Attitude Synchronization of Rigid Spacecraft with Inertia Uncertainties and Environmental Disturbances

Abstract

The problem of attitude synchronization for a group of rigid spacecraft is investigated in this chapter under the general directed communication topology. Combining the strategies of finite-time control, fast terminal sliding mode (FTSM) control, and adaptive control, a novel decentralized finite-time control law is proposed in the presence of inertia uncertainties and environmental disturbances. The new control scheme ensures that each spacecraft can attain the desired time-varying attitude and angular velocity in finite time while maintaining attitude synchronization with other spacecraft in the formation. The feasibility of the control algorithm is investigated by an illustrative simulation example.

1 Introduction

Various techniques and results have been proposed for the attitude synchronization of rigid spacecraft, which can be classified as leader–follower [80, 255], virtual structure [119, 278], behavior-based [1, 120], and graph-theoretical approach [86, 236]. Especially, the graph-theoretical approach has been introduced to study the cooperative control of multi-agent system using limited local interaction [205] and also has been applied to attitude synchronization [236], but most of the results assumed that the communication links of the interspacecraft are undirected, i.e., bidirectional. In the actual application, the communication topology of the interspacecraft may be directed, such as in unidirectional satellite laser communication system. Furthermore, comparing with the undirected communication topology, the control problem of attitude synchronization under directed communication topology is more challenging.

The key feature of the finite-time control is to control the system states to the equilibrium in finite time and to keep them there then after, which is arisen in time-optimal control. In recent years, the finite-time control has drawn an increasing attention, numerous finite-time controllers have been investigated for a variety of systems [17, 209], especially the nonlinear systems [17, 247]. It would be sufficient for many engineering applications to achieve asymptotic/exponential stability, and for some very demanding applications, such as fast response, high tracking precision, and disturbance-rejection properties [194, 265], the finite-time stability can offer an effective alternative way. Therefore, the finite-time control for attitude synchronization and tracking has more practical application value.

In this chapter, a novel decentralized finite-time control law is proposed to ensure that each spacecraft can attain the desired time-varying attitude and angular velocity in finite time while maintaining attitude synchronization with other spacecraft in the formation, even in the presence of model uncertainties and external disturbances.

2 Problem Statement

2.1 Spacecraft Attitude Kinematics and Dynamics

In this chapter, the spacecraft is assumed to be a rigid body, and the rigid spacecraft attitude tracking error dynamics is described as follows [29]:

$$\begin{aligned} {{J}_{i}}{{\dot{\tilde{\omega }}}_{i}}= & {} -\omega _{i}^{\times }{{J}_{i}}{{\omega }_{i}}+{{J}_{i}}\left( \tilde{\omega }_{i}^{\times }R_i\omega _{i}^{d}-R_i\dot{\omega }_{i}^{d} \right) +{{u}_{i}}+{{d }_{i}} \end{aligned}$$

(13.1)

$$\begin{aligned} {{\dot{q}}_{i}}= & {} \frac{1}{2}\left( q _{i}^{\times }+{{q }_{0,i}}I \right) {{\tilde{\omega }}_{i}} \end{aligned}$$

(13.2)

$$\begin{aligned} {{\dot{q}}_{0,i}}= & {} -\frac{1}{2}{{q}_{i}}^{T}{{\tilde{\omega }}_{i}},\ \ \ \ \ i=1,\dots , n, \end{aligned}$$

(13.3)

where superscript i denotes the ith spacecraft in the formation, ${{J}_{i}}=J_{i}^{T}$ denotes a positive definite inertia matrix of the ith spacecraft. ${{\omega }_{i}}\in \mathbb {R}^{3}$ denotes the body angular velocity of the ith spacecraft with respect to an inertial frame $\mathcal {I}$. Let $\omega _{i}^{d}=\omega _{i}^{d} \left( t \right) \in \mathbb {R}^{3}$ be the desired angular velocity of the ith spacecraft of frame $\mathcal {D}$ with respect to frame $\mathcal {I}$, which is time-varying. ${{\tilde{\omega }}_{i}}={{\omega }_{i}}-R_i\omega _{i}^{d}$, $i=1,\dots , n$ denotes the angular velocity error of the ith spacecraft. $R_i=R_i\left( {{q }_{i}}\, {{q }_{0,i}}\right) \in SO\left( 3 \right)$ is the corresponding rotation matrix, it is given by $R_i=\left( q _{0,i}^{2}-q _{i}^{T}{{q }_{i}} \right) I+2{{q }_{i}}q _{i}^{T}-2{{q }_{0,i}}q _{i}^{\times }$. $\left( {{q }_{i}}\, {{q }_{0,i}}\right) \in {\mathbb {R}}^{3}\times \mathbb {R}$ denotes the error quaternion representing the ith spacecraft’s orientation of frame $\mathcal {B}$ with respect to frame $\mathcal {D}$. $I\in {\mathbb {R}^{3\times 3}}$ denotes the identity matrix. $u_i\in \mathbb {R}^{3}$ denotes the control torque. ${{d }_{i}}\in {\mathbb {R}^{3}}$ is the disturbance torque.

The following assumptions are made for the further analysis in Sect. 13.3.

Assumption 13.1

There exist two known constants ${{\bar{\omega }}_{1}}>0$ and ${{\bar{\omega }}_{2}}>0$ such that $\left| \omega _{i}^{d}\left( t \right) \right| \le {{\bar{\omega }}_{1}}$ and $\left| \dot{\omega }_{i}^{d}\left( t \right) \right| \le {{\bar{\omega }}_{2}}$ for all $t\ge 0$.

Assumption 13.2

Let ${{J}_{i}}={{\bar{J}}_{i}}+{{\tilde{J}}_{i}}$, where ${{\bar{J}}_{i}}$ and ${{\tilde{J}}_{i}}$ are the nominal part and uncertain part of the inertia matrix of the ith spacecraft, respectively. The inertia matrix uncertainty ${{\tilde{J}}_{i}}$ is assumed to be bounded and satisfy $\left\| {{{\tilde{J}}}_{i}} \right\| \le {{\alpha }_{i, 0}}$, where ${{\alpha }_{i, 0}}$ is an unknown nonnegative constant.

