2. Dynamic model and problem formulation

The dynamic model of the hybrid mechanism in joint space can be expressed as

(1) \begin{equation} M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)=\tau \end{equation}

where $q=(q_{1},\cdots, q_{N})^{T}$ is the joint posture vector, $\dot{q}=(\dot{q}_{1},\cdots, \dot{q}_{N})^{T}$ is the joint velocity vector, $M(q)\in R^{n\times n}$ is the inertia matrix, $C(q,\dot{q})$ is the Gauche and centrifugal force item, $G(q)$ is the gravity term, and $\tau$ is the joint driving force/torque.

Taking into account the effects of modeling errors, joint friction, external disturbances, and load variation during actual operation, the dynamic model (1) can be described as

(2) \begin{equation} [\hat{M}(q)+\Delta M(q)]\ddot{q}+[\hat{C}(q,\dot{q})+\Delta C(q,\dot{q})]\dot{q}+[\hat{G}(q)+\Delta G(q)]+N(t)=\tau +d \end{equation}

where $\Delta M(q)$ , $\Delta C(q,\dot{q})$ , $\Delta G(q)$ denote the uncertain terms in $M(q)$ , $C(q,\dot{q})$ , $G(q)$ , respectively. $M(q)=\hat{M}(q)+\Delta M(q)$ , $C(q,\dot{q})=\hat{C}(q,\dot{q})+\Delta C(q,\dot{q})$ , $G(q)=\hat{G}(q)+\Delta G(q)$ , $N(t)$ are the frictional force and $d$ is the external random disturbances.

Further, the dynamic model (2) of the hybrid mechanism can be rewritten as

(3) \begin{equation} \ddot{q}=\hat{M}^{-1}[\tau -\hat{C}(q,\dot{q})\dot{q}-\hat{G}(q)]+D(t) \end{equation}

where $D(t)=\hat{M}^{-1}[d-\Delta M(q)\ddot{q}-\Delta C(q,\dot{q})\dot{q}-\Delta G(q)-N(t)]$ denotes the set total disturbances term containing modeling errors, joint friction, external disturbances, and load variation.

Let

(4) \begin{equation} \begin{split} &{x}_{1}=q\\ &{x}_{2}=\dot{{x}_{1}}=\dot{q} \end{split} \end{equation}

then

(5) \begin{equation} \begin{split} &\dot{x}_{1}={x}_{2}\\ &\dot{x}_{2}=f(x)+g(x)u+D(x) \end{split} \end{equation}

where $f(x)=-\hat{M}^{-1}[\hat{C}(q,\dot{q})\dot{q}+\hat{G}(q)]$ , $g(x)=\hat{M}^{-1}$ , $u=\tau$ .

Let the desired trajectory of the joints be $x_{1d}$ , in order to realize that the actual trajectory $x_{1}$ can be tracked to the desired trajectory $x_{1d}$ in fixed-time, that is, the tracking error $e=x_{1}-x_{1d}$ converges to $0$ in fixed-time; the following assumptions and lemmas are given.

Lemma 1. [Reference Zuo, Han, Ning, Ge and Zhang34] Consider the system

(6) \begin{equation} \dot{x}(t)=f(t,x),\,\,\,{x}(t_{0})={x}(t_{0}) \end{equation}

where ${x}(t)\in R^{n}$ is the state variable of the system, $f(t,x)$ denotes a smooth nonlinear function, and the origin is assumed to be the equilibrium point of the system. If there is a continuous radial bounded function $V:R^{n}\rightarrow R_{+}\cup \{0\}$ satisfying

\begin{equation*} V(x)\leq -\varphi _{1}V^{P}(x)-\varphi _{2}V^{Q}(x) \end{equation*}

where $\varphi _{1},\varphi _{2}\gt 0$ , $P\gt 1$ , $0\lt Q\lt 1$ , then the system ( 6 ) is said to be globally fixed-time stable, and its convergence time $T$ satisfies

\begin{equation*} T\leq T_{max}\,:\!=\,\frac {1}{\varphi _{1}(P-1)}+\frac {1}{\varphi _{2}(1-Q)} \end{equation*}

Lemma 2. [Reference Ba, Li and Tong35] If there is a continuous radial bounded function $V:R^{n}\rightarrow R_{+}\cup \{0\}$ satisfying

\begin{equation*} V(x)\leq -\varphi _{1}V^{P}(x)-\varphi _{2}V^{Q}(x)+\eta _{0} \end{equation*}

where $\varphi _{1},\varphi _{2}\gt 0$ , $P\gt 1$ , $0\lt Q\lt 1$ , then the system ( 6 ) is said to be real fixed-time stable, and its convergence time $T$ satisfies

\begin{equation*} T\leq T_{max}\,:\!=\,\frac {1}{\varphi _{1}\bar {\phi }(P-1)}+\frac {1}{\varphi _{2}\bar {\phi }(1-Q)} \end{equation*}

where $\bar{\phi }$ is a positive constant and satisfies $0\lt \bar{\phi }\lt 1$ , the residual set of the solution of system ( 6 ) is

\begin{equation*} x\in \bigg \{x|V(x)\leq \min \left\{\left(\frac {\eta _{0}}{(1-\bar {\phi })\varphi _{1}}\right)^{\frac {1}{P}},\left(\frac {\eta _{0}}{(1-\bar {\phi })\varphi _{2}}\right)^{\frac {1}{Q}}\right\}\bigg \} \end{equation*}

Figure 2.

