2.1. Coordinate system definition

The structural schematic of the Stewart parallel platform is shown below. The mechanism consists of an upper platform, a lower platform, and 6 connecting legs. The legs are connected to the upper platform using spherical joints and to the lower platform using Hooke joints. Hydraulic cylinders are used to drive the motion of the upper platform. Moving coordinate system $O_{\textrm{b}}-X_{b}Y_{b}Z_{b}$ is established at the center of mass $O_{\textrm{b}}$ of the upper platform, and static coordinate system $O_{a}-X_{a}Y_{a}Z_{a}$ is established at the center of mass $O_{\textrm{a}}$ of the lower platform. $b_{i}$ represents the position of the upper joint in the moving coordinate system, and $a_{i}$ represents the position of the upper joint in the static coordinate system Figure 1.

Figure 1.

Schematic diagram of the Stewart parallel platform.

2.2. The dynamic model of the Stewart platform

The dynamic equations of the Stewart platform can be derived using the Newton-Euler equations as [Reference Dasgupta and Mruthyunjaya22]:

(1) \begin{equation}M(X)\ddot{X}+C(X,\dot{X})\dot{X}+G(X)+F_{d}+\tau _{d}=J^{T}\tau\end{equation}

where $M(X)$ is the inertia matrix in the joint space, $C(X,\dot{X})$ is the centrifugal and Coriolis force matrix, $G(X)$ is the gravity matrix, $\tau _{\textrm{d}}$ is the friction force matrix, $\tau \in R^{6}$ is the actuator input vector, $F_{\textrm{d}}$ is the external disturbance matrix, and $X=[x,y,z,\alpha ,\beta ,\gamma ]^{T}$ represents the pose of the upper platform. Where $M(X)$ is the inertia matrix in the joint space, $C(X,\dot{X})$ is the centrifugal and Coriolis force.

The uncertainty in the system dynamics can be represented as the nominal and deviation as follows:

(2) \begin{equation} M=M_{0}+{\unicode[Arial]{x0394}} M,C=C_{0}+{\unicode[Arial]{x0394}} C,G=G_{0}+{\unicode[Arial]{x0394}} G \end{equation}

By combining the dynamic Eq. (1), we can obtain:

(3) \begin{equation}M_{0}\ddot{X}+C_{0}\dot{X}+G_{0}=J^{T}\tau +d\end{equation}

where $d=-(\Delta M\ddot{X}+\Delta C\dot{X}+\Delta G+F_{d}+\tau _{d})$ is caused by external disturbances, unmodeled dynamics, and other factors. In this study, all uncertainties and disturbances are collectively considered in the form of a lumped uncertainty term.

Assumption 1. The lumped uncertainty term $d$ has an upper bound $\delta$ , that is $\| d\| \leq \delta$ , where $\delta$ is a known constant. Its first-order derivative $\rho$ is also bounded by a normal constant, that is, $\| \dot{d}\| \leq \rho$ .

To facilitate subsequent analysis, the Stewart dynamic model is represented in state-space form as:

(4) \begin{eqnarray} \left\{\begin{array}{l} \dot{x}_{1} = x_{2} \\[2pt] \dot{x}_{2} = \Phi (x_{1})u+f(x_{1},x_{2})+D \end{array}\right. \end{eqnarray}

Among them, $x_{1}=X\in R^{6}$ , $x_{2}=\dot{X}\in R^{6}$ , and other related functions $f(x_{1},x_{2})=-M_{0}^{-1}(C_{0}x_{2}+G_{0}) , {\Phi} (x_{1})=M_{0}^{-1}J^{T} , u=\tau$ . Based on Assumption 1, it can be deduced that the lumped uncertainty term D is Lipschitz continuous and bounded, i.e., $\| D\| \leq \delta _{d}$ , and its first derivative is also bounded, $\| \dot{D}\| \leq \rho _{d}$ .

3.1. Non-singular composite integral sliding mode surface design

First, define the position tracking error and its derivative as:

(7) \begin{equation}e=x_{1}-x_{d}\quad \dot{e}=x_{2}-\dot{x}_{d}\end{equation}

Introduce an intermediate variable:

(8) \begin{align} v &=e+\alpha Z \end{align}

(9) \begin{align} \dot{Z} &=G\left(e\right)=\left[g(e_{1}),g(e_{2}),g(e_{3}),g(e_{4}),g(e_{5}),g(e_{6})\right]^{T} \end{align}

(10) \begin{align} g\left(e_{i}\right) &=\left\{\begin{array}{c} K_{1}sig^{r}\left(e_{i}\right)+K_{2}\kappa ^{\left| e_{i}\right| }e_{i},\left| e_{i}\right| \lt \kappa \\[4pt] sig^{m}\left(e_{i}\right),\left| e_{i}\right| \geq \kappa \end{array}\right. \end{align}

(11) \begin{align} g'\left(e_{i}\right)=\frac{dg\left(e_{i}\right)}{de_{i}} &=\left\{\begin{array}{c} K_{1}r\left| e_{i}\right| ^{r-1}+K_{2}\left(\left| e_{i}\right| \ln \kappa +1\right)\kappa ^{\left| e_{i}\right| },\left| e_{i}\right| \lt \kappa \\[4pt] m\left| e_{i}\right| ^{m-1},\left| e_{i}\right| \geq \kappa \end{array}\right. \end{align}

where $\alpha \gt 0$ , $r\gt 1$ , $m=1-\kappa$ , $0\lt \kappa \lt e^{-1}$ , $K_{1}=\frac{-1-\ln \kappa }{m-\kappa \ln \kappa }$ , $K_{2}=\frac{\kappa ^{2m-2}}{m-\kappa \ln \kappa }$ , $r=1+m$ . When $v_{i}\neq 0$ and $e_{i}$ enter the $| e_{i}| \lt \kappa$ region, the terminal sliding manifold switches to a non-singular sliding manifold, allowing the system to move in a more controllable way as it approaches the equilibrium point, avoiding the potentially infinite derivatives that may be caused by the terminal sliding mode surface, thereby overcoming the singularity issues that occur at $v_{i}\neq 0$ and $e_{i}=0$ .

