2.1 Notations and lemmas

For an n-dimensional vector $ x=[{x}_{1},{x}_{2}, \ldots ,{x}_{n}{]}^{\textrm{T}}$ with the $ i$ th element $ {x}_{i}$ $ (i=\textrm{1,2}, \ldots ,n)$ . The symbols $ \parallel \cdot \parallel $ and $ \parallel \cdot {\parallel }_{1}$ represent the Euclidean norm and 1-norm, respectively. For a constant $ \gamma \gt 0$ , define $ \textrm{s}\textrm{i}{\textrm{g}}^{\gamma }\left(x\right)={\left[|{x}_{1}{|}^{\gamma }\textrm{s}\textrm{i}\textrm{g}\textrm{n}\left({x}_{1}\right),|{x}_{2}{|}^{\gamma }\textrm{s}\textrm{i}\textrm{g}\textrm{n}\left({x}_{2}\right),\dots ,|{x}_{n}{|}^{\gamma }\textrm{s}\textrm{i}\textrm{g}\textrm{n}\left({x}_{n}\right)\right]}^{\textrm{T}}$ and the vector $ \textrm{s}\textrm{g}\textrm{n}\left(x\right)=\left[\textrm{s}\textrm{i}\textrm{g}\textrm{n}\right({x}_{1}),\textrm{s}\textrm{i}\textrm{g}\textrm{n}({x}_{2}), \ldots ,\textrm{s}\textrm{i}\textrm{g}\textrm{n}({x}_{n}){]}^{\textrm{T}}$ , where $ \textrm{s}\textrm{i}\textrm{g}\textrm{n}(\cdot )$ denotes the sign function. $ {I}_{n}\in {\mathbb{R}}^{n\times n}$ is n-dimensional identity matrix.

Lemma 1. [Reference Zuo and Tie42] Let $ {A}_{i}\ge 0(i=\textrm{1,2}, \ldots ,n)$ , for $ 0 \lt {p}_{1}\le 1$ and $ {p}_{2}\gt 1$ , the following inequalities hold

(1) \begin{align}\mathop \sum \limits_{i = 1}^n A_i^{{p_1}} \ge {\left( {\mathop \sum \limits_{i = 1}^n {A_i}} \right)^{{p_1}}},\mathop \sum \limits_{i = 1}^n A_i^{{p_2}} \ge {n^{1 - {p_2}}}{\left( {\mathop \sum \limits_{i = 1}^n {A_i}} \right)^{{p_2}}}\end{align}

Definition 1 (Fixed-time stable). [Reference Polyakov17] Consider the nonlinear system

(2) \begin{align} \dot{\textbf{z}}(t)=g(\textbf{z}(t)),\textbf{z}(\boldsymbol{0})={\textbf{z}}_{0},g(\boldsymbol{0})=\boldsymbol{0},\textbf{z}\in {\mathbb{R}}^{n}\end{align}

The origin of the system (2) is said to be fixed-time stable if it is finite-time stable and the convergence time $ T\left({\textbf{z}}_{0}\right)$ is bounded for any initial states, that is, $ \exists {T}_{max}\gt 0$ such that $ T\left({\textbf{z}}_{0}\right)\le {T}_{max},\forall {\textbf{z}}_{0}\in {\mathbb{R}}^{n}$ .

2.2 Space manipulator model

In the presence of disturbances, the dynamic equation of the free-floating space manipulator shown in Fig. 1 can be expressed as [Reference Shao, Sun, Xue and Li15]:

(3) \begin{align} \textbf{H}\left(\textbf{q}\right)\ddot{\textbf{q}}+\textbf{C}\left(\textbf{q},\dot{\textbf{q}}\right)\dot{\textbf{q}}=\textbf{u}+\textbf{d}(t)\end{align}

Figure 1.

Space manipulator system model.

where, $ \textbf{H}\left(\textbf{q}\right)\in {\mathbb{R}}^{n\times n}$ is the positive definite symmetric inertia matrix. $ \textbf{C}(\textbf{q},\dot{\textbf{q}})\in {\mathbb{R}}^{n\times n}$ denotes the Coriolis and centrifugal matrix. $ \textbf{q}$ , $ \dot{\textbf{q}}$ , $ \ddot{\textbf{q}}\in {\mathbb{R}}^{n}$ represent the generalised position, velocity and acceleration vector, respectively. $ \textbf{u}\in {\mathbb{R}}^{n}$ is the input control torque, and $ \textbf{d}(t)\in {\mathbb{R}}^{n}$ represents the bounded disturbances caused by internal factors, satisfying $ \parallel \textbf{d}\parallel \lt {d}_{0}$ with $ {d}_{0}$ is an unknown positive constant.

Noticed that matrices $ \textbf{H}\left(\textbf{q}\right)$ and $ \textbf{C}(\textbf{q},\dot{\textbf{q}})$ always contain uncertainties, which can be further described as:

(4) \begin{align} \textbf{H}\left(\textbf{q}\right)={\textbf{H}}_{0}\left(\textbf{q}\right)+{\Delta }\textbf{H}\left(\textbf{q}\right),\end{align}

(5) \begin{align} \textbf{C}\left(\textbf{q},\dot{\textbf{q}}\right)={\textbf{C}}_{0}(\textbf{q},\dot{\textbf{q}})+{\Delta }\textbf{C}(\textbf{q},\dot{\textbf{q}})\end{align}

where, $ {\textbf{H}}_{0}\left(\textbf{q}\right)$ and $ {\textbf{C}}_{0}(\textbf{q},\dot{\textbf{q}})$ represent the nominal parts, $ {\Delta }\textbf{H}\left(\textbf{q}\right)$ and $ {\Delta }\textbf{C}(\textbf{q},\dot{\textbf{q}})$ denote the uncertainty parts.

Define $ {\textbf{x}}_{1}=\textbf{q}$ and $ {\textbf{x}}_{2}=\dot{\textbf{q}}$ , the modified dynamic model can be reformulated as:

(6) \begin{align} \left\{\begin{array}{c}{\dot{\textbf{x}}}_{1}={\textbf{x}}_{2}\\[6pt] {\dot{\textbf{x}}}_{2}=\textbf{F}({\textbf{x}}_{1},{\textbf{x}}_{2})+\textbf{B}\left({\textbf{x}}_{1}\right)\textbf{u}+{\textbf{f}}_{dis}\end{array}\right.\end{align}

where $ \textbf{F}({\textbf{x}}_{1},{\textbf{x}}_{2})=-{\textbf{H}}_{0}^{-1}\left({\textbf{x}}_{1}\right){\textbf{C}}_{0}({\textbf{x}}_{1},{\textbf{x}}_{2}){\textbf{x}}_{2}$ , $ \textbf{B}\left({\textbf{x}}_{1}\right)={\textbf{H}}_{0}^{-1}\left({\textbf{x}}_{1}\right)$ , and $ {\textbf{f}}_{dis}$ stands for an unknown lumped uncertain term including unknown disturbances and model uncertainties, which is expressed as

(7) \begin{align} {\textbf{f}}_{dis}={\textbf{H}}_{0}^{-1}\left({\textbf{x}}_{1}\right)\left(\textbf{d}(t)-{\Delta }\textbf{C}({\textbf{x}}_{1},{\textbf{x}}_{2}){\textbf{x}}_{2}-{\Delta }\textbf{H}\left({\textbf{x}}_{1}\right){\dot{\textbf{x}}}_{2}\right)\end{align}

Owing to the physical limitations of the actuators, the control torque $ \textbf{u}$ is restricted by the saturation value, which can be expressed as

(8) \begin{align} \textbf{u}=\textrm{s}\textrm{a}\textrm{t}\left({\textbf{u}}_{c}\right)\end{align}

where $ {\textbf{u}}_{c}={\left[{u}_{c1},{u}_{c2},\dots ,{u}_{cn}\right]}^{\textrm{T}}\in {\mathbb{R}}^{n}$ is the desired torque command, and the saturation function $ \textrm{s}\textrm{a}\textrm{t}(\cdot )$ represents saturation nonlinear characteristics of the actuator with $ \textrm{s}\textrm{a}\textrm{t}\left({u}_{ci}\right)=\textrm{s}\textrm{i}\textrm{g}\textrm{n}\left({u}_{ci}\right)\cdot \textrm{m}\textrm{i}\textrm{n}\left\{\left|{u}_{{c}_{i}}\right|,{u}_{{c}_{i\textrm{m}\textrm{a}\textrm{x}}}\right\}$ , where $ {u}_{{c}_{i\textrm{m}\textrm{a}\textrm{x}}}$ is the allowable maximum input torque of the $ i$ th actuator. Obviously, the saturation nonlinear term $ \textrm{s}\textrm{a}\textrm{t}\left({\textbf{u}}_{c}\right)$ can be abbreviated as the following form:

(9) \begin{align} \textrm{s}\textrm{a}\textrm{t}\left({\textbf{u}}_{c}\right)={\textbf{u}}_{c}{+{\delta }}_{c}\end{align}

where, $ {{\delta }}_{c}={\left[{\delta }_{{c}_{1}},{\delta }_{{c}_{2}},\dots ,{\delta }_{{c}_{n}}\right]}^{\textrm{T}}\in {\mathbb{R}}^{n}$ denotes the saturation degree of the actuators, satisfying $ \left||{\boldsymbol\delta}_{c}\right||\le {l}_{\delta }$ with $ {l}_{\delta }\gt 0$ , and $ {\delta }_{{c}_{i}}$ is defined as

