## 2 Design and Modelling of a 3D Printer with five Degrees of Freedom
### Dynamic Analysis  
#### 2.3.1 Inverse Dynamics  
The primary objective of inverse dynamics analysis for any mechanism is to ascertain the requisite forces or torques at the end actuator of the mechanism in order to precisely track a certain trajectory. Essentially, these forces or torques are seen as a mathematical function that relies on the location, velocity, and acceleration of the system's joints, as well as the position of the mobile platform. The Euler technique was used in this work to deduce the analytical dynamic model of the specified system. The Euler approach stands out from other techniques, such as Lagrangian, due to its ability to address the computational time issue in real-time applications and provide a closed-form model in [37; 38; 39; 40; 41]. The Euler approach, applicable to any kinematic chain, treats each link as a rigid body. These connections may be individually examined and then merged. By using this method, the force or torque expressions that arise between the legs may be readily exposed. Consequently, this investigation included analysing the connections of each kinematic chain, beginning with the parallelogram and extending to the moving table, individually and sequentially. Figure 4 displays a three-dimensional perspective of the mechanism and the geometric indices used in dynamic modelling. Given that the Euler technique was used to study each leg separately, it is necessary to assign each leg to its own local coordinate system. In order to address this issue, individual coordinate frames are denoted as \(\{L_{i}\}\) being positioned inside each parallelogram structure. The connecting points of the connectors and parallelograms for each chain are represented by \(A_{i},B_{i},C_{i}\), and \(D_{i}\) as illustrated in Figure 4. Furthermore, the \(E_{i}\) and \(D_{i}\) points denote the central points of the ball joints used for mounting the parallelogram onto the carriages utilized in the mechanism. On the other hand, the \(G_{i}\) and \(M_{i}\) points indicate the central points of the attachments utilized for connecting the parallelogram to the moving table in the mechanism. Figure 4: Freehand diagram of the mechanism. Figure 4 displays the separate kinematic chains, with each component represented by circled integers ranging from one to eight. \(G_{j}^{i}\) denotes the centroid of each element, with the upper symbol \(i\) denoting the chain number and the lower sign \(j\) denoting the element number. Figure 5 displays the intricate free body diagram that is used in the dynamic analysis of the fully designed mechanism. Given the similarity between the first and third kinematic chains, as well as the third and second kinematic chains, Figure 5 only provides detailed information on the first and fourth kinematic chains, together with the forces and torques acting on them. The spherical joints, \({}^{i}F_{j,k}\) (\(j=2\), \(6\), \(7\)) seen in Figure 5, indicate the resultant force exerted on each leg of the parallelogram. In this context, the superscript \(i\) denotes the quantity of kinematic chains, the subscript \(j\) denotes the quantity of components, and \(k\) denotes the quantity of the joints used for connection in the parallelogram's legs. The forces and torques applied to the parallelogram connections via the connectors employed in the mechanism are denoted as \({}^{i}F_{j,k}\), \({}^{i}M_{j,k}\) (\(j=3\), \(5\)) respectively. The forces and torques applied to the center of mass of each connecting component are denoted as \({}^{i}F_{G,}\), \({}^{i}M_{G,}\), respectively. The terms \({}^{i}F_{R,}\), \({}^{i}M_{R,}\) (\(j=6\)) represent the rotational forces and torques that occur in the joints. In addition, \(u_{i,k}\), \(v_{i,k}\), as seen in Figure 5, denote the vectors that extend towards the connecting points \(A_{i},\ B_{i},\ C_{i}\), and \(D_{i}\) of the parallelograms. The symbol \(\gamma_{i}\) denotes the vector that spans from the spherical joints to the center of mass of each leg of the parallelogram.