By introducing the intermediate variable, the proposed non-singular composite integral sliding mode surface is:

(12) \begin{equation} \left\{\begin{array}{l} S=\dot{v}+\beta W\\[3pt] v=e+\alpha Z\\[3pt] \dot{W}=Sig^{{\lambda _{1}}}\left(v\right)+Sig^{{\lambda _{2}}}\left(\dot{v}\right) \end{array}\right. \end{equation}

where $\beta \gt 0$ , $0\lt \lambda _{1}\lt 1$ , $\lambda _{2}=\frac{2\lambda _{1}}{\lambda _{1}+1}$ .

It can be rearranged as:

(13) \begin{eqnarray} \begin{array}{ll} S & =\dot{v}+\beta \int _{0}^{t}\left(Sig^{{\lambda _{1}}}\left(v\right)+Sig^{{\lambda _{2}}}\left(\dot{v}\right)\right)d\tau \\[3pt] & =\dot{e}+\alpha G\left(e\right)+\beta \int _{0}^{t}\left(Sig^{{\lambda _{1}}}\left(v\right)+Sig^{{\lambda _{2}}}\left(\dot{v}\right)\right)d\tau \end{array} \end{eqnarray}

When the initial values are $W\left(0\right)=-\frac{\left(\dot{e}\left(0\right)+\alpha G\left(e\left(0\right)\right)\right)}{\beta }$ and $Z\left(0\right)=-\frac{e\left(0\right)}{\alpha }$ , that is, at the initial time $S=0$ and $v=0$ , the sliding mode reaching phase of the system can be eliminated, thereby improving the system’s robustness.

Theorem 1: Using the above non-singular composite integral sliding mode surface, when $S=0$ is satisfied, the intermediate variables $v$ and $\dot{v}$ approach zero, ensuring finite-time stability. At the same time, based on the finite-time stable intermediate variable $v$ , the position tracking error converges to a neighborhood of zero within a finite time.

Proof: When $S=0$ is satisfied, the non-singular composite integral sliding mode surface can be written as:

(14) \begin{equation} \ddot{v}=-\beta Sig^{{\lambda _{1}}}\left(v\right)-\beta Sig^{{\lambda _{2}}}\left(\dot{v}\right) \end{equation}

Choose the Lyapunov function $V_{1}$ as follows:

(15) \begin{align} V_{1} &=\frac{1}{1+\lambda _{1}}\beta \sum _{i=1}^{6}\left| v_{i}\right| ^{{\lambda _{1}}+1}+\frac{1}{2}\dot{v}^{T}\dot{v} \end{align}

(16) \begin{align} \dot{V}_{1} &=\beta \sum _{i=1}^{6}sign\left(v_{i}\right)\left| v_{i}\right| ^{{\lambda _{1}}}\dot{v}_{i}+\sum _{i=1}^{6}\dot{v}_{i}\ddot{v}_{i} \nonumber\\[6pt] &=\beta \sum _{i=1}^{6}sign\left(v_{i}\right)\left| v_{i}\right| ^{{\lambda _{1}}}\dot{v}_{i}-\beta \sum _{i=1}^{6}\dot{v}_{i}\left(sig^{{\lambda _{1}}}\left(v_{i}\right)+sig^{{\lambda _{2}}}\left(\dot{v}_{i}\right)\right) \nonumber\\[6pt] &=-\beta \sum _{i=1}^{6}\left| \dot{v}_{i}\right| ^{{\lambda _{2}}+1} \end{align}

where it can be concluded that $\dot{V}_{1}$ is negative semi-definite. According to LaSalle’s invariance principle, the closed-loop system is asymptotically stable. Based on the proof process of Theorem 1 in reference [Reference Li, Hu and Yang24], the system has homogeneity degree $k=\frac{\lambda _{1}-1}{2}\lt 0$ . From Lemma 5, it is known that the system is finite-time stable, and the system state variables $v$ and $\dot{v}$ converge to 0 within the finite time $T_{1}$ .

Case 1. When $| e_{i}| \geq \kappa , e_{i}=-\alpha Z_{i}$ .

(17) \begin{equation} \dot{Z}_{i}=g\left(e_{i}\right)=sig^{m}\left(e_{i}\right)=\left| -\alpha Z_{i}\right| ^{m}sign\left(-\alpha Z_{i}\right) \end{equation}

It follows that $Z_{i}$ reaches the origin within $T_{2}$ time.

(18) \begin{equation} T_{2}=\frac{\left| Z_{i}\left(0\right)\right| ^{1-r}}{a^{r}\left(1-r\right)}=\frac{\left| e\left(0\right)\right| ^{1-r}}{a\left(1-r\right)} \end{equation}

Case 2. When $| e_{i}| \lt \kappa , e_{i}=-aZ_{i}$ .

