2.1 Notations and lemmas

Define $\left\| \cdot \right\|$ as the induced norm of a matrix or the Euclidean norm of a vector. ${\boldsymbol{{I}}_n}$ denotes the $n \times n$ identity matrix. $ \otimes $ is defined as the Kronecker product. ${\lambda _{\min }}\!\left( \cdot \right)$ and ${\lambda _{\max }}\!\left( \cdot \right)$ represent the minimum and maximum eigenvalues of a symmetric matrix. The basic operations of the dual number, quaternion and dual quaternion are defined in the appendix. Given a vector $\boldsymbol{{x}} = {\left[ {{x_1},{x_2}, \ldots ,{x_n}} \right]^{\rm{T}}} \in {\mathbb{R}^n}$ and $\alpha \gt 0$ , we denote ${\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol{{x}}) = {\left[ {{\rm{sign}}\!\left( {{x_1}} \right){{\left| {{x_1}} \right|}^\alpha },{\rm{sign}}\!\left( {{x_2}} \right){{\left| {{x_2}} \right|}^\alpha }, \ldots ,{\rm{sign}}\!\left( {{x_n}} \right){{\left| {{x_n}} \right|}^\alpha }} \right]^{\rm{T}}}$ and $|\boldsymbol{{x}}| = {\left[ {\left| {{x_1}} \right|,\left| {{x_2}} \right|, \ldots ,\left| {{x_n}} \right|} \right]^{\rm{T}}}$ , where ${\rm{sign}}( \cdot )$ is the signum function. ${\rm{diag}}\!\left( {{x_1},{x_2}, \ldots ,{x_n}} \right)$ denotes a block-diagonal matrix. ${\rm{max}}\!\left( {{x_1},{x_2}, \ldots ,{x_n}} \right)$ and ${\rm{min}}\!\left( {{x_1},{x_2}, \ldots ,{x_n}} \right)$ denote the maximum and minimum value in $\left( {{x_1},{x_2}, \ldots ,{x_n}} \right)$ , respectively.

In order to get the main results of this paper, the following lemmas are introduced.

2.2 Mathematical model

The coordinate system used in this paper is shown in Fig. 1, which defines the Earth-centred inertial frame ${O_E}{X_I}{Y_I}{Z_I}$ and the jth spacecraft body frame ${O_{Bj}}{X_{Bj}}{Y_{Bj}}{Z_{Bj}}$ . ${O_E}$ is the centre of the Earth and ${O_{Bj}}$ is the centre of mass of the jth deputy.

Figure 1. Coordinate system.

Based on dual quaternion, the states of ith spacecraft can be described as ${\hat{\boldsymbol{{q}}}_i}$ and ${\hat{\boldsymbol\omega}_i}$ , where ${\hat{\boldsymbol{{q}}}_i} = {\boldsymbol{{q}}_i} + \varepsilon \frac{1}{2}{\boldsymbol{{q}}_i} \circ {\boldsymbol{{r}}_i}$ , ${\hat{\boldsymbol\omega}_i} = {\boldsymbol\omega _i} + \varepsilon\!\left( {{{\dot{\boldsymbol{{r}}}}_i} + {\boldsymbol\omega _i} \times {\boldsymbol{{r}}_i}} \right)$ , $\varepsilon $ is the dual unit that satisfies ${\varepsilon ^2} = 0,\varepsilon \ne 0$ . ${\boldsymbol{{q}}_i}$ is the quaternion describing the rotation of the ith spacecraft body frame relative to the inertial frame. ${\boldsymbol{{r}}_i} = {\left[ {0,{{\left( {{{\vec{\boldsymbol{{r}}}}_i}} \right)}^{\rm{T}}}} \right]^{\rm{T}}} \in {\mathbb{R}^4}$ , ${\vec{\boldsymbol{{r}}}_i} \in {\mathbb{R}^3}$ denotes the position of the ith spacecraft expressed in the spacecraft body frame. ${\boldsymbol\omega _i} = {\left[ {0,{{\left( {{{\vec{\boldsymbol\omega}}_i}} \right)}^{\rm{T}}}} \right]^{\rm{T}}} \in {\mathbb{R}^4}$ , ${\vec{\boldsymbol\omega}_i} \in {\mathbb{R}^3}$ is the angular velocity of the ith spacecraft body frame relative to the inertial frame expressed in the body frame. The dynamic equations of six-DOF motion are [Reference Wu, Liu and Han32]

(1) \begin{align}{\dot{\hat{\boldsymbol{{q}}}}_i} = \frac{1}{2}{\hat{\boldsymbol{{q}}}_i} \circ {\hat{\boldsymbol\omega}_i}\end{align}

(2) \begin{align}\dot{\hat{\boldsymbol\omega}}_{i} = M_{i}^{-1}{\ast}\!\left\{\hat{\boldsymbol{{F}}}_{ig} + \boldsymbol{{sat}}\!\left[ \boldsymbol{{Q}}\!\left( \hat{\boldsymbol{{F}}}_{ic} \right) \right] + \hat{\boldsymbol{{F}}}_{ip} - \hat{\boldsymbol\omega}_{i} \times \left( \boldsymbol{{M}}_{i}\,{\ast}\,\hat{\boldsymbol\omega}_{i} \right) \right\}\end{align}

where ${\hat{\boldsymbol{{F}}}_{ig}} = {\boldsymbol{{f}}_{ig}} + \varepsilon {\boldsymbol\tau _{ig}} = - \mu {m_i}{\boldsymbol{{r}}_i}/{\left\| {{\boldsymbol{{r}}_i}} \right\|^3} + \varepsilon 3\mu {\boldsymbol{{r}}_i} \times \left( {{{\overline{\boldsymbol{{J}}} }_i}{\boldsymbol{{r}}_i}} \right)/{\left\| {{\boldsymbol{{r}}_i}} \right\|^5}$ denotes the gravity dual force, ${\hat{\boldsymbol{{F}}}_{ic}} = {\boldsymbol{{f}}_{ic}} + \varepsilon {\boldsymbol\tau _{ic}}$ denotes the control dual force that is yet to be designed, $\boldsymbol{{sat}}\!\left[ {\boldsymbol{{Q}}\!\left( {{{\hat{\boldsymbol{{F}}}}_{ic}}} \right)} \right]$ denotes the control dual force with actuator saturation and input quantisation, ${\hat{\boldsymbol{{F}}}_{ip}} = {\boldsymbol{{f}}_{ip}} + \varepsilon {\boldsymbol\tau _{ip}}$ denotes the disturbing dual force. $\mu $ is the gravitational parameter of Earth. Define that ${m_i}$ and ${\boldsymbol{{J}}_i} \in {\mathbb{R}^{3 \times 3}}$ are the mass and inertia matrix of the ith deputy, respectively. The matrix ${\boldsymbol{{M}}_i}$ is

(3) \begin{align}{\boldsymbol{{M}}_i} = \left[ {\begin{array}{c@{\quad}c}{{{\bf{0}}_{4 \times 4}}} & {{{\overline{\boldsymbol{{m}}} }_i}}\\[5pt] {{{\overline{\boldsymbol{{J}}}}_i}} & {{{\bf{0}}_{4 \times 4}}}\end{array}} \right] \in {\mathbb{R}^{8 \times 8}}\end{align}

where

\begin{align*}{\bar{\boldsymbol{{J}}}_i} = \left[ {\begin{array}{c@{\quad}c}1 & {{{\bf{0}}_{1 \times 3}}}\\[5pt] {{{\bf{0}}_{3 \times 1}}} & {{\boldsymbol{{J}}_i}}\end{array}} \right] \in {\mathbb{R}^{4 \times 4}},{\overline{\boldsymbol{{m}}}_i} = \left[ {\begin{array}{c@{\quad}c}1 & {{{\bf{0}}_{1 \times 3}}}\\[5pt] {{{\bf{0}}_{3 \times 1}}} & {{m_i}{\boldsymbol{{I}}_{3 \times 3}}}\end{array}} \right] \in {\mathbb{R}^{4 \times 4}}\end{align*}

Equation (1) and Equation (2) can be rewritten as

(4) \begin{align}\left\{ {\begin{array}{l}{{{\dot{\boldsymbol{{x}}}}_i} = {\boldsymbol{{v}}_i}}\\[5pt] {{{\dot{\boldsymbol{{v}}}}_i} = {\boldsymbol{{f}}_i}\!\left( {{\boldsymbol{{x}}_i},{\boldsymbol{{v}}_i}} \right) + {\boldsymbol{{g}}_i}\!\left( {{\boldsymbol{{x}}_i}} \right)\boldsymbol{{sat}}\!\left[ {\boldsymbol{{Q}}\!\left( {{\boldsymbol{{u}}_i}} \right)} \right] + {\boldsymbol{{d}}_i}}\end{array}} \right.\end{align}

where ${\boldsymbol{{x}}_i} = {\hat{\boldsymbol{{q}}}_i} = {\left[ {\begin{array}{l}{{{\hat{\boldsymbol{{q}}}}_{ir}}^{\rm{T}}}\quad {{{\hat{\boldsymbol{{q}}}}_{id}}^{\rm{T}}}\end{array}} \right]^{\rm{T}}},{\boldsymbol{{v}}_i} = {\dot{\hat{\boldsymbol{{q}}}}_i} = {\left[ {\begin{array}{l}{{{\dot{\hat{\boldsymbol{{q}}}}}_{ir}}^{\rm{T}}}\quad {{{\dot{\hat{\boldsymbol{{q}}}}}_{id}}^{\rm{T}}}\end{array}} \right]^{\rm{T}}}$ , ${\boldsymbol{{f}}_i}\!\left( {{\boldsymbol{{x}}_i},{\boldsymbol{{v}}_i}} \right) = \dot{\boldsymbol\Gamma}\!\left( {{\boldsymbol{{x}}_i}} \right){\boldsymbol\Gamma ^{ - 1}}\!\left( {{\boldsymbol{{x}}_i}} \right){\boldsymbol{{v}}_i} - \boldsymbol\Gamma\!\left( {{\boldsymbol{{x}}_i}} \right)\boldsymbol{{M}}_i^{ - 1}\boldsymbol{{Z}} ({{\boldsymbol{{x}}_i}, {\boldsymbol{{v}}_i}}) \boldsymbol\Gamma\!\left( {{\boldsymbol{{x}}_i}} \right)\boldsymbol{{M}}_i^{ - 1}\boldsymbol{{G}}$ , ${\boldsymbol{{g}}_i}\!\left( {{\boldsymbol{{x}}_i}} \right) = {\boldsymbol\Gamma}\!\left( {{\boldsymbol{{x}}_i}} \right)\boldsymbol{{M}}_i^{ - 1}$ , ${\boldsymbol{{u}}_i} = {\left[ {\begin{array}{l}{{\boldsymbol{{f}}_{ic}}^{\rm{T}}}\quad {{\boldsymbol\tau _{ic}}^{\rm{T}}}\end{array}} \right]^{\rm{T}}}$ , ${\boldsymbol{{d}}_i} = {\boldsymbol{{g}}_i}\!\left( {{\boldsymbol{{x}}_i}} \right){\left[ {\begin{array}{l}{{\boldsymbol{{f}}_{ip}}^{\rm{T}}}\quad {{\boldsymbol\tau _{ip}}^{\rm{T}}}\end{array}} \right]^{\rm{T}}}$ , $\boldsymbol{{Z}}\!\left( {{\boldsymbol{{x}}_i},{\boldsymbol{{v}}_i}} \right) = \boldsymbol\Gamma \left[ {{\boldsymbol\Gamma ^{ - 1}}\!\left( {{\boldsymbol{{x}}_i}} \right){\boldsymbol{{v}}_i}} \right]{\boldsymbol{{M}}_i}{\boldsymbol\Gamma ^{ - 1}}\!\left( {{\boldsymbol{{x}}_i}} \right){\boldsymbol{{v}}_i} - \boldsymbol\Gamma \left\{ {{{\left[ {{\boldsymbol{{M}}_i}{\Gamma ^{ - 1}}\!\left( {{\boldsymbol{{x}}_i}} \right){\boldsymbol{{v}}_i}} \right]}^*}} \right\}{\left[ {{\boldsymbol\Gamma ^{ - 1}}\!\left( {{\boldsymbol{{x}}_i}} \right){\boldsymbol{{v}}_i}} \right]^*}$ , $G = {\left[ {\begin{array}{l}{{\boldsymbol{{f}}_{ig}}^{\rm{T}}}\quad {{\boldsymbol\tau _{ig}}^{\rm{T}}}\end{array}} \right]^{\rm{T}}}$ , the transformation matrix $\boldsymbol\Gamma\!\left( {{\boldsymbol{{x}}_i}} \right)$ is defined in Equation (63). ${\boldsymbol{{g}}_i}\!\left( {{\boldsymbol{{x}}_i}} \right)$ is invertible because $\boldsymbol\Gamma ({\boldsymbol{{x}}_i})$ and ${\boldsymbol{{M}}_i}$ are invertible.

