2.1 Control objective

The control block diagram of the proposed 6-DOF fixed-time anti-saturation control scheme is illustrated in Fig. 2. It is necessary to establish an error model between the actual and the desired trajectories to achieve tracking of the desired trajectory. The desired position and attitude can be expressed as $\boldsymbol{r}_{\boldsymbol{t}}$ $ \in {\mathbb{R}^3}$ and $\boldsymbol{\sigma}_{\boldsymbol{t}} \in {\mathbb{R}^3}$ . The actual states of the spacecraft are ${\boldsymbol{r}}$ and $\boldsymbol{\sigma} $ , and the tracking error states are expressed by $\boldsymbol{e}_{\boldsymbol{l}} = {[{({\boldsymbol{r}}_{\boldsymbol{e}})^T},{({\boldsymbol{\sigma}}_{\boldsymbol{e}})^T}]^T}$ , where ${\boldsymbol{r}}_{\boldsymbol{e}}$ and ${\boldsymbol{\sigma}}_{\boldsymbol{e}}$ are the position error and the attitude error, respectively. The control objective is to converge tracking errors to zero $\left( {\boldsymbol{e}_{\boldsymbol{l}} \to {\textbf{0}}} \right)$ in a fixed time. As illustrated in Fig. 2, ${\boldsymbol{u}}_{\boldsymbol{i}}$ is the actual controller output, and $\hat{\boldsymbol{d}}_{\boldsymbol{s}}$ is the estimate of the modified lumped disturbance, which includes the influence caused by input overshoot.

2.2 Definitions and notations

The coordinate systems of orbital and attitude motion are shown in Fig. 3. $\left\{ {O - XYZ} \right\}$ is the Earth Centered Inertial (ECI) coordinate system. The origin $O$ is located at the centre of the Earth. The $O - XY$ plane coincides with the equatorial plane of the Earth. The $X$ -axis points to the vernal equinox, the $Z$ -axis points to the North Pole, and the $Y$ -axis, $X$ -axis and $Z$ -axis form a right-handed coordinate system. $\left\{ {B - {x_0}{y_0}{z_0}} \right\}$ is the RTN (Radial-Transverse-Normal) coordinate system, which is used as the reference coordinate system of attitude. The origin $B$ is located at the centre of mass of the spacecraft. The direction of $B - {x_0}$ points from the Earth towards the spacecraft, while the $B - {y_0}$ direction lies within the orbital plane, perpendicular to the $B - {x_0}$ axis, and points towards the direction of spacecraft motion. The $B - {z_0}$ axis forms a right-handed coordinate system with the $B - {x_0}$ and $B - {y_0}$ axes. ${S_b} {\rm{\Delta }} = \left\{ {B - xyz} \right\}$ is the spacecraft body-fixed frame, whose axes are aligned with the orthogonal inertial principal axes to form a right-handed coordinate system.

Figure 3. Coordinate system (ECI coordinate system $O - XYZ$ , RTN coordinate system $B - {x_0}{y_0}{z_0}$ , spacecraft body-fixed coordinate system $B - xyz$ ).

Definition 1. For a vector $\boldsymbol{\chi} = {[{\chi _1}\;\;\;\;{\chi _2}\;\;\;\;{\chi _3}]^T}$ , $\psi \!\left( \boldsymbol{\chi} \right) \in {\mathbb{R}^{3 \times 3}}$ is defined as

(1) \begin{align}\psi \!\left( \boldsymbol{\chi} \right) = \left[ {\begin{array}{c@{\quad}c@{\quad}c}0 & { - {\chi _3}} & {{\chi _2}}\\[5pt] {{\chi _3}} & 0 & { - {\chi _1}}\\[5pt] { - {\chi _2}}& {{\chi _1}} &0\end{array}} \right]\end{align}

Then, the cross product of $\boldsymbol{\chi} $ and any vector ${\boldsymbol{\Upsilon }} \in {\mathbb{R}^3}$ can be expressed as $\psi \!\left( \chi \right){\boldsymbol{\Upsilon }}$ .

Notation 1. $\| \cdot\| $ is the 2-norm of a vector or matrix. $diag\!\left( \cdot \right)$ represents the diagonal matrix. ${O_n}$ and ${I_n}$ represent the zero matrix and the identity matrix of $n \times n$ , respectively.

Notation 2. For a vector $\boldsymbol{\chi} = {[{\chi _1}\;\;\;\;{\chi _2}\;\;\;\; \ldots \;\;\;\;{\chi _n}]^T}$ , $si{g^z}\!\left( \boldsymbol{\chi} \right) = [{\left| {{\chi _1}} \right|^z}sign\!\left( {{\chi _1}} \right),{\left| {{\chi _2}} \right|^z}sign\!\left( {{\chi _2}} \right), \ldots ,{\left| {{\chi _n}} \right|^z} sign\!\left( {{\chi _n}} \right)]^{T}$ , where $sign\!\left( \cdot \right)$ is the Signum function.

Notation 3. The vectors $\boldsymbol{X}_{\textbf{1}} = [1,\underbrace {0,0...0}_{n - 1}],\boldsymbol{X}_{\textbf{2}} = [0,1,\underbrace {0...0}_{n - 2}], \ldots ,{\boldsymbol{X}}_{\boldsymbol{n}} = [\underbrace {0,0 \ldots 0}_{n - 1},1]$ and matrices ${\boldsymbol{E}}_{\textbf{1}} = {\left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}1 & 0 & \cdots & 0\\[5pt] 0& 0& \cdots& 0\\[5pt] 0& 0& \ddots & \vdots \\[5pt] 0 & 0 & \cdots & 0\end{array}} \right]_{n \times n}}$ , ${\boldsymbol{E}}_{\textbf{2}} = {\left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}0 & 0& \cdots & 0\\[5pt] 0& 1 & \cdots& 0\\[5pt] 0& 0 & \ddots & \vdots \\[5pt] 0& 0& \cdots & 0\end{array}} \right]_{n \times n}}$ , …, ${\boldsymbol{E}}_{\boldsymbol{n}} = {\left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}0& 0 & \cdots & 0\\[5pt] 0& 0& \cdots & 0\\[5pt] 0& 0& \ddots &\vdots \\[5pt] 0& 0& \cdots &1\end{array}} \right]_{n \times n}}$ are defined. For a vector $\boldsymbol{\chi} = \left[ {{\chi _1}\;\;\;\;{\chi _2}\;\;\;\; \ldots \;\;\;\;{\chi _n}} \right]$ , the diagonal matrix formed by the elements of vector $\boldsymbol{\chi} $ can be expressed as $diag\!\left( \boldsymbol{\chi} \right) = \sum \nolimits_{m = 1}^n \!\left( {\boldsymbol{\chi} *{\boldsymbol{X}}_{\boldsymbol{m}}^T*{\boldsymbol{E}}_{\boldsymbol{m}}} \right) = \left[ \begin{array}{c@{\quad}c@{\quad}c}{{\chi _1}} & 0 & 0 \\[5pt] 0 & \ddots & 0\\[5pt] 0 & 0 & {{\chi _n}} \end{array} \right]$ .

2.3 Spacecraft model

2.3.1 Spacecraft dynamics with external disturbance and parameter uncertainties

In this paper, the orbital dynamic equations are expressed in the ECI coordinate system, and the attitude kinematic and dynamic equations are expressed in the ${S_b}$ coordinate system. To streamline the control law derivation and mitigate singularity issues in attitude kinematics, Modified Rodrigues Parameters (MRPs) are used to represent the attitude. Consequently, the satellite dynamics is formulated as [Reference Kristiansen, Groli, Nicklasson and Gravdahl35]

(2) \begin{align}\begin{array}{*{20}{l}}{\left\{ {\begin{array}{*{20}{l}}\dot{\boldsymbol{r}} = {\boldsymbol{v}}\\[5pt] {m\dot{\boldsymbol{v}} + m\mu {\boldsymbol{r}} = {\boldsymbol{f}} + {\boldsymbol{d}}_{\boldsymbol{f}}}\\[5pt] {\dot{\boldsymbol\sigma} = G\!\left( \boldsymbol{\sigma} \right)\!{\boldsymbol{w}}}\\[5pt] {{\boldsymbol{J}}\dot{\boldsymbol{w}} + \psi \!\left( {\boldsymbol{w}} \right){{\boldsymbol{J}}{\boldsymbol{w}}} = \tau + \boldsymbol{d}_{\boldsymbol\tau}}\end{array}} \right.} {} {}\end{array}\end{align}

where $G\!\left( \boldsymbol{\sigma} \right)$ is defined as $G\!\left( \boldsymbol{\sigma} \right) = \left[ {\left( {1 - \boldsymbol{\sigma}^{\boldsymbol{T}}{\boldsymbol\sigma} } \right){I_3} + 2\psi \!\left( \boldsymbol{\sigma} \right) + 2\boldsymbol{\sigma {\sigma ^T}}} \right]/4$ , and it is nonsingular. $\boldsymbol{r} \in {\mathbb{R}^3}$ and $\boldsymbol{v} \in {\mathbb{R}^3}$ are the position and velocity vectors, respectively. $m$ is the mass of the spacecraft. $\mu = {\mu _g}/\|{{\boldsymbol{r}\|}^3}$ , where ${\mu _g}$ is the geocentric gravitational constant. $\boldsymbol{\sigma} = {[{\sigma _1},{\sigma _2},{\sigma _3}]^T}$ is the MRPs. $\boldsymbol{w} \in {\mathbb{R}^3}$ is the angular velocity. ${\boldsymbol{f}}$ and $\boldsymbol{\tau} $ are the control force and torque. ${\boldsymbol{d}}_{\boldsymbol{f}},\boldsymbol{d}_{\boldsymbol\tau}$ $ \in {\mathbb{R}^3}$ are the external disturbance force and torque. The external disturbances mainly include the forces and moments caused by non-spherical Earth, atmospheric drag, lunisolar gravitational perturbation and solar radiation pressure. ${\boldsymbol{J}} \in {\mathbb{R}^{3 \times 3}}$ is the inertia matrix of the spacecraft.

