## 2 Dynamic Model of Delta Robot
### Inverse Kinematics  
Inverse kinematics stands out as one of the most widely adopted algorithms for modeling the Delta robot. Solving the inverse kinematics involves deriving the geometrical relationships between its links and joints. From Figure 2, the relationship between the link positions and angles \(\theta_{i}\) can be given as  
\[\theta_{i}=\arctan\left(\frac{Z_{ji}}{Y_{Fi}-Y_{ji}}\right) \tag{1}\]  
where \(i=1,2,3\); \(Y_{Fi}\) and \(Y_{ji}\) are the positions of the points \(F_{i}\) and \(J_{i}\) in the \(Y\) direction; and \(Z_{Fi}\) and \(Z_{ji}\) are the positions of the points \(F_{i}\) and \(J_{i}\) in the \(Z\) direction. The positions of the joints \(Y_{Fi}\) and \(Y_{ji}\) satisfy the geometric constraints of the Delta robot as described in [17]. The inverse kinematics describes how the angles of the joints relate to the positions of the end effector. The joint velocities \(\dot{\theta}_{i}\) are not considered in Equation (1). In this study, the control inputs include angles and joint velocities \([\mathbf{\theta},\dot{\mathbf{\theta}}]\) of the stepper motors. Hence, the dynamic input-output model of the Delta robot should involve joint velocities. One way to compute the joint velocities is by using the Jacobian matrix \(J\) that relates to the joint velocities \(\dot{\mathbf{\theta}}=[\dot{\theta}_{1},\dot{\theta}_{2},\dot{\theta}_{3}]^{T}\) and the end effector position velocities \(\mathbf{v}=[\dot{x},\dot{y},\dot{z}]^{T}\)[20]:  
\[\dot{\mathbf{\theta}}=J(\mathbf{r})\mathbf{v} \tag{2}\]  
where \(\mathbf{r}=(x,y,z)^{T}\) describes the centroid position of the end effector. Equations (1) and (2) suggest that the joint velocities are highly nonlinear and complex functions of the positions and its velocities of the end effector. In the data-driven study, the joint velocities can be computed using finite difference approximation as, for example,  
\[\dot{\theta}_{i}(k)=(\theta_{i}(k)-\theta_{i}(k-1))/\Delta t \tag{3}\]  
where \(k\) is the discrete time, \(i\) represents the stepper motor's index number and \(\Delta t\) is the sample time. However, as discussed in the Section 1, this information alone is not enough to help us build a dynamic input-output model for the Delta robot. Furthermore, if the finite difference method or Jacobian matrix method is used to compute joint velocities, errors in the computed velocities can accumulate over time, leading to a poor position prediction. This affects the performance of trajectory tracking control. It is commonly known that the control-affine model of a system provides a convenient platform for control design. In this study, we propose to develop a control-affine nonlinear model of the Delta robot by utilizing neural networks. Neural networks can learn and model the underlying physical relationship between the inputs and outputs. A rich set of input-output data from the forward model of the Delta robot can be used to train neural networks in order to build the control-affine nonlinear dynamic model of the Delta robot without the need for the Jacobian matrix, finite difference, and inverse kinematics. The proposed control-affine model is completely data-driven and is intended for control design consideration. The \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Notation** & **Value** \\ \hline Radius of the fixed platform & \(R\) & 0.325 m \\ \hline Radius of the moving platform & \(r\) & 0.075 m \\ \hline Length of the active arm & \(r_{f}\) & 0.5 m \\ \hline Length of the passive arm & \(r_{e}\) & 0.25 m \\ \hline Mass of the active arm & \(m_{f}\) & 0.205 kg \\ \hline Mass of the passive arm & \(m_{e}\) & 0.153 kg \\ \hline Mass of the end effector & \(m_{b}\) & 0.653 kg \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the Delta robot. model is not derivable from [PERSON]'s law or [PERSON]'s equation. The input-output data for training the neural networks are randomly sampled in a sufficiently large workspace.