The quantiser in Equation (4) is given as $\boldsymbol{{Q}}\!\left( {{\boldsymbol{{u}}_i}} \right) = {\left[ {Q\!\left( {{u_{i1}}} \right),Q\!\left( {{u_{i2}}} \right), \ldots ,Q\!\left( {{u_{i8}}} \right)} \right]^{\rm{T}}}$ . In our work, $Q\!\left( u \right)$ denotes the hysteretic quantiser, as shown in Equation (5). The subscript i and j are omitted for brevity.

(5) \begin{align}Q(u(t)) = \left\{ {\begin{array}{l@{\quad}l}{{u_p}{\rm{sign}}(u),} & {\dfrac{{{u_p}}}{{1 + \delta }} \lt |u| \le {u_p},\dot u \lt 0,{\rm{or}}}\\[10pt]{} & {{u_i} \lt |u| \le \dfrac{{{u_p}}}{{1 - \delta }},\dot u \gt 0}\\[10pt] {{u_p}(1 + \delta ){\rm{sign}}(u),} & {{u_i} \lt |u| \le \dfrac{{{u_p}}}{{1 - \delta }},\dot u \lt 0,{\rm{or}}}\\[10pt] {} & {\dfrac{{{u_i}}}{{1 - \delta }} \lt |u| \le \dfrac{{{u_i}(1 + \delta )}}{{(1 - \delta )}},\dot u \gt 0}\\[10pt] {0,} & {0 \le |u| \lt \dfrac{{{u_{\min }}}}{{1 + \delta }},\dot u \lt 0,{\rm{or}}}\\[10pt] {} & {\dfrac{{{u_{\min }}}}{{1 + \delta }} \le |u| \le {u_{\min }},\dot u \gt 0,}\\[14pt] {Q\!\left( {u\left( {{t^ - }} \right)} \right)} & {\dot u = 0}\end{array}} \right.\end{align}

where ${u_p} = {\rho ^{1 - p}}{u_{\min }}(p = 1,2, \ldots )$ , $\delta = (1 - \rho )/(1 + \rho )$ , ${u_{\min }} \gt 0$ , $0 \lt \rho \lt 1$ , and $Q(u(t)) \in \left\{ {0, \pm {u_p}, \pm {u_p}(1 + \delta ),p = 1,2, \ldots } \right\}$ . Besides, ${u_{\min }} \gt 0$ denotes the range of the dead-zone for $Q(u(t))$ , and $\rho \gt 0$ represents a measure of quantisation density. We decompose the quantiser Equation (5) into $Q\!\left( {{u_{ij}}} \right) = {\chi _{qij}}\!\left( {{u_{ij}}} \right){u_{ij}} + {d_{qij}},j = 1,2, \cdots ,8$ , where

\begin{align*} \begin{array}{l@{\quad}l}{{\chi _{qij}}\!\left( {{u_{ij}}} \right) = \left\{ {\begin{array}{l@{\quad}l}{\dfrac{{Q\!\left( {{u_{ij}}} \right)}}{{{u_{ij}}}},} & {Q\!\left( {{u_{ij}}} \right) \ne 0,}\\[10pt] {1,} & {Q\!\left( {{u_{ij}}} \right) = 0,}\end{array}} \right.}\\[22pt] {{d_{qij}}\!\left( {{u_{ij}}} \right) = \left\{ {\begin{array}{l@{\quad}l}{0,} & {q\!\left( {{u_{ij}}} \right) \ne 0,}\\[5pt] { - {u_{ij}},} & {Q\!\left( {{u_{ij}}} \right) = 0.}\end{array}} \right.}\end{array}\end{align*}

It can be easily proven that the control coefficient ${\chi _{qij}}$ and the disturbance-like term ${d_{qij}}$ satisfy $1 - \delta \le {\chi _{qij}} \le 1 + \delta ,\left| {{d_{qij}}} \right| \le {u_{\min }}$ . Thus, we can rewritten the quantiser as

(6) \begin{align}\boldsymbol{{Q}}\!\left( {{\boldsymbol{{u}}_i}} \right) = {\boldsymbol\chi _q}\!\left( {{\boldsymbol{{u}}_i}} \right){\boldsymbol{{u}}_i} + {\boldsymbol{{d}}_q}\end{align}

where ${\boldsymbol\chi _q}\!\left( {{\boldsymbol{{u}}_i}} \right) = {\rm{diag}}{\left[ {{\chi _{qi1}}\!\left( {{u_{i1}}} \right)\!,{\chi _{qi2}}\!\left( {{u_{i2}}} \right)\!, \cdots ,{\chi _{qi8}}\!\left( {{u_{i8}}} \right)} \right]^{\rm{T}}}$ , ${\boldsymbol{{d}}_q}\!\left( {{\boldsymbol{{u}}_i}} \right) = {\rm{diag}}\big[ {{d_{qi1}}\!\left( {{u_{i1}}} \right)}, {d_{qi2}}\!\left( {{u_{i2}}} \right), \cdots , {d_{qi8}} \left( {{u_{i8}}} \right)\big]^{\rm{T}}$ .

Define $[\underline M ,\overline M ]$ as the magnitude constraint of the actuator. The saturation nonlinearity in Equation (4) is given as

(7) \begin{align}\boldsymbol{{sat}}\!\left( \boldsymbol{{x}} \right) = {\boldsymbol\chi _s}\!\left( \boldsymbol{{x}} \right)\boldsymbol{{x}} = {\left[ {sat({x_1}),sat({x_2}), \cdots ,sat({x_k})} \right]^{\rm{T}}}\end{align}

where ${\boldsymbol\chi _s}\!\left( \boldsymbol{{x}} \right) = {\rm{diag}}{\left[ {{\chi _{s1}}\!\left( {{x_1}} \right)\!,{\chi _{s2}}\!\left( {{x_2}} \right)\!, \cdots ,{\chi _{sk}}\!\left( {{x_k}} \right)} \right]^{\rm{T}}}$ with

\begin{align*}{\chi _{sk}}\!\left( x \right) = \left\{ {\begin{array}{l@{\quad}l}{\overline{M} /x,} & {{\rm{if}}\;x \geq {{\overline{M}}_u}}\\[5pt] {1,} & {{\rm{if}}\;{{\underline{M}}_u} \le x \le {{\overline{M}}_u}}\\[5pt] {{{\underline{M}}_u}/x,} & {{\rm{if}}\;x \leq {{\underline{M}}_u}}\end{array}} \right.\end{align*}

such that $0 \lt {\chi _{sk}} \lt 1$ . We can rewritten the term $\boldsymbol{{sat}}\!\left[ {\boldsymbol{{Q}}\!\left( {{\boldsymbol{{u}}_i}} \right)} \right]$ in Equation (4) as

(8) \begin{align}\boldsymbol{{sat}}\!\left[ {\boldsymbol{{Q}}\!\left( {{\boldsymbol{{u}}_i}} \right)} \right] = \boldsymbol\chi\!\left( {{\boldsymbol{{u}}_i}} \right){\boldsymbol{{u}}_i} + {\boldsymbol{{d}}_{sq}}\end{align}

where $\boldsymbol\chi\!\left( {{\boldsymbol{{u}}_i}} \right) = {\boldsymbol\chi _s}\!\left( {{\boldsymbol{{u}}_i}} \right){\boldsymbol\chi _q}\!\left( {{\boldsymbol{{u}}_i}} \right),{\boldsymbol{{d}}_{sq}} = {\boldsymbol\chi _s}{\boldsymbol{{d}}_q}$ .

2.3 Graph theory

We use an undirected graph $\boldsymbol{{G}} = \left\{ {\upsilon ,\boldsymbol\varsigma ,\boldsymbol{{A}}} \right\}$ to describe the communication topology of a formation with $n$ spacecrafts, where the node set $\upsilon = \left\{ {{\upsilon _1},{\upsilon _2}, \cdots ,{\upsilon _n}} \right\}$ , the edge set $\varsigma \subseteq \upsilon \times \upsilon $ , and the adjacency matrix $\boldsymbol{{A}} = \left[ {{a_{ij}}} \right] \in {\mathbb{R}^{n \times n}}(i = 1,2, \cdots ,n;\;j = 1,2, \cdots ,n)$ . If node ${\upsilon _i}$ can directly obtain the information of node ${\upsilon _j}$ , there is an edge in the graph from ${\upsilon _j}$ points to ${\upsilon _i}$ , denoted as $({\upsilon _i},{\upsilon _j}) \in \boldsymbol\varsigma $ , and ${\upsilon _j}$ is the child node of ${\upsilon _i}$ . For an undirected graph, if an edge connects ${\upsilon _i}$ and ${\upsilon _j}$ , then the two are parent-child nodes, that is, if $({\upsilon _i},{\upsilon _j}) \in \boldsymbol\varsigma $ , then $({\upsilon _j},{\upsilon _i}) \in \boldsymbol\varsigma $ . In the adjacency matrix $\boldsymbol{{A}}$ , ${a_{ij}} \gt 0$ if $({\upsilon _i},{\upsilon _j}) \in \boldsymbol\varsigma $ ; otherwise, ${a_{ij}} = 0$ . Generally, it is assumed that ${a_{ii}} = 0$ . The Laplace matrix $\boldsymbol{{L}}$ of graph $\boldsymbol{{G}}$ is defined as $\boldsymbol{{L}} = {\rm{diag}}\!\left\{ {\sum\limits_{j = 1}^n {a_{1j}}, \cdots ,\sum\limits_{j = 1}^n {a_{nj}}} \right\} - \boldsymbol{{A}}$ . For undirected graphs, the Laplace matrix $\boldsymbol{{L}}$ is symmetric.

Additionally, denote the virtual leader as spacecraft 0, of which states are given by ${\boldsymbol{{x}}_0}$ and ${\boldsymbol{{v}}_0}$ . A leader-following graph $\overline{\boldsymbol{{G}}}$ contains the virtual leader and the original graph $\boldsymbol{{G}}$ of n follower spacecraft. ${b_i} = 1$ if the ith follower spacecraft can access the virtual leader spacecraft directly, and ${b_i} = 0$ , otherwise. The graph $\overline{\boldsymbol{{G}}}$ is connected if there exists a path in $\overline{\boldsymbol{{G}}}$ from the leader node ${\upsilon _0}$ to every node ${\upsilon _i}$ . If $\overline{\boldsymbol{{G}}} $ is connected, the matrix $\boldsymbol{{L}} + \boldsymbol{{B}}$ associated with $\overline{\boldsymbol{{G}}}$ is symmetric and positive definite [Reference Hong, Hu and Gao35], where $\boldsymbol{{B}} = {\rm{diag}}({b_1}, \cdots ,{b_2})$ .

2.4 Problem statement

This paper committed to developing a distributed control scheme such that all follower spacecraft can track the virtual leader with the desired formation shape considering limited communication, i.e., $\mathop {{\rm{lim}}}\limits_{t \to T} \left\| {{\boldsymbol{{x}}_i} - {\boldsymbol\Delta _{i,0}} - {\boldsymbol{{x}}_0}} \right\| \le {o_1},\mathop {{\rm{lim}}}\limits_{t \to T} \left\| {{\boldsymbol{{v}}_i} - {{\dot{\boldsymbol\Delta} }_{i,0}} - {\boldsymbol{{v}}_0}} \right\| \le {o_2}$ , where ${\boldsymbol\Delta _{i,0}}$ is the ith follower spacecraft’s expected deviation formation vector relative to the virtual leader, and ${o_1},{o_2}$ are compact sets around zero.