Remark 1: As per the Euler theorem, the relative attitude between two coordinate systems can be described by rotating around an axis $\boldsymbol{X} = {[{X_1},{X_2},{X_3}]^T}$ by a certain angle $\Phi $ . There exists a relationship between MRPs and the vector $\boldsymbol{X}$ , given by $\boldsymbol{\sigma} = \boldsymbol{X}\;tan\frac{\Phi }{4}$ . The MRPs become singular when $\Phi = \pm 2\pi $ rad. However, when $\boldsymbol{\sigma}^{\boldsymbol{T}}{\boldsymbol\sigma} \gt 1$ , the original MRPs $\sigma $ can be switched to its shadow set ${\boldsymbol{\sigma} ^s} = - \boldsymbol{\sigma} /{\boldsymbol{\sigma} ^T}\boldsymbol{\sigma} $ to ensure a non-singular global attitude rotation. In this paper, to avoid using shadow sets and simplify the derivation of attitude, the range of motion for Euler angles $\Phi $ is $\left[ { - \pi ,\pi } \right]$ , which guarantees $\boldsymbol{\sigma}^{\boldsymbol{T}}{\boldsymbol\sigma} \le 1$ at all times [Reference Zhang, Ma, Sun and Li36].

During spacecraft operations, the mass and moment of inertia usually have uncertainties. The actual values are typically composed of the nominal values and uncertainties, expressed as $m = {m_o} + {\rm{\Delta }}m$ and ${\boldsymbol{J}} = {\boldsymbol{J}}_{\boldsymbol{o}} + \boldsymbol{\Delta}\boldsymbol{J}$ , respectively. Here, ${m_o}$ and ${{\boldsymbol{J}}_{{\boldsymbol{o}}}}$ represent the nominal values of the mass and moment of inertia matrices. ${\rm{\Delta }}m$ represents the mass uncertainty caused by fuel consumption, while ${\boldsymbol{\Delta }J}$ is the uncertainty of the moment of inertia caused by the displacement of the centre of mass and the expansion and contraction of solar panels. To simplify equation derivation, when representing the inverse matrix of mass and moment of inertia, the influence caused by parameter uncertainty makes it expressed as two parts. Then, we can define ${m^{ - 1}} \buildrel \Delta \over = {m_o}^{ - 1} + {\rm{\Delta }}{m^{ - 1}}$ , ${{\boldsymbol{J}}^{ - 1}} \buildrel \Delta \over = {{\boldsymbol{J}}_{{\boldsymbol{o}}}}^{ - 1} + {\rm{\Delta }}{{\boldsymbol{J}}^{ - 1}}$ [Reference Wu, Chen, Zhang, Chen and Shao24]. Combined with Equation (2), the 6-DOF dynamical system with external disturbance and parameter uncertainties is [Reference Wu, Chen, Zhang, Chen and Shao24]

(3) \begin{align}\ddot{\boldsymbol{r}} &= - \mu {\boldsymbol{r}} + m_o^{ - 1}{\boldsymbol{f}} + {\rm{\Delta }}{m^{ - 1}}{\boldsymbol{f}} + {m^{ - 1}}{\boldsymbol{d}}_{\boldsymbol{f}}, {}\nonumber\\[5pt] \boldsymbol{\ddot \sigma} &= \dot G\!\left( \boldsymbol{\sigma} \right){\boldsymbol{w}} - G\!\left( \boldsymbol{\sigma} \right){{\boldsymbol{J}}_{\boldsymbol{o}}}^{ - 1}\psi \!\left( {\boldsymbol{w}} \right)\boldsymbol{J}_{\boldsymbol{o}}{\boldsymbol{w}} + G\!\left( \boldsymbol{\sigma} \right)\boldsymbol{J}_{\boldsymbol{o}}^{ - 1}\tau + G\!\left( \boldsymbol{\sigma} \right)\boldsymbol{\Delta}{\boldsymbol{J}}^{ - \textbf{1}}\tau {}\nonumber\\[5pt] &\quad - G\!\left( \boldsymbol{\sigma} \right){{\boldsymbol{\Delta} }\boldsymbol{J}^{-\textbf{1}}}\psi \!\left( {\boldsymbol{w}} \right)\boldsymbol{Jw} - G\!\left( \boldsymbol{\sigma} \right)\boldsymbol{J}_{\boldsymbol{o}}^{ - 1}\psi \!\left( {\boldsymbol{w}} \right){\boldsymbol{\Delta }}\boldsymbol{Jw} + G\!\left( \boldsymbol{\sigma} \right)\boldsymbol{J}^{-\textbf{1}}\boldsymbol{d}_{\boldsymbol\tau} \end{align}

2.3.2 Spacecraft tracking error dynamics

The attitude error and the rotation matrix are [Reference Sun, Huo and Jiao37]

(4) \begin{align}\left\{ \begin{array}{l}{\boldsymbol{{\sigma}}}_{\boldsymbol{e}} = \frac{{{\boldsymbol{\sigma}_{\boldsymbol{t}}}({{\boldsymbol{\sigma}} ^T}{\boldsymbol{\sigma}} - 1) + {\boldsymbol{\sigma}} (1 - \boldsymbol{\sigma}_{\boldsymbol{t}}^T{\boldsymbol{\sigma}_{\boldsymbol{t}}}) - 2\psi ({\boldsymbol{\sigma}_{\boldsymbol{t}}}){\boldsymbol{\sigma}} }}{{1 + {\boldsymbol{\sigma}_{\boldsymbol{t}}}^T{\boldsymbol{\sigma}_{\boldsymbol{t}}{\boldsymbol\sigma} ^T}{\boldsymbol{\sigma}} + 2{\boldsymbol{\sigma}_{\boldsymbol{t}}}^T{\boldsymbol{\sigma}} }}\\[5pt]T({\boldsymbol{{\sigma}}}_{\boldsymbol{e}}) = \frac{8}{{{{(1 + {\boldsymbol{{\sigma}}}_{\boldsymbol{e}}^T{\boldsymbol{{\sigma}}}_{\boldsymbol{e}})}^2}}}{\psi ^2}({\boldsymbol{{\sigma}}}_{\boldsymbol{e}}) - \frac{{4(1 - {\boldsymbol{{\sigma}}}_{\boldsymbol{e}}^T{\boldsymbol{{\sigma}}}_{\boldsymbol{e}})}}{{{{(1 + {\boldsymbol{{\sigma}}}_{\boldsymbol{e}}^T{\boldsymbol{{\sigma}}}_{\boldsymbol{e}})}^2}}}\psi ({\boldsymbol{{\sigma}}}_{\boldsymbol{e}}) + {I_3}\end{array} \right. \end{align}

By the way, there are

(5) \begin{align}\begin{array}{*{20}{l}}{\left\{ {\begin{array}{*{20}{l}}{{\dot{\boldsymbol\sigma }_e} = G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{w}}_{\boldsymbol{e}}}\\[5pt] {{\boldsymbol{w}}_{\boldsymbol{e}} = \boldsymbol{w} - T(\boldsymbol\sigma _{\boldsymbol{e}}){\boldsymbol{w}_{\boldsymbol{t}}}}\end{array}} \right.} {}\end{array}\end{align}

and

(6) \begin{align}\begin{array}{*{20}{l}}{\dot T = - \psi \!\left( {{\boldsymbol{w}}_{\boldsymbol{e}}} \right)T} {}\\[5pt] {} {}\end{array}\end{align}

Then, the 6-DOF tracking error dynamics with external disturbance and parameter uncertainties can be expressed as a second-order tracking error dynamic equation

(7) \begin{align}\begin{array}{l}{\ddot{\boldsymbol{r}}_{\boldsymbol{e}}} = - \mu {\boldsymbol{r}} - {\ddot{\boldsymbol{r}}_t} + m_o^{ - 1}{\boldsymbol{f}} + {\rm{\Delta }}{m^{ - 1}}{\boldsymbol{f}} + {m^{ - 1}}{\boldsymbol{d}}_{\boldsymbol{f}},\\[5pt] {\ddot{\boldsymbol\sigma }_{\boldsymbol{e}}} = \dot G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{w}}_{\boldsymbol{e}} - G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{J}}_{\boldsymbol{o}}^{ - 1}\psi \!\left( {\boldsymbol{w}} \right)\boldsymbol{J}_{\boldsymbol{o}}{\boldsymbol{w}} + G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right)\psi \!\left( {{\boldsymbol{w}}_{\boldsymbol{e}}} \right)T\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{w}}_{\boldsymbol{t}} + G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{J}_{\boldsymbol{o}}^{ - \textbf{1}}\tau} \\[5pt] \qquad - G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{\Delta }}{J^{ - 1}}\psi \!\left( {\boldsymbol{w}} \right)\boldsymbol{Jw} - G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{J}}_{\boldsymbol{o}}^{ - 1}\psi \!\left( {\boldsymbol{w}} \right){\boldsymbol{\Delta }}\boldsymbol{Jw} + G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{\Delta }}\boldsymbol{J}^{-\textbf{1}}\boldsymbol{\tau} + G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right)\boldsymbol{J}^{ - \textbf{1}}\boldsymbol{d}_{\boldsymbol\tau}\end{array}\end{align}

The first derivative of $G\!\left( \boldsymbol{\sigma}_{\boldsymbol{e}} \right)$ is

(8) \begin{align} \dot{G}\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right) = \frac{1}{4}\left[ { - 2{\boldsymbol{\sigma}}_{\boldsymbol{e}}^T{\dot{{\boldsymbol \sigma }}_e}\boldsymbol{I}{_3} + 2\psi \!\left( {{\dot{\boldsymbol\sigma }_e}} \right) + 2\boldsymbol{{{\dot \sigma }_e}{\sigma _{{\boldsymbol{e}}}}}^T + 2\boldsymbol{{\sigma _e}{{\dot \sigma }_e}}^T} \right]\end{align}

Thus, the 6-DOF model can be expressed as

(9) \begin{align} {\ddot{\boldsymbol{r}}_{\boldsymbol{e}}} &= \boldsymbol{h}_{\boldsymbol{b}} + m_o^{ - 1}{\boldsymbol{f}} + \boldsymbol{d}_{\boldsymbol{b}}, {}\nonumber \\[5pt] {\boldsymbol{\ddot \sigma }_{\boldsymbol{e}}} &= {\boldsymbol{h}}_{\boldsymbol{a}} + G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{J}}_{\boldsymbol{o}}{}^{ - 1}\boldsymbol{\tau} + {\boldsymbol{d}}_{\boldsymbol{a}} {} \end{align}

where

(10) \begin{align}\boldsymbol{h}_{\boldsymbol{b}} &= - \mu {\boldsymbol{r}} - {\ddot{\boldsymbol{r}}_t}, {}\nonumber\\[5pt] \boldsymbol{d}_{\boldsymbol{b}} &= {\rm{\Delta }}{m^{ - 1}}{\boldsymbol{f}} + {m^{ - 1}}{\boldsymbol{d}}_{\boldsymbol{f}}, {}\nonumber\\[5pt] {\boldsymbol{h}}_{\boldsymbol{a}} &= \dot G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{w}}_{\boldsymbol{e}} - G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{J}}_{\boldsymbol{o}}^{ - 1}\psi \!\left( {\boldsymbol{w}} \right)\boldsymbol{J}_{\boldsymbol{o}}{\boldsymbol{w}} + G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right)\psi \!\left( {{\boldsymbol{w}}_{\boldsymbol{e}}} \right)T\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{w}}_{\boldsymbol{t}}, {}\nonumber\\[5pt] {\boldsymbol{d}}_{\boldsymbol{a}} &= - G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{\Delta }}\boldsymbol{J}^{ - \textbf{1}}\psi \!\left( {\boldsymbol{w}} \right)\boldsymbol{Jw} - G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{J}}_{\boldsymbol{o}}^{ - 1}\psi \!\left( {\boldsymbol{w}} \right){\boldsymbol{\Delta }Jw} + G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{\Delta }}\boldsymbol{J}^{ - \textbf{1}}\tau + G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){{\boldsymbol{J}}^{ - 1}}\boldsymbol{d}_{\boldsymbol\tau} {} \end{align}

