## 2 Problem Formulations  
### Dynamic Model of Parallel Robot Manipulator  
The general dynamic equation for a 3-DOF parallel robotic manipulator is depicted as in References [55; 56]:  
\[M_{\alpha}\ddot{\theta}_{\alpha}+Q_{\alpha}\dot{\theta}_{\alpha}+f_{\alpha}+ \delta_{\alpha}=\tau_{\alpha} \tag{1}\]  
where \(\theta_{\alpha}=\left[\begin{array}{cc}\theta_{\alpha 1},&\theta_{\alpha 2},& \theta_{\alpha 3}\end{array}\right]^{T}\), \(\dot{\theta}_{\alpha}=\left[\begin{array}{cc}\dot{\theta}_{\alpha 1},&\dot{ \theta}_{\alpha 2},&\dot{\theta}_{\alpha 3}\end{array}\right]^{T}\), and \(\ddot{\theta}_{\alpha}=\left[\begin{array}{cc}\ddot{\theta}_{\alpha 1},& \ddot{\theta}_{\alpha 2},&\ddot{\theta}_{\alpha 3}\end{array}\right]^{T}\) are the position vector, velocity vector, and acceleration vector at each active joint, respectively. \(M_{\alpha}=\dot{M}_{\alpha}+\Delta M_{\alpha}\in R^{3\times 3}\) represents the real inertia matrix and \(Q_{\alpha}=\dot{Q}_{\alpha}+\Delta Q_{\alpha}\in R^{3\times 3}\) represents the real Coriolis and centrifugal force matrix. \(\dot{M}_{\alpha}\in R^{3\times 3}\) represents the estimated inertia matrix and \(\dot{Q}_{\alpha}\in R^{3\times 3}\) represents the estimated Coriolis and centrifugal force matrix. \(f_{\alpha\alpha}\in R^{3\times 1}\) and \(\delta_{\alpha}\in R^{3\times 1}\) are the friction vector and disturbance vector at the active joints, respectively. \(\Delta M_{\alpha}\in R^{3\times 3}\) and \(\Delta Q_{\alpha}\in R^{3\times 3}\) are the errors of the real dynamic model. The detail elements of \(M_{\alpha}\), \(Q_{\alpha}\), \(f_{\alpha\alpha}\) and \(\delta_{\alpha}\)can be found in Reference [55].  
Consequently, the real dynamic equation of the 3-DOF parallel robot manipulator is achieved as  
\[\dot{M}_{\alpha}\ddot{\theta}_{\alpha}+\dot{Q}_{\alpha}\dot{\theta}_{\alpha}+ \Delta U_{\alpha}=\tau_{\alpha} \tag{2}\]  
The vector of the lumped uncertain terms \(\Delta U_{\alpha}\) from Equation (2) is stated as follows:  
\[\Delta U_{\alpha}=\Delta M_{\alpha}\ddot{\theta}_{\alpha}+\Delta Q_{\alpha} \dot{\theta}_{\alpha}+f_{\alpha}+\delta_{\alpha} \tag{3}\]  
Accordingly, the robot dynamic Equation (1) is rewritten with the following expression:  
\[\begin{array}{c}\ddot{\theta}_{\alpha}=\dot{M}_{\alpha}^{-1}\left[\tau_{ \alpha}-\dot{Q}_{\alpha}\dot{\theta}_{\alpha}\right]-\begin{array}{c}\dot{M }_{\alpha}^{-1}\left[\Delta U_{\alpha}\right]\\ =\dot{M}_{\alpha}^{-1}\left[\tau_{\alpha}-\dot{Q}_{\alpha}\dot{\theta}_{\alpha }\right]-\begin{array}{c}\dot{\Xi}(\theta_{\alpha},\Delta U_{\alpha})\end{array} \end{array} \tag{4}\]  
where \(\begin{array}{c}\dot{\Xi}(\theta_{\alpha},\Delta U_{\alpha})=\dot{M}_{\alpha }^{-1}\;\Delta U_{\alpha}\).  
The control objective of this research is to further increase the precision in the path tracking control for an uncertain 3-DOF parallel robotic manipulator (Equations (1) or (4)) such that the position tracking errors and synchronization errors can approach zero at the same time, regardless of the influences of uncertain dynamics in the robotic system.  
In order to obtain this control objective, the cross-coupling errors will be used to design the sliding surfaces and develop the proposed control system.  
**Assumption 1**.: _The lumped uncertain terms are a limited function, which needs to satisfy the constrained condition:_  
\[\left|\;\dot{\Xi}(\theta_{\alpha},\Delta U_{\alpha})\right|\leq G \tag{5}\]  
where \(G\) is a known positive constant, that is, the first-order differentiable.