Assumption 13.3

All the environmental disturbances due to gravitation, solar radiation pressure, magnetic forces, and aerodynamic drag are assumed to be bounded. Thus, the external disturbances ${{d }_{i}}$ are assumed to satisfy $\left\| {{d }_{i}} \right\| \le {{\alpha }_{i}}$, where ${{\alpha }_{i}}$ are unknown nonnegative constants.

Assumption 13.4

([29]) The control law of each spacecraft may use angular velocity errors and error quaternions of its neighboring spacecraft in the cooperative attitude control problem, and error quaternion is bounded from its definition. Thus, the control torque ${{u}_{i}}$ is assumed to satisfy

$$\left\| {{u}_{i}}\right\| \le {{\zeta }_{i, 0}}+{{\zeta }_{i, 1}} \sum \limits _{j\in {{N}_{i}}}{{{\left\| {{{\tilde{\omega }}}_{j}} \right\| }_{1}}}+{{\zeta }_{i, 2}}\sum \limits _{j\in {{N}_{i}}}{\left\| {{{\tilde{\omega }}}_{j}} \right\| _{1}^{2}},$$

where ${{\zeta }_{i, j}}\left( i=1,\ldots n, j=0,1,2 \right)$ are unknown nonnegative constants, ${{N}_{i}}$ represents the ith spacecraft and all the spacecrafts with which the ith spacecraft can communicate.

2.2 Lemmas of Algebraic Graph Theory

Suppose ${{G}_{n}}$ is a directed graph, it consists of a finite set of vertices V and a set of arcs ${A}\subset {{{V}}^{2}}$, where ${A}=\left\{ \left( \alpha ,\beta \right) \left| \alpha ,\beta \in {V} \right. \right\}$, in our research, the arc $\left( \alpha ,\beta \right)$ denotes that spacecraft $\beta$ can obtain the information of spacecraft $\alpha$. It is assumed that the graph has no self-loops, i.e., $\left( \alpha ,\beta \right) \in {A}$ implies $\alpha \ne \beta$. Let A be the adjacency matrix of ${{G}_{n}}$.

Lemma 13.1

([185]) For a directed graph ${{G}_{n}}$ with N vertices, all the eigenvalues of the weighted Laplacian L have a nonnegative real part (follows from Gershgorin’s theorem).

Lemma 13.2

([287]) Consider the error system Eqs. (13.1)–(13.3) for sliding surface $\sigma _i={{\tilde{\omega }}_{i}}+c_1{{q }_{i}}+c_2{{q }_{i}}^r$, where $0<r<1$, $c_1>0$, $c_2>0$, for $i=1,\dots , n$. If $\sigma _i=0$, then ${\tilde{\omega }}_{i}=0$, ${q }_{0,i}=1$ and ${q }_{i}=0$ can be reached in finite time, respectively.

2.3 FTSM Surface Design

Using the information of quaternion, angular velocity, and the angular velocity error and error quaternion of the neighboring spacecraft, and motivated by the work of [17, 29, 287], a novel multispacecraft FTSM is designed as follows:

$$\begin{aligned}&S={{\left[ {{s}_{1}},\ldots ,{{s}_{n}} \right] }^{T}}, \end{aligned}$$

(13.4)

where ${{s}_{i}}\in {\mathbb {R}^{3\times 1}}$, and it is given by

$$\begin{aligned} {{s}_{i}}= & {} {{b}_{i}}\bar{J}_{i}\left( {{{\tilde{\omega }}}_{i}}+\kappa _1{{q }_{i}}+\kappa _2{\alpha _{i}({q }_{i}})\right) \nonumber \\&+\sum \limits _{j=1,j\ne i}^{n}{{{a}_{i, j}}\left[ \left( \bar{J}_{i}{{{\tilde{\omega }}}_{i}}-\bar{J}_{j}{{{\tilde{\omega }}}_{j}} \right) +\left( \kappa _1\bar{J}_{i}{{q }_{i}}-\kappa _1\bar{J}_{j}{{q }_{j}} \right) \right. }\nonumber \\&\left. +\left( \kappa _2\bar{J}_{i}{\alpha _{i}({q }_{i})}-\kappa _2{\bar{J}_{j}\alpha _{j}({q }_{j})}\right) \right] \end{aligned}$$

(13.5)

with $\alpha _{i}({q }_{i})=[\alpha _{i, 1}({q }_{i, 1}),\alpha _{i, 2}({q }_{i, 2}),\alpha _{i, 3}({q }_{i, 3})]^{T}\in {{{\mathbb {R}}}}^{3\times 1}$,

$$\begin{aligned} \alpha _{i,j}({q }_{i,j})=\left\{ \begin{array}{ccc} \text{ sig }^r({q }_{i,j}),\,mathrm{{if}}\ \bar{s}_{i, j}=0\ \mathrm {{or}}\ \bar{s}_{i, j}\ne 0,\ \left| {q }_{i, j}\right| >\phi \\ l_1{q }_{i, j}+l_2\text{ sig }^2({q }_{i,j}),\ \mathrm {{if}}\ \bar{s}_{i, j}\ne 0,\ \left| {q }_{i, j}\right| \le \phi \\ \end{array} \right. \end{aligned}$$