(19) \begin{equation} \begin{array}{ll} \dot{Z}_{i} & =K_{1}sig^{r}\left(e_{i}\right)+K_{2}\kappa ^{\left| e_{i}\right| }e_{i}\\[6pt] & =K_{1}\left| -aZ_{i}\right| ^{r}sign\left(-aZ_{i}\right)-aK_{2}\kappa ^{\left| e_{i}\right| }Z_{i} \end{array} \end{equation}

Choose the Lyapunov function $V_{2}$ as follows:

(20) \begin{align} V_{2} &=\frac{1}{2}Z_{i}^{2} \end{align}

(21) \begin{align} \dot{V}_{2} & =Z_{i}\dot{Z}_{i} \nonumber\\[6pt] & =-\frac{K_{1}}{\alpha }\left(-aZ_{i}\right)\left| -aZ_{i}\right| ^{r}sign\left(-\alpha Z_{i}\right)-aK_{2}\kappa ^{\left| e_{i}\right| }Z_{i}^{2}\nonumber\\[6pt] & =-\frac{K_{1}}{\alpha }\left| -aZ_{i}\right| ^{r+1}-\alpha K_{2}\kappa ^{\left| e_{i}\right| }Z_{i}^{2}\leq 0 \end{align}

According to Lyapunov stability theory, the position tracking error converges to a neighborhood of $| e_{i}| \lt \kappa$ within the finite time $T_{f}=T_{1}+T_{2}$ . The proof is complete.

3.2. Non-singular composite integral sliding mode surface design

The traditional expansion state observer sets the first derivative of the lumped uncertainty to zero and cannot converge in finite time, which affects the controller’s performance. Therefore, in this paper, based on the proposed sliding mode surface, feedback is applied to the output estimation error through nonlinear and switching terms, considering the derivative of the lumped uncertainty. A fast finite-time expansion state observer is designed to estimate the lumped uncertainty, improving the system’s disturbance rejection capability and reducing chattering.

The derivative of the proposed non-singular composite integral sliding mode surface is derived as:

(22) \begin{equation} \begin{array}{ll} \dot{S} & =\ddot{e}+\alpha G'\left(e\right)\dot{e}+\beta \left(Sig^{{\lambda _{1}}}\left(v\right)+Sig^{{\lambda _{2}}}\left(\dot{v}\right)\right)\\[6pt] & ={\Phi} \left(x_{1}\!\right)u+f\left(x_{1},x_{2}\right)+D-\ddot{x}_{d}+\alpha G'\left(e\right)\!\dot{e}+\beta \left(Sig^{{\lambda _{1}}}\left(v\right)+Sig^{{\lambda _{2}}}\left(\dot{v}\right)\right) \end{array} \end{equation}

First, a new state variable $z_{1}=S$ is defined, and an expanded state variable $z_{2}=D$ and $\dot{z}_{2}=\dot{D}$ are defined, where $\dot{D}$ is unknown but bounded, i.e., $\| \dot{D}\| \leq \rho _{d}$ . The augmented dynamic model of the platform is as follows:

(23) \begin{equation} \left\{\begin{array}{l} \dot{z}_{1}={\Phi} \left(x_{1}\right)u+{\Gamma} +z_{2}\\[6pt] \dot{z}_{2}=\dot{D} \end{array}\right. \end{equation}

where ${\Gamma} =f(x_{1},x_{2})-\ddot{x}_{d}+\beta (Sig^{{\lambda _{1}}}(v)+Sig^{{\lambda _{2}}}(\dot{v}))+\alpha G'(e)\dot{e}$ .

The variables $\hat{z}_{1}$ and $\hat{z}_{2}$ represent the estimated values of $z_{1}$ and $z_{2}$ , respectively. Defining the error $\xi _{1}=z_{1}-\hat{z}_{1}$ , the fast finite-time expansion state observer is designed as follows:

(24) \begin{equation} \left\{\begin{array}{l} \dot{\hat{z}}_{1}={\Phi} \left(x_{1}\right)u+{\Gamma} +\hat{z}_{2}+\eta _{1}\left(Sig^{{\alpha _{1}}}\left(\xi _{1}\right)+\xi _{1}\right)\\[6pt] \dot{\hat{z}}_{2}=\eta _{2}\left(Sig^{{\alpha _{2}}}\left(\xi _{1}\right)+2Sig^{{\alpha _{1}}}\left(\xi _{1}\right)+\xi _{1}\right) \end{array}\right. \end{equation}

where $\eta _{1}\gt 0 , \eta _{2}\gt 0 , \frac{1}{2}\lt \alpha _{1}\lt 1 , \alpha _{2}=2\alpha _{1}-1$ .

Define another error variable $\xi _{2}=z_{2}-\hat{z}_{2}$ , and the error dynamics of the proposed FFESO can be expressed as follows:

(25) \begin{equation} \left\{\begin{array}{l} \dot{\xi }_{1}=\xi _{2}-\eta _{1}\left(Sig^{{\alpha _{1}}}\left(\xi _{1}\right)+\xi _{1}\right)\\[6pt] \dot{\xi }_{2}=\dot{D}-\eta _{2}\left(Sig^{{\alpha _{2}}}\left(\xi _{1}\right)+2Sig^{{\alpha _{1}}}\left(\xi _{1}\right)+\xi _{1}\right) \end{array}\right. \end{equation}

Theorem 2: Consider the Stewart system in Eq. (4). If the fast finite-time expansion state observer is designed as in Eq. (23), with appropriate gains that satisfy the specified constraints, then the observation error $\xi =[\xi _{1},\xi _{2}]^{T}$ will converge to the residual set $U_{1}$ within a finite time $T_{3}$ .

