k}}_1$ and ${\mathit{k}}_2$ . If Assumption 1 $(\left|\Delta {\stackrel{·}{\stackrel{~}{\mathit{d}}}}_i\right|\le {\mathbf{\delta }}_{2\mathbf{i}})$ , and ${\mathit{k}}_{1i}>0,{\mathit{k}}_{2i}>0,i=1,2,\dots ,6$ are satisfied, then the estimate errors in equation (52) converge in finite time to the region

$${\sigma }_i=\{\Vert {\ell }_i\Vert \le {\left(\frac{{\mathbf{\Delta }}_{2i}\Vert \mathbf{\Psi }\Vert }{{\lambda }_{\min }(\mathbf{\Theta })}\right)}^{\beta /(2\beta -1)}\}$$

(53)

where ${\ell }_i=[\mathrm{si}{g}^{\beta }({\mathit{e}}_{1i}),{\mathit{e}}_{2i}]$ , the matrices $\mathbf{\Theta }$ and $\mathbf{\Psi }$ given by

$$\{\begin{array}{c}\mathbf{\Theta }\mathbf{=}\frac{{\mathit{k}}_{1\mathbf{i}}}{2}\mathbf{\left[}\begin{array}{cc}2{\mathit{k}}_{2\mathbf{i}}\mathbf{+}2{\mathit{k}}_{1\mathbf{i}}^2\mathbf{\beta }\hfill & -2{\mathit{k}}_{1\mathbf{i}}\mathbf{\beta }\hfill \\ \multicolumn{1}{c}{-2{\mathit{k}}_{1\mathbf{i}}\mathbf{\beta }}& 2\mathbf{\beta }\hfill \end{array}\mathbf{\right]}\hfill \\ \multicolumn{1}{c}{\mathbf{\Psi }\mathbf{=}\mathbf{\left[}\begin{array}{c}{\mathit{k}}_{1\mathbf{i}}\hfill \\ \multicolumn{1}{c}{-2}\end{array}\mathbf{\right]}}\end{array}$$

(54)

Proof

By extending the ideas of Moreno and Osorio, ²⁸ we candidate the following Lyapunov function

$$V_3=\frac{{\mathit{k}}_{2i}}{\beta }\left|{\mathit{e}}_{1i}\right|+\frac{1}{2}{\mathit{e}}_{2i}^2+\frac{1}{2}({\mathit{k}}_{1i}\mathrm{si}{g}^{\beta }({\mathit{e}}_{1i})-{\mathit{e}}_{2i})$$

(55)

which can be written as

$$V_3={\ell }_i^{\mathrm{T}}\mathit{P}{\ell }_i$$

(56)

where

$$\mathit{P}=\frac{1}{2}\left[\begin{array}{cc}\frac{2{\mathit{k}}_{2i}}{\beta }+{\mathit{k}}_{1i}^2\hfill & -{\mathit{k}}_{1i}\hfill \\ \multicolumn{1}{c}{-{\mathit{k}}_{1i}}& 2\hfill \end{array}\right]$$

Thus, we have

$${\lambda }_{\min }(\mathit{P})\Vert {\ell }_i{\Vert }^2\le V_3\le {\lambda }_{\max }(\mathit{P})\Vert {\ell }_i{\Vert }^2$$

(57)

where $\Vert {\ell }_i{\Vert }^2={\left|{\mathit{e}}_{1i}\right|}^{2\beta }+{\mathit{e}}_{2i}^2$ , and $V_3$ is continuous but is not differentiable at ${\mathit{e}}_{1i}=0$ . Since ${\mathit{k}}_{1i}>0\mathrm{and}{\mathit{k}}_{2i}>0$ , it is obviously that $V_3$ is positive definite and radially unbounded.