Let ${\boldsymbol{e}}_{\textbf{1}} = {[\boldsymbol{r}_{\boldsymbol{e}}{}^T,\boldsymbol{\sigma}_{\boldsymbol{e}}{}^T]^T}$ and $\boldsymbol{e}_{\textbf{2}} = {[{\boldsymbol{v}}_{\boldsymbol{e}}{}^T,{\boldsymbol{w}}_{\boldsymbol{e}}{}^T]^T}$ . The 6-DOF tracking error dynamics with parameter uncertainties and external disturbance can be formulated as

(11) \begin{align}\dot{\boldsymbol{e}}_{\textbf{1}} &= \boldsymbol{e}_{\textbf{2}}, {}\nonumber\\[5pt] \dot{\boldsymbol{e}}_{\textbf{2}} &= {\boldsymbol{h}} + \boldsymbol{M}_{\boldsymbol{C}}{\boldsymbol{u}} + {\boldsymbol{d}} {} \end{align}

where ${{\boldsymbol{h}}} = {[{{{\boldsymbol{h}}}_{{\boldsymbol{b}}}}^T,{{{\boldsymbol{h}}}_{{\boldsymbol{a}}}}^T]^T}$ is the inherent nonlinearity of the system. ${{\boldsymbol{u}}} = {[{{{\boldsymbol{f}}}^T},{\boldsymbol{\tau} ^T}]^T}$ is the control input and ${{\boldsymbol{u}}} = {[{u_1},{u_2}, \ldots ,{u_6}]^T}$ . ${{\boldsymbol{d}}} = {[{{{\boldsymbol{d}}}_{{\boldsymbol{b}}}}^T,{{{\boldsymbol{d}}}_{{\boldsymbol{a}}}}^T]^T}$ is the lumped disturbance term, including external disturbance and parameter uncertainties. ${{\boldsymbol{M}}_{\boldsymbol{C}}}$ is

(12) \begin{align}\boldsymbol{M}_{\boldsymbol{C}} = \left[ {\begin{array}{c@{\quad}c}{m_o^{ - 1}{I_3}} & {{O_3}}\\[5pt] {{O_3}}& {G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{J}}_{\boldsymbol{o}}^{ - 1}}\end{array}} \right]\end{align}

The input saturation of the spacecraft system is

(13) \begin{align}{u_k} = {\rm{sat}}\!\left( {{u_{ik}}} \right) = \left\{ \begin{array}{l@{\quad}l}{\rm{sign}}\!\left( {{u_{ik}}} \right){u_{ik{\rm{max}}}},& \left| {{u_{ik}}} \right| \geq {u_{ik{\rm{max}}}}\\[5pt] {u_{ik}}, & \left| {{u_{ik}}} \right| \lt {u_{ik{\rm{max}}}} \end{array} \right. \quad k = 1,2,3 \ldots 6.\end{align}

where $\boldsymbol{u}_{\boldsymbol{i}}$ is the actual controller output, ${u_{ik{\rm{max}}}}$ is the bound of ${u_{ik}}$ and ${u_{ik{\rm{max}}}} \gt 0$ .

Before designing the observer and controller, there are two points to note.

1. (1) The modeling and control problems of the spacecraft tracking mission discussed in this paper are conducted in the continuous-time domain. The position and attitude of a controlled spacecraft can be accurately measured through onboard sensors. Since the measurement signals from various onboard sensors are usually filtered, this paper assumes that the state and its uncertainty are second-order differentiable [Reference Dong and Yang5]. Moreover, the characteristics of the spacecraft (such as shape and mass) change slowly over time due to fuel consumption and vibrations [Reference Li, Zou, Shi, Song and Wang4]. Therefore, the derivatives of mass uncertainty and moment of inertia uncertainty are bounded, meaning they are Lipschitz continuous.

2. (2) The disturbance forces acting on the spacecraft mainly include Earth oblateness perturbation, atmospheric drag perturbation, solar radiation pressure perturbation and the perturbation from the third-body gravitation. The main disturbance torques include gravity gradient torque, aerodynamic torque, solar radiation torque and magnetic torque [Reference Wu, Chen, Zhang, Chen and Shao24]. These disturbances forces and torques change slowly over time. Thus, they are assumed to be Lipschitz continuous [Reference Li, Zou, Shi, Song and Wang4]. In other words, disturbance forces and torques can be differentiated, and their differential values have an upper bound.

Based on the above two points, the following assumptions can be made.

Assumption 1. The spacecraft’s dynamical system is detectable and can be stabilised [Reference Zhang, Wang, Zhang and Shao38].

Assumption 2. The parameter uncertainties ${\rm{\Delta }}m$ and $\boldsymbol{\Delta}{\boldsymbol{J}}$ are bounded and satisfy $\left| {{\rm{\Delta }}m} \right| \le {{\Delta \bar{m}}}$ and $\|{\boldsymbol{\Delta}\boldsymbol{J}}\| \le {\rm{\Delta }}\bar J$ , where $\Delta \bar{m}$ and ${\rm{\Delta }}\bar J$ are nonnegative constants. The disturbance force and torque ${\boldsymbol{d}}_{\boldsymbol{f}},\boldsymbol{d}_{\boldsymbol\tau} \in {\mathbb{R}^3}$ are bounded. The lumped disturbance ${\boldsymbol{d}}$ is bounded, and its derivative satisfies $||\dot{\boldsymbol{d}}|| \le \delta d$ , where $\delta d$ is a nonnegative constant.

Assumptions 1 and 2 mean that the spacecraft system is controllable and observable. It is reasonable because spacecraft always install enough sensors and actuators to provide enough measurement of control force and torque. These two assumptions make it possible to design the corresponding controller and observer of the spacecraft system.

3.1 Modified lumped disturbance

Due to the physical limitations, the actuator exhibits a saturation phenomenon. Input saturation nonlinearity will reduce the system performance and even break the stability of the closed-loop system. Therefore, the influence of input saturation should be considered in the controller design. An auxiliary variable is introduced in this section to represent the input overshoot. The adverse effects are directly transformed into a lumped disturbance. The fixed-time disturbance observer is designed to suppress the lumped disturbance, including parameter uncertainties, external disturbance and input saturation. Compared with the traditional method of adding an anti-saturation structure, the system structure of this method will be concise.

In the input saturation formula (13), when ${u_{ik}} = {u_{ik}}_{{\rm{max}}}\!\left( {k = 1,2 \ldots 6} \right)$ , there is a sharp corner, which makes the output of the actuator have strong nonlinearity. To avoid this strong nonlinearity, a smooth function is used to approximate the input saturation function [Reference Wen, Zhou, Liu and Su39]

(14) \begin{align}\begin{array}{*{20}{l}}{{\rm{sat}}\!\left( {{u_{ik}}} \right) \approx {u_{ik{\rm{max}}}} \cdot {\rm{tanh}}\!\left( {\frac{{{u_{ik}}}}{{{u_{ik{\rm{max}}}}}}} \right)} {}\end{array}\end{align}

Furthermore, define an auxiliary variable ${\boldsymbol{\Delta }}u = {[{\rm{\Delta }}{u_1},{\rm{\Delta }}{u_2}, \cdots ,{\rm{\Delta }}{u_6}]^T},$ with

(15) \begin{align}\begin{array}{*{20}{l}}{{\rm{\Delta }}{u_k} = {\rm{sat}}\!\left( {{u_{ik}}} \right) - {u_{ik}},k = 1,2,3 \ldots 6} {}\end{array}\end{align}

Equations (14) and (15) yield to ${\rm{sat}}\!\left( {\boldsymbol{u}_{\boldsymbol{i}}} \right) = \boldsymbol{u}_{\boldsymbol{i}} + {\boldsymbol{\Delta}{\boldsymbol{u}}}$ . Then replace ${\boldsymbol{u}}$ in Equation (11) with ${\rm{sat}}\!\left( {{\boldsymbol{u}}} \right)$ , and the tracking error system can be rewritten as

(16) \begin{align}\eqalign{ & \dot{\boldsymbol{e}}_{\textbf{1}} = \boldsymbol{e}_{\textbf{2}}, \cr & \dot{\boldsymbol{e}}_{\textbf{2}} = {\boldsymbol{h}} + \boldsymbol{M}_{\boldsymbol{c}}{\rm{sat}}\!\left( {\boldsymbol{u}_{\boldsymbol{i}}} \right) + \bar{\boldsymbol{d}} = {\boldsymbol{h}} + \boldsymbol{M}_{\boldsymbol{c}}{\boldsymbol{u}} + \boldsymbol{M}_{\boldsymbol{c}}{{\boldsymbol\Delta}\boldsymbol{u}} + \bar{\boldsymbol{d}} \cr} \end{align}

where $\bar{\boldsymbol{d}} = {[\bar{\boldsymbol{d}}_{\boldsymbol{b}}{}^T,\bar{\boldsymbol{d}}_{\boldsymbol{a}}{}^T]^T}$ and

