## 2 Dynamic Model
### Dynamic Model without Friction  
The total kinetic energy \(T\) of the mechanism is the sum of the kinetic energy corresponding to all the moving components of the mechanism, i.e., the actuators (sliders), the legs, and the moving platform:  
\[T=\sum\
olimits_{i=1}^{3}T_{si}+\sum\
olimits_{i=1}^{3}T_{Li}+T_{p}, \tag{2}\]  
\[T_{si}=\frac{1}{2}m_{si}\dot{x_{i}}^{2}, \tag{3}\]  
\[T_{Li}=\frac{1}{2}m_{Li}V_{Gi}.V_{Gi}+\frac{1}{2}I_{Gi}\omega_{1}{}^{2}, \tag{4}\]  
\[T_{p}=\frac{1}{2}m_{p}\Big{(}\dot{x_{p}}^{2}+\dot{y_{p}}^{2}\Big{)}+\frac{1}{ 2}I_{Gp}\omega_{p}{}^{2}, \tag{5}\]  
where \(m_{si}\), \(m_{Li}\), and \(m_{p}\) denote the masses of slider \(i\), leg \(i\), and the moving platform, respectively; \(\dot{x}_{i}\) is the translational velocity of the slider \(i\); \(V_{Gi}\) is the translational velocity of the COM of leg \(i\) relative to the inertial frame; \(\omega_{p}\) and \(\omega_{i}\) are the angular velocities of the platform and leg \(i\), respectively; \(I_{Gp}\) and \(I_{Gi}\) are the moments of inertia of the platform and leg \(i\) relative to their COMs, respectively; and \(\dot{x}_{p}\) and  
Figure 2: Centers of masses (COMs) of a 3 PRR parallel kinematics mechanism. \(\dot{y}_{p}\) are the translational velocities of the COM of the moving platform in \(x\) and \(y\) directions, respectively. The COM of the moving platform is given by:  
\[x_{p}=x+x_{pG(x)}\cos\theta-x_{pG(y)}\sin\theta;\ y_{p}=y+x_{pG(x)}\sin\theta+x_{ pG(y)}\cos\theta\] (6a,6b)  
It can be observed in both Equations (4) and (5) that the kinetic energy of the legs and the moving platform both have translational and rotational components due to the translation and rotation relative to the inertial frame. On the other hand, it can be observed in Equation (3) that the sliders only have kinetic energy from translations as they do not undergo rotation relative to the inertial frame. Similarly, the total potential energy \(V\) is the sum of the potential energy of the legs and the moving platform. The potential energy considered here comes from the altitude of the mechanism components in the direction of gravity. Let us first assume that the mechanism as shown in Figures 1 and 2 is standing vertically, i.e., the Y-axis is vertical and therefore parallel with the direction of gravity. By considering the sliding path of the sliders as the datum in the potential energy evaluation, it can be observed that the sliding path of the sliders is coincident with the datum. As a result, the potential energy of the sliders vanishes and accordingly does not make any contribution to the total potential energy. The potential energy of the legs and the moving platform are respectively:  
\[V_{Li}=m_{Li}gL_{Ci}\sin\alpha_{i}, \tag{7}\]  
\[V_{p}=m_{p}gy_{p}, \tag{8}\]  
where \(g\) denotes the gravitational acceleration, \(L_{Ci}\) denotes the altitude of the COM of leg \(i\) relative to the datum, and \(y_{p}\) denotes the altitude of the COM of the moving platform relative to the datum. The potential energy expressions in Equations (7) and (8) hold when the mechanism is standing vertically, as described earlier. When the mechanism is oriented on a horizontal plane such that all of the COMs have an identical altitude in the direction of gravity, the equations are no longer valid. In such a case, when the datum of the potential energy evaluation has an identical altitude to all of the COMs, then the potential energy due to gravity vanishes. In this case, the rigid body motion of the mechanism is not affected by gravity. A practical way to suppress the effect of gravity in a general model is by imposing a zero value to the gravitational acceleration constant \(g\). Using the Euler-Lagrange approach, the equations of motion are given by:  
\[\frac{d}{dt}\bigg{(}\frac{\partial L}{\partial\dot{q}_{i}}\bigg{)}-\frac{ \partial L}{\partial\dot{q}_{i}}=Q_{i}+\sum\
olimits_{k=1}^{m}{}_{Ak}\frac{ \partial\Gamma_{k}}{\partial\dot{q}_{i}}, \tag{9}\]  
\[q=\big{[}