Taking the derivative of $V_3$ with respect to time, and substituting equation (52) into it yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3=2{\mathit{k}}_{2i}\mathrm{si}{g}^{2\beta -1}({\mathit{e}}_{1i}){\stackrel{·}{\mathit{e}}}_{1i}+{\mathit{e}}_{2i}{\stackrel{·}{\mathit{e}}}_{2i}}\\ +({\mathit{k}}_{1i}\mathrm{si}{g}^{\beta }({\mathit{e}}_{1i})-{\mathit{e}}_{2i})({\mathit{k}}_{1i}\beta {\left|{\mathit{e}}_{1i}\right|}^{\beta -1}{\stackrel{·}{\mathit{e}}}_{1i}-{\stackrel{·}{\mathit{e}}}_{2i})\hfill \\ \multicolumn{1}{c}{=(2{\mathit{k}}_{2i}+{\mathit{k}}_{1i}^2\beta ){\left|{\mathit{e}}_{1i}\right|}^{2\beta -1}\mathrm{sgn}({\mathit{e}}_{1i})}\\ \times ({\mathit{e}}_{2i}-{\mathit{k}}_{1i}\mathrm{si}{g}^{\beta }({\mathit{e}}_{1i}))\hfill \\ +2{\mathit{e}}_{2i}(-{\mathit{k}}_{2i}\mathrm{si}{g}^{2\beta -1}({\mathit{e}}_{1i})-\mathit{g}(t))\hfill \\ -{\mathit{k}}_{1i}\mathrm{si}{g}^{\beta }({\mathit{e}}_{1i})(-{\mathit{k}}_{2i}\mathrm{si}{g}^{2\beta -1}({\mathit{e}}_{1i})-\mathit{g}(t))\hfill \\ -{\mathit{k}}_{1i}\beta {\mathit{e}}_{2i}{\left|{\mathit{e}}_{1i}\right|}^{\beta -1}({\mathit{e}}_{2i}-{\mathit{k}}_{1i}\mathrm{si}{g}^{\beta }({\mathit{e}}_{1i}))\hfill \\ \multicolumn{1}{c}{=-{\left|{\mathit{e}}_{1i}\right|}^{\beta -1}[({\mathit{k}}_{1i}{\mathit{k}}_{2i}+{\mathit{k}}_{1i}^2\beta ){\left|{\mathit{e}}_{1i}\right|}^{2\beta }-2{\mathit{k}}_{1i}^2\beta {\mathit{e}}_{2i}\mathrm{si}{g}^{\beta }({\mathit{e}}_{1i})+{\mathit{k}}_{1i}\beta {\mathit{e}}_{2i}^2]}\\ -2\mathit{g}(t){\mathit{e}}_{2i}+{\mathit{k}}_{1i}\mathrm{si}{g}^{\beta }({\mathit{e}}_{1i})\mathit{g}(t)\hfill \end{array}$$

(58)

With the definitions in equation (54), after some algebraic manipulation, we have

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3=-{\left|{\mathit{e}}_{1i}\right|}^{\beta -1}{\ell }_i^{\mathrm{T}}\mathbf{\Theta }{\ell }_{\mathbf{i}}\mathbf{+}\mathit{g}\mathbf{(}\mathbf{t}\mathbf{)}{\mathbf{\Psi }}^{\mathrm{T}}{\ell }_{\mathbf{i}}}\\ \mathbf{\le }-{\mathbf{\left|}{\mathit{e}}_{1\mathbf{i}}\mathbf{\right|}}^{\mathbf{\beta }-1}{\mathbf{\lambda }}_{\mathbf{min}}\mathbf{(}\mathbf{\Theta }\mathbf{)}\mathbf{\Vert }{\ell }_{\mathbf{i}}{\mathbf{\Vert }}^2\mathbf{+}{\mathbf{\delta }}_{2\mathbf{i}}\mathbf{\Vert }\mathbf{\Psi }\mathbf{\Vert }\mathbf{\Vert }{\ell }_{\mathbf{i}}\mathbf{\Vert }\hfill \end{array}$$

(59)

Using ${\left|{\mathit{e}}_{1i}\right|}^{\beta -1}\ge \Vert {\ell }_i{\Vert }^{\frac{\beta -1}{\beta }}$ , we obtain