(17) \begin{align}\eqalign{ & \bar{\boldsymbol{d}}_{\boldsymbol{b}} = \Delta {m^{ - 1}}{\rm{sat}}\!\left( {\boldsymbol{f}} \right) + {m^{ - 1}}{\boldsymbol{d}}_{\boldsymbol{f}}, \cr & \bar{\boldsymbol{d}}_{\boldsymbol{a}} = - G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right)\boldsymbol{\Delta}{\boldsymbol{J}}^{ - \textbf{1}}\psi \!\left( {\boldsymbol{w}} \right)\boldsymbol{Jw} - G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right){\boldsymbol{J}}_{\boldsymbol{o}}^{ - 1}\psi \!\left( {\boldsymbol{w}} \right)\boldsymbol{\Delta}{\boldsymbol{Jw}} + G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right)\boldsymbol{\Delta}{\boldsymbol{J}}^{ - \textbf{1}}{\rm{sat}}\!\left( \boldsymbol{\tau} \right) + G\!\left( {\boldsymbol{\sigma}}_{\boldsymbol{e}} \right)\boldsymbol{J}^{-\textbf{1}}\boldsymbol{d}_{\boldsymbol\tau} \cr} \end{align}

Define ${\boldsymbol{d}}_{\boldsymbol{s}} = \boldsymbol{M}_{\boldsymbol{c}}{{\boldsymbol\Delta}\boldsymbol{u}} + \bar{\boldsymbol{d}}$ . The modified lumped disturbance ${\boldsymbol{d}}_{\boldsymbol{s}}$ is bounded. Since ${\rm{sat}}\!\left( {\boldsymbol{u}_{\boldsymbol{i}}} \right) = \boldsymbol{u}_{\boldsymbol{i}} + \boldsymbol{\Delta}{\boldsymbol{u}}$ , ${\boldsymbol{d}}_{\boldsymbol{s}} = \boldsymbol{M}_{\boldsymbol{c}}{{\boldsymbol\Delta}\boldsymbol{u}} + \bar{\boldsymbol{d}}$ , ${\boldsymbol{d}}_{\boldsymbol{s}} = \boldsymbol{M}_{\boldsymbol{c}}\!\left[ {{\rm{sat}}\!\left( {\boldsymbol{u}_{\boldsymbol{i}}} \right) - \boldsymbol{u}_{\boldsymbol{i}}} \right] + \bar{\boldsymbol{d}}$ and $\dot{\boldsymbol{d}}_{\boldsymbol{s}} = \dot{\boldsymbol{M}}_{\boldsymbol{c}}\!\left[ {{\rm{sat}}\!\left( {\boldsymbol{u}_{\boldsymbol{i}}} \right) - \boldsymbol{u}_{\boldsymbol{i}}} \right] + \boldsymbol{M}_{\boldsymbol{c}}\!\left[ {{\rm{sat}}\!\left( \dot{\boldsymbol{u}_{\boldsymbol{i}}} \right) - \dot{\boldsymbol{u}}_{\boldsymbol{i}}} \right] + \dot{\bar{\boldsymbol{d}}}$ hold. The system is controllable, so the control input $u$ is bounded. Since a smooth function replaces the traditional saturation function, ${\rm{sat}}\!\left( {\boldsymbol{u}_{\boldsymbol{i}}} \right)$ and ${{sat}}\!\left( \dot{\boldsymbol{u}_{\boldsymbol{i}}} \right)$ are bounded. In addition, since the uncertain parameters ${\rm{\Delta }}m$ and ${\boldsymbol{\Delta }}J$ and the external disturbance ${\boldsymbol{d}}_{\boldsymbol{f}}$ and $\boldsymbol{d}_{\boldsymbol\tau}$ are bounded, we can know ${\boldsymbol{d}}_{\boldsymbol{s}}$ and $\dot{\boldsymbol{d}}_{\boldsymbol{s}}$ are bounded. Therefore, ${\boldsymbol{d}}_{\boldsymbol{s}}$ is bounded, and its derivative satisfies $||\dot{\boldsymbol{d}}_{\boldsymbol{s}}|| \le \delta {d_s}$ , where $\delta {d_s}$ is a nonnegative constant.

3.2 FT-DO design

In this section, an FT-DO with fewer parameters is proposed for estimating the modified lumped disturbances. The convergence of the estimation error subsystem does not depend on the initial state of the system. First of all, introduce the following lemma.

Lemma 1. [Reference Basin, Panathula and Shtessel34] For the following dynamical system (18) with a bounded disturbance $\boldsymbol{\xi}(\boldsymbol{t})$ , both state vectors $\boldsymbol{x}(\boldsymbol{t})$ and $\boldsymbol{y}(\boldsymbol{t})$ converge to the origin uniformly in a fixed time ${T_f}$ calculated by Equation (19).

(18) \begin{align}\dot{\boldsymbol{x}}\!\left( \boldsymbol{t} \right) &= - {\lambda _1}\frac{{{\boldsymbol{x}}\!\left( t \right)}}{\|{{\boldsymbol{x}}{{\!\left( t \right)}\|^{\frac{1}{2}}}}} - {\lambda _2}\boldsymbol{x}\!\left( t \right)||\boldsymbol{x}{{\!\left( t \right)}||^{p - 1}} + \boldsymbol{y}\!\left( \boldsymbol{t} \right)\nonumber\\[5pt] \dot{\boldsymbol{y}}\!\left( \boldsymbol{t} \right) &= - \ell \frac{{{\boldsymbol{x}}\!\left( \boldsymbol{t} \right)}}{\|{{\boldsymbol{x}}\!\left( t \right)}\|} + \boldsymbol{\xi} \!\left( \boldsymbol{t} \right)\end{align}

(19) \begin{align}{T_f} \le \left( {\frac{1}{{{\lambda _2}\left( {p - 1} \right){\varpi ^{p - 1}}}} + \frac{{2{{\left( {\sqrt n \varpi } \right)}^{1/2}}}}{{{\lambda _1}}}} \right)\left( {1 + \frac{M}{{m\!\left( {1 - \sqrt {2\ell } /{\lambda _1}} \right)}}} \right)\end{align}

where $n$ is the dimension of the state vector. The parameters satisfy $\varpi \gt 0$ , ${\lambda _1},{\lambda _2},\ell \gt 0$ , and the bound of $\|\boldsymbol\xi(t)\|$ is $L$ . The relationships $M = \ell + L$ , $m = \ell - L$ , $\ell \gt 4L$ , and ${\lambda _1} \gt \sqrt {2\ell } $ hold.

The state vector of the observer is defined as $\boldsymbol{\vartheta} = {[\boldsymbol{\vartheta _1}^T,\boldsymbol{\vartheta _2}^T]^T}$ , where $\boldsymbol{\vartheta _1},\boldsymbol{\vartheta _2} \in {\mathbb{R}^6}$ . The fixed-time disturbance observer can be expressed as

(20) \begin{align} {\dot{{\boldsymbol \vartheta }_{\textbf{1}}}} &= - {\lambda _1}\frac{{{\boldsymbol{e}_{\boldsymbol{o}\textbf{1}}}}}{{{\|e_{o1}\|}^{1/2}}} - {\lambda _2}\!\left( {\boldsymbol{\vartheta _1} - \boldsymbol{e}_{\textbf{2}}} \right)||{\boldsymbol{\vartheta _1}} - \boldsymbol{e}_{\textbf{2}}||^{p - 1} + \boldsymbol{\vartheta _2} + {\boldsymbol{h}} + \boldsymbol{M}_{\boldsymbol{c}}{\boldsymbol{u}}\nonumber \\[5pt] \dot{{\boldsymbol \vartheta }_{\textbf{2}}} &= - {\lambda _3}\frac{{{\boldsymbol{e}_{\boldsymbol{o}\textbf{1}}}}}{{{\|e_{o1}\|}}} \end{align}

where $\boldsymbol{e}_{\boldsymbol{o}1} = \boldsymbol{\vartheta _1} - \boldsymbol{e}_{\textbf{2}}$ , $\boldsymbol{e}_{\boldsymbol{o}2} = \boldsymbol{\vartheta _2} - {\boldsymbol{d}}_{\boldsymbol{s}}$ , and they are estimated errors, $p \in \mathbb{R}$ and $p \gt 1$ . The parameters ${\lambda _1},{\lambda _2},{\lambda _3}$ satisfy ${\lambda _1} \gt \sqrt {2{\lambda _3}} $ , ${\lambda _2} \gt 0$ , and ${\lambda _3} \gt \delta {d_s}$ , where $\delta {d_s}$ is the bound of $||\dot{\boldsymbol{d}}_{\boldsymbol{s}}||$ .

Proof. Since $\boldsymbol{e}_{\boldsymbol{o}1} = \boldsymbol{\vartheta_1} - \boldsymbol{e}_{\textbf{2}}$ and $\boldsymbol{e}_{\boldsymbol{o}2} = \boldsymbol{\vartheta _2} - {\boldsymbol{d}}_{\boldsymbol{s}}$ , combining Equations (16) and (20), $\dot{\boldsymbol{e}}_{\boldsymbol{o}1}$ can be derived as follows

(21) \begin{align}\dot{\boldsymbol{e}}_{\boldsymbol{o}1} = - {\lambda _1}\frac{{\boldsymbol{e}_{\boldsymbol{o}1}}}{||{\boldsymbol{e}_{\boldsymbol{o}1}||^{1/2}}} - {\lambda _2}\boldsymbol{e}_{\boldsymbol{o}1}||\boldsymbol{e}_{\boldsymbol{o}1}||^{p - 1} + \boldsymbol{e}_{\boldsymbol{o}2}\end{align}

Taking the derivative of ${{{\boldsymbol{e}}}_{{{\boldsymbol{o}}}2}}$ , the estimated error dynamics can be established as

(22) \begin{align} \dot{\boldsymbol{e}}_{\boldsymbol{o}1} &= - {\lambda _1}\frac{{\boldsymbol{e}_{\boldsymbol{o}1}}}{||{\boldsymbol{e}_{\boldsymbol{o}1}||^{1/2}}} - {\lambda _2}\boldsymbol{e}_{\boldsymbol{o}1}||\boldsymbol{e}_{\boldsymbol{o}1}||^{p - 1} + \boldsymbol{e}_{\boldsymbol{o}2}\nonumber\\[5pt] \dot{\boldsymbol{e}}_{\boldsymbol{o}2} &= - {\lambda _3}\frac{{\boldsymbol{e}_{\boldsymbol{o}1}}}{||{\boldsymbol{e}_{\boldsymbol{o}1}||}} - \dot{\boldsymbol{d}}_{\boldsymbol{s}} \end{align}

According to Lemma 1, the state vectors ${\boldsymbol{e}_{o\textbf{1}}}$ and ${\boldsymbol{e}_{o\textbf{2}}}$ of the estimated error dynamical system (22) with a bounded disturbance ${\boldsymbol{d}_s}$ will uniformly converge to the origin within a fixed time ${T_o}$ .