(26) \begin{equation} U_{1}=\left\{\xi |\gamma _{1}V_{1}\left(e\right)^{\frac{1}{2}-\frac{1}{2\alpha _{1}}}+\gamma _{2}V_{1}\left(e\right)^{\frac{1}{2}}\lt \gamma _{3}\right\} \end{equation}

Proof: Introduce an auxiliary state variable $\delta$ .

(27) \begin{align} \delta &=\left[\left(Sig^{{\alpha _{1}}}\left(\xi _{1}\right)+\xi _{1}\right)^{T},\xi _{2}^{T}\right]^{T} \end{align}

(28) \begin{align} \dot{\delta } & =\left[\begin{array}{l} \alpha _{1}diag\left(\left| \xi _{1}\right| ^{{\alpha _{1}}-1}\right)\dot{\xi }_{1}\nonumber\\[3pt] -\eta _{2}\left(Sig^{{\alpha _{1}}}\left(\xi _{1}\right)+Sig^{{\alpha _{1}}}\left(\xi _{1}\right)\right) \end{array}\right]+\left[\begin{array}{l} \dot{\xi }_{1}\nonumber\\[3pt] -\eta _{2}\left(Sig^{{\alpha _{1}}}\left(\xi _{1}\right)+\xi _{1}\right) \end{array}\right]+\left[\begin{array}{l} 0_{6}\nonumber\\[3pt] \dot{D} \end{array}\right]\\[10pt] & =diag\left(\left| \xi _{1}\right| ^{{\alpha _{1}}-1},\left| \xi _{1}\right| ^{{\alpha _{1}}-1}\right)H_{1}\delta +H_{1}\delta +M \end{align}

where $M=[0_{6};\,\dot{D}] , H_{1}=[-\alpha _{1}\beta _{1}I_{6},\alpha _{1}I_{6};-\eta _{2},0_{6}] , H_{2}=[-\beta _{1}I_{6},I_{6};-\eta _{2},0_{6}]$ ,. The coefficient matrices $H_{1}$ and $H_{2}$ are Hurwitz matrices. There exists a symmetric positive definite matrix $P=P^{T}$ such that $H_{1}^{T}P+PH_{1}=-Q_{1}$ and $H_{2}^{T}P+PH_{2}=-Q_{2}$ hold, and $Q_{1}$ and $Q_{2}$ are symmetric positive definite matrices.

Choose the Lyapunov function $V_{3}(\xi )$ as follows:

(29) \begin{align} V_{3}\left(\xi \right)& =\delta ^{T}P\delta \end{align}

(30) \begin{align} \dot{V}_{3}\left(\xi \right) &=\delta ^{T}\left[diag\left(\left| \xi _{1}\right| ^{{\alpha _{1}}-1},\left| \xi _{1}\right| ^{{\alpha _{1}}-1}\right)\left(H_{1}^{T}P+PH_{1}\right)\right]\delta +\delta ^{T}\left(H_{2}^{T}P+PH_{2}\right)\delta +M^{T}P\delta +\delta ^{T}PM \nonumber\\[6pt] &\leq -\left(\left| \xi _{1}\right| _{\max }\right)^{{\alpha _{1}}-1}\delta ^{T}Q_{1}\delta -\delta ^{T}Q_{2}\delta +2\left\| \delta \right\| \left\| P\right\| \left\| M\right\| \end{align}

where $| \xi _{1}| _{\max }=\max \{| \xi _{1i}| \} , i=1,2,\cdots ,6$ and $\alpha _{1}\in (0.5,1)$ are known, $\left| \xi _{1}\right| _{\max }\leq \left\| \xi _{1}\right\| \leq \left\| \delta \right\| ^{\frac{1}{\alpha _{1}}}$ can be obtained, and $\lambda _{\max }(P)^{-1}V_{3}\leq \| \delta \| ^{2}\leq \lambda _{\min }(P)^{-1}V_{3}$ and $\| M\| =\| \dot{D}\| \leq \rho _{d}$ hold.

(31) \begin{eqnarray} \dot{V}_{3}\left(\xi \right) &\leq& -\lambda _{\min }\left(Q_{1}\right)\left\| \delta \right\| ^{3-\frac{1}{\alpha _{1}}}-\lambda _{\min }\left(Q_{1}\right)\left\| \delta \right\| ^{2}+2\left\| \delta \right\| \left\| P\right\| \left\| M\right\| \nonumber\\[10pt] &\leq& -\lambda _{\min }\left(Q_{1}\right)\lambda _{\max }\left(P\right)^{\frac{1}{2\alpha _{1}}-\frac{3}{2}}{V_{3}}^{\frac{3}{2}-\frac{1}{2\alpha _{1}}}-\lambda _{\min }\left(Q_{1}\right)\lambda _{\max }\left(P\right)^{-1}V_{3}+2\rho _{d}\lambda _{\min }\left(P\right)^{-1}\left\| P\right\| {V_{3}}^{\frac{1}{2}} \nonumber\\[10pt] &=& -\lambda _{1}{V_{3}}^{\frac{3}{2}-\frac{1}{2\alpha _{1}}}-\lambda _{2}V_{3}+\lambda _{3}{V_{3}}^{\frac{1}{2}} \end{eqnarray}

According to Lemma 3, the proposed FFESO trajectory is fast finite-time uniformly ultimately bounded stable. The state error $\xi _{1}$ and $\xi _{2}$ will converge to a small region around the origin within the finite time $T_{3}$ . The proof is complete.