$i=1,\dots , n$, $j=1,2,3$, $\bar{s}_i=[\bar{s}_{i, 1},\bar{s}_{i, 2},\bar{s}_{i, 3}]^{T}$ and

$$\begin{aligned} {\bar{s}_{i}}= & {} {{{\tilde{\omega }}}_{i}}+\kappa _1{{q }_{i}}+\kappa _2{\text{ sig }^r({q }_{i}}), \end{aligned}$$

where $\kappa _1$ and $\kappa _2$ are positive constants, $r\in \left( 0,1\right)$, $l_1=(2-r)\phi ^{r-1}$, $l_2=(r-1)\phi ^{r-2}$, $\phi$ denotes a small positive constant. For this kind of functions, it follows from [217] that $\frac{d\text {sig}^{\alpha }\left( x\right) }{dx}=\alpha \left| x\right| ^{\alpha -1}$. Define $\text {sig}^{r}\left( q_{i}\right) \!=[\text{ sig }^{r}\left( q_{i, 1}\right) ,\text{ sig }^{r}\left( q_{i, 2}\right) ,\text{ sig }^{r}\left( q_{i, 3}\right) ]^{T}$, $\text{ sig }^{2}\left( q_{i}\right) =[\text{ sig }^{2}\left( q_{i, 1}\right) ,\text {sig}^{2} \left( q_{i, 2}\right) ,\text{ sig }^{2}\left( q_{i, 3}\right) ]^{T}$. Scalar ${{b}_{i}}>0$ is the control weight parameter for attitude tracking which is used to keep the ith spacecraft’s station behavior, scalar ${{a}_{i, j}}\ge 0$ is the control weight parameter for interspacecraft attitude synchronization between the ith and jth spacecraft which is used to keep the formation behavior.

Using the Kronecker product, we can rewrite the sliding mode vector Eq. (13.4) as follows:

$$\begin{aligned} S=\left[ \left( L+B \right) \otimes {{I}_{3}} \right] \bar{J}\left( \tilde{\varOmega }+{\kappa _1}q+{\kappa _2}\alpha (q)\right) , \end{aligned}$$

(13.6)

where L is the weighted Laplacian matrix which is respected to the interspacecraft directed communication topology, $B=\text{ diag }\{ {{b}_{1}},\ldots ,{{b}_{n}} \}$, $\bar{J}=\text{ diag }\{ \bar{J}_{1},\dots ,\bar{J}_{n} \}$, $\tilde{\varOmega }={{\left[ {{{\tilde{\omega }}}_{1}},\ldots ,{{{\tilde{\omega }}}_{n}} \right] }^{T}}$, $q ={{\left[ {{q }_{1}},\ldots ,{{q }_{n}} \right] }^{T}}$ and $\alpha (q)=[\alpha _1(q_1),\ \dots , \alpha _n(q_n)]^{T}$.

Remark 13.3

From Lemma 13.1, the third result in Lemma 1.7, and the definition of B, it follows that $\left( L+B \right) \otimes {{I}_{3}}$ has full rank, $\bar{J}$ has full rank. Consequently, we obtain that $[\left( L+B \right) \otimes {{I}_{3}}]\bar{J}$ has full rank. Thus, if the sliding mode surface $S=0$ reached, then $\tilde{\varOmega }+{\kappa _1}q+{\kappa _2}\alpha (q)=0$, i.e., ${{\tilde{\omega }}_{i}}+\kappa _1{{q }_{i}}+\kappa _2\alpha _{i}\left( q_i\right) =0\left( i=1,\ldots , n \right)$ will be satisfied.

2.4 Control Objective

In this chapter, the control objective is to design a decentralized finite-time control law such that the states of the closed-loop system (13.1)–(13.3) can reach the sliding mode surface (13.4) in finite time. Furthermore, the angular velocity errors ${\tilde{\omega }}_{i}$ and the error quaternions ${q }_{i}$ $(i=1,\dots , n)$ can converge to small regions in finite time, respectively.

3 Decentralized Finite-Time Control Law Design

In order to develop the control law, the following equations are derived from Eqs. (13.1)–(13.2)

$$\begin{aligned}&\bar{{{J}}_{i}}\left( {{{\dot{\tilde{\omega }}}}_{i}}+\kappa _1{{{\dot{q }}}_{i}}+\kappa _2\dot{\alpha }_i\left( q_i\right) \right) \nonumber \\&={{{z}}_{i}}\left( t \right) +\delta _{i}+{{u}_{i}}\ \ i=1,\ldots , n \end{aligned}$$

(13.7)

with

$$\begin{aligned} {{{z}}_{i}}\left( t \right)= & {} -\omega _{i}^{\times }{{\bar{J}}_{i}}{{\omega }_{i}}+{{\bar{J}}_{i}}\left( \tilde{\omega }_{i}^{\times }R_i\omega _{i}^{d}-R_i\dot{\omega }_{i}^{d} \right) \nonumber \\&+\frac{1}{2}\kappa _1{{\bar{J}}_{i}}\left( q _{i}^{\times }+{{q }_{i, 0}}I \right) {{\tilde{\omega }}_{i}}+\kappa _2{{\bar{J}}_{i}}\dot{\alpha }_i\left( q_i\right) \end{aligned}$$

(13.8)

$$\begin{aligned} \dot{\alpha }_{i}\left( q_{i}\right)= & {} \left\{ \begin{array}{l} r\text{ diag }\left( \left| {q }_{i, j}\right| ^{r-1}\right) {\dot{q }}_{i},\ \mathrm {{if}}\ \bar{s}_{i, j}=0\ \mathrm {{or}}\ \bar{s}_{i, j}\ne 0,\ \left| {q }_{i, j}\right| >\phi \\ l_1{\dot{q }}_{i}+2l_2{q }_{i}\text{ sgn }({q }_{i}){\dot{q }}_{i}, \mathrm {{if}}\ \bar{s}_{i, j}\ne 0,\ \left| {q }_{i, j}\right| \le \phi \\ \end{array} \right. \end{aligned}$$

(13.9)

for $i=1,\dots , n,\ j=1,2,3$.

$$\begin{aligned} {{\delta }_{i}}\left( t \right)= & {} {{d}_{i}}-{{\tilde{J}}_{i}}{{\dot{\tilde{\omega }}}_{i}}-\omega _{i}^{\times }{{\tilde{J}}_{i}}{{\omega }_{i}}+{{\tilde{J}}_{i}}\left( \tilde{\omega }_{i}^{\times }R_i\omega _{i}^{d}-R_i\dot{\omega }_{i}^{d} \right) . \end{aligned}$$

