2.1 Notations and lemmas

Notations. ${\mathbf{z}} = {[{z_1},{z_2}, \ldots, {z_n}]^{\mathrm{T}}}$ is an n-dimensional vector with the $i$ th element ${z_i}$ . The symbols $\parallel \cdot \parallel $ denotes the Euclidean norm of a vector or the induced norm of a matrix. For a positive constant $a \gt 0$ , define the vector ${\mathrm{si}}{{\mathrm{g}}^a}({\mathbf{z}}) = {\left[ {\left| {{z_1}{|^a}{\mathrm{sign}}\left( {{z_1}} \right),} \right|{z_2}{|^a}{\mathrm{sign}}\left( {{z_2}} \right), \ldots, |{z_n}{|^a}{\mathrm{sign}}\left( {{z_n}} \right)} \right]^{\mathrm{T}}}$ and the vector ${\mathrm{sgn}}({\mathbf{z}}) = {[{\mathrm{sign}}\left( {{z_1}} \right),{\mathrm{sign}}\left( {{z_2}} \right), \ldots, {\mathrm{sign}}\left( {{z_n}} \right)]^{\mathrm{T}}}$ .

Definition 1. ([Reference Polyakov35]) Consider a nonlinear system

(1) \begin{align}\dot{\boldsymbol{y}}(t) = g\left( {\boldsymbol{y}(t)} \right),\boldsymbol{y}\left( \textbf{0} \right) = {\boldsymbol{y}_0},\boldsymbol{y} \in {\mathbb{R}^n}\end{align}

If the system (1) is finite-time convergent and the settling time $t\left( {{\boldsymbol{y}_0}} \right)$ is bounded for all initial states, then the system’s equilibrium is said to be fixed-time stable, i.e., $\exists {t_{max}} \gt 0$ such that $t\left( {{y_0}} \right) \le {t_{max}},\forall\boldsymbol{y}_{0} \in {\mathbb{R}^n}$ .

Lemma 1. ([Reference Jiang, Hu and Friswell36, Reference Cao, Xiao, Golestani and Ran37]) Let ${v_j} \geq 0\left( {j = 1,2, \ldots, N} \right)$ , for two constants ${w_1}$ and ${w_2}$ satisfying $0 \lt {w_1} \le 1$ and ${w_2} \gt 1$ , the following inequalities hold

(2) \begin{align}\mathop \sum \limits_{j = 1}^N v_j^{{w_1}} \geq {\left( {\mathop \sum \limits_{j = 1}^N {v_j}} \right)^{{w_1}}},\mathop \sum \limits_{j = 1}^N v_j^{{w_2}} \geq {N^{1 - {w_2}}}{\left( {\mathop \sum \limits_{j = 1}^N {v_j}} \right)^{{w_2}}}\end{align}

2.2 Mathematical modelling

For the final approaching phase of target capture, the primary objective is to precisely manoeuver the joint in accordance with the planned trajectory. To describe the space manipulator ’s trajectory tracking issue, the reference coordinate systems, which include the inertial coordinate system ${O_I}{X_I}{Y_I}{Z_I}$ and the body coordinate system ${O_b}{X_b}{Y_b}{Z_b}$ , are depicted in Fig. 1. In the free-floating mode, the spacecraft has its control system turned off and there is no external force or torques acting on the system. Therefore, motion of the manipulator will result in translation and rotation of the spacecraft. Considering the actual working environment of FFSM, the following reasonable assumptions are given [Reference Shao, Sun, Xue and Li14]:

Figure 1. System model for a free-floating space manipulator.

Based on the relative relations in Fig. 1, the position vector from ${B_i}$ to ${O_I}$ can be represented as:

(3) \begin{align}\boldsymbol{r}_{i} = \boldsymbol{r}_{b} + \boldsymbol{b}_{0} + \mathop \sum \limits_{j = 1}^{i - 1} \left( {\boldsymbol{a}_{j} + \boldsymbol{b}_{j}} \right) + \boldsymbol{a}_{i}\quad i = 1,2, \ldots, n\end{align}

The line and angular velocity of ${B_i}$ can be expressed as

(4) \begin{align}\boldsymbol{v}_{i} = \boldsymbol{v}_{b} + \boldsymbol{\omega} _{b} \times \left( {\boldsymbol{r}_{i} - \boldsymbol{r}_{b}} \right) + \mathop \sum \limits_{j = 1}^i \left( {{z_j} \times \left( {\boldsymbol{r}_{i} - \boldsymbol{\rho} _{j}} \right)} \right){\dot q_j}\end{align}

(5) \begin{align}\boldsymbol{\omega} _{i} = \boldsymbol{\omega} _{b} + \mathop \sum \limits_{j = 1}^i {\dot q_j}\boldsymbol{z}_{j}\end{align}

For zero initial momentum, the conservation equation of an FFSM system can be deduced as

(6) \begin{align}\left[ {\begin{array}{*{20}{l}}\boldsymbol{P}\\[3pt]\boldsymbol{L}\end{array}} \right] = \left[ {\begin{array}{l@{\quad}l}{M_g}{\boldsymbol{E}_3} & {M_g}{{\left( {\boldsymbol{r}_{bg}^ \times } \right)}^T}\\[3pt]{M_g}\boldsymbol{r}_{bg}^ \times & {\boldsymbol{H}_\omega }\end{array}} \right]\left[ {\begin{array}{*{20}{l}}{{\boldsymbol{v}_b}}\\[3pt]{{\omega _b}}\end{array}} \right] + \left[ {\begin{array}{*{20}{l}}{{\boldsymbol{J}_{T\omega }}}\\[3pt]{{\boldsymbol{H}_{\omega \phi }}}\end{array}} \right]\dot{\boldsymbol{q}} = \textbf{0}\end{align}

where the system total linear momentum $\boldsymbol{P} = {M_b}\boldsymbol{v}_{b} + \mathop \sum \nolimits_{i = 1}^n {M_i}\boldsymbol{v}_{i}$ , and the system total angular momentum $\boldsymbol{L} = {I_b}\boldsymbol{\omega} _{b} + \boldsymbol{r}_{b} \times {M_b}\boldsymbol{v}_{b} + \mathop \sum \nolimits_{i = 1}^n \boldsymbol{I}_{i}{\omega _i} + \mathop \sum \nolimits_{i = 1}^n \left( {r_i} \times {\boldsymbol{M}}_{i}\boldsymbol{r}_{i} \right)$ , the total mass ${M_g} = {M_b} + \mathop \sum \nolimits_{i = 1}^n {M_i}$ , $\boldsymbol{r}_{{bg}} = \boldsymbol{r}_{g} - \boldsymbol{r}_{b}$ , $\boldsymbol{H}_{\omega } = \mathop \sum \nolimits_{i = 1}^n \left( {\boldsymbol{I}_{i} - {M_i}{{(\boldsymbol{r}_{i} - \boldsymbol{r}_{b})}^ \times }{{(\boldsymbol{r}_{i} - \boldsymbol{r}_{b})}^ \times }} \right) + \boldsymbol{I}_{b}$ , $\boldsymbol{H}_{\omega \phi} = \mathop \sum \nolimits_{i = 1}^n \left( \boldsymbol{I}_{i}\boldsymbol{J}_{\omega i} + {M_i}\left( \boldsymbol{r}_{i} - \boldsymbol{r}_{b} \right) \times \boldsymbol{J}_{Ti} \right)$ , $\boldsymbol{J}_{T\omega } = \mathop \sum \nolimits_{i = 1}^n \left( {M_i}{\boldsymbol{J}_{{Ti}}} \right)$ , $\boldsymbol{J}_{{\omega i}} = \left[ {{z_1}\;\;\;\;{z_2}\;\;\;\; \ldots \;\;\;\;{\textbf{0}_{3 \times \left( {n - i} \right)}}} \right]$ , and ${\boldsymbol{J}}_{Ti} = \left[ {\mathbf{z}}_1 \times \left( \mathbf{r}_i - {\rho _1} \right) \ldots {{\mathbf{z}}_i} \times \left( \mathbf{r}_i - {\rho _i} \right)\textbf{0}_{3 \times \left( {n - i} \right)} \right]$ . Moreover, the total kinetic energy of the FFSM can be derived

(7) \begin{align}T = \frac{1}{2}\left( {\boldsymbol{\omega} _b^T{I_b}{\omega _b} + {M_b}\boldsymbol{v}_b^T{\boldsymbol{v}_b} + \mathop \sum \limits_{i = 1}^n \left( {\omega _i^T{I_i}{\omega _i} + {M_i}\boldsymbol{v}_i^T{\boldsymbol{v}_i}} \right)} \right)\end{align}

Substituting (4)–(6) into (7) and rearranging it, ones eventually have

(8) \begin{align}T = \frac{1}{2}{\dot{\boldsymbol{q}}^{\mathrm{T}}}\boldsymbol{H}\dot{\boldsymbol{q}}\end{align}

where $\boldsymbol{H}$ is called the generalised inertia tensor of the space manipulator, which can expressed as $\boldsymbol{H} = \boldsymbol{H}_m + \boldsymbol{H}_{bm}^{\mathrm{T}}\boldsymbol{J}_{bm}$ . And the matrices $\boldsymbol{H}_m$ is the inertia matrix of the manipulator, $\boldsymbol{J}_{bm}$ is the Jacobian matrix between the base spacecraft and the manipulator, and $\boldsymbol{H}_{bm}$ is the coupling matrix between the base spacecraft and the manipulator.

