2.1 Preliminaries

The control objective in this study is to design the TRNNO-TFTSMC strategy to guarantee that the real state $ [x,y,z,\phi, \theta, \psi {]}^{T}$ tracks the desired trajectory $ [{x}_{d},{y}_{d},{z}_{d},{\phi }_{d},{\theta }_{d},{\psi }_{d}{]}^{T}$ . To design a robust controller, analyse the stability, and simplify the design procedure of the control scheme, a virtual position controller is proposed as follows:

(5) \begin{align} \left[\begin{array}{l}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array}\right]=\left[\begin{array}{c}\left(\left(cos\!\left({x}_{7}\right)\!sin\!\left({x}_{9}\right)os\!\left({x}_{11}\right)+sin\!\left({x}_{7}\right)\!sin\!\left({x}_{11}\right)\right)\!/m\right)\\[2pt] \left(\left(cos\!\left({x}_{7}\right)\!sin\!\left({x}_{9}\right)\!sin\!\left({x}_{11}\right)-sin\!\left({x}_{7}\right)\!sin\!\left({x}_{11}\right)\right)\!/m\right)\\[2pt] \left(cos\!\left(\phi \right)cos\!\left(\theta \right)\right)\!/m\end{array}\right]{u}_{1}\end{align}

The following definition and lemma are given:

Lemma 1. [Reference Lu, Li and Li29] For the nonlinear system:

(6) \begin{align} \dot{x}=f(x),f(0)=0,x\subset {\mathbb{R}}^{n}\end{align}

Consider a continuous differentiable positive definite Lyapunov candidate function $ V(x)$ , if the differential of $ V(x)$ satisfies the inequality:

(7) \begin{align} \dot{V}(x)\leqslant -{\rho }_{m}V(x)-{\vartheta }_{m}{V}^{\frac{\mathrm{\aleph }+1}{2}}(x)\end{align}

where $ {\rho }_{m}\gt 0,{\vartheta }_{m}\gt \mathrm{0,0}\lt \aleph \lt 1$ . Then, the nonlinear system is finite-time stable, and the expression for the reaching time $ T$ is given by:

(8) \begin{align} T\leqslant \frac{2}{{\rho }_{m}(1-\mathrm{\aleph })}ln\frac{{\vartheta }_{m}+{\rho }_{m}{V}^{\frac{1-\mathrm{\aleph }}{2}}(x)}{{\vartheta }_{m}}\end{align}

3.1 Design of the position controller

In this part, a new controller is proposed for the position subsystem. The virtual control inputs $ ({u}_{x},{u}_{y},{u}_{z})$ have been given in (5) that the position subsystem is written as:

(9) \begin{align} \left\{\begin{array}{l}{\dot{x}}_{1}={x}_{2}\\ {\dot{x}}_{2}={u}_{x}-{c}_{1}{x}_{2}+{d}_{x}\\ {\dot{x}}_{3}={x}_{4}\\ {\dot{x}}_{4}={u}_{y}-{c}_{2}{x}_{4}+{d}_{y}\\ {\dot{x}}_{5}={x}_{6}\\ {\dot{x}}_{6}={u}_{z}-g-{c}_{3}{x}_{6}+{d}_{z}\end{array}\right.\end{align}

Inspired by literature [Reference Du and Song30], the external disturbances in (9) are estimated by designing the following observer:

(10) \begin{align} \left\{\begin{array}{l}{\widehat{T}}_{j}={\eta }_{j}({\dot{x}}_{j}-{T}_{j+1})\\[5pt] {\dot{T}}_{j+1}={\widehat{T}}_{j}+{\mathrm{\Pi }}_{i}+{F}_{p}\\ \end{array}\right.\end{align}

where $ j=\mathrm{1,3},5$ , $ i=x,y,z$ , $ 0\lt {\eta }_{j}\lt \frac{1}{2}$ . $ \mathrm{\Pi }\in {\mathbb{R}}^{3}$ , $ [{\mathrm{\Pi }}_{x},{\mathrm{\Pi }}_{y},{\mathrm{\Pi }}_{z}{]}^{T}=[{u}_{x},{u}_{y},{u}_{z}{]}^{T}+{I}_{g}$ ,where $ {I}_{g}=[\mathrm{0,0},-g{]}^{T}$ and $ {F}_{p}=[\!-\!{c}_{1}{x}_{2},-{c}_{2}{x}_{4},-{c}_{3}{x}_{6}{]}^{T}$ , The $ [{\widehat{T}}_{1},{\widehat{T}}_{3},{\widehat{T}}_{5}{]}^{T}$ are approximations of $ {d}_{p}=[{d}_{},{d}_{y},{d}_{z}{]}^{T}$ .

Theorem 1. Considering the disturbance observer in (10), the estimation error $ ({\tilde{d}}_{p}={d}_{p}-{\widehat{T}}_{j})$ converges to a neighbourhood of zero in finite time, which implies the following approximation can be obtained:

(11) \begin{align} {\widehat{T}}_{1}={d}_{x},\!{\widehat{T}}_{5}={d}_{y},\!{\widehat{T}}_{5}={d}_{z}.\end{align}

Proof of Theorem 1. To ensure the clarity of the proof process, we introduce an auxiliary constant $ {\gamma }_{1}$ , which is a positive constant and exists in the positive neighbourhood of zero. Then, assuming that $|{\tilde{d}}_{p}({t}_{0})|\gt {\gamma }_{1}$ , where $ {t}_{0}$ is the initial phase of the system. The reaching time $ {t}_{1}$ is defined as: $ |{\tilde{d}}_{p}({t}_{1})|={\gamma }_{1}$ .

According to the $ {\tilde{d}}_{p}={d}_{p}-{\widehat{T}}_{j}$ , $ d$ is calculated as:

(12) \begin{equation} \begin{aligned} {\dot{\tilde{d}}}_{p}& ={\dot{d}}_{p}-{\dot{\widehat{T}}}_{j}\\ & ={\dot{d}}_{p}-{\eta }_{j}({\ddot{x}}_{\mathrm{j}}-{\dot{T}}_{j+1})\\ & ={\dot{d}}_{p}-{\eta }_{j}{\tilde{d}}_{p} \end{aligned} \end{equation}

where $ j=\mathrm{1,3},5$ .

To show the finite-time convergence of $ {\tilde{d}}_{p}$ , the Lyapunov function candidate is chosen as:

(13) \begin{align} {V}_{i,d}=\frac{1}{2}{\tilde{d}}_{p}^{2}\end{align}

where $ i=x,y,z$ , the derivative of $ {V}_{i,d}$ is calculated as:

(14) \begin{align} {\dot{V}}_{i,d}={\tilde{d}}_{p}{\dot{\tilde{d}}}_{p}={\tilde{d}}_{p}{\dot{d}}_{p}-{\tilde{d}}_{p}{\eta }_{j}{\tilde{d}}_{p}\end{align}

Simplify the $ {V}_{i,d}$ , we can get:

(15) \begin{equation} \begin{aligned} {\dot{V}}_{i,d}& ={\tilde{d}}_{p}{\dot{d}}_{p}-{\tilde{d}}_{p}{\eta }_{j}{\tilde{d}}_{p}\\ & \leqslant -\bigg({\eta }_{j}-\frac{1}{2}\bigg){\tilde{d}}_{p}^{2}+\frac{3}{2}{{d}^{\mathrm{*}}}^{2}\\ & \leqslant -{a}_{0}{V}_{i,d}+{b}_{0}\end{aligned} \end{equation}

with $ {a}_{0}=-({\eta }_{j}-\frac{1}{2})$ and $ {b}_{0}=\frac{3}{2}{{d}^{\mathrm{*}}}^{2}$ .

Multiply both sides of the inequality by an integrating factor $ {e}^{{a}_{0}t}$ , we can obtain:

(16) \begin{align} {e}^{{a}_{0}t}{\dot{V}}_{i,d}\leqslant -{a}_{0}{e}^{{a}_{0}t}{V}_{i,d}+{b}_{0}{e}^{{a}_{0}t}\end{align}

By the chain rule, the (16) is written:

(17) \begin{align} \frac{d}{dt}\left({e}^{{a}_{0}t}{V}_{i,d}\right)-{a}_{0}{e}^{{a}_{0}t}{V}_{i,d}\leqslant -{a}_{0}{e}^{{a}_{0}t}{V}_{i,d}+{b}_{0}{e}^{{a}_{0}t}\end{align}

By integrating of (17), we obtain:

(18) \begin{align} {e}^{{a}_{0}t}{V}_{i,d}(t)-{e}^{{a}_{0}{t}_{0}}{V}_{i,d}({t}_{0})& \leqslant \frac{{b}_{0}}{{a}_{0}}({e}^{{a}_{0}t}-{e}^{{a}_{0}{t}_{0}}) \end{align}

Furthermore, dividing each term by $ {e}^{{a}_{0}t}$ :

(19) \begin{align} {V}_{i,d}(t)& \leqslant {e}^{-{a}_{0}({t}_{1}-{t}_{0})}{V}_{i,d}({t}_{0})+\frac{{b}_{0}}{{a}_{0}}(1-{e}^{-{a}_{0}({t}_{1}-{t}_{0})}) \end{align}

Using the definition of $ {V}_{i,d}$ in (13):

(20) \begin{align} |{\tilde{d}}_{p}(t)|\lt |{\tilde{d}}_{p}({t}_{0})|{e}^{-\frac{{a}_{0}}{2}({t}_{1}-{t}_{0})}+\sqrt{\frac{{b}_{0}}{{a}_{0}}(1-{e}^{-{a}_{0}({t}_{1}-{t}_{0})})}\end{align}

From (20), the upper bound of $ {t}_{1}$ is obtained as:

(21) \begin{align} {t}_{1}={t}_{0}+\frac{2}{{a}_{0}}\left(ln\right|{\tilde{d}}_{p}({t}_{0})|-ln|{\tilde{d}}_{p}(t)-{{{\gimel}}}_{p}\left|\right)\end{align}

where $ {\mathrm{{\gimel}}}_{p}=\sqrt{\frac{{b}_{0}}{{a}_{0}}(1-{e}^{-{a}_{0}({t}_{1}-{t}_{0})})}$ is bounded. By selecting a sufficiently large $ {\eta }_{j}$ , the $ {\mathrm{{\gimel}}}_{p}$ is arbitrarily small. Then, the disturbance observer guarantees an accurate estimation in a finite time.