Under Assumptions 13.1–13.4, it can be shown that

$$\begin{aligned} {{\left\| \left( L+B \right) \otimes {{I}_{3}} \right\| }_{1}}{{\left\| {{\delta }_{i}} \right\| }_{1}}\le & {} {{\theta }_{i, 1}}+{{\theta }_{i, 2}}\sum \limits _{j\in {{N}_{i}}}{{{\left\| {{{\tilde{\omega }}}_{j}} \right\| }_{1}}}\nonumber \\&+\,{{\theta }_{i, 3}}\sum \limits _{j\in {{N}_{i}}}{\left\| {{{\tilde{\omega }}}_{j}} \right\| _{1}^{2}} \end{aligned}$$

(13.10)

where ${{\theta }_{i, 1}}$, ${{\theta }_{i, 2}}$, and ${{\theta }_{i, 3}}$ are nonnegative constant numbers. Let ${{\hat{\theta }}_{i, 1}}$, ${{\hat{\theta }}_{i, 2}}$, and ${{\hat{\theta }}_{i, 3}}$ denote the estimates of ${{\theta }_{i, 1}}$, ${{\theta }_{i, 2}}$, and ${{\theta }_{i, 3}}$, respectively. Let the adaptive upper bound of the norm ${{\left\| \left( L+B \right) \otimes {{I}_{3}} \right\| }_{1}}{{\left\| {{\delta }_{i}} \right\| }_{1}}$ be

$$\begin{aligned}&{{\hat{\delta }}_{i}}= {{\hat{\theta }}_{i, 1}}+{{\hat{\theta }}_{i, 2}}\sum \limits _{j\in {{N}_{i}}}{{{\left\| {{{\tilde{\omega }}}_{j}} \right\| }_{1}}}+{{\hat{\theta }}_{i, 3}}\sum \limits _{j\in {{N}_{i}}}{\left\| {{{\tilde{\omega }}}_{j}} \right\| _{1}^{2}} \end{aligned}$$

(13.11)

with $i=1,\ldots , n$. Then, the parameter adaptation errors can be written as ${{\tilde{\theta }}_{i, 1}}={{\hat{\theta }}_{i, 1}}-{{\theta }_{i, 1}}$, ${{\tilde{\theta }}_{i, 2}}={{\hat{\theta }}_{i, 2}}-{{\theta }_{i, 2}}$, and ${{\tilde{\theta }}_{i, 3}}={{\hat{\theta }}_{i, 3}}-{{\theta }_{i, 3}}$.

Theorem 13.4

Consider the spacecraft formation attitude tracking dynamics described by Eqs. (13.1)–(13.3) satisfied Assumptions 13.1–13.4. Suppose that the decentralized finite-time control law is designed as

$$\begin{aligned} {{u}_{i}}= & {} -{{z}_{i}}+{{\left( \sum \limits _{j=1,j\ne i}^{n}{{{a}_{i, j}}}+{{b}_{i}} \right) }^{-1}}\left[ \sum \limits _{j=1,j\ne i}^{n}{{{a}_{i, j}}}\left( {{u}_{j}}+{{z}_{j}} \right) \right. \nonumber \\&\left. -H_i{s_i}-M_i{s_{i}^{r}}-{{K}_{i}}\text{ sgn }\left( {{s}_{i}} \right) -v_{i}\right] , \end{aligned}$$

(13.12)

where $v_{i}={\hat{\delta }_{i}}\text{ sgn } \left( {{s}_{i}} \right)$, and the adaptation laws are chosen as

$$\begin{aligned}&{{\dot{\hat{\theta }}}_{i, 1}}={{\gamma }_{i, 1}}{{\left\| {{s}_{i}} \right\| }_{1}} \end{aligned}$$

(13.13)

$$\begin{aligned}&{{\dot{\hat{\theta }}}_{i, 2}}={{\gamma }_{i, 2}}{{\left\| {{s}_{i}} \right\| }_{1}}\sum \limits _{j\in {{N}_{i}}}{{{\left\| {{{\tilde{\omega }}}_{j}} \right\| }_{1}}}\end{aligned}$$

(13.14)

$$\begin{aligned}&{{\dot{\hat{\theta }}}_{i, 3}}={{\gamma }_{i, 3}}{{\left\| {{s}_{i}} \right\| }_{1}}\sum \limits _{j\in {{N}_{i}}}{\left\| {{{\tilde{\omega }}}_{j}} \right\| _{1}^{2}}. \end{aligned}$$

(13.15)

Then, the trajectory of the closed-loop system in Eqs.(13.1)–(13.3) can be driven onto the multispacecraft sliding mode surface $S=0$ in finite time. Furthermore, the angular velocity errors ${\tilde{\omega }}_{i}$ and the error quaternions ${q }_{i}$ $(i=1,\dots , n)$ will converge to small regions in finite time, respectively, where ${{H}_{i}}$, ${{M}_{i}}$, ${{K}_{i}}\in {{{R}}^{3\times 3}}$ are positive definite matrixes, $\gamma _{i, j}$ $(i=1,\dots ,n, j=1,2,3)$ are positive constants, $s^{r}_{i}=[s^{r}_{i, 1},s^{r}_{i, 2},s^{r}_{i, 3}]^{T}$.