The control schematic diagrams of FTSTSMC and IFTSTSMC for the uncertain hybrid mechanism.

Figure 2 presents the control schematic diagrams. As shown in Figure 2, first of all, the hybrid mechanism system is controlled under the designed FTSTSMC controller (Part A). Then, on this basis, RBF neural network is introduced to construct IFTSTSMC controller (Part B) to completely replace FTSTSMC controller (Part A).

3. Design of control algorithms

3.1. Fixed-time super-twisting sliding mode control

Let the sliding surface be

(7) \begin{equation} s=\dot{e}+\text{sig}^{\alpha }(e)+\text{sig}^{\beta }(e) \end{equation}

where $0\lt \alpha \lt 1$ , $1\lt \beta$ , $s=(s_{1},\cdots, s_{6})^{T}$ .

To compute the derivative of $s$

(8) \begin{equation} \dot{s}=\ddot{e}+\alpha |e|^{\alpha -1}\dot{e}+\beta |e|^{\beta -1}\dot{e} \end{equation}

since

(9) \begin{equation} \begin{split} \ddot{e}&=\ddot{x}_{1}-\ddot{x}_{1d}\\ &=\dot{x}_{2}-\ddot{x}_{1d}\\ &=f(x)+g(x)u+D(x)-\ddot{x}_{1d} \end{split} \end{equation}

then

(10) \begin{equation} \dot{s}=f(x)+g(x)u+D(x)-\ddot{x}_{1d}+\alpha |e|^{\alpha -1}\dot{e}+\beta |e|^{\beta -1}\dot{e} \end{equation}

Based on the super-twisting algorithm, the following control law is designed

(11) \begin{equation} \begin{split} \left \{ \begin{array}{ll} &\dot{s}=-k_{1}\text{sig}^{\frac{1}{2}}(s)-k_{3}\text{sig}^{\frac{3}{2}}(s)+w \\[5pt] &\dot{w}=-k_{2}\text{sig}^{\frac{1}{2}}(s)+k_{4}\text{sig}^{\frac{3}{2}}(s) \end{array} \right . \end{split} \end{equation}

Combining (5), (9), (10), and the following controller is designed

(12) \begin{equation} u= -g^{-1}(x)[f(x)-\ddot{x}_{1d}+\alpha |e|^{\alpha -1}\dot{e}+\beta |e|^{\beta -1}\dot{e} +\rho \text{sign}(s)+k_{1}\text{sig}^{\frac{1}{2}}(s)+k_{3}\text{sig}^{\frac{3}{2}}(s)-w ] \end{equation}

where $\rho \geq \delta$ , $k_{1},k_{2},k_{3},k_{4}\gt 0$ is the design parameters to be determined.

Theorem 1. If Assumption 1 is true, the super-twisting second-order sliding mode dynamic control law designed by ( 12 ) is adopted for the dynamic control system of hybrid mechanism ( 5 ), then the sliding mode surface $s$ and $\dot{s}$ designed based on tracking error can converge to the origin within a fixed-time.

Proof. Selecting Lyapunov function

(13) \begin{equation} V=\eta ^{T}\mathbf{B}\eta \end{equation}

where $\eta =[|s|^{\frac{3}{2}}\text{sign}(s),s,w]$ , $\mathbf{B}$ is a symmetric positive definite matrix, defined as

\begin{equation*} \mathbf {B}=\frac {1}{2} \left [ \begin {matrix} 4k_{2}+k_{1}^{2} & \quad k_{1}k_{3} & \quad -k_{1} \\[5pt] k_{1}k_{3} & \quad 2k_{4}+k_{3}^{2} & \quad -k_{3} \\[5pt] -k_{1} & \quad -k_{3} & \quad \dfrac {k_{1}k_{3}}{k_{2}} \end {matrix} \right ] \end{equation*}

