## 4 Boundary Conditions for the Elastic Coordinates  
What meant by the boundary conditions here are some reference conditions exclusively corresponding with the elastic coordinates only. They are also commonly called the reference conditions. These boundary conditions should be defined to eliminate the redundancy in the resulting linear system so that a unique solution, that is, unique displacement, can be obtained. Hence, these boundary conditions are not defined to constrain the manipulator in order to achieve the expected rigid body motions which are already constrained by the constraint equations representing the joints in the manipulator. In other words, the elastic coordinates should be reduced through the boundary conditions in order to convert the system from a singular system to a non-singular one and therefore  
Figure 4: Assignment of the frames. Figure 3: The (**a**) schematic and (**b**) prototype of the hybrid five-axis machine developed by conjugating two nonsymmetric planar 3 PRR PMs. make it solvable. This reduction is conducted by suppressing some of the elastic coordinates based on the boundary conditions. The determination of the boundary conditions depends on the expected behavior of the elastic displacement in a multibody system. In general, different boundary conditions give different solutions. These boundary conditions include fixed-pinned, fixed-free (cantilever), pinned-pinned, pinned-rolled (simply supported) and pinned-free boundary conditions. The fixed-pinned boundary conditions do not allow any type of elastic displacement at the initial end of the body \(i\) but allow the slope at the final end. The fixed-rolled boundary conditions do not allow any type of elastic displacement at the initial end of the body \(i\) but allow the axial elastic displacement and the slope at the final end. The fixed-free boundary conditions do not allow any type of elastic displacement at the initial end of the body \(i\) but allow all types of elastic displacement at the final end. The pinned-pinned boundary conditions imply that only slopes are allowed at the two ends of the body \(i\). The pinned-rolled boundary conditions imply that the axial elastic displacement is allowed at the final end of the body \(i\) beside the slopes at both the ends. The pinned-free boundary conditions indicate that the slope at the initial end of the body \(i\) is allowed while the axial elastic displacement, transversal elastic displacement and slope are all allowed at the final end. Since the translational elastic displacement at the final end of each body is of interest, at least one translational DOF at the final end should be allowed. If fixed-pinned or pinned-pinned boundary conditions are used, no translational elastic displacement is allowed at the final end. As the result, there will be no elastic displacement at the end-effector of the manipulator at hand as it is defined at a link final end. This will be useless as our interest is the elastic displacement at the end-effector. However, in some other applications where one is interested in the elastic displacement at a certain intermediate point along the link, the fixed-pinned or pinned-pinned boundary conditions can be used. Using either the fixed-rolled or the pinned-rolled boundary conditions, the axial displacement at the final point is allowed. However, the transversal elastic displacement is not allowed. For slender links such as the links of the manipulator at hand, the transversal elastic displacement is typically much more significant than the axial displacement. Using either the fixed-free or the pinned-free boundary conditions, both the axial and transversal displacements at the final point is allowed. If the pinned-free boundary condition is used, the system is still singular, which physically means underconstrained, unless an additional constraining force is added. Based on this consideration, the fixed-free boundary conditions are used in this work.