(32) \begin{equation} T_{3}\leq \frac{\ln \left(1+\left(\lambda _{2}-\gamma _{2}\right)V_{3}\left(\xi _{0}\right)^{\frac{3}{2}-\frac{1}{2\alpha _{1}}}/\left(\lambda _{1}-\gamma _{1}\right)\right)}{\left(\lambda _{2}-\gamma _{2}\right)\left(\frac{1}{2\alpha _{1}}-\frac{1}{2}\right)} \end{equation}

3.3. Non-singular composite integral sliding mode surface design

This section designs a Stewart platform finite-time adaptive sliding mode controller that adopts a nonsingular composite integral terminal sliding mode surface. It compensates for lumped uncertainties through a fast finite-time expansion state observer and designs an adaptive reaching law to reduce sliding mode chattering. The control system block diagram is shown in Figure 3.

Based on the proposed FFESO, the state-space expression of the Stewart platform can be rewritten as follows:

(33) \begin{equation} \left\{\begin{array}{l} \dot{x}_{1}=x_{2}\\ \dot{x}_{2}={\Phi} \left(x_{1}\right)u+f\left(x_{1},x_{2}\right)+z_{2}+\xi _{2} \end{array}\right. \end{equation}

where $\xi _{2}$ is the observation error. According to Theorem 2, $\xi _{2}$ converges to the stable region within a finite time $T_{3}$ . Therefore, it can be assumed that there exists a small normal constant $\xi _{d}$ that satisfies $\xi _{2}\leq \xi _{d}$ .

The proposed control scheme consists of three components: equivalent control, switching control, and lumped uncertainty compensation. It can be expressed as:

(34) \begin{equation} u=u_{eq}+u_{asw}+u_{d} \end{equation}

(35) \begin{equation} u_{eq}={\Phi} ^{-1}\left(x_{1}\right)\left(\begin{array}{l} \ddot{x}_{d}-f\left(x_{1},x_{2}\right)-\alpha G'\left(e\right)\dot{e}-\\[3pt] \beta \left(Sig^{{\lambda _{1}}}\left(v\right)+Sig^{{\lambda _{2}}}\left(\dot{v}\right)\right)-\xi _{d}b\left(S\right) \end{array}\right) \end{equation}

(36) \begin{equation} u_{d}=-{\Phi} ^{-1}\left(x_{1}\right)\hat{z}_{2} \end{equation}

(37) \begin{equation} u_{asw}={\Phi} ^{-1}\left(x_{1}\right)\left(-\frac{k_{1}}{\varphi \left(S\right)}b\left(S\right)-k_{2}h\left(S\right)S\right) \end{equation}

where $b\left(S\right)=\left\{\begin{array}{l} \frac{S}{\left\| S\right\| },\,\left\| S\right\| \neq 0\\ 0, \left\| S\right\| =0 \end{array}\right.$ is the norm-normalized sign function [Reference Li, Yu, Tee, Wu, Ge and Lee26], and $\varphi (S)=\varepsilon +(1-\varepsilon )e^{-c(| S| -1)} , h(S)=1-e^{-\lambda | S| } , k_{1}$ ‵ $k_{2}$ ‵ $\varepsilon$ ‵ $\mathrm{c}$ ‵ $\lambda$ are normal constants, as well as $k_{4}=k_{2}h(S)\gt 0$ and $k_{3}=\frac{k_{1}}{\varphi (S)}\gt 0$ .

It can be seen that when the system $| S_{i}|$ is far from the sliding surface, $\varphi (S_{i})\lt 1$ and $\nu (S_{i})$ approach 1, and at this point, $k_{3}=\frac{k_{1}}{\varphi (S)}\gt k_{1}$ and $k_{4}=k_{2}h(S)$ approach $k_{2}$ . The system will approach the sliding surface at a faster rate. Conversely, when $| S_{i}|$ approaches 0, $h(S_{i})$ approaches 0, and at this point, $k_{3}=\frac{k_{1}}{\varphi \left(S\right)}\lt k_{1}$ and $k_{4}=k_{2}h(S)$ approach 0, which reduces the gain and consequently reduces chattering.

A comparison of the boundary layer sizes between the traditional exponential reaching law and the adaptive reaching law is shown in the following Figure 2. The proposed adaptive reaching law can dynamically adjust the approaching and sliding phase rates in real-time with changes in the sliding mode surface, solving the contradiction between approaching speed and chattering in the traditional exponential reaching law.

Figure 2.

Phase plane graphs with different reaching laws. (a) Conventional exponential approach rate. (b) Adaptive approach rate.

Theorem 3: For the Stewart platform (4) with aggregated uncertainties, and by choosing the finite-time non-singular composite integral sliding mode surface (13) as mentioned above, using the proposed control rate (34), the entire closed-loop system is finite-time stable, and the position tracking error can converge to a small region near the origin within a finite time.

Proof: Consider the following Lyapunov function $V_{4}$ .