(10) \begin{align} {\delta }_{ci}=\left\{\begin{array}{l@{\quad}l}0 , & \left|{u}_{{c}_{i}}\right|\le {u}_{{c}_{i\textrm{m}\textrm{a}\textrm{x}}}\\[5pt] \textrm{s}\textrm{i}\textrm{g}\textrm{n}\left({u}_{{c}_{i}}\right)\cdot {u}_{{c}_{i\textrm{m}\textrm{a}\textrm{x}}}-{u}_{{c}_{i}} ,& \left|{u}_{{c}_{i}}\right|\gt {u}_{{c}_{i\textrm{m}\textrm{a}\textrm{x}}}\end{array}\right.\end{align}

2.3 Problem formulation

To address the trajectory tracking issue of the space manipulator, the trajectory tracking error $ {\textbf{e}}_{1}$ and velocity tracking error $ {\textbf{e}}_{2}$ are defined as

(11) \begin{align} {\textbf{e}}_{1}={\textbf{x}}_{1}-{\textbf{q}}_{d},{\textbf{e}}_{2}={\textbf{x}}_{2}-{\dot{\textbf{q}}}_{d}\end{align}

where $ {\textbf{q}}_{d}\in {\mathbb{R}}^{n}$ and $ {\dot{\textbf{q}}}_{d}\in {\mathbb{R}}^{n}$ are the desired motion and corresponding time derivative, respectively.

Considering the input saturation, the tracking error equation of the space manipulator can be derived:

(12) \begin{align} \left\{\begin{array}{c}{\dot{\textbf{e}}}_{1}={\textbf{e}}_{2}\\[5pt] {\dot{\textbf{e}}}_{2}=\textbf{F}({\textbf{x}}_{1},{\textbf{x}}_{2})+\textbf{B}\left({\textbf{x}}_{1}\right){\textbf{u}}_{c}+\textbf{B}{\left({\textbf{x}}_{1}\right){\delta }}_{c}+{\textbf{f}}_{dis}-{\ddot{\textbf{q}}}_{d}\end{array}\right.\end{align}

The main control objective of this paper is to design a tracking control command $ {\textbf{u}}_{c}$ , such that

1. 1. All signals in the closed-loop system are bounded.

2. 2. Trajectory tracking is fulfilled within a specific time regardless of initial states.

3. 3. The position tracking error is always confined within the prescribed performance bounds despite the presence of unknown disturbances, parameter uncertainties and input saturation.

3.1 Non-singular fast terminal slide mode surface

To improve the convergence property, the following non-singularity fast terminal sliding surface (NFTSS) is designed as

(13) \begin{align} \textbf{s}={\textbf{e}}_{2}+N\left({\textbf{e}}_{1}\right)\left({\textbf{K}}_{1}\textrm{s}\textrm{i}{\textrm{g}}^{1+{\sigma }_{1}}\left({\textbf{e}}_{1}\right)+{\textbf{K}}_{2}{\textbf{s}}_{\rho }\left({\textbf{e}}_{1}\right)\right)\end{align}

where sliding surface $ \textbf{s}=[{s}_{1},{s}_{2}, \ldots ,{s}_{n}{]}^{\textrm{T}}\in {\mathbb{R}}^{n}$ . $ N\left({\textbf{e}}_{1}\right)=1+2{s}_{m}\textrm{a}\textrm{r}\textrm{c}\textrm{t}\textrm{a}\textrm{n}\left({s}_{n}\parallel {\textbf{e}}_{1}{\parallel }^{{s}_{r}}\right)/\pi $ with adjustable positive constants $ {s}_{m}$ , $ {s}_{n}$ and $ {s}_{r}$ . $ {\sigma }_{1}=\frac{{m}_{1}}{2{n}_{1}}(1+\textrm{s}\textrm{g}\textrm{n}(\left|\right|{\textbf{e}}_{1}\left|\right|-1\left)\right)$ with positive odd integers $ {m}_{1}$ , $ {n}_{1}$ satisfying $ {m}_{1}\gt {n}_{1}$ . $ {\textbf{K}}_{1}=\textrm{d}\textrm{i}\textrm{a}\textrm{g}({K}_{11},{K}_{12}, \ldots ,{K}_{1n})$ and $ {\textbf{K}}_{2}=\textrm{d}\textrm{i}\textrm{a}\textrm{g}({K}_{21},{K}_{22}, \ldots ,{K}_{2n})$ are positive definite matrices. $ {\textbf{s}}_{\rho }\left({\textbf{e}}_{1}\right)=[{s}_{{\rho }_{1}},{s}_{{\rho }_{2}}, \ldots ,{s}_{{\rho }_{n}}{]}^{\textrm{T}}\in {\mathbb{R}}^{n}$ with $ {\textbf{s}}_{{\rho }_{i}}$ is defined as

(14) \begin{align} {s}_{{\rho }_{i}}=\left\{\begin{array}{l@{\quad}l}\textrm{s}\textrm{i}{\textrm{g}}^{p/q}\left({e}_{1i}\right), & {\stackrel{-}{s}}_{i}=0\cup {\stackrel{-}{s}}_{i}\ne 0,\left|{e}_{1i}\right|\ge {\epsilon }_{0}\\[5pt] {l}_{1}{e}_{1i}+{l}_{2}{e}_{1i}^{2}\textrm{s}\textrm{g}\textrm{n}\left({e}_{1i}\right), & {\stackrel{-}{s}}_{i}\ne 0,\left|{e}_{1i}\right| \lt {\epsilon }_{0}\end{array}\right.\end{align}

where $ {\stackrel{-}{s}}_{i}={e}_{2i}+N\left({\textbf{e}}_{1}\right)\left({K}_{1i}\textrm{s}\textrm{i}{\textrm{g}}^{1+{\sigma }_{1}}\left({e}_{1i}\right)+{K}_{2i}\textrm{s}\textrm{i}{\textrm{g}}^{p/q}\left({e}_{1i}\right)\right)$ . $ p$ , $ q$ are positive odd integers with $ p \lt q$ , and $ {\epsilon }_{0}$ is a small positive constant. To assure the sliding surface and its time derivative smooth and continuous, $ {l}_{1}$ and $ {l}_{2}$ are selected as

(15) \begin{align} \left\{\begin{array}{ll}{l}_{1}=\left(2-\frac{p}{q}\right){\epsilon }_{0}^{p/q-1} \\[5pt] {l}_{2}=\left(\frac{p}{q}-1\right){\epsilon }_{0}^{p/q-2} \end{array}\right.\end{align}

Lemma 4. Consider the space manipulator system (12). Once the NFTSS (13) satisfies $ \boldsymbol{s}=0$ , then the system states $ {\textit{e}}_{1}$ and $ {\textit{e}}_{2}$ convergence to zero along the sliding surface in a fixed time for an arbitrary initial condition. The upper bound of settling time is given by

(16) \begin{align} {T}_{s}\le \underset{1\le i\le n}{\textrm{m}\textrm{a}\textrm{x}}\left\{\frac{q}{{K}_{1i}(q-p)}\textrm{l}\textrm{n}\left(1+\frac{{K}_{1i}}{{K}_{2i}}\right)+\frac{{n}_{1}}{{K}_{1i}{m}_{1}}\right\}\end{align}

Proof. Once the MFTSS is reached, i.e., $ \textbf{s}=0$ , a series of the following differential equations can be obtained

(17) \begin{align} {e}_{2i}+N\left({\textbf{e}}_{1}\right)\left({K}_{1i}\textrm{s}\textrm{i}{\textrm{g}}^{1+{\sigma }_{1}}\left({e}_{1i}\right)+{K}_{2i}\textrm{s}\textrm{i}{\textrm{g}}^{p/q}\left({e}_{1i}\right)\right)=0, \quad i=\textrm{1,2}, \ldots ,n\end{align}

Define a new variable as $ Q=|{e}_{1i}{|}^{\frac{q-p}{q}}$ , and the corresponding time-derivative is given by

(18) \begin{align} \dot{Q} & =\frac{q-p}{q}\textrm{s}\textrm{i}{\textrm{g}}^{-\frac{p}{q}}\left({e}_{1i}\right){e}_{2i}\nonumber\\[3pt]& =-\frac{(q-p)N\left({\textbf{e}}_{1}\right)}{q}\textrm{s}\textrm{i}{\textrm{g}}^{-\frac{p}{q}}\left({e}_{1i}\right)\left({K}_{1i}\textrm{s}\textrm{i}{\textrm{g}}^{1+{\sigma }_{1}}\left({e}_{1i}\right)+{K}_{2i}\textrm{s}\textrm{i}{\textrm{g}}^{p/q}\left({e}_{1i}\right)\right)\nonumber\\[3pt]& =-\frac{q-p}{q}N\left({\textbf{e}}_{1}\right)\left({K}_{1i}|{e}_{1i}{|}^{{\sigma }_{1}+\frac{q-p}{q}}+{K}_{2i}\right)\\[3pt]& =-\frac{q-p}{q}N\left({\textbf{e}}_{1}\right)\left({K}_{1i}{Q}^{1+\frac{{\sigma }_{1}q}{q-p}}+{K}_{2i}\right)\nonumber\end{align}

