## 3 Sliding Mode Controller Design  
### Friction Compensation  
In order to research dynamical characteristic of the end-effector during normal orientation adjusting, considering that feed unit and compressing unit maintain resting state during the normal adjusting process, Equation (23) is rewritten as:  
\[\mathbf{D}(\mathbf{\Theta})\ddot{\mathbf{\Theta}}+\mathbf{H}\Big{(}\mathbf{ \Theta},\dot{\mathbf{\Theta}}\Big{)}+\mathbf{G}(\mathbf{\Theta})=\boldsymbol{\tau} \tag{25}\]  
where \(\mathbf{\Theta}=\left[\begin{array}{cc}\theta_{1}&\theta_{2}\end{array} \right]^{T}\) and \(\boldsymbol{\tau}=\left[\begin{array}{cc}\tau_{1}&\tau_{2}\end{array} \right]^{T}\).  
Equation (25) is an ideal dynamical model, because it didn't take into account the effects of friction torque and disturbing torque on the joints. Friction is a phenomenon with obvious nonlinearity that remains ubiquitous in mechanical systems. It always causes untoward consequences such as system steady-state error, limit cycle, static-slip and oscillation. Therefore, the friction torque and disturbing torque should be considered to obtain comparative control quality. The dynamical model needs to be modified as:  
\[\mathbf{D}(\mathbf{\Theta})\ddot{\mathbf{\Theta}}+\mathbf{H}\Big{(}\mathbf{ \Theta},\dot{\mathbf{\Theta}}\Big{)}+\mathbf{G}(\mathbf{\Theta})+\mathbf{F} \Big{(}\mathbf{\Theta},\dot{\mathbf{\Theta}},\ddot{\mathbf{\Theta}}\Big{)}+ \boldsymbol{\tau_{\mathrm{d}}}=\boldsymbol{\tau} \tag{26}\]  
where \(\mathbf{F}\Big{(}\mathbf{\Theta},\dot{\mathbf{\Theta}},\ddot{\mathbf{\Theta}} \Big{)}\) denotes friction torque while \(\boldsymbol{\tau_{\mathrm{d}}}\) denotes disturbing torque.  
Therefore, in order to compensate the friction torque, it is necessary to establish a friction model to estimate the friction torque. Stribeck model is a classical friction model which has components that are either linear in velocity or constant and describes the mechanical characteristics of various friction stages. The resulting friction force \(F_{f}\) can be expressed as [20]:  
\[F_{f}=\left\{\begin{array}{ll}F(v)&\text{if }v\
eq 0\\ F_{e}&\text{if }v=0\text{ and }|F_{e}|<F_{s}\\ F_{s}\text{sgn}(F_{e})&\text{other}\end{array}\right. \tag{27}\]  
\[F(v)=F_{c}+(F_{s}-F_{c})e^{-|v|/v_{g}|^{2}}+F_{v}v \tag{28}\]  
where \(v\) is motion velocity; \(v_{s}\) is named Stribeck velocity; \(F_{e}\) denotes driving force; \(F_{s}\) denotes maximum static friction force; \(F_{c}\) denotes Coulomb friction force; \(F_{v}\) denotes viscous frictional torque coefficient. It is easy to identify the expression for \(F_{f}\) by measuring friction force in condition of uniform motion.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameters** & \(\boldsymbol{i=1}\) & \(\boldsymbol{i=2}\) & \(\boldsymbol{i=3}\) \\ \hline \(I_{xxi}\) (m\({}^{4}\)) & 0.03 & 0.04 & 0.03 \\ \(I_{xyi}\) (m\({}^{4}\)) & 0 & 0.01 & 0 \\ \(I_{zzi}\) (m\({}^{4}\)) & 0 & \(-\)0.01 & 0 \\ \(I_{yzi}\) (m\({}^{4}\)) & 0 & 0.01 & 0 \\ \(I_{yzi}\) (m\({}^{4}\)) & 0.09 & 0.11 & 0.04 \\ \(I_{yzi}\) (m\({}^{4}\)) & 0 & 0 & 0 \\ \(I_{zxi}\) (m\({}^{4}\)) & 0 & \(-\)0.11 & 0 \\ \(I_{zxi}\) (m\({}^{4}\)) & 0 & 0 & 0 \\ \(I_{zzi}\) (m\({}^{4}\)) & 0.06 & 0.11 & 0.01 \\ \(n_{tt}\) (kg) & 5.8 & 7.6 & 8.6 \\ \(c_{xi}\) (m) & 0.05 & 0.02 & 0 \\ \(c_{yi}\) (m) & 0 & 0.02 & 0 \\ \(c_{zi}\) (m) & 0 & \(-\)0.01 & 0.14 \\ \hline \end{tabular}
\end{table}
Table 1: Joint parameters of the end-effector.