## 2 Nonlinear Dynamics of Delta Robot Model
### Delta Robot Manipulator
To guarantee the stability of angular motor positions without load, we can choose \(\mathbf{K}_{P}=\mathbf{\Lambda}\mathbf{K}_{D}\) as:_  
\[\mathbf{\Gamma}_{\text{PD}}=\mathbf{\Gamma}_{\text{M}}=\mathbf{K}_{P}\mathbf{e}+\mathbf{K}_{D}\dot {\mathbf{e}}=\mathbf{K}_{D}\mathbf{\Psi} \tag{36}\]  
where \(\mathbf{K}_{P}=\text{diag}\{\text{k}_{\text{P1}},\text{k}_{\text{P2}},\text{k}_{ \text{P3}}\}\in\mathbb{R}^{3\times 3}\), \(\mathbf{K}_{D}=\text{diag}\{\text{k}_{\text{D1}},\text{k}_{\text{D2}},\text{k}_{ \text{D3}}\}\in\mathbb{R}^{3\times 3}\), and \(\mathbf{\Lambda}=\text{diag}\{\lambda_{1},\lambda_{2},\lambda_{3}\}\in\mathbb{R}^{ 3\times 3}\) are positive-definite diagonal matrices, respectively. \(\mathbf{\hat{f}}_{\text{PD}}\in\mathbb{R}^{3}\) is the torque vector to control the three motor without torques of load. Proposition 2: _To overcome the nominal dynamics of the Delta manipulator, the finite time synergetic control scheme is proposed._  
\[\mathbf{\Gamma}_{\text{FS}}=\mathbf{\hat{M}}(\mathbf{q})\Bigg{(}\ddot{\mathbf{q}}_{d}+\Big{(} \frac{\mathbf{\Psi}}{\beta}\Big{)}^{\frac{p}{p}}+\mathbf{\Lambda}\dot{\mathbf{e}}\Bigg{)}+ \mathbf{\hat{C}}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\dot{\mathbf{q}}+\mathbf{\hat{f}}_{S}(\bm {q}) \tag{37}\]_where \(\mathbf{\Gamma}_{\text{FS}},\hat{\mathbf{M}}(\mathbf{q})\), \(\hat{\mathbf{C}}(\mathbf{q},\dot{\mathbf{q}})\), and \(\hat{\mathbf{\Gamma}}_{g}(\mathbf{q})\) are calculated as in Section 2.2._  
**Proposition 3**. : _The unknown parameters and external disturbance of the practical Delta robot system consist of disturbances in the motors, the Delta manipulator from the hypothesis in Section 2.2, and an external load. They all affect the tracking errors of joint positions and velocities by applying an unknown torque value to each motor. For simplifying disturbance models, a generalized disturbance model \(\hat{\mathbf{\Gamma}}_{D}=\left[\hat{\tau}_{D1}\quad\hat{\tau}_{D2}\quad\hat{\tau} _{D3}\right]^{T}\) with \(\hat{\mathbf{\Gamma}}_{D}\in\mathbb{R}^{3}\) is proposed to estimate online three RBF neural networks [39; 40] as:_  
\[\hat{\tau}_{Di}=\hat{\mathbf{W}}_{i}^{*\text{T}}\mathbf{h}_{i}\big{(}e_{i},\dot{e}_{i} \big{)} \tag{38}\]  
_The adaptive law is designed as:_  
\[\dot{\hat{W}}_{i}=\gamma\mathbf{h}_{i}\big{(}e_{i},\dot{e}_{i}\big{)}\psi_{i} \tag{39}\]  
From descriptions of RBF networks in [39; 40], we choose the tracking error and its derivative value as the input vector in this paper. \[h_{ij}\big{(}e_{i},\dot{e}_{i}\big{)}=\Phi\big{(}\|\mathbf{E}_{i}-\mathbf{c}_{j}\|,b_ {i}\big{)}=\exp\left(-\frac{\|\mathbf{E}_{i}-\mathbf{c}_{j}\|^{2}}{2b_{i}^{2}}\right) \tag{40}\]  
where \(\mathbf{\Phi}(. )\) is the Gaussian function of the \(j\)-th neural network of the \(i\)-th joint, \(\mathbf{E}_{i}=\big{[}e_{i}\quad\dot{e}_{i}\big{]}^{\text{T}}\in\mathbb{R}^{2}\) is the input vector of the \(i\)-th joint, \(\mathbf{c}= \begin{bmatrix}c_{q1}&c_{q2}&c_{qj}&\dots&c_{qm}\\ c_{\dot{q}1}&c_{\dot{q}2}&c_{\dot{q}j}&\dots&c_{qm}\end{bmatrix} \in\mathbb{R }^{2\times m}\) represents the center point matrix, \(\mathbf{W}_{i},\mathbf{W}_{i}^{*}\in\mathbb{R}^{3}\) are the weight vector and updated weight vector of joint \(i\). \(j=1,\,2,\,\dots,\,m\) is hidden neural net number in the hidden layer, \(\mathbf{h}_{i}\big{(}e_{i},\dot{e}_{i}\big{)}= \begin{bmatrix}h_{i1}&h_{i2}&h_{ij} &\dots&h_{im}\end{bmatrix} ^{\text{T}}\in\mathbb{R}^{m}\) is the output vector of hidden layer, with \(i=1,\,2,\,3\) as the number order of joint, the learning rate is denoted by \(\gamma\). The structure of RBF network is presented in Figure 2.  
The optimal weights are found by using the Gradient descent algorithm and the adaptive law (39)-; the weight vector of the \(i\)-th joint is:  
\[\mathbf{W}_{i}(t)=\mathbf{W}_{i}(t-\Delta t_{s})-\dot{\hat{\mathbf{W}}}_{i}(t)\mathbf{W}_{i}^ {*}=\mathbf{W}_{i}-\dot{\hat{\mathbf{W}}}_{i} \tag{41}\]  
In total, the proposal controller of the nonlinear Delta robot is proposed as:  
Figure 2: The RFB neural network in this paper. \[