According to $ N\left({\textbf{e}}_{1}\right)\ge 1$ , the settling time $ {T}_{si}$ from $ {s}_{i}=0$ to $ {e}_{1i}=0$ along the sliding surface is determined by

(19) \begin{align} {T}_{si} & =\frac{q}{(q-p)}{\int }_{0}^{Q(0)} \frac{1}{N\left({\textbf{e}}_{1}\right)\left({K}_{1i}{Q}^{1+\frac{{\sigma }_{1}q}{q-p}}+{K}_{2i}\right)}dQ\nonumber\\[4pt] & =\frac{q}{(q-p)}{\int }_{0}^{1} \frac{1}{N\left({\textbf{e}}_{1}\right)\left({K}_{1i}{Q}^{1+\frac{{\sigma }_{1}q}{q-p}}+{K}_{2i}\right)}dQ\nonumber\\[4pt] &\quad +\frac{q}{(q-p)}{\int }_{1}^{Q(0)} \frac{1}{N\left({\textbf{e}}_{1}\right)\left({K}_{1i}{Q}^{1+\frac{{\sigma }_{1}q}{q-p}}+{K}_{2i}\right)}dQ\nonumber\\[4pt] & =\frac{q}{(q-p)}{\int }_{0}^{1} \frac{1}{N\left({\textbf{e}}_{1}\right)\left({K}_{1i}Q+{K}_{2i}\right)}dQ\\[4pt] &\quad +\frac{q}{(q-p)}{\int }_{1}^{Q(0)} \frac{1}{N\left({\textbf{e}}_{1}\right)\left({K}_{1i}{Q}^{\psi }+{K}_{2i}\right)}dQ\nonumber\\[8pt] & \le \frac{q}{q-p}\left[\frac{1}{{K}_{1i}}\textrm{l}\textrm{n}\left(1+\frac{{K}_{1i}}{{K}_{2i}}\right)+\frac{1-Q(0{)}^{1-\psi }}{{K}_{1i}(\psi -1)}\right]\nonumber \end{align}

where $ \psi =1+\frac{{m}_{1}/{n}_{1}}{1-p/q}$ . Due to $ \psi \gt 1$ and $ Q(0)\gt 1$ , $ {T}_{si}$ is further bounded by

(20) \begin{align} {T}_{si}\le \frac{q}{{K}_{1i}(q-p)}\textrm{l}\textrm{n}\left(1+\frac{{K}_{1i}}{{K}_{2i}}\right)+\frac{{n}_{1}}{{K}_{1i}{m}_{1}}\end{align}

Based on Equation (20), the upper bound of the convergence time $ {T}_{s}$ can be calculated by

(21) \begin{align} {T}_{s}=\underset{1\le i\le n}{\textrm{m}\textrm{a}\textrm{x}}\left\{{T}_{si}\right\}\le \underset{1\le i\le n}{\textrm{m}\textrm{a}\textrm{x}}\left\{\frac{q}{{K}_{1i}(q-p)}\textrm{l}\textrm{n}\left(1+\frac{{K}_{1i}}{{K}_{2i}}\right)+\frac{{n}_{1}}{{K}_{1i}{m}_{1}}\right\}\end{align}

From Equation (21), the upper bound of convergence time is independent of the system state values. It means that the tracking error $ {\textbf{e}}_{1}$ and its derivative $ {\textbf{e}}_{2}$ can converge to the origin within a refined time along the sliding surface. The proof is completed.

3.2 Prescribed performance function

Appropriate constraints on the system states are necessary to obtain the desired dynamic performance. With this in mind, the PPC is investigated to confine the position tracking error within a predefined permissible range here. First, describe such a constraint via a prescribed performance function, defined as follows.

Generally, a prescribed performance function is designed as

(22) \begin{align} \rho (t)=\left({\rho }_{0}-{\rho }_{{\infty }}\right){\textrm{e}}^{-{l}_{0}t}+{\rho }_{{\infty }}\end{align}

where $ {\rho }_{0}\gt {\rho }_{{\infty }}\gt 0$ and $ {l}_{0}\gt 0$ are design parameters and should be set appropriately to obtain the desired time-domain characteristics, such as steady-state offset, overshoot and rising time. As stated in Ref. [Reference Ik Han and Lee46], the prescribed performance is attained provided that the state is limited to the area bounded by the decaying function of time. According to the concept, achieving the prescribed error constraints is equivalent to satisfy the following relationship:

(23) \begin{align} -{\rho }_{i}(t) \lt {e}_{1i}(t) \lt {\rho }_{i}(t)\end{align}

where $ {\rho }_{i}$ denotes the prescribed performance function of $ {s}_{i}$ .

Based on the condition (23), one concludes that as long as $ \left|{e}_{1i}(0)\right| \lt {\rho }_{i}(0)$ is satisfied, the tracking error is always never exceed the predefined boundary.

3.3 Fixed-time tracking controller design

Take the time derivative of $ \textbf{s}$ in Equation (13) and using Equation (12), one obtains

(24) \begin{align} \dot{\textbf{s}} & =\textbf{F}({\textbf{x}}_{1},{\textbf{x}}_{2})+\textbf{B}\left({\textbf{x}}_{1}\right){\textbf{u}}_{c}+\textbf{B}{\left({\textbf{x}}_{1}\right){\delta }}_{c}+{\textbf{f}}_{dis}-{\ddot{\textbf{q}}}_{d}\nonumber\\[4pt] & \quad +\dot{N}\left({\textbf{e}}_{1}\right)\left({\textbf{K}}_{1}\textrm{s}\textrm{i}{\textrm{g}}^{1+{\sigma }_{1}}\left({\textbf{e}}_{1}\right)+{\textbf{K}}_{2}{\textbf{s}}_{\rho }\left({\textbf{e}}_{1}\right)\right)\\[4pt] & \quad +N\left({\textbf{e}}_{1}\right)\left((1+{\sigma }_{1}){\textbf{K}}_{1}\textrm{d}\textrm{i}\textrm{a}\textrm{g}\left(\right|{e}_{1i}{|}^{{\sigma }_{1}}){\textbf{e}}_{2}+{\textbf{K}}_{2}{\dot{\textbf{s}}}_{\rho }({\textbf{e}}_{1})\right)\nonumber\end{align}

where the $ i$ th element of $ {\dot{\textbf{s}}}_{\rho }$ is given by

(25) \begin{align} {\dot{s}}_{i}=\left\{\begin{array}{l@{\quad}l}\frac{p}{q}|{e}_{1i}{|}^{p/q-1}{e}_{2i}, & {\stackrel{-}{s}}_{i}=0\cup {\stackrel{-}{s}}_{i}\ne 0,\left|{e}_{1i}\right|\ge {\epsilon }_{0}\\[5pt] {l}_{1}{e}_{2i}+2{l}_{2}\left|{e}_{1i}\right|{e}_{2i}, & {\stackrel{-}{s}}_{i}\ne 0,\left|{e}_{1i}\right| \lt {\epsilon }_{0}\end{array}\right.\end{align}

Aiming to attain trajectory tracking with high precision, the following adaptive nonsingular fixed-time sliding mode control with prescribed performance constraints is developed

(26) \begin{align} {\textbf{u}}_{c}={\textbf{u}}_{eq}+{\textbf{u}}_{r}+{\textbf{u}}_{s}\end{align}

where,

(27) \begin{align} {\textbf{u}}_{eq} & = -{\textbf{B}}^{-1}\left({\textbf{x}}_{1}\right)\left(\textbf{F}\left({\textbf{x}}_{1},{\textbf{x}}_{2}\right)-{\ddot{\textbf{q}}}_{d}\right)\nonumber\\[5pt] & -{\textbf{B}}^{-1}\left({\textbf{x}}_{1}\right)\left((1+{\sigma }_{1}){\textbf{K}}_{1}\textrm{d}\textrm{i}\textrm{a}\textrm{g}\left(\right|{e}_{1i}{|}^{{\sigma }_{1}}){\textbf{e}}_{2}+{\textbf{K}}_{2}{\dot{\textbf{s}}}_{\rho }({\textbf{e}}_{1})\right)\\[5pt] & -{\textbf{B}}^{-1}\left({\textbf{x}}_{1}\right)\dot{N}\left({\textbf{e}}_{1}\right)\left({\textbf{K}}_{1}\textrm{s}\textrm{i}{\textrm{g}}^{1+{\sigma }_{1}}\left({\textbf{e}}_{1}\right)+{\textbf{K}}_{2}{\textbf{s}}_{\rho }\left({\textbf{e}}_{1}\right)\right)\qquad\quad\nonumber \end{align}

(28) \begin{align} {\textbf{u}}_{r} & = -{\textbf{B}}^{-1}\left({\textbf{x}}_{1}\right)G\left(\textbf{s}\right)\left({\gamma }_{1}\textrm{s}\textrm{i}{\textrm{g}}^{1+{\sigma }_{2}}\left(\textbf{s}\right)+{\gamma }_{2}\textrm{s}\textrm{i}{\textrm{g}}^{{\beta }_{1}}\left(\textbf{s}\right)+{\gamma }_{3}\textbf{s}\right)\nonumber\\[5pt] & -{\textbf{B}}^{-1}\left({\textbf{x}}_{1}\right){{\Gamma }}_{4}\textrm{d}\textrm{i}\textrm{a}\textrm{g}\left\{{z}_{i}\textrm{t}\textrm{a}\textrm{n}\left(\frac{\pi {z}_{i}}{2}\right)\right\}\textbf{s}-{\textbf{B}}^{-1}\left({\textbf{x}}_{1}\right)\frac{{\widehat{\textbf{c}}}_{m}^{\textrm{T}}{\boldsymbol\Theta }}{2{\epsilon }_{N}^{2}}\textbf{s} \end{align}