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3\le -{\lambda }_{\min }(\mathbf{\Theta }\mathbf{)}\mathbf{\Vert }{\ell }_{\mathbf{i}}{\mathbf{\Vert }}^{3-1\mathbf{/}\mathbf{\beta }}\mathbf{+}{\mathbf{\delta }}_{2\mathbf{i}}\mathbf{\Vert }\mathbf{\Psi }\mathbf{\Vert }\mathbf{\Vert }{\ell }_{\mathbf{i}}\mathbf{\Vert }}\\ \mathbf{=}-\mathbf{(}{\mathbf{\lambda }}_{\mathbf{min}}\mathbf{(}\mathbf{\Theta }\mathbf{)}{\mathbf{\Vert }{\ell }_{\mathbf{i}}\mathbf{\Vert }}^{2-1\mathbf{/}\mathbf{\beta }}-{\mathbf{\delta }}_{2\mathbf{i}}\mathbf{\Vert }\mathbf{\Psi }\mathbf{\Vert }\mathbf{)}\mathbf{\Vert }{\ell }_{\mathbf{i}}\mathbf{\Vert }\hfill \end{array}$$

(60)

From equation (57), we know that $\Vert {\ell }_i{\Vert }^2\ge V_3/{\lambda }_{\max }(\mathit{P})$ , then we have

$${\stackrel{·}{V}}_3\le -({\lambda }_{\min }(\mathbf{\Theta }){\Vert {\ell }_i\Vert }^{2-1/\beta }-{\mathbf{\delta }}_{2\mathbf{i}}\Vert \mathbf{\Psi }\Vert )\frac{{(V_3)}^{1/2}}{\sqrt{{\lambda }_{\max }(\mathit{P})}}$$

(61)

Let $a={\lambda }_{\min }(\mathbf{\Theta }\mathbf{)}\mathbf{\Vert }{\ell }_{\mathbf{i}}{\mathbf{\Vert }}^{2-1\mathbf{/}\mathbf{\beta }}-{\mathbf{\delta }}_{2\mathbf{i}}\mathbf{\Vert }\mathbf{\Psi }\mathbf{\Vert }$ , then equation (61) can be expressed as

$${\stackrel{·}{V}}_3\le -\frac{a}{\sqrt{{\lambda }_{\max }(\mathit{P})}}{\left(V_3\right)}^{1/2}$$

(62)

Then, using Lemma 2, we know that $\Vert {\ell }_i\Vert$ will approach to the region ${\sigma }_i$ in finite time.

Based on the disturbance estimates of ESO, the control law equation (41) can be modified as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{\Gamma }}_c=-\mathit{\Xi }({\mathit{C}}_1\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathit{V}}+\alpha {\mathit{C}}_2\mathrm{si}{g}^{\alpha -1}(\tilde{\mathbf{\eta }})\mathit{G}(\tilde{\mathbf{\eta }})\tilde{\mathit{V}})}\\ -{\mathit{\Gamma }}_{\mathrm{g}}-\mathit{\Xi }\mathit{H}-{\mathit{K}}_1\mathrm{sgn}(\mathit{S})-{\mathit{K}}_2\mathit{S}-\mathit{\Xi }{\mathit{Z}}_2\hfill \end{array}$$

(63)

Remark 6

The stability proof process of the modified control law equation (63) is the same as equations (43)–(45), which is omitted here. But, we should notice that the constraint for ${\mathit{K}}_1$ in equation (42) has been changed to

$${\mathit{K}}_{1i}>\left|{\left(\mathit{\Xi }{\mathit{e}}_2\right)}_i\right|,i=1,2,\dots ,6$$

(64)

It is important to mention that from equation (53), if we select suitable ESO gains ${\mathit{k}}_1\mathrm{and}{\mathit{k}}_2$ , the region ${\sigma }_i$ can be small enough, which means the ESO designed in this article is effective to guarantee the output ${\mathit{Z}}_2$ converge to $\Delta \tilde{\mathit{d}}$ in finite time with high accuracy. That is to say, the follower trajectory can converge to the reference state with respect to the leader in finite time with a small ${\mathit{K}}_1$ , and combined with the saturation function in equation (48), the chattering phenomenon can be eliminated better with high control accuracy.