The desired position signals are referred as $ {D}^{t}=[{x}_{1d},{x}_{3d},{x}_{5d}{]}^{T}=[{x}_{d},{y}_{d},{z}_{d}{]}^{T}$ . And the tracking errors of position are defined as follows:

(22) \begin{align} \begin{array}{l}\left\{\begin{array}{l}{e}_{x}={x}_{1}-{x}_{1d}=x-{x}_{d}\\ {e}_{y}={x}_{3}-{x}_{3d}=y-{y}_{d}\\ {e}_{z}={x}_{5}-{x}_{5d}=z-{z}_{d}\\ \end{array}\right.\end{array}\end{align}

where $ [{x}_{1},{x}_{3},{x}_{5}{]}^{T}=[x,y,z{]}^{T}$ are real states.

Inspired by Ref. [Reference Derrouaoui, Bouzid and Doula31], defining the fast terminal sliding surfaces as follows:

(23) \begin{align} \begin{array}{l}\left\{\begin{array}{l}{s}_{x}={\dot{e}}_{x}+{k}_{1}{{e}_{x}}^{{\beta }_{1}}+{k}_{2}si{g}^{{\beta }_{2}}({e}_{x})\\ {s}_{y}={\dot{e}}_{y}+{k}_{3}{{e}_{y}}^{{\beta }_{3}}+{k}_{4}si{g}^{{\beta }_{4}}({e}_{y})\\ {s}_{z}={\dot{e}}_{z}+{k}_{5}{{e}_{z}}^{{\beta }_{5}}+{k}_{6}si{g}^{{\beta }_{6}}({e}_{z})\\ \end{array}\right.\end{array}\end{align}

where $ {k}_{i}$ $ (\forall i=\mathrm{1,2}, \ldots, 6)$ and $ {\beta }_{i}$ $ (\forall i=\mathrm{1,2}, \ldots, 6)$ are positive constants, and the control inputs of the position subsystem are obtained as:

(24) \begin{equation} \begin{aligned} {u}_{x}& ={c}_{1}{x}_{2}-{k}_{1}{\beta }_{1}{{e}_{x}}^{{\beta }_{1}-1}{\dot{e}}_{x}-{k}_{2}{\beta }_{2}|{e}_{x}{|}^{{\beta }_{2}-1}{\dot{e}}_{x}+{p}_{x}+{u}_{swx}\\[4pt] {u}_{y}& ={c}_{2}{x}_{4}-{k}_{3}{\beta }_{3}{{e}_{y}}^{{\beta }_{3}-1}{\dot{e}}_{y}-{k}_{4}{\beta }_{4}|{e}_{y}{|}^{{\beta }_{4}-1}{\dot{e}}_{y}+{p}_{y}+{u}_{swy}\\[4pt] {u}_{z}& ={c}_{3}{x}_{6}-{k}_{5}{\beta }_{5}{{e}_{z}}^{{\beta }_{5}-1}{\dot{e}}_{z}-{k}_{6}{\beta }_{6}|{e}_{z}{|}^{{\beta }_{6}-1}{\dot{e}}_{z}+{p}_{z}+{u}_{swz} \end{aligned} \end{equation}

with $ {p}_{x}={\ddot{x}}_{1d}+{\widehat{T}}_{1},{p}_{y}={\ddot{x}}_{3d}+{\widehat{T}}_{3},{p}_{z}={\ddot{\mathrm{x}}}_{5d}+{\widehat{T}}_{5}+g$ . And $ {u}_{swx},{u}_{swy},{u}_{swz}$ are switching control laws that guarantee the robustness of the position subsystem and reject the influence of external factors. Specifically, they are indicated as follows:

(25) \begin{equation} \begin{aligned} {u}_{swx}& =-{k}_{x1}{s}_{x}-{k}_{x2}si{g}^{{\delta }_{x}}({s}_{x})\\[4pt] {u}_{swy}& =-{k}_{y1}{s}_{y}-{k}_{y2}si{g}^{{\delta }_{y}}({s}_{y})\\[4pt] {u}_{swz}& =-{k}_{z1}{s}_{z}-{k}_{z2}si{g}^{{\delta }_{z}}({s}_{z}) \end{aligned} \end{equation}

where $ {k}_{x1},{k}_{x2},{k}_{y1},{k}_{y2},{k}_{z1},{k}_{z2}$ are positive constants.

Proof of Theorem 2. The Lyapunov function candidate of the position subsystem is given as:

(26) \begin{align} {V}_{i}=\frac{1}{2}{s}_{i}^{2},(\forall i=x,y,z)\end{align}

By calculating the time derivative of (26) and (23), the following result is obtained:

(27) \begin{equation} \begin{aligned} {\dot{V}}_{i}& ={s}_{i}{\dot{s}}_{i}\\ & ={s}_{i}\big({\ddot{e}}_{i}+{k}_{j}{\beta }_{j}{{e}_{i}}^{{\beta }_{j}-1}{\dot{e}}_{i}+{k}_{j+1}{\beta }_{j+1}|{e}_{i}{|}^{{\beta }_{j+1}-1}{\dot{e}}_{i}\big)\end{aligned} \end{equation}

in which $ j=\mathrm{1,3},5$ , when $ i=x,y,z$ . Substituting (9) and the second-order derivative of (22), the $ {\dot{V}}_{i}$ can be written as:

(28) \begin{equation} \begin{aligned} {\dot{V}}_{i} ={s}_{i}\big({I}_{m}-{c}_{m}{x}_{m+1}+{d}_{i}-{{D}^{t}}_{m}+{k}_{j}{\beta }_{j}{{e}_{i}}^{{\beta }_{j}-1}{\dot{e}}_{i}+{k}_{j+1}{\beta }_{j+1}|{e}_{i}{|}^{{\beta }_{j+1}-1}{\dot{e}}_{i}\big)\end{aligned} \end{equation}

where $ m=\mathrm{1,2},3$ , when $ i=x,y,z$ . And $ [{I}_{1},{I}_{2},{I}_{3}{]}^{T}=[{u}_{x},{u}_{y},{u}_{z}-g{]}^{T}$ . By substituting (24) into (28), and external disturbances are estimated by $ {T}_{1},{T}_{3},{T}_{5}$ , we will get:

(29) \begin{equation} \begin{aligned} {\dot{V}}_{i}& ={s}_{i}\left(\!-{k}_{i1}{s}_{i}-{k}_{i2}si{g}^{{\delta }_{i}}({s}_{i})\right)\\[4pt] & =-{k}_{i1}{s}_{i}^{2}-{k}_{i2}{s}_{i}si{g}^{{\delta }_{i}}\left({s}_{i}\right)\\[4pt] & =-{k}_{i1}{s}_{i}^{2}-{k}_{i2}|{s}_{i}{|}^{{\delta }_{i}+1}\\[4pt] & \leqslant -{k}_{i1}{V}_{i}-{2}^{\frac{{\delta }_{i}-1}{2}}{k}_{i2}{V}_{i}^{\frac{{\delta }_{i}+1}{2}}\\[4pt] & \leqslant -{{\hbar }}_{1}{V}_{i}-{{\hbar }}_{2}{V}_{i}^{\frac{{\delta }_{i}+1}{2}}\end{aligned} \end{equation}

where $ {{\hbar }}_{1}={k}_{i1}$ and $ {{\hbar }}_{2}={2}^{\frac{{\delta }_{i}-1}{2}}{k}_{i2}$ . According to Lemma 1, the following result is obtained:

(30) \begin{align} {T}_{s}\leqslant \frac{2}{{{\hbar }}_{1}(1-{\delta }_{i})}ln\frac{{{\hbar }}_{2}+{{\hbar }}_{1}{V}_{i}^{\frac{1-{\delta }_{i}}{2}}}{{{\hbar }}_{2}}\end{align}

To reduce chattering, the sign function is replaced by a smooth nonlinear function, specifically the hyperbolic tangent function (tanh). Furthermore, the tanh function ensures a faster convergence rate compared to the sign function. This approach avoids performance degradation caused by excessive control inputs. Accordingly, a new two-stage fast terminal sliding surface is proposed that incorporates the tanh function while preserving the sign function, using the absolute value of the error variable as a critical threshold:

(31) \begin{align} \begin{array}{ll}{s}_{x}& =\begin{array}{l}\left\{\begin{array}{l@{\quad}l}{\dot{e}}_{x}+{k}_{1}{{e}_{x}}^{{\beta }_{1}}+{k}_{2}si{g}^{{\beta }_{2}}({e}_{x}), & if\left|{e}_{x}\right|\ge {e}_{p}\\[3pt] {\dot{e}}_{x}+{\lambda }_{1}{e}_{x}^{{\alpha }_{1}}+tanh({e}_{x}), & else\\[3pt] \end{array}\right.\end{array}\\[12pt] {s}_{y}& =\begin{array}{l}\left\{\begin{array}{l@{\quad}l}{\dot{e}}_{y}+{k}_{3}{{e}_{y}}^{{\beta }_{3}}+{k}_{4}si{g}^{{\beta }_{4}}({e}_{y}), & if\left|{e}_{y}\right|\ge {e}_{p}\\[3pt] {\dot{e}}_{y}+{\lambda }_{2}{e}_{y}^{{\alpha }_{2}}+tanh({e}_{y}), & else\\[3pt] \end{array}\right.\end{array}\\[12pt] {\mathrm{s}}_{z}& =\begin{array}{l}\left\{\begin{array}{l@{\quad}l}{\dot{e}}_{z}+{k}_{5}{{e}_{z}}^{{\beta }_{5}}+{k}_{6}si{g}^{{\beta }_{6}}({e}_{z}), & if\left|{e}_{z}\right|\ge {e}_{p}\\[3pt] {\dot{e}}_{z}+{\lambda }_{3}{e}_{z}^{{\alpha }_{3}}+tanh({e}_{z}), & else\\[3pt] \end{array}\right.\end{array} \end{array}\end{align}

where $ {k}_{i}$ $ (\forall i=\mathrm{1,2}, \ldots, 6)$ and $ {\beta }_{i}$ $ (\forall i=\mathrm{1,2}, \ldots, 6)$ take the same value as in (23), $ {\alpha }_{1},{\alpha }_{2},{\alpha }_{3}$ and $ {\lambda }_{1},{\lambda }_{2},{\lambda }_{3}$ are positive constants. $ {e}_{p}\gt 0$ represents the convergence accuracy of the position subsystem.