(29) \begin{align} {\textbf{u}}_{s}=-{\textbf{B}}^{-1}\left({\textbf{x}}_{1}\right){\gamma }_{5}{\Omega }-\frac{1}{2}{\textbf{B}}^{-1}\left({\textbf{x}}_{1}\right)\textbf{s}\qquad\qquad\qquad\qquad\end{align}

(30) \begin{align} G\left(s\right)=1+\chi -\chi {\textrm{e}}^{-{v}_{1}\left|\right|\textbf{s}|{|}^{k}}\qquad\qquad\qquad\qquad\quad\end{align}

where $ {\sigma }_{2}=\frac{{\alpha }_{1}}{2}\left(1+\textrm{s}\textrm{g}\textrm{n}\left(\right|\left|\textbf{s}\right||-1)\right)$ with $ {\alpha }_{1}\gt 1$ , and $ {\beta }_{1}\in \left(\textrm{0,1}\right)$ . $ \chi \gt 0$ , $ {v}_{1}\gt 0$ and $ k$ is a positive integer, $ {\gamma }_{1},{\gamma }_{2},{\gamma }_{3},{\gamma }_{5}$ are positive design parameters, $ {{\Gamma }}_{4}=\textrm{d}\textrm{i}\textrm{a}\textrm{g}\left({\gamma }_{41},{\gamma }_{42}, \ldots ,{\gamma }_{4n}\right)$ is a positive constant matrix, which is used as the tuning gain to force the tracking error to remain within the defined boundary. $ {\widehat{\textbf{c}}}_{m}$ is the estimation of $ {\textbf{c}}_{m}$ with $ {\textbf{c}}_{m}={\left[{c}_{1}^{2},{c}_{2}^{2},{c}_{3}^{2}\right]}^{\textrm{T}}$ , and $ {\boldsymbol\Theta }={\left[1,\parallel {\textbf{x}}_{1}{\parallel }^{2},\parallel {\textbf{x}}_{2}{\parallel }^{4}\right]}^{\textrm{T}}$ . The adaptive update law is designed as

(31) \begin{align} {\dot{\widehat{\textbf{c}}}}_{m}={\boldsymbol\eta}\left(\frac{{{\boldsymbol\Theta}||\textbf{s}||}^{2}}{2{\epsilon }_{N}^{2}}-\boldsymbol{\zeta}{\widehat{\textbf{c}}}_{m}\right)\end{align}

where $ \boldsymbol{\eta} =\textrm{d}\textrm{i}\textrm{a}\textrm{g}({\eta }_{1},{\eta }_{2},{\eta }_{3})$ and $ \boldsymbol{\zeta} =\textrm{d}\textrm{i}\textrm{a}\textrm{g}({\zeta }_{1},{\zeta }_{2},{\zeta }_{3})$ are two positive matrices, and $ {\epsilon }_{N}$ is an arbitrary small positive constant.

To compensate for input saturation, the auxiliary variable $ {\boldsymbol\Omega }$ is given by

(32) \begin{align} \dot{{\boldsymbol\Omega }}=\left\{\begin{array}{l@{\quad}l}\boldsymbol{0}, & \parallel {\boldsymbol\Omega }\parallel \le {{\Omega }}_{0}\\[4pt] -{b}_{1}{{\boldsymbol\Omega }}^{1+{\sigma }_{2}}-{b}_{2}{{\boldsymbol\Omega }}^{{\beta }_{1}}-{b}_{3}{\boldsymbol\Omega }- & \\[8pt] \dfrac{{||{\textbf{s}}^{\textrm{T}}\textbf{B}{\left({\textbf{x}}_{1}\right){\delta }}_{c}||}_{1}+0.5{{{b}_{4}{\delta }}_{c}^{\textrm{T}}{\delta }}_{c}}{\parallel {\boldsymbol\Omega }{\parallel }^{2}}{\boldsymbol\Omega }+{{b}_{4}} {{\delta }c}, & \parallel {\boldsymbol\Omega }\parallel \gt {{\Omega }}_{0}\end{array}\right.\end{align}

where $ {b}_{1},{b}_{2},{b}_{3},{b}_{4}$ and $ {{\Omega }}_{0}$ are positive constants.

Remark 5. It is worth noting that when $ \left|{e}_{1i}\right|$ approaches the constraint boundary $ {\rho }_{i}$ , then $ {z}_{i}\to 1$ , and the value of $ tan\left(\frac{\pi {z}_{i}}{2}\right)$ tends to infinity. The control gain is thus increased to suppress the evolution of the tracking error. Figure 2 illustrates this constraint control scheme in graphical form. If the required control law is unreasonable or the evolution of tracking error violates the constraint boundaries, the values of $ {\rho }_{0}$ , $ {\rho }_{\infty }$ , $ {l}_{0}$ and $ {{\Gamma }}_{4}$ must be adjusted according to practical application.

Figure 2.

The description of the constraint control scheme.

For a deeper understanding of the overall structure of the suggested method, Fig. 3 shows the closed-loop tracking control system.

Figure 3.

The structure of the proposed control scheme.

3.4 Stability analysis

To prove Lemma 5, the following will analyse and demonstrate in two steps.

1. Step 1. Fixed time convergence property of the sliding surface $ \boldsymbol{s}$ is verified.

Substituting Equations (26)–(29) into Equation (24), one obtains

(33) \begin{align} \dot{\textbf{s}} & =-G\left(\textbf{s}\right)\left({\gamma }_{1}\textrm{s}\textrm{i}{\textrm{g}}^{1+{\sigma }_{2}}\left(\textbf{s}\right)+{\gamma }_{2}\textrm{s}\textrm{i}{\textrm{g}}^{{\beta }_{1}}\left(\textbf{s}\right)+{\gamma }_{3}\textbf{s}\right)+\textbf{B}{\left({\textbf{x}}_{1}\right){\delta }}_{c}\nonumber\\[5pt]& -{{\Gamma }}_{4}\textrm{d}\textrm{i}\textrm{a}\textrm{g}\left\{{z}_{i}\textrm{t}\textrm{a}\textrm{n}\left(\frac{\pi {z}_{i}}{2}\right)\right\}\textbf{s}-\left(\frac{{\widehat{\textbf{c}}}_{m}^{\textrm{T}}{\Theta }}{2{\epsilon }_{N}^{2}}+\frac{1}{2}\right)\textbf{s}-{\gamma }_{5}{\Omega }+{\textbf{f}}_{dis} \end{align}

To validate the stability of the closed-loop system, define the following Lyapunov function:

(34) \begin{align} {V}_{2}=\frac{1}{2}{\textbf{s}}^{\textrm{T}}\textbf{s}+\frac{1}{2}{{\tilde{\textbf{c}}}_{m}^{\textrm{T}}} \boldsymbol{\eta}^{-1}{\tilde{\textbf{c}}}_{m}+\frac{1}{2}{{\Omega }}^{\textrm{T}}{\Omega }\end{align}

where $ \tilde{\textbf{c}}_{m}={\textbf{c}}_{m}-{\widehat{\textbf{c}}}_{m}$ denotes the estimate error of $ {\textbf{c}}_{m}$ .

The derivative of $ {V}_{2}$ with respect to time yields

(35) \begin{align} \dot{{V}}_{2}=\textbf{s}^{\text{T}} \dot{\textbf{s}} - \tilde{\textbf{c}}_{m}^{\text{T}} \boldsymbol{\eta}^{-1} \dot{\hat{\textbf{c}}}_{m} +\boldsymbol{\Omega}^{\text{T}} \dot{\boldsymbol{\Omega}}\end{align}

