## 2 Nonlinear Dynamics of Delta Robot Model
### Delta Robot Manipulator
\begin{split}\Gamma&=\widehat{M}(q)\left(\ddot{q}_{d}+ \left(\frac{\mathbf{\Psi}}{\beta}\right)^{\frac{r}{p}}+\Lambda\dot{e}\right)+\hat{ C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\dot{\mathbf{q}}+\hat{\Gamma}_{g}(\mathbf{q})+ \Gamma_{\text{PD}}+\hat{\Gamma}_{D}\\ &=\widehat{M}(\mathbf{q})\left[\begin{array}{c}\ddot{q}_{d1}+ \left(\frac{\mathbf{\Psi}_{1}}{\beta_{1}}\right)^{\frac{r_{1}}{p_{1}}}+\lambda_{1} \dot{e}_{1}\\ \ddot{q}_{d2}+\left(\frac{\mathbf{\Psi}_{2}}{\beta_{2}}\right)^{\frac{r_{2}}{p_{2}} }+\lambda_{2}\dot{e}_{2}\\ \ddot{q}_{d3}+\left(\frac{\mathbf{\Psi}_{3}}{\beta_{3}}\right)^{\frac{r_{3}}{p_{3} }}+\lambda_{3}\dot{e}_{3}\end{array}\right]+\hat{\mathbf{C}}(\mathbf{q},\dot{\mathbf{q}}) \left[\begin{array}{c}\dot{q}_{1}\\ \ddot{q}_{2}\\ \dot{q}_{3}\end{array}\right]+\hat{\Gamma}_{g}(\mathbf{q})+\left[\begin{array}{ c}k_{d1}\dot{e}_{1}+k_{p1}e_{1}\\ k_{d2}e_{2}+k_{p2}e_{2}\\ k_{d3}e_{3}+k_{p3}e_{3}\end{array}\right]+\left[\begin{array}{c}W_{1}^{\text {T}}h_{1}\big{(}e_{1},\dot{e}_{1}\big{)}\\ W_{2}^{\text{T}}h_{2}\big{(}e_{2},\dot{e}_{2}\big{)}\\ W_{3}^{\text{T}}h_{3}\big{(}e_{3},\dot{e}_{3}\big{)}\end{array}\right]\end{split}  \tag{42}\]  
The system errors and error rates converge to zero in the finite time with the rate convergence depending on the parameters \(\beta\), \(p\), and \(r\) if the control law is selected as Theorem 1 and Section 3.1. The disturbance estimation \(\hat{\Gamma}_{D}\) in (38) is used to approximate the external disturbance and model error compensation in order to minimize the tracking errors. In the following, we prove the stability of the Delta robot model controlled by the PDAFS controller. **Assumption 1**. : _The uncertainty parameters and external disturbance are unknown but bounded. The characteristics of the rigid robotic manipulator associated with the nominal matrices (16)-(18) are used in the stability analysis and are described in [41, 42] as follows:_  
1. The inertia matrix \(\mathbf{M}(\mathbf{q})\) is a positive definite symmetric matrix. 2. The matrix \(\hat{\mathbf{M}}(\mathbf{q})-2\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\) is a skew symmetric matrix. **Theorem 2**. : _Assume that the desired trajectories \(\mathbf{q}_{d},\dot{\mathbf{q}}_{d},\ \ddot{\mathbf{q}}_{d}\) are bounded. Let the control input be given by (40) with the estimated torques \(\hat{\mathbf{\Gamma}}_{\Delta}\) calculated as (38). Then, we ensure that \(\mathbf{\Psi}\) is uniformly ultimately bounded._  
**Proof:** If a Lyapunov candidate function is defined as:  
\[V=\frac{1}{2}\mathbf{\Psi}^{\text{T}}\mathbf{M}(\mathbf{q})\mathbf{\Psi}+\sum\limits_{i=1}^{3 }\frac{1}{2}\widetilde{\mathbf{W}}_{i}^{\text{T}}\gamma^{-1}\widetilde{\mathbf{W}}_{i} \tag{43}\]  
Then we can have its derivative as:  
\[ \begin{split}\dot{V}&=\frac{1}{2}\mathbf{\Psi}^{\text{T} }\hat{M}(\mathbf{q})\mathbf{\Psi}+\mathbf{\Psi}^{\text{T}}\mathbf{M}(\mathbf{q})\dot{\mathbf{\Psi}}+ \sum\limits_{i=1}^{3}\widetilde{\mathbf{W}}_{i}^{\text{T}}\gamma^{-1}\dot{\widetilde {\mathbf{W}}}_{i}\\ &=\frac{1}{2}\mathbf{\Psi}^{\text{T}}\Big{[}\hat{M}(\mathbf{q})-2\mathbf{C} \big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\Big{]}\mathbf{\Psi}+\mathbf{\Psi}^{\text{T}}\mathbf{C} \big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\mathbf{\Psi}+\mathbf{\Psi}^{\text{T}}\mathbf{M}(\mathbf{q}) \dot{\mathbf{\Psi}}+\sum\limits_{i=1}^{3}\widetilde{\mathbf{W}}_{i}^{\text{T}}\gamma^{ -1}\dot{\widetilde{\mathbf{W}}}_{i}\end{split}  \tag{44}\]  
If we consider structural property 2 in Assumption 1, then one can have \(\mathbf{\Psi}^{\text{T}}\Big{[}\hat{M}(\mathbf{q})-2\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}} \big{)}\Big{]}\mathbf{\Psi}=\mathbf{0}\) and:  
\[\dot{V}=\mathbf{\Psi}^{\text{T}}\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\mathbf{\Psi}+ \mathbf{\Psi}^{\text{T}}\mathbf{M}(\mathbf{q})\dot{\mathbf{\Psi}}+\sum\limits_{i=1}^{3} \widetilde{\mathbf{W}}_{i}^{\text{T}}\gamma^{-1}\dot{\widetilde{\mathbf{W}}}_{i} \tag{45}\]  
Substituting the manifold surface in (23) and the robot system model in (34) into (24), we have the equation below.