## 2 Dynamic Model
### Friction Modeling  
Since the friction forces cannot be derived from the energy expressions, their contribution to the dynamics is presented separately here. Although the friction can be modeled as being either linear or nonlinear and either symmetric or asymmetric, it is commonly modeled as symmetric and linear. Furthermore, the friction at the prismatic joints is typically much higher than that at the revolute joints. This is particularly true when roller or ball bearings are used as the revolute joints since the friction in such bearings is very small. Therefore, when the friction should be considered, one can either consider the friction at all of the joints or the friction at only the prismatic joints, while neglecting the friction at the revolute joints. The mechanism at hand uses roller bearings as the revolute joints. As a result, the friction at the revolute joints is not significant and therefore neglected. Figure 3: Joint reaction forces corresponding to the Lagrange multipliers. In this work, the Stribeck friction model, as shown in Figure 4b, is used. This friction model is modified from the static-viscous-Coulomb friction model shown in Figure 4a by adding the Stribeck friction effect to the static, Coulomb, and viscous friction components. This can be written as:  
\[f\left\{ \begin{array}{c}=f_{s}sign(\dot{q})\text{ for }\dot{q}=0^{+}\text{ or }\dot{q}=0^{-},\\ =f_{c}sign(\dot{q})+(f_{s}-f_{c})e^{-(\frac{\dot{q}}{\hbar_{\text{S}}})}sign( \dot{q})+f_{c}\dot{q}\text{ for }\dot{q}\
eq 0,\end{array} \right. \tag{23}\]  
where \(f_{s}\), \(f_{c}\), and \(f_{v}\) denote diagonal matrices containing the static, Coulomb, and viscous friction parameters, respectively. For the case of a nonzero velocity in Equation (23), the first and the third terms in the right-hand side are the Coulomb and viscous friction components, whereas the second term is the Stribeck-effect friction component. Furthermore, some values of \(\delta\) have been recommended. In this work, the value of 1 was used. Using this friction model, there are four friction parameters to be identified: \(f_{s}\), \(f_{c}\), \(f_{v}\), and \(v_{s}\). In the case of varying normal forces, the static and Coulomb friction forces vary accordingly. As a result, the static and Coulomb friction cannot be identified as forces in the usual manner. In such a case, which is the case in this work, the static and Coulomb frictions should be identified as coefficients, i.e., \(\mu_{s}\) and \(\mu_{c}\). The static and Coulomb friction parameters in the case of a translational joint, i.e., prismatic joint, are given by multiplying each of a dimensionless static friction coefficient \(\mu_{s}\) and a dimensionless Coulomb friction coefficient \(\mu_{c}\) by the normal force \(f_{n}\) exerted on the contact surface at the joint:  
\[f_{s}=\mu_{s}f_{n}\, \tag{24a}\] \[f_{c}=\mu_{c}f_{n} \tag{24b}\]  
The static and Coulomb friction coefficients typically depend on the types of materials in contact with each other and the lubrication condition. In general, the normal force in a posture-changing mechanism, such as a PKM like the mechanism at hand, is not constant. In the prismatic joint of the mechanism at hand, as illustrated in Figure 5, three force components are acting, namely, the weight of the prismatic joint \(m_{si}\delta\), which is given by the total weight of the slider and the moving part of the linear motor; the input force \(Q_{xi}\) exerted on the prismatic joint by the linear motor; and the force \(R_{i}\) exerted to the prismatic joint due to the weight of the other components mounted on the slider, as well as the external payload and the external force, if any. The exerted force \(R_{i}\) can be decomposed into horizontal and vertical components, namely, \(R_{xi}\) and \(R_{yi}\). At any time, the total force in the horizontal direction is given by the sum of \(R_{xi}\) and \(Q_{xi}\). In a static case, \(R_{xi}\) is overcome by \(Q_{xi}\) such that the total force in the horizontal direction is zero.