Proof

Define the candidate Lyapunov function as follows:

$$\begin{aligned}&V={{V}_{1}}+{{V}_{2}} \end{aligned}$$

(13.16)

with

$${{V}_{1}}=\frac{1}{2}{{S}^{T}}S$$

$${{V}_{2}}=\frac{1}{2}\sum \limits _{i=1}^{n}{\left( \gamma _{i, 1}^{-1}\tilde{\theta }_{i, 1}^{2}+\gamma _{i, 2}^{-1}\tilde{\theta }_{i, 2}^{2}+\gamma _{i, 3}^{-1}\tilde{\theta }_{i, 3}^{2} \right) .}$$

Using Eqs. (13.14)–(13.17), it leads to the derivative of $V_1$ as

$$\begin{aligned} {{\dot{V}}_{1}}= & {} {{S}^{T}}\dot{S}\nonumber \\= & {} {{S}^{T}}\left[ \left( L+B \right) \otimes {{I}_{3}} \right] \left( \bar{J} \dot{\tilde{\varOmega }}+\kappa _{1}\bar{J}\dot{q } +\kappa _{2}\bar{J}\dot{\alpha }\left( q\right) \right) \nonumber \\= & {} {{S}^{T}}\left[ \left( L+B \right) \otimes {{I}_{3}} \right] \left( Z+\delta +U \right) , \end{aligned}$$

(13.17)

where $\dot{\tilde{\varOmega }}={{\left[ \dot{\tilde{\omega }}_{1},\dots ,\dot{\tilde{\omega }}_{n} \right] }^{T}}$, $\dot{q} ={{\left[ {{\dot{q} }_{1}},\ldots ,{{\dot{q }}_{n}} \right] }^{T}}$, $\dot{\alpha }(q)=[\dot{\alpha }_1(q_1),\dots ,\dot{\alpha }_n(q_n)]^{T}$, $\bar{J}=\text{ diag }\{\bar{J}_{1},\dots ,\bar{J}_{n}\}\in \mathbb {R}^{3n\times 3n}$, $Z=\left[ z_{1},\dots , z_{n}\right] ^{T}$, $\delta ={{\left[ {{\delta }_{1}},\ldots ,{{\delta }_{n}} \right] }^{T}}$ and

$$\begin{aligned} U= & {} -Z-{{\left\{ {{I}_{3n}}-\left[ {{\left( D+B \right) }^{-1}}\otimes {{I}_{3}} \right] \left( A\otimes {{I}_{3}} \right) \right\} }^{-1}}\times \nonumber \\&\left[ {{\left( D+B \right) }^{-1}}\otimes {{I}_{3}} \right] \left[ HS+MS^{r}+ K\text{ sgn } \left( S \right) \right. \nonumber \\&\left. +\,\hat{\delta }\text{ sgn } \left( S \right) \right] \end{aligned}$$

(13.18)

$$\begin{aligned}= & {} -Z-{{\left[ \left( D+B \right) \otimes {{I}_{3}} \right] }^{-1}}\nonumber \\&\times \,\left[ HS+MS^{r}+ K\text{ sgn } \left( S \right) +\hat{\delta }\text{ sgn } \left( S \right) \right] , \end{aligned}$$

(13.19)

where $H=\text{ diag }\{H_1,\dots , H_n\}$, $M=\text{ diag }\{M_1,\dots , M_n\}$, $K=\text{ diag }\{K_1,\dots , K_n\}$, $\hat{\delta }=\text{ diag }\{\hat{\delta }_1\otimes I_{3},\dots ,\hat{\delta }_n\otimes I_{3}\}$.

Substituting Eq. (13.18) into Eq. (13.17), it leads to

$$\begin{aligned} {{\dot{V}}_{1}}= & {} {{S}^{T}}\left[ \left( L+B \right) \otimes {{I}_{3}} \right] \delta -{{S}^{T}}HS-{{S}^{T}}MS^{r}\\&-{{S}^{T}}K \text{ sgn } \left( S \right) -{{S}^{T}}\hat{\delta }\text{ sgn } \left( S \right) \\\le & {} \sum \limits _{i=1}^{n}{{{\left\| \left( L+B \right) \otimes {{I}_{3}} \right\| }_{1}}\cdot {{\left\| {{\delta }_{i}} \right\| }_{1}}\cdot {{\left\| {{s}_{i}} \right\| }_{1}}}\\&-{{S}^{T}}HS-{{S}^{T}}MS^{r}-k\sum \limits _{i=1}^{n}{{{\left\| {{s}_{i}} \right\| }_{1}}}-\sum \limits _{i=1}^{n}{{{{\hat{\delta }}}_{i}}{{\left\| {{s}_{i}} \right\| }_{1}}}\\\le & {} -\sum \limits _{i=1}^{n}{ {{{\tilde{\theta }}}_{i, 1}}{\left\| {{s}_{i}} \right\| }_{1}-\sum \limits _{i=1}^{n}\sum \limits _{j\in {{N}_{i}}}{{{\tilde{\theta }}}_{i, 2}}{{{\left\| {{{\tilde{\omega }}}_{j}} \right\| }_{1}}}{\left\| {{s}_{i}} \right\| }_{1}}-k\sum \limits _{i=1}^{n}{{{\left\| {{s}_{i}} \right\| }_{1}}}\\&-{\sum \limits _{i=1}^{n}\sum \limits _{j\in {{N}_{i}}}{{{\tilde{\theta }}}_{i, 3}}{\left\| {{{\tilde{\omega }}}_{j}} \right\| _{1}^{2}} {{\left\| {{s}_{i}} \right\| }_{1}}}-{{S}^{T}}HS-{{S}^{T}}MS^{r}, \end{aligned}$$

where $k=\min \{{{k}_{1}},\ldots ,{{k}_{n}}\}$ $(i=1,\dots , n)$ and $k_{i}$ is the minimum eigenvalue of ${{K}_{i}}$.

Using Eqs. (13.13)–(13.20), it obtains the derivative of ${{V}_{2}}$ as follows:

$$\begin{aligned} {{\dot{V}}_{2}}= & {} \sum \limits _{i=1}^{n}{\left( \gamma _{i, 1}^{-1}{{{\tilde{\theta }}}_{i, 1}}{{{\dot{\tilde{\theta }}}}_{i, 1}}+\gamma _{i, 2}^{-1}{{{\tilde{\theta }}}_{i, 2}}{{{\dot{\tilde{\theta }}}}_{i, 2}}+\gamma _{i, 3}^{-1}{{{\tilde{\theta }}}_{i, 3}}{{{\dot{\tilde{\theta }}}}_{i, 3}} \right) }\\= & {} \sum \limits _{i=1}^{n}{\left( {{{\tilde{\theta }}}_{i, 1}}+{{{\tilde{\theta }}}_{i, 2}}\sum \limits _{j\in {{N}_{i}}}{{{\left\| {{{\tilde{\omega }}}_{j}} \right\| }_{1}}}+{{{\tilde{\theta }}}_{i, 3}}\sum \limits _{j\in {{N}_{i}}}{\left\| {{{\tilde{\omega }}}_{j}} \right\| _{1}^{2}} \right) {{\left\| {{s}_{i}} \right\| }_{1}}}. \end{aligned}$$