Then, one can derive

(36) \begin{align} \dot{V}_{2}=&-\textbf{s}^{\text{T}}\left\{G \left(\textbf{s}\right) \left(\gamma_{1}{\textrm{sig}}^{1+\sigma_2}(\textbf{s}) + \gamma_{2} {\textrm{sig}}^{\beta_1}(\textbf{s}) + \gamma_{3} \textbf{s}\right)-\textbf{B}\left(\textbf{x}_{1}\right)\boldsymbol{\delta}_{c}\right. \nonumber\\ & \left.+ \boldsymbol{\Gamma_{4}} \textrm{diag}\left\{ z_{i} \tan\left(\frac{\pi z_{i}}{2}\right) \right\} \textbf{s} +\left(\frac{\hat{\textbf{c}}_{m}^{\text{T}} \boldsymbol{\Theta}}{2 \varepsilon_N^{2}} + \frac{1}{2}\right) \textbf{s}+\gamma_{5}\boldsymbol{\Omega}-\textbf{f}_{dis}\right\}\nonumber\\ &- \tilde{\textbf{c}}_{m}^{\text{T}} \left( \frac{\boldsymbol{\Theta}\left\|{\textbf{s}}\right\|^{2}}{2 \varepsilon_N^{2}}-\boldsymbol{\zeta} \hat{\textbf{c}}_{m} \right) +\boldsymbol{\Omega}^{\text{T}} \dot{\boldsymbol{\Omega}}\nonumber\\ =& - G \left(\textbf{s}\right) \left(\gamma_{1}\|\textbf{s}\|^{\sigma_2+2}+ \gamma_{2}\|\textbf{s}\|^{\beta_1+1}+ \gamma_{3}\|\textbf{s}\|^{2}\right)\\ & - \textbf{s}^{\text{T}} \boldsymbol{\Gamma_{4}} \textrm{diag}\left\{ z_{i} \tan\left(\frac{\pi z_{i}}{2}\right) \right\} \textbf{s} - \frac{\hat{\textbf{c}}_{m}^{\text{T}} \boldsymbol{\Theta}}{2 \varepsilon_N^{2}}\|\textbf{s}\|^{2} + \textbf{s}^{\text{T}} \textbf{f}_{dis} -\frac{\tilde{\textbf{c}}_{m}^{\text{T}}\boldsymbol{\Theta}}{2\varepsilon_N^{2}}\left\|{\textbf{s}}\right\|^{2} \nonumber\\ & + \tilde{\textbf{c}}_{m}^{\text{T}} \boldsymbol{\zeta} \hat{\textbf{c}}_{m} -\frac{1}{2}\textbf{s}^{\text{T}}\textbf{s} -\gamma_{5}\textbf{s}^{\text{T}}\boldsymbol{\Omega}+\textbf{s}^{\text{T}}\textbf{B}\left(\textbf{x}_1\right)\boldsymbol{\delta}_{c}+\boldsymbol{\Omega}^{\text{T}}\dot{\boldsymbol{\Omega}}\nonumber \end{align}

Considering the auxiliary system Equation (32), the above equation leads to

(37) \begin{align} \dot{V}_{2}=& - G \left(\textbf{s}\right) \left(\gamma_{1}\|\textbf{s}\|^{\sigma_2+2}+ \gamma_{2}\|\textbf{s}\|^{\beta_1+1}+ \gamma_{3}\|\textbf{s}\|^{2}\right)\nonumber\\[5pt] & - \textbf{s}^{\text{T}}\boldsymbol{\Gamma_{4}} \textrm{diag} \left\{ z_{i} \tan\left(\frac{\pi z_{i}}{2}\right)\right\} \textbf{s} - \frac{\hat{\textbf{c}}_{m}^{\text{T}} \boldsymbol{\Theta}}{2 \varepsilon_N^{2}}\| \textbf{s}\|^{2} + \textbf{s}^{\text{T}} \textbf{f}_{dis} -\frac{\tilde{\textbf{c}}_{m}^{\text{T}}\boldsymbol{\Theta}}{2\varepsilon_N^{2}}\left\|{\textbf{s}}\right\|^{2} \nonumber\\[5pt] & + \tilde{\textbf{c}}_{m}^{\text{T}} \boldsymbol{\zeta} \hat{\textbf{c}}_{m} -\frac{1}{2}\textbf{s}^{\text{T}}\textbf{s} -\gamma_{5}\textbf{s}^{\text{T}}\boldsymbol{\Omega}+\textbf{s}^{\text{T}}\textbf{B}\left(\textbf{x}_1\right)\boldsymbol{\delta}_{c} -b_{1}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}^{1+\sigma_{2}}\nonumber\\[5pt] & -b_{2}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}^{\beta_{1}} -b_{3}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega} -\left\|\textbf{s}^{\text{T}} \textbf{B}\left(\textbf{x}_{1}\right) \boldsymbol{\delta}_{c}\right\|_{1} -\frac{1}{2} b_{4} \boldsymbol{\delta}_{c}^{\text{T}} \boldsymbol{\delta}_{c} +b_{4}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\delta}_{c}\nonumber\\[5pt] \leq & - G \left(\textbf{s}\right) \left(\gamma_{1}\|\textbf{s}\|^{\sigma_2+2}+ \gamma_{2}\|\textbf{s}\|^{\beta_1+1}+ \gamma_{3}\|\textbf{s}\|^{2}\right)\\[5pt] & - \boldsymbol{\Gamma_{4}} \textrm{diag} \left\{z_{i}\tan\left(\frac{\pi z_{i}}{2}\right)\right\} \|\textbf{s}\|^2 - \frac{{\textbf{c}}_{m}^{\text{T}} \boldsymbol{\Theta}}{2 \varepsilon_N^{2}}\|\textbf{s}\|^{2} + \textbf{c}_{z}^{\text{T}}\boldsymbol{theta}\|\textbf{s}\| +\tilde{\textbf{c}}_{m}^{\text{T}}\boldsymbol{ \zeta }\hat{\textbf{c}}_{m}\nonumber\\[5pt] & -\frac{1}{2}\textbf{s}^{\text{T}}\textbf{s} -\gamma_{5}\textbf{s}^{\text{T}}\boldsymbol{\Omega} -b_{1}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}^{1+\sigma_2}-b_{2}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}^{\beta_{1}} -b_{3}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}\nonumber\\[5pt] & -\frac{1}{2} b_{4} \boldsymbol{\delta}_{c}^{\text{T}} \boldsymbol{\delta}_{c} +b_{4}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\delta}_{c}\nonumber \end{align}

According to Young’s inequality, for $ \forall {\lambda }_{j}\gt \frac{1}{2}$ , $ j=(\textrm{1,2},3)$ , the following inequality relations hold:

(38) \begin{align} \textbf{c}_{z}^{\text{T}}\boldsymbol{theta}\|\textbf{s}\|=c_1\|\textbf{s}\|+c_2\|\textbf{x}_1\|\|\textbf{s}\|+c_3\|\textbf{x}_2\|^2\|\textbf{s}\| \leq \frac{\textbf{c}_{m}^{\text{T}}\boldsymbol{\Theta}}{2\varepsilon_N^{2}}\|\textbf{s}\|^{2} +\frac{3}{2}\varepsilon_{N}^{2}\end{align}

(39) \begin{align} \tilde{\textbf{c}}_{m}^{\text{T}}\boldsymbol{\zeta}\hat{\textbf{c}}_{m}&=\sum_{j=1}^{3}\zeta_j\tilde{c}_{m_j}\hat{c}_{m_j}=\sum_{j=1}^{3}\zeta_j\tilde{c}_{m_j}\left(c_{m_j}-\tilde{c}_{m_j} \right)\nonumber\\ & \leq\sum_{j=1}^{3}\zeta_j\left(\frac{\lambda_{j}}{2}c_{m_j}^{2} -\frac{2\lambda_{j}-1}{2\lambda_{j}}\tilde{c}_{m_j}^{2} \right)\end{align}

(40) \begin{align} -\gamma_{5}\textbf{s}^{\text{T}}\boldsymbol{\Omega}\leq\frac{1}{2}\textbf{s}^{\text{T}}\textbf{s}+\frac{1}{2}\gamma_{5}^{2}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}\end{align}

(41) \begin{align} \boldsymbol{\Omega}^{\text{T}}\boldsymbol{\delta}_{c}\leq\frac{1}{2}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}+\frac{1}{2}\boldsymbol{\delta}_{c}^{\text{T}}\boldsymbol{\delta}_{c}\end{align}

Using the above inequalities, Equation (37) can be written as

(42) \begin{align} \dot{V}_{2}\leq &- G \left(\textbf{s}\right) \left(\gamma_{1}\|\textbf{s}\|^{\sigma_2+2}+ \gamma_{2}\|\textbf{s}\|^{\beta_1+1}+ \gamma_{3}\|\textbf{s}\|^{2}\right)\nonumber\\ & +\frac{3}{2}\varepsilon_{N}^{2} +\sum_{j=1}^{3}\zeta_{j}\left(\frac{\lambda_{j}}{2}c_{m_j}^{2}\right) -\sum_{j=1}^{3}\zeta_{j}\left(\frac{2\lambda_{j}-1}{2\lambda_{j}}\tilde{c}_{m_j}^{2} \right)\\ &-b_{1}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}^{1+\sigma_2} -b_{2}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}^{\beta_{1}} -\left( b_{3}-\frac{1}{2}\gamma_{5}^2-\frac{1}{2}b_{4}\right) \boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}\nonumber\end{align}

Choose parameters $ {b}_{3}$ , $ {b}_{4}$ and $ {\gamma }_{5}$ , such that $ {b}_{3}-\frac{1}{2}{\gamma }_{5}^{2}-\frac{1}{2}{b}_{4}\gt 0$ . Thus, one has

(43) \begin{align} \dot{V}_{2}\leq & - \gamma_{1} G \left(\textbf{s}\right) 2^{\frac{\sigma_2+2}{2}} \left(\frac{1}{2} \textbf{s}^{\text{T}} \textbf{s}\right)^{\frac{\sigma_2+2}{2}} - \gamma_{2} G \left(\textbf{s}\right) 2^{\frac{\beta_1+1}{2}} \left(\frac{1}{2} \textbf{s}^{\text{T}} \textbf{s}\right)^{\frac{\beta_1+1}{2}}\nonumber \\ & -\sum_{j=1}^{3} \left(\frac{y_j}{2\eta_j} \tilde{c}_{m_j}^{2} \right)^{\frac{\sigma_2+2}{2}} -\sum_{j=1}^{3} \left(\frac{y_j}{2\eta_j} \tilde{c}_{m_j}^{2} \right)^{\frac{\beta_1+1}{2}} -2^{\frac{\sigma_2+2}{2}}b_{2}\left(\frac{1}{2}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}\right) ^{\frac{\sigma_2+2}{2}}\\ & -2^{\frac{\beta_1+1}{2}}b_{3}\left(\frac{1}{2}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}\right) ^{\frac{\beta_1+1}{2}}+\Delta\nonumber \end{align}

where $ {y}_{j}={\eta }_{j}{\zeta }_{j}\frac{2{\lambda }_{j}-1}{2{\lambda }_{j}}$ , and $ {\Delta }={\sum }_{j=1}^{3} {\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)}^{\frac{{\sigma }_{2}+2}{2}}+{\sum }_{j=1}^{3} {\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)}^{\frac{{\beta }_{1}+1}{2}}+{\sum }_{j=1}^{3} \left(\frac{{\zeta }_{j}{\lambda }_{j}}{2}{c}_{{m}_{j}}^{2}\right)-{\sum }_{j=1}^{3} \left(\frac{{y}_{j}}{{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)+\frac{3}{2}{\epsilon }_{N}^{2}$ .