The derivative of equation (13) is obtained

(14) \begin{align} \dot{V} & =2\eta ^{T}\mathbf{B}\dot{\eta }\nonumber\\ &=|s|^{\frac{5}{2}}\left[-\frac{3}{2}k_{1}\left(4k_{2}+k_{1}^{2}\right)-k_{3}\left(2k_{4}+k_{3}^{2}\right)-k_{3}k_{4}\right]+|s|^{2}\left[-\frac{5}{2}k_{1}^{2}k_{3}+k_{1}k_{2}\right]\nonumber\\[4pt] &+|s|w\left[\frac{3}{2}k_{1}+\left(2k_{4}+k_{3}^{2}\right)\right]-\frac{3}{2}k_{3}\left(4k_{2}+k_{1}^{2}\right)|s|^{\frac{7}{2}}+|s|^{3}\left[-\frac{3}{2}k_{1}k_{3}-k_{1}k_{3}^{2}-k_{1}k_{4}\right]\nonumber\\[4pt] &+|s|^{2}w\left[\frac{3}{2}k_{1}k_{3}+\frac{3}{2}\left(4k_{2}+k_{1}^{2}\right)\right]+|s|^{\frac{3}{2}}w\left[\frac{3}{2}k_{1}k_{3}+k_{3}^{2}+k_{1}k_{3}+2k_{4}\right]\nonumber\\[4pt]&-\frac{3}{2}k_{1}|s|^{\frac{1}{2}}w^{2}+|s|^{\frac{3}{2}}\left[-k_{1}\left(2k_{4}+k_{3}^{2}\right)+k_{2}k_{3}\right]-k_{3}w^{2}\nonumber\\[1pt]&=-|s|^{-\frac{5}{2}}\left(\eta ^{T}\mathbf{T_{1}}\eta \right)\left(\eta ^{T}\mathbf{T_{1}}\eta \right)^{T}-|s|^{-\frac{3}{2}}\eta ^{T}\mathbf{T_{2}}\eta -|s|^{\frac{1}{2}}\eta ^{T}\mathbf{T_{3}}\eta -\eta ^{T}\mathbf{T_{4}}\eta \end{align}

where $\mathbf{T_{1}}$ , $\mathbf{T_{2}}$ , $\mathbf{T_{3}}$ , $\mathbf{T_{4}}$ are symmetric positive definite matrices.

Since $|s|^{\frac{3}{2}}\leq \|\eta \|_{2}\leq \left(\frac{V}{\lambda _{\min }(\mathbf{B})}\right)^{\frac{1}{2}}$ , $|s|^{\frac{5}{2}}\leq \|\eta \|^{\frac{5}{3}}_{2}\leq \left(\frac{V}{\lambda _{\min }(\mathbf{B})}\right)^{\frac{5}{6}}$ . Therefore, it follows that

(15) \begin{equation} \begin{split} \dot{V}\leq &-\dfrac{(\lambda _{\min }(\mathbf{B}))^{\frac{5}{6}}(\lambda _{\min }(\mathbf{T_{1}}))^{2}}{V^{\frac{5}{6}}}\dfrac{V^{2}}{(\lambda _{\max }(\mathbf{B}))^{2}}-\dfrac{(\lambda _{\min }(\mathbf{B}))^{\frac{1}{2}}\lambda _{\min }(\mathbf{T_{2}})}{V^{\frac{1}{2}}}\dfrac{V}{\lambda _{\max }(\mathbf{B})}\\[4pt] =&-\dfrac{(\lambda _{\min }(\mathbf{B}))^{\frac{5}{6}}(\lambda _{\min }(\mathbf{T_{1}}))^{2}}{(\lambda _{\max }(\mathbf{B}))^{2}}V^{\frac{7}{6}}-\dfrac{(\lambda _{\min }(\mathbf{B}))^{\frac{1}{2}}\lambda _{\min }(\mathbf{T_{2}})}{\lambda _{\max }(\mathbf{B})}V^{\frac{1}{2}} \end{split} \end{equation}

According to Lemma1, the sliding surface $s$ and $\dot{s}$ can converge to the origin in fixed-time $T$ , that is, the system (5) can achieve stability in fixed-time $T$ , and the convergence time is estimated as

(16) \begin{equation} T\leq \dfrac{6(\lambda _{\max }(\mathbf{B}))^{2}}{(\lambda _{\min }(\mathbf{B}))^{\frac{5}{6}}(\lambda _{\min }(\mathbf{T_{1}}))^{2}}+\dfrac{2\lambda _{\max }(\mathbf{B})}{(\lambda _{\min }(\mathbf{B}))^{\frac{1}{2}}\lambda _{\min }(\mathbf{T_{2}})} \end{equation}

Remark 1. The convergence time of finite-time sliding mode control (FTSMC) is related to the initial conditions of the system. However, the initial conditions of the actual hybrid mechanism may be difficult to obtain accurately in advance. In order to solve this problem, sliding mode surface (7) and control law (11) are designed on the basis of Lemma 1. The fixed-time performance parameters satisfying the conditions $P\gt 1$ and $0\lt Q\lt 1$ are introduced into sliding variables and the transcendental function of the super-twisting algorithm respectively, so that the contraction of Lyapunov function can be applied to the form of lemma. Therefore, the global fixed-time convergence of the system is guaranteed.