Considering the disturbance observer in (10), the control inputs will be divided into two phases, with the absolute value of the error acting as the critical threshold.

Case(a): when $ \left|{e}_{i}\right|\ge {e}_{p}(i=x,y,z)$ , the control inputs of the position subsystem arereferenced to the form of (24).

Case(b): when $ \left|{e}_{i}\right|\lt {e}_{p}(i=x,y,z)$ , the control inputs of the position subsystem are represented as follows:

(32) \begin{align} \begin{array}{ll}{u}_{x}& ={c}_{1}{x}_{2}-{\lambda }_{1}{\alpha }_{1}{{e}_{x}}^{{\alpha }_{1}-1}{\dot{e}}_{\mathrm{x}}-(1-\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}({e}_{x}){)}^{2}{\dot{e}}_{x}+{p}_{x}+{u}_{swx}\\[4pt] {u}_{y}& ={c}_{2}{x}_{4}-{\lambda }_{2}{\alpha }_{2}{{e}_{y}}^{{\alpha }_{2}-1}{\dot{e}}_{y}-(1-\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}({e}_{y}){)}^{2}{\dot{e}}_{y}+{p}_{y}+{u}_{swy}\\[4pt] {u}_{z}& ={c}_{3}{x}_{6}-{\lambda }_{3}{\alpha }_{3}{{e}_{z}}^{{\alpha }_{3}-1}{\dot{e}}_{z}-(1-\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}({e}_{z}){)}^{2}{\dot{e}}_{z}+{p}_{z}+{u}_{swz}\end{array}\end{align}

in which $ {p}_{x}={\ddot{x}}_{1d}+{\widehat{T}}_{1},{p}_{y}={\ddot{x}}_{3d}+{\widehat{T}}_{3},{p}_{z}={\ddot{x}}_{5d}+{\widehat{T}}_{5}+g$ . And $ {u}_{swx},{u}_{s\mathrm{w}y},{u}_{swz}$ are designed in (25).

Proof of Theorem 3. The Lyapunov function candidate of the position subsystem is given as:

(33) \begin{align} {V}_{i}=\frac{1}{2}{s}_{i}^{2},(\forall i=x,y,z)\end{align}

Case(a): when $ \left|{e}_{i}\right|\ge {e}_{p}(i=x,y,z)$ , the proof based on Theorem 2 is satisfied.

Case(b): when $ \left|{e}_{i}\right|\lt {e}_{p}(i=x,y,z)$ , substituting (31) into the derivative of $ {V}_{i}$ , the following result is obtained:

(34) \begin{equation} \begin{aligned} {\dot{V}}_{i}& ={s}_{i}{\dot{s}}_{i}\\[5pt] & ={s}_{i}({\ddot{e}}_{i}+{\lambda }_{m}{\alpha }_{m}{{e}_{i}}^{{\alpha }_{m}-1}{\dot{e}}_{i}-(1-tanh({e}_{i}))^{2}{\dot{e}}_{i})\end{aligned} \end{equation}

where $ m=\mathrm{1,2},3$ , when $ i=x,y,z$ . Substituting (9) and the second-order derivative of (22), the $ {\dot{V}}_{i}$ can be written as:

(35) \begin{align} {\dot{V}}_{i} ={s}_{i}({P}_{m}-{c}_{m}{x}_{m+1}+{d}_{i}-{{D}^{t}}_{m}+{\lambda }_{m}{\alpha }_{m}{{e}_{i}}^{{\alpha }_{m}-1}{\dot{e}}_{i}-(1-tanh({e}_{i}){)}^{2}{\dot{e}}_{i}) \end{align}

where $ m=\mathrm{1,2},3$ , when $ i=x,y,z$ . And $ [{P}_{1},{P}_{2},{P}_{3}{]}^{T}=[{u}_{x},{u}_{y},{u}_{z}-g{]}^{T}$ . Substituting (32) into (35), we will get:

(36) \begin{equation} \begin{aligned} {\dot{V}}_{i}& ={s}_{i}(\!-{k}_{i1}{s}_{i}-{k}_{i2}si{g}^{{\delta }_{i}}({s}_{i}))\\[5pt] & =-{k}_{i1}{s}_{i}^{2}-{k}_{i2}{s}_{i}si{g}^{{\delta }_{i}}\left({s}_{i}\right)\\[5pt] & \leqslant -{{\hbar }'}_{\!\!1}{V}_{i}-{{\hbar }'}_{\!\!2}{V}_{i}^{\frac{{\delta }_{i}+1}{2}}\end{aligned} \end{equation}

where $ {{\hbar }'}_{\!\!1}={k}_{i1}$ , and $ {{\hbar }'}_{\!\!2}={2}^{\frac{{\delta }_{i}-1}{2}}{k}_{i2}$ . According to Lemma 1, the system states will converge in a finite time.

3.2 Design of the attitude controller

This part is devoted to the design of a new controller for the attitude subsystem. The attitude subsystem is written as:

(37) \begin{align} \left\{\begin{array}{l}{\dot{x}}_{7}={x}_{8}\\[5pt] {\dot{x}}_{8}={a}_{1}{x}_{10}{x}_{12}+{b}_{1}\overline{\omega }{x}_{10}-{c}_{4}{x}_{8}+{f}_{1}{u}_{2}+{d}_{\phi }\\[5pt] {\dot{x}}_{9}={x}_{10}\\[5pt] {\dot{x}}_{10}={a}_{2}{x}_{8}{x}_{12}+{b}_{2}\overline{\omega }{x}_{8}-{c}_{5}{x}_{10}+{f}_{2}{u}_{3}+{d}_{\theta }\\[5pt] {\dot{x}}_{11}={x}_{12}\\[5pt] {\dot{x}}_{12}={a}_{3}{x}_{8}{x}_{12}-{c}_{6}{x}_{12}+{f}_{3}{u}_{4}+{d}_{\psi }\end{array}\right.\end{align}

Similar to (10), the external disturbances in (37) are estimated by designing the following observer:

(38) \begin{align} \left\{\begin{array}{l}{T}_{j}={\eta }_{j}({\dot{x}}_{j}-{T}_{j+1})\\[5pt] {\dot{T}}_{j+1}={T}_{j}+{\mathrm{\Pi }}_{i}+{F}_{a}\\ \end{array}\right.\end{align}

where $ j=\mathrm{7,9},11$ , $ i=\phi, \theta, \psi $ . $ \mathrm{\Pi }\in {\mathbb{R}}^{3}$ and $ [{\mathrm{\Pi }}_{\phi },{\mathrm{\Pi }}_{\theta },{\mathrm{\Pi }}_{\psi }{]}^{T}=f\diamond [{u}_{2},{u}_{3},{u}_{4}{]}^{T}$ , where $ f=[\,{f}_{1},{f}_{2},{f}_{3}{]}^{T}$ . And $ {F}_{a}$ = $ [{a}_{1}{x}_{10}{x}_{12}+{b}_{1}\overline{\omega }{x}_{10}-{c}_{4}{x}_{8},{a}_{2}{x}_{8}{x}_{12}+{b}_{2}\overline{\omega }{x}_{8}-{c}_{5}{x}_{10},{a}_{3}{x}_{8}{x}_{12}-{c}_{6}{x}_{12}{]}^{T}$ . $ {\eta }_{j}$ and $ {\eta }_{j+1}$ are positive values. The $ {\widehat{T}}_{7},{\widehat{T}}_{9},{\widehat{T}}_{11}$ are approximations of $ {d}_{\phi },{d}_{\theta },{d}_{\psi }$ , respectively.

From Theorem 1, the estimation errors $ ({\tilde{d}}_{a}={d}_{a}-{\widehat{T}}_{j})$ of external disturbances will converge to a neighbourhood of zero within a finite time.

Following the approach in (31), this part proposes a design incorporating sliding mode surfaces.

(39) \begin{align} \begin{array}{ll}{s}_{\phi}& =\begin{array}{l}\left\{\begin{array}{l@{\quad}l}{\dot{e}}_{\phi }+{k}_{7}{{e}_{\phi }}^{{\beta }_{7}}+{k}_{8}si{g}^{{\beta }_{8}}({e}_{\phi}), & if\left|{e}_{\phi }\right|\ge {e}_{a}\\[3pt] {\dot{e}}_{\phi }+{\lambda }_{4}{e}_{\phi }^{{\alpha }_{4}}+tanh({e}_{\phi}), & else\\[3pt] \end{array}\right.\end{array}\\[12pt] {s}_{\theta }& =\begin{array}{l}\left\{\begin{array}{l@{\quad}l}{\dot{e}}_{\theta }+{k}_{9}{{e}_{\theta }}^{{\beta }_{9}}+{k}_{10}si{g}^{{\beta }_{10}}({e}_{\theta}), & if\left|{e}_{\theta }\right|\ge {e}_{a}\\[3pt] {\dot{e}}_{\theta }+{\lambda }_{5}{e}_{\theta }^{{\alpha }_{5}}+tanh({e}_{\theta}), & else\\[3pt] \end{array}\right.\end{array}\\[12pt] {s}_{\psi }& =\begin{array}{l}\left\{\begin{array}{l@{\quad}l}{\dot{e}}_{\psi }+{k}_{11}{{e}_{\psi }}^{{\beta }_{11}}+{k}_{12}si{g}^{{\beta }_{12}}({e}_{\psi}), & if\left|{e}_{\psi }\right|\ge {e}_{a}\\[3pt] {\dot{e}}_{\psi }+{\lambda }_{6}{e}_{\psi }^{{\alpha }_{6}}+tanh({e}_{\psi}), & else\\[3pt] \end{array}\right.\end{array}\end{array}\end{align}

where $ {k}_{i}$ $ (\forall i=\mathrm{7,8}, \ldots, 12)$ and $ {\beta }_{i}$ $ (\forall i=\mathrm{7,8}, \ldots, 12)$ are positive constants, $ {\alpha }_{4},{\alpha }_{5},{\alpha }_{6}$ and $ {\lambda }_{4},{\lambda }_{5},{\lambda }_{6}$ are positive constants. $ {e}_{a}\gt 0$ represents the convergence accuracy of the attitude subsystem.

