## 3 Adaptive Robust Control Design
### Fuzzy Dynamic Modeling of Delta Robot  
A 3-DOF Delta robot, with three common RRPaR (R denotes the rotation pair, Pa is the parallelogram configuration) topology mechanisms connecting the fixed platform and the moving platform, is taken into consideration. The sub-chains with active arms and passive arms are located on the three sides of an equilateral triangle, which are presented in Figure 1a. Assign a base frame \(O\{X,Y,Z\}\) at the geometric center of the fixed platform in Figure 1b, and denote the parameters in the Table 1\((i=1,2,3)\):  
Select \(\delta=[\delta_{1},\delta_{2},\delta_{3}]^{\mathrm{T}}\) as the generalized coordinate vector. Based on the dynamics constructed in [31], the motion equation of Delta robot with uncertainty can be formulated as  
\[M(\delta(t),\varsigma(t),t)\ddot{\delta}(t) +C(\delta(t),\dot{\delta}(t),\varsigma(t),t)\dot{\delta}(t)+G( \delta(t),\varsigma(t),t)\] \[+F_{\varsigma}(\delta(t),\dot{\delta}(t),\varsigma(t),t)+M_{f}( \delta(t),\dot{\delta}(t),\varsigma(t),t)=\tau(t), \tag{3}\]  
where \(t\in\mathbf{R}\) is time, \(\dot{\delta}(t)\in\mathbf{R}^{n}\) is the velocity, \(\ddot{\delta}(t)\in\mathbf{R}^{n}\) is the acceleration. \(\varsigma(t)\in\Sigma\subset\mathbf{R}^{p}\) is the bounded uncertain parameters, \(\Sigma\) represents the possible bound of \(\varsigma\), which is compact but unknown. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Notation** & **Units** \\ \hline Angle of the \(i\)-th active joint & \(\delta_{i}\) & rad \\ Length of the \(i\)-th active arm & \(l_{ai}\) & m \\ Length of the \(i\)-th passive arm & \(l_{pi}\) & m \\ Radius of the fixed platform & \(r_{f}\) & m \\ Radius of the moving platform & \(r_{m}\) & m \\ Mass of the \(i\)-th active arm & \(m_{ai}\) & kg \\ Mass of the \(i\)-th passive arm & \(m_{pi}\) & kg \\ Mass of the moving platform & \(m_{o^{\prime}}\) & kg \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of Delta robot. Figure 1: (a) The 3-DOF Delta robot; (b) Three-dimensional model of Delta robot. \(M(\delta(t),\varsigma(t),t)\) is the inertial matrix, \(C(\delta(t),\dot{\delta}(t),\varsigma(t),t)\) is the Coriolis/centrifugal force, \(G(\delta(t),\varsigma(t),t)\) is the gravitation force, \(F_{e}(\delta(t),\dot{\delta}(t),\varsigma(t),t)\) stands for the external disturbances, \(M_{f}(\delta(t),\dot{\delta}(t),\varsigma(t),t)\) represents the joint frictional moment, and \(\tau(t)\in\mathbf{R}^{n}\) is the control torques. For simplicity, the argument will be omitted without ambiguity. The detail expressions of \(M(\cdot)\), \(C(\cdot)\) and \(G(\cdot)\) are shown in Appendix A. For a Delta robot, the actuators are installed in the active joints, which are generally composed of drive motors and harmonic reducers. The addition of the harmonic reducers not only leads to the flexibility of the joints, but also increases the complexity of internal joints friction. Therefore, in this paper, the friction of the active joints is considered. The frictional moment during the transmission process of the harmonic reducer is mainly related to the joint speed and direction. A Stribeck friction model, which conforms to the friction characteristics in the transmission process, is used to model the joints friction of Delta robot, and the model is given by [32]  
\[M_{f} =\left[F_{c}+(F_{s}-F_{c})e^{-(\delta/\delta_{s})^{2}}\right] \operatorname{sgn}(\dot{\delta})r_{a}+\mu_{v}\dot{\delta}r_{a},\] \[F_{c} =\mu_{d}F_{n},\quad F_{s}=\mu_{s}F_{n}, \tag{4}\]  
where \(F_{c}\) is Coulomb friction, \(F_{s}\) is the stiction friction, \(\delta_{s}\) is Stribeck velocity, \(\mu_{v}\) is the coefficient of the viscous friction, \(\mu_{d}\) is the coefficient of Coulomb friction, \(\mu_{s}\) is the coefficient of static friction, \(F_{n}\) is the normal force, \(r_{a}\) is the friction arm of the active joints. **Remark 2**. : _The dimension of the uncertain factors \(\varsigma\in\Sigma\subset\mathbf{R}^{p}\) is determined by the number of the uncertain terms in Delta robot. Each of them is compact and bounded.