Remark 2. The existence of sliding mode must be guaranteed during the implementation of SMC. It is worth noting that a sliding mode does exist on a discontinuity surface whenever the distances to this surface and the velocity of its change $\dot{s}$ are of opposite signs [Reference Barbashin36], that is, $\lim _{s\rightarrow 0^+}\dot{s}\lt 0$ , $\lim _{s\rightarrow 0^-}\dot{s}\gt 0$ . Therefore, Utkin was the first to propose sufficient conditions for the existence of sliding modes: $\lim _{s\rightarrow 0^+}\dot{s}\leq 0$ , $\lim _{s\rightarrow 0^-}\dot{s}\geq 0$ , also known as the local reachability condition [Reference Utkin37]. Further, if the condition $s\dot{s}\leq 0$ is satisfied, it is called the global reachability condition of the sliding mode. According to the analysis of Theorem 1, the SMC designed in this paper satisfies the global reachability condition. Therefore, controller $u$ needs to find $u^-$ and $u^+$ to ensure that the system state variables can continue sliding along $s=0$ when it reaches the sliding mode surface $s$ . According to (5), $u^-$ and $u^+$ should be satisfied that $g(x)u^+\lt -\min (x_{2}+D(x)+f(x))$ , $g(x)u^-\gt -\max (x_{2}+D(x)+f(x))$ . It can be seen from the above that the boundary of the system uncertainty $\delta$ determines the realization of $u$ to ensure the robustness of the SMC for the uncertain hybrid mechanism. Therefore, when the uncertainty parameter of the system exceeds $\delta$ , the SMC of the system will collapse. In order to avoid the limitation of this control, this paper will further introduce RBF neural network to deal with the control problem of the system with a wide range of uncertainties.

Remark 3. From the design viewpoint of the FTSTSMC, the method of literature [Reference Wang, Jahanshahi, Volos, Bekiros, Yusuf, Agarwal and Aly33] (FTSTSMC-1) can only achieve fixed-time convergence in the reaching stage of SMC. When it is applied to the hybrid mechanism, the sliding phase of SMC is greatly affected by the initial information of the system. Compared with FTSTSMC-1, the method designed in this paper (FTSTSMC) achieves fixed-time convergence in both reaching stage and sliding stage of SMC by adding the exponential power terms with the fixed-time performance parameters in sliding variables and the transcendental function of the super-twisting algorithm. Therefore, FTSTSMC achieves the global convergence performance of the system and overcomes the limitation that the sliding phase of FTSTSMC-1 is greatly affected by the initial information of the system.

3.2. Intelligent fixed-time super-twisting sliding mode control

RBF neural network [Reference Broomhead and Lowe38] is a neural network for local approximation proposed by J. Moody and C. Darken in the late 1980s. It can approximate any nonlinear function and deal with the law that is difficult to analyze in the system. In (5), $F(x)=f(x)+D(x)$ is designed to be an unknown nonlinear function of the system, where $r(x)$ represents the load variation in $D(x)$ . The uncertainty and unknown nature of $F(x)$ make it challenging to ensure the realization of the control law $u$ . Therefore, RBF neural network is introduced to improve the adaptability of the system to the uncertainties with a wide range of changes.

RBF neural network is a forward-oriented network composed of three-layer networks.

The first layer: input layer, $x=[x_{1},\cdots, x_{n}]^{T}$ is the input of the network, and $n$ represents the number of dimensions of the input.

The second layer: the implicit layer, the output is $h(x)=[h_{1},\cdots, h_{m}]^{T}$ . Using the Gaussian basis function as the membership function of the input layer, that is,

\begin{equation*} h_{j}(x)=exp\left(-\frac {\|x-c\|^{2}}{2b^{2}}\right) \end{equation*}

where $c$ is the coordinate vector of the centroid of the Gaussian basis function in the implicit layer, $b$ is the width of the Gaussian basis function in the implicit layer, $j=1,\cdots, m$ . The RBF neural network weight is $W=[W_{1},\cdots, W_{m}]^{T}$ .

The third layer: the output layer, the output of the neural network is $y=W^{T}h(x)$ .

The RBF neural network is used to approximate $F(x)$ , and the ideal output of the neural network is given as

\begin{equation*} F(x)={W^{*}}^{T}h(x)-\varepsilon \end{equation*}

where $W^{*}$ is the ideal weight of the RBF neural network, $h(x)$ is the output of the Gaussian function, and $\varepsilon$ is the approximation error of the network. Let the actual output of the neural network be

\begin{equation*} \hat {f}(x)=\hat {W}^{T}h(x) \end{equation*}

where $\hat{W}$ is the actual weight of the neural network. Defining $\tilde{W}=W^{*}-\hat{W}$ , $\tilde{f}(x)=F(x)-\hat{f}(x)$ , where $\tilde{W}$ is the error between the ideal weight and the actual weight, and $\tilde{f}(x)$ is the error between the ideal output and the actual output, so the fixed-time sliding mode controller based on RBF neural network can be designed as