Considering the time-varying disturbances, the mathematical model of FFSM through the application of the Lagrange equation is given by:

(9) \begin{align}\boldsymbol{H}(\boldsymbol{q})\ddot{\boldsymbol{q}} + \boldsymbol{C}({\boldsymbol{q},\dot{\boldsymbol{q}}})\dot{\boldsymbol{q}} = \boldsymbol{u} + \boldsymbol{d}(t)\end{align}

where, $\boldsymbol{q}$ , $\dot{\boldsymbol{q}}$ , $\ddot{\boldsymbol{q}} \in {\mathbb{R}^n}$ are the joint generalised position, velocity and acceleration vectors, respectively. The inertia matrix $\boldsymbol{H}(\boldsymbol{q}) \in {\mathbb{R}^{n \times n}}$ is described by $\boldsymbol{H}(\boldsymbol{q}) = {\boldsymbol{H}}_0(\boldsymbol{q}) + \boldsymbol{\Delta} H(\boldsymbol{q})$ , and the Coriolis and Centrifugal matrix $\boldsymbol{C}(\boldsymbol{q},\dot{\boldsymbol{q}}) \in {\mathbb{R}^{n \times n}}$ is described by ${\boldsymbol{C}({\boldsymbol{q},\dot{\boldsymbol{q}}})} = \boldsymbol{C}_{0}({\boldsymbol{q},\dot{\boldsymbol{q}}}) + \Delta \boldsymbol{C}({\boldsymbol{q},\dot{\boldsymbol{q}}})$ , in which $\boldsymbol{H}_{0}{(\boldsymbol{q})}$ and $\boldsymbol{C}_{0} {(\boldsymbol{q},\dot{\boldsymbol{q}})}$ indicate the nominal matrices, ${{\Delta }}\boldsymbol{H}(\boldsymbol{q})$ and ${{\Delta }} \boldsymbol{C}({\boldsymbol{q},\dot{\boldsymbol{q}}})$ represent the deviations caused by the parametric uncertainty, $\boldsymbol{d}(t) \in {\mathbb{R}^n}$ denotes the time-varying disturbance acting on the space manipulator system, and $\boldsymbol{u} \in {\mathbb{R}^n}$ represents the control input.

Considering the desired position and velocity $\boldsymbol{q}_{d}$ and $\dot{\boldsymbol{q}}_{d}$ , define $\boldsymbol{e}_{1} = \boldsymbol{q}_{1} - \boldsymbol{q}_{d}$ and $\boldsymbol{e}_{2} = \dot{\boldsymbol{q}}_{1} - \dot{\boldsymbol{q}}_{d}$ as position and velocity tracking errors, respectively. The system (9) can be then rewritten as

(10) \begin{align}\left\{ {\begin{array}{*{20}{l}}{\dot{\boldsymbol{e}}_{1} = \boldsymbol{e}_{2}}\\[5pt]{\dot{\boldsymbol{e}}_{2} = \boldsymbol{H}_0^{ - 1}(\boldsymbol{q})\boldsymbol{u} - \boldsymbol{H}_0^{ - 1}(\boldsymbol{q}){\boldsymbol{C}_{0}} {({\boldsymbol{q},\dot{\boldsymbol{q}}})\dot{\boldsymbol{q}}} + \boldsymbol{f}_{{dis}} - \ddot{\boldsymbol{q}}_{d}}\end{array}} \right.\end{align}

where $\boldsymbol{f}_{{dis}}$ is lumped uncertainty denoted by $\boldsymbol{f}_{{dis}} = \boldsymbol{H}_0^{ - 1}(\boldsymbol{q})\left( {\boldsymbol{d} - \Delta \boldsymbol{H}(\boldsymbol{q})\ddot{\boldsymbol{q}} - \Delta \boldsymbol{C}({\boldsymbol{q},\dot{\boldsymbol{q}}})\dot{\boldsymbol{q}}} \right)$ .

3.1 A novel fast fixed-time stability system

Theorem 1. When a system satisfies

(11) \begin{align}\dot x = - N\left( x \right)\left( {{c_1}si{g^{1 + {\sigma _0}}}\left( x \right) + {c_2}si{g^{{\lambda _0}}}\left( x \right)} \right)\end{align}

where ${\sigma _0} = \frac{{{m_0}}}{{2{n_0}}}\left( {1 + sgn\left( {\left| x \right| - 1} \right)} \right) + \left( {\frac{{{p_0}}}{{2{q_0}}} - \frac{1}{2}} \right)\left( {1 - sgn\left( {\left| x \right| - 1} \right)} \right)$ , ${\lambda _0} = \left( {\frac{{{p_0}}}{{{q_0}}} + \frac{{{m_0}}}{{2{n_0}}}} \right) + \left( {1 - \frac{{{p_0}}}{{{q_0}}} + \frac{{{m_0}}}{{2{n_0}}}} \right)$ $sgn\left( {\left| x \right| - 1} \right)$ , ${c_1} \gt 0$ and ${c_2} \gt 0$ are two scalars, ${m_0} \gt 0$ , ${n_0} \gt 0$ , ${p_0} \gt 0$ , and ${q_0} \gt 0$ are odd integers, which satisfy ${m_0} \gt {n_0}$ and $\frac{{{q_0}}}{2} \lt {p_0} \lt {q_0}$ . $N\left( x \right) = 1 + {2_a}arctan\left( {{s_b}|x{|^{{s_c}}}} \right)/\pi $ with ${s_a} \gt 0$ , ${s_b} \gt 0$ and ${s_c} \gt 0$ satisfying ${s_c} = \left\{ {\begin{array}{*{20}{l}}{{s_c}\;\;\;\;} {}{\left| x \right| \geq 1}\\{1\;\;\;\;} {}{\left| x \right| \lt 1}\end{array}} \right.$ . Then the system (11) is fixed-time stable and the convergence time ${T_c}$ is bounded by

(12) \begin{align}{T_c} \le \frac{{{q_0}}}{{\left( {{q_0} - {p_0}} \right){c_1}}}\left( {1 - \frac{{{c_2}}}{{{c_1}}}{\mathrm{ln}}\left( {1 + \frac{{{c_1}}}{{{c_2}}}} \right)} \right) + \frac{{{n_0}}}{{\left( {{c_1} + {c_2}} \right){m_0}}}\end{align}

Proof. Introduce a variable ${{\Psi }} = |x{|^{1 - \frac{{{p_0}}}{{{q_0}}}}}$ and its time derivative can be obtained

(13) \begin{align}{{{\dot \Psi }}} {} & = \left( {1 - \frac{{{p_0}}}{{{q_0}}}} \right){\mathrm{si}}{{\mathrm{g}}^{ - \frac{{{p_0}}}{{{q_0}}}}}\left( x \right)\dot{x}\nonumber\\[3pt]& = - \left( {1 - \frac{{{p_0}}}{{{q_0}}}} \right)N\left( x \right)\left( {{c_1}\left| {x{|^{1 + {\sigma _0} - \frac{{{p_0}}}{{{q_0}}}}} + {c_2}} \right|x{|^{{\lambda _0} - \frac{{{p_0}}}{{{q_0}}}}}} \right)\nonumber\\[3pt]& = - \left( {1 - \frac{{{p_0}}}{{{q_0}}}} \right)N\left( x \right)\left( {{c_1}\left| {{{\Psi }}{|^\varepsilon } + {c_2}} \right|{{\Psi }}{|^\gamma }} \right)\end{align}

where $\varepsilon = 1 + \frac{{{\sigma _0}{q_0}}}{{{q_0} - {p_0}}}$ and $\gamma = \frac{{{\lambda _0}{q_0} - {p_0}}}{{{q_0} - {p_0}}}$ .