Hence, the control inputs for the attitude subsystem will be divided into two phases:

Case(a): when $ \left|{e}_{i}\right|\ge {e}_{a}(i=\phi,\theta, \psi )$ , the control inputs are written:

(40) \begin{equation} \begin{aligned} {u}_{2} & ={f}_{1}^{-1}\big({c}_{4}{x}_{8}-{a}_{1}{x}_{10}{x}_{12}-{b}_{1}\overline{\omega }{x}_{10}+{\ddot{\phi }}_{d}+{\widehat{T}}_{7}\\[5pt] & \quad-{k}_{7}{\beta }_{7}{{e}_{\phi }}^{{\beta }_{7}-1}{\dot{e}}_{\phi }-{k}_{8}{\beta }_{8}|{e}_{\phi }{|}^{{\beta }_{8}-1}{\dot{e}}_{\phi }+{u}_{sw\phi }\big)\\[5pt] {u}_{3}& ={f}_{2}^{-1}\big({c}_{5}{x}_{10}-{a}_{1}{x}_{8}{x}_{12}-{b}_{2}\overline{\omega }{x}_{8}+{\ddot{\theta }}_{d}+{\widehat{T}}_{9}\\[5pt] & \quad -{k}_{9}{\beta }_{9}{{e}_{\theta }}^{{\beta }_{9}-1}{\dot{e}}_{\theta }-{k}_{10}{\beta }_{10}|{e}_{\theta }{|}^{{\beta }_{10}-1}{\dot{e}}_{\theta }+{u}_{sw\theta }\big)\\[5pt] {u}_{4}& ={f}_{3}^{-1}\big({c}_{6}{x}_{12}-{a}_{3}{x}_{8}{x}_{12}-{k}_{11}{\beta }_{11}{{e}_{\psi }}^{{\beta }_{11}-1}{\dot{e}}_{\psi } \\[5pt] &\quad +{\ddot{\psi }}_{d}+{\widehat{T}}_{11}-{k}_{12}{\beta }_{12}|{e}_{\psi }{|}^{{\beta }_{12}-1}{\dot{e}}_{\psi }+{u}_{sw\psi }\big) \end{aligned} \end{equation}

in which $ {u}_{sw\phi },{u}_{sw\theta },{u}_{sw\psi }$ are switching control laws:

(41) \begin{align} \begin{array}{ll}{u}_{sw\phi }& =-{k}_{\phi 1}{s}_{\phi }-{k}_{\phi 2}si{g}^{{\delta }_{\phi }}({s}_{\phi })\\[3pt] {u}_{sw\theta }& =-{k}_{\theta 1}{s}_{\theta }-{k}_{\theta 2}si{g}^{{\delta }_{\theta }}({s}_{\theta})\\[3pt] {u}_{sw\psi }& =-{k}_{\psi 1}{s}_{\psi }-{k}_{\psi 2}si{g}^{{\delta }_{\psi }}({s}_{\psi}) \end{array}\end{align}

where $ {k}_{\phi 1},{k}_{\phi 2},{k}_{\theta 1},{k}_{\theta 2},{k}_{\psi 1},{k}_{\psi 2}$ are positive constants. Case(b): when $ \left|{e}_{i}\right|\lt {e}_{a}(i=\phi, \theta, \psi )$ , the control inputs are written:

(42) \begin{equation} \begin{aligned} {u}_{2}& ={f}_{1}^{-1}({c}_{4}{x}_{8}-{a}_{1}{x}_{10}{x}_{12}-{b}_{1}\overline{\omega }{x}_{10}+{\ddot{\phi }}_{d}+{\widehat{T}}_{7}\\[3pt] & \quad -{\lambda }_{4}{\alpha }_{4}{{e}_{\phi }}^{{\alpha }_{4}-1}{\dot{e}}_{\phi }-(1-tanh({e}_{\phi }){)}^{2}{\dot{e}}_{\phi }+{u}_{sw\phi })\\[3pt] {u}_{3}& ={f}_{2}^{-1}({c}_{5}{x}_{10}-{a}_{1}{x}_{8}{x}_{12}-{b}_{2}\overline{\omega }{x}_{8}+{\ddot{\theta }}_{d}+{\widehat{T}}_{9}\\[3pt] &\quad -{\lambda }_{5}{\alpha }_{5}{{e}_{\theta }}^{{\alpha }_{5}-1}{\dot{e}}_{\theta }-(1-tanh({e}_{\theta }){)}^{2}{\dot{e}}_{\theta }+{u}_{sw\theta })\\[3pt] {u}_{4}& ={f}_{3}^{-1}({c}_{6}{x}_{12}-{a}_{3}{x}_{8}{x}_{12}+{\ddot{\psi }}_{d}+{\widehat{T}}_{11}\\[3pt] & \quad-{\lambda }_{6}{\alpha }_{6}{{e}_{\psi }}^{{\alpha }_{6}-1}{\dot{e}}_{\psi }-(1-tanh({e}_{\psi }){)}^{2}{\dot{e}}_{\psi }+{u}_{sw\psi }) \end{aligned} \end{equation}

where $ {u}_{sw\phi },{u}_{sw\theta },{u}_{w\psi }$ are designed in (41).

The proof of Theorem 4 is divided into two parts:

Case(a): when $ \left|{e}_{i}\right|\ge {e}_{a}(i=\phi, \theta, \psi )$ , replace the position subsystem with the attitude subsystem, and the proof of Theorem 2 is satisfied.

Case(b): when $ \left|{e}_{i}\right|\lt {e}_{a}(i=\phi, \theta, \psi )$ , replace the position subsystem with the attitude subsystem, and the proof of Case(b) in Theorem 3 is satisfied.

Therefore, Theorem 4 is valid.

4.1 Design of controller for quadrotor position based on TRNNO-TFTSMC method

The unknown nonlinear function $ {f}_{p}$ is estimated by the output of the TRNN. Therefore, ${f}_{p}=[\,{f}_{x},\,{f}_{y},$ ${f}_{z}{]}^{T}=[\!-{c}_{1}{x}_{2},-{c}_{2}{x}_{4},-{c}_{3}{x}_{6}{]}^{T}$ in the position subsystem can be approximated as:

(47) \begin{align} {\widehat{f}}_{p}={\widehat{W}}_{p}^{T}{\widehat{\mathrm{\Phi }}}_{p}\big({\widehat{w}}_{p}^{I},{\widehat{w}}_{p}^{D},{\widehat{w}}_{p}^{oI},{\widehat{w}}_{p}^{oH},{\widehat{w}}_{p}^{HI}\big)\end{align}

Hence, the approximation error $ (\,{\tilde{f}}_{p}={f}_{p}-{\widehat{f}}_{p}$ ) is calculated as follows:

(48) \begin{align} \begin{array}{ll}{\tilde{f}}_{p}& ={f}_{p}-{\widehat{f}}_{p}\\[5pt] & ={W}_{p}^{T}\mathrm{\Phi }-{\widehat{W}}_{p}^{T}\widehat{\mathrm{\Phi }}+{\varepsilon }_{p}\end{array}\end{align}

where $ {\varepsilon }_{p}$ is an approximation error term.

To derive the adaptive laws of TRNN, the Taylor expansion linearisation method is utilised. We can get:

(49) \begin{align} \begin{array}{ll}\tilde{\mathrm{\Phi }}& =\dfrac{\partial \tilde{\mathrm{\Phi }}}{\partial {w}_{p}^{I}}{|}_{|{w}_{p}^{I}={\widehat{w}}_{p}^{I}}\big({w}_{p}^{I}-{\widehat{w}}_{p}^{I}\big)+\dfrac{\partial \tilde{\mathrm{\Phi }}}{\partial {w}_{p}^{D}}{|}_{|{w}_{p}^{D}={\widehat{w}}_{p}^{D}}\big({w}_{p}^{\mathrm{D}}-{\widehat{w}}_{p}^{D}\big)\\[10pt] & \quad +\dfrac{\partial \tilde{\mathrm{\Phi }}}{\partial {w}_{p}^{OI}}{|}_{|{w}_{p}^{OI}={\widehat{w}}_{p}^{OI}}\big({w}_{p}^{OI}-{\widehat{w}}_{p}^{OI}\big)\\[10pt] & \quad +\dfrac{\partial \tilde{\mathrm{\Phi }}}{\partial {w}_{p}^{OH}}{|}_{|{w}_{p}^{OH}={\widehat{w}}_{p}^{OH}}\big({w}_{p}^{OH}-{\widehat{w}}_{p}^{OH}\big)\\[10pt] & \quad +\dfrac{\partial \tilde{\mathrm{\Phi }}}{\partial {w}_{p}^{HI}}{|}_{|{w}_{p}^{HI}={\widehat{w}}_{p}^{HI}}\big({w}_{p}^{HI}-{\widehat{w}}_{p}^{HI}\big)+O\\[10pt] & ={D}_{w}^{I}\cdot {\tilde{w}}_{p}^{I}+{D}_{w}^{D}\cdot {\tilde{w}}_{p}^{D}+{D}_{w}^{OI}\cdot {\tilde{w}}_{p}^{OI}+{D}_{w}^{OH}\cdot {\tilde{w}}_{p}^{OH}+{D}_{\mathrm{w}}^{HI}\cdot {\tilde{w}}_{p}^{HI}+O\end{array}\end{align}

where $ O$ is a high-order term.