Then, we get the derivative of V as

$$\begin{aligned} \dot{V}= & {} {{\dot{V}}_{1}}+{{\dot{V}}_{2}}\nonumber \\\le & {} -{{S}^{T}}HS-{{S}^{T}}MS^{r}-k\sum \limits _{i=1}^{n}{{{\left\| {{s}_{i}} \right\| }_{1}}}\le 0. \end{aligned}$$

(13.20)

From the discussion above, we get $S\in \mathcal {L}^{\infty }$, $\tilde{\theta }_{i, j}\in \mathcal {L}^{\infty }$. Consequently, $u_{i}\in \mathcal {L}^{\infty }$. It follows that $\dot{\tilde{\omega }}_{i}$, $\dot{q}_{i}$, and hence $\dot{s}_{i}$ are all bounded. Integrating $\dot{V}$ leads the results that ${s}_{i}\in \mathcal {L}^{2}$. Hence, using the corollary of Barbalat’s lemma, it gets that $\lim \limits _{t\rightarrow \infty }s_i(t)=0$. From the definition of S, it obtains that $\lim \limits _{t\rightarrow \infty }\tilde{\omega }_{i}=0$, $\lim \limits _{t\rightarrow \infty }q_{i}=0$. Denote $\epsilon _{i}={ |{{{\tilde{\theta }}}_{i, 1}}|+|{{{\tilde{\theta }}}_{i, 2}}|\sum \limits _{j\in {{N}_{i}}}{{{\left\| {{{\tilde{\omega }}}_{j}} \right\| }_{1}}}+|{{{\tilde{\theta }}}_{i, 3}}|\sum \limits _{j\in {{N}_{i}}}{\left\| {{{\tilde{\omega }}}_{j}} \right\| _{1}^{2}} }$, we have $\epsilon _{i}\in \mathcal {L}^{\infty }$. Let $\epsilon =\max \left\{ \epsilon _{1},\dots ,\epsilon _{n}\right\}$, then there exist a positive constant $\epsilon _{0}$, such that $\epsilon _{0}\ge \epsilon$. Choose the appropriate $K_{i}$, such that the minimum eigenvalue $k_{i}>\epsilon _{0}$.

Then invoking Lemma 1.6, it is not difficult to obtain that

$$\begin{aligned} \dot{V}_{1}\le & {} -\mu _1\left( \frac{1}{2}S^{T}S\right) -\mu _2\left( \frac{1}{2}\sum \limits _{i=1}^{n}{\left\| {{s}_{i}} \right\| _{1}^{2}}\right) ^{(1+r)/2}\nonumber \\\le & {} -\mu _1 {{V_1}}-\mu _2 {{V_1}^{(1+r)/2}}, \end{aligned}$$

(13.21)

where $\mu _1=2h$, $\mu _2 =2^{(1+r)/2}m$, $h=\min \{{{h}_{1}},\ldots ,{{h}_{n}}\}$, $m=\min \{{{m}_{1}},\ldots ,{{m}_{n}}\}$, $k=\min \{{{k}_{1}},\ldots ,{{k}_{n}}\}$ $(i=1,\dots , n)$, $h_{i}$, $m_{i}$ and $k_{i}$ are the minimum eigenvalues of ${{H}_{i}}$, ${M_{i}}$ and ${{K}_{i}}$, respectively. From Eq. (13.21) and Lemma 1.4, it concludes that the sliding manifold S converges to 0 in finite time

$$\begin{aligned} T_{r}\le \frac{2}{\mu _{1}}\ln {\frac{\mu _{1}V_{1}(0)^{(1+r)/2}+\mu _{2}}{\mu _{2}}}, \end{aligned}$$

where $V_{1}(0)$ is the initial value of $V_{1}$.

When $S=0$ is reached, three cases should be considered as follows.

$Case\ 1$ If $\overline{s}_{i, j}=0\ (i=1,\dots ,n, j=1,2,3)$ is achieved, then we obtain

$$\begin{aligned} {{\tilde{\omega }}_{i, j}}+\kappa _1{{q }_{i, j}}+\kappa _2\text{ sig }^{r}(q_{i, j})=0. \end{aligned}$$

(13.22)

By Lemma 13.2, we can obtain that $\tilde{\omega }_{i, j}\rightarrow 0$, $q_{i, j}\rightarrow 0$ in finite time, i.e., the closed-loop system Eqs. (13.1)–(13.3) can achieve finite-time stable.

$Case\ 2$ If $\overline{s}_{i, j}\ne 0$ and $\left| {q }_{i, j}\right| \le \phi \ (i=1,\dots ,n, j=1,2,3)$, which implies that ${q }_{i, j}$ has converged to the region $\left| {q }_{i, j}\right| \le \phi$ in finite time, then from ${s}_{i, j}=0$ and Remark 2.1, we have

$$\begin{aligned} {{\tilde{\omega }}_{i, j}}+\kappa _1{{q }_{i, j}}+\kappa _2\left( l_1{q }_{i, j}+l_2\text{ sig }^2({q }_{i, j})\right) =0. \end{aligned}$$

(13.23)

Therefore, $\tilde{\omega }_{i, j}$ will converge to the region

$$\begin{aligned} |{{\tilde{\omega }}_{i, j}}|\le & {} \kappa _1\left| {{q }_{i, j}}\right| +\kappa _2\left| l_1{q }_{i, j}+l_2\text{ sig }^2({q }_{i, j})\right| \nonumber \\\le & {} \kappa _1\phi +\kappa _2\phi ^{r} \end{aligned}$$

(13.24)

in finite time.

