## 4 Integrated Fixed Time Sliding Mode Control for Motion and Vibration  
### Fixed Time Sliding Mode Control  
For the fully driven slow rigid subsystem Equation (18), it is considered that the complete information of the model is difficult to obtain for practical applications, but a nominal model can usually be obtained. Let the nominal models of system Equation (18) be \(M_{0}\) and \(\mathbf{C}_{0}\), \(\Delta\mathbf{M}=\mathbf{M}_{0}-\mathbf{M}\), \(\Delta\mathbf{C}=\mathbf{C}_{0}-\mathbf{C}\). Then, Equation (18) can be rewritten as:  
\[M_{0}\ddot{\mathbf{q}}+\mathbf{C}_{0}\dot{\mathbf{q}}=\mathbf{\tau}_{\rm oj}+\varepsilon, \tag{20}\]  
where \(\varepsilon=\Delta\mathbf{M}\ddot{\mathbf{q}}+\Delta\mathbf{C}\dot{\mathbf{q}}-\mathbf{c}_{1}+\mathbf{d}\). Taking the state variable \(\mathbf{x}_{1}=\mathbf{q},\mathbf{x}_{2}=\dot{\mathbf{q}}\). From Equation (20), we have:  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}_{1}=\mathbf{x}_{2}\\ \dot{\mathbf{x}}_{2}=\mathbf{\Gamma}+\mathbf{\Phi}+\mathbf{h}\end{array} \right., \tag{21}\]  
where \(\mathbf{\Gamma}=M_{0}^{-1}\mathbf{\tau}_{\rm oj}=[\Gamma_{1},\Gamma_{2},\Gamma_{3}]^{ \sf T}\), \(\Phi=M_{0}^{-1}\mathbf{\varepsilon}=[\mathbf{\Phi}_{1},\mathbf{\Phi}_{2},\mathbf{\Phi}_{3}]^{ \sf T}\), \(\mathbf{h}=-M_{0}^{-1}\mathbf{C}_{0}\dot{\mathbf{q}}=[\mathbf{h}_{1},\mathbf{h}_{2},\mathbf{h}_{3}]^{ \sf T}\). Consider the nonlinear system as follows:  
\[\dot{\mathbf{x}}(\mathbf{t})=\mathbf{f}(\mathbf{t},\mathbf{x}),\,\mathbf{x}(\mathbf{t}_{0})=\mathbf{x}_{0}, \tag{22}\]  
where \(\mathbf{x}(t)\in\mathbf{R}^{n}\) is the system state variable, \(\mathbf{f}(t,\mathbf{x})\) is a smooth nonlinear function, and the origin is assumed to be the equilibrium point of the system (22). **Definition 1**. : _Suppose that system (22) is globally asymptotically stable if there exists a finite convergence time \(t_{\rm fc}(\mathbf{x}_{0})\), which holds for all \(t\geq t_{\rm fc}\) satisfying \(\mathbf{x}(t)=0\), then system (22) is globally finite time stable [21]._  
**Definition 2**. : _The system (22) is globally fixed-time stable if it is globally finite-time stable while there exists a definite upper bound on the convergence time and the value of the upper bound is independent of the system state variables [22]._  
The following lemma gives a sufficient condition for the fixed time convergence of system (22). **Lemma 1**. : _If there is a continuous radial bounded positive definite function\(V\), it satisfies the relationship as follows:_  
\[\dot{V}\leq-\kappa_{1}V^{p_{1}}(\mathbf{x})-\kappa_{2}V^{p_{2}}(\mathbf{x}), \tag{23}\]  
_where \(\kappa_{1},\ \ \kappa_{2}>0\), \(p_{1}>1\), \(0<p_{2}<1\)._  
Then, the system (22) is globally fixed and time stable, and its convergence time satisfies the equation as follows [23]:  
\[t_{\rm fc}\leq t_{\rm max}:=\frac{1}{\kappa_{1}(p_{1}-1)}+\frac{1}{\kappa_{2}(1 -p_{2})}. \tag{24}\]  
For the dynamic model described in equation (21), the purpose of robot control is to make the base attitude angle and joint angle vector \(\mathbf{q}\) track the desired trajectory \(\mathbf{q}_{\rm d}=[q_{\rm d0},q_{\rm d1},q_{\rm d2}]^{T}\). The tracking error \(\mathbf{e}=\mathbf{q}-\mathbf{q}_{\rm d}\) is defined, \(\mathbf{e}=[e_{0},e_{1},e_{2}]^{\rm T}\), where \(e_{0}\) is the base attitude tracking error, and \(e_{1}\) and \(e_{2}\) are the trajectory tracking errors of 1 joint and 2 joint, respectively. The switching function is selected as follows:  
\[s_{i}=\dot{e}_{i}+\gamma_{1}\dot{e}_{i}^{a_{1}}+\beta_{1}\dot{e}_{i}^{a_{2}}, \ i=0,1,2, \tag{25}\]  
where \(s_{0}\) is the switching function of the base, \(s_{1}\) and \(s_{2}\) are the switching functions of 1 and 2 joints, respectively. \(a_{1}=m_{1}/n_{1}\), \(a_{2}=p_{1}/q_{1}\gamma_{1}\) and \(\beta_{1}\) are normal numbers, \(m_{1},n_{1},p_{1},q_{1}\) are positive odd number and satisfies \(m_{1}>n_{1}\) and \(p_{1}<q_{1}<2p_{1}\).