According the definitions of ${\sigma _0}$ and ${\lambda _0}$ , it can be known that

(14) \begin{align}\left\{ \begin{array}{l@{\quad}l}{\varepsilon = \gamma = 1 + \dfrac{{{m_0}{q_0}}}{{{n_0}\left( {{q_0} - {p_0}} \right)}}} {}& {\left| x \right| \geq 1}\\[10pt]{\varepsilon = 0,\gamma = - 1} {}& {\left| x \right| \lt 1}\end{array} \right.\end{align}

Solving (13), the convergence time is derived

(15) \begin{align}{T_c} & = \frac{{{q_0}}}{{{q_0} - {p_0}}}\mathop \int \nolimits_0^{{{\Psi }}\left( 0 \right)} \frac{{d{{\Psi }}}}{{N\left( x \right)\left( {{c_1}\left| {{{\Psi }}{|^\varepsilon } + {c_2}} \right|{{\Psi }}{|^\gamma }} \right)}}\nonumber\\[5pt]& = \frac{{{q_0}}}{{{q_0} - {p_0}}}\left( {\mathop \int \nolimits_0^1 \frac{{d{{\Psi }}}}{{N\left( x \right)\left( {{c_1}\left| {{{\Psi }}{|^\varepsilon } + {c_2}} \right|{{\Psi }}{|^\gamma }} \right)}} + \mathop \int \nolimits_1^{{{\Psi }}\left( 0 \right)} \frac{{d{{\Psi }}}}{{N\left( x \right)\left( {{c_1}\left| {{{\Psi }}{|^\varepsilon } + {c_2}} \right|{{\Psi }}{|^\gamma }} \right)}}} \right)\nonumber\\[5pt]& = \frac{{{q_0}}}{{{q_0} - {p_0}}}\left( {\mathop \int \nolimits_0^1 \frac{{d{{\Psi }}}}{{N\left( x \right)\left( {{c_1} + {c_2}|{{\Psi }}{|^{ - 1}}} \right)}} + \mathop \int \nolimits_1^{{{\Psi }}\left( 0 \right)} \frac{{d{{\Psi }}}}{{N\left( x \right)\left( {{c_1} + {c_2}} \right)|{{\Psi }}{|^\rho }}}} \right) \end{align}

where $\rho = 1 + \frac{{{m_0}{q_0}}}{{{n_0}\left( {{q_0} - {p_0}} \right)}}$ .

Since $1 \le N\left( x \right) \lt 1 + {s_a}$ , then one has

(16) \begin{align}{{T_c}} {}& \le \frac{{{q_0}}}{{{q_0} - {p_0}}}\left( {\mathop \int \nolimits_0^1 \frac{{d{{\Psi }}}}{{{c_1} + {c_2}|{{\Psi }}{|^{ - 1}}}} + \mathop \int \nolimits_1^{{{\Psi }}\left( 0 \right)} \frac{{d{{\Psi }}}}{{\left( {{c_1} + {c_2}} \right)|{{\Psi }}{|^\rho }}}} \right)\nonumber\\[6pt]& \le \frac{{{q_0}}}{{{q_0} - {p_0}}}\left\{ {\frac{1}{{{c_1}}}\left( {1 - \frac{{{c_2}}}{{{c_1}}}{\mathrm{ln}}\left( {1 + \frac{{{c_1}}}{{{c_2}}}} \right)} \right) + \frac{{1 - {{\Psi }}{{(0)}^{1 - \rho }}}}{{\left( {{c_1} + {c_2}} \right)\left( {\rho - 1} \right)}}} \right\}\end{align}

Invoking $\rho = 1 + \frac{{{m_0}{q_0}}}{{{n_0}\left( {{q_0} - {p_0}} \right)}} \gt 1$ and ${{\Psi }}\left( 0 \right) \gt 0$ , the settling time ${T_c}$ is given by

(17) \begin{align}{T_c} \le \frac{{{q_0}}}{{\left( {{q_0} - {p_0}} \right){c_1}}}\left( {1 - \frac{{{c_2}}}{{{c_1}}}{\mathrm{ln}}\left( {1 + \frac{{{{\mathrm{c}}_1}}}{{{c_2}}}} \right)} \right) + \frac{{{n_0}}}{{\left( {{c_1} + {c_2}} \right){m_0}}}\end{align}

Remark 2. Zuo et al. [Reference Zuo33] constructed a fixed-time stable system (named FTSS1) $\dot x = - {c_1}{x^{{m_0}/{n_0}}} - {c_2}{x^{{p_0}/{q_0}}}$ . Ni et al. [Reference Ni, Liu, Liu, Hu and Li38] investigated a fast fixed-time stable system (called FTSS2) $\dot x = - {c_1}{x^{\frac{1}{2} + \frac{{{m_0}}}{{2{n_0}}} + \left( {\frac{{{m_0}}}{{2{n_0}}} - \frac{1}{2}} \right)sgn\left( {\left| x \right| - 1} \right)}} - {c_2}{x^{{p_0}/{q_0}}}$ . By observation, the convergence time of FTSS1 and FTSS2 can be uniformly calculated as

(18) \begin{align}{T_F} = \frac{{{q_0}}}{{{q_0} - {p_0}}}\left( {\mathop \int \nolimits_0^1 \frac{1}{{{c_1}W + {c_2}}}dW + \mathop \int \nolimits_1^{W\left( 0 \right)} \frac{1}{{{c_1}{W^\zeta } + {c_2}}}dW} \right)\end{align}

where $\zeta = 1$ for FTSS1 and $\zeta = 1 + \frac{{\left( {{m_0} - {n_0}} \right){q_0}}}{{{n_0}\left( {{q_0} - {p_0}} \right)}}$ for FTSS2. Due to the fact the inequality $\frac{1}{{{c_1}W + {c_2}}} \gt \frac{1}{{{c_1} + {c_2}{W^{ - 1}}}}$ holds for $W \in ({0,1})$ , and the inequality $\frac{1}{{{c_1}{W^\zeta } + {c_2}}} \gt \frac{1}{{\left( {{c_1} + {c_2}} \right){W^\rho }}}$ holds for $W \in \left( {1, + \infty } \right)$ , it is concluded that the proposed fixed-time stable system (11) attains a faster convergence rate than FTSS1 and FTSS2.

3.2 Nonlinear disturbance observer design

The system (10) can be represented as

(19) \begin{align}\dot{\boldsymbol{e}}_2 = - {l_d}{\boldsymbol{e}_2} + \boldsymbol{H}_0^{ - 1}(\boldsymbol{q})\boldsymbol{u} + {\boldsymbol{F}_d}\end{align}

where ${l_d}$ is a positive gain, and ${\boldsymbol{F}_d} = - \boldsymbol{H}_0^{ - 1}(\boldsymbol{q}){\boldsymbol{C}_0}({\boldsymbol{q},\dot{\boldsymbol{q}}})\dot{\boldsymbol{q}} + {\boldsymbol{f}_{dis}} - \ddot{\boldsymbol{q}}_d + {l_d}{\boldsymbol{e}_2}$ .

For system (19), introduce an auxiliary system as

(20) \begin{align}\dot{z}_d = - {l_d}{z_d} + \boldsymbol{H}_0^{ - 1}(\boldsymbol{q})\boldsymbol{u}\end{align}

where ${z_d} \in {\mathbb{R}^{n \times n}}$ is the state of the auxiliary system.

Define ${x_d}$ as the deviation between ${e_2}$ and ${{\mathbf{z}}_d}$ , which is expressed as ${x_d} = {e_2} - {z_d}$ . Then, the time derivative of ${x_d}$ yields

(21) \begin{align}\dot{x}_d = - {l_d}{x_d} + {F_d}\end{align}

Theorem 2. Construct a nonlinear disturbance observer as

(22) \begin{align}\dot{\hat{\boldsymbol{x}}}_d & = {}{ - {l_k}{k_d}{{\hat{\boldsymbol{x}}}_d} + l_k^{ - 1}{\dot{{\boldsymbol{y}}}_d} + {k_d}{\boldsymbol{y}_d}}\nonumber\\&\quad + \frac{{{n^{\frac{{{\delta _d}}}{4}}}\pi }}{{2{\delta _d}{T_d}\sqrt {{\lambda _{d1}}{\lambda _{d2}}} }}\left( {{\lambda _{d1}}si{g^{1 + {\delta _d}}}\left( {{\boldsymbol{e}_d}} \right) + {\lambda _{d2}}si{g^{1 - {\delta _d}}}\left( {{\boldsymbol{e}_d}} \right)} \right)\end{align}

with its output provided by

(23) \begin{align}{\,\hat{\!\boldsymbol{f}}_{dis}} = \hat{\boldsymbol{F}}_d + \boldsymbol{H}_0^{ - 1}(\boldsymbol{q}){\boldsymbol{C}_0}({\boldsymbol{q},\dot{\boldsymbol{q}}})\dot{\boldsymbol{q}} + \ddot{\boldsymbol{q}}_d - {l_d}{\boldsymbol{e}_2}\end{align}

with

(24) \begin{align}\hat{\boldsymbol{F}}_d = l_k^{ - 1}{{\boldsymbol{y}}_d} + {l_d}{\hat{\boldsymbol{x}}_d}\end{align}

where $\hat{\boldsymbol{x}}_d$ and $\,\hat{\!\boldsymbol{f}}_{dis}$ denote the estimations of ${\boldsymbol{x}_d}$ and ${\boldsymbol{f}_{dis}}$ , respectively. ${\boldsymbol{y}_d} = {l_k}\boldsymbol{x}_{\boldsymbol{d}}$ , ${l_k} \gt 0$ , ${k_d} \gt 0$ , $0 \lt {\delta _d} \lt 1$ , ${\lambda _{d1}}$ and ${\lambda _{d2}}$ are two positive constants, ${T_d}$ is an adjustable parameter that characterises the convergence time. Then, the observer error ${\boldsymbol{e}_d} = {\boldsymbol{x}_d} - \hat{\boldsymbol{x}}_d$ and the disturbance estimation error ${\boldsymbol{e}_f} = {\boldsymbol{f}_{dis}} - \,\hat{\!\boldsymbol{f}}_{dis}$ are fixed-time stable, and the convergence time ${T_e}$ is bounded by ${T_d}$ .

