## 2 Dynamic Model
### Friction Modeling
However, when the prismatic joint is in motion, the magnitude of \(Q_{xi}\) is different from the magnitude of \(R_{xi}\). If the magnitude of \(R_{xi}\) is larger, the slider is moving toward the equilibrium state of the mechanism, i.e., the mechanism is collapsing to the base in the case of mechanism working in XY plane. Conversely, if the magnitude of \(Q_{xi}\) is larger, this means that the slider is moving due to an actuation point. Figure 4: (**a**) Static-viscous-Coulomb friction and (**b**) Stribeck friction models. The normal force \(f_{n}\) is equal but in the opposite direction to the sum of all the vertical force components. This can be written as follows:  
\[f_{n}=m_{si}g+R_{yi}. \tag{25}\]  
Although the weight of the prismatic joint, i.e., the first term on the right-hand side of Equation (25), is constant, the force component \(R_{yi}\) varies in general, depending on the mechanism posture. As a result, the normal force \(f_{n}\), and accordingly the static and Coulomb friction parameters, \(f_{s}\) and \(f_{c}\), are not constant. Since the varying components of the normal forces are mainly due to the effect of gravity and the external force, the posture dependency of the static and Coulomb friction parameters as described above is more significant when the rigid body motion of the mechanism is against a gravitational load, i.e., the rigid body motion is in the XY plane and/or under an external force. However, when the mechanism is free from an external force and working in the XY plane while it is statically balanced by a gravitational compensation mechanism or the mechanism is working in XZ plane, i.e., horizontally, then the posture dependency is less significant. The external force can be excluded during the dynamic parameter identification by simply moving the mechanism without any working load. In such a case, only the gravitational effect remains, if any. When the mechanism working in the XY plane is statically balanced, the gravity effect is eliminated. When the mechanism is working in the XZ plane, the gravitational effect is working in the off-plane direction and most of the weight of all the mechanism components is borne by the structure of the mechanism. This does not mean that the gravitational effect is eliminated from the joints. In fact, the prismatic and revolute joints are still subject to the gravity-induced load but it works in the off-plane direction. At the prismatic joint, this gravity-induced load can be seen as a vertical reaction force that is exerted on the base of a cantilever due to the weight of the cantilever. In addition to this vertical reaction force, there are also a horizontal reaction force and a reaction moment at the prismatic joint due to the weight of the cantilever. If the external force is excluded, the vertical reaction force will be the main force component, which results in an off-plane normal force working at the prismatic joint. In addition to this normal force component, there is also an in-plane normal force component that results from the motion of the mechanism. All of these normal forces lead to a friction force, which should be overcome during the actuation of the mechanism. Similarly, the off-plane normal forces that are equal but in the opposite direction to the bearing thrust forces also arise in the revolute joints. In addition to this, there are also in-plane normal forces at the revolute joints resulting from the motion of the mechanism. Furthermore, if an in-plane external force exists, it will be additionally exerted to the prismatic and revolute joints. If the external force does not have a varying vertical (off-plane) component, the vertical (off-plane) normal forces working at the three prismatic and revolute joints may be assumed to be uniform. Accordingly, the static and Coulomb friction parameters may be  
Figure 5: Free-body diagram at the slider \(i\), including the translational friction\(f_{trans}\). assumed to be constant. A more detailed friction formulation for the mechanism working in the XZ plane can be seen in Section 5.