(17) \begin{equation} u= -g^{-1}(x)\left[\hat{f}(x)-\ddot{x}_{1d}+\alpha |e|^{\alpha -1}\dot{e}+\beta |e|^{\beta -1}\dot{e}+k_{1}\text{sig}^{\frac{1}{2}}(s)+k_{3}\text{sig}^{\frac{3}{2}}(s)-w \right]\end{equation}

Then, the control law based on the super-twisting algorithm is

(18) \begin{equation} \left \{ \begin{array}{ll} &\dot{s}=-k_{1}\text{sig}^{\frac{1}{2}}(s)-k_{3}\text{sig}^{\frac{3}{2}}(s)+w+\tilde{W}^{T}h(x)-\varepsilon \\[4pt] &\dot{w}=k_{2}\text{sig}^{\frac{1}{2}}(s)-k_{4}\text{sig}^{\frac{3}{2}}(s) \end{array} \right . \end{equation}

Theorem 2. If Assumption 1 is true, for the dynamic control system of hybrid mechanism ( 5 ), the intelligent super-twisting second-order sliding mode dynamic control law designed by ( 17 ) is adopted. Then the sliding mode surface $s$ and $\dot{s}$ designed based on tracking error can converge to the origin within a real fixed-time.

Proof. Selecting Lyapunov function

(19) \begin{equation} V_{2}=\eta ^{T}\mathbf{B}\eta \end{equation}

where $\eta =[|s|^{\frac{3}{2}}\text{sign}(s),w,\tilde{W}]$ , $\mathbf{B}$ is a symmetric positive definite matrix, defined as

\begin{equation*} \mathbf {B}=\frac {1}{2} \left [ \begin {matrix} 4k_{2}+k_{1}^{2} & \quad k_{1}k_{3} & \quad -k_{1} \\[4pt] k_{1}k_{3} & \quad 2k_{4}+k_{3}^{2} & \quad -k_{3} \\[4pt] -k_{1} & \quad -k_{3} & \quad 2 \end {matrix} \right ] \end{equation*}

The derivative of equation (19) is obtained

(20) \begin{align} \dot{V}_{2}=&2\eta ^{T}\mathbf{B}\dot{\eta }\nonumber\\ =&-6k_{1}\left(k_{2}+k_{1}^{2}\right)|s|^{\frac{5}{2}}-6k_{3}\left(k_{2}+k_{1}^{2}\right)|s|^{\frac{7}{2}}-6\left(k_{2}+k_{1}^{2}\right)|s|^{2}w+6\left(k_{2}+k_{1}^{2}\right)|s|^{2}\nonumber\\ &\times \tilde{W}^{T}h(x)-6\left(k_{2}+k_{1}^{2}\right)|s|^{2}\varepsilon -\frac{3}{2}k_{1}^{2}k_{3}|s|w-\frac{3}{2}k_{1}k_{3}^{2}|s|^{2}w+\frac{3}{2}k_{1}k_{3}|s|^{\frac{1}{2}}w^{2}\nonumber\\ &+\frac{3}{2}k_{1}k_{3}|s|^{\frac{1}{2}}w\tilde{W}^{T}h(x)-\frac{3}{2}k_{1}k_{3}|s|^{\frac{1}{2}}w\varepsilon +\frac{3}{2}k_{1}^{2}\tilde{W}|s|+\frac{3}{2}k_{1}k_{3}\tilde{W}|s|^{2}-\frac{3}{2}k_{1}\\ &\times |s|^{\frac{1}{2}}w\tilde{W}-\frac{3}{2}k_{1}|s|^{\frac{1}{2}}\tilde{W}\tilde{W}^{T}h(x)+\frac{3}{2}k_{1}|s|^{\frac{1}{2}}\tilde{W}\varepsilon +k_{1}k_{2}k_{3}|s|^{2}-k_{1}k_{3}k_{4}|s|^{3}\nonumber\\ &+k_{2}(2k_{4}+k_{3}^{2})|s|^{\frac{1}{2}}w-k_{4}(2k_{4}+k_{3}^{2})|s|^{\frac{3}{2}}w-k_{2}k_{3}\tilde{W}|s|^{\frac{1}{2}}+k_{3}k_{4}\tilde{W}|s|^{\frac{3}{2}}\nonumber\\ &-k_{1}|s|^{\frac{3}{2}}\dot{\tilde{W}}-k_{3}w\dot{\tilde{W}}+2\tilde{W}\dot{\tilde{W}}\nonumber\\ \leq &-|s|^{-\frac{5}{2}}(\eta ^{T}\mathbf{T_{5}}\eta )(\eta ^{T}\mathbf{T_{5}}\eta )^{T}-|s|^{-\frac{1}{2}}\eta ^{T}\mathbf{T_{6}}\eta -|s|^{\frac{1}{2}}\eta ^{T}\mathbf{T_{7}}\eta -\eta ^{T}\mathbf{T_{8}}\eta +\eta _{0}\nonumber \end{align}

where $\varepsilon =0$ is negligible, $\eta _{0}\gt 0$ , $\dot{\tilde{W}}=-3\left(k_{2}+k_{1}^{2}\right)|s|^{2}h(x)$ , and $\mathbf{T_{5}}$ , $\mathbf{T_{6}}$ , $\mathbf{T_{7}}$ , $\mathbf{T_{8}}$ are symmetric positive definite matrices.