(38) \begin{equation} V_{4}=\frac{1}{2}S^{T}S \end{equation}

Take the derivative and substitute into the control rate (34):

(39) \begin{eqnarray} \dot{V}_{4} &=& S^{T}\left(\xi _{2}-\xi _{d}b\left(S\right)\right)-k_{3}\sum _{i=1}^{6}\left| S_{i}\right| -k_{4}S^{T}S\nonumber\\[0pt] &\leq& -k_{3}\sqrt{2}\left(\frac{1}{2}S^{T}S\right)^{\frac{1}{2}}-2k_{2}\frac{1}{2}S^{T}S\nonumber\\[0pt] &\leq& -\sqrt{2}k_{3}V_{4}^{\frac{1}{2}}-2k_{2}V_{4} \end{eqnarray}

where $\sqrt{2}k\gt 0$ and $2k_{2}\gt 0$ , according to Lemma 2, the sliding mode surface will converge to zero within a finite time $T_{4}$ . When $S=0$ , from Theorem 1, it is known that the position tracking error will converge to a neighborhood of zero within a finite time $T_{f}$ . Therefore, the position tracking error can converge within a finite time $T_{5}$ , denoted as $T_{5}\lt T_{\max }=T_{3}+T_{4}+T_{f}$ .

(40) \begin{equation} T_{4}\leq \frac{1}{k_{2}}\ln \frac{\mu _{1}V^{0.5}\left(S\left(0\right)\right)+\sqrt{3}k_{3}}{\sqrt{3}k_{3}} \end{equation}

3.4. Controller parameter selection

Unmodeled dynamics, external disturbances, and measurement noise in practical systems require an integrated selection of appropriate control parameters to balance control speed, accuracy, and signal smoothness.

1. 1) Selection of Sliding Surface Parameters $\alpha , \beta , \lambda _{1} , \lambda _{2} , r , \kappa , m$ : The parameter $\alpha$ affects the convergence rate of the tracking error, a larger $\alpha$ leads to faster error convergence. However, an excessively large $\alpha$ may result in increased control effort. An integral term in the sliding variable is introduced to suppress steady-state error, but an overly large integral gain $\beta$ may induce steady-state deviation. Parameters $\lambda _{1} , \lambda _{2}$ also influence the convergence rate and should satisfy $0\lt \lambda _{1}\lt 1$ and $\lambda _{2}=\frac{2\lambda _{1}}{\lambda _{1}+1}$ . Parameters $r$ and $\kappa$ are introduced to address the singularity problem, satisfy $r\gt 1$ and $m=1-\kappa$ , where $\kappa$ is a small positive constant near the origin, satisfying $0\lt \kappa \lt e^{-1}$ . In the experimental implementation, the following parameters were chosen: $\alpha =10 , \beta =15 , \lambda _{1}=0.8 , \lambda _{2} , r=1.7 , \kappa =0.3 , m=0.7$ .

2. 2) Selection of Parameters $\eta _{1} , \eta _{2} , \alpha _{1} , \alpha _{2}$ : The parameters $\eta _{1} , \eta _{2} , \alpha _{1} , \alpha _{2}$ determine the performance of the observer. In particular, $\eta _{1}$ and $\eta _{2}$ need to be sufficiently large to ensure satisfactory observation accuracy and convergence speed. The selection of $\alpha _{1}$ and $\alpha _{2}$ must also satisfy the conditions derived in the proof of Theorem 2 to guarantee system stability and estimation performance. In the experiments, the following parameter values were adopted: $\eta _{1}=1500 , \eta _{2}=2000 , \alpha _{1}=0.8 , \alpha _{2}=0.6$ .

3. 3) Selection of Parameters $\varepsilon , c , \lambda , k_{1} , k_{2}$ : The parameters $\varepsilon , c , \lambda , \lambda _{2} , k_{1} , k_{2}$ are used to define the reaching law dynamics. Specifically, $\varepsilon$ determines the convergence region of the tracking error. A smaller value of $\varepsilon$ implies higher control precision; however, if $\varepsilon$ is too small, it may cause control input saturation and induce chattering. Parameters $k_{1}$ and $k_{2}$ influence the convergence rate of the sliding motion, overly small values lead to slow convergence, while excessively large values may result in chattering. Therefore, proper tuning is essential. Parameters $c$ and $\lambda$ determine the adaptation rate in the adaptive law and must satisfy certain conditions $0\lt c\lt 1$ and $0\lt \lambda \lt 1$ . In the experimental implementation, the following values were selected: $\varepsilon =0.1 , c=0.2 , \lambda =0.2 , k_{1}=10 , k_{2}=15$ .

Figure 3.

Control System Block Diagram.

4.1. Simulated analysis

Finite-time control is a strategy aimed at driving the system from any initial state to the target state within a finite time. The initial state of the system often affects the control time, especially when the initial deviation is large, requiring more time for the controller to adjust the system. This paper proposes a globally finite-time convergent sliding mode surface composed of intermediate variables and integral terms. By designing appropriate initial values for the intermediate variables, the sliding mode reaching phase can be eliminated, reducing the impact of initial conditions on the controller.

The tracking trajectories of the z-axis are set to 41, 42, and 43 cm, with other conditions remaining the same. The tracking performance of the traditional integral sliding mode surface [Reference Asif, Khan and Memon27] and the proposed nonsingular composite integral sliding mode surface are shown in Figure 5, respectively. For the same tracking trajectory, by selecting $\alpha =10$ and $\beta =15$ , the proposed nonsingular composite integral sliding mode surface can eliminate the sliding mode reaching phase by setting the initial values $W(0)$ and $Z(0)$ in Eq. (13), thus resulting in a shorter convergence time than the traditional integral sliding mode surface.