In order to get the main results of this paper, the following assumptions are adopted.

3.1 Event-triggered fixed-time observer design

This subsection develops the event-triggered fixed-time observer with continuous and without continuous communications to reconstruct the virtual leader spacecraft’s information.

Define a auxiliary variable as ${\boldsymbol\zeta _i} = \sum\nolimits_{j = 1}^n {a_{ij}}\!\left( {{{\hat{\boldsymbol{{v}}}}_{i,0}} - {{\hat{\boldsymbol{{v}}}}_{j,0}}} \right) + {b_i}\!\left( {{{\hat{\boldsymbol{{v}}}}_{i,0}} - {\boldsymbol{{v}}_0}} \right)$ where ${\hat{\boldsymbol{{v}}}_{i,0}}$ denotes the estimation of ${\boldsymbol{{v}}_0}$ of the ith follower spacecraft, ${\widetilde{\boldsymbol{{v}}}_i} = {\hat{\boldsymbol{{v}}}_{i,0}} - {\boldsymbol{{v}}_0}$ is the estimation error, $\widetilde{\boldsymbol{{v}}} = {\left[ {{{\widetilde{\boldsymbol{{v}}}}_1}^{\rm{T}},{{\widetilde{\boldsymbol{{v}}}}_2}^{\rm{T}}, \ldots ,{{\widetilde{\boldsymbol{{v}}}}_n}^{\rm{T}}} \right]^{\rm{T}}}$ , $\boldsymbol\zeta = \boldsymbol{{H}}\widetilde{\boldsymbol{{v}}} = {\left[ {{\boldsymbol\zeta _1}^{\!\!\!\rm{T}},{\boldsymbol\zeta _2}^{\!\!\!\rm{T}}, \ldots ,{\boldsymbol\zeta _n}^{\!\!\!\rm{T}}} \right]^{\rm{T}}}$ .

3.1.1 Event-triggered fixed-time observer with continuous communications

The event-triggered fixed-time observer with continuous communications for the ith follower spacecraft is described as:

(9) \begin{align}{{{\dot{\hat{\boldsymbol{{v}}}}}_{i,0}} = {{\dot{\hat{\boldsymbol{{v}}}}}_{i,0}}({t_k})} &= { - {o_1}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left[ {{\boldsymbol\zeta _i}({t_k})} \right] - {o_2}{\rm{si}}{{\rm{g}}^\beta }\!\left[ {{\boldsymbol\zeta _i}({t_k})} \right]}\nonumber \\[5pt] & \quad { - {o_3}{\rm{si}}{{\rm{g}}^\alpha }\!\left[ {{\boldsymbol\zeta _i}({t_k})} \right] - {o_4}{\rm{obsat}}\!\left[ {{\boldsymbol\zeta _i}({t_k})} \right]}\end{align}

(10) \begin{align}{t_{k + 1}} = \inf\!\left\{ {t \gt {t_k}\;:\;{h_{ij}} \gt 0,\forall j,j = 1, \cdots ,8} \right\}\end{align}

where

(11) \begin{align}\begin{array}{l}{{h_{ij}}\!\left( t \right) = } {\left| {{E_{ij}}\!\left( t \right)} \right| - \iota \left| {{v_{uij}}} \right| - {k_{o1}}\sqrt {{e^{1 + {k_{o2}}\tanh \left( {{k_{o3}}\left| {{\zeta _{ij}}(t)} \right|} \right)}}} }\end{array}\end{align}

With $0 \lt \alpha \lt 1$ , $\beta \gt 1$ , $\iota \in (0,1),{o_1},{o_2},{o_3},{o_4},{k_{o1}},{k_{o2}},{k_{o3}}$ are positive observer parameters, which will be chosen later. ${\boldsymbol{{E}}_i}\!\left( t \right) = {\left[ {{E_{i1}}, \cdots ,{E_{ij}}, \cdots } \right]^{\rm{T}}} = {o_1}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left[ {{\boldsymbol\zeta _i}\!\left( {{t_k}} \right)} \right] + {o_2}{\rm{si}}{{\rm{g}}^\beta }\!\left[ {{\boldsymbol\zeta _i}\!\left( {{t_k}} \right)} \right] + {o_3}{\rm{si}}{{\rm{g}}^\alpha }\!\left[ {{\boldsymbol\zeta _i}\!\left( {{t_k}} \right)} \right] + {o_4}{\rm{obsat}}\!\left[ {{\boldsymbol\zeta _i}\!\left( {{t_k}} \right)} \right] - {o_1}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right] - {o_2}{\rm{si}}{{\rm{g}}^\beta }\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right] - {o_3}{\rm{si}}{{\rm{g}}^\alpha }\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right] - {o_4}{\rm{obsat}}\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right] = - {\dot{\hat{\boldsymbol{{v}}}}_{i,0}}({t_k}) - {\boldsymbol{{v}}_{ui}}(t)$ where ${\boldsymbol{{v}}_{ui}}(t) = {o_1}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right] + {o_2}{\rm{si}}{{\rm{g}}^\beta }\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right] + {o_3}{\rm{si}}{{\rm{g}}^\alpha }\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right] + {o_4}{\rm{obsat}}\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right]$ . ${\rm{obsat}}(\boldsymbol{{x}}) = [{\rm{obsat}}({x_1}), \cdots , {\rm{obsat}}({x_n}{)]^{\rm{T}}}$ where ${\rm{obsat}}(x)$ is designed as:

(12) \begin{align}{\rm{obsat}}(x) = \left\{ {\begin{array}{l@{\quad}l}{x/{\delta _o},} & {|x| \le {\delta _o}}\\[5pt] {{\rm{sign}}(x),} & {|x| \gt {\delta _o}}\end{array}} \right.\end{align}

Theorem 1. Consider the system Equation (4) with the designed observer Equation (9) and the event-triggered updated rule Equation (10). Suppose Assumption 1 holds. ${\hat{\boldsymbol{{v}}}_i}$ will estimate ${\boldsymbol{{v}}_0}$ in fixed time ${T_1}$ if the parameters satisfy $(1 - \iota ){o_4} \gt {B_3} + {k_{o1}}\sqrt {{e^{1 + {k_{o2}}}}} $ . Meanwhile, there always exists a non-zero bound time for any two triggers, which means ${t_{k + 1}} - {t_k} \gt \tau \gt 0$ , and $\tau $ is a positive constant.

Proof. Consider the following Lyapunov function candidate:

(13) \begin{align}{V_\zeta } = \frac{1}{2}{\widetilde{\boldsymbol{{v}}}^{\rm{T}}}\boldsymbol{{H}}\widetilde{\boldsymbol{{v}}}\end{align}

where $\boldsymbol{{H}} = (\boldsymbol{{L}} + \boldsymbol{{B}}) \otimes {\boldsymbol{{I}}_8}$ describes the communication topology of the formation. The time derivative of Equation (13) along with Equation (9) is given by

(14) \begin{align}{{{\dot V}_{\boldsymbol\zeta} }} &= {{\boldsymbol\zeta ^{\rm{T}}}\dot{\tilde{\boldsymbol{{v}}}}(t) = {\boldsymbol\zeta ^{\rm{T}}}\!\left[ {\dot{\hat{\boldsymbol{{v}}}}(t) - {{\dot{\boldsymbol{{v}}}}_{n0}}} \right]}\nonumber \\[8pt] &= {- {\boldsymbol\zeta ^{\rm{T}}}\!\left[ {\boldsymbol{{E}} + {o_1}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}(\boldsymbol\zeta ) + {o_2}{\rm{si}}{{\rm{g}}^\beta }(\boldsymbol\zeta ) + {o_3}{\rm{si}}{{\rm{g}}^\alpha }\!\left( \boldsymbol\zeta \right) + {o_4}{\rm{obsat}}\boldsymbol\zeta ) + {{\dot{\boldsymbol{{v}}}}_{n0}}} \right]}\nonumber \\[8pt] & \quad \le { |\boldsymbol\zeta {|^{\rm{T}}}|\boldsymbol{{E}}| - {o_1}{\boldsymbol\zeta ^{\rm{T}}}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}(\boldsymbol\zeta ) - {o_2}{\boldsymbol\zeta ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\beta }(\boldsymbol\zeta )}\nonumber \\[8pt] & \quad { - {o_3}{\zeta ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\alpha }\!\left( \boldsymbol\zeta \right) - {o_4}{\boldsymbol\zeta ^{\rm{T}}}{\rm{obsat}}(\boldsymbol\zeta ) - {\boldsymbol\zeta ^{\rm{T}}}{{\dot{\boldsymbol{{v}}}}_{n0}}}\end{align}

where $E = {\left[ {{E_1},{E_2}, \cdots ,{E_n}} \right]^{\rm{T}}}$ , ${v_{n0}} = {\left[ {{v_0},{v_0}, \cdots ,{v_0}} \right]^{\rm{T}}} \in {\mathbb{R}^{8n \times 1}}$ .

Considering the event-triggered update rule Equation (11) and Assumption 1 , we can imply the following relationship from Equation (14):

(15) \begin{align}{{{\dot V}_{\boldsymbol\zeta} }} & \le { \boldsymbol\zeta {|^{\rm{T}}}|\boldsymbol{{E}}| - {o_1}{\boldsymbol\zeta ^{\rm{T}}}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}(\boldsymbol\zeta ) - {o_2}{\boldsymbol\zeta ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\beta }(\boldsymbol\zeta )}\nonumber \\[5pt]& { - {o_3}{\boldsymbol\zeta ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol\zeta ) - {o_4}{\boldsymbol\zeta ^{\rm{T}}}{\rm{obsat}}(\boldsymbol\zeta ) + {B_3}\sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 \left| {{\boldsymbol\zeta _{ij}}} \right|}\nonumber\\[5pt]& \le { - {o_1}(1 - \iota ){\boldsymbol\zeta ^{\rm{T}}}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}(\boldsymbol\zeta ) - {o_2}(1 - \iota ){\boldsymbol\zeta ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\beta }(\boldsymbol\zeta ) - {o_3}(1 - \iota ){\boldsymbol\zeta ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol\zeta )}\nonumber\\[5pt]& { - {o_4}(1 - \iota )\sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 {\zeta _{ij}}{\rm{obsat}}\!\left( {{\zeta _{ij}}} \right) + \left( {{B_3} + {k_{o1}}\sqrt {{e^{1 + {k_{o2}}}}} } \right)\sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 \left| {{\zeta _{ij}}} \right|}\nonumber\\[5pt]& \le { - {o_1}(1 - \iota )(8n{)^{\frac{{\alpha - 1}}{{2\alpha }}}}\parallel \boldsymbol\zeta {\parallel ^{1 + \frac{1}{\alpha }}} - {o_2}(1 - \iota )(8n{)^{\frac{{1 - \beta }}{2}}}\parallel \boldsymbol\zeta {\parallel ^{1 + \beta }} - {o_3}(1 - \iota )\parallel \boldsymbol\zeta {\parallel ^{1 + \alpha }}}\nonumber\\[5pt]& { - {o_4}(1 - \iota )\sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 {\zeta _{ij}}{\rm{obsat}}\!\left( {{\zeta _{ij}}} \right) + \left( {{B_3} + {k_{o1}}\sqrt {{e^{1 + {k_{o2}}}}} } \right)\sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 \left| {{\zeta _{ij}}} \right|}\end{align}

where $ - {\boldsymbol\zeta ^T}{\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol\zeta ) = - \sum\nolimits_{i = 1}^n \sum\nolimits_{j = 1}^8 {\left( {\zeta _{ij}^2} \right)^{\frac{{1 + \alpha }}{2}}} \le - \parallel \boldsymbol\zeta {\parallel ^{1 + \alpha }}$ and $ - {\boldsymbol\zeta ^T}{\rm{si}}{{\rm{g}}^\beta }(\boldsymbol\zeta ) = - \sum\nolimits_{i = 1}^n \sum\nolimits_{j = 1}^8 {\left( {\zeta _{ij}^2} \right)^{\frac{{1 + \beta }}{2}}} \le - {(8n)^{\frac{{1 - \beta }}{2}}}\parallel \boldsymbol\zeta {\parallel ^{1 + \beta }}$ are derived by Lemma 6 and Lemma 7 .