Substituting (49) into (48), the following result can be obtained:

(50) \begin{equation} \begin{aligned} {\tilde{f}}_{p}& ={f}_{p}-{\widehat{f}}_{p}\\[5pt] & ={W}_{p}^{T}\mathrm{\Phi }-{\widehat{W}}_{p}^{T}\widehat{\mathrm{\Phi }}+{\varepsilon }_{p}\\[5pt] & ={\tilde{W}}^{T}\widehat{\mathrm{\Phi }}+{\widehat{W}}^{T}\tilde{\mathrm{\Phi }}+{\tilde{W}}^{T}\tilde{\mathrm{\Phi }}+{\varepsilon }_{p}\\[5pt] & ={\widehat{W}}^{T}\big({D}_{w}^{I}{\tilde{w}}_{p}^{I}+{D}_{w}^{D}{\tilde{w}}_{p}^{D}+{D}_{w}^{oI}{\tilde{w}}_{p}^{oI}+{D}_{w}^{oH}{\tilde{w}}_{p}^{oH}\\[5pt] & \quad +{D}_{w}^{HI}{\tilde{w}}_{p}^{HI}\big)+{\tilde{W}}^{T}\big(\widehat{\mathrm{\Phi }}-{D}_{w}^{I}{\widehat{w}}_{p}^{I}-{D}_{w}^{D}{\widehat{w}}_{p}^{D}-{D}_{w}^{oI}{\widehat{w}}_{p}^{oI}\\[5pt] & \quad -{D}_{w}^{oH}{\widehat{w}}_{p}^{oH}-{D}_{w}^{HI}{\widehat{w}}_{p}^{HI}\big)+{\mathrm{\Delta }}_{p} \end{aligned} \end{equation}

with $ {\mathrm{\Delta }}_{p}={\tilde{W}}^{T}({D}_{w}^{I}{w}_{p}^{I}+{D}_{w}^{D}{w}_{p}^{D}+{D}_{w}^{oI}{w}_{p}^{oI}+{D}_{w}^{oH}{w}_{p}^{oH}+{D}_{w}^{HI}{w}_{p}^{HI}+O)+{\varepsilon }_{p},(p=x,y,z)$ denotes the lumped approximation error that is assumed to be bounded: $ {\mathrm{\Delta }}_{p}\leqslant {\mathrm{\Delta }}_{0}$ , where $ {\mathrm{\Delta }}_{0}$ is a position constant.

Designing based on the neural network, the adaptive weights of the TRNN are updated online, respectively, as follows:

(51) \begin{align} \left\{\begin{array}{ll}& {\dot{\tilde{W}}}_{p}=-{\eta }^{v}\big(\widehat{\mathrm{\Phi }}-{D}_{w}^{I}{\widehat{w}}^{I}-{D}_{w}^{D}{\widehat{w}}^{D}-{D}_{w}^{oI}{\widehat{w}}^{oI}\\[9pt] & \!\!\! \quad \qquad -{D}_{w}^{oH}{\widehat{w}}^{oH}-{D}_{w}^{HI}{\widehat{w}}^{HI}\big){s}_{p}^{T}\\[9pt] & {\dot{\tilde{w}}}_{p}^{I}=-{\eta }^{I}{D}_{w}^{I}{\widehat{W}}_{p}^{T}{s}_{P}^{T}\\[9pt] & {\dot{\tilde{w}}}_{p}^{D}=-{\eta }^{D}{D}_{w}^{D}{\widehat{W}}_{p}^{T}{s}_{P}^{T}\\[9pt] & {\dot{\tilde{w}}}_{p}^{oI}=-{\eta }^{oI}{D}_{w}^{oI}{\widehat{W}}_{p}^{T}{s}_{P}^{T}\\[9pt] & {\dot{\tilde{w}}}_{p}^{oH}=-{\eta }^{oH}{D}_{w}^{oH}{\widehat{W}}_{p}^{T}{s}_{P}^{T}\\[9pt] & {\dot{\tilde{w}}}_{p}^{HI}=-{\eta }^{HI}{D}_{w}^{HI}{\widehat{W}}_{p}^{T}{s}_{P}^{T}\end{array}\right.\end{align}

where $ {\eta }^{v},{\eta }^{I},{\eta }^{D},{\eta }^{OI},{\eta }^{OH},{\eta }^{HI}$ are positive learning rates, and all weights are ensured to remain bounded during tracking.

To address the unavailability of detailed information about $ {\mathrm{\Delta }}_{0}$ , a novel disturbance observer is proposed for real-time estimation of the combined effect of the external disturbance $ {d}_{p}$ and $ {\mathrm{\Delta }}_{p}$ .

Define $ {Z}_{}={d}_{p}+{\mathrm{\Delta }}_{p}$ , the observer is obtained as:

(52) \begin{align} \left\{\begin{array}{l}{\widehat{Z}}_{j}={\eta }_{j}({\dot{x}}_{j}-{Z}_{j+1})\\[6pt] {\dot{Z}}_{j+1}={\widehat{Z}}_{j}+{\mathrm{\Pi }}_{p}+{\widehat{W}}_{p}^{T}\widehat{h} \end{array}\right.\end{align}

where $ j=\mathrm{1,3},5$ , $ p=x,y,z$ , $ 0\lt {\eta }_{j}\lt \frac{1}{2}$ . $ \mathrm{\Pi }\in {\mathbb{R}}^{3}$ , $ [{\mathrm{\Pi }}_{x},{\mathrm{\Pi }}_{y},{\mathrm{\Pi }}_{z}{]}^{T}=[{u}_{x},{u}_{y},{u}_{z}{]}^{T}+{I}_{g}$ ,where $ {I}_{g}=[\mathrm{0,0},$ $-g{]}^{T}$ , and $ {\widehat{W}}_{p}^{T}\widehat{h}$ is an approximation of the unknown nonlinear function in the position subsystem. Hence, the $ [{\widehat{Z}}_{1},{\widehat{Z}}_{3},{\widehat{Z}}_{5}{]}^{T}$ are approximations of $ [{Z}_{x},{Z}_{y},{Z}_{z}{]}^{T}$ .

Considering the disturbance observer in (52), the estimation error $ ({\tilde{Z}}_{p}={Z}_{p}-{\widehat{Z}}_{j})$ will converge to a neighbourhood of zero, that is to say: $ {\widehat{Z}}_{1}={Z}_{x},{\widehat{Z}}_{3}={Z}_{y},{\widehat{Z}}_{5}={Z}_{z}$ . This conclusion can be derived using a proof method similar to that of Theorem 1.

Considering the application of TRNN and observer in the position subsystem (9), then the control inputs are changed as:

(53) \begin{align} {u}_{x} & ={l}\left\{\begin{array}{l@{\quad}l}{\widehat{f}}_{x}-{k}_{1}{\beta }_{1}{{e}_{x}}^{{\beta }_{1}-1}{\dot{e}}_{x} -{k}_{2}{\beta }_{2}|{e}_{x}{|}^{{\beta }_{2}-1}{\dot{e}}_{x}+{p}_{x}+{u}_{swx}, & if|{e}_{x}|\ge {e}_{p}\\[6pt] {\widehat{f}}_{x}-{\lambda }_{1}{\alpha }_{1}{{e}_{x}}^{{\alpha }_{1}-1}{\dot{e}}_{x}-(1-tanh({e}_{x}))^{2}{\dot{e}}_{x}+{p}_{x}+{u}_{swx}, & else\end{array} \right.\nonumber\\[5pt] {u}_{y}& =\left\{\begin{array}{l@{\quad}l}{\widehat{f}}_{y}-{k}_{3}{\beta }_{3}{{e}_{y}}^{{\beta }_{3}-1}{\dot{e}}_{y} -{k}_{4}{\beta }_{4}|{e}_{y}{|}^{{\beta }_{4}-1}{\dot{e}}_{y}+{p}_{y}+{u}_{swy}, & if|{e}_{y}|\ge {e}_{p}\\[6pt] {\widehat{f}}_{y}-{\lambda }_{2}{\alpha }_{2}{{e}_{y}}^{{\alpha }_{2}-1}{\dot{e}}_{y}-(1-tanh({e}_{y}))^{2}{\dot{e}}_{y}+{p}_{y}+{u}_{swy}, & else \end{array}\right.\\[5pt] {u}_{z}& =\left\{\begin{array}{l@{\quad}l}{\widehat{f}}_{z}-{k}_{5}{\beta }_{5}{{e}_{z}}^{{\beta }_{5}-1}{\dot{e}}_{z} -{k}_{6}{\beta }_{6}|{e}_{z}{|}^{{\beta }_{6}-1}{\dot{e}}_{z}+{p}_{z}+{u}_{swz}, & if|{e}_{z}|\ge {e}_{p}\\[6pt] {\widehat{f}}_{z}-{\lambda }_{3}{\alpha }_{3}{{e}_{z}}^{{\alpha }_{3}-1}{\dot{e}}_{z}-(1-tanh({e}_{z}))^{2}{\dot{e}}_{z}+{p}_{z}+{u}_{swz}, & else \end{array}\nonumber\right. \end{align}

where $ {p}_{x}={\ddot{x}}_{1d}+{\widehat{Z}}_{1},{p}_{y}={\ddot{x}}_{3}+{\widehat{Z}}_{3},{p}_{z}={\ddot{x}}_{5d}+{\widehat{Z}}_{5}+g$ . And $ {u}_{swx},{u}_{swy},{u}_{swz}$ are designed in (25).