Proof. The observer error dynamics can be given by

(25) \begin{align}\dot{\boldsymbol{e}}_d & = \dot{\boldsymbol{x}}_d - \dot{\hat{\boldsymbol{x}}}_d\nonumber\\& = \dot{\boldsymbol{x}}_d + {l_k}{k_d}\hat{\boldsymbol{x}}_d - l_k^{ - 1}\dot{\boldsymbol{y}}_d - {k_d}{\boldsymbol{y}_d}\nonumber\\& \quad - \frac{{{n^{\frac{{{\delta _d}}}{4}}}\pi }}{{2{\delta _d}{T_d}\sqrt {{\lambda _{d1}}{\lambda _{d2}}} }}\left( {{\lambda _{d1}}si{g^{1 + {\delta _d}}}\left( {{\boldsymbol{e}_d}} \right) + {\lambda _{d2}}si{g^{1 - {\delta _d}}}\left( {{\boldsymbol{e}_d}} \right)} \right)\nonumber\\ & = - {l_k}{k_d}{\boldsymbol{e}_d} - \frac{{{n^{\frac{{{\delta _d}}}{4}}}\pi }}{{2{\delta _d}{T_d}\sqrt {{\lambda _{d1}}{\lambda _{d2}}} }}\left( {{\lambda _{d1}}si{g^{1 + {\delta _d}}}\left( {{\boldsymbol{e}_d}} \right) + {\lambda _{d2}}si{g^{1 - {\delta _d}}}\left( {{\boldsymbol{e}_d}} \right)} \right) \end{align}

Choose a Lyapunov function as ${V_d} = e_d^{\mathrm{T}}{e_d}$ , and its time derivation can be obtained

(26) \begin{align}{\dot V}_d {}& = 2e_d^T{\dot{\boldsymbol{e}}_d}\nonumber\\& = - 2{l_k}{k_d}e_d^T{e_d} - \frac{{{n^{\frac{{{\delta _d}}}{4}}}\pi }}{{{\delta _d}{T_d}\sqrt {{\lambda _{d1}}{\lambda _{d2}}} }}\left( \mathop \sum \limits_{i = 1}^n {\lambda _{d1}} | {e_{di}}|^{2 + {\delta _d}} + \mathop \sum \limits_{i = 1}^n {\lambda _{d2}} |{e_{di}}|{^{2 - {\delta _d}}} \right)\end{align}

According to Lemma 1, ${\dot V_d}$ has the following inequality relation

(27) \begin{align}{{\dot V}_d} \le - \frac{{{n^{\frac{{{\delta _d}}}{4}}}\pi }}{{{\delta _d}{T_d}\sqrt {{\lambda _{d1}}{\lambda _{d2}}} }}\left\{ {{n^{ - \frac{{{\delta _d}}}{2}}}{\lambda _{d1}}V_d^{{\delta _d}} + {\lambda _{d2}}} \right\}V_d^{\frac{{2 - {\delta _d}}}{2}}\end{align}

Define ${\chi _d} = {n^{ - \frac{{{\delta _d}}}{4}}}\sqrt {\frac{{{\lambda _{d1}}}}{{{\lambda _{d2}}}}} V_d^{\frac{{{\delta _d}}}{2}}$ , then $d{\chi _d} = \frac{{{\delta _d}}}{2}{n^{ - \frac{{{\delta _d}}}{4}}}\sqrt {\frac{{{\lambda _{d1}}}}{{{\lambda _{d2}}}}} V_d^{\frac{{{\delta _d}}}{2} - 1}d{V_d}$ . Then, solving the inequality (27) yields ${V_d}(t) \equiv 0$ for $t \geq {T_e}$ , and ${T_e}$ is bounded by

(28) \begin{align}{{T_e}} {}& \le \frac{{{\delta _d}{T_d}\sqrt {{\lambda _{d1}}{\lambda _{d2}}} }}{{{n^{\frac{{{\delta _d}}}{4}}}\pi }}\mathop \int \nolimits_0^{{V_d}\left( 0 \right)} \frac{{V_d^{\frac{{{\delta _d}}}{2} - 1}}}{{\left\{ {{n^{ - \frac{{{\delta _d}}}{2}}}{\lambda _{d1}}V_d^{{\delta _d}} + {\lambda _{d2}}} \right\}}}d{V_d}\nonumber\\[4pt]& \le \frac{{2{T_d}}}{\pi }\mathop \int \nolimits_0^{{\chi _d}\left( 0 \right)} \frac{{d{\chi _d}}}{{\left\{ {\chi _d^2 + 1} \right\}}}\nonumber\\[4pt]& \le \frac{{2{T_d}}}{\pi }{\mathrm{arctan}}\left( {{\chi _d}\left( 0 \right))} \right)\end{align}

Since $0 \lt arctan\left( {{\chi _d}\left( 0 \right)} \right) \lt \frac{\pi }{2}$ , one has

(29) \begin{align}{T_e} \le {T_d}\end{align}

Consequently, it is shown that the observer error ${e_d}$ is fixed-time stable. Applying (23), the disturbance estimation error ${\boldsymbol{e}_f}$ is derived

(30) \begin{align}{{\boldsymbol{e}_f}} {} & = {\boldsymbol{f}_{dis}} - \,\hat{\!\boldsymbol{f}}_{dis}\nonumber\\[3pt]& = {\boldsymbol{F}_d} + \boldsymbol{H}_0^{ - 1}(\boldsymbol{q}){\boldsymbol{C}_0}({\boldsymbol{q},\dot{\boldsymbol{q}}})\dot{\boldsymbol{q}} + \ddot{\boldsymbol{q}}_d - {l_d}{\boldsymbol{e}_2}\nonumber\\[3pt]& \quad - \hat{\boldsymbol{F}}_d - \boldsymbol{H}_0^{ - 1}(\boldsymbol{q}){\boldsymbol{C}_0}({\boldsymbol{q},\dot{\boldsymbol{q}}})\dot{\boldsymbol{q}} - \ddot{\boldsymbol{q}}_d + {l_d}{\boldsymbol{e}_2}\nonumber\\[3pt]& = {\boldsymbol{F}_d} - \hat{\boldsymbol{F}}_d\end{align}

From (21), it follows ${\boldsymbol{F}_d} = \dot{\boldsymbol{x}}_d + {l_d}{\boldsymbol{x}_d}$ . Applying (24), Equation (30) can be simplified as

(31) \begin{align}{{\boldsymbol{e}_f}} {}& = \dot{\boldsymbol{x}}_d + {l_d}{\boldsymbol{x}_d} - l_k^{ - 1}\dot{\boldsymbol{y}}_d - {l_d}\hat{\boldsymbol{x}}_d\nonumber\\& = {l_d}{\boldsymbol{e}_d}\end{align}

As a result, it can be conclude that ${\boldsymbol{e}_f} = \textbf{0}$ is obtained for $t \geq {T_e}$ . This means that the lumped uncertainty ${\boldsymbol{f}_{dis}}$ can be reconstructed by $\,\hat{\!\boldsymbol{f}}_{dis}$ after the specific time ${T_d}$ .