If $ \frac{{}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\ge 1$ , the following inequality holds

(44) \begin{align} {\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)}^{\frac{{\sigma }_{2}+2}{2}}+{\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)}^{\frac{{\beta }_{1}+1}{2}}-\left(\frac{{y}_{j}}{{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right) & \le {\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)}^{\frac{{\sigma }_{2}+2}{2}}-\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)\nonumber\\[5pt]& \le {\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)}^{\frac{{\sigma }_{2}+2}{2}}-1\end{align}

On the other hand, for the case $ \frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2} \lt 1$ , it yields

(45) \begin{align} {\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)}^{\frac{{\sigma }_{2}+2}{2}}+{\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)}^{\frac{{\beta }_{1}+1}{2}}-\left(\frac{{y}_{j}}{{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)\le {\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)}^{\frac{{\beta }_{1}+1}{2}}-\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)\le 1\end{align}

According to the theory of uniform boundedness, $ \textbf{s}$ and $ {\tilde{\textbf{c}}}_{m}$ are uniformly bounded. Therefore, there are always some positive constants $ {\vartheta }_{j}(j=\textrm{1,2},3)$ such that $ \left|{\tilde{c}}_{{m}_{j}}\right| \lt {\vartheta }_{j}$ . Then,

(46) \begin{align} {\Delta } & = \sum _{j=1}^{3} \left\{{\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)}^{\frac{{\sigma }_{2}+2}{2}}+{\left(\frac{{y}_{j}}{2{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)}^{\frac{{\beta }_{1}+1}{2}}-\left(\frac{{y}_{j}}{{\eta }_{j}}{\tilde{c}}_{{m}_{j}}^{2}\right)\right\}\nonumber\\& +\sum _{j=1}^{3} \left(\frac{{\zeta }_{j}{\lambda }_{j}}{2}{c}_{{m}_{j}}^{2}\right)+\frac{3}{2}{\epsilon }_{N}^{2}\\ & \le \sum _{j=1}^{3} \textrm{m}\textrm{a}\textrm{x}\left\{{\left(\frac{{y}_{j}}{2{\eta }_{j}}{\vartheta }_{j}^{2}\right)}^{\frac{{\sigma }_{2}+2}{2}}-\textrm{1,1}\right\}+\sum _{j=1}^{3} \left(\frac{{\zeta }_{j}{\lambda }_{j}}{2}{c}_{{m}_{j}}^{2}\right)+\frac{3}{2}{\epsilon }_{N}^{2} \lt \infty \nonumber \end{align}

Using Lemma 1, the previous inequality Equation (43) is further simplified as

(47) \begin{align} \dot{{V}}_{2} \leq & - \gamma_{1} G \left(\textbf{s}\right) 2^{\frac{\sigma_2+2}{2}} \left(\frac{1}{2} \textbf{s}^{\text{T}} \textbf{s}\right)^{\frac{\sigma_2+2}{2}} -{y}_{m}\left(\frac{1}{2}{\tilde{\textbf{c}}}_{m}^{\text{T}}\boldsymbol{\eta}^{-1} {\tilde{\textbf{c}}}_{m} \right)^{\frac{\sigma_2+2}{2}}\nonumber\\ & -2^{\frac{\sigma_2+2}{2}}b_{2}\left(\frac{1}{2}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}\right) ^{\frac{\sigma_2+2}{2}} - \gamma_{2} G \left(\textbf{s}\right) 2^{\frac{\beta_1+1}{2}} \left(\frac{1}{2} \textbf{s}^{\text{T}} \textbf{s}\right)^{\frac{\beta_1+1}{2}} \\ & - {y}_{m}\left(\frac{1}{2} {\tilde{\textbf{c}}}_{m}^{\text{T}}\boldsymbol{\eta}^{-1} {\tilde{\textbf{c}}}_{m} \right)^{\frac{\beta_1+1}{2}} -2^{\frac{\beta_1+1}{2}}b_{3}\left(\frac{1}{2}\boldsymbol{\Omega}^{\text{T}}\boldsymbol{\Omega}\right) ^{\frac{\beta_1+1}{2}}+\Delta\nonumber\\ \leq & -a V_{2}^{\frac{\sigma_2+2}{2}} - b V_{2}^{\frac{\beta_1+1}{2}}+\Delta\nonumber \end{align}

where ${y_m} = {\rm{min}}\left\{ {y_1^{\frac{{{\sigma _2} + 2}}{2}},y_2^{\frac{{{\sigma _2} + 2}}{2}},y_3^{\frac{{{\sigma _2} + 2}}{2}}} \right\}$ , $a = {3^{ - \frac{{{\sigma _2}}}{2}}}{\rm{min}}\left\{ {{2^{\frac{{{\sigma _2} + 2}}{2}}}{\gamma _1}G\left( {\bf{s}} \right),{y_m},{2^{\frac{{{\sigma _2} + 2}}{2}}}{b_2}} \right\}$ , $b = {\rm{min}}\{ {2^{\frac{{{\beta _1} + 1}}{2}}}{\gamma _2}G\left( {\bf{s}} \right),$ ${y_m},{2^{\frac{{{\beta _1} + 1}}{2}}}{b_3}\} $ .

Therefore, according to Lemma 3, it can be concluded that the NFTSS (13) is practically fixed-time stable and converges to the following residual set ${\rm{\Pi }}$ after a bounded time ${T_r}$ .

(48) \begin{align}{\rm{\Pi }}: = \left\{ {\mathop {{\rm{lim}}}\limits_{t \to {T_r}} {\bf{s}}(t)|\parallel {\bf{s}}(t)\parallel \le {\rm{\Phi }}} \right\}\end{align}

where for $0 \lt {o_1},{o_2} \le 1$ ,

(49) \begin{align}{\rm{\Phi }} = {\rm{min}}\left\{ {{{\left( {\frac{{\rm{\Delta }}}{{a\left( {1 - {o_1}} \right)}}} \right)}^{\frac{2}{{{\sigma _2} + 2}}}},{{\left( {\frac{{\rm{\Delta }}}{{b\left( {1 - {o_2}} \right)}}} \right)}^{\frac{2}{{{\beta _1} + 1}}}}} \right\}\end{align}

(50) \begin{align}{T_r} = \frac{2}{{a{\sigma _2}}} + \frac{2}{{b\left( {1 - {\beta _1}} \right)}}\end{align}

1. Step 2. Once $s$ reaches the region $\Pi $ , the fixed-time convergence property of the system states should be demonstrated by the following analysis.

Case 2. In the case of $\overline {{s_i}} \ne 0$ and $\left| {{e_{1i}}} \right| \ge $ ${\varepsilon _0}$ , let ${s_i} = {\phi _i} \in {\rm{\Pi }}$ and one has

(51) \begin{align}s_{i}=e_{2i}+ N\left(\textbf{e}_1\right) \left( K_{1i}\textrm{sig}^{1+\sigma_{1}}(e_{1i}) +K_{2i}\textrm{sig}^{\frac{p}{q}}(e_{1i}) \right) =\phi_{i}\end{align}

it follows that

(52) \begin{align}{{e_{2i}}} & + N\left( {{{\bf{e}}_1}} \right)\left( {{K_{1i}} - \frac{{{\phi _i}}}{{N\left( {{{\bf{e}}_1}} \right){\rm{si}}{{\rm{g}}^{1 + {\sigma _1}}}\left( {{e_{1i}}} \right)}}} \right){\rm{si}}{{\rm{g}}^{1 + {\sigma _1}}}\left( {{e_{1i}}} \right)\nonumber\\[4pt]& + N\left( {{{\bf{e}}_1}} \right){K_{2i}}{\rm{si}}{{\rm{g}}^{\frac{p}{q}}}\left( {{e_{1i}}} \right) = 0\end{align}

(53) \begin{align}{{e_{2i}}} & + N\left( {{{\bf{e}}_1}} \right){K_{1i}}{\rm{si}}{{\rm{g}}^{1 + {\sigma _1}}}\left( {{e_{1i}}} \right)\nonumber\\[4pt]& + N\left( {{{\bf{e}}_1}} \right)\left( {{K_{2i}} - \frac{{{\phi _i}}}{{N\left( {{{\bf{e}}_1}} \right){\rm{si}}{{\rm{g}}^{\frac{p}{q}}}\left( {{e_{1i}}} \right)}}} \right){\rm{si}}{{\rm{g}}^{\frac{p}{q}}}\left( {{e_{1i}}} \right) = 0\end{align}

Choosing ${K_{1i}}$ such that ${K_{1i}} - \dfrac{{{\phi _i}}}{{N\left( {{{\bf{e}}_1}} \right){\rm{si}}{{\rm{g}}^{1 + {\sigma _1}}}\left( {{e_{1i}}} \right)}} \gt 0$ or ${K_{2i}} - \dfrac{{{\phi _i}}}{{N\left( {{{\bf{e}}_1}} \right){\rm{si}}{{\rm{g}}^{\dfrac{p}{q}}}\left( {{e_{1i}}} \right)}} \gt 0$ , then the fixed-time stability of the system can still be guaranteed. Therefore, the tracking error ${e_{1i}}$ ultimately converges to the region $\left| {{e_{1i}}} \right| \le {{\rm{\Psi }}_{e1}}$ after a fixed time. At this time, it is easy to derive that the tracking velocity ${e_{2i}}$ converges to the region $\left| {{e_{2i}}} \right| \le {{\rm{\Psi }}_{e2}}$ after a fixed time.