From (53), we can get the derivative of sliding mode surfaces:

(54) \begin{align} \begin{array}{ll}{\dot{s}}_{x}& ={f}_{x}-{\widehat{f}}_{x}-{\mathrm{\Delta }}_{x}-{k}_{x1}{s}_{x}-{k}_{x2}si{g}^{{\delta }_{x}}({s}_{x})\\[4pt] {\dot{s}}_{y}& ={f}_{y}-{\widehat{f}}_{y}-{\mathrm{\Delta }}_{y}-{k}_{y1}{s}_{y}-{k}_{y2}si{g}^{{\delta }_{y}}({s}_{y})\\[4pt] {\dot{s}}_{z}& ={f}_{z}-{\widehat{f}}_{z}-{\mathrm{\Delta }}_{z}-{k}_{z1}{s}_{z}-{k}_{z2}si{g}^{{\delta }_{z}}({s}_{z})\end{array}\end{align}

with $ {k}_{x1},{k}_{x2},{k}_{y1},{k}_{y2},{k}_{z1},{k}_{z2}$ are positive constants.

Proof of Theorem 5. Select the Lyapunov function candidate as:

(55) \begin{equation} \begin{aligned} {V}_{P}& =\frac{1}{2}{s}_{p}^{T}{s}_{p}+\frac{1}{2{\eta }^{v}}tr\!\left({\tilde{W}}_{p}^{T}{\tilde{W}}_{p}\right)+\frac{1}{2{\eta }^{I}}\big({\tilde{w}}_{p}^{I}\big)^{T}{\tilde{w}}_{p}^{I}+\frac{1}{2{\eta }^{D}}\big({\tilde{w}}_{p}^{D}\big)^{T}{\tilde{w}}_{p}^{D}\\[4pt] &\quad +\frac{1}{2{\eta }^{oI}}\big({\tilde{w}}_{p}^{oI}\big)^{T}{\tilde{w}}_{p}^{oI}+\frac{1}{2{\eta }^{oH}}\big({\tilde{w}}_{p}^{oH}\big)^{T}{\tilde{w}}_{p}^{oH}+\frac{1}{2{\eta }^{HI}}\big({\tilde{w}}_{p}^{HI}\big)^{T}{\tilde{w}}_{p}^{HI} \end{aligned}\end{equation}

where $ p=x,y,z$ . The derivative of (55) is calculated as:

(56) \begin{equation} \begin{aligned} {\dot{V}}_{p}& ={s}_{p}^{T}{\dot{s}}_{p}+\frac{1}{{\eta }^{v}}tr\big({\tilde{W}}_{p}^{T}{\dot{\tilde{W}}}_{p}\big)+\frac{1}{{\eta }^{I}}({\tilde{w}}^{I})_{p}^{T}{\dot{\tilde{w}}}_{p}^{I}+\frac{1}{{\eta }^{D}}({\tilde{w}}^{D}{)}_{p}^{T}{\dot{\tilde{w}}}_{p}^{\mathrm{D}}\\[4pt] & \quad +\frac{1}{{\eta }^{oI}}\big({\tilde{w}}_{p}^{oI}\big)^{T}{\dot{\tilde{w}}}_{p}^{oI}+\frac{1}{{\eta }^{oH}}\big({\tilde{w}}_{p}^{oH}\big)^{T}{\dot{\tilde{w}}}_{p}^{oH}+\frac{1}{{\eta }^{HI}}\big({\tilde{w}}_{p}^{HI}\big)^{T}{\dot{\tilde{w}}}_{p}^{HI}\\[4pt] & ={s}_{p}^{T}\big(\,{f}_{p}-{\widehat{f}}_{p}-{\mathrm{\Delta }}_{p}-{k}_{p1}{s}_{p}-{k}_{p2}si{g}^{{\delta }_{p}}({s}_{p})\big)+\frac{1}{{\eta }^{v}}{\tilde{W}}_{p}^{T}{\dot{\tilde{W}}}_{p}\\[4pt] & \quad +\frac{1}{{\eta }^{I}}\big({\tilde{w}}_{p}^{I}\big)^{T}{\dot{\tilde{w}}}_{p}^{I}+\frac{1}{{\eta }^{D}}\big({\tilde{w}}_{p}^{D}\big)^{T}{\dot{\tilde{w}}}_{p}^{D}+\frac{1}{{\eta }^{oI}}\big({\tilde{w}}_{p}^{oI}\big)^{T}{\dot{\tilde{w}}}_{p}^{oI}\\[4pt] & \quad +\frac{1}{{\eta }^{oH}}\big({\tilde{w}}_{p}^{oH}\big)^{T}{\dot{\tilde{w}}}_{p}^{oH}+\frac{1}{{\eta }^{HI}}\big({\tilde{w}}_{p}^{HI}\big)^{T}{\dot{\tilde{w}}}_{p}^{HI}\end{aligned}\end{equation}

From (56), we can get:

(57) \begin{equation} \begin{aligned} {\dot{V}}_{p}&= {s}_{p}^{T}\big({\widehat{W}}^{T}\big({D}_{w}^{I}{\tilde{w}}_{p}^{I}+{D}_{w}^{D}{\tilde{w}}_{p}^{D}+{D}_{w}^{oI}{\tilde{w}}_{p}^{oI}+{D}_{w}^{oH}{\tilde{w}}_{p}^{oH}\\[4pt] & \quad +{D}_{w}^{HI}{\tilde{w}}_{p}^{HI}\big)+{\tilde{W}}^{T}\big(\widehat{\mathrm{\Phi }}-{D}_{w}^{I}{\widehat{w}}_{p}^{I}-{D}_{w}^{D}{\widehat{w}}_{p}^{D}-{D}_{w}^{oI}{\widehat{w}}_{p}^{oI}\\[4pt] & \quad-{D}_{w}^{oH}{\widehat{w}}_{p}^{oH}-{D}_{w}^{HI}{\widehat{w}}_{p}^{HI}\big)+{\mathrm{\Delta }}_{p}\big)+\frac{1}{{\eta }^{v}}{\tilde{W}}_{p}^{T}{\dot{\tilde{W}}}_{p}\\[4pt] & \quad+\frac{1}{{\eta }^{I}}\big({\tilde{w}}_{p}^{I}\big)^{T}{\dot{\tilde{w}}}_{p}^{I}+\frac{1}{{\eta }^{D}}\big({\tilde{w}}_{p}^{D}\big)^{T}{\dot{\tilde{w}}}_{p}^{D}+\frac{1}{{\eta }^{oI}}\big({\tilde{w}}_{p}^{oI}\big)^{T}{\dot{\tilde{w}}}_{p}^{oI}\\[4pt] & \quad+\frac{1}{{\eta }^{oH}}\big({\tilde{w}}_{p}^{oH}\big)^{T}{\dot{\tilde{w}}}_{p}^{oH}+\frac{1}{{\eta }^{HI}}\big({\tilde{w}}_{p}^{HI}\big)^{T}{\dot{\tilde{w}}}_{p}^{HI}\\[4pt] & \quad+{s}_{p}^{T}\big(\!-{\mathrm{\Delta }}_{p}-{k}_{p1}{s}_{p}-{k}_{p2}si{g}^{{\delta }_{p}}({s}_{p})\big) \end{aligned}\end{equation}

Furthermore, the $ {\dot{V}}_{p}$ in (57) can be written as:

(58) \begin{equation} \begin{aligned} {\dot{V}}_{p}& =\bigg({D}_{w}^{I}{\widehat{W}}_{p}^{T}{s}_{p}^{T}+\frac{1}{{\eta }^{I}}{\dot{\tilde{w}}}_{p}^{I}\bigg)\big({\tilde{w}}_{p}^{I}\big)^{T}+\bigg({D}_{w}^{D}{\widehat{W}}_{p}^{T}{s}_{p}^{T}+\frac{1}{{\eta }^{D}}{\dot{\tilde{w}}}_{p}^{D}\bigg)\big({\tilde{w}}_{p}^{D}\big)^{T}\\[4pt] & \quad +\bigg({D}_{w}^{OI}{\widehat{W}}_{p}^{T}{s}_{p}^{T}+\frac{1}{{\eta }_{p}^{OI}}{\dot{\tilde{w}}}_{}^{OI}\bigg)({\tilde{w}}^{OI})^{T}\\[4pt] & \quad +\bigg({D}_{w}^{OH}{\widehat{\mathrm{W}}}_{p}^{T}{s}_{p}^{T}+\frac{1}{{\eta }^{OH}}{\dot{\tilde{w}}}_{p}^{OH}\bigg)\big({\tilde{w}}_{p}^{OH}\big)^{T}\\[4pt] & \quad +\bigg({D}_{w}^{HI}{\widehat{W}}_{p}^{T}{s}_{p}^{T}+\frac{1}{{\eta }^{HI}}{\dot{\tilde{w}}}_{p}^{HI}\bigg)\big({\tilde{w}}_{p}^{HI}\big)^{T}\\[4pt] & \quad + \bigg(\big(\widehat{\mathrm{\Phi }}-{D}_{w}^{I}{\widehat{w}}_{p}^{I}-{D}_{w}^{D}{\widehat{w}}_{p}^{D}-{D}_{w}^{oI}{\widehat{w}}_{p}^{oI}\\[4pt] & \quad -{D}_{w}^{oH}{\widehat{w}}_{p}^{oH}-{D}_{w}^{HI}{\widehat{w}}_{p}^{HI}\big){s}_{p}^{T}+\frac{1}{{\eta }^{v}}{\dot{\tilde{W}}}_{p}\bigg){\tilde{W}}_{p}^{T}\\[4pt] & \quad +{s}_{p}\big(\!-{k}_{p1}{s}_{p}-{k}_{p2}si{g}^{{\delta }_{p}}({s}_{p})\big) \end{aligned}\end{equation}

Substituting the adaptive weights (51) into (58), $ {\dot{V}}_{p}$ is simplified as follows:

(59) \begin{equation} \begin{aligned} {\dot{V}}_{p}& ={s}_{p}(\!-{k}_{p1}{s}_{p}-{k}_{p2}si{g}^{{\delta }_{p}}({s}_{p}))\\[5pt] & =-{k}_{p1}{s}_{p}^{2}-{k}_{p2}{s}_{p}si{g}^{{\delta }_{p}}({s}_{p})\\[5pt] & =-{k}_{p1}{s}_{p}^{2}-{k}_{p2}|{s}_{p}{|}^{{\delta }_{p}+1}\\[5pt] & \leqslant -{k}_{p1}{V}_{p}-{2}^{\frac{{\delta }_{p}-1}{2}}{k}_{p2}{V}_{p}^{\frac{{\delta }_{p}+1}{2}}\\[5pt] & \leqslant -{{{\eth}}}_{1}{V}_{p}-{{{\eth}}}_{2}{V}_{p}^{\frac{{\delta }_{p}+1}{2}}\end{aligned}\end{equation}

where $ {{{\eth}}}_{1}={k}_{p1}$ and $ {\mathrm{{\eth}}}_{2}={2}^{\frac{{\delta }_{p}-1}{2}}{k}_{p2}$ . According to Lemma 1, the tracking error of the position subsystem converges in finite time.