(23) \begin{align} {{T_o} \le \left( {\frac{1}{{{\lambda _2}\left( {p - 1} \right){\varsigma ^{p - 1}}}} + \frac{{2{{(\sqrt 6 \varsigma )}^{1/2}}}}{{{\lambda _1}}}} \right)\left( {1 + \frac{{{\lambda _3} + \delta d}}{{\left( {{\lambda _3} - \delta d} \right)\left( {1 - 2\sqrt {{\lambda _3}} /{\lambda _1}} \right)}}} \right)} \end{align}

where $\varsigma $ is a small constant. When $\varsigma = {({6^{1/4}}{\lambda _1}/{\lambda _2})^{1/\left( {p + 0.5} \right)}}$ , the upper bound of ${T_o}$ is minimised. Therefore, the lumped disturbance ${\boldsymbol{d}}_{\boldsymbol{s}}$ can be estimated by $\boldsymbol{\vartheta _2}$ within a fixed time.

The designed FT-DO can observe various complex disturbances. Especially in this paper, the modified lumped disturbance includes the adverse effects caused by input saturation. The FT-DO designed in this paper can still suppress this lumped disturbance. Compared with the traditional anti-saturation scheme (Fig. 1), the design of the FT-DO avoids adding additional sub-dynamics to the system, making the control system structure concise.

4.1 FT-NTSM controller design

Since the design of the sliding mode surface is based on the phase plane, the position and attitude tracking error vector $\boldsymbol{e}_{\textbf{1}} = {[{e_{11}},{e_{12}}, \ldots ,{e_{16}}]^T}$ and its first-order derivative ${\boldsymbol{e}_{\textbf{2}}} = {[{e_{21}},{e_{22}}, \ldots ,{e_{26}}]^T}$ are selected as state variables to construct the phase plane. The parameters ${p_k}$ , ${q_k}$ , ${m_k}$ and ${n_k}$ , $\left( {k = 1,2} \right)$ are positive odd integers and satisfy ${p_1} \lt {q_1} \lt 2{p_1}$ , ${p_2} \lt {q_2}$ , ${m_k} \gt {n_k}$ and $\frac{{{m_1}}}{{{n_1}}} - \frac{{{p_1}}}{{{q_1}}} \gt 1$ . In order to avoid singularity, inspired by [Reference Zuo19], the non-singular terminal sliding mode surface $s$ is designed as

(26) \begin{align}\begin{array}{*{20}{l}}{{\boldsymbol{s}} = {\boldsymbol{e}}_{\textbf{1}} + {{[\boldsymbol{\kappa} \!\left( {{\boldsymbol{e}}_{\textbf{1}}} \right) \boldsymbol{\cdot}\boldsymbol{e}_{\textbf{2}}]}^{\frac{{{q_1}}}{{{p_1}}}}}}\end{array}\end{align}

where $s = {[{s_1},{s_2}, \cdots ,{s_6}]^T}$ . $\boldsymbol\kappa \in {\mathbb{R}^{6 \times 6}}$ is a diagonal matrix. Its diagonal element ${\kappa _k}$ ( ${\kappa _k}$ : $\mathbb{R} \to {\mathbb{R}^ + },k = 1, \ldots 6$ ) is a scalar positive function and can be expressed as

(27) \begin{align}\begin{array}{*{20}{l}}{{\kappa _k}\!\left( {{e_{1k}}} \right) = \frac{1}{{{\nu _1}{e_{1k}}^{\frac{{{m_1}}}{{{n_1}}} - \frac{{{p_1}}}{{{q_1}}}} + {\eta _1}}},k = 1,2 \ldots 6}\end{array}\end{align}

where ${\nu _1} \gt 0$ and ${\eta _1} \gt 0$ . Combined with the disturbance observer, the first derivative of the sliding mode surface is

(28) \begin{align}\dot s &= \frac{{{q_1}}}{{{p_1}}}\!\left[ {\boldsymbol\kappa \cdot {\boldsymbol{e}_{\textbf{2}}}{]^{\frac{{{q_1}}}{{{p_1}}} - 1}}} \right[ - {\nu _1}\!\left( {\frac{{{m_1}}}{{{n_1}}} - \frac{{{p_1}}}{{{q_1}}}} \right){\rm{diag}}{{({\boldsymbol{e}_{\textbf{1}}})}^{\frac{{{m_1}}}{{{n_1}}} - \frac{{{p_1}}}{{{q_1}}} - 1}}{\boldsymbol\kappa ^2}{\rm{diag}}\!\left( {{\boldsymbol{e}_{\textbf{2}}}} \right){\boldsymbol{e}_{\textbf{2}}}\nonumber\\[5pt] & + \boldsymbol\kappa \!\left( {\boldsymbol{h} + {\boldsymbol{M}_{\boldsymbol{C}}}\boldsymbol{u} + {\boldsymbol{d}_{\boldsymbol{s}}}} \right)] + {\boldsymbol{e}_{\boldsymbol{2}}} \end{align}

Considering all the possibilities of the initial state of the spacecraft error system, the fixed-time non-singular control law base on FT-DO can be expressed as

(29) \begin{align}\begin{array}{*{20}{l}}\left\{ \begin{array}{l}{\boldsymbol{u}} = {(\boldsymbol{M}_{\boldsymbol{C}}{\boldsymbol\kappa} )^{ - 1}}\left[ {\begin{array}{*{20}{l}}{{\nu _1}(\frac{{{m_1}}}{{{n_1}}} - \frac{{{p_1}}}{{{q_1}}}){{\rm diag}}{{(\boldsymbol{e}_{\textbf{1}})}^{\frac{{{m_1}}}{{{n_1}}} - \frac{{{p_1}}}{{{q_1}}} - 1}}{{\boldsymbol{{\kappa}}} ^2}{{\rm diag}}(\boldsymbol{e}_{\textbf{2}})\boldsymbol{e}_{\textbf{2}} - \frac{{{p_1}}}{{{q_1}}}{{\boldsymbol{{\kappa}}} ^{1 - \frac{{{q_1}}}{{{p_1}}}}}\boldsymbol{e}_{\textbf{2}}^{2 - \frac{{{q_1}}}{{{p_1}}}}}\end{array}} \right]\\[5pt]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \boldsymbol{M}_{\boldsymbol{C}}{}^{ - 1}({\boldsymbol{h}} + \hat{\boldsymbol{d}}_{\boldsymbol{s}} + \gamma {{\rm sign}}({\boldsymbol{s}})) - \boldsymbol{M}_{\boldsymbol{C}}{}^{ - 1}\frac{{{p_1}}}{{{q_1}}}{{\boldsymbol{{\kappa}}} ^{ - \frac{{{q_1}}}{{{p_1}}}}}\boldsymbol{e}_{\textbf{2}}^{1 - \frac{{{q_1}}}{{{p_1}}}}{{\rm diag}}({\boldsymbol{{\phi _\tau }}})({\nu _2}{{\boldsymbol{s}}^{\frac{{{m_2}}}{{{n_2}}}}} + {\eta _2}{{\boldsymbol{s}}^{\frac{{{p_2}}}{{{q_2}}}}}),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \boldsymbol{e}_{\textbf{2}}(0) \ne {\boldsymbol{{0}}}\\[5pt]{\boldsymbol{u}} = - \boldsymbol{M}_{\boldsymbol{C}}{}^{ - 1}({\boldsymbol{h}} + \hat{\boldsymbol{d}}_{\boldsymbol{s}} + \gamma {{\rm sign}}({\boldsymbol{s}})) - \boldsymbol{M}_{\boldsymbol{C}}{}^{ - 1}\frac{{{p_1}}}{{{q_1}}}{{\boldsymbol{{\kappa}}} ^{ - \frac{{{q_1}}}{{{p_1}}}}}({\nu _2}{{\boldsymbol{s}}^{\frac{{{m_2}}}{{{n_2}}}}} + {\eta _2}{{\boldsymbol{s}}^{\frac{{{p_2}}}{{{q_2}}}}}),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \boldsymbol{e}_{\textbf{2}}(0) = {\boldsymbol{{0}}}\end{array} \right.\end{array}\end{align}

where $\gamma $ ( $\gamma \gt \|{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}\|$ ) is the designed switching gain, and the parameters satisfy ${\nu _2} \gt 0$ and ${\eta _2} \gt 0$ . $\hat{\boldsymbol{d}}_{\boldsymbol{s}}$ ( $\hat{\boldsymbol{d}}_{\boldsymbol{s}} = {\boldsymbol\vartheta _{\textbf{2}}}$ ) denotes the estimated value of $\boldsymbol{d}_{\boldsymbol{s}}$ . Due to $1 \lt \frac{{{q_1}}}{{{p_1}}} \lt 2$ , the nonlinear function ${\phi _{\tau k}}\!\left( \cdot \right)\;:\;\left[ {0, + \infty } \right) \to \!\left[ {0,1} \right]$ is introduced to avoid the singularity of $\boldsymbol{e}_{\boldsymbol{2}}^{1 - \frac{{{q_1}}}{{{p_1}}}}$ in (29). ${\boldsymbol\phi _{\boldsymbol\tau} } = {[{\phi _{\tau 1}},{\phi _{\tau 2}}, \cdots {\phi _{\tau k}}]^T}\!\left( {k = 1,2 \ldots 6} \right)$ , and ${\phi _{\tau k}}$ is given by

(30) \begin{align}\begin{array}{*{20}{l}}{\phi _{\tau k}(f_k)} = \left\{ {\begin{array}{*{20}{c}} {\sin \!\left( {\frac{{\pi {f_k}}}{{2\varepsilon }}} \right)}&\;\;\;\;{,{f_k} \leqslant \varepsilon } \\[5pt] 1&\;\;\;\;{,{f_k} > \varepsilon }\end{array}} \right.\end{array}\end{align}

where ${f_k} = {\left| {{e_{2k}}} \right|^{\frac{{{q_1}}}{{{p_1}}} - 1}}$ , and $\varepsilon $ is a positive constant.

4.2 System stability analysis

Proof. The candidate Lyapunov function for the entire system is defined as

(31) \begin{align}\begin{array}{*{20}{l}}{V = \frac{1}{2}{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}^T{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}} + \frac{1}{2}{\boldsymbol{s}^T}\boldsymbol{s}}\end{array}\end{align}

To facilitate subsequent derivations, we denote the term $\frac{1}{2}{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}^T{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}$ as ${V_1}$ and the term $\frac{1}{2}{{{\boldsymbol{s}}}^T}{{\boldsymbol{s}}}$ as ${V_2}$ . The derivative of $V$ is

(32) \begin{align}\dot V = {\dot V_1} + {\dot V_2}\end{align}

For ${\dot V_1}$ , the following inequality holds.