Noting that $\sum\nolimits_{i = 1}^n \sum\nolimits_{j = 1}^8 {\zeta _{ij}}{\rm{obsat}}\!\left( {{\zeta _{ij}}} \right) = \frac{1}{{{\delta _o}}}\sum \left( {{{\bar \zeta }_{ij}}^2} \right) + \sum \left| {{{\bar \zeta }_{ij}}} \right|$ where ${\bar \zeta _{ij}} = \left\{ {{\zeta _{ij}}:\left| {{\zeta _{ij}}} \right| \le {\delta _o}} \right\},{\bar \zeta _{ij}} = \left\{ {{\zeta _{ij}}:\left| {{\zeta _{ij}}} \right| \gt {\delta _o}} \right\}$ , $k \in [0,8n]$ .

For Equation (15), one has

(16) \begin{align}&\quad { - {o_4}(1 - \iota )\sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 {\zeta _{ij}}{\rm{obsat}}\!\left( {{\zeta _{ij}}} \right) + \left( {{B_3} + {k_{o1}}\sqrt {{e^{1 + {k_{02}}}}} } \right)\sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 \left| {{\zeta _{ij}}} \right|}\nonumber \\[5pt] &\le { - {o_4}(1 - \iota )\left[ {\sum \left| {{\zeta _{ij}}} \right| + \frac{1}{{{\delta _o}k}}{{\left( {\sum \left| {{\zeta _{ij}}} \right|} \right)}^2}} \right]}\nonumber \\[5pt] & \quad + { \left( {{B_3} + {k_{o1}}\sqrt {{e^{1 + {k_{02}}}}} } \right)\sum \left| {{\zeta _{ij}}} \right| + \left( {{B_3} + {k_{o1}}\sqrt {{e^{1 + {k_{o2}}}}} } \right)\sum \left| {{\zeta _{ij}}} \right|}\nonumber \\[5pt] & \le { - {o_4}(1 - \iota )\frac{1}{{{\delta _o}k}}{{\left( {\sum \left| {{\zeta _{ij}}} \right|} \right)}^2} + \left( {{B_3} + {k_{o1}}\sqrt {{e^{1 + {k_{o2}}}}} } \right)\sum \left| {{\zeta _{ij}}} \right|}\nonumber {}\\[5pt] & = { - {o_4}(1 - \iota )\sum \left| {{\zeta _{ij}}} \right| + \left( {{B_3} + {k_{o1}}\sqrt {{e^{1 + {k_{02}}}}} } \right)\sum \left| {{\zeta _{ij}}} \right|}\nonumber \\[5pt] & \quad { - {o_4}(1 - \iota )\frac{1}{{{\delta _o}k}}{{\left( {\sum \left| {{\zeta _{ij}}} \right|} \right)}^2} + {o_4}(1 - \iota )\sum \left| {{\zeta _{ij}}} \right|}\nonumber \\[5pt] & \le {\frac{{{o_4}(1 - \iota ){\delta _o}k}}{4}}\end{align}

when $k \ne 0$ , where the inequalities $k\sum \left( {{{\bar \zeta }_{ij}}^2} \right) \geqslant {\left( {\sum \left| {{{\bar \zeta }_{ij}}} \right|} \right)^2}$ which are derived by Lemma 6 has been used. It should be noticed that $ - {o_4}(1 - \iota )\sum\nolimits_{i = 1}^n \sum\nolimits_{j = 1}^8 {\zeta _{ij}}{\rm{obsat}}\!\left( {{\zeta _{ij}}} \right) + \left( {{B_3} + {k_{o1}}\sqrt {{e^{1 + {k_{02}}}}} } \right)\sum\nolimits_{i = 1}^n \sum\nolimits_{j = 1}^8 \left| {{\zeta _{ij}}} \right| \le 0$ when $k = 0$ . It seems that the discussion of whether $k = 0$ in Equation (16) is redundant from the results, but the application of Lemma 6 introduces a singular term $\frac{1}{k}$ . Therefore, the discussion above is necessary. We can rewrite Equation (15) as

(17) \begin{align}{{{\dot V}_{\boldsymbol\zeta}} } &\le { - {o_1}(1 - \iota )(8n{)^{\frac{{\alpha - 1}}{{2\alpha }}}}\parallel \boldsymbol\zeta {\parallel ^{1 + \frac{1}{\alpha }}} - {o_2}(1 - \iota )(8n{)^{\frac{{1 - \beta }}{2}}}\parallel \boldsymbol\zeta {\parallel ^{1 + \beta }}}\nonumber \\[5pt] & \quad { - {o_3}(1 - \iota )\parallel \boldsymbol\zeta {\parallel ^{1 + \alpha }} + \frac{{{o_4}(1 - \iota ){\delta _o}k}}{4}}\end{align}

Noting that ${V_\zeta } = \frac{1}{2}{\tilde{\boldsymbol{{v}}}^{\rm{T}}}H\tilde{\boldsymbol{{v}}} = \frac{1}{2}{\boldsymbol\zeta ^{\rm{T}}}{H^{ - 1}}\boldsymbol\zeta \le \frac{1}{{2{\lambda _{{\rm{min}}}}\!\left( H \right)}}\parallel \boldsymbol\zeta {\parallel ^2}$ , Equation (17) becomes

(18) \begin{align}{{{\dot V}_{\boldsymbol\zeta} } } &\le { - {o_2}(1 - \iota )(8n{)^{\frac{{1 - \beta }}{2}}}{{\left[ {2{\lambda _{\min }}(\boldsymbol{{H}})} \right]}^{\frac{{1 + \beta }}{2}}}V_{\boldsymbol\zeta}^{\frac{{1 + \beta }}{2}}}\nonumber \\[5pt] &\quad { - {o_3}(1 - \iota ){{\left[ {2{\lambda _{\min }}(\boldsymbol{{H}})} \right]}^{\frac{{1 + \alpha }}{2}}}V_{\boldsymbol\zeta}^{\frac{{1 + \alpha }}{2}} + \frac{{{o_4}(1 - \iota ){\boldsymbol\delta _o}k}}{4}}\end{align}

which implies that ${V_{\boldsymbol\zeta} }$ converges to a small neighbourhood of zero in fixed time ${T_1} \le 1/\left[ {{\kappa _1}\!\left( {1 - \frac{{1 + \alpha }}{2}} \right)} \right] + 1/\left[ {{\kappa _2}\!\left( {\frac{{1 + \beta }}{2} - 1} \right)} \right]$ where ${\kappa _1} = {o_3}(1 - \iota ){\left[ {2{\lambda _{\min }}(\boldsymbol{{H}})} \right]^{\frac{{1 + \alpha }}{2}}}$ , ${\kappa _2} = {o_2}(1 - \iota )(8n{)^{\frac{{1 - \beta }}{2}}}{\left[ {2{\lambda _{\min }}(\boldsymbol{{H}})} \right]^{\frac{{1 + \beta }}{2}}}$ . Thus, we conclude that the estimation error $\widetilde {{\boldsymbol{{v}}_i}} = {\hat{\boldsymbol{{v}}}_i} - {\boldsymbol{{v}}_0}$ converges to a small neighbourhood of zero in fixed time.

For ${\boldsymbol{{E}}_i}\!\left( t \right) = {o_1}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left[ {{\boldsymbol\zeta _i}\!\left( {{t_k}} \right)} \right] + {o_2}{\rm{si}}{{\rm{g}}^\beta }\!\left[ {{\boldsymbol\zeta _i}\!\left( {{t_k}} \right)} \right] + {o_3}{\rm{si}}{{\rm{g}}^\alpha }\!\left[ {{\boldsymbol\zeta _i}\!\left( {{t_k}} \right)} \right] + {o_4}{\rm{sign}}\!\left[ {{\boldsymbol\zeta _i}\!\left( {{t_k}} \right)} \right] - {o_1}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right] - {o_2}{\rm{si}}{{\rm{g}}^\beta }\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right] - {o_3}{\rm{si}}{{\rm{g}}^\alpha }\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right] - {o_4}{\rm{sign}}\!\left[ {{\boldsymbol\zeta _i}\!\left( t \right)} \right]$ , we have

(19) \begin{align}{\left| {{{\dot E}_{ij}}(t)} \right|} &= \left| \left[ {{o_1}{\rm{si}}{\rm{g}}^{\frac{1}{\alpha }}\!\left(\zeta _{ij}\right) + {o_2}{\rm{si}}{\rm{g}^{\beta}}\!\left(\zeta _{ij} \right) + {o_3}{\rm{si}}{{\rm{g}}^{\alpha }}\!\left( \zeta _{ij} \right) + {o_4}{\rm{obsat}}\!\left( {{\zeta _{ij}}} \right)} \right]' \right|\nonumber \\[5pt] & \le { \left| {{o_1}\frac{1}{\alpha }{\rm{si}}{{\rm{g}}^{\frac{{1 - \alpha }}{\alpha }}}\!\left( {{\zeta _{ij}}} \right) + {o_2}\beta {\rm{si}}{{\rm{g}}^{\beta - 1}}\!\left( {{\zeta _{ij}}} \right) + {o_3}\alpha {\rm{si}}{{\rm{g}}^{\alpha - 1}}\!\left( {{\zeta _{ij}}} \right) + \frac{1}{{{\delta _o}}}} \right|\left| {{{\dot \zeta }_{ij}}} \right|}\end{align}

From Equation (18) we know that ${\zeta _{ij}}$ is bounded, then we define the upper bound of $\left| {{o_1}\frac{1}{\alpha }{\rm{si}}{{\rm{g}}^{\frac{{1 - \alpha }}{\alpha }}}\!\left( {{\zeta _{ij}}} \right) + {o_2}\beta {\rm{si}}{{\rm{g}}^{\beta - 1}}\!\left( {{\zeta _{ij}}} \right) + {o_3}\alpha {\rm{si}}{{\rm{g}}^{\alpha - 1}}\!\left( {{\zeta _{ij}}} \right) + \frac{1}{{{\delta _o}}}} \right|$ as ${\eta _1}$ . Equation (19) can be transformed into:

(20) \begin{align}{\left| {{{\dot E}_{ij}}(t)} \right|} &\le { {\eta _1}\left| {{{\dot \zeta }_{ij}}} \right| \le {\eta _1}{h_{\max }}\!\left[ {{{\dot{\hat{v}}}_{ij}}(t) - {{\dot v}_{0ij}}\!\left( {{t_k}} \right)} \right]}\nonumber \\[5pt] & \le { {\eta _1}{h_{\max }}\left| {{\eta _{2ij}}\!\left( {{t_k}} \right) + {B_3}} \right|}\end{align}

where ${h_{max}}$ is the largest element in matrix $\boldsymbol{{H}}$ , ${\eta _{2ij}}\!\left( {{t_k}} \right) = {o_1}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left[ {{\zeta _{ij}}\!\left( {{t_k}} \right)} \right] + {o_2}{\rm{si}}{{\rm{g}}^\beta }\!\left[ {{\zeta _{ij}}\!\left( {{t_k}} \right)} \right] + {o_3}{\rm{si}}{{\rm{g}}^\alpha }\!\left[ {{\zeta _{ij}}\!\left( {{t_k}} \right)} \right] + {o_4}{\rm{sign}}\!\left[ {{\zeta _{ij}}\!\left( {{t_k}} \right)} \right]$ . Since ${\dot E_{ij}}({t_k}) = 0$ , it follows when $t \gt {t_k}$ :

(21) \begin{align}\left| {{E_{ij}}\!\!\left( t \right)} \right| \le \int_{{t_k}}^t {\eta _1}{h_{\max }}\left| {{\eta _{2ij}}\!\left( {{t_k}} \right) + {B_3}} \right|{\rm{d}}s\end{align}