$Case\ 3$ If $\overline{s}_{i, j}\ne 0$ and $\left| {q }_{i, j}\right| >\phi \ (i=1,\dots ,n, j=1,2,3)$, then we obtain ${s}_{i, j}\ne 0$. From ${s}_{i, j}=0$, we can see that this case will not occur.

From the above discussion, we conclude that ${q }_{i, j}$ and $\tilde{\omega }_{i, j}$, $i=1,\dots , n,\ j=1,2,3$ will converge to the regions

$$\begin{aligned} \left| {q }_{i,j}\right|\le & {} \phi \\ |{{\tilde{\omega }}_{i, j}}|\le & {} \kappa _1\phi +\kappa _2\phi ^{r} \end{aligned}$$

in finite time, i.e., $\Vert {q }_{i}\Vert _{1}\le 3\phi$, and $\Vert {\tilde{\omega }}_{i}\Vert _{1}\le 3\kappa _1\phi +3\kappa _2\phi ^{r}$, for $i=1,\dots , n$. This completes the proof.

Remark 13.5

The proposed control law is discontinuous across the surface, which will lead to control chattering. We can remedy this situation by smoothing out the control discontinuity in a thin boundary layer [110]. Though the boundary layer leads to small terminal tracking error, the practical advantages may be significant. The modified decentralized finite-time control law is given by

${{u}_{i}}=-{{z}_{i}}+{{\left( \sum \limits _{j=1,j\ne i}^{n}{{{a}_{i, j}}}+{{b}_{i}} \right) }^{-1}}\times$

$$\begin{aligned} \left[ \sum \limits _{j=1,j\ne i}^{n}{{{a}_{i, j}}}\left( {{u}_{j}}+{{z}_{j}} \right) -H_i{s'_i}-M_i{s'^{r}_i}-{{K}_{i}}\text{ sat }\left( {{s}_{i}} \right) -v'_{i}\right] , \end{aligned}$$

(13.25)

where $v'_{i}={\hat{\delta }}_{i}\text{ sat }\left( {{s}_{i}} \right)$, and the adaptive laws in Eqs. (13.13)–(13.20) can be modified as

$$\begin{aligned}&{{\dot{\hat{\theta }}}_{i, 1}}={{\gamma }_{i, 1}}{{\left\| {{{{s}'}}_{i}} \right\| }_{1}}\end{aligned}$$

(13.26)

$$\begin{aligned}&{{\dot{\hat{\theta }}}_{i, 2}}={{\gamma }_{i, 2}}{{\left\| {{{{s}'}}_{i}} \right\| }_{1}}\sum \limits _{j\in {{N}_{i}}}{{{\left\| {{{\tilde{\omega }}}_{j}} \right\| }_{1}}}\end{aligned}$$

(13.27)

$$\begin{aligned}&{{\dot{\hat{\theta }}}_{i, 3}}={{\gamma }_{i, 3}}{{\left\| {{{{s}'}}_{i}} \right\| }_{1}}\sum \limits _{j\in {{N}_{i}}}{\left\| {{{\tilde{\omega }}}_{j}} \right\| _{1}^{2}} , \end{aligned}$$

(13.28)

where $S'=[{{s}'}_{1},\dots ,{{s}'}_{n}]^{T}$, ${{{s}'}_{i}}={{\left[ \begin{matrix} {{{{s}'}}_{i, 1}},{{{{s}'}}_{i, 2}},{{{{s}'}}_{i, 3}} \\ \end{matrix} \right] }^{T}}$, ${{{s}'}_{i,j}}={{s}_{i,j}}-\varphi _{i,j} \text{ sat }\left( {{s}_{i, j}} \right)$, $\varphi _{i, j}>0$ is the boundary-layer thickness, for $i=1,\ldots , n$, $j=1,2,3$, and

$$\begin{aligned}&\text{ sat }\left( {{s}_{i, j}} \right) =\left\{ \begin{array}{ccc} &{}1,\ \ \ &{}\mathrm {{if}}\ \frac{{{s}_{i,j}}}{\varphi _{i, j}}\ge 1\\ &{}\frac{{{s}_{i,j}}}{\varphi _{i, j}},\ \ \ &{}\mathrm {{if}}\ -1<\frac{{{s}_{i,j}}}{\varphi _{i, j}}<1\\ &{}-1,\ \ \ &{}\mathrm {{if}}\ \frac{{{s}_{i,j}}}{\varphi _{i, j}}\le 1.\\ \end{array} \right. \end{aligned}$$

From [110], the convergence to the boundary layer can be easily shown.

4 Illustrative Examples

In this section, a ring topology is considered for circular-like formation as shown in Fig. 13.1.

Give the corresponding weighted Laplacian matrix as

$$\begin{aligned} {{L}}=\left[ \begin{array}{cccc} 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 1 &{} -1 &{} 0 \\ 0 &{} 0 &{} 1 &{} -1 \\ -1 &{} 0 &{} 0 &{} 1 \\ \end{array} \right] . \end{aligned}$$

Fig. 13.1

Interspacecraft directed communication topology

The actual inertia matrices are assumed to be [29] (with unit expressed in $kg.{{m}^{2}}$)

$$\begin{aligned}&{{J}_{1}}=\left[ \begin{array}{ccc} 20 &{} 2 &{} 0.9 \\ 2 &{} 17 &{} 0.5 \\ 0.9 &{} 0.5 &{} 15 \\ \end{array} \right] ,\ \ \ {{J}_{2}}=\left[ \begin{array}{ccc} 22 &{} 1 &{} 0.9 \\ 1 &{} 19 &{} 0.5 \\ 0.9 &{} 0.5 &{} 15 \\ \end{array} \right] \\&{{J}_{3}}=\left[ \begin{array}{ccc} 18 &{} 1 &{} 1.5 \\ 1 &{} 15 &{} 0.5 \\ 1.5 &{} 0.5 &{} 17 \\ \end{array} \right] ,\ \ \ {{J}_{4}}=\left[ \begin{array}{ccc} 18 &{} 1 &{} 1 \\ 1 &{} 20 &{} 0.5 \\ 1 &{} 0.5 &{} 15 \\ \end{array} \right] .\\ \end{aligned}$$

