## 3 Controller Design  
### The Design of CE-SOSMC  
The synchronization error \(\mathbf{\varepsilon}_{g}(t)\) of each active joint is defined by (4):  
\[\mathbf{\varepsilon}_{g}(t)=\mathbf{H}_{1}\mathbf{e}_{x}(t) \tag{7}\]  
in which \(\mathbf{e}_{x}(t)=[e_{x1}(t),e_{x2}(t),e_{x3}(t),e_{x4}(t),e_{\varphi 1}(t),e_{ \varphi 2}(t)]^{\mathrm{T}}\) is the tracking errors of UBSHR's active joints and \(\mathbf{H}_{1}\) is the transfer matrix. According to (6), the synchronization error \(\mathbf{\varepsilon}(t)\) of UBSHR's end-effector could be defined as follows:  
\[\mathbf{\varepsilon}(t)=\mathbf{e}_{1}(t)-\mathbf{e}_{2}(t)=\mathbf{H}_{2}\mathbf{e}_{x}(t) \tag{8}\]  
where \(\mathbf{H}_{2}= \begin{bmatrix}\frac{x_{2}-x_{1}}{4x_{1}}&\frac{x_{1}-x_{2}}{4x_{1 }}&\frac{x_{3}-x_{4}}{4x_{2}}&\frac{x_{4}-x_{3}}{4x_{2}}&0&0\\ 0&0&0&0&\frac{x_{4}}{r_{d}}&-\frac{r_{d}}{r_{d}}\end{bmatrix} \), \(x_{i}(i=1,2,3,4)\) is the \(i_{th}\) slider position for the hybrid robot in the \(X\)-axis direction, \(z_{i}(i=1,2)\) denotes the position of the hybrid robot end-effector in the \(Z\)-axis, and \(r_{a}\) and \(r_{d}\) are the radius of the driving wheel and driven wheel, respectively. In order to achieve precise tracking control for the hybrid robot and to ensure synchronization performance of the active joints and end-effector, by further combining \(\mathbf{e}(t)\) with \(\mathbf{\varepsilon}(t)\) of UBSHR's end-effector, a novel composite error \(\mathbf{e}^{\alpha\prime}(t)\) is designed as follows:  
\[\mathbf{e}^{\alpha\prime}(t)=\mathbf{e}(t)+\mathbf{\lambda}\int_{0}^{t}\mathbf{\varepsilon} (t)dt \tag{9}\]  
in which \(\mathbf{\lambda}=\text{diag}(\lambda_{1},\lambda_{2})\). Figure 2: The control schematic diagram. On the basis of the novel composite error \(\mathbf{\varepsilon}^{\mathbf{c}\varepsilon}(t)\), the sliding surface \(\mathbf{S}=[S_{1},S_{2}]^{\rm T}\) of CE-SOSMC is constructed as follows:  
\[\mathbf{S}=\mathbf{\varepsilon}^{\mathbf{c}\varepsilon}(t)+\mathbf{\alpha}\mathbf{\varepsilon}^{ \mathbf{c}\varepsilon}(t)^{\tau}+\mathbf{\varepsilon}\mathbf{\varepsilon}^{\mathbf{c} \varepsilon}(t)^{p/q} \tag{10}\]  
where \(\mathbf{\alpha}=\text{diag}(\alpha_{1},\alpha_{2})\), \(\mathbf{\varepsilon}=\text{diag}(\varepsilon_{1},\varepsilon_{2})\); \(\alpha_{i}\in R^{+}\), \(\varepsilon_{i}\in R^{+}\); \(r>1\), \(q<p<2q\); and \(p.q\) are odd numbers. The non-singularity of \(\mathbf{S}=[S_{1},S_{2}]^{\rm T}\) is guaranteed in theory due to \(1<p/q<2\). Then, the CE-SOSMC law for a hybrid robot is  
\[\mathbf{\tau}= \mathbf{M}(\mathbf{q})[\mathbf{\tilde{q}}_{d}(t)+\mathbf{\sigma}^{-1}(\mathbf{\lambda }|S|^{1/2}\text{sgn}(\mathbf{S})-\mathbf{u}+\mathbf{\varepsilon}\mathbf{\varepsilon}^{\mathbf{c} \varepsilon}(t))]+\mathbf{C}(\mathbf{q},\mathbf{q})\mathbf{\dot{q}}+\mathbf{G}(\mathbf{q})+\mathbf{F}(\mathbf{ \dot{q}}) \tag{11}\] \[\mathbf{\dot{u}}= -\mathbf{W}\text{sgn}(\mathbf{S})\]  
in which \(\mathbf{W}=\text{diag}(W_{1},W_{2})\); \(\mathbf{\lambda}=\text{diag}(\lambda_{1},\lambda_{2})\); \(\mathbf{\xi}=\text{diag}(1+r\alpha_{1}\mathbf{\varepsilon}_{1}^{\mathbf{c}\varepsilon}(t) ^{(r-1)}\), \(1+r\alpha_{2}\mathbf{\varepsilon}_{2}^{\mathbf{c}\varepsilon}(t)^{(r-1)})\); and \(\mathbf{\sigma}=\text{diag}(\sigma_{1},\sigma_{2})=\text{diag}(\frac{p}{q}\mathbf{ \varepsilon}_{1}\mathbf{\varepsilon}_{1}^{\mathbf{c}\varepsilon}(t)^{(p-q)/q},\frac{p }{q}\mathbf{\varepsilon}_{2}\mathbf{\varepsilon}_{2}^{\mathbf{c}\varepsilon}(t)^{(p-q)/q})\). The first result is provided below. Theorem 1: _For the nonlinear system (1), if the CE-SOSMC algorithm presented in (11) with \(W_{1}>0\), \(W_{2}>0\); \(\lambda_{1}>0\), \(\lambda_{2}>0\); and \(sgn(\mathbf{S})=(sgn(S_{1}),sgn(S_{2}))^{\rm T}\) is adopted, the stabilization of sliding variable \(\mathbf{S}\) will be guaranteed in a finite time.