According to the triggering condition Equation (11), one has that the next event of state j of agent i will not be triggered before ${h_{ij}}(t) = 0$ or equivalently $\left| {{E_{ij}}\!\!\left( t \right)} \right| = \iota {o_2}{\rm{si}}{{\rm{g}}^\beta }\!\left( {\left| {{\zeta _{ij}}\!\left( t \right)} \right|} \right) + \iota {o_3}{\rm{si}}{{\rm{g}}^\alpha }\!\left( {\left| {{\zeta _{ij}}\!\left( t \right)} \right|} \right) + {k_{o1}}\sqrt {{e^{1 + {k_{o2}}\tanh \!\left( {\left| {{\zeta _{ij}}(t)} \right|} \right)}}} $ . From Equation (21), one has:

(22) \begin{align}{\left| {{E_{ij}}\!\!\left( {{t_{k + 1}}} \right)} \right|} &= { \iota {o_2}{\rm{si}}{{\rm{g}}^\beta }\!\left( {\left| {{\zeta _{ij}}\!\left( {{t_{k + 1}}} \right)} \right|} \right) + \iota {o_3}{\rm{si}}{{\rm{g}}^\alpha }\!\left( {\left| {{\zeta _{ij}}\!\left( {{t_{k + 1}}} \right)} \right|} \right)}\nonumber \\[5pt] & \quad + { {k_{o1}}\sqrt {{e^{1 + {k_{o2}}\tanh \!\left( {\!\left( {{\xi _{ij}}\!\left( {{t_{k + 1}}} \right)|} \right)} \right.}}} }\nonumber \\[5pt] & \le { \int_{{t_k}}^{{t_{k + 1}}} {\eta _1}{h_{\max }}\left| {{\eta _{2ij}}\!\left( {{t_k}} \right) + {B_3}} \right|{\rm{d}}s}\nonumber \\[5pt] & \le { \int_{{t_k}}^{{t_{k + 1}}} {\eta _1}{h_{\max }}\left| {{\eta _{2ij\max }} + {B_3}} \right|{\rm{d}}s}\end{align}

where ${\eta _{2ij\max }}$ is the upper bound of ${\eta _{2ij}}\!\left( {{t_k}} \right)$ . Equation (22) yields that ${t_{k + 1}} - {t_k} \geqslant {k_{o1}}\sqrt e / \!\left( {{\eta _1}{h_{\max }}\left| {{\eta _{2ij\max }} + {B_3}} \right|} \right)$ , which is shown that Zeno-free is guaranteed.

3.1.2 Event-triggered fixed-time observer without continuous communications

This subsection proposes the event-triggered fixed-time observer without continuous communications based on subsection 3.1.1.

For the ith follower spacecraft, $\dot{\hat{v}}_{i}$ is the same as Equation (9). The triggering condition is described as:

(23) \begin{align}{t_{k + 1}} = \inf\!\left\{ {t \gt {t_k}\;:\;{g_{ij}} \gt 0,\forall j,j = 1, \cdots ,8} \right\}\end{align}

where

(24) \begin{align}{{g_{ij}}(t)} & = { \int_{{t_k}}^t {\eta _1}{h_{\max }}\left| {{\eta _{2ij}}\!\left( {{t_k}} \right) + {B_3}} \right|{\rm{d}}s}\nonumber \\[5pt] & { - \frac{\iota }{{1 + \iota }}\left| { - {{\dot{\hat{v}}}_{ij}}\!\left( {{t_k}} \right) + \frac{{{k_{o1}}}}{\iota }\sqrt {{e^{1 + {k_{o2}}\tanh \!\left( {{k_{o3}}\left| {{\xi _{ij}}\!\left( {{t_k}} \right)} \right|} \right)}}} } \right|}\end{align}

Theorem 2. Consider the system Equation (4) with the designed observer Equation (9) and the event-triggered updated rule Equation (24). Suppose Assumption 1 holds. ${\hat{\boldsymbol{{v}}}_i}$ will estimate ${\boldsymbol{{v}}_0}$ in fixed time ${T_1}$ if the parameters satisfy $(1 - \iota ){o_4} \gt {B_3} + {k_{o1}}\sqrt {{e^{1 + {k_{o2}}}}} $ .. Meanwhile, there always exists a non-zero bound time for any two triggers, which means ${t_{k + 1}} - {t_k} \gt \tau \gt 0$ , and $\tau $ is a positive constant.

Proof. According to Equation (21), we can obtain $\left| {{E_{ij}}\!\!\left( t \right)} \right| \le \int_{{t_k}}^t {\eta _1}{h_{\max }}\left| {{\eta _{2ij}}\!\left( {{t_k}} \right) + {B_3}} \right|ds$ , $t \in \left[ {{t_k},{t_{k + 1}}} \right)$ . Further, the triggering function Equation (24) enforces

(25) \begin{align}\left| {{E_{ij}}\!\!\left( t \right)} \right| \le \frac{\iota }{{1 + \iota }}\left| { - {{\dot{\hat{v}}}_{ij}}\!\left( {{t_k}} \right) + \frac{{{k_{o1}}}}{\iota }\sqrt {{e^{1 + {k_{o2}}\tanh \!\left( {\left| {{\zeta _{ij}}\!\left( {{t_k}} \right)} \right|} \right)}}} } \right|\end{align}

Noting that ${E_{ij}} = - {\dot{\hat{v}}_{ij}}\!\left( {{t_k}} \right) - {v_{uij}}$ , Equation (25) can be transformed into

(26) \begin{align}\left| {{E_{ij}}(t)} \right| + \iota \left| {{{\dot{\hat{v}}}_{ij}}\!\left( {{t_k}} \right) + {v_{ui}}} \right| \le \iota \left| {{{\dot{\hat{v}}}_{ij}}\!\left( {{t_k}} \right)} \right| + {k_{o1}}\sqrt {{e^{1 + {k_{o2}}\tanh \!\left( {\left| {{\zeta _{ij}}\!\left( {{t_k}} \right)} \right|} \right)}}} \end{align}

which is is a sufficient condition for

(27) \begin{align}\begin{array}{l}{} {\left| {{E_{ij}}\!\!\left( t \right)} \right| \le \iota \left| {{v_{uij}}} \right| + {k_{o1}}\sqrt {{e^{1 + {k_{o2}}\tanh \!\left( {\left| {{\zeta _{ij}}(t)} \right|} \right)}}} }\end{array}\end{align}

Equation (27) is the same as the event-triggered updated rule Equation (11). Hence, similar to the proof of Theorem 1 , the estimation error $\widetilde {{\boldsymbol{{v}}_i}} = {\hat{\boldsymbol{{v}}}_i} - {\boldsymbol{{v}}_0}$ converges to a small neighbourhood of zero in fixed time ${T_1} \le 1/\left[ {{\kappa _1}\!\left( {1 - \frac{{1 + \alpha }}{2}} \right)} \right] + 1/\left[ {{\kappa _2}\!\left( {\frac{{1 + \beta }}{2} - 1} \right)} \right]$ , and Zeno-free is guaranteed.

Remark 4. Compared with the distributed fixed-time observer in [Reference Han, Yuanqing, Zhang and Zhang12], an additional term ${\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left[ {{\boldsymbol\zeta _i}({t_k})} \right]$ is adopted in Equation (9) to handle certain nonlinear term generated by the fixed-time controller. As shown in Equation (43), we guarantee the fixed time stability of the entire closed-loop system without recourse to the separation principle. Besides, the observer introduces an ETM without continuous communications, reducing energy consumption significantly.

Remark 5. The sign function results in adverse chattering. Differing from the work of [Reference Han, Yuanqing, Zhang and Zhang12], the saturation function is used in place of the sign function for the alleviation of chattering. The constant term $\frac{{{o_4}(1 - \iota )\delta k}}{4}$ is thus introduced in Equation (18), implying that ${V_{\boldsymbol\zeta} }$ converges to a region. Nonetheless, the region can be as small as desired by selecting parameter $\delta $ . The region is correspondingly scaled down to a smaller neighbourhood of the origin and the error caused by the approximation is negligible.

Remark 6. The ETM without continuous communications in Equation (24) is inspired by [Reference Liu, Zhang, Yu and Sun17], where the event-triggered mechanism is used in a consensus control scheme. Differently, Equation (24) embeds a bounded nonlinear term $\frac{{{k_{o1}}}}{\iota }\sqrt {{e^{1 + {k_{o2}}\tanh \!\left( {{k_{o3}}\left| {{\xi _{ij}}\!\left( {{t_k}} \right)} \right|} \right)}}} $ , which realises a more reasonable trigger threshold with characteristics of the exponential function and the hyperbolic tangent function in the transient and steady state.

Remark 7. In view of Equation (24), the key rules in selecting the parameters are: The trigger interval can be increased by increasing the parameter ${k_{o1}}$ . Subsequently, tuning the parameter ${k_{o2}}$ can adjust the exponential convergence rate of the triggering condition during the adjustment period. Meanwhile, the selection of ${k_{o3}}$ should ensure that ${k_{o3}}\left| {{\xi _{ij}}\!\left( {{t_k}} \right)} \right|$ can cover the unsaturated area of the hyperbolic tangent function.

3.2 Fixed-time controller design

In this subsection, a fixed-time distributed control scheme is designed for each follower spacecraft, handling the actuator saturation and the input quantisation problems ingeniously without embedding auxiliary systems. Define a auxiliary vector for ith follower spacecraft as

(28) \begin{align}{{{\boldsymbol\sigma}_i}} &= { \sum\limits_{j = 1}^n {a_{ij}}\!\left[ {({\boldsymbol{{x}}_i} - {\boldsymbol\Delta _{i,0}}) - ({\boldsymbol{{x}}_j} - {\boldsymbol\Delta _{j,str}})} \right] + {b_i}\!\left( {{\boldsymbol{{x}}_i} - {\boldsymbol\Delta _{i,0}} - {\boldsymbol{{x}}_0}} \right)}\nonumber \\[5pt] & = { \sum\limits_{j = 1}^n {l_{ij}}{\boldsymbol{{e}}_{xj}} + {b_i}{\boldsymbol{{e}}_{xi}}}\end{align}

where ${\boldsymbol{{e}}_{xi}} = {\boldsymbol{{x}}_i} - {\boldsymbol\Delta _{i,0}} - {\boldsymbol{{x}}_0}$ . Denote $\boldsymbol\sigma = {\left[ {\boldsymbol\sigma _1^{\rm{T}},\boldsymbol\sigma _2^{\rm{T}}, \ldots ,\boldsymbol\sigma _n^{\rm{T}}} \right]^{\rm{T}}}$ , ${\boldsymbol{{e}}_x} = \left[ {\boldsymbol{{e}}_{x1}^{\rm{T}},\boldsymbol{{e}}_{x2}^{\rm{T}},} \right.$ ${\left. { \ldots ,\boldsymbol{{e}}_{xn}^{\rm{T}}} \right]^{\rm{T}}}$ , then we have $\boldsymbol\sigma = \boldsymbol{{H}}{\boldsymbol{{e}}_x}$ . Based on the backstepping technique, the controller is designed as follows:

Step 1. Design of a virtual control scheme for ${\boldsymbol\gamma _i} = {\boldsymbol{{v}}_i} - {\hat{\boldsymbol{{v}}}_{i,0}} - {\dot{\boldsymbol\Delta}_{i,0}} + {k_1}{\rm{si}}{{\rm{g}}^\beta }\!\left( {{\boldsymbol\sigma _i}} \right)$ , where ${k_1}$ is a positive parameter.