With the existence of model uncertainties and external disturbances, the nominal inertia matrices of the spacecraft are given by

${{\bar{J}}_{1}}={{\bar{J}}_{2}}={{\bar{J}}_{3}}={{\bar{J}}_{4}}=\text{ diag }\left( {{\left[ \begin{array}{ccc} 20 &{} 20 &{} 20 \\ \end{array} \right] }^{T}} \right) kg.{{m}^{2}}$

The sinusoidal-wave disturbances are introduced as follows (with unit expressed in N.m):

$$\begin{aligned} d_1\left( t\right)= & {} \left[ 0.03\sin \left( 0.4t\right) , 0.06\cos \left( 0.5t\right) , 0.09\cos \left( 0.7t\right) \right] ^{T}\\ d_2\left( t\right)= & {} \left[ 0.07\cos \left( 0.4t\right) , 0.11\sin \left( 0.5t\right) , 0.08\sin \left( 0.7t\right) \right] ^{T}\\ d_3\left( t\right)= & {} \left[ 0.09\sin \left( 0.4t+\pi /4\right) , 0.07\cos \left( 0.5t+\pi /4\right) ,\right. \\&\left. 0.10\cos \left( 0.7t+\pi /4\right) \right] ^{T}\\ d_4\left( t\right)= & {} \left[ 0.08\cos \left( 0.4t+\pi /4\right) , 0.09\cos \left( 0.5t+\pi /4\right) ,\right. \\&\left. 0.12\sin \left( 0.7t+\pi /4\right) \right] ^{T}. \end{aligned}$$

Choose the initial angular velocity errors of all spacecraft to be zeros, and the initial attitude tracking errors are given as ${{q }_{1}}\left( 0 \right) ={{\left[ \begin{array}{cccc} 0.8276 &{} 0.5 &{} -0.2 &{} 0.3 \\ \end{array} \right] }^{T}}$, ${{q }_{2}}\left( 0 \right) ={{\left[ \begin{array}{cccc} 0.8918 &{} -0.3 &{} 0.4 &{} 0.5 \\ \end{array} \right] }^{T}}$, ${{q }_{3}}\left( 0 \right) ={{\left[ \begin{array}{cccc} 0.8352 &{} 0.3 &{} -0.2 &{} 0.4 \\ \end{array} \right] }^{T}}$, ${{q }_{4}}\left( 0 \right) ={{\left[ \begin{array}{cccc} 0.8806 &{} -0.3 &{} -0.1 &{} 0.2 \\ \end{array} \right] }^{T}}$.

The initial desired quaternion is given by ${{\xi }_{i}}\left( 0 \right) ={{\left[ \begin{array}{cccc} 1 &{} 0 &{} 0 &{} 0 \\ \end{array} \right] }^{T}}$, $i=1,2,3,4$.

Assume that the time-varying desired angular velocities of the spacecraft are identical and given them as follows: $\omega _{i}^{d}\left( t \right) \!=\!\left[ \begin{array}{cccc} 0.1\cos \left( \frac{t}{10}\right) &{} -0.1\sin \left( \frac{t}{10}\right) &{}\! -0.1\cos \left( \frac{t}{10}\right) \\ \end{array}\right] ^{T}$.

The controller parameters are chosen with ${\kappa }_{1}{=}1$, ${\kappa }_{2}{=}0.4$, ${{b}_{i}}=1$, ${{a}_{1,2}}={{a}_{2,3}}={{a}_{3,4}}=a_{4,1}=1$, $r=0.6$, $\varphi _{i, j} =0.13$, ${{H}_{i}}{=}{{0.1I}_{3}}$, ${{M}_{i}}{=}{{I}_{3}}$, ${{K}_{i}}{=}{{6I}_{3}}$, and the parameters of the adaptation laws in Eqs. (13.26)–(13.28) are chosen with ${{\gamma _{i, j}}=0.1}$ for $i=1,2,3,4,j=1,2,3$. The initial values of ${{\hat{\theta }}_{i, j}}$ are given by $\hat{\theta }_{i, j}^{0}=0.1$.

Fig. 13.2

Control torques

Figure 13.2 shows the control torque $u_{i}$, $i=1,2,3,4$ of the spacecraft 1–4. The adaptive parameter $\hat{\delta }_{i}$, $i=1,2,3,4$, which is defined in (13.16), is bounded as shown in Fig. 13.3, and thus the efficacy of the adaptation laws in Eqs. (13.26)–(32) are verified. The sliding surfaces $s_i$ converge to the boundary layer in 2 s as shown in Fig. 13.4. Figure 13.5 shows the attitude errors $q_{i}$, $i=1,2,3,4$ of the spacecraft 1–4 using the control law (13.25). For ease of interpretation, attitude errors are expressed in Euler angles converted from unit quaternion. The angular velocity errors $\tilde{\omega }_{i}$ of the spacecraft 1–4 are shown in Fig. 13.6. Figure 13.7 depicts the relative attitude errors between the first and second, the second and third, the third and fourth, and the fourth and first spacecraft, respectively.

Fig. 13.3

$\hat{\delta }_{1}$, $\hat{\delta }_{2}$, $\hat{\delta }_{3}$, $\hat{\delta }_{4}$

Fig. 13.4

Sliding surfaces

Fig. 13.5

Attitude tracking errors

Fig. 13.6

Angular velocity errors

Fig. 13.7

Relative attitude errors

As observed from the simulation results, the feasibility of the control algorithm presented in this chapter has been illustrated sufficiently.

5 Summary

In this chapter, combining the finite-time control, FTSM control, and adaptive control techniques, a novel decentralized finite-time control law is proposed to ensure that each spacecraft attains the desired time-varying attitude and angular velocity in finite time while maintaining attitude synchronization with other spacecraft in the formation. Simulation results illustrate the feasibility of this control algorithm. Further work includes extending the results in this chapter to cases when there exist communication delays between spacecrafts, and the angular velocity is not available.