## 2 Nonlinear Dynamics of Delta Robot Model
### Delta Robot Manipulator  
Consider the experimental Delta robot and its parameters illustrated in Figure 1 and Table 2, which consists of three symmetric branches constrained in a kinematic manner by the universal or spherical joint at the traveling plate. The complexity of the model arises mainly due to the movement of the forearms in branches. The forearm is constructed of two lightweight aluminum rods and attached rod end bearings at their extremities. The rod end bearings of the forearms transfer the motion of the upper arms to the traveling plate, giving it three degrees of freedom. By eliminating the forearms' rotational inertia, we may reduce the intricacy of their motion. This assumption is not very restricting due to the weight distribution of the forearm to the extremities. Moreover, a generalized disturbance of the Delta robot is analyzed in Section 3, so we can neglect the effects of joint friction and elasticity in this section to simplify the dynamic model of the Delta manipulator. The inverse dynamic model of the Delta robot is developed as [10; 11] based on the Newton-Euler method with the following simplifying hypotheses:  
* The rotational inertia of forearms is neglected. * For analytical purposes, the masses of forearms are optimally separated into portions and at their extremities: a two-thirds majority part at its upper extremity and the other part at its lower extremity, which contributes to the traveling plate mass. * Friction effects and elasticity are neglected. With the above-mentioned simplifying hypotheses, the robot can be reduced to only four bodies: the traveling plate and the three upper arms. We choose \(r=2/3\) as the optimal \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Description** & **Units** & **Values** \\ \hline \(L_{aa}\) & \(\text{}\) & \(\text{}\) & \(\text{}\) \\ \(L_{fa}\) & \(\text{}\) & \(\text{}\) & \(\text{}\) \\ \(R_{p}\) & \(\text{}\) & \(\text{}\) & \(\text{}\) \\ \(R_{b}\) & \(\text{}\) & \(\text{}\) & \(\text{}\) \\ \(m_{aa}\) & \(\text{}\) & \(\text{}\) & \(\text{}\) \\ \(m_{fa}\) & \(\text{}\) & \(\text{}\) & \(\text{}\) \\ \(m_{c}\) & \(\text{}\) & \(\text{}\) & \(\text{}\) \\ \(m_{p}\) & \(\text{}\) & \(\text{}\) & \(\text{}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Kinematic and dynamic parameters of Delta manipulator. Figure 1: The Delta robot in 3D view (**a**) and top-view (**b**). distribution of the three forearm masses, the total virtual mass \(m_{pt}\) of the traveling plate based on the hypotheses is calculated as:  
\[m_{pt}=m_{p}+m_{L}+3(1-r)m_{fa} \tag{3}\]  
At the upper arm level, let \(m_{a}=m_{ua}+m_{c}+rm_{fa}\) denote the mass of each virtual arm. We can define the position of the center of mass in the arm based on the hypothesis as:  
\[L_{C}=L_{ua}\left(\frac{\frac{1}{2}m_{ua}+m_{c}+rm_{fa}}{m_{a}}\right) \tag{4}\]  
where \(L_{ua}\) is the length of upper arm, \(m_{p}\) is the mass of the traveling plate and connectors at this plate, \(m_{L}\), \(m_{fa}\), \(m_{c}\), \(m_{ua}\) are the mass of external load, a forearm, and connectors at an elbow and an upper arm, respectively. The inertia matrix \(\mathbf{I}_{a}\) of the arms in joint space is a diagonal matrix, and the inertia of each arm \(l_{a1}=l_{a2}=l_{a3}=l_{ai}\), \(i=1,\ 2,\ 3\), can be defined as:  
\[\mathbf{I}_{a}=\text{diag}\{l_{a1},l_{a2},l_{a3}\}\text{ with }l_{ai}=\left(\frac{ 1}{3}m_{ua}+m_{c}+rm_{fa}\right)L_{ua}^{2} \tag{5}\]  
There are two kinds of forces that act on the traveling plate: the gravity force \(\mathbf{G}_{p}\in\mathbb{R}^{3}\) and the inertia force \(\mathbf{F}_{p}\in\mathbb{R}^{3}\). They are respectively given by:  
\[\mathbf{G}_{p}=m_{pt}\big{[}0\quad 0\quad-g\big{]}^{\text{T}} \tag{6}\]  
\[\mathbf{F}_{p}=m_{pt}\ddot{\mathbf{X}} \tag{7}\]  
where \(g\) is the gravitational acceleration, \(\ddot{\mathbf{X}}\) is the Cartesian acceleration vector of the traveling plate, with \(\ddot{\mathbf{X}}\in\mathbb{R}^{3}\).