Consider ${\boldsymbol\gamma _i}$ as a virtual control input for the system $\dot{\boldsymbol\sigma} = \boldsymbol{{H}}{\dot{\boldsymbol{{e}}}_x} = \boldsymbol{{H}}{\boldsymbol{{e}}_v} = \boldsymbol{{H}}\!\left( {{\boldsymbol{{v}}_i} - {{\hat{\boldsymbol{{v}}}}_{i,0}} - {{\dot{\boldsymbol\Delta}}_{i,0}} + {{\widetilde{\boldsymbol{{v}}}}_i}} \right)$ , A virtual control scheme for ${\boldsymbol\gamma _i}$ is designed as ${\boldsymbol\gamma _{di}} = - {k_2}{\rm{si}}{{\rm{g}}^\alpha }\!\left( {{\boldsymbol\sigma _i}} \right)$ , where ${k_1}$ is a positive parameter, ${\boldsymbol{{e}}_v} = {\left[ {\boldsymbol{{e}}_{v1}^{\rm{T}},\boldsymbol{{e}}_{v2}^{\rm{T}}, \ldots ,\boldsymbol{{e}}_{vn}^{\rm{T}}} \right]^{\rm{T}}}$ , ${\boldsymbol{{e}}_{vi}} = {\boldsymbol{{v}}_i} - {\dot{\boldsymbol\Delta}_{i,0}} - {\boldsymbol{{v}}_0}$ .

Consider the following Lyapunov function candidate

(29) \begin{align}{V_1} = \frac{1}{2}{\boldsymbol\sigma ^{\rm{T}}}{\boldsymbol{{H}}^{ - 1}}\boldsymbol\sigma \end{align}

and the time derivative of ${V_1}$ is given by

(30) \begin{align}{{{\dot V}_1}} &= { {\boldsymbol\sigma ^{\rm{T}}}{\boldsymbol{{e}}_v} = {\boldsymbol\sigma ^{\rm{T}}}\!\left( {\boldsymbol{{v}} - \hat{\boldsymbol{{v}}} - {{\dot{\boldsymbol\Delta}}_{str}} + \tilde{\boldsymbol{{v}}}} \right) = {\boldsymbol\sigma ^T}(\boldsymbol\gamma + \tilde{\boldsymbol{{v}}})}\nonumber \\[5pt] & = { {\boldsymbol\sigma ^{\rm{T}}}\!\left( {\boldsymbol\gamma - {\boldsymbol\gamma _d}} \right) + {\boldsymbol\sigma ^{\rm{T}}}\tilde{\boldsymbol{{v}}} - {k_1}{\boldsymbol\sigma ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\beta }(\boldsymbol\sigma ) - {k_2}{\boldsymbol\sigma ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol\sigma )}\end{align}

where $\boldsymbol\gamma = {\left[ {{\boldsymbol\gamma _1}^{\rm{T}},{\boldsymbol\gamma _2}^{\rm{T}}, \ldots ,{\boldsymbol\gamma _n}^{\rm{T}}} \right]^{\rm{T}}}$ , ${\boldsymbol\gamma _d} = {\left[ {{\boldsymbol\gamma _{d1}}^{\rm{T}},{\boldsymbol\gamma _{d2}}^{\rm{T}}, \ldots ,{\boldsymbol\gamma _{dn}}^{\rm{T}}} \right]^{\rm{T}}}$ .

According to Lemma 2 and Lemma 5 , we can imply the following unequal relationship:

(31) \begin{align}{{\boldsymbol\sigma ^{\rm{T}}}\!\left( {\boldsymbol\gamma - {\boldsymbol\gamma _d}} \right)} & \le { \sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 \left| {{\sigma _{ij}}} \right|\left| {{\gamma _{ij}} - {\gamma _{dij}}} \right| \le \sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 {2^{1 - \alpha }}{{\left| {{\sigma _{ij}}\parallel {\kappa _{ij}}} \right|}^\alpha }}\nonumber \\[5pt] & \le { \frac{{{2^{1 - \alpha }}}}{{1 + \alpha }}{\boldsymbol\sigma ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol\sigma ) + \frac{{{2^{1 - \alpha }}\alpha }}{{1 + \alpha }}{\boldsymbol\kappa ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol\kappa )}\end{align}

(32) \begin{align}{\boldsymbol\sigma ^{\rm{T}}}\tilde{\boldsymbol{{v}}} \le \sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 \left| {{\sigma _{ij}}} \right|{\!\left( {{{\left| {{{\tilde v}_{ij}}} \right|}^{\frac{1}{\alpha }}}} \right)^\alpha } \le \frac{1}{{1 + \alpha }}{\boldsymbol\sigma ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol\sigma ) + \frac{\alpha }{{1 + \alpha }}{\tilde{\boldsymbol{{v}}}^{\rm{T}}}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}(\tilde{\boldsymbol{{v}}})\end{align}

where ${\kappa _{ij}} = {\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left( {{\gamma _{ij}}} \right) - {\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left( {{\gamma _{dij}}} \right)$ .

Substitution of Equation (31) and Equation (32) into Equation (30) yields that

(33) \begin{align}{{{\dot V}_1}} & \le { - {k_1}{\boldsymbol\sigma ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\beta }(\boldsymbol\sigma ) - \!\left( {{k_2} - \frac{{1 + {2^{1 - \alpha }}}}{{1 + \alpha }}} \right){\boldsymbol\sigma ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol\sigma )}\nonumber \\[5pt] & + { \frac{{{2^{1 - \alpha }}\alpha }}{{1 + \alpha }}{\boldsymbol\kappa ^{\rm{T}}}{\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol\kappa ) + \frac{\alpha }{{1 + \alpha }}{{\tilde{\boldsymbol{{v}}}}^{\rm{T}}}{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}(\tilde{\boldsymbol{{v}}})}\nonumber \\[5pt] & \le { - {k_1}\parallel \boldsymbol\sigma {\parallel ^{1 + \beta }} - \!\left( {{k_2} - \frac{{1 + {2^{1 - \alpha }}}}{{1 + \alpha }}} \right)\parallel \boldsymbol\sigma {\parallel ^{1 + \alpha }}}\nonumber\\[5pt] & + { \frac{{{2^{1 - \alpha }}\alpha }}{{1 + \alpha }}\parallel \boldsymbol\kappa {\parallel ^{1 + \alpha }} + \frac{\alpha }{{1 + \alpha }}\tilde{\boldsymbol{{v}}}{\parallel ^{1 + \frac{1}{\alpha }}}}\end{align}

Step 2. Design of the control scheme for ${\boldsymbol{{u}}_i}$ .

The distributed fixed-time control scheme is described as:

(34) \begin{align}{{\boldsymbol{{u}}_i}} &= { {\boldsymbol{{g}}^{ - 1}}\!\left( {{\boldsymbol{{x}}_i}} \right)\left[ { - \frac{{{{\left\| {{\boldsymbol{{F}}_i}} \right\|}^2}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)}}{{2{\iota _f}}} - {k_3}{\rm{si}}{{\rm{g}}^{2\alpha - 1}}\!\left( {{\boldsymbol\kappa _i}} \right)} \right]}\nonumber \\[5pt] & \quad +{ {\boldsymbol{{g}}^{ - 1}}\!\left( {{\boldsymbol{{x}}_i}} \right)\left[ { - {k_4}{\rm{si}}{{\rm{g}}^{\beta - 1 + \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right) - {k_5}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)} \right]}\end{align}

where ${\boldsymbol{{F}}_i} = {\boldsymbol{{f}}_i}\!\left( {{\boldsymbol{{x}}_i},{\boldsymbol{{v}}_i}} \right) - {\dot{\hat{\boldsymbol{{v}}}}_{i,0}} - {\ddot{\boldsymbol\Delta}_{i,0}} + {k_1}\beta {\rm{si}}{{\rm{g}}^{\beta - 1}}\!\left( {{\boldsymbol\sigma _i}} \right){\dot{\boldsymbol\sigma}_i}$ , ${k_3},{k_4},{k_5}$ and ${\iota _f}$ are positive parameters.

Proof. Consider the following Lyapunov function candidate [Reference Qian and Lin30]:

(35) \begin{align}{V_2} = \sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 \int\limits_{{\gamma _{dij}}}^{{\gamma _{ij}}} {\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left( s \right) - {\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left( {{\gamma _{dij}}} \right)} \right){\rm{d}}s\end{align}

then the time derivative of ${V_2}$ is given by

(36) \begin{align}{\dot V_2} = {\dot \gamma ^T}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( \boldsymbol\kappa \right) + \sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 \int\limits_{{\gamma _{dij}}}^{{\gamma _{ij}}} \frac{{\rm{d}}}{{{\rm{d}}t}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left( s \right) - {\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left( {{\gamma _{dij}}} \right)} \right){\rm{d}}s\end{align}

For the second term in the Equation (36), we could rewrite it as

(37) \begin{align}\sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 \int\limits_{{\gamma _{dij}}}^{{\gamma _{ij}}} \frac{{\rm{d}}}{{{\rm{d}}t}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left( s \right) - {\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left( {{\gamma _{dij}}} \right)} \right){\rm{d}}s = \sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 \!\left( {2 - \alpha } \right){k_2}^{\frac{1}{\alpha }}{\dot \sigma _{ij}}{\Upsilon _{ij}}\end{align}

where

(38) \begin{align}{\Upsilon _{ij}} = \int\limits_{{\gamma _{ij}}}^{{\gamma _{ij}}} {\rm{si}}{{\rm{g}}^{1 - \alpha }}\!\left( {{\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left( s \right) - {\rm{si}}{{\rm{g}}^{\frac{1}{\alpha }}}\!\left( {{\gamma _{dij}}} \right)} \right){\rm{d}}s \le {2^{1 - \alpha }}\left| {{\kappa _{ij}}} \right|\end{align}

which is obtained by Lemma 2 and Lemma 5 .

Substituting Equation (37) and Equation (38) into Equation (36), it can yield

(39) \begin{align}{{{\dot V}_2}} &\le { {{\dot{\boldsymbol\gamma}}^{\rm{T}}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}(\boldsymbol\kappa ) + (2 - \alpha )k_2^{\frac{1}{\alpha }}{2^{1 - \alpha }}{{\dot{\boldsymbol\sigma}}^{\rm{T}}}|\kappa |}\nonumber \\[5pt] &\le { {{\dot{\boldsymbol\gamma}}^{\rm{T}}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}(\kappa )}\nonumber\\[5pt] & +{ (2 - \alpha )k_2^{\frac{1}{\alpha }}{\lambda _{\max }}(\boldsymbol{{H}})\parallel \boldsymbol\kappa \parallel \left\| {\boldsymbol\gamma - {\boldsymbol\gamma _d} + \tilde{\boldsymbol{{v}}} - {k_1}{\rm{si}}{{\rm{g}}^\beta }(\boldsymbol\sigma ) - {k_2}{\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol\sigma )} \right\|}\end{align}

Noting that $\left\| {\boldsymbol\gamma - {\boldsymbol\gamma _d}} \right\| \le \sum\limits_{i = 1}^n \sum\limits_{j = 1}^8 {2^{1 - \alpha }}{\left| {{\boldsymbol\kappa _{ij}}} \right|^\alpha } \le {2^{1 - \alpha }}{\!\left( {8n} \right)^{\frac{{1 - \alpha }}{2}}}{\left\| \boldsymbol\kappa \right\|^\alpha }$ , $\left\| {{k_1}{\rm{si}}{{\rm{g}}^\beta }\!\left( \boldsymbol\sigma \right)} \right\| \le {k_1}{\left\| \boldsymbol\sigma \right\|^\beta }$ , $\left\| {{k_2}{\rm{si}}{{\rm{g}}^\alpha }\!\left( \boldsymbol\sigma \right)} \right\| \le {k_2}{(8n)^{\frac{{1 - \alpha }}{2}}}{\left\| \boldsymbol\sigma \right\|^\alpha }$ where Lemma 2 , Lemma 5 and Lemma 6 are used, we have