Since $|s|^{\frac{3}{2}}\leq \|\eta \|_{2}\leq \left(\dfrac{V_{2}}{\lambda _{\min }(\mathbf{B})}\right)^{\frac{1}{2}}$ , $|s|^{\frac{1}{2}}\leq \|\eta \|^{\frac{1}{3}}_{2}\leq \left(\dfrac{V_{2}}{\lambda _{\min }(\mathbf{B})}\right)^{\frac{1}{6}}$ , $|s|^{\frac{5}{2}}\leq \|\eta \|^{\frac{5}{3}}_{2}\leq \left(\dfrac{V_{2}}{\lambda _{\min }(\mathbf{B})}\right)^{\frac{5}{6}}$ . Therefore, it follows that

(21) \begin{align} \dot{V}_{2}\leq &-\frac{(\lambda _{\min }(\mathbf{B}))^{\frac{5}{6}}}{V_{2}^{\frac{5}{6}}}(\lambda _{\min }(\mathbf{T_{5}}))^{2}\frac{V_{2}^{2}}{(\lambda _{\max }(\mathbf{B}))^{2}}-\frac{(\lambda _{\min }(\mathbf{B}))^{\frac{1}{6}}}{V_{2}^{\frac{1}{6}}}\lambda _{\min }(\mathbf{T_{6}})\frac{V_{2}}{\lambda _{\max }(\mathbf{B})}+\eta _{0}\nonumber\\[5pt] &=-\frac{(\lambda _{\min }(\mathbf{B}))^{\frac{5}{6}}(\lambda _{\min }(\mathbf{T_{5}}))^{2}}{(\lambda _{\max }(\mathbf{B}))^{2}}V_{2}^{\frac{7}{6}}-\frac{(\lambda _{\min }(\mathbf{B}))^{\frac{1}{6}}\lambda _{\min }(\mathbf{T_{6}})}{\lambda _{\max }(\mathbf{B})}V_{2}^{\frac{5}{6}}+\eta _{0} \end{align}

According to Lemma2, the sliding mode surface $s$ and $\dot{s}$ can converge to the origin in the real fixed-time $\bar{T}$ , that is, the system (5) can achieve stability in the real fixed-time $\bar{T}$ , and the estimated convergence time is

(22) \begin{equation} \bar{T}\leq \frac{6(\lambda _{\max }(\mathbf{B}))^{2}}{\lambda _{\min }(\mathbf{B}))^{\frac{5}{6}}(\lambda _{\min }(\mathbf{T_{5}}))^{2}\bar{\phi }}+\frac{6\lambda _{\max }(\mathbf{B})}{(\lambda _{\min }(\mathbf{B}))^{\frac{1}{6}}\lambda _{\min }(\mathbf{T_{6}})\bar{\phi }} \end{equation}

And the system states converge to the following compact set $\Omega$ within the real fixed-time $\bar{T}$ .

\begin{align*} \Omega \in \left\{x|V_{2}(x)\leq \min \left\{\left(\frac{\eta _{0}(\lambda _{\max }(\mathbf{B}))^{2}}{(1-\bar{\phi })(\lambda _{\min }(\mathbf{B}))^{\frac{5}{6}}(\lambda _{\min }(\mathbf{T}_{5}))^{2}}\right)^{\frac{6}{7}},\left(\frac{\eta _{0}\lambda _{\max }(\mathbf{B})}{(1-\bar{\phi })(\lambda _{\min }(\mathbf{B}))^{\frac{1}{6}}\lambda _{\min }(\mathbf{T}_{6})}\right)^{\frac{6}{5}}\right\}\right\} \end{align*}

Figure 3.

Structure of hybrid mechanism system (single side).

4. Simulations

To verify the effectiveness of the proposed methods, the proposed method FTSTSMC is compared with FTSMC, and the proposed method IFTSTSMC is compared with FTSTSMC. Simulation comparisons are conducted with Matlab/Simulink.

The unilateral structure of the bilaterally symmetrical hybrid mechanism system is shown in Figure 3. The forward and backward movement of the hybrid mechanism system is driven by the walking wheels. The up-down and turnover movement is achieved by two symmetrical parallel mechanisms. The rack of the lifting and turning mechanism (LTM) is the motion part of the walking mechanism. Two sets of LTM are connected via a connecting rod and a fixed mount used to place the car body. The LTM consists of a lifting part fixed on the walking mechanism and a turning part fixed on the connecting rod. Via the combination of the walking mechanism and the LTM, the car body is driven to realize the forward and backward, up and down, turnover, and compound movement.