The overall structure of the LF system with the proposed ESO-based controller in this research can be illustrated in Figure 2. The trajectory tracking errors $\tilde{\eta }$ and $\tilde{\mathit{V}}$ between the leader and the follower, and the controller output ${\mathit{\Gamma }}_{\mathbf{c}}$ are used to update the ESO, then the output ${\mathit{Z}}_2$ is applied to form the modified controller equation (63). Combining the analysis in the proof of Theorems 1 and 2, if the following conditions:

1. ${\mathit{K}}_{1i}>\left|{(\mathit{\Xi }{\mathit{e}}_2\mathbf{)}}_i\right|,i=1,2,\dots ,6$ ;

2. ${\mathit{k}}_{1i}>0,{\mathit{k}}_{2i}>0$ , and $\beta \in (0.5,1)$ ;

3. Choosing suitable ${\mathit{k}}_{1i},{\mathit{k}}_{2i},\mathrm{and}\beta$ to guarantee that $({\delta }_{2i}\Vert \mathbf{\Psi }\Vert /{\lambda }_{min}(\mathbf{\Theta }))\ll 1$ and $(\beta /(2\beta -1))$ are sufficiently large.

are satisfied and then we can conclude that the proposed ESO-based controller in this article has strong robustness and disturbance-rejection ability to make our control objective come true in poor environment conditions.

Figure 2.

Structure of the ESO-based control system.

Simulation results

In this section, to verify the effectiveness and performance of the proposed ESO in equation (51) and the trajectory tracking control scheme in equation (63), numerical simulations are presented for a LF system with the leader specified to follow a circle orbit with a 400-km orbit altitude, and its body-fixed frame is always perfectly aligned with the orbit frame. The initial configuration and velocity of the virtual leader are expressed in $\{I\}$ and $\{B\}$ , respectively

$$\begin{array}{l}{\chi }_l=\left[\begin{array}{cccc}-0.7660& -0.6428& 0& -5268.96\\ 0.4545& -0.5417& -0.7071& 3126.25\\ 0.4545& -0.5416& 0.7071& 3126.25\\ 0& 0& 0& 1\end{array}\right]\\ V_l={\left[0,0,0.001107,0,7.612607,0\right]}^{\mathrm{T}}\end{array}$$

(65)

where the displacements are in kilometers, the translational velocities are in kilometers per second, and the angular velocities are in radians per second.

In the simulations, the nominal parts of the mass and inertia matrix of the leader and the follower are assumed to be identical and are chosen the same as ³

$$m=100\mathrm{kg}\mathrm{and}\mathit{J}=\left[\begin{array}{ccc}22\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 20\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 23\hfill \end{array}\right]\mathrm{kg}{\mathrm{m}}^2$$

(66)

Thus, we have $\mathit{\Xi }\mathbf{=}\mathrm{diag}\mathbf{(}22\mathbf{,}20\mathbf{,}23\mathbf{,}100\mathbf{,}100\mathbf{,}100\mathbf{)}$ , and the model uncertainties are chosen as $\Delta \mathit{\Xi }\mathbf{=}\mathrm{diag}\mathbf{(}0\mathbf{.}15\mathit{J}\mathbf{,}0\mathbf{.}03\mathbf{m}\mathit{I}\mathbf{)}$ .

The formation flying mission is the desired relative position of the follower spacecraft with respect to the leader $([-5,0,0{]}^{\mathrm{T}}\mathrm{m})$ and is expressed in the leader’s body-fixed frame while synchronizing the leader’s attitude, the translational velocity, and the angular velocity.

