## 3 Description of the 2 PRU-1 PRS Parallel Manipulator and Its Kinematic Problem  
The parallel manipulator analyzed in this work is a MAST for automobile pieces testing purposes, introduced by [PERSON] et al. (2018) and shown in Figure 4.  
As Figure 4 shows that the 2 PRU-1 PRS PM consists of a mobile and a fixed platform connected by three limbs. The mobile and fixed platforms are both isosceles triangles. The height of the mobile platform is R and its base is 2R, while the fixed platform has a height of H and a base of 2H. The first and the third limbs are identical chains composed by a prismatic joint, a revolute joint and a universal joint (PRU). The second limb is formed by a prismatic joint, a revolute joint and a spherical joint (PRS). Additionally, the first and third limb planes are coplanar, and the second one is perpendicular to them. The 2 PRU-1 PRS parallel manipulator in this configuration has three pure degrees of freedom: a vertical translation and two rotations about two perpendicular axes intersecting at the fixed platform center. According to the reference system defined in Figure 4, the 3-DOFs are a translation along the Z-axis and two rotations about the \(X\)-ais and \(Y\)-axis (\(\psi\) and \(\theta\), respectively). Apart from these three pure degrees of freedom, the 2 PRU-1 PRS parallel  
Figure 3: Optimization of GP to minimize the power consumption. manipulator has a parasitic motion along the \(X\)-axis, given by Equation (1), as explained by [PERSON] et al. (2018). \[x=R\cdot(sin\theta\cdot sin\psi) \tag{1}\]  
In 2018, [PERSON] et al. (2018) described the full kinematics, singularities and dynamic studies of the manipulator. In 2019, they also presented the study of its stiffness by using the matrix structural method ([PERSON] et al., 2018). Since the equations of the kinematic problem are important to understand the following section, we summarize and reproduce them here. The position of all points of the manipulator for given values of the rotations about X-axis and Y-axis and the translation along Z-axis can be obtained by solving the inverse position problem. The value of the displacement of the linear actuators is given by Equation (2). \[\rho_{1} =z-R\cdot s\theta-\sqrt{L^{2}-(R\cdot(c\theta-s\theta\cdot s\psi)-H) ^{2}} \tag{2}\] \[\rho_{2} =z+R\cdot c\theta\cdot s\psi-\sqrt{L^{2}-(R\cdot c\psi-H)^{2}}\] \[\rho_{3} =z+R\cdot s\theta-\sqrt{L^{2}-(-R\cdot(c\theta+s\theta\cdot s\psi )+H)^{2}}\]  
The loop equation of the manipulator is given by Equation (3). By differentiating it, we obtain the expression of the velocity of the mobile platform in vectorial notation as Equation (4):  
\[\mathbf{OP} =\mathbf{OC}_{i}+\mathbf{CB}_{i}+\mathbf{BA}_{i}+\mathbf{A}_{i} \mathbf{P} \tag{3}\] \[\mathbf{v}_{p} =\rho_{i}\mathbf{k}+\mathbf{\Omega}_{i}\times(\mathbf{B}_{i} \mathbf{A}_{i})+\mathbf{\Omega}_{p}\times(\mathbf{A}_{i}\mathbf{P}) \tag{4}\]  
where \(\mathbf{k}\) is a vertical unit vector, \(\mathbf{\Omega}_{p}\) is the angular velocity of the mobile platform, \(\mathbf{\Omega}_{i}\) is the angular velocity of each leg and \(\mathbf{v}_{p}\) is the linear velocity of the platform. In order to express this equation in terms of the Jacobian matrix of the platform, we can multiply Equation (4) by vector \(\mathbf{s}_{i}\), which is a unit vector in the direction of the limb, to obtain Equation (5) and multiply Equation (4) by \(\mathbf{u}_{i}\), which is a unit vector perpendicular to the limb plane, to obtain Equation (6). \[\mathbf{s}_{i}\cdot\mathbf{v}_{p}+\mathbf{\Omega}_{i}\cdot( \mathbf{PA}_{i}\times\mathbf{s}_{i})=\rho_{i}\cdot(\mathbf{s}_{i}\mathbf{k}) \tag{5}\] \[\mathbf{u}_{i}\cdot\mathbf{v}_{p}+\mathbf{\Omega}_{p}\cdot( \mathbf{PA}_{i}\times\mathbf{u}_{i})=0 \tag{6}\]  
Figure 3:0 Sketch of the manipulator. We express Equations (5) and (6) in matrix notation, and we obtain Equation (7) which, written in a compact way, is equivalent to Equation (8):  
\[