(33) \begin{align} {{{\dot V}_1} = {\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}^T\dot{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}} \le \left\| {{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}^T\dot{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}} \right\| = \left\| {{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}^T} \right\|\left\| {{\lambda _3}{{{\boldsymbol{e}_{\boldsymbol{o}\textbf{1}}}} \over {{\boldsymbol{e}_{\boldsymbol{o}\textbf{1}}}}} + {\dot{\boldsymbol{d}}}_{\boldsymbol{s}}} \right\| \le {\lambda _3}\left\| {{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}} \right\| + \left\| {{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}^T} \right\|\left\| {\delta {d_s}} \right\|} \end{align}

Substituting the controller $u\!\left( {\boldsymbol{e}_{\boldsymbol{2}}\!\left( 0 \right) \ne 0} \right)$ in (29) into ${\dot{\boldsymbol{s}}}$ yields

(34) \begin{align} {{\dot V}_2} &= {\boldsymbol{s}^T}\dot{\boldsymbol{s}} \nonumber\\[5pt] & = {- {\nu _2}\!\left( {\mathop \sum \limits_{k = 1}^n \!\left( {{\phi _{\tau k}}{{\!\left( {\!\left| {{s_k}} \right|} \right)}^{{{{m_2} + {n_2}} \over {{n_2}}}}}} \right)} \right) - {\eta _2}\!\left( {\mathop \sum \limits_{k = 1}^n \!\left( {{\phi _{\tau k}}{{\!\left( {\left| {{s_k}} \right|} \right)}^{{{{p_2} + {q_2}} \over {{q_2}}}}}} \right)} \right) - {\boldsymbol{s}^T}\!\left( {{{{q_1}} \over {{p_1}}}{\boldsymbol\kappa ^{{{{q_1}} \over {{p_1}}}}}{\boldsymbol{e}_{\textbf{2}}}^{{{{q_1}} \over {{p_1}}} - 1}\!\left( {{\boldsymbol{e}_{{{\boldsymbol{o}}}\textbf{2}}} + \gamma {\rm{sign}}\!\left( \boldsymbol{s} \right)} \right)} \right)} \end{align}

Since the existence of the sinusoidal function ${\phi _{\tau k}}$ , the phase plane will be divided into two parts: ${P_1} = \left\{ {\left. {\left( {\boldsymbol{e}_{\boldsymbol{1}},\boldsymbol{e}_{\boldsymbol{2}}} \right)} \right|{f_k} \gt \varepsilon } \right\}$ and ${P_2} = \left\{ {\left. {\left( {\boldsymbol{e}_{\boldsymbol{1}},\boldsymbol{e}_{\boldsymbol{2}}} \right)} \right|{f_k} \le \varepsilon } \right\}$ , as shown in Fig. 4.

Figure 4. The phase plane diagram of the system.

Part 1: In area ${P_1}$ , ${\phi _{\tau k}} = 1$ . Since $\frac{{{m_2} + {n_2}}}{{2{n_2}}} \gt 1,\frac{1}{2} \lt \frac{{{p_2} + {q_2}}}{{2{q_2}}} \lt 1$ , based on Lemma 2, the following inequality can be obtained.

(35) \begin{align}\eqalign{ & {{\dot V}_2} \leqslant \left( {{\nu _2}\left( {{n^{{{{n_2} - {m_2}} \over {2{n_2}}}}}{{\left( {\mathop \sum \limits_{k = 1}^n {{\left| {{s_k}} \right|}^2}} \right)}^{{{{m_2} + {n_2}} \over {2{n_2}}}}}} \right) + {\eta _2}\left( {\mathop \sum \limits_{k = 1}^n {{\left( {{{\left| {{s_k}} \right|}^2}} \right)}^{{{{p_2} + {q_2}} \over {2{q_2}}}}}} \right)} \right) - {\boldsymbol{s}^T}\!\left( {{{{q_1}} \over {{p_1}}}{\boldsymbol\kappa ^{{{{q_1}} \over {{p_1}}}}}\boldsymbol{e}_{\boldsymbol{2}}^{{{{q_1}} \over {{p_1}}} - 1}\!\left( {{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}} + \gamma {\rm{sign}}\!\left( \boldsymbol{s} \right)} \right)} \right) \cr & \qquad\leqslant \left( {{\nu _2}\!\left( {{n^{{{{n_2} - {m_2}} \over {2{n_2}}}}}{{\left( {\mathop \sum \limits_{k = 1}^n {{\left| {{s_k}} \right|}^2}} \right)}^{{{{m_2} + {n_2}} \over {2{n_2}}}}}} \right) + {\eta _2}{{\!\left( {\mathop \sum \limits_{k = 1}^n {{\left| {{s_k}} \right|}^2}} \right)}^{{{{p_2} + {q_2}} \over {2{q_2}}}}}} \right) - {\boldsymbol{s}^T}\!\left( {{{{q_1}} \over {{p_1}}}{\boldsymbol\kappa ^{{{{q_1}} \over {{p_1}}}}}\boldsymbol{e}_{\boldsymbol{2}}^{{{{q_1}} \over {{p_1}}} - 1}\!\left( {{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}} + \gamma {\rm{sign}}\!\left( \boldsymbol{s} \right)} \right)} \right) \cr & \quad= - \left( {{\nu _2}\!\left( {{n^{{{{n_2} - {m_2}} \over {2{n_2}}}}} \cdot {2^{{{{m_2} + {n_2}} \over {2{n_2}}}}}{{\!\left( {{1 \over 2}\mathop \sum \limits_{k = 1}^n {{\left| {{s_k}} \right|}^2}} \right)}^{{{{m_2} + {n_2}} \over {2{n_2}}}}}} \right) + {\eta _2} \cdot {2^{{{{p_2} + {q_2}} \over {2{q_2}}}}}{{\left( {{1 \over 2}\mathop \sum \limits_{k = 1}^n {{\left| {{s_k}} \right|}^2}} \right)}^{{{{p_2} + {q_2}} \over {2{q_2}}}}}} \right) \cr & \qquad - {\boldsymbol{s}^T}\left( {{{{q_1}} \over {{p_1}}}{\boldsymbol\kappa ^{{{{q_1}} \over {{p_1}}}}}\boldsymbol{e}_{\boldsymbol{2}}^{{{{q_1}} \over {{p_1}}} - 1}\left( {{{{\boldsymbol{e}}}_{{{\boldsymbol{o}}}\textbf{2}}} + \gamma {\rm{sign}}\left( \boldsymbol{s} \right)} \right)} \right) \cr & \quad = - \left( {{n^{{{{n_2} - {m_2}} \over {2{n_2}}}}} \cdot {2^{{{{m_2} + {n_2}} \over {2{n_2}}}}}{\nu _2}{{\left( {{V_2}} \right)}^{{{{m_2} + {n_2}} \over {2{n_2}}}}} + {2^{{{{p_2} + {q_2}} \over {2{q_2}}}}}{\eta _2}{{\left( {{V_2}} \right)}^{{{{p_2} + {q_2}} \over {2{q_2}}}}}} \right) - {s^T}\left( {{{{q_1}} \over {{p_1}}}{\boldsymbol\kappa ^{{{{q_1}} \over {{p_1}}}}}\boldsymbol{e}_{\boldsymbol{2}}^{{{{q_1}} \over {{p_1}}} - 1}\left( {{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}} + \gamma {\rm{sign}}\left( \boldsymbol{s} \right)} \right)} \right) \cr} \end{align}

where $n = 6$ . Therefore, $\dot V$ satisfies

(36) \begin{align}\eqalign{ & \dot V = {{\dot V}_1} + {{\dot V}_2} \cr & {\rm{\;}} \le - \left( {{n^{{{{n_2} - {m_2}} \over {2{n_2}}}}} \cdot {2^{{{{m_2} + {n_2}} \over {2{n_2}}}}}{\nu _2}{{\left( {{V_2}} \right)}^{{{{m_2} + {n_2}} \over {2{n_2}}}}} + {2^{{{{p_2} + {q_2}} \over {2{q_2}}}}}{\eta _2}{{\left( {{V_2}} \right)}^{{{{p_2} + {q_2}} \over {2{q_2}}}}}} \right) - {\boldsymbol{s}^T}\left( {{{{q_1}} \over {{p_1}}}{\boldsymbol\kappa ^{{{{q_1}} \over {{p_1}}}}}\boldsymbol{e}_{\boldsymbol{2}}^{{{{q_1}} \over {{p_1}}} - 1}\left( {{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}} + \gamma {\rm{sign}}\left( {{\boldsymbol{s}}} \right)} \right)} \right) \cr & \quad + {\lambda _3}||{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}|| + ||{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}^T||\,\,||\delta {d_s}|| \cr & \le - {n^{{{{n_2} - {m_2}} \over {2{n_2}}}}} \cdot {2^{{{{m_2} + {n_2}} \over {2{n_2}}}}}{\nu _2}{\left( {{V_2}} \right)^{{{{m_2} + {n_2}} \over {2{n_2}}}}} - {2^{{{{p_2} + {q_2}} \over {2{q_2}}}}}{\eta _2}{\left( {{V_2}} \right)^{{{{p_2} + {q_2}} \over {2{q_2}}}}} - {n^{{{{n_2} - {m_2}} \over {2{n_2}}}}} \cdot {2^{{{{m_2} + {n_2}} \over {2{n_2}}}}}{\nu _2}{({V_1})^{{{{m_2} + {n_2}} \over {2{n_2}}}}} - {2^{{{{p_2} + {q_2}} \over {2{q_2}}}}}{\eta _2}{\left( {{V_1}} \right)^{{{{p_2} + {q_2}} \over {2{q_2}}}}} \cr & \quad+ {n^{{{{n_2} - {m_2}} \over {2{n_2}}}}} \cdot {2^{{{{m_2} + {n_2}} \over {2{n_2}}}}}{\nu _2}{({V_1})^{{{{m_2} + {n_2}} \over {2{n_2}}}}} + {2^{{{{p_2} + {q_2}} \over {2{q_2}}}}}{\eta _2}{\left( {{V_1}} \right)^{{{{p_2} + {q_2}} \over {2{q_2}}}}} + {\lambda _3}||{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}|| + ||{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}^T||\,\,||\delta {d_s}|| \cr & = - {n^{{{{n_2} - {m_2}} \over {2{n_2}}}}} \cdot {2^{{{{m_2} + {n_2}} \over {2{n_2}}}}}{\nu _2}\left( {{{\left( {{V_2}} \right)}^{{{{m_2} + {n_2}} \over {2{n_2}}}}} + {{\left( {{V_1}} \right)}^{{{{m_2} + {n_2}} \over {2{n_2}}}}}} \right) - {2^{{{{p_2} + {q_2}} \over {2{q_2}}}}}{\eta _2}\left( {{{\left( {{V_2}} \right)}^{{{{p_2} + {q_2}} \over {2{q_2}}}}} + {{\left( {{V_1}} \right)}^{{{{p_2} + {q_2}} \over {2{q_2}}}}}} \right) \cr & \quad + {n^{{{{n_2} - {m_2}} \over {2{n_2}}}}} \cdot {2^{{{{m_2} + {n_2}} \over {2{n_2}}}}}{\nu _2}{\left( {{1 \over 2}||{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}|{|^2}} \right)^{{{{m_2} + {n_2}} \over {2{n_2}}}}} + {2^{{{{p_2} + {q_2}} \over {2{q_2}}}}}{\eta _2}{\left( {{1 \over 2}||{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}|{|^2}} \right)^{{{{p_2} + {q_2}} \over {2{q_2}}}}} + {\lambda _3}||{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}|| + ||{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}^T||\,\,||\delta {d_s}|| \cr} \end{align}