Initially, the follower is assumed to be performing a different flying task, with its initial states which are given as follows: the relative position of the follower in the body-fixed frame of the leader is $[-15,5,5{]}^{\mathrm{T}}\mathrm{m}$ and the initial orientation of the follower spacecraft is assumed to be different from the initial orientation of the leader through [3,1,3] with Euler angles $[\pi /4,\pi /4,\pi /4]$ . The translational velocity and angular velocity expressed in its body-fixed frame are

$$\begin{array}{c}{\mathbf{\omega }}_1(0)=[0.00076,0.000743,0.00031{]}^{\mathrm{T}}\mathrm{rad}/\mathrm{s}\hfill \\ {\mathit{v}}_1(0)=[3.6254,-0.93764,-6.62791{]}^{\mathrm{T}}\mathrm{km}/\mathrm{s}\hfill \end{array}$$

(67)

The time-varying bounded disturbances ${\mathit{\Gamma }}_d$ are assumed to be

$${\mathit{\Gamma }}_d=\left[\begin{array}{c}0.005\sin (0.2t+1)\mathrm{N}\mathrm{m}\hfill \\ \multicolumn{1}{c}{0.005\cos (0.25t+1)\mathrm{N}\mathrm{m}}\\ \multicolumn{1}{c}{-0.005\sin (0.15t+1)\mathrm{N}\mathrm{m}}\\ \multicolumn{1}{c}{0.05\sin (0.1t+1)\mathrm{N}}\\ \multicolumn{1}{c}{0.05\cos (0.15t+1)\mathrm{N}}\\ \multicolumn{1}{c}{-0.05\sin (0.2t+1)\mathrm{N}}\end{array}\right]$$

(68)

In order to make our control objective come true in finite time with fine performance, the parameters of the observer equation (51) and the controller equation (63) are presented in Table 1. The numerical simulations are performed using the MATLAB software over $150\mathrm{s}$ with $0.1\mathrm{s}$ step size. With the application of the proposed control schemes, simulation results are presented in Figures 3–10.

Table 1.

Control parameters for numerical simulation.

Control schemes	Parameter values
ESO	$\beta =2/3$
	${\mathit{k}}_1=\mathrm{diag}(0.2,0.2,0.2,0.5,0.5,0.5)$
	${\mathit{k}}_2=\mathrm{diag}(0.04,0.04,0.04,0.06,0.06,0.06)$
	$\mathit{k}=\mathrm{diag}(0.02,0.02,0.02,0.04,0.04,0.04)$
Controller	$\alpha =2/3$ ; ${\tau }_{\max }=0.8Nm$ ; $u_{\max }=5N$
	${\mathit{C}}_1=\mathrm{diag}(0.1,0.1,0.1,0.02,0.02,0.02)$
	${\mathit{C}}_2=\mathrm{diag}(0.1,0.1,0.1,0.05,0.05,0.05)$
	${\mathit{K}}_1=\mathrm{diag}(5,5,5,10,10,10)$
	${\mathit{K}}_2=\mathrm{diag}(0.2,0.2,0.2,0.4,0.4,0.4)$

ESO: extended state observer.

Figure 3.

Time response of the observer errors e ₁ .

Figure 4.

Time response of the observer errors e ₂ .

Figure 5.

Time response of the attitude tracking errors with respect to the leader.

Figure 6.

Time response of the angular velocity tracking errors.

Figure 7.

Time response of the position tracking errors with respect to desired relative position.

Figure 8.

Time response of the translational velocity tracking errors.

Figure 9.

Control torques.

Figure 10.

Control forces.

