## 2 Nonlinear Dynamics of Delta Robot Model
### Delta Robot Manipulator
: _Consider a nonlinear system as (21) with the control law in (26), the tracking errors converge to zero in a finite time with a convergence rate depending on the selected parameter \(\beta\)._  
Proof of Theorem 1.: To demonstrate the asymptotic stability of the system, we define a Lyapunov candidate function \(V=\frac{1}{2}\mathbf{\
abla}^{\text{T}}\mathbf{\
abla}\) and we have its derivative as:  
\[\dot{V}=\mathbf{\
abla}^{\text{T}}\dot{\mathbf{\Psi}} \tag{30}\]Substituting the constraint in (25) into (30), and assuming \(r_{1}=r_{2}=r_{3}=r,\ p_{1}=p_{2}=p_{3}=p\), then:  
\[\dot{V} =\mathbf{\Psi}^{\mathrm{T}}\big{(}-\beta^{-1}\mathbf{\Psi}\big{)}^{\frac{p}{ p}}=\big{(}-\beta^{-1}\big{)}^{\frac{p}{p}}\big{(}\mathbf{\Psi}^{\mathrm{T}}\big{)}^{ \frac{p}{p}}\mathbf{\Psi}^{\frac{p}{p}} \tag{31}\] \[=-\big{(}\beta^{-1}\big{)}^{\frac{p}{p}}\|\mathbf{\[PERSON]}\|^{\frac{p+r}{ p}}\] \[\leq-\big{(}\beta^{-1}\big{)}^{\frac{p}{p}}2^{\frac{p+r}{2p}}\Big{(} \frac{1}{2}\|\mathbf{\Psi}\|^{2}\Big{)}^{\frac{p+r}{2p}}\] \[\dot{V}(t) \leq-\beta_{1}^{-1}V^{\frac{p+r}{2p}}(t_{0})\xrightarrow[]{\text{ if }1<\frac{p}{p}<2,\ \beta>0}\dot{V}\leq 0 \tag{32}\]  
where \(\beta_{1}^{-1}=(\beta^{-1})^{\frac{p}{p}}2^{\frac{p+r}{2p}}\geq 0\), if \(\beta\) and \(\eta=\frac{p+r}{2p}\) satisfies Lemma 1. The synergetic manifold \(\mathbf{\Psi}\) can converge to zero in the finite time \(t_{1}\), given by:  
\[t_{1}=\frac{V^{(1-\frac{p+r}{2p})}(t_{0})}{\beta_{1}^{-1}\Big{(}1-\frac{p+r}{ 2p}\Big{)}} \tag{33}\]  
The PD Adaptive Finite-Time Nonsingular Synergetic Controller (PDAFS) Based on Radial Basis Function Neural Networks  
Based on the Delta manipulator (21) and motor dynamics (2), the general equation describing the robot system incorporating the external disturbance is presented as:  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}_{1}=\mathbf{x}_{2}\\ \dot{\mathbf{x}}_{2}=-M(\mathbf{q})^{-1}\big{(}\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)} \dot{\mathbf{q}}+\mathbf{\Gamma}_{S}(\mathbf{q})+\mathbf{\Gamma}_{M}+\mathbf{\Gamma}_{D}\big{)}+ \mathbf{M}(\mathbf{q})^{-1}\mathbf{\Gamma}\\ \mathbf{q}=\mathbf{x}_{1}\end{array} \right. \tag{34}\]  
where \(\mathbf{M}(\mathbf{q})\in\mathbb{R}^{3\times 3}\), \(\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\in\mathbb{R}^{3\times 3}\) are the actual mass matrix and actual centrifugal and Coriolis matrix. \(\mathbf{\Gamma}_{S}(\mathbf{q})\), \(\mathbf{\Gamma}_{M}\), and \(\mathbf{\Gamma}_{D}\in\mathbb{R}^{3}\) are the actual gravitational, actual motor dynamic, and actual disturbance torque, respectively. From the dynamic model in (34), we can see that if the dynamic motions of the robot satisfy the desired trajectory, the torque control signal must be calculated precisely. In discrete-time control, the signal controls in (2) and (34) are discrete, quantized, and extremely difficult to achieve accurate torque values. Based on three object-model controls for the motor model, the manipulator model, and a generalized disturbance model in the Delta robot, this paper proposes the PDAFS controller as a combination of PD, finite-time synergetic, and RBF neural network controller. \[\mathbf{\Gamma}_{\text{PDAFS}}=\mathbf{\Gamma}_{\text{PD}}+\mathbf{\Gamma}_{\text{FS}}+ \mathbf{\hat{f}}_{D} \tag{35}\]  
Throughout this paper, the controller structure is designed with the following propositions, used directly in the convergence analysis. Proposition 1: _A proportional derivative control (PD) estimates the torques, matching the nominal dynamics of DC motors.