Let $\varphi = {\nu _3}{\left( {\frac{1}{2}\|{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}\|^2} \right)^{\frac{{{m_2} + {n_2}}}{{2{n_2}}}}} + {\eta _3}{\left( {\frac{1}{2}\|{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}\|^2} \right)^{\frac{{{p_2} + {q_2}}}{{2{q_2}}}}} + {\lambda _3}\|{\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}\| +\| {\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}}^T\|\|\delta {d_s}\|$ , where ${\nu _3} = {n^{\frac{{{n_2} - {m_2}}}{{2{n_2}}}}} \cdot {2^{\frac{{{m_2} + {n_2}}}{{2{n_2}}}}}{\nu _2}$ , ${\eta _3} = {2^{\frac{{{p_2} + {q_2}}}{{2{q_2}}}}}{\eta _2}$ , Equation (36) can be rewritten as

(37) \begin{align}\dot V \le - {\nu _3}\left( {{{\left( {{V_2}} \right)}^{{{{m_2} + {n_2}} \over {2{n_2}}}}} + {{\left( {{V_1}} \right)}^{{{{m_2} + {n_2}} \over {2{n_2}}}}}} \right) - {\eta _3}\left( {{{\left( {{V_2}} \right)}^{{{{p_2} + {q_2}} \over {2{q_2}}}}} + {{\left( {{V_1}} \right)}^{{{{p_2} + {q_2}} \over {2{q_2}}}}}} \right) + \varphi \end{align}

According to Lemma 2, there are ${\left( {{V_2}} \right)^{{{{m_2} + {n_2}} \over {2{n_2}}}}} + {\left( {{V_1}} \right)^{{{{m_2} + {n_2}} \over {2{n_2}}}}} \ge {2^{{{{n_2} - {m_2}} \over {2{n_2}}}}}{V^{{{{m_2} + {n_2}} \over {2{n_2}}}}}$ and ${\left( {{V_2}} \right)^{{{{p_2} + {q_2}} \over {2{q_2}}}}} + {\left( {{V_1}} \right)^{{{{p_2} + {q_2}} \over {2{q_2}}}}} \ge {V^{{{{p_2} + {q_2}} \over {2{q_2}}}}}$ . Then

(38) \begin{align}\dot V \le - {\eta _3}\left( {{V^{{{{p_2} + {q_2}} \over {2{q_2}}}}}} \right) - {2^{{{{n_2} - {m_2}} \over {2{n_2}}}}}{\nu _3}\left( {{V^{{{{m_2} + {n_2}} \over {2{n_2}}}}}} \right) + \varphi \end{align}

Based on Theorem 1, the estimation error will converge to a small bounded value near zero. Therefore, $\varphi $ is a small positive number. Let ${\nu _4} = {2^{\frac{{{n_2} - {m_2}}}{{2{n_2}}}}}{\nu _3}$ , based on Lemma Reference Dony and Dong3, the whole system is fixed-time stable. The convergence time is bounded by

(39) \begin{align}{T_s} \le \left( {1/\left( {{\nu _4}{\theta _s}\left( {{{{m_2}} \over {{n_2}}} - 1} \right)} \right) + 1/\left( {{\eta _3}{\theta _s}\left( {1 - {{{p_2}} \over {{q_2}}}} \right)} \right)} \right) + {T_o}\end{align}

where ${\theta _s} \in \left( {0,1} \right)$ , and ${T_o}$ is the convergence time of the observer.

Part 2: In area ${P_2}$ , there is $0 \lt {\phi _{\tau k}} \lt 1$ . When ${{{\boldsymbol{e}}}_{\textbf{2}}} \boldsymbol\ne \textbf{0}$ , similar to the steps in Part 1, it is not difficult to prove that $\boldsymbol{s} = \textbf{0}$ is still a domain of attraction. In this part, we need to prove that when $\boldsymbol{e}_{\boldsymbol{2}} \to \textbf{0}$ , only the origin $\left( {\boldsymbol{e}_{\boldsymbol{1}},\boldsymbol{e}_{\boldsymbol{2}}} \right) = \left( {\textbf{0},\textbf{0}} \right)$ is an attractor.

When $\boldsymbol{e}_{\boldsymbol{2}} \to \textbf{0}$ , ${\phi _{\tau k}}{e_{2k}}^{1 - \frac{{{q_1}}}{{{p_1}}}} \to 1$ holds. The control law is

(40) \begin{align}\boldsymbol{u} = - {\boldsymbol{M}_{\boldsymbol{C}}}^{ - 1}\left( {\boldsymbol{h} + {{\hat{\boldsymbol{d}}}_{\boldsymbol{s}}} + \gamma {\rm{sign}}\left( s \right)} \right) - {\boldsymbol{M}_{\boldsymbol{C}}}^{ - 1}{{{p_1}} \over {{q_1}}}{\boldsymbol\kappa ^{ - {{{q_1}} \over {{p_1}}}}}\left( {{\nu _2}{\boldsymbol{s}^{{{{m_2}} \over {{n_2}}}}} + {\eta _2}{\boldsymbol{s}^{{{{p_2}} \over {{q_2}}}}}} \right)\end{align}

Then substitute (40) into ${{\dot{\boldsymbol{e}}}_{\textbf{2}}}$ , and we can get

(41) \begin{align} \dot{\boldsymbol{e}}_{\textbf{2}} &= {\boldsymbol{d}_{\boldsymbol{s}}} - \hat{\boldsymbol{d}}_{\boldsymbol{s}} - \gamma {\rm{sign}}\left( \boldsymbol{s} \right) - {{{p_1}} \over {{q_1}}}{\boldsymbol\kappa ^{ - {{{q_1}} \over {{p_1}}}}}\left( {{\nu _2}{\boldsymbol{s}^{{{{m_2}} \over {{n_2}}}}} + {\eta _2}{\boldsymbol{s}^{{{{p_2}} \over {{q_2}}}}}}\right) \nonumber \\[5pt] &= - {\boldsymbol{e}_{\boldsymbol{o}\textbf{2}}} - \gamma {\rm{sign}}\left( \boldsymbol{s} \right) - \frac{{{p_1}}}{{{q_1}}}{\boldsymbol\kappa ^{ - \frac{{{q_1}}}{{{p_1}}}}}\left( {{\nu _2}{{{\boldsymbol{s}}}^{\frac{{{m_2}}}{{{n_2}}}}} + {\eta _2}{{{\boldsymbol{s}}}^{\frac{{{p_2}}}{{{q_2}}}}}} \right)\end{align}

Since $\gamma \gt \|{{{\boldsymbol{e}}}_{{{\boldsymbol{o}}}\textbf{2}}}\|$ , then $\boldsymbol{s} \gt \textbf{0},\dot{\boldsymbol{e}}_{\textbf{2}} \lt \textbf{0}$ and $\boldsymbol{s}\left\langle {\textbf{0},{{\dot{\boldsymbol{e}}}_{\textbf{2}}}} \right\rangle \textbf{0}$ hold. Therefore, if $\boldsymbol{e}_{\boldsymbol{2}}$ is in area ${P_2}$ , it will increase or decrease monotonically until it leaves area ${P_2}$ . That is, all states in area ${P_2}$ will enter ${P_1}$ , as shown in Fig. 4. The $\boldsymbol{e}_{\boldsymbol{1}}$ -axis is not an attractor. The crossing time is denoted as ${T_\varepsilon }\left( \varepsilon \right)$ , which is very small and can be ignored.

Therefore, for the tracking error dynamics system (16), if the fixed-time disturbance observer (20) and the fixed-time terminal sliding mode controller (29) are designed, the states of the tracking error system can converge to the neighbourhood of zero in a fixed-time bounded by ${T_c} = {T_s} + {T_\varepsilon }$ .

5.1 Simulation parameters

For the orbit control of spacecraft, the scenario is set as manoeuvring a circular low Earth orbit spacecraft to another circular orbit within the same orbital plane. In this scenario, adjusting the spacecraft’s semi-major axis is sufficient. Therefore, the initial and desired values of the spacecraft’s orbital elements $\left[ {a\left( {{\rm{km}}} \right),\bar e,i\left( {{\rm{deg}}} \right),{\rm{\Omega }}\left( {{\rm{deg}}} \right),\omega \left( {{\rm{deg}}} \right),{M_o}\left( {{\rm{deg}}} \right)} \right]$ can be set as $\left[ {7000,0,28.5,100,0,130} \right]$ and $\left[ {7000.5,0,28.5,100,0,130} \right]$ , respectively. Simultaneously, it is also desired for the spacecraft’s attitude to effectively track the desired values. The initial tracking errors of orbit and attitude of the spacecraft are given in Table 1.

Table 1. The initial tracking errors of orbit and attitude

Figure 5. The estimation error convergence curves of the proposed FT-DO.

Figure 6. Comparison of sliding mode surface convergence curves for position tracking error in three methods.

Figure 7. Comparison of sliding mode surface convergence curves for velocity tracking error in three methods.