Case 3. For the case ${\bar s_i} \ne 0$ and $\left| {{e_{1i}}} \right| \lt $ ${\varepsilon _0}$ . Since the tracking error ${e_{1i}}$ has entered the region ${{\rm{\Psi }}_{e1}}$ , according to the NFTSS (13), one obtains

(54) \begin{align}{s_i} = {e_{2i}} + N\left( {{{\bf{e}}_1}} \right)\left( {{K_{1i}}{\rm{si}}{{\rm{g}}^{1 + {\sigma _1}}}\left( {{e_{1i}}} \right) + {K_{2i}}\left( {{l_1}{e_{1i}} + {l_2}e_{1i}^2{\rm{sgn}}\left( {{e_{1i}}} \right)} \right)} \right) = {\phi _i}\end{align}

It follows that

(55) \begin{align}\left| {{e_{2i}}} \right| \le {\phi _i} + N\left( {{{\bf{e}}_1}} \right)\left( {{K_{1i}}\left| {{e_1}{|^{1 + {\sigma _1}}} + {K_{2i}}} \right|{l_1}{e_{1i}} + {l_2}e_{1i}^2{\rm{sgn}}\left( {{e_{1i}}} \right)|} \right) \le {{\rm{\Psi }}_{{e_2}}}\end{align}

The above analysis shows that the system states ${e_{1i}}$ and ${e_{2i}}$ will converge to the sets $\left| {{e_{1i}}} \right| \le {{\rm{\Psi }}_{e1}}$ and $\left| {{e_{2i}}} \right| \le {{\rm{\Psi }}_{e2}}$ after a fixed time, respectively. It is concluded that the closed-loop tracking control system is practically fixed-time stable. The proof is completed.

4.1 Performance evaluation with healthy actuators

In this part, the input saturation is ignored, and the main purpose is to investigate the effect of the PPC constraint on the tracking performance. The proposed control scheme is compared with an adaptive fixed-time sliding mode controller (AFSMTC) that does not take into account performance constraints. The AFSMTC has the same form as the proposed scheme, except for the tan-type PPC term. For a fair comparison, the control parameters chosen for the AFSMTC are identical to those of the proposed scheme. The simulation results are given in Figs 5–8.

Figure 5.

Time response of position tracking error under the AFSMTC.

Figure 6.

Time response of position tracking error under the proposed controller.

Figure 7.

Time response of velocity tracking error.

Figure 8.

Time response of control torques.

Figures 5–7 illustrate the time responses of the position and velocity tracking errors. These results validate the stability analysis in Section 3 comfortably. Although both controllers enable fixed-time convergence, the proposed controller possesses a faster transient response with higher steady-state accuracy. Furthermore, with the help of tan-type PPC, the position tracking errors are always strictly limited to the predefined boundary in both transient and steady-state phases, whereas those of the AFSMTC violate the constraint. As can be seen, since the constraint boundary is decaying with time, the norm of tracking error is decreasing, which increases the tracking precision. It is the effect of the prescribed performance function in the proposed control scheme. Control torques are depicted in Fig. 8, from which the maximum torque required by the proposed controller is slightly larger than that of the AFSMTC. This is because the control gain is increased to counteract the evolution of the tracking error when it approaches the constraint boundary. It is worth noting that to obtain fast trajectory tracking, both controllers have a large transient response at the beginning of the movement, which can make them impossible for practical applications. Therefore, the influence of actuator saturation needs to be considered during the controller design process.

4.2 Performance evaluation with input saturation

In this part, the problem of input saturation is investigated and the capability of the proposed controller is evaluated in the following two simulation sections. To this end, the allowable maximum input control torques are selected as ${[60,50,35,25,25,20,30]^{\rm{T}}}{\rm{N}} \cdot {\rm{m}}$ .

4.2.1 Performance validation of the saturation compensator

In this simulation, the main objective is to illustrate the effectiveness of the proposed saturation compensator in the proposed control method, two cases are discussed for comparison. One is the tracking control with saturation compensation, and the other is the tracking control without saturation compensation.

The simulation results without and with saturation compensation are presented in Figs 9 and 10, respectively. As shown in Fig. 9, the convergence time of tracking error and tracking speed error is finite, but the error constraint boundary is no longer satisfied. It indicates that input saturation has greatly attenuated the convergence performance of trajectory tracking. From Fig. 10, the predicted compensation parameter ${\bf{\Omega }}$ converges quickly to a constant, which means a strong compensation role when starting to track the desired trajectory. Under the action of the saturation compensator, the tracking error is not only always strictly remained within the predefined constraint bounds, but also a superior control performance is obtained with shorter settling time and faster convergence speed. By comparison, the actual control torques for both controllers are guaranteed to be within their maximum allowable limits. It should be noted that due to the influence of input saturation, the torque produces a strong chattering phenomenon. Whereas with the help of the saturation compensator, the control torque obtained is smoother with significantly less chattering. To sum up, conclusions can be drawn the proposed saturation compensator can eliminate the effect of input saturation to a certain extent and accelerate the convergence rate of trajectory tracking in the proposed control scheme.

Figure 9.

Simulation results without the saturation compensator.

Figure 10.

Simulation results with the saturation compensator.

4.2.2 Performance comparison with existing mehtods

To further assess the proposed controller’s performance, comparisons are performed with the existing fixed-time control strategies proposed in Refs [Reference Zuo18, Reference Zhang, Tang and Guo47]. To ensure the rationality of the comparison, all simulation experiments are conducted under the same initial conditions. The FTSMC can be described as follows [Reference Zuo18]:

(56) \begin{align} \textbf{u} & ={\textbf{u}}_{1}+{\textbf{u}}_{2}+{\textbf{u}}_{3}\nonumber\\[-2pt] {\textbf{u}}_{1} & =-{\textbf{B}}^{-1}\left({\textbf{x}}_{1}\right)\left(\textbf{F}\right({\textbf{x}}_{1},{\textbf{x}}_{2})-{\ddot{\textbf{q}}}_{d})\nonumber\\[-2pt] & -{\textbf{B}}^{-1}\left({\textbf{x}}_{1}\right){\sigma }^{-1}{\Phi }{\textbf{e}}_{2}-\frac{{p}_{1}}{{q}_{1}}\left(\sigma \textbf{B}\right({\textbf{x}}_{1}\left){)}^{-1}\textrm{d}\textrm{i}\textrm{a}\textrm{g}\right(\left|\sigma {\textbf{e}}_{2}{|}^{1-{q}_{1}/{p}_{1}}\right){\textbf{e}}_{2}\nonumber\\[-2pt] {\textbf{u}}_{2} & =-\frac{{p}_{1}}{{q}_{1}}\left(\sigma \textbf{B}\right({\textbf{x}}_{1}\left){)}^{-1}\textrm{d}\textrm{i}\textrm{a}\textrm{g}\right(\textrm{s}\textrm{i}{\textrm{g}}^{2-{q}_{1}/{p}_{1}}\left(\sigma {\textbf{e}}_{2}\right)){\mu }_{\tau }\nonumber\\[-2pt] & \left({\alpha }_{2}\textrm{s}\textrm{i}{\textrm{g}}^{{m}_{2}/{n}_{2}}\left(\textbf{s}\right)+{\beta }_{2}\textrm{s}\textrm{i}{\textrm{g}}^{{p}_{2}/{q}_{2}}\left(\textbf{s}\right)\right)\\[-2pt] {\textbf{u}}_{3} & =-{\textbf{B}}^{-1}\left({\textbf{x}}_{1}\right)({c}_{1}+{c}_{2}|\left|{\textbf{x}}_{1}\right||+{c}_{3}|\left|{\textbf{x}}_{2}\right|{|}^{2})\frac{\textbf{s}}{\left|\right|\textbf{s}\left|\right|}\nonumber\\[-2pt] \textbf{s} & ={\textbf{e}}_{1}+\textrm{s}\textrm{i}{\textrm{g}}^{q1/p1}\left(\sigma {\textbf{e}}_{2}\right)\nonumber \end{align}