Figure 4.

Stewart experimental Platform.

To validate the performance of the proposed fast finite-time observer (FFESO), its observation performance was compared with that of the traditional ESO and the finite-time extended state observer (FESO) proposed in ref. [Reference He, Wu and Li28]. To test the observer’s observation accuracy and time, disturbances of 100N and 200N were added to the z-axis at 1s and 4s, respectively. The observer parameters $\eta _{1}=1500 , \eta _{2}=2000 , \alpha _{1}=0.8$ , and $\alpha _{2}=0.6$ were set, and verification was carried out.

The FFESO presented in this paper is designed based on a nonsingular composite integral sliding mode surface, enabling global finite-time convergence. Therefore, its convergence speed is faster than that of traditional observers, allowing for quicker estimation of total uncertainties. The results, as shown in Figure 6, indicate that the traditional ESO has poor observation performance and cannot accurately observe disturbances. The FESO can observe disturbances within 2 s, while the proposed FFESO can observe disturbances within 0.5 s, achieving a 75% improvement in convergence speed and more accurately estimating the magnitude of total uncertainty.

Figure 5.

Comparison of initial conditions.(a) Traditional integral sliding surface (b) Non-singular composite integral sliding mode.

In order to fully demonstrate the superiority of the proposed sliding mode controller in terms of system convergence speed, stability, robustness, and control accuracy, a comparative analysis was conducted with traditional torque control (CTC) [Reference Dixon, de Queiroz, Zhang and Dawson29], neural network-based adaptive control (NNC) [Reference Chen, Ming and Qin30], and robust sliding mode control (RC) [Reference Xie, Yu and Chen31]. Firstly, to verify the control performance of the proposed method, step response comparisons of the z-axis were conducted for the four controllers. Time-varying disturbances $d=0.1\sin (20t-\pi /3)+0.2\cos (10\pi t)$ were introduced in the simulation to emulate unknown uncertainties, aiming to assess the effectiveness of the system in practical application scenarios. The proposed controller eliminates steady-state errors through NCISMS and quickly estimates disturbances using FFESO, resulting in higher convergence speed and accuracy compared to other control methods.

To evaluate the control performance of the proposed method, step response comparisons of the Z-axis under four different controllers were conducted. Time-varying disturbances were introduced in the simulation to emulate unknown uncertainties, aiming to assess the effectiveness of the system in practical application scenarios.

The tracking results are shown in Figure 7(a). The proposed controller exhibits no overshoot, with a convergence time of only 0.5 s, which is reduced by 75%, 67%, and 47% compared to the other three methods, respectively. The tracking error is shown in Figure 7. Due to the presence of the integral term, it significantly reduces the steady-state error, resulting in better control accuracy.

Figure 6.

Comparison of Observation Results from Different Observers.

To further verify the robustness of the proposed method, a 300N disturbance is applied to the upper platform at 5 s to simulate an abrupt load.

The tracking results are shown in Figure 8. Compared to other methods, the proposed method has an overshoot of only 0.0002 cm, which is reduced by 99%, 80%, and 50% compared to CTC, NNC, and RC, respectively. The tracking error is shown in Figure 8, with a convergence time of 0.4 s, which is reduced by 84%, 67%, and 50%, respectively. This indicates that the proposed method has stronger robustness, as the fast finite-time observer and adaptive approach rate can effectively compensate for the impact of concentrated uncertainties on the system.

Figure 7.

Step response of different controller. (a) tracking curve. (b) tracking error.

A further comparison of the combined tracking trajectory effect of the spiral ascent was conducted, considering 20% model uncertainty, i.e., $M_{0}=0.8M , C_{0}=0.8C$ and $G_{0}=0.8G$ . The tracking trajectories of different controllers are shown in Figure 9. The results indicate that for complex trajectories, the proposed sliding mode controller has the smallest tracking error, still maintaining high-precision tracking and overcoming the impact of model uncertainties.

4.2. Comparison of tracking performance of different sliding mode controllers

To examine the trajectory tracking capability of the Stewart platform with six degrees of freedom, the reference trajectory is given as follows:

(41) \begin{equation} X_{d}\left(t\right)=\left\{\begin{array}{l} \sin \left(\pi t\right)cm\\[2pt] \cos \left(\pi t\right)\,cm\\[2pt] 0.2t+40\,cm\\[2pt] 0.1\sin \left(\pi t\right)\,rad\\[2pt] 0.1\cos \left(\pi t\right)\,rad\\[2pt] 0.1\sin \left(\pi t\right)\,rad \end{array}\right. \end{equation}

To verify the effectiveness of the proposed improved sliding mode control strategy, this section compares the proposed finite-time adaptive non-singular composite integral sliding mode control (FFESO-FANCISMC) based on a fast finite-time observer with traditional sliding mode control (SMC), the switching-type non-singular terminal sliding mode control (NTSMC) proposed in reference [Reference Fang, Cheng, Cheng and Fan32], and the non-singular composite integral sliding mode control with adaptive rate (ANCISMC) from reference [Reference Vo and Kang33], and performs experimental verification.

