# Dynamic Analysis of a Delta Parallel Robot with Flexible Links and Joint Clearances
## 2 Problem Formulation  
The effect of element flexibilities, frictions and clearances can be studied numerically by developing a CAD model of the analyzed robot in the first stage, for example, using the CATIA software, followed in the second stage by the ADAMS analysis. Since ADAMS does not easily allow the creation of elements with complex shapes, it was decided to develop the CAD model in CATIA and then export the 3D bodies (in an IGES format) to ADAMS. Thus, a CAD model can be obtained in ADAMS that reflects the properties of the existing physical robot as accurately as possible. In this analysis, we consider the case study of a Delta Siax D3-1600 parallel robot (Figure 1a, [35]). Its simplified CAD model (without motors) at a 1:1 scale was represented in CATIA (Figure 1b) and then transferred to ADAMS (Figure 1c). The Delta Siax D3-1600 is a three degree-of-freedom (3-DOF) robot (the end-effector performs three independent translations). It is composed of a fixed platform (0) and a mobile platform (4), interconnected by three arms, A, B and C, each of them with a driving element (1) connected to the base by a motor drive (R). Each arm has a parallelogram-type kinematic chain with two flexible elements (2 and 3) and four passive spherical couples each (\(S_{2k1}\), \(S_{2k2}\) and \(S_{3k1}\), \(S_{3k2}\), where \(k=A\), \(B\), C--Figure 1d). The three arms are equiangularly distributed (Figure 1e) in relation to the global coordinate system (\(X_{0}Y_{0}Z_{0}\)) of the robot, with the origin located at point O (Figure 1d). The Delta robot has attached to the end-effector (4) a payload in the form of a cylinder (5) with a mass of 5 kg. The characteristic point P is the origin of the mobile coordinate system of the end-effector and it travels a spatial trajectory established so as to reach the maximum velocity and acceleration according to the values specified in Table 1.  
The geometrical, mass and material properties of the robot's component bodies are key factors in determining its dynamic behavior. In the proposed analysis, the following assumptions have been made:  
- Elements 0, 1, 4 and 5 are rigid solid bodies;  
- Rod Elements 2 and 3 have higher elastic characteristics than the other elements due their dimensions, see Table 2;  
- All bodies are made of steel. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Body** & **CAD Model** & **Parameters** \\ \hline Fixed platform (0) & & \\ \hline Crank (1) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometrical and mass details of the Delta robot bodies. Figure 1: Delta parallel robot: (**a**) Delta Siax D3-1600 type robot; (**b**) CATIA model without payload; (**c**) ADAMS model with payload; (**d**) ADAMS model parametrisation; (**e**) arm arrangement. The reference model of the Delta robot is based on the assumptions of an ideal mechanism, where all bodies are rigid solids, all kinematic joints are ideal (no clearance, no friction), and the characteristic point P follows a trajectory that reaches maximum allowed values of speed and acceleration. Thus, the fifth degree polynomial function was chosen to generate the movement trajectory in the joint space as well as a short trajectory travel time of 0.2 s, the time resulting from the simultaneous provision of the conditions for the robot to touch the \(P_{0}P_{1}\) trajectory (Figure 2), the maximum speed of the end-effector \(v_{Pmax}=8\) m/s and the maximum acceleration \(a_{Pmax}=120\) m/s\({}^{2}\) (see Table 1). The starting position of the Cartesian trajectory (\(P_{0}\)) corresponds to the initial position of the robot where all three motor torques \(R_{A}\), \(R_{B}\) and \(R_{C}\) are in the zero position and Elements 1 are arranged in a horizontal plane (parallel to \(X_{0}Y_{0}\), see Figure 1d). The trajectory in the Cartesian space \(P_{0}P_{1}\) is a spatial curve obtained by applying an angular