3.3 Tracking controller design and stability analysis

Based on Theorem 1, a novel non-singular fixed-time terminal sliding mode surface (NFTSMS) is presented as

(32) \begin{align}\boldsymbol{s} = {\boldsymbol{e}_2} + N({{\boldsymbol{e}}_1})\left( {{k_a}{\boldsymbol{S}_c} + {k_b}{\boldsymbol{S}_z}} \right)\end{align}

where ${k_a} \gt 0$ and ${k_b} \gt 0$ are two scalars, $N\left( \boldsymbol{e}_1 \right) = 1 + 2{s_m}{\mathrm{arctan}}\left( {s_n}{e_1}^{s_r} \right)/\pi $ with ${s_n} \gt 0$ , ${s_m} \gt 0$ and ${s_r} \gt 0$ satisfying ${s_r} = \left\{ {\begin{array}{l@{\quad}l}{{s_r}} {}& {\left| {\left| {{{\boldsymbol{e}}}_1} \right|} \right| \geq 1}\\[3pt]{1} {}& {\left| {\left| {{{\boldsymbol{e}}}_1} \right|} \right| \lt 1}\end{array}} \right.$ . ${S_{ci}}$ and ${S_{zi}}$ are respectively the $i$ th elements of ${\boldsymbol{S}_c}$ and ${\boldsymbol{S}_z}$ , and have the following forms

(33) \begin{align}{S_{ci}} & = \left\{ \begin{array}{l@{\quad}l}\mathrm{sig}^{{1 + 2{\sigma _1}}}\left( {{e_{1i}}} \right) & {\mathrm{if}}\quad {{{\bar{s}}}_i} = 0\ {\mathrm{or}}\ \overline{{s}_i} \ne 0,\left| {{{\mathrm{e}}_{1i}}} \right| \geq \delta \\[4pt]{l_1}{e_{1i}} + {l_2}e_{1i}^2\mathrm{sgn}\left( {{e_{1i}}} \right) + {l_3}e_{1i}^3 & {\mathrm{if}}\quad {{{\bar{s}}}_i} \ne 0,\left| {{{\mathrm{e}}_{1i}}} \right| \lt \delta \end{array} \right.\nonumber\\[5pt]{S_{zi}} & = \left\{ \begin{array}{l@{\quad}l}\mathrm{sig}{^{2{\lambda _1} - 1}}\left( {{{\mathrm{e}}_{1i}}} \right) & {\mathrm{if}}\quad \overline{{s}_i} = 0\quad {\mathrm{or}}\quad \overline{{s}_i} \ne 0,\left| {{{\mathrm{e}}_{1i}}} \right| \geq \delta \\[4pt]{{g_1}{e_{1i}} + {g_2}e_{1i}^2\ \mathrm{sgn}\left( {{e_{1i}}} \right) + {g_3}e_{1i}^3} & {}{{\mathrm{if}}\quad \overline{{s}_i} \ne 0,\left| {{{\mathrm{e}}_{1i}}} \right| \lt \delta }\end{array} \right.\end{align}

where $i = 1,2, \ldots, n$ , ${\sigma _1} = \frac{{{m_1}}}{{2{n_1}}}\left( {1 + \mathrm{sgn}\left( {\left| {\left| {{\boldsymbol{e}_1}} \right|} \right| - 1} \right)} \right) + \left( {\frac{{{p_1}}}{{2{q_1}}} - \frac{1}{2}} \right)\left( {1 - \mathrm{sgn}\left( {\left| {\left| {{\boldsymbol{e}_1}} \right|} \right| - 1} \right)} \right)$ , ${\lambda _1} = \left( {\frac{{{p_1}}}{{{q_1}}} + \frac{{{m_1}}}{{2{n_1}}}} \right) + \left( {1 - \frac{{{p_1}}}{{{q_1}}} + \frac{{{m_1}}}{{2{n_1}}}} \right)\mathrm{sgn}\left( {\left| {\left| {{\boldsymbol{e}_1}} \right|} \right| - 1} \right)$ with ${m_1} \gt {n_1}$ and $\frac{3}{4}{q_1} \lt {p_1} \lt {q_1}$ , $0 \lt \delta \lt 1$ is a constant. To make the functions ${S_{ci}}$ and ${S_{zi}}$ , and their time derivative continuous, the values of ${l_1}$ , ${l_2}$ , ${l_3}$ , ${g_1}$ , ${g_2}$ , ${g_3}$ are chosen as [Reference Li and Cai41]

(34) \begin{align}{l_1} & = \left( {2{p_1}/{q_1} - 3} \right)\left( {{p_1}/{q_1} - 2} \right){\delta ^{2{p_1}/{q_1} - 2}}\nonumber\\[4pt]{l_2} & = - \left( {2{p_1}/{q_1} - 2} \right)\left( {2{p_1}/{q_1} - 4} \right){\delta ^{2{p_1}/{q_1} - 3}}\nonumber\\[4pt]{l_3} & = \left( {{p_1}/{q_1} - 1} \right)\left( {2{p_1}/{q_1} - 3} \right){\delta ^{2{p_1}/{q_1} - 4}}\nonumber\\[4pt]{g_1} & = \left( {4{p_1}/{q_1} - 5} \right)\left( {2{p_1}/{q_1} - 3} \right){\delta ^{4{p_1}/{q_1} - 4}}\nonumber\\[4pt]{g_2} & = - \left( {4{p_1}/{q_1} - 4} \right)\left( {4{p_1}/{q_1} - 6} \right){\delta ^{4{p_1}/{q_1} - 5}}\nonumber\\[4pt]{g_3} & = \left( {2{p_1}/{q_1} - 2} \right)\left( {4{p_1}/{q_1} - 5} \right){\delta ^{4{p_1}/{q_1} - 6}}\nonumber\\[4pt]\bar{\boldsymbol{s}} & = {\boldsymbol{e}_2} + N\left( {{\boldsymbol{e}_1}} \right)\left( {{k_a}si{g^{1 + 2{\sigma _1}}}\left( {{\boldsymbol{e}_1}} \right) + {k_b}si{g^{2{\lambda _1} - 1}}\left( {{\boldsymbol{e}_1}} \right)} \right)\end{align}

Let $\boldsymbol{G}\left( {{\boldsymbol{e}}}_1 \right) = N\left( {{\boldsymbol{e}}}_1 \right)\left( {{k_a}{\boldsymbol{S}_c} + {k_b}{\boldsymbol{S}_z}} \right)$ . And taking the time derivative of the NFTSMS by using (10) yields

(35) \begin{align}\dot{\boldsymbol{s}} {}& = \dot{\boldsymbol{e}}_2 + \dot{\boldsymbol{G}}\nonumber\\[4pt]& = \boldsymbol{H}_0^{ - 1}(\boldsymbol{q})\boldsymbol{u} - \boldsymbol{H}_0^{ - 1}(\boldsymbol{q}){\boldsymbol{C}_0}({\boldsymbol{q},\dot{\boldsymbol{q}}})\dot{\boldsymbol{q}} + {\boldsymbol{f}_{dis}} - \ddot{\boldsymbol{q}}_d + \dot{\boldsymbol{G}}\end{align}

To obtain accurate and fast trajectory tracking, design an observer-based fixed-time SMC strategy as

(36) \begin{align}\boldsymbol{u} = {\boldsymbol{u}}_1 + {\boldsymbol{u}_2}\end{align}

where

(37) \begin{align}{\boldsymbol{u}}_1 = - {\boldsymbol{H}_0}(\boldsymbol{q})\,\,\hat{\!\boldsymbol{f}}_{dis} + {\boldsymbol{C}_0}({\boldsymbol{q},\dot{\boldsymbol{q}}})\dot{\boldsymbol{q}} + {\boldsymbol{H}_0}(\boldsymbol{q})\ddot{\boldsymbol{q}}_d - {\boldsymbol{H}_0}\dot{\boldsymbol{G}}\end{align}

(38) \begin{align}{\boldsymbol{u}_2} = - {\boldsymbol{H}_0}(\boldsymbol{q})N(\boldsymbol{s})\left( {{\gamma _1}{\mathrm{si}}{{\mathrm{g}}^{1 + 2{\sigma _2}}}(\boldsymbol{s}) + {\gamma _2}{\mathrm{si}}{{\mathrm{g}}^{2{\lambda _2} - 1}}(\boldsymbol{s}) + {\gamma _3}\boldsymbol{s}} \right)\end{align}

where ${\sigma _2} = \frac{{{m_2}}}{{2{n_2}}}\left( {1 + \mathrm{sgn}\left( {\parallel s\parallel - 1} \right)} \right) + \left( {\frac{{{p_2}}}{{2{q_2}}} - \frac{1}{2}} \right)\left( {1 - {\mathrm{sgn}}\left( {\parallel s\parallel - 1} \right)} \right)$ , ${\lambda _2} = \left( {\frac{{{p_2}}}{{{q_2}}} + \frac{{{\sigma _2}}}{2}} \right) + \left( {1 - \frac{{{p_2}}}{{{q_2}}} + \frac{{{\sigma _2}}}{2}} \right)$ $\mathrm{sgn}\left( {\parallel s\parallel - 1} \right)$ with ${m_2} \gt {n_2}$ and $\frac{{{q_2}}}{2} \lt {p_2} \lt {q_2}$ . ${\gamma _1},{\gamma _2},{\gamma _3}$ are positive control gains, $N(s) = 1 + 2s{s_m}\ {\mathrm{arctan}}\left( {s{s_n}\parallel s{\parallel ^{s{s_r}}}} \right)/\pi $ with $s{s_m} \gt 0,s{s_n} \gt 0$ and $s{s_m} \gt 0$ satisfying $s{s_r} = \left\{ {\begin{array}{l@{\quad}l}{s{s_r}} {}& {\left| {\left| \boldsymbol{s} \right|} \right| \gt 1}\\[4pt]{1} {}& {\left| {\left| \boldsymbol{s} \right|} \right| \lt 1}\end{array}} \right.$ .