Figures 3 and 4 show time response of the observer errors obtained from the ESO in equation (51). ${\mathit{e}}_{1i}\mathrm{and}{\mathit{e}}_{2i},i=1,2,3$ are the observer errors of the rotational motion parts and ${\mathit{e}}_{1i}\mathrm{and}{\mathit{e}}_{2i},i=4,5,6$ are the observer errors of the translational motion parts, with the upper rows of Figures 3 and 4 represent the initial response and the lower rows represent the steady-state behavior of the observer errors. We can see that the presented ESO has estimated the actual composed system disturbances with good performance. The finite-time convergence of ${\mathit{e}}_1$ and ${\mathit{e}}_2$ achieved no more than $15\mathrm{s}$ . Moreover, high accuracy of the observer for ${\mathit{e}}_1\mathrm{and}{\mathit{e}}_2$ was guaranteed with $\left|{\mathit{e}}_{1i}\right|<5\times {10}^{-6},\left|{\mathit{e}}_{2i}\right|<5\times {10}^{-5},i=1,2,3$ and $\left|{\mathit{e}}_{1i}\right|<3\times {10}^{-6},\left|{\mathit{e}}_{2i}\right|<1.5\times {10}^{-4},i=4,5,6$ . These results verify the analysis in Remark 6 that by adjusting suitable observer gains, the observed states ${\mathit{Z}}_2$ can precisely estimate the actual states $\Delta \tilde{\mathit{d}}$ in finite time with high accuracy.

Figure 5 shows the time response of the attitude tracking errors in terms of the exponential coordinates with respect to the leader, and the angular velocity tracking errors with respect to the leader expressed in the follower’s body-fixed frame are shown in Figure 6. It can be seen that under the bounded control torques in Figure 9, the follower can track the leader’s attitude and angular velocity with a settling time less than 25 s as shown in the upper parts of these figures. Since the lower parts of these figures present the steady response, we know that the desired high tracking accuracy is achieved with the attitude tracking errors less than $2\times {10}^{-5}(\mathrm{deg})$ and the angular velocity tracking errors less than $5\times {10}^{-4}(\mathrm{deg}/\sec )$ .

Figures 7 and 8 show the time response of the relative position and translational velocity tracking errors of the follower with respect to the leader, with position tracking errors expressed by the exponential coordinates and velocity tracking errors expressed in the follower’s body-fixed frame. It can been seen that under the bounded control forces in Figure 10, the tracking errors fall to the tolerance within 100 s with the position tracking errors in steady-state less than $0.015\mathrm{m}$ and velocity tracking errors less than $3.5\times {10}^{-3}\mathrm{m}/\sec$ .

Figures 9 and 10 show the control torques and forces acting on the follower, respectively. Initially, large control torques and forces are required to drive the follower to the desired configuration with respect to the leader quickly. Once the consensus has been achieved, only less control energy is required to compensate the total composed disturbances and then the ideal relative configuration will be guaranteed. In these two figures, we see that the large chattering due to the use of the signum function was eliminated using the modified controller equation (63).

By the above analysis of the simulation results in Figures 3–10, we conclude that the proposed extended observer-based finite-time controller equation (63) with the control parameters in Table 1 can guarantee the ESO errors converge to a really small region with zero included, and the follower spacecraft tracks the ideal relative configuration with respect to the leader in finite time in the presence of external disturbances and model uncertainties, while meeting the constraints on the actuator output. To the best of the authors’ knowledge, it’s the first study to develop a finite-time controller based on the 6-DOF model described by Lie group SE(3) considering all the above drawbacks. While, compared with the existing literature, the chattering phenomenon has been eliminated in the novel ESO, which means the control scheme proposed in this research provides the theoretical basis for spacecraft formation control system design in practice.

Conclusion

In this article, we solved the dynamical synchronization tracking control problem of a LF system, based on the 6-DOF relative-coupled dynamics derived on Lie group SE(3) in the presence of model uncertainties, external disturbances, and input saturation. A FTSM surface was designed based on the derived relative-coupled dynamics, and then a robust control scheme was proposed to guarantee the follower tracks the relative configuration with respect to the leader in finite time. To eliminate the chattering phenomenon and reduce the conservation of the controller, a modified robust control law incorporated with the ESO was proposed. The finite-time stability of all the proposed schemes was guaranteed by Lyapunov methods. The simulation results with fine performance illustrated the effectiveness of the proposed controllers. However, communication delays are not considered in this article, based on the ideas in Wei et al.,^29–31 developing the relative dynamics model via Markov jump systems and extending the result to the model will be the authors’ future work.