## 2 Nonlinear Dynamics of Delta Robot Model
### Delta Robot Manipulator
\[\mathbf{M}(\mathbf{q})\dot{\mathbf{\Psi}}=\mathbf{M}(\mathbf{q})\big{(}\ddot{\mathbf{q}}_{d}+\lambda\dot {e}\big{)}+\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\big{(}\ddot{\mathbf{q}}_{d}+ \lambda e\big{)}+\mathbf{\Gamma}_{g}(\mathbf{q})-\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)} \mathbf{\Psi}+\mathbf{\Gamma}_{M}+\mathbf{\Gamma}_{D}-\mathbf{\Gamma} \tag{46}\]The reference states \(\mathbf{q}_{r}\), \(\dot{\mathbf{q}}_{r}\) and torque \(\mathbf{f}_{r}\big{(}\mathbf{q},\dot{\mathbf{q}},\dot{\mathbf{q}}_{d},\ddot{\mathbf{q}}_{d}\big{)}\) to control the generalized Delta robot system are described by Equations (47)-(49):  
\[\mathbf{q}_{r} =\dot{\mathbf{q}}_{d}+\Lambda\mathbf{e} \tag{47}\] \[\dot{\mathbf{q}}_{r} =\ddot{\mathbf{q}}_{d}+\Lambda\dot{\mathbf{e}}\] (48) \[\mathbf{f}_{r}\big{(}\mathbf{q},\dot{\mathbf{q}},\mathbf{q}_{r},\dot{\mathbf{q}}_{ r}\big{)}=\mathbf{M}(\mathbf{q})\dot{\mathbf{q}}_{r}+\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)} \mathbf{q}_{r}+\mathbf{\Gamma}_{g}(\mathbf{q})+\mathbf{\Gamma}_{M}+\mathbf{\Gamma}_{D} \tag{49}\]  
Therefore, Formula (46) can be simplified. \[\mathbf{M}(\mathbf{q})\ddot{\mathbf{\Psi}}=\mathbf{f}_{r}-\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}} \big{)}\mathbf{\Psi}-\mathbf{\Gamma} \tag{50}\]  
Substituting (50) into the Lyapunov function derivative in (45), we can get the following formula:  
\[\dot{V}=\mathbf{\Psi}^{\mathrm{T}}(\mathbf{f}_{r}-\mathbf{\Gamma})+\sum_{i=1}^{3} \widetilde{\mathbf{W}}_{i}^{\mathrm{T}}\gamma^{-1}\ddot{\mathbf{W}}_{i} \tag{51}\]  
We can rewrite the computed torque vector for the Delta robot from the PDAFS controller (42) as:  
\[\mathbf{\Gamma}=\hat{\mathbf{f}}_{r}+\hat{\mathbf{M}}(\mathbf{q})\bigg{(}\frac{\mathbf{\Psi}} {\beta}\bigg{)}^{\frac{r}{\beta}} \tag{52}\]  
where:  
\[\hat{\mathbf{f}}_{r}\big{(}\mathbf{q},\dot{\mathbf{q}},\mathbf{q}_{r},\dot{\mathbf{q}}_{r} \big{)}=\hat{\mathbf{M}}(\mathbf{q})\dot{\mathbf{q}}_{r}+\hat{\mathbf{C}}\big{(}\mathbf{q},\dot{ \mathbf{q}}\big{)}\mathbf{q}_{r}+\hat{\mathbf{\Gamma}}_{g}(\mathbf{q})-\hat{\mathbf{C}}\big{(}\bm {q},\dot{\mathbf{q}}\big{)}\mathbf{\Psi}+\mathbf{\Gamma}_{\mathrm{PD}}+\hat{\mathbf{\Gamma}}_{D}\]  
With the ideal torque to control the Delta robot as \(\mathbf{f}_{r}=\hat{\mathbf{f}}+\mathbf{\Gamma}_{D}^{*}\), the computed torque \(\hat{\mathbf{f}}_{r}=\hat{\mathbf{f}}+\hat{\mathbf{\Gamma}}_{D}\), \(\hat{\mathbf{f}}\) is the torque vector matching the nominal models with the computed torque vector for controlling the robot system. For any \(\mathbf{\Psi}\), we have the difference between \(\mathbf{f}_{r}\) and \(\hat{\mathbf{f}}_{r}\) caused by the model errors and external disturbances [13; 43], denoted as \(\hat{\mathbf{\Gamma}}_{D}\), which is set as follows:  
\[\hat{\mathbf{\Gamma}}_{D}=\mathbf{f}_{r}-\hat{\mathbf{f}}_{r}=\mathbf{\Gamma}_{D}^{*} -\hat{\mathbf{\Gamma}}_{D} \tag{53}\]  
Since the disturbances are always bounded with the desired trajectory, assume that the ideal disturbance at joint \(i\) is \(\tau_{Di}^{*}\) with \(i=1,\ 2,\ 3\), calculated as:  
\[\tau_{Di}^{*}=\mathbf{W}_{  
According to the Lyapunov theory and the proof in (32), the proposed PDAFS is stable in the finite time for the tracking control of the Delta robot. The equality \(\dot{V}=0\) is satisfied if and only if \(\mathbf{e}=\dot{\mathbf{e}}=0\). Since the Lyapunov function \(V\) is positive definite, and \(\dot{V}\) is negative definite, which leads to the conclusion that the proposed controller is globally asymptotically stable from the Lyapunov stability criterion. **Remark 1**. : _Using the PD controller is realistic, where the PD controller is implemented directly on a microcontroller, and it can stabilize the robot system in the short-term control and guarantee the error boundary of the system and the adaptive finite-time synergetic control (AFS) is calculated on the computer. The torque signals of the AFS can compensate for PD control signals on the microcontroller to ensure the accuracy and stability of the robot in long-term control._  
**Remark 2**.