## 2 Design and Modelling of a 3D Printer with five Degrees of Freedom  
The 3D printer system, seen in Figures 1 and 2, comprises a stationary platform and a movable platform that are interconnected with four kinematic chains. This platform consists of two PR (Pa)U and two PR (Pa)R kinematic chains. In the system, the terms R, P, U, and Pa correspond to rotary, prismatic, universal joint, and parallelograms, respectively. The fundamental element of kinematic chains is the planar, four-bar parallelogram structure. The parallelogram construction utilizes four separate linear rail systems to provide accurate control. The carriages, which form the parallelogram structure, are joined to each link via ball joints. The second and third arms are attached to the moving platform with ball joints, whereas the first and fourth arms have an indirect connection to the moving platform. Each connecting part is connected to the moving platform using a rotary joint. The present manipulator's moving platform has a construction that enables three degrees of freedom for translational movement and one DOF for rotating movement in the x-direction. This is shown in Figures 1 and 2. Utilizing a turntable enables the provision of rotating motion along the z-axis. representation of the system, which is used for the study. The index 'i' denotes a single kinematic chain out of several chains that are identical to each other. Figure 3 shows that the global coordinate system has an origin denoted as {O}, and it is determined by the \(x_{o},~{}y_{o}\), and \(z_{o}\) components. For each prismatic joint, the vector \(a_{i}\) is used, connecting the origin of the global coordinate system and the beginning of the rail. The local starting point of the rail is the vector \(d_{i}\), which is connected to the prismatic joint, \(i\), on the car. The magnitude of the \(l_{i}\) vector seen in the figure is equal to the length of the \(i\)th connecting arm. Additionally, the vector \(c_{i}\) is a vector connected to the \(i\)th connector, starting from the midpoint of the spherical joints and ending at the rotary axis of the connector. However, for cases \(i=2.4\), the vector \(c_{i}\) is equal to \(0\) due to the structure of the mechanism. The vector \(b_{i}\) is fixed at the centre of the moving platform {P}, as indicated in Figure 3. At the same time, the vector \(b_{i}\) expresses the position of the parallelograms determined by the local frame. All the kinematic relationships are defined considering the fixed platform attached to the global frame {O}. Based on Figure 3, the vector equation for the \(i\)th kinematic chain can be expressed as Equation (1). \[{}^{O}p+{}^{O}R_{P}{}^{P}b_{i}={}^{O}a_{i}+{}^{O}d_{i}+{}^{O}l_{i}+{}^{O}c_{i}, \tag{1}\]  
The vector must be defined according to the global coordinate system using the transformation formula provided in the following equation:  
\[{}^{O}b_{i}={}^{O}R_{P}{}^{P}b_{i}, \tag{2}\]  
Figure 3: Vector notation for the kinematic modeling of the proposed manipulator. The expression denoted as \({}^{O}R_{p}\) in Equation (2) is a rotation matrix that is only defined in the x-direction, as shown in Equation (3), owing to the specific attributes of the mechanical system. \[{}^{O}R_{p}=R_{x}(\theta)= \begin{bmatrix}1&0&0\\ 0&cos\theta_{x}&-sin\theta_{x}\\ 0&sin\theta_{x}&cos\theta_{x}\end{bmatrix} , \tag{3}\]  
In Equation (1), the vectors \({}^{O}d_{i}\) and \({}^{O}c_{i}\) are consistently aligned in the same direction. Due to this rationale, given that \(d_{i}\) represents the unit vector of vector \(d_{i}\), Equation (1) may be restated as Equation (4) as follows:  
\[\left(p+b_{i}-a_{i}-c_{i}\hat{d_{i}}\right)-L_{i}=d_{i}\hat{d_{i}}, \tag{4}\]  
In Equation (4), \(c_{i}\) and \(d_{i}\) denote the magnitudes of the vectors \(c_{i}\) and \(d_{i}\), respectively. The vector expression \(L_{i}\) is represented as \(l_{i}\) and its value is provided.