The other parameters involved in the control process are as follows. The geocentric gravitational constant is ${u_g} = 3.986 \times {10^{14}}$ m ${^3}$ s ${^{ - 2}}$ . The nominal mass and inertial matrix are ${m_o} = 600$ kg and ${\boldsymbol{J}}_{\boldsymbol{o}} = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{166.5} & {4.44} & {3.33}\\[5pt] {4.44}& {74}& {5.18}\\[5pt] {3.33}& {5.18}& {62.9}\end{array}} \right]$ kgm ${^2}$ respectively. The external disturbance force and torque are ${\boldsymbol{d}}_{\boldsymbol{f}} = {[{\rm{sin}}\left( {0.1t} \right),{\rm{cos}}\left( {0.2t} \right),{\rm{sin}}\left( {0.3t} \right)]^T}\;{\rm{*}}\;{10^{ - 5}}$ N and $\boldsymbol{d}_{\boldsymbol\tau} = {[2{\rm{sin}}\left( {0.1t} \right),2{\rm{cos}}\left( {0.2t} \right),2{\rm{sin}}\left( {0.3t} \right)]^T}\;{\rm{*}}\;{10^{ - 6}}$ Nm. The uncertainties are: ${\boldsymbol{\Delta}\boldsymbol{J}} = \left[ \begin{array}{ccc}{0.1 + 0.01{\;\rm{*\;sin}}\left( {0.1t} \right)}& 0& 0\\[5pt] 0 & {0.1 + 0.01{\;\rm{*\;sin}}\left( {0.2t} \right)} & 0\\[5pt] 0& 0 &{0.1 + 0.01{\;\rm{*\;sin}}\left( {0.3t} \right)}\end{array} \right]$ kgm ${^2}$ and ${\rm{\Delta }}m = 1 + 0.1{\;\rm{*\;sin}}\left( {0.1t} \right)$ kg. The initial conditions of the fixed-time disturbance observer are $\boldsymbol{\vartheta _1}\left( 0 \right) = {[0.00001,0.00002,0.00004,0.00002,0.00001,0.00002]^T}$ and ${\boldsymbol\vartheta _{\textbf{2}}}\left( 0 \right) = {[1,1,1,0.2, - 0.4,0.4]^T}\;{\rm{*}}\;{10^{ - 6}}$ . The focus of this paper revolves around the scenario of performing orbit and attitude adjustments within a small range on a small satellite. Therefore, the maximum control force and torque are preset as $2$ N and $1$ Nm, respectively.

The designed parameters of the observer are $p = 1.2$ , ${\lambda _1} = 0.5$ , ${\lambda _2} = 0.1$ and ${\lambda _3} = 0.1$ . The designed parameters of the controller (29) are ${\nu _1} = 0.001$ , ${\eta _1} = 0.04$ , ${\nu _2} = 0.01$ , ${\eta _2} = 0.001$ , $\frac{{{m_1}}}{{{n_1}}} = 2$ , $\frac{{{p_1}}}{{{q_1}}} = 0.9$ , $\frac{{{m_2}}}{{{n_2}}} = 1.1$ , $\frac{{{p_2}}}{{{q_2}}} = 0.6$ , $\varepsilon = 0.01$ and $\gamma = 0.05$ .

Figure 8. Comparison of tracking errors convergence curves for position in three methods.

Figure 9. Comparison of tracking errors convergence curves for velocity in three methods.

Figure 10. Comparison of tracking errors convergence curves for MRPs in three methods.

Figure 11. Comparison of tracking errors convergence curves for angular velocity in three methods.

Figure 12. Comparisons of input force.

Figure 13. Comparisons of input torque.

5.2 Simulation implementation and results

5.2.2 Simulation cases and results

The following two cases have been implemented to verify the effectiveness of the proposed FT-AS control method.

Case 1: Compare the proposed FT-AS controller (29) with the traditional controller to verify its effectiveness in handling input saturation. The traditional controller is [Reference Wu, Chen, Zhang, Chen and Shao24]

\begin{align*}{{{\boldsymbol{u}}}_{\boldsymbol{T}}} = {({{\boldsymbol{M}}_{\boldsymbol{C}}}\boldsymbol{\kappa} )^{ - 1}}\left[ {{\nu _1}\left( {\frac{{{m_1}}}{{{n_1}}} - \frac{{{p_1}}}{{{q_1}}}} \right){\rm{diag}}{{({{\boldsymbol{e}}_1})}^{\frac{{{m_1}}}{{{n_1}}} - \frac{{{p_1}}}{{{q_1}}} - 1}}{\boldsymbol{\kappa} ^2}{\rm{diag}}\left( {\boldsymbol{e}_{\textbf{2}}} \right)\boldsymbol{e}_{\textbf{2}} - \frac{{{p_1}}}{{{q_1}}}{\boldsymbol{\kappa} ^{1 - \frac{{{q_1}}}{{{p_1}}}}}\boldsymbol{e}_{\textbf{2}}^{2 - \frac{{{q_1}}}{{{p_1}}}}} \right]\end{align*}

(42) \begin{align} - \boldsymbol{M}_{\boldsymbol{C}}{}^{ - 1}\left( {{\boldsymbol{h}} + \hat{\boldsymbol{d}} + \gamma {\rm{sign}}\left( \boldsymbol{s} \right)} \right) - \boldsymbol{M}_{\boldsymbol{C}}{}^{ - 1}{{{p_1}} \over {{q_1}}}{\boldsymbol{\kappa} ^{ - {{{q_1}} \over {{p_1}}}}}\boldsymbol{e}_{\textbf{2}}^{1 - {{{q_1}} \over {{p_1}}}}{\rm{diag}}\left( {\boldsymbol{\phi}_{\boldsymbol\tau }} \right)\left( {{\nu _2}{{\boldsymbol{s}}^{{{{m_2}} \over {{n_2}}}}} + {\eta _2}{{\boldsymbol{s}}^{{{{p_2}} \over {{q_2}}}}}} \right)\end{align}

where $\hat{\boldsymbol{d}}$ in (42) is the disturbance estimate that does not consider the adverse effects of input saturation.

Case 2: To demonstrate the effectiveness of the proposed FT-AS control system in reducing tracking error convergence time, it is compared with traditional sliding mode AS control systems with additional anti-saturation structure. The controller still uses (42).

The simulation results are shown in Figs. 5–13.

Figure 5 shows the estimation error convergence curve of the proposed FT-DO. The convergence time is within $5$ s, which is very short. Thus, the influence of the convergence time of the proposed FT-DO on the overall system convergence time can be negligible. The estimation errors converge to $0.05$ m/s ${^2}$ and $2\;{\rm{*}}\;{10^{ - 4}}$ , respectively, indicating that the proposed FT-DO can effectively estimate lumped disturbance. The observations of the disturbances will be fed back into the controller and suppressed using controller gains.

The comparisons between the proposed scheme and the above two cases are shown in Figs. 6–13.

Figures 6 and 7 compare the convergence of the sliding surfaces for the three mentioned control methods. ${s_1},{s_2}$ and ${s_3}$ represent the sliding surfaces for position tracking errors, while ${s_4}$ , ${s_5}$ and ${s_6}$ represent the sliding surfaces for attitude tracking errors. As can be seen from Fig. 6, the proposed FT-AS method and the method used in Case 1 exhibit almost no difference in the convergence time of sliding surfaces ${s_1}$ , ${s_2}$ and ${s_3}$ . Therefore, the position tracking errors of these two methods reach the sliding surfaces almost simultaneously. However, in Case 2, which involves an additional saturation structure, the time it takes for the position tracking errors to reach the sliding surfaces is the longest, indicating that its performance is significantly inferior to the previous two methods. The sliding surfaces ${s_4}$ , ${s_5}$ , and ${s_6}$ are shown in Fig. 7. The method in Case 1 exhibits the shortest convergence time, followed by the proposed FT-AS method. The worst performance is still Case 2. This indicates that the additional anti-saturation structure significantly affects the convergence time of the sliding surface. Therefore, it is meaningful to design a more concise control loop without the additional anti-saturation subsystem.

Figures 8 and 9 demonstrate the convergence of position and velocity tracking errors for the three methods. For the position tracking errors ${\boldsymbol{r}_{\boldsymbol{e}}}$ shown in Fig. 8, Case 2 exhibits the poorest convergence time and accuracy. The method in Case 1 and the proposed FT-AS method have similar convergence times for position tracking errors, but the FT-AS method outperforms in terms of convergence accuracy. The convergence accuracy by the proposed FT-AS method is within $2\;{\rm{*}}\;{10^{ - 4}}$ m, Case 1 is within $5\;{\rm{*}}\;{10^{ - 4}}$ m, and Case 2 is within $0.5$ m. For velocity tracking errors ${\boldsymbol{v}_{\boldsymbol{e}}}$ shown in Fig. 9, although the three methods have similar convergence accuracy, both Case 1 and the proposed FT-AS method outperform Case 3 in terms of convergence time. Therefore, the proposed FT-AS method exhibits advantages in convergence time and accuracy of position and velocity tracking errors.

The attitude tracking errors of the three methods are shown in Figs. 10 and 11. For the tracking errors of the MRPs and the angular velocity, the shortest convergence time is Case 1, followed by the proposed FT-AS method, both of which have a shorter convergence time than Case 2. In addition, for the convergence accuracy of attitude tracking errors, the MRPs and angular velocity of the proposed method and Case 1 converge to $1\;{\rm{*}}\;{10^{ - 4}}$ and $2\;{\rm{*}}\;{10^{ - 7}}$ rad $/$ s, respectively. However, the MRPs and angular velocity of Case 2 converge to $2\;{\rm{*}}\;{10^{ - 3}}$ and $2\;{\rm{*}}\;{10^{ - 5}}$ rad $/$ s, respectively. Therefore, the proposed FT-AS method and the method used in Case 1 perform well.

Figures 12 and 13 illustrate the control force and control moment curves. Both the proposed FT-AS method and Case 2 method ensure that the control force and control moment remain within the specified limits. In Case 1, however, without saturation limits, the control force and moment initially exceed the actuator’s limits. Such large control energy will accelerate the convergence of the sliding mode surface, position tracking error, and attitude tracking error. Nonetheless, this poses a danger in practical applications as it may cause damage to the spacecraft’s actuators. At the beginning of the simulation, all three methods exhibit high control force and moment, as large forces and moments are required for orbital and attitude manoeuvres. In the later stages, the control force and torque stabilise within the desired range, which is utilised to overcome spatial disturbances.

Nevertheless, it should be noted that the proposed FT-AS method still has some drawbacks, such as incomplete elimination of control force and moment chattering. Furthermore, the estimation of the system’s convergence time is relatively conservative.

In conclusion, the FT-DO proposed in this paper can effectively suppress the adverse effects of input saturation. The proposed FT-AS control system has higher control accuracy and shorter convergence time. Compared to traditional control schemes with additional saturation structures, this scheme simplifies the entire control system structure and has good control performance.