Proof. Substituting the controller (36) into (35) yields

(39) \begin{align}\boldsymbol{s} = - N(\boldsymbol{s})\left( {{\gamma _1}{\mathrm{si}}{{\mathrm{g}}^{1 + 2{\sigma _2}}}(\boldsymbol{s}) + {\gamma _2}{\mathrm{si}}{{\mathrm{g}}^{2{\lambda _2} - 1}}(\boldsymbol{s}) + {\gamma _3}\boldsymbol{s}} \right) - \,\hat{\!\boldsymbol{f}}_{dis} + {\boldsymbol{f}_{dis}}\end{align}

Choose a Lyapunov function as ${V_1} = \frac{1}{2}\boldsymbol{s}^{\mathrm{T}}\boldsymbol{s}$ , and one obtains

(40) \begin{align}{{{\dot V}_1}} {}& = {\boldsymbol{s}^{\mathrm{T}}}\dot{\boldsymbol{s}}\nonumber\\[2pt]& = - N({\boldsymbol{s}})\left( {\gamma _1}\parallel {\mathrm{s}}{\parallel ^{2 + 2{\sigma _2}}} + {\gamma _2}\parallel \boldsymbol{s}{\parallel ^{2{\lambda _2}}} + {\gamma _3}\parallel \boldsymbol{s}{\parallel ^2} \right) + {\boldsymbol{s}^{\mathrm{T}}}{\boldsymbol{e}_f}\nonumber\\[2pt]& \le - N(\boldsymbol{s}){\gamma _1}{2^{1 + {\sigma _2}}}\left( \frac{1}{2}{\boldsymbol{s}^{\mathrm{T}}}\boldsymbol{s} \right)^{1 + {\sigma _2}} - N(\boldsymbol{s}){\gamma _2}{2^{{\lambda _2}}}{{\left( {\frac{1}{2}{\boldsymbol{s}^{\mathrm{T}}}\boldsymbol{s}} \right)}^{{\lambda _2}}} + {\boldsymbol{s}^{\mathrm{T}}}{\boldsymbol{e}_f}\end{align}

Since ${\boldsymbol{e}_f} = 0$ is obtained for $t \geq {T_d}$ , the above inequality is simplified as

(41) \begin{align}{\dot V_1} \le - N(\boldsymbol{s})\left( {{\rho _1}V_1^{1 + {\sigma _2}} - {\rho _2}V_1^{{\lambda _2}}} \right)\end{align}

where ${\rho _1} = {\gamma _1}{2^{1 + {\sigma _2}}}$ , ${\rho _2} = {\gamma _2}{2^{{\lambda _2}}}$ .

Similar to the analysis and demonstration of Theorem 1, the sliding surface NFTSMS is fixed-time stable. Noted that the sliding surface converges to the origin only if it is guaranteed that both the observation system and the sliding surface converge. Taking this into account, the convergence time ${t_r}$ satisfies ${t_r} \geq {\mathrm{max}}\left\{ {{T_d},{T_r}} \right\}$ , where ${T_r} \le \frac{{{n_2}}}{{\left( {{\rho _1} + {\rho _2}} \right){m_2}}} + \frac{{{q_2}}}{{\left( {{q_2} - {p_2}} \right){\rho _1}}}\left( {1 - \frac{{{\rho _2}}}{{{\rho _1}}}{\mathrm{ln}}\left( {1 + \frac{{{\rho _1}}}{{{\rho _2}}}} \right)} \right).$ As a result, the system states are capable of reaching the sliding surface under the proposed controller after fixed-time ${t_r}$ .

Once the system states reach the NFTSMS, i.e., $\boldsymbol{s} = \textbf{0}$ , the ideal sliding motion satisfies the following differential equation

(42) \begin{align}{{\boldsymbol{e}_2}} {}& = - N({{\boldsymbol{e}}_1})\left( {{k_a}{\boldsymbol{S}_c} + {k_b}{\boldsymbol{S}_z}} \right)\nonumber\\[4pt]& = - N({{\boldsymbol{e}}_1})\left( {{k_a}{\mathrm{si}}{{\mathrm{g}}^{1 + 2{{{\sigma }}_1}}}({{\boldsymbol{e}}_1}) + {k_b}{\mathrm{si}}{{\mathrm{g}}^{2{{{\lambda }}_1} - 1}}\left( {{\boldsymbol{e}}}_1 \right)} \right)\end{align}

Choose a Lyapunov function candidate ${V_2} = e_1^{\mathrm{T}}{e_1}$ , its time derivative yields

(43) \begin{align}{{{\dot V}_2}} & = - 2N({{\boldsymbol{e}}_1})\boldsymbol{e}_1^T\left( {{k_a}{\mathrm{si}}{{\mathrm{g}}^{1 + 2{{{\sigma }}_1}}}({{\boldsymbol{e}}_1}) + {k_b}{\mathrm{si}}{{\mathrm{g}}^{2{{{\lambda }}_1} - 1}}({{\boldsymbol{e}}_1})} \right)\nonumber\\[5pt]& = - 2N({{\boldsymbol{e}}_1})\left( {{k_a}V_1^{1 + {\sigma _1}} + {k_b}V_1^{{\lambda _1}}} \right)\end{align}

Invoking Theorem 1, the tracking errors ${{\boldsymbol{e}}_1}$ and ${{\boldsymbol{e}}_2}$ are proved to converge to the origin along the proposed NFTSMS (32) within a fixed-time ${t_s}$ , which is given by

(44) \begin{align}{t_s} \le \frac{{{n_1}}}{{2\left( {{k_a} + {k_b}} \right){m_1}}} + \frac{{{q_1}}}{{2\left( {{q_1} - {p_1}} \right){k_a}}}\left( {1 - \frac{{{k_b}}}{{{k_a}}}{\mathrm{ln}}\left( {1 + \frac{{{k_a}}}{{{k_b}}}} \right)} \right)\end{align}

Through the above demonstrations, it is concluded that the system states are fixed-time stable with the convergence time ${T_s}$ satisfying ${T_s} \le {t_r} + {t_s}$ .

Remark 4. Noted that ${\dot{\boldsymbol{y}}_d}$ and $\dot{\boldsymbol{G}}$ are respectively contained in the disturbance observer (22) and the control law ${\boldsymbol{u}}_1$ (37), which are required to adopt the presented methodology. To satisfy this requirement, an exact fixed-time estimation of the input signal’s derivatives is obtained by utilising the following uniform robust exact differentiator (URED) [Reference Cruz-Zavala, Moreno and Fridman42]

(45) \begin{align}{{\dot{\hat{\xi}} }_{1i}} & = - {h_{1i}}\left( {{\mathrm{si}}{{\mathrm{g}}^{\frac{1}{2}}}\left( {{{\hat{\xi}}_{1i}} - {v_i}} \right) + {\mu _i}{\mathrm{si}}{{\mathrm{g}}^{\frac{3}{2}}}\left( {{{\hat{\xi}}_{1i}} - {v_i}} \right)} \right) + {{\hat{\xi}}_{2i}}\nonumber\\[4pt]{{\dot{\hat{\xi}} }_{2i}} & = - {h_{2i}}\left( {\frac{1}{2}{\mathrm{sign}}\left( {{{\hat{\xi}}_{1i}} - {v_i}} \right) + 2{\mu _i}\left( {{{\hat{\xi}}_{1i}} - {v_i}} \right) + \frac{3}{2}\mu _i^2{\mathrm{si}}{{\mathrm{g}}^2}\left( {{{\hat{\xi}}_{1i}} - {v_i}} \right)} \right)\end{align}

where $i = 1,2, \ldots, n$ , ${\xi _{1i}}$ and ${\xi _{2i}}$ are the estimations of input ${v_i}$ and its derivative ${\dot v_i}$ , respectively, ${h_{1i}}$ , ${h_{2i}}$ and ${\mu _i}$ are positive gains. According to Ref. [Reference Cruz-Zavala, Moreno and Fridman42], the differentiator (45) enables to guarantee that the states ${\hat{\xi}_{1i}}$ and ${\hat{\xi}_{2i}}$ converge to ${v_i}$ and its derivative within a fixed time with respect to the parameters ${h_{1i}},{h_{2i}}$ and ${\mu _i}$ .

4.1 Verification of the suggested disturbance observer

This section primarily aims to demonstrate the estimation performance of the suggested observer (22). Considering various complex disturbances that may be in the space environment, the reference signal ${x_d} = 3 + {\mathrm{sin}}\left( {0.5t} \right) + 0.3{\mathrm{sin}}({2t}) + 0.2{\mathrm{cos}}({0.1t}) + 0.1{\mathrm{sin}}({5t}) + 0.2{{\mathrm{e}}^{ - t}} + 0.01{\mathrm{rand}}(1)$ is simulated, and no reconstruction of the lumped disturbance is involved here, which will be discussed in Section 4.2. The simulation results are verified from the following two scenarios.