Figure 4.

Tracking trajectory curves of sliders and driving wheel.

Figure 5.

Tracking error curves of sliders and driving wheel.

In FTSTSMC, the parameters are respectively designed as $k_{1}=1.80$ , $k_{2}=1.90$ , $k_{3}=18.70$ , $k_{4}=6.40$ , $\rho =2.00$ , $\alpha =0.66$ , $\beta =2.00$ . The simulation results of FTSTSMC and FTSMC are as follows. Since the hybrid mechanism system has the characteristics of bilateral symmetry, the tracking trajectory curves and tracking error curves of sliders 1, 2 and driving wheel 1 are shown in Figure 4 and Figure 5, respectively. As seen in the partial block of Figure 4, the trajectory tracking curves of slider 1 and slider 2 for the hybrid mechanism system show the abrupt changes caused by the fluid resistance when entering into the tank. The values of the abrupt changes are $0.3916 m$ and $0.3913 m$ , respectively, which affect the tracking accuracy and the control performance of the system. Compared to FTSMC, due to the fixed-time control with faster convergence performance and better robustness, FTSTSMC can reduce the abrupt changes caused when the hybrid mechanism system enters into the tank. In addition, the root mean square error (RMSE) and the maximum absolute steady-state error (MASSE) of the active joints are shown in Table I; it can be seen that the RMSE of the active joints for FTSTSMC is about $9.93 \%$ , $10.32 \%$ , and $11.03 \%$ reduced than FTSMC, respectively. And the MASSE of the active joints for FTSTSMC is about $83.47 \%$ , $63.14 \%$ , and $4.81 \%$ reduced than FTSMC, respectively. Therefore, FTSTSMC has better tracking accuracy.

Table I.

The RMSE and MASSE of the active joints for simulation experimental results.

Figure 6.

Tracking trajectory curves of sliders and driving wheel.

Figure 7.

Tracking error curves of sliders and driving wheel.

From the design viewpoint of the FTSTSMC, compared with literature [Reference Wang, Jahanshahi, Volos, Bekiros, Yusuf, Agarwal and Aly33] (FTSTSMC-1), the method designed in this paper (FTSTSMC) has variable fixed-time performance parameters, and FTSTSMC achieves fixed-time convergence in both reaching stage and sliding stage of the SMC. Figure 6 and Figure 7 show the comparison results of the above two simulation experiments. Table I shows the RMSE and MASSE of the active joints for the controllers. It shows that the RMSE of the active joints for FTSTSMC is about $0.09 \%$ , $0.12 \%$ , and $1.98 \%$ reduced than FTSTSMC-1, respectively. The MASSE of the active joints for FTSTSMC is about $65.66 \%$ , $52.45 \%$ , and $40.87 \%$ reduced than FTSTSMC-1, respectively. Therefore, the FTSTSMC designed in this paper has better tracking effect.

However, it can be seen from Figure 4 and Figure 5 that the tracking effect of FTSTSCM has not been greatly improved and local chattering still exists. Therefore, IFTSTSCM is proposed in this paper. In the IFTSTSMC, the parameters are, respectively, designed as $k_{1}=3.00$ , $k_{2}=2.30$ , $k_{3}=56.69$ , $k_{4}=229.54$ , $\alpha =0.52$ , $\beta =1.01$ . The simulation results are shown in Figure 8 and Figure 9. It can be seen from the partial block of Figure 8 that the trajectory tracking curves for IFTSTSCM converges about $57.47 \%$ faster than FTSTSMC. Table I shows that the RMSE of the active joints for IFTSTSCM is about $33.21 \%$ , $32.88 \%$ , and $40.16 \%$ reduced than FTSTSMC, respectively. And the MASSE of the active joints for IFTSTSCM is about $69.54 \%$ , $78.60 \%$ , and $1.34 \%$ reduced than FTSTSMC, respectively. Therefore, compared with FTSTSMC, IFTSTSMC has better tracking effect. When the uncertainties of the hybrid mechanism change in a wide range, the existence condition of SMC is destroyed, resulting in the failure of the robustness for the FTSTSMC. IFTSTSMC introduces RBF neural network, which deals with nonlinear functions containing the inherent dynamics and uncertainties of the system. In order to break through the uncertainties range of FTSTSCM, the uncertainties are amplified by 10 times and 100 times for simulation experiments. Figure 10 and Figure 11 show the tracking curves and error tracking curves of the system after the uncertainties amplification. It can be seen that FTSTSCM can achieve system stability in response to uncertainties within a certain range. However, when faced with a wide range of uncertainties, it may lead to control failures. IFTSTSMC has a good adjustment effect on a wide range of uncertainties. It can not only make the system converge in real fixed-time but also improve the adaptability of the hybrid mechanism system to the uncertainties with a wide range of changes.