4.2 Design of controller for quadrotor attitude based on TRNNO-TFTSMC method

In this part, a controller for the attitude subsystem is proposed based on the TRNNO-TFTSMC method. The adaptive laws generated by this subsystem guarantee the stability of the attitude in the closed loop. The unknown nonlinear function $ {f}_{a}=[\,{f}_{\phi },{f}_{\theta },{f}_{\phi }{]}^{T}=[{a}_{1}{x}_{10}{x}_{12}+{b}_{1}\overline{\omega }{x}_{10}-{c}_{4}{x}_{8},{a}_{2}{x}_{8}{x}_{12}+{b}_{2}\overline{\omega }{x}_{8}-{c}_{5}{x}_{10},{a}_{3}{x}_{8}{x}_{12}-{c}_{6}{x}_{12}{]}^{T}$ is approximated by $ {\widehat{f}}_{a}={\widehat{W}}_{a}^{T}\widehat{h}({\widehat{\mathrm{\Gamma }}}_{a},{\widehat{c}}_{a},{\widehat{b}}_{a},{\widehat{W}}_{a}^{r})$ . Hence, the adaptive weights are given as:

(60) \begin{align} \left\{\begin{array}{ll}& {\dot{\tilde{W}}}_{a}=-{\zeta }^{v}\big(\widehat{\mathrm{\Phi }}-{D}_{w}^{I}{\widehat{w}}_{a}^{I}-{D}_{w}^{D}{\widehat{w}}_{a}^{D}-{D}_{w}^{o}{\widehat{w}}_{a}^{oI}\\[5pt] & \qquad\quad\!\! -{D}_{w}^{oH}{\widehat{w}}_{a}^{oH}-{D}_{w}^{HI}{\widehat{w}}_{a}^{HI}\big){s}_{a}^{T}\\[5pt] & {\dot{\tilde{w}}}_{a}^{I}=-{\zeta }^{I}{D}_{w}^{I}{\widehat{W}}_{a}^{T}{s}_{a}^{T}\\[5pt] & {\dot{\tilde{w}}}_{a}^{D}=-{\zeta }^{D}{D}_{w}^{D}{\widehat{W}}_{a}^{T}{s}_{a}^{T}\\[5pt] & {\dot{\tilde{w}}}_{a}^{oI}=-{\zeta }^{oI}{D}_{w}^{oI}{\widehat{W}}_{a}^{T}{s}_{a}^{T}\\[5pt] & {\dot{\tilde{w}}}_{a}^{oH}=-{\zeta }^{oH}{D}_{w}^{oH}{\widehat{W}}_{a}^{T}{s}_{a}^{T}\\[5pt] & {\dot{\tilde{w}}}_{a}^{HI}=-{\zeta }^{HI}{D}_{w}^{HI}{\widehat{W}}_{a}^{T}{s}_{a}^{T}\end{array}\right.\end{align}

where $ {\zeta }^{v},{\zeta }^{I},{\zeta }^{D},{\zeta }^{OI},{\zeta }^{OH},{\zeta }^{HI}$ are positive learning rates, and all weights are ensured to remain bounded during tracking.

Define $ {Z}_{a}={d}_{a}+{\mathrm{\Delta }}_{a}$ , the observer is designed as:

(61) \begin{align} \left\{\begin{array}{l}{\widehat{Z}}_{j}={\eta }_{j}({\dot{x}}_{j}-{Z}_{j+1})\\[4pt] {\dot{Z}}_{j+1}={\widehat{Z}}_{j}+{\mathrm{\Pi }}_{a}+{\widehat{W}}_{a}^{T}\widehat{h}\\ \end{array}\right.\end{align}

where $ j=\mathrm{7,9},11$ , $ a=\phi, \theta, \psi $ , $ 0\lt {\eta }_{j}\lt \frac{1}{2}$ . $ \mathrm{\Pi }\in {\mathbb{R}}^{3}$ , $ [{\mathrm{\Pi }}_{\phi },{\mathrm{\Pi }}_{\theta },{\mathrm{\Pi }}_{\psi }{]}^{T}=f\diamond [{u}_{2},{u}_{3},{u}_{4}{]}^{T}$ , where $ f=[\,{f}_{1},{f}_{2},{f}_{3}{]}^{T}$ , and $ {\widehat{W}}_{a}^{T}\widehat{h}$ is an approximation of the unknown nonlinear function in the attitude subsystem. The $ [{\widehat{Z}}_{7},{\widehat{Z}}_{9},{\widehat{Z}}_{11}{]}^{T}$ are approximations of $ [{Z}_{\phi },{Z}_{\theta },{Z}_{\psi }{]}^{T}$ .

Furthermore, the control inputs are changed as:

(62) \begin{align} u_{2} & = \begin{cases} f_1^{ - 1}(\,{{\hat f}_\phi } + {{\ddot \phi }_d} + {{\hat T}_7} - {k_7}{\beta _7}{e_\phi }^{{\beta _7} - 1}{{\dot e}_\phi }\\ \quad \quad - {k_8}{\beta _8}|{e_\phi }{|^{{\beta _8} - 1}}{{\dot e}_\phi } + {u_{sw\phi }}),\;\;\;\;if\left| {{e_\phi }} \right| \ge {e_a}\\[6pt] f_1^{ - 1}(\,{{\hat f}_\phi } + {{\ddot \phi }_d} + {{\hat T}_7} - {\lambda _4}{\alpha _4}{e_\phi }^{{\alpha _4} - 1}{{\dot e}_\phi }\\ \quad \quad - ( {1 - \textit{tanh}({{e_\phi }}){)^2}{{\dot e}_\phi } + {u_{sw\phi }}} ),\;\;\;\;else \end{cases} \nonumber \\[6pt] u_{3} & = \begin{cases} f_2^{ - 1}(\,{{\hat f}_\theta } + {{\ddot \theta }_d} + {{\hat T}_9} - {k_9}{\beta _9}{e_\theta }^{{\beta _9} - 1}{{\dot e}_\theta }\\ \quad \quad - {k_{10}}{\beta _{10}}|{e_\theta }{|^{{\beta _{10}} - 1}}{{\dot e}_\theta } + {u_{sw\theta }}),\;\;\;\;if\left| {{e_\theta }} \right| \ge {e_a}\\[6pt] f_2^{ - 1}(\,{{\hat f}_\theta } + {{\ddot \theta }_d} + {{\hat T}_9} - {\lambda _5}{\alpha _5}{e_\theta }^{{\alpha _5} - 1}{{\dot e}_\theta } \\ \quad \quad - ( {1 - \textit{tanh} ( {{e_\theta }} ){)^2}{{\dot e}_\theta } + {u_{sw\theta }}} ),\;\;\;\;else \end{cases}\\[6pt] u_{4} & = \begin{cases} f_3^{ - 1}(\,{{\hat f}_\psi } - {k_{11}}{\beta _{11}}{e_\psi }^{{\beta _{11}} - 1}{{\dot e}_\psi } + {{\ddot \psi }_d} + {{\hat T}_{11}}\\ \quad \quad- {k_{12}}{\beta _{12}}|{e_\psi }{|^{{\beta _{12}} - 1}}{{\dot e}_\psi } + {u_{sw\psi }}),\;\;\;\;if\left| {{e_\psi }} \right| \ge {e_a}\\[6pt] f_3^{ - 1}(\,{{\hat f}_\psi } + {{\ddot \psi }_d} + {{\hat T}_{11}} - {\lambda _6}{\alpha _6}{e_\psi }^{{\alpha _6} - 1}{{\dot e}_\psi } \\ \quad \quad - ( {1 - \textit{tanh} ( {{e_\psi }} ){)^2}{{\dot e}_\psi } + {u_{sw\psi }}}),\;\;\;\;else \end{cases}\nonumber \end{align}

where $ {u}_{sw\phi },{u}_{sw\theta },{u}_{sw\psi }$ are designed in (41).