1) The design of the SMC is as follows:

(42) \begin{equation} S =\dot{e}+ce \end{equation}

(43) \begin{equation} u={\Phi} ^{-1}\left(x_{1}\right)\left(\begin{array}{l} \ddot{x}_{d}-f\left(x_{1},x_{2}\right)+c\dot{e}\\ -k_{1}sign\left(s\right)-k_{2}s \end{array}\right) \end{equation}

2) The design of the NTSMC is as follows:

(44) \begin{equation} S=\dot{e}+c_{1}G\left(e\right)+c_{2}Sig^{\beta }\left(e\right) \end{equation}

(45) \begin{equation} u={\Phi} ^{-1}\left(x_{1}\right)\left(\begin{array}{l} \ddot{x}_{d}-f\left(x_{1},x_{2}\right)-c_{1}G'\left(e\right)-\\[2pt] c_{2}\beta \left| e\right| ^{\beta -1}-k_{1}sign\left(s\right)-k_{2}s \end{array}\right) \end{equation}

3) The design of the ANCISMC is as follows:

(46) \begin{equation} u={\Phi} ^{-1}\left(x_{1}\right)\left(\begin{array}{l} \ddot{x}_{d}-f\left(x_{1},x_{2}\right)-\beta \left(Sig^{{\lambda _{1}}}\left(u\right)+Sig^{{\lambda _{2}}}\left(\dot{u}\right)\right)\\[2pt] -\alpha G'\left(e\right)-\left(\hat{\varphi }+\eta \right)sign\left(s\right)-ks \end{array}\right) \end{equation}

(47) \begin{equation} \hat{\varphi }=\frac{1}{\kappa }\left| S\right| \end{equation}

Table II.

Controller parameters.

Figure 8.

Step response of different controller(Load at 5s). (a) tracking curve. (b) tracking error.

Figure 9.

Comparison of spiral upward tracking of different controllers.

Figure 13 shows the relationship between sliding mode switching gain and chattering. Apart from the switching gain, all other parameters are the same as those in the proposed method. It can be seen that chattering increases as the switching gain increases. In this experiment, parameters $k_{1}$ and $k_{2}$ were selected, with the other controller parameters specified in Table II

SMC exhibits significant oscillation and has a slow convergence rate. NTSMC improves the convergence speed through switching terms, and its transient response is better than SMC, but it still has steady-state error. ANCISMC uses the proposed NCISMS to eliminate steady-state error and adjusts the gain through adaptive coefficients, reducing oscillation to some extent, but sacrificing some convergence speed. The proposed FFESO-FANCISMC eliminates steady-state error through NCISMS, sets initial values to eliminate the sliding mode reaching phase, and improves system robustness through FFESO and adaptive rates. Figure 10 shows the trajectory tracking results of four sliding mode controllers, indicating that all controllers can smoothly complete the required position tracking. Figure 11 displays the tracking errors of different controllers. Compared to others, ANCISMC and the proposed controller have smaller steady-state tracking errors. The proposed controller has faster error convergence, smaller oscillations, and higher stability accuracy.

Figure 10.

Relationship between the switching gain and chattering.

To better evaluate the performance of the proposed control scheme, this study uses two error-based performance metrics: Integral of Squared Error (ISE) and Integral of Time-weighted Absolute Error (ITAE), defined as follows: $ISE=\int _{0}^{t}e^{2}(t)d\tau , ITAE=\int _{0}^{t}t| e(t)| d\tau$ . The smaller the metric value, the better the control performance. The ISE and ITAE metric results for the trajectory tracking of different controllers are shown in Tables III and IV.

Table III.

ISE of different controllers.

Table IV.

ITAE of different controllers.

Figure 11.

Stewart experimental Platform.

By comparing the metric values in Tables III and IV, it can be easily concluded that the proposed controller has smaller ISE and ITAE values than other methods. For example, for axis tracking, the ISE value of the proposed controller is 12.3% of the SMC, 33.9% of the NTSMC, and 37.2% of the ANCISMC, representing a reduction of 87.7%, 66.1%, and 62.8%, respectively. The ITAE value of the proposed controller is 25.4% of the SMC, 42.4% of the NTSMC, and 44.5% of the ANCISMC, representing a reduction of 74.6%, 57.6%, and 55.5%, respectively. In both transient and steady-state responses, the proposed finite-time adaptive sliding mode controller can achieve superior tracking performance compared to other controllers.

Figure 12 shows the control inputs for tracking the trajectories of the six legs. It can be seen that the chattering amplitude of SMC is 3N, while the switching-type nonsingular terminal sliding mode control and the nonsingular composite integral sliding mode control with adaptive rates can reduce the chattering of SMC to some extent, with chattering amplitudes of 2.5N and 1.5N, respectively. The method proposed in this paper uses a fast finite-time extended state observer, which can more accurately estimate the magnitude of both internal and external lumped uncertainties. At the same time, the use of adaptive approaching rates significantly reduces the chattering phenomenon, with a chattering amplitude of only 0.2N. Compared to other methods, this reduces the chattering by 93.3%, 92%, and 86.7%, respectively. Therefore, the designed controller has smaller chattering and can track the desired trajectory more accurately, achieving high-precision tracking control for the Stewart platform.

Figure 12.

6-DOF trajectory tracking error.

To fully verify the robustness and finite-time convergence of the proposed method, trajectory tracking tests were conducted under three different conditions: no load, 10 kg load, and 30 kg load, following a given reference trajectory $X_{d}(t)=\{2\sin (\pi t),2\cos (\pi t),41,0,0,0\}$ . As shown in Figure 14, the proposed control strategy demonstrates excellent trajectory tracking performance under all load conditions, further confirming its strong robustness and fast response capability.

Figure13.

Control Input.