(40) \begin{align}\dot{V}_{2}&\le{ {{\dot{\boldsymbol\gamma}}^{\rm{T}}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}(\boldsymbol\kappa ) + {c_1}{c_2}\parallel \boldsymbol\kappa {\parallel ^{1 + \alpha }} + {c_1}\parallel \boldsymbol\kappa \parallel \parallel \tilde{\boldsymbol{{v}}}\parallel + {c_1}{k_1}\parallel \boldsymbol\kappa \parallel \parallel \boldsymbol\sigma {\parallel ^\beta } + {c_3}\parallel \boldsymbol\kappa \parallel \parallel \boldsymbol\sigma {\parallel ^\alpha }}\nonumber \\[5pt]&\le { {{\dot{\boldsymbol\gamma}}^{\rm{T}}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}(\boldsymbol\kappa ) + {c_1}{c_2}\parallel \boldsymbol\kappa {\parallel ^{1 + \alpha }} + {c_1}\!\left( {\frac{1}{{1 + \alpha }}\parallel \boldsymbol\kappa {\parallel ^{1 + \alpha }} + \frac{\alpha }{{1 + \alpha }}\parallel \tilde{\boldsymbol{{v}}}{\parallel ^{1 + \frac{1}{\alpha }}}} \right)}\nonumber \\[5pt] &+{ {c_1}{k_1}\!\left( {\frac{1}{{1 + \beta }}\parallel \boldsymbol\kappa {\parallel ^{1 + \beta }} + \frac{\beta }{{1 + \beta }}\parallel \boldsymbol\sigma {\parallel ^{1 + \beta }}} \right) + {c_3}\!\left( {\frac{1}{{1 + \alpha }}\boldsymbol\kappa {\parallel ^{1 + \alpha }} + \frac{\alpha }{{1 + \alpha }}\parallel \boldsymbol\sigma {\parallel ^{1 + \alpha }}} \right)}\nonumber \\[5pt] &= { {{\dot{\boldsymbol\gamma }}^{\rm{T}}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}(\boldsymbol\kappa ) + \!\left( {{c_1}{c_2} + \frac{{{c_1} + {c_3}}}{{1 + \alpha }}} \right)\parallel \boldsymbol\kappa {\parallel ^{1 + \alpha }} + \frac{{{c_1}{k_1}}}{{1 + \beta }}\parallel \boldsymbol\kappa {\parallel ^{1 + \beta }}}\nonumber \\[5pt] &+{ \frac{{{c_3}\alpha }}{{1 + \alpha }}\parallel \boldsymbol\sigma {\parallel ^{1 + \alpha }} + \frac{{{c_1}{k_1}\beta }}{{1 + \beta }}\parallel \boldsymbol\sigma {\parallel ^{1 + \beta }} + \frac{{{c_1}\alpha }}{{1 + \alpha }}\parallel \tilde{\boldsymbol{{v}}}{\parallel ^{1 + \frac{1}{\alpha }}}}\end{align}

where ${c_1} = \!\left( {2 - \alpha } \right){k_2}^{\frac{1}{\alpha }}{\lambda _{\max }}\!\left( \boldsymbol{{H}} \right)$ , ${c_2}= { 2^{1 - \alpha }}{\!\left( {8n} \right)^{\frac{{1 - \alpha }}{2}}}$ , ${c_3} = {c_1}{k_2}{\left( {8n} \right)^{\frac{{1 - \alpha }}{2}}}$ .

Noting that ${\boldsymbol\gamma _i} = {\boldsymbol{{v}}_i} - {\hat{\boldsymbol{{v}}}_i} - {\dot{\boldsymbol\Delta}_{i,0}} + {k_1}{\rm{si}}{{\rm{g}}^\beta }\!\left( {{\boldsymbol\sigma _i}} \right)$ , we substitute ${\boldsymbol\gamma _i}$ , Equation (4) and control scheme Equation (34) into the first term in the left hand of Equation (40) and we can get

(41) \begin{align}& \quad {\dot{\boldsymbol\gamma}_i^{\rm{T}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)}\nonumber\\[5pt] & = { - {{\left[ {{k_3}\boldsymbol\chi \!\left( {{\boldsymbol{{u}}_i}} \right){\rm{si}}{{\rm{g}}^{2\alpha - 1}}\!\left( {{\boldsymbol\kappa _i}} \right) + {k_4}\boldsymbol\chi \!\left( {{u_i}} \right){\rm{si}}{{\rm{g}}^{\beta - 1 + \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)} \right]}^{\rm{T}}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)}\nonumber \\[5pt] & \quad + {} { {{\left[ {{k_5}\boldsymbol\chi \!\left( {{\boldsymbol{{u}}_i}} \right){\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)} \right]}^{\rm{T}}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)}\nonumber \\[5pt] & \quad+ { {{\left[ {{\boldsymbol{{F}}_i} + {\boldsymbol{{d}}_i} + \boldsymbol{{g}}({x_i}){\boldsymbol{{d}}_{sqi}} - \boldsymbol\chi \!\left( {{u_i}} \right)\frac{{{{\left\| {{F_i}} \right\|}^2}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)}}{{2\iota _f^2}}} \right]}^{\rm{T}}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)}\nonumber \\[5pt] & \le { - {k_3}{\chi _{min}}(1 - \delta )\boldsymbol\kappa _i^{\rm{T}}{\rm{si}}{{\rm{g}}^\alpha }\!\left( {{\boldsymbol\kappa _i}} \right) - {k_4}{\boldsymbol\chi _{min}}(1 - \delta )\boldsymbol\kappa _i^{\rm{T}}{\rm{si}}{{\rm{g}}^\beta }\!\left( {{\boldsymbol\kappa _i}} \right) - {k_5}{\boldsymbol\chi _{min}}(1 - \delta ){{\left\| {{\boldsymbol\kappa _i}} \right\|}^{4 - 2\alpha }}}\nonumber \\[5pt] & \quad - { {\chi _{min}}(1 - \delta )\frac{{{{\left\| {{\boldsymbol{{F}}_i}} \right\|}^2}{{\left\| {{\boldsymbol\kappa _i}} \right\|}^{4 - 2\alpha }}}}{{2{\iota _f}}} + {{\left| {{\boldsymbol{{F}}_i}} \right|}^{\rm{T}}}\left| {{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)} \right| + {{\!\left( {{\boldsymbol{{d}}_i} + \boldsymbol{{g}}({\boldsymbol{{x}}_i}){\boldsymbol{{d}}_{sqi}}} \right)}^{\rm{T}}}\!\left[ {{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)} \right]}\nonumber {}\\[5pt] &\le { - {k_3}{\chi _{min}}\!\left( {1 - \delta } \right){\boldsymbol\kappa _i}^{\rm{T}}{\rm{si}}{{\rm{g}}^\alpha }\!\left( {{\boldsymbol\kappa _i}} \right) - {k_4}{\chi _{min}}\!\left( {1 - \delta } \right){\boldsymbol\kappa _i}^{\rm{T}}{\rm{si}}{{\rm{g}}^\beta }\!\left( {{\boldsymbol\kappa _i}} \right)}\nonumber \\[5pt] & \quad + { \frac{{{\iota _f}}}{{2{\chi _{min}}(1 - \delta )}} + \frac{{d_{\max }^2 + 8({g_{max}}{u_{\min }}{)^2}}}{{2{\iota _d}}}}\end{align}

where the inequality ${\left| {{\boldsymbol{{F}}_i}} \right|^{\rm{T}}}\left| {{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\boldsymbol\kappa _i}} \right)} \right| \le {\chi _{min}}\!\left( {1 - \delta } \right)\frac{{{{\left\| {{\boldsymbol{{F}}_i}} \right\|}^2}{{\left\| {{\boldsymbol\kappa _i}} \right\|}^{4 - 2\alpha }}}}{{2{\iota _f}}} + \frac{{{\iota _f}}}{{2{\chi _{min}}\!\left( {1 - \delta } \right)}}$ and Assumption 2 have been used.

Noting that ${\dot{\boldsymbol\gamma}^{\rm{T}}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( \boldsymbol\kappa \right) = \sum\limits_{i = 1}^n {\dot{\boldsymbol\gamma}_i}^{\rm{T}}{\rm{si}}{{\rm{g}}^{2 - \alpha }}\!\left( {{\kappa _i}} \right)$ , we can rewrite Equation (40) as

(42) \begin{align}{{{\dot V}_2}} &\le{ - \left[ {{k_3}{\chi _{min}}(1 - \delta ) - {c_1}{c_2} - \frac{{{c_1} + {c_3}}}{{1 + \alpha }}} \right]\parallel \boldsymbol\kappa {\parallel ^{1 + \alpha }}}\nonumber \\[5pt] & { - \left[ {{k_4}{\chi _{min}}(1 - \delta )(8n{)^{\frac{{1 + \beta }}{2}}} - \frac{{{c_1}{k_1}}}{{1 + \beta }}} \right]\parallel \boldsymbol\kappa {\parallel ^{1 + \beta }}}\nonumber \\[5pt] &+ { \frac{{{c_3}\alpha }}{{1 + \alpha }}\parallel \boldsymbol\sigma {\parallel ^{1 + \alpha }} + \frac{{{c_1}{k_1}\beta }}{{1 + \beta }}\parallel \boldsymbol\sigma {\parallel ^{1 + \beta }} + \frac{{{c_1}\alpha }}{{1 + \alpha }}\parallel \tilde{\boldsymbol{{v}}}{\parallel ^{1 + \frac{1}{\alpha }}}}\nonumber \\[5pt] &+{ \frac{{n{\iota _f}}}{{2{\chi _{min}}(1 - \delta )}} + \frac{{nd_{\max }^2 + 8n{{({g_{max}}{u_{\min }})}^2}}}{{2{\iota _d}}}}\end{align}

Consider the overall Lyapunov function $V = {V_{\zeta 1}} + {V_1} + {V_2}$ and the time derivative of $V$ is given by the sum of Equation (17), Equation (33) and Equation (42).

(43) \begin{align}{\dot V } &\le{ - {p_{\sigma \beta }}\parallel \boldsymbol\sigma {\parallel ^{1 + \beta }} - {p_{\sigma \alpha }}\parallel \boldsymbol\sigma {\parallel ^{1 + \alpha }} - {p_{\kappa \beta }}\parallel \boldsymbol\kappa {\parallel ^{1 + \beta }} - {p_{\kappa \alpha }}\parallel \boldsymbol\kappa {\parallel ^{1 + \alpha }}}\nonumber\\[5pt] & \quad - {{p_{\zeta \beta }}\parallel \boldsymbol\zeta {\parallel ^{1 + \beta }} - {p_{\zeta \alpha }}\parallel \boldsymbol\zeta {\parallel ^{1 + \alpha }} - {p_\zeta }\parallel \boldsymbol\zeta {\parallel ^{1 + \frac{1}{\alpha }}} + {p_b}}\nonumber \\[5pt] & \le { - {l_\alpha }{{\!\left( {\parallel \boldsymbol\zeta {\parallel ^2} + \parallel \boldsymbol\sigma {\parallel ^2} + \parallel \boldsymbol\kappa {\parallel ^2}} \right)}^{\frac{{1 + \alpha }}{2}}} - {l_\beta }{{\!\left( {\parallel \boldsymbol\zeta {\parallel ^2} + \parallel \boldsymbol\sigma {\parallel ^2} + \parallel \boldsymbol\kappa {\parallel ^2}} \right)}^{\frac{{1 + \beta }}{2}}} + {p_b}}\end{align}

where ${p_{\sigma \beta }} = {k_1} - \frac{{{c_1}{k_1}\beta }}{{1 + \beta }}$ , ${p_{\sigma \alpha }} = {k_2} - \frac{{1 + {2^{1 - \alpha }}}}{{1 + \alpha }} - \frac{{{c_3}\alpha }}{{1 + \alpha }}$ , ${p_{\kappa \beta }} = {k_4}{\chi _{min}}(1 - \delta )(8n{)^{\frac{{1 + \beta }}{2}}} - \frac{{{c_1}{k_1}}}{{1 + \beta }}$ , ${p_{\kappa \alpha }} = {k_3}{\chi _{min}}(1 - \delta ) - {c_1}{c_2} - \frac{{{c_1} + {c_3}}}{{1 + \alpha }} - \frac{{{2^{1 - \alpha }}\alpha }}{{1 + \alpha }}$ , ${p_{\zeta \beta }} = {o_2}(1 - \iota )(8n{)^{\frac{{1 - \beta }}{2}}}$ , ${p_{\zeta \alpha }} = {o_3}(1 - \iota )$ , ${p_\zeta } = {o_1}(1 - \iota )(8n{)^{\frac{{\alpha - 1}}{{2\alpha }}}} - \frac{{({c_1} + 1)\alpha }}{{(1 + \alpha ){\lambda _{\min }}{{(H)}^{1 + \frac{1}{\alpha }}}}}$ , ${p_b} = \frac{{n{\iota _f}}}{{2{\chi _{min}}(1 - \delta )}} + \frac{{nd_{\max }^2 + 8n{{({g_{max}}{u_{\min }})}^2}}}{{2{\iota _d}}} + \frac{{{o_4}(1 - \iota )\delta k}}{4}$ , ${l_\alpha } = {\rm{max}}\!\left( {{p_{\sigma \alpha }},{p_{\kappa \alpha }},{p_{\zeta \alpha }}} \right)$ , ${l_\beta } = {\rm{max}}\!\left( {{p_{\sigma \beta }},{p_{\kappa \beta }},{p_{\zeta \beta }}} \right)$ are positive constants.