Figure 8.

Tracking trajectory curves of sliders and driving wheel.

Figure 9.

Tracking error curves of sliders and driving wheel.

Figure 10.

Tracking trajectory curves of sliders and driving wheel with RBF neural network.

Figure 11.

Tracking error curves of sliders and driving wheel with RBF neural network.

Figure 12.

The prototype of hybrid mechanism system.

Figure 13.

Time-series images of the hybrid mechanism pose in operation.

Figure 14.

The pose of the hybrid mechanism system in operation with the electrophoretic tank.

5. Prototype experiment

The prototype of hybrid mechanism system is shown in Figure 12. Figure 13 and Figure 14 show the time-series images of the hybrid mechanism pose in operation and the pose of the hybrid mechanism system in operation with the electrophoretic tank, respectively. The physical parameters of the prototype include the length of the connecting rod is $0.495m$ , the length of the connecting rod is $0.72m$ , the lead of the lead screw is $0.01m$ , the radius of the active driving wheel is $0.025m$ , and the radius of the driven driving wheel is $0.075m$ . The control system is divided into three layers: upper computer (PC), lower computer (UMAC), and servo drive system. Among them, the PC uses Intel Core i7-4790 3.60 GHz processor to complete the functions of the human-computer interaction, code initialization and compilation, and condition monitoring. The UMAC mainly includes a TURBO PMAC2 OPT-5C0 CPU main board, tow AC-24E2A axis board, one AC-65E I/O board, and one AC-E1 power board. It is used to accomplish data calculation, digital to analog conversion, and motion control. The servo drive system is composed of MRJ4-70A(30W), MR-J4-100A(30W) servo drivers, and HG-KR73BJ(750W), HG-SR102BJ(30W) AC servo motors. The reducer is selected as EVH-115-20-K7-28HF24, and its reduction ratio is 1:20. To realize the feedback control loop of the system, the 22-bit (4194304pulses/rev) high-resolution absolute position encoder is selected. Based on the above setup, an experimental platform of the hybrid mechanism system for the automobile electro-coating conveying is constructed. The control algorithm is implemented as follows: opening the Pewin32 software to enable the PC to communicate with the UMAC; the AC servo motor is closed looped via Pewin32 software instruction and the motion trajectory program is downloaded to UMAC; opening the PC interface via VC++ 6.0(SP6) and downloading the control algorithm program into the UMAC; and running the designed control algorithm program.

Figure 15.

Experimental results of tracking errors for FTSTSCM and FTSCM.

Figure 16.

Experimental results of tracking errors for FTSTSCM-1 and FTSTSCM.

Three contrast motion control experiments are carried out on the prototype. The control algorithm is realized by establishing the connection between PC and UMAC to open the closed loop of the system. Download the required tracks and user-defined motion programs precompiled by PeWin32pro software to the UMAC buffer. The results of the comparison experiments between FTSTSCM and FTSCM, FTSTSCM-1 and FTSTSCM, and FTSTSCM and IFTSTSCM are shown in Figure 15, Figure 16, and Figure 17, respectively. And the RMSE and MASSE of the comparison experiments are shown in Table II, where the RMSE of the active joints for FTSTSCM is about $59.26 \%$ , $59.28 \%$ , and $59.87 \%$ reduced than FTSMC; the MASSE of the active joints for FTSTSCM is about $60.55 \%$ , $62.25 \%$ , and $60.02 \%$ reduced than FTSMC; the RMSE of the active joints for FTSTSCM is about $32.50 \%$ , $32.49 \%$ , and $33.29 \%$ reduced than FTSTSCM-1; the MASSE of the active joints for FTSTSCM is about $33.19 \%$ , $34.51 \%$ , and $34.69 \%$ reduced than FTSTSCM-1; the RMSE of the active joints for IFTSTSCM is about $49.25 \%$ , $49.37 \%$ , and $49.89 \%$ reduced than FTSTSCM; and the MASSE of the active joints for IFTSTSCM is about $51.46 \%$ , $49.81 \%$ , and $50.65 \%$ reduced than FTSTSCM. Therefore, in general, the proposed control method can obtain better tracking control performance. As a nonlinear control method, IFTSTSMC can effectively deal with the uncertainties of the system, so it is excellent in enhancing the robustness of hybrid mechanism system to the uncertainties with a wide range of changes. In addition to the uncertain hybrid mechanisms, IFTSTSMC can be used in mechanical systems, power systems, aerospace systems, industrial production systems, and other practical systems. By designing the sliding mode surface and control law reasonably, IFTSTSMC can effectively achieve the stability and convergence of the system. In general, the method proposed in this paper is not only beneficial to the hybrid mechanism, but also has universal applicability to other practical systems. Therefore, this method is of great significance in engineering practice.

Table II.

The RMSE and MASSE of the active joints for prototype experimental results.

Figure 17.

Experimental results of tracking errors for FTSTSCM and IFTSTSCM.