The proposed observer’s fixed-time convergence characteristic is evaluated in the first scenario. This simulation tests six possibilities with various initial states. The observer gains are selected as ${k_d} = 0.1,{l_d} = 2,{T_d} = 1,{\delta _d} = 0.2,{\lambda _{d1}} = 4,{\lambda _{d2}} = 3,{h_1} = 10,{h_2} = 50,\mu = 2$ . The observer’s output for the reference signal is illustrated in Fig. 4. As illustrated, it is apparent that the settling time of the observer consistently remains approximately the same, regardless of the distinct values chosen for the initial conditions. In addition, the actual convergence time is observed to be significantly less than the predetermined time ${T_d}$ which is attributed to the fact that ${T_d}$ represents the upper bound of the convergence time. This implies that, regardless of the initial conditions, the precise estimate is always achieved within the predefined time. This fixed-time convergence feature provides increased flexibility and predictability in the observation phase.

Figure 4. Observer output for the reference signal with different initial states.

In the second scenario, the convergence time tunability of the suggested observer is assessed via selecting multiple predetermined settling times for an identical beginning state. The observer gains are the same as the first scenario expect that ${T_d}$ is different. For a given initial state value of $2$ , the simulation chooses ${T_d}$ to be $0.5$ , $1.5,3,5,8$ and $10$ seconds, respectively. Figure 5 manifests the observer’s output for the reference signal under various predefined times. The observation process for the reference signal adjusts in conjunction with ${T_d}$ changes. More detailed, the larger ${T_d}$ is, the more time it takes to accurately estimate the reference signal. Additionally, it can be concluded from Fig. 5, that the observer can realise the precise estimation of the reference signal within the predefined time ${T_d}$ . The predefined time allows straightforward and simplistic adjustment in (22), which is the intention of the observer proposed in the present research.

Figure 5. Observer output for the reference signal with different predefined times.

4.2 Verification of the suggested fixed-time tracking controller

This section focuses on the effectiveness of the proposed DOBFTSMC strategy against parametric uncertainty and time-varying disturbance. Table 2 lists the selected control parameters. The simulation results ensured by the suggested approach are displayed in Figs 6 and 7. The analysis in Section 3 is well-verified by those results. As depicted in Figs 6(a) and (b), the space manipulator under the suggested control strategy can accomplish the trajectory tracking manoeuver after a brief amount of time (approximately 1.2 seconds), even in the presence of uncertain parameter and time-varying disturbance. Figure 6(c) depicts the sliding surface’s time response. With the application of the proposed control law, the sliding surface is fixed-time stable, which is consistent with Theorem 3. The requested control torque is shown in Figs 6(d). Figure 7 gives the estimation performance of lumped disturbance under the proposed controller. As shown, the lumped disturbance can be precisely estimated in an extremely short period of time. This indicates that the suggested disturbance observer may accurately observe and quickly reconstruct the lumped disturbance within a predetermined time though the information of disturbance is unknown. Based on the feed-forward compensation of the constructed disturbance observer, the suggested controller achieves outstanding disturbance attenuation and satisfying tracking performance, simultaneously.

Table 2. Parameters of the proposed controller

Figure 6. Simulation results under DOBFTSMC.

Figure 7. Esitimation performance of lumped disturbance under DOBFTSMC.

To further highlight the superiority of the suggested fixed-time control strategy, comparisons are carried out with fixed-time terminal sliding mode control (FTSMC) in Ref. [Reference Zuo33] and adaptive nonsingular fast terminal sliding mode control (ANFTSMC) in Ref. [Reference Zhang, Tang and Guo34]. The corresponding controllers for space manipulator can be formulated as follows:

The FTSMC is written as

(46) \begin{align}\boldsymbol{u} & = {}{{\boldsymbol{u}_1} + {\boldsymbol{u}_2}}\nonumber\\{\boldsymbol{u}_1} & = {}{{\boldsymbol{C}_0}({\boldsymbol{q},\dot{\boldsymbol{q}}})\dot{\boldsymbol{q}} + {\boldsymbol{H}_0}(\boldsymbol{q})\ddot{\boldsymbol{q}}_d - {\boldsymbol{H}_0}(\boldsymbol{q})\zeta {\mathrm{sgn}}(\boldsymbol{s})}\nonumber\\& \quad - {\boldsymbol{H}_0}(\boldsymbol{q}){\boldsymbol{\sigma} ^{ - 1}}\boldsymbol{\Phi} {\boldsymbol{e}_2} - \frac{{{p_1}}}{{{q_1}}}{\boldsymbol{H}_0}(\boldsymbol{q}){\boldsymbol{\sigma} ^{ - 1}}{\mathrm{diag}}\left( {{{\left| {{\sigma _i}{e_{2i}}} \right|}^{1 - {q_1}/{p_1}}}} \right){e_2}\nonumber\\{\boldsymbol{u}_2} & = {}{ - \frac{{{p_1}}}{{{q_1}}}{\boldsymbol{H}_0}(\boldsymbol{q}){\mathrm{diag}}\left( {\sigma _i^{ - {q_1}/{p_1}}} \right)\left( {{\gamma _\alpha }{\mathrm{si}}{{\mathrm{g}}^{{{\mathrm{m}}_2}/{{\mathrm{n}}_2}}}(\boldsymbol{s}) + {\gamma _\beta }{\mathrm{si}}{{\mathrm{g}}^{{{\mathrm{p}}_2}/{{\mathrm{q}}_2}}}(\boldsymbol{s})} \right)}\nonumber\\\boldsymbol{s} & = {{\boldsymbol{e}_1} + {\mathrm{si}}{{\mathrm{g}}^{{\mathrm{q}}1/{\mathrm{p}}1}}\left( {\boldsymbol{\sigma} {\boldsymbol{e}_2}} \right)}\end{align}

where ${\boldsymbol{\Phi }} = {\mathrm{diag}}\left( {{{{\Phi }}_1},{{{\Phi }}_2}, \ldots, {{{\Phi }}_n}} \right)$ with ${{{\Phi }}_i} = - {\alpha _{1i}}\left( {\frac{{{m_1}}}{{{n_1}}} - \frac{{{p_1}}}{{{q_1}}}} \right)e_{1i}^{{m_1}/{n_1} - {p_1}/{q_1} - 1}\sigma _i^2{e_{2i}}$ , $\boldsymbol{\sigma} = diag\left( {{\sigma _1},{\sigma _2}, \ldots, {\sigma _n}} \right)$ with ${\sigma _i} = \frac{1}{{{\alpha _1}\left( i \right)e_{1i}^{{m_1}/{n_1} - {p_1}/{q_1}} + {\beta _1}\left( i \right)}}$ , ${\gamma _\alpha }$ , ${\gamma _\beta }$ , $\zeta $ are three positive constants, ${\boldsymbol{\alpha}_1} = {\left[ {{\alpha _{11}},{\alpha _{12}}, \ldots, {\alpha _{1n}}} \right]^{\mathrm{T}}}$ and ${\boldsymbol{\beta} _1} = {\left[ {{\beta _{11}},{\beta _{12}}, \ldots, {\beta _{1n}}} \right]^{\mathrm{T}}}$ with ${\alpha _{1i}} \gt 0$ and ${\beta _{1i}} \gt 0$ , positive odd integers ${m_1},{m_2},{n_1},{n_2},{p_1},{p_2},{q_1},{q_2}$ satisfy the relationships ${m_1}/{n_1} \gt 1,{m_2}/{n_2} \gt 1,\frac{1}{2} \lt {p_1}/{q_1} \lt 1,0 \lt {p_2}/{q_2} \lt 1$ , and ${m_1}/{n_1} - {p_1}/{q_1} \gt 1$ . To reduce the chattering phenomena, the saturation function is utilised in place of the function ${\mathrm{sgn}}\left( \cdot \right)$ in the control design. The definition of the elements of ${\mathrm{sat}}\left( \cdot \right)$ are

(47) \begin{align}{\mathrm{sat}}({s_i}) = \left\{ {\begin{array}{*{20}{l}}{{\mathrm{sgn}}({s_i}),\left| {{s_i}} \right| \geq {\varepsilon _0},}\\[6pt]{{s_i}/{\varepsilon _0},\left| {{s_i}} \right| \lt {\varepsilon _0},}\end{array}i = 1,2, \ldots, 7} \right.\end{align}

with a positive small constant ${\varepsilon _0} \gt 0$ .