Figure 3. Communication topology.

Similar to the method used in Equation (38), we can derive that ${V_2} \le {2^{1 - \alpha }}\parallel \boldsymbol\kappa {\parallel ^2}$ . Then, $V = {V_{**\zeta 1}} + {V_1} + {V_2}$ can be rewritten as

(44) \begin{align}V &\le{ \frac{1}{{2{\lambda _{{\rm{min}}}}(\boldsymbol{{H}})}}\parallel \boldsymbol\zeta {\parallel ^2} + \frac{1}{{2{\lambda _{{\rm{max}}}}(\boldsymbol{{H}})}}\parallel \boldsymbol\sigma {\parallel ^2} + {2^{1 - \alpha }}\parallel \boldsymbol\kappa {\parallel ^2}}\nonumber \\[5pt] &\le { l\!\left( {\parallel \boldsymbol\zeta {\parallel ^2} + \parallel \boldsymbol\sigma {\parallel ^2} + \parallel \boldsymbol\kappa {\parallel ^2}} \right)}\end{align}

where $l = {\rm{max}}\!\left( {\frac{1}{{2{\lambda _{{\rm{min}}}}(\boldsymbol{{H}})}}{{,2}^{1 - \alpha }}} \right)$ . Substituting Equation (44) into Equation (43), it can yield

(45) \begin{align}\dot V \le - \frac{{{l_\alpha }}}{{{l^{\frac{{1 + \alpha }}{2}}}}}{V^{\frac{{1 + \alpha }}{2}}} - \frac{{{l_\beta }}}{{{l^{\frac{{1 + \beta }}{2}}}}}{V^{\frac{{1 + \beta }}{2}}} + {p_b}\end{align}

Table 1. Initial orbital elements of the virtual leader

Table 2. Desired states of each follower spacecraft

Table 3. Initial states of each follower spacecraft

By using Lemma 1 , we can conclude that $\boldsymbol\sigma $ and $\boldsymbol\kappa $ converge to a region within fixed time ${T_2} \le 1/\left[ {\frac{{{l_\alpha }}}{{{l^{\frac{{1 + \alpha }}{2}}}}}\!\left( {1 - \frac{{1 + \alpha }}{2}} \right)} \right] + 1/\left[ {\frac{{{l_\beta }}}{{{l^{\frac{{1 + \alpha }}{2}}}}}\!\left( {\frac{{1 + \beta }}{2} - 1} \right)} \right]$ , which in turn implies that $\boldsymbol\gamma $ and ${\boldsymbol\gamma _d}$ converge to zero within fixed time as well. Since $\boldsymbol\sigma = \boldsymbol{{He}}_{x}$ and ${\boldsymbol{{e}}_v} = \boldsymbol\gamma - {\boldsymbol\gamma _d} + \tilde{\boldsymbol{{v}}} - {k_1}{\rm{si}}{{\rm{g}}^\beta }(\boldsymbol\sigma ) - {k_2}{\rm{si}}{{\rm{g}}^\alpha }(\boldsymbol\sigma )$ , we can obtain that the tracking errors ${\boldsymbol{{e}}_x}$ and ${\boldsymbol{{e}}_v}$ converge to the origin within fixed time.

4.1 Evaluation of control performance and convergence

To display the observation errors intuitively, we depict ${\widetilde{\boldsymbol{{q}}}_i},{\widetilde{\boldsymbol{{r}}}_i}$ in Fig. 4 instead of ${\widetilde{\boldsymbol{{v}}}_i}$ , where ${\widetilde{\boldsymbol{{q}}}_i}$ and ${\widetilde{\boldsymbol{{r}}}_i}$ are observation errors of the virtual leader’s vector quaternion and position. It can be observed that all follower spacecrafts can estimate the virtual leader’s velocity in a fixed time.

The time responses of the attitude quaternion and position tracking errors ${\boldsymbol\theta _e},{\boldsymbol{{r}}_e}$ of each follower spacecraft are given in Fig. 5. The trajectory of the spacecraft’s position in the uncontrolled and the controlled case during an orbital period of 5,578 seconds is shown in Fig. 6. It is important to note that Fig. 6 uses an inhomogeneous coordinate system for more visual comparison. In addition, the minimum value, maximum value, and standard deviation of the controller convergence errors from 250 to 300 seconds are explicitly described in Table 4, which is needed while implementing the engineering solution of the mathematical treatment.

Figure 7 illustrate the control force and the control torque bounded by $0.52N$ and $0.02N \cdot m$ . As shown in these figures, the control output in the initial stage is saturated. Obviously, even in the presence of actuator saturation and input quantisation, the SFF system can estimate the states of the virtual leader effectively and track the trajectory of the virtual leader while maintaining the desired formation configuration.

Table 4. Desired states of each follower spacecraft

Figure 7. Control torque and force.

4.2 Comparison

To highlight the advantages of the proposed triggering condition Equation (24), we use the trigger conditions in [Reference Liu, Zhang, Yu and Sun17] as a comparison, which can be expressed as

(46) \begin{align}{{g_{ij}}(t)} = { \int_{{t_k}}^t {\eta _1}{h_{\max }}\left| {{\eta _{2ij}}\!\left( {{t_k}} \right) + {B_3}} \right|{\rm{d}}s - \frac{\iota }{{1 + \iota }}\left| {{{\dot{\hat{v}}}_{ij}}\!\left( {{t_k}} \right) + \frac{{{k_{o4}}}}{\iota }} \right|}\end{align}

where ${k_{o4}}$ is positive parameter. Meanwhile, to obtain a fair comparison, take the two cases of ${k_{o4}} = {k_{o1}}\sqrt e $ and ${k_{o4}} = {k_{o1}}\sqrt {{e^{1 + {k_{o2}}}}} $ for comparison, where ${k_{o1}}\sqrt e $ and ${k_{o1}}\sqrt {{e^{1 + {k_{o2}}}}} $ are the upper and lower bounds of the nonlinear term ${k_{o1}}\sqrt {{e^{1 + {k_{o2}}\tanh \!\left( {{k_{o3}}\left| {{\xi _{ij}}\!\left( {{t_k}} \right)} \right|} \right)}}} $ in Equation (24), respectively. The two cases are expressed as

(47) \begin{align}{{g_{ij}}(t)} = { \int_{{t_k}}^t {\eta _1}{h_{\max }}\left| {{\eta _{2ij}}\!\left( {{t_k}} \right) + {B_3}} \right|{\rm{d}}s - \frac{\iota }{{1 + \iota }}\left| {{{\dot{\hat{v}}}_{ij}}\!\left( {{t_k}} \right) + \frac{{{k_{o1}}}}{\iota }\sqrt e } \right|}\end{align}

(48) \begin{align}{{g_{ij}}(t)} = { \int_{{t_k}}^t {\eta _1}{h_{\max }}\left| {{\eta _{2ij}}\!\left( {{t_k}} \right) + {B_3}} \right|{\rm{d}}s - \frac{\iota }{{1 + \iota }}\left| {{{\dot{\hat{v}}}_{ij}}\!\left( {{t_k}} \right) + \frac{{{k_{o1}}}}{\iota }\sqrt {{e^{1 + {k_{o2}}}}} } \right|}\end{align}

According to the statistical analysis of the trigger simulation data, Figs. (8)–(9) can be obtained. Figure (8) plots the comparison results of the trigger threshold. The triggering intervals of the SFF system is presented in Fig. (9).

Figure 8. Curves of triggering condition thresholds.

Figure 9. Curves of triggering condition thresholds.

We divide the convergence process of the observation errors into the transient-state phase and the steady-state phase for discussion, aiming to demonstrate the superiority of the proposed ETM in detail. (A). In the transient-state phase. Equation (24) has a larger trigger threshold than Equation (47), but almost the same as Equation (48). Hence, the trigger times of Equation (24) is in the middle of the trigger times of Equation (47) and Equation (48). (B). In the steady-state phase. Equation (24) has a smaller trigger threshold than Equation (48), but almost the same as Equation (47). Therefore, the trigger times of Equation (24) and Equation (47) are almost the same. Further, The norm distribution of the observation errors of Equation (24) and Equation (47) are almost the same.

As can be observed in these discussions and figures, Equation (24) achieves the same steady-state performance as Equation (47) with less communication penalty. It is because that the lower limit of the trigger threshold of Equation (24) fluctuates with the accuracy compared with the stable lower limit of the trigger threshold of Equation (46). In other words, Equation (24) realises the amplification of the lower limit of trigger threshold when the error is large and the reduction of the lower limit of trigger threshold when the error is small, that is the proposed ETM achieves the same steady-state performance with less communication penalty.

However, there is no win-win situation in control systems. In fact, although Equation (24) has reduced communication penalty compared to Equation (47), it has increased fuel consumption. Figure (10) illustrates this with an example ${\widetilde{\boldsymbol\theta}_x}$ and ${\dot{\hat{\boldsymbol\theta}}_x}$ of the follower spacecraft 1. For an observer, the fuel it consumes is its control input ${\dot{\hat{\boldsymbol\theta}}_x}$ . The quantitative analysis of the different ETMs is given in Table 5, where errors distribution denotes the range of $\sum\nolimits_{i = 1}^4 \sum\nolimits_{j = 1}^6 |{z_{ij}}|/24$ with observer error ${z_{ij}}$ , and total input denotes $\sum\nolimits_{i = 1}^4 \sum\nolimits_{j = 1}^6 \int o{u_{ij}}(t)$ with observer control input $o{u_{ij}}$ . It can be calculated that ETM Equation (24) achieves a 8.98% saving in communication resources at a cost of 2.62% fuel consumption compared to ETM Equation (47).

Figure 10. Comparison of observer performance under different ETMs.

Please note that ETM is only embedded in the observer and is not mounted on the controller. Figure (11) illustrates the impact of different ETMs on the controller’s performance with an example ${\boldsymbol{\theta}_{ex}}$ of the follower spacecraft 1. It can be seen that the observer embedding different ETMs has no effect on the transient performance of the controller, which is due to the fact that the observer converges much faster than the controller converges. Equation (24) and (47) have the same effect on the steady-state performance of the controller, and Equation (48) causes the steady-state performance of the controller to deteriorate. This is because the observation error of Equation (48) is higher than that of Equation (24) and Equation (47) (as shown in Fig. 10), and the observation error affects the convergence error of the controller.

Table 5. Communication analysis of different ETMs for all spacecrafts in 10 seconds

Figure 11. Impact of different ETMs on the controller’s performance.

Besides, to emphasise the advantages of the proposed control scheme, we reproduced the asymptotic controller [Reference Fan, Chen, Liu and Cao40] and the finite-time controller [Reference Zhang and Li41] in numerical simulations for comparison. We consider two performance indices for an intuitive comparison: (A) The Attitude Station-Keeping Absolute Error (ASKAE) $\sum\limits_{i = 1}^n \left\| {{\boldsymbol{{q}}_{e,i}}} \right\|$ . (B) The Position Station-Keeping Absolute Error (PSKAE) $\sum\limits_{i = 1}^n \left\| {{\boldsymbol{{r}}_{e,i}}} \right\|$ . Figure 12 gives the two performance indicators’ comparison curves for each control scheme. Obviously, the proposed fixed time controllers demonstrate faster convergence rates in both performance indices.

Figure 12. The comparison curves of the two performance indicators.