From (62), the derivative of sliding mode surfaces is written as:

(63) \begin{align} \begin{array}{ll}{\dot{s}}_{\phi }& ={f}_{\phi }-{\widehat{f}}_{\phi }-{\mathrm{\Delta }}_{\phi }-{k}_{\phi 1}{s}_{\phi }-{k}_{\phi 2}si{g}^{{\delta }_{\phi }}({s}_{\phi })\\[4pt] {\dot{s}}_{\theta }& ={f}_{\theta }-{\widehat{f}}_{\theta }-{\mathrm{\Delta }}_{\theta }-{k}_{\theta 1}{s}_{\theta }-{k}_{\theta 2}si{g}^{{\delta }_{\theta }}({s}_{\theta})\\[4pt] {\dot{s}}_{\psi }& ={f}_{\psi }-{\widehat{f}}_{\psi }-{\mathrm{\Delta }}_{\psi }-{k}_{\psi 1}{s}_{\psi }-{k}_{\psi 2}si{g}^{{\delta }_{\psi }}({s}_{\psi})\end{array}\end{align}

Proof of Theorem 6. Select the Lyapunov function candidate as:

(64) \begin{equation} \begin{aligned} {V}_{a}& =\frac{1}{2}{s}_{a}^{T}{s}_{a}+\frac{1}{2{\zeta }^{v}}tr\left({\tilde{W}}_{a}^{T}{\tilde{W}}_{a}\right)+\frac{1}{2{\zeta }^{I}}\big({\tilde{w}}_{a}^{I}\big)^{T}{\tilde{w}}_{a}^{I}+\frac{1}{2{\zeta }^{D}}\big({\tilde{w}}_{a}^{D}\big)^{T}{\tilde{w}}_{a}^{D}\\[4pt] &\quad +\frac{1}{2{\zeta }^{oI}}\big({\tilde{w}}_{a}^{oI}\big)^{T}{\tilde{w}}_{a}^{oI}+\frac{1}{2{\zeta }^{oH}}\big({\tilde{w}}_{a}^{oH}\big)^{T}{\tilde{w}}_{a}^{oH}+\frac{1}{2{\zeta }^{HI}}\big({\tilde{w}}_{a}^{HI}\big)^{T}{\tilde{w}}_{a}^{HI} \end{aligned} \end{equation}

With the application of the adaptive weights (60), (64) is simplified as follows:

(65) \begin{equation} \begin{aligned} {\dot{V}}_{a}& ={s}_{a}\big(\!-{k}_{a1}{s}_{a}-{k}_{a2}si{g}^{{\delta }_{a}}({s}_{a})\big)\\[4pt] & \leqslant -{{{\eth}}'}_{\!\!1}{V}_{a}-{{{\eth}}'}_{\!\!2}{V}_{a}^{\frac{{\delta }_{a}+1}{2}}\end{aligned} \end{equation}

where $ {{{\eth}}'}_{\!\!1}={k}_{a1}$ and $ {{{\eth}}'}_{\!\!2}={2}^{\frac{{\delta }_{a}-1}{2}}{k}_{a2}$ . By using a procedure similar to Theorem 5, the finite-time stability of tracking errors in the attitude subsystem can be guaranteed.

5.1 Comparative experiments under different disturbances

To construct a more physically realistic testing scenario within the simulation environment and thereby thoroughly evaluate the robustness and control efficacy of the proposed control scheme in the face of complex and varying external disturbances, this experiment introduces a class of external disturbances with stochastic statistical characteristics. This disturbance simulates the continuous random fluctuations caused by gusts, turbulence or airflow variations in actual flight environments, rather than simple periodic signals, aiming to validate the dynamic response performance and disturbance rejection capability of the controller under non-ideal operating conditions. Consider a bounded external disturbance $ d$ with randomness, designed as a sinusoidal function with a random amplitude r, where r is uniformly distributed within the range [0.03,0.05]. Hence, the $ d$ is expressed as ${d}_{i}=rsin(\pi t/0.02),$ $\forall i=x,y,z,\phi, \theta, \psi $ . To evaluate the controller’s robustness against different external disturbances, simulation experiments are conducted under the following two cases.

5.1.1 Case (a): take the value of $ \mathit{d}$ as the external disturbance

In this case, the quadrotor tracks the desired trajectory in the presence of external disturbances $ d$ . Comparative simulation results in Figs 4(a)–7 demonstrate the performance of the proposed TRNNO-TFTSMC against four benchmark controllers: (i) conventional FTSMC, (ii) ANN-TFTSMC, (iii) RBFNN-based control from Ref. [Reference Ranjan and Majhi23] and (iv) FTDO method in Ref. [Reference Sanwale, Dahiya and Trivedi26].

Figure 4.

(a) Quadrotor position ( $ x,y,z$ ). (b) Quadrotor attitude ( $ \phi, \theta, \psi $ ).

Figure 5.

(a) Tracking errors ( $ {e}_{x},{e}_{y},{e}_{z}$ ) of the position. (b) Tracking errors ( $ {e}_{\phi },{e}_{\theta },{e}_{\psi}$ ) of the attitude angles.

Figure 6.

The control inputs of the quadrotor.

Figure 7.

The desired and real trajectory of the quadrotor.

Figure 4(a) illustrates the performance of the quadrotor in position tracking. The figure presents a comparative analysis of the position tracking performance under different control schemes. As illustrated, all three position variables ( $ x,y,z$ ) converge smoothly and stably from the initial position of 0.1 $ m$ to the desired trajectory. To more clearly illustrate the differences among the various control schemes, the local details of the convergence process are enlarged in the figure. This further demonstrates that, at the moments when the step signal changes, the proposed approach attains faster trajectory tracking convergence compared with the other methods. Figure 4(b) presents the tracking results of the three attitude angles ( $ \phi, \theta, \psi $ ) in response to the desired step trajectory. Under identical control conditions, the attitude angles converge gradually from an initial value of 0.4 $ rad$ to the target value, exhibiting a response time similar to that of the position loop. A detailed comparison in the zoomed-in plots reveals that the proposed control scheme achieves a significantly faster convergence rate than the other comparative schemes, which fully demonstrates its distinct advantage in enhancing the rapidity of the system.

Figure 5(a) and (b) show the tracking errors of position and attitude, respectively. As observed from the figures, even though the errors exhibit nonlinear variations at certain moments, the proposed control scheme ensures convergence to zero in the shortest time. This fast convergence characteristic is attributed to the introduced finite-time control mechanism, which enables rapid suppression of uncertainties and external disturbances while ensuring system stability. Furthermore, the error trajectories remain smooth and free of significant chattering after convergence, further validating the effectiveness of the proposed method in enhancing both the transient performance and robustness of the system.

Figure 6 illustrates the control inputs of the quadrotor, which converge to steady-state values. Comparative experiments with conventional FTSMC demonstrate that the proposed control scheme effectively suppresses controller output amplitude. The convergence of $ {u}_{1}$ is around 7 $ N$ , with a maximum amplitude of approximately 7.50687 $ N$ . The input $ {u}_{2}$ converges to a neighbourhood around zero, with an amplitude of approximately 0.00344438 $ N$ . By comparing the performance index curves of control inputs $ {u}_{3}$ and $ {u}_{4}$ , it can be clearly observed that the controller generated by the proposed scheme exhibits a smaller control effort. This fully demonstrates the significant effectiveness of the proposed approach in optimising control input. While effectively reducing unnecessary control actions, it can also ensure an improvement in the smoothness of the control process.

Figure 7 presents a comparison between the desired trajectory and the actual tracking trajectory of the quadrotor UAV. It can be observed from the overall trajectory trend that, even in the presence of persistent external disturbances, the actual trajectory closely follows the desired one without significant deviation. The enlarged view of the intermediate moment further quantifies the tracking accuracy: the UAV trajectory reaches the coordinates $ X=0.299996m,Y=0.0999966m,Z=0.299955m$ , with tracking errors within the millimetre range compared to the desired position (0.3 $ m$ , 0.1 $ m$ , 0.3 $ m$ ) at the same moment. In contrast, the other control schemes evidently result in larger flight errors. These results demonstrate that the proposed control scheme possesses excellent robustness and high-precision trajectory tracking capability.

To further illustrate the performance of the TRNNO-TFTSMC method, two error-related performance indices (IAE and ISE) are employed for comparative analysis. Figure 8 presents the histograms of the two error terms, where subfigure (a) shows the comparison of IAE, and subfigure (b) depicts the comparison of ISE. The comparative results clearly demonstrate that all six error terms of the proposed control scheme are the smallest, which directly evidences that the proposed method achieves more stable control performance.

Figure 8.

(a) Integral absolute error (IAE). (b) Integral square error (ISE).

Figure 9.

(a) Quadrotor position (X, Y, Z). (b) Quadrotor attitude ( $ \phi, \theta, \psi $ ).

Figure 10.

The desired and real trajectory of the quadrotor.

5.2 Analysis of experimental feasibility

Based on the proposed TRNNO-TFTSMC scheme and the simulation experiments described above, the analysis of experimental feasibility can be summarised in the following aspects:

• • The proposed sliding mode control scheme exhibits strong robustness against system parameter perturbations and model uncertainties, making it highly suitable for quadrotor UAVs, which are strongly coupled underactuated systems.

• • Sinusoidal disturbances with random amplitudes are introduced in the simulation, and disturbance amplification tests (by a factor of 25) are conducted, demonstrating that the controller can effectively cope with uncertain disturbances encountered in actual flight.

• • A more saturated and smoother tanh function-based control logic is adopted in the experiment, which effectively reduces chattering and mitigates physical actuator tremors during quadrotor UAV flight.

• • The neural network control logic is optimised, with initial weights selected as small values to prevent drastic control inputs during the initial stage when stabilising the airframe.

• • Through feedforward compensation by the disturbance observer, the neural network approximation error is reduced while the robustness in actual flight is enhanced.

Table 5.

Comparative analysis of IAE

Table 6.

Comparative analysis of ISE

Table 7.

Comparative analysis of ITAE

5.3 Practical implementation platform

This note describes an experimental validation programme for the proposed control methodology. To validate and test the proposed scenario, we have compiled a complete list of required equipment and, based on this, established a drone test bench. The UAV experimental platform’s hardware configuration is illustrated in Fig. 11. It is a quadrotor of version X450 based on the CUAV V5+ flight control system and is equipped with a ground control station (GCS). The flight control system is based on the STM32F765 as the main functional mock-up unit (FMU) processor, integrates a 6-axis inertial measurement unit (IMU) (including a 3-axis gyroscope and a 3-axis accelerometer), a magnetometer and a barometer. Velocity and position in the horizontal plane are measured by the Global Positioning System (GPS) module and transmitted to the flight control system. The quadrotor system is powered by a lithium-polymer battery pack and governed by a remote control. In addition, GCS ensures communication with the quadrotor via Gigabit Power over Ethernet (PoE). Wind disturbances are generated by a fan and introduced to the quadrotor system to evaluate its disturbance rejection capabilities. Finally, actuators execute the flight controller’s commands, driving the propellers to generate the required aerodynamic forces for achieving the specified flight state.

Figure 11.

The quadrotor’s hardware structure.