where ${\alpha _2} \gt 0,{\beta _2} \gt 0$ , ${c_1},{c_2},{c_3}$ are three positive constants, $\sigma = {\rm{diag}}\left( {{\sigma _1},{\sigma _2}, \ldots ,{\sigma _n}} \right)$ with ${\sigma _i} = \frac{1}{{{\alpha _1}\left( i \right)e_{1i}^{{m_1}/{n_1} - {p_1}/{q_1}} + {\beta _1}\left( i \right)}}$ , ${\alpha _1} = \left[ {{\alpha _{11}},{\alpha _{12}}, \ldots ,{\alpha _{1n}}} \right]$ and ${\beta _1} = \left[ {{\beta _{11}},{\beta _{12}}, \ldots ,{\beta _{1n}}} \right]$ with ${\alpha _{1i}} \gt 0$ and ${\beta _{1i}} \gt 0$ , ${\bf{\Phi }} = {\rm{diag}}\left( {{{\rm{\Phi }}_1},{{\rm{\Phi }}_2}, \ldots ,{{\rm{\Phi }}_n}} \right)$ with ${{\rm{\Phi }}_i} = - {\alpha _{1i}}\left( {\frac{{{m_1}}}{{{n_1}}} - \frac{{{p_1}}}{{{q_1}}}} \right)e_{1i}^{{m_1}/{n_1} - {p_1}/{q_1} - 1}\sigma _i^2{e_{2i}}$ , ${m_1}$ , ${m_2}$ , ${n_1}$ , ${n_2}$ , ${p_1}$ , ${p_2}$ , ${q_1}$ , ${q_2}$ are positive odd integers satisfying ${m_1} \gt {n_1},{m_2} \gt {n_2},{p_1} \lt {q_1} \lt 2{p_1},{p_2} \lt {q_2}$ and ${m_1}/{n_1} - {p_1}/{q_1} \gt 1$ . ${\mu _\tau } = {\rm{diag}}\left( {{\mu _{\tau 1}},{\mu _{\tau 2}}, \ldots ,{\mu _{\tau n}}} \right)$ with ${\mu _{\tau i}}$ given by:

(57) \begin{align}{\mu _{\tau i}} = \left\{ {\begin{array}{l@{\quad}l}{{\rm{sin}}\left( {\frac{\pi }{2} \cdot \frac{{e_{2i}^{{q_1}/{p_1} - 1}}}{{{\tau _i}}}} \right)} & {{\rm{if}}e_{2i}^{\frac{{{q_1}}}{{{p_1}}} - 1} \le {\tau _i}}\\1 & {{\rm{otherwise}}}\end{array}} \right.\end{align}

The value of ${\mu _{\tau i}}$ is 1 as the initial tracking speed ${{\bf{e}}_2}\left( 0 \right) = 0$ .

The adaptive nonsingular fast terminal sliding mode control (ANFTSMC) scheme proposed by Ref. [Reference Zhang, Tang and Guo47] is written as

(58) \begin{align}\textbf{u}&=\textbf{u}_1+\textbf{u}_{2}+\textbf{u}_{3}\nonumber\\ \textbf{u}_{1}&=-\textbf{B}^{-1}\left(\textbf{x}_{1}\right) \left(\textbf{F}\left(\textbf{x}_{1},\textbf{x}_{2}\right) -\ddot{\textbf{q}}_{d} \right)\nonumber\\ &\quad -\textbf{B}^{-1}\left(\textbf{x}_{1}\right) \left( k_1 \textbf{K}_{a} \textrm{diag}(\mid e_{1i}\mid^{k_1-1}) \textbf{e}_{2} +\textbf{K}_{b} \dot{\textbf{s}}_\rho(\textbf{e}_{1}) \right)\nonumber\\ \textbf{u}_{2}&= -\textbf{B}^{-1}\left(\textbf{x}_{1}\right) \left( \alpha_{2} {\textrm{sig}}^{k_2}(\textbf{s}) + \beta_{2} {\textrm{sig}}^{p_2/q_2}(\textbf{s}) \right)\nonumber\\ \textbf{u}_{3}&=-\textbf{B}^{-1}\left(\textbf{x}_{1}\right) \left(\hat{c}_{1}+\hat{c}_{2}\|{\boldsymbol{\boldsymbol{x}}_{1}}\|+\hat{c}_{3}\|\boldsymbol{\textbf{x}}_{2}\|^{2}\right) \tanh (\boldsymbol{s} / \varepsilon)\\\dot{\hat{c}}_{1}&=\eta_{11}\left(\boldsymbol{s}^{T} \tanh (\boldsymbol{s} / \varepsilon)-\eta_{21} \hat{c}_{1}\right)\nonumber\\ \dot{\hat{c}}_{2}&=\eta_{12}\left(\|{\boldsymbol{x}_1}\| \boldsymbol{s}^{T} \tanh (\boldsymbol{s} / \varepsilon)-\eta_{22} \hat{c}_{2}\right)\nonumber\\\dot{\hat{c}}_{3}&=\eta_{13}\left(\|\boldsymbol{x}_{2}\|^{2} \boldsymbol{s}^{T} \tanh (\boldsymbol{s} / \varepsilon)-\eta_{23} \hat{b}_{3}\right)\nonumber\\\textbf{s}&=\textbf{e}_{2}+\textbf{K}_{a}\textrm{sig}^{k_1 }(\textbf{e}_{1})+ \textbf{K}_{b}\textbf{s}_{\rho}(\textbf{e}_{1})\nonumber\end{align}

where ${{\bf{K}}_a},{{\bf{K}}_b}$ are positive definite matrices, ${\alpha _2} \gt 0,{\beta _2} \gt 0,$ ${k_1} = \frac{1}{2} + \frac{{{m_1}}}{{2{n_1}}} + \left( {\frac{{{m_1}}}{{2{n_1}}} - \frac{1}{2}} \right){\rm{sign}}\left( {\left| {\left| {{{\bf{e}}_1} - 1} \right|} \right|} \right)$ , ${k_2} = \frac{1}{2} + \frac{{{m_2}}}{{2{n_2}}} + \left( {\frac{{{m_2}}}{{2{n_2}}} - \frac{1}{2}} \right){\rm{sign}}\left( {\left| {\left| {{\bf{s}} - 1} \right|} \right|} \right)$ , ${m_1},{m_2},{p_1}$ and ${p_2}$ are positive odd integers satisfying ${m_1} \gt {n_1},{m_2} \gt {n_2}$ and ${p_2} \lt {q_2}$ , ${\rm{tanh}}\left( {\frac{{\bf{s}}}{\varepsilon }} \right) = {[{\rm{tanh}}\left( {\frac{{{s_1}}}{\varepsilon }} \right),{\rm{tanh}}\left( {\frac{{{s_2}}}{\varepsilon }} \right), \ldots ,{\rm{tanh}}\left( {\frac{{{s_n}}}{\varepsilon }} \right)]^T}$ with $\varepsilon $ is a arbitrary small positive constant, ${\eta _{1j}}$ and ${\eta _{2j}}$ are positive constants.

Table 3.

Controller parameters for FTSMC and ANFTSMC

Generally, the faster the system responses, the larger control torques generates, resulting in more energy consumption. Therefore, the selection of control parameters requires a trade-off between control performances and input torques. The control parameters for FTSMC and ANFTSMC are listed in Table 3. Figures 11–12 give the response curves of the position and speed tracking errors under these different controllers. The convergence time of all controllers is observed to be bounded, and the suggested controller takes less time to track the desired trajectory than the other two controllers. Meanwhile, the transient performance requirement is consistently guaranteed only under the proposed control, implying that a quicker convergence performance is realised. Figure 13 clearly depicts the comparison of joint torques. In the presence of input saturation, the proposed controller provides a smoother, more continuous and less chattering control input.

Figure 11.

Position tracking error performance.

Figure 12.

Velocity tracking error performance.

Figure 13.

Control torque.

To quantitatively evaluate the performance of the various controllers, for the total number of sampling times $N$ , three metrics including the integrated absolute error (IAE) ${\rm{IA}}{{\rm{E}}_{{e_{1i}}}} = {1 \over N}\mathop \sum \nolimits_{j = 1}^N \left| {{e_{1i}}\left( j \right)} \right|$ , the integrated time absolute error (ITAE) ${\rm{ITA}}{{\rm{E}}_{{e_{1i}}}} = {1 \over N}\mathop \sum \nolimits_{j = 1}^N t\left( j \right)\left| {{e_{1i}}\left( j \right)} \right|$ , and the energy consumption (EC) ${\rm{E}}{{\rm{C}}_{{u_i}}} = {1 \over N}\mathop \sum \nolimits_{j = 1}^N \left| {{\tau _i}\left( j \right)} \right|$ are introduced, as shown in Tables 4–6.

Table 4.

IAE index comparison of different controllers

Table 5.

ITAE index comparison of different controllers

Table 6.

EC index comparison of different controllers

The lower the values of the above three indicators, the better the control performance of the closed-loop tracking system. According to the comparison, the proposed control method can achieve faster tracking convergence and higher tracking precision with less energy consumption than the other two referenced methods. The simulation results demonstrate the noticeable superiority of the proposed controller in terms of tracking performance, energy consumption and chattering suppression.

In summary, it can be concluded that the proposed controller realises fast and accurate trajectory tracking of the space manipulator in a bounded time, while satisfying the transient and steady-state performance constraints of tracking error, even in the presence of unknown disturbances, inertial uncertainty and input saturation.