The ANFTSMC scheme is described as

(48) \begin{align}\boldsymbol{u} {} & = {{\boldsymbol{u}_1} + {\boldsymbol{u}_2}}\nonumber\\{{\boldsymbol{u}_1}} {} & = { {\boldsymbol{C}_0}({\boldsymbol{q},\dot{\boldsymbol{q}}})\dot{\boldsymbol{q}} + {\boldsymbol{H}_0}(\boldsymbol{q})\ddot{\boldsymbol{q}}_d}\nonumber\\&- {\boldsymbol{H}_0}(\boldsymbol{q})\left( {{\alpha _1}{\boldsymbol{K}_a}\mathrm{diag}(|{e_{1i}}{|^{{\alpha _1} - 1}}){\boldsymbol{e}_2} + {\boldsymbol{K}_b}\dot{\boldsymbol{s}}_\rho \left( {{\boldsymbol{e}_1}} \right)} \right)\nonumber\\{{\boldsymbol{u}_2}} {} & = {- {\boldsymbol{H}_0}(\boldsymbol{q})\left( {{\gamma _a}\mathrm{sig}^{{{\alpha _2}}}(\boldsymbol{s}) + {\gamma _b}\mathrm{sig}^{{{p_2}/{q_2}}}(\boldsymbol{s})} \right) - {\boldsymbol{H}_0}(\boldsymbol{q})k\hat{\eta} \mathrm{tanh}( {\boldsymbol{s}/\varepsilon })}\\{\dot{\hat{\eta}}} {}& = {k{\boldsymbol{s}^T}\mathrm{tanh}( {\boldsymbol{s}/\varepsilon })}\nonumber\\\boldsymbol{s} {}& = { {\boldsymbol{e}_2} + {\boldsymbol{K}_a}\mathrm{sig}^{{{\alpha _1}}}\left( {{\boldsymbol{e}_1}} \right) + {\boldsymbol{K}_b}{\boldsymbol{s}_\rho }\left( {{\boldsymbol{e}_1}} \right)}\nonumber\end{align}

with the $i$ th element of ${{\mathbf{s}}_\rho }$ , which is expressed as

(49) \begin{align}{s_{{\rho _i}}} = \left( {\begin{array}{l@{\quad}l}{\mathrm{sig}^{{{p_1}/{q_1}}}\left( {{e_{1i}}} \right),} {}& {{{\bar s}_i} = 0 \cup \bar s \ne 0,\left| {{e_{1i}}} \right| \geq {\varepsilon _0}}\\[6pt]{{l_1}{e_{1i}} + {l_2}e_{1i}^2\ \mathrm{sgn}\left( {{e_{1i}}} \right),} {}& {\bar s \ne 0,\left| {{e_{1i}}} \right| \lt {\varepsilon _0}}\end{array}} \right.\end{align}

where for a small positive constant ${\varepsilon _0}$ , ${l_1} = \left( {2 - \frac{{{p_1}}}{{{q_1}}}} \right)\varepsilon _0^{{p_1}/{q_1} - 1},{l_2} = \left( {\frac{{{p_1}}}{{{q_1}}} - 1} \right)\varepsilon _0^{{p_1}/{q_1} - 2}$ , ${\gamma _a},{\gamma _b},k$ are positive constants, ${\boldsymbol{K}_a},{\boldsymbol{K}_b}$ are two positive definite matrices, ${\alpha _1} = \frac{1}{2} + \frac{{{m_1}}}{{2{n_1}}} + \left( {\frac{{{m_1}}}{{2{n_1}}} - \frac{1}{2}} \right){\mathrm{sign}}\left( {\left| {\left| {{{\mathbf{e}}_1} - 1} \right|} \right|} \right),$ ${\alpha _2} = \frac{1}{2} + \frac{{{m_2}}}{{2{n_2}}} + \left( {\frac{{{m_2}}}{{2{n_2}}} - \frac{1}{2}} \right){\mathrm{sign}}\left( {\left| {\left| {{\mathbf{s}} - 1} \right|} \right|} \right)$ , ${\mathrm{tanh}}\left( {\frac{{\mathbf{s}}}{\varepsilon }} \right) = {[{\mathrm{tanh}}\left( {\frac{{{s_1}}}{\varepsilon }} \right),{\mathrm{tanh}}\left( {\frac{{{s_2}}}{\varepsilon }} \right), \ldots, {\mathrm{tanh}}\left( {\frac{{{s_n}}}{\varepsilon }} \right)]^{\mathrm{T}}}$ with a constant $\varepsilon \gt 0$ , ${m_1},{m_2},{n_1},{n_2},{p_1},{q_1}$ are positive odd integers satisfying ${m_1}/{n_1} \gt 1,{m_2}/{n_2} \gt 1$ and $0 \lt {p_2}/{q_2} \lt 1$ .

Table 3. Control parameters for compared controllers

Figure 8. Simulation results under FTSMC.

Figure 9. Simulation results under ANFTSMC.

Table 4. IAEs of the different controllers

Table 5. ITAEs of the different controllers

Table 6. ECs of the different controllers

Figure 10. Norms of tracking position and velocity errors.

For a fair and rational comparison, the simulations are conducted under the identical conditions. The selected control parameters for the two comparison controllers mentioned earlier are provided in Table 3. Figures 8 and 9 present the simulation results using FTSMC and ANFTSMC, respectively. In comparison with Fig. 6, all three controllers have finite convergence time, however, the suggested DOBFTSMC exhibits faster convergence. The norms of tracking position and velocity errors are shown in Fig. 10, demonstrating that the proposed controller has improved convergence and higher steady-state accuracy in more detail. Furthermore, three critical metrics $ - $ integrated absolute errors (IAEs), integrated time absolute errors (ITAEs), and energy consumptions (ECs) $ - $ are introduced to quantitatively evaluate the tracking performance of these controllers. Comparing the obtained indices displayed in Tables 4–6, it is observed that the suggested DOBFTSMC obtains the lower IAEs, ITAEs and ECs values than the other two referenced controllers. It is evident from this that the suggested controller outperforms the other two controllers with regard to tracking accuracy and convergence speed while consuming less energy.

Figure 11. Time response of the position tracking errors under the Monte Carlo tests.

Figure 12. Time response of the velocity tracking errors under the Monte Carlo tests.

4.3 Monte Carlo tests

In this section, a total of 30 times Monte Carlo tests are executed to evaluate the robustness of the proposed DOBFTSMC method against parametric uncertainty and time-varying disturbance. Each trial subjects the space manipulator to a unique set of parametric uncertainties and time-varying disturbances. Specifically, the actual mass of each system component is set to ${M_i}' = {M_i}\left( {1 + 0.1{\mathcal{N}}({0,1})} \right)$ , and the time-varying disturbances acting on the FFSM are chosen as $\boldsymbol{d}(t) = \left( {2{\mathcal{N}}({0,1}) - 1} \right){[0.03{\mathrm{cos}}(t),0.01{\mathrm{cos}}({2t}),0.01{\mathrm{sin}}({2t}),0.03{\mathrm{sin}}(t),0.02 + 0.01{\varphi _d}(t),0.01{\varphi _d}(t),0.01]^{\mathrm{T}}}$ Nm, where ${\varphi _d}(t) = \mathop \sum \nolimits_{r = 0}^N {b^{ - r{v_c}}}{\mathrm{sin}}\left( {{b^r}t} \right)$ is continuous but nowhere differentiable and limited, which attains continuous extended Caputo derivatives of any order $v \lt {v_c}$ [Reference Dou and Yue44]. Additionally, ${\mathcal{N}}({0,1})$ is a randomised number following a standard Gaussian distribution. In the simulation, the values of $N = 200,b = 6$ and ${v_c} = 0.7$ are chosen. Besides, the remaining simulation conditions and the control parameter settings remain identical to those in Section 4.2.

The simulation results are illustrated in Figs 11–14. Figures 11 and 12 display the time responses of the position and velocity tracking errors under the Monte Carlo tests, respectively. Figure 13 shows the disturbance estimation of the proposed observer under the Monte Carlo tests. As evident from the figures, the change curves of position and velocity tracking errors under various test conditions are almost coincident. This suggests that the proposed controller effectively mitigates the adverse effects of parametric uncertainties and time-varying disturbances, thereby ensuring stable tracking performance within a fixed time. Figure 14 represents the time response of the control torques under the Monte Carlo tests. Notably, at the beginning of the simulation, the control input is comparatively large, which is primarily attributed to a substantial initial tracking error coupled with an overestimation of the lumped disturbances. Based on the above analysis, it can be inferred that thanks to the excellent disturbance compensation capabilities of the observer, the proposed controller exhibits outstanding robustness against parametric uncertainties and time-varying disturbances. This guarantees the successful implementations of the specific on-orbit servicing missions, such as approaching or grasping a non-cooperative target.

Figure 13. The disturbance estimation of the proposed observer under the Monte Carlo tests.

Figure 14. Time response of control torques under the Monte Carlo tests.

Concluding from the aforementioned simulation results, the suggested DOBFTSMC scheme successfully solves the fixed-time trajectory tracking issue for space manipulator with parametric uncertainty and time-varying disturbance. Compared with the existing fixed-time control methods, the suggested method is verified to provide a better tracking performance when it comes to steady-state precision, convergence speed and control energy consumption.