## 1 Introduction  
No machine components are perfectly rigid. Flexibility exists in the machine components regardless of the magnitude. Components with high rigidity are commonly assumed to be rigid. A flexible body assumption becomes significant for components with low rigidity, such as those with slender or thin components. Parallel manipulators (PMs) are typically built by using slender or thin components to maintain their low weight advantage while they provide an acceptable stiffness. When a parallel manipulator is used for a high-precision application such as machining, a small elastic displacement matters and should be taken care of in order to achieve the expected accuracy. The dynamics of a flexible multibody system undergoing small elastic displacements can be represented by using either an analytical (exact) model, discretized model or finite segment (lumped parameter) model. The analytical model employs a continuous model which only applies to very simple geometry, that is, bar or beam. The continuous model results in an infinite number of degrees of freedom which is not feasible for control purpose. The discretized models include the Rayleigh-Ritz model, sometimes called the assumed modes model (AMM), and the finite element model (FEM). The drawback of the Rayleigh-Ritz model is its inconvenience to handle a complex geometry. On the other, the finite element model can handle a complex geometry by using an appropriate element. The finite segment model represents a link as several rigid body links interconnected by springs and dampers to incorporate the flexibility of the link. A list of works utilizing the aforementioned models for the flexible multibody dynamics of manipulators can be found in References [1; 2]. The finite element model of a flexible multibody dynamics can be categorized broadly into two approaches--the incremental finite element modeling [3; 4] and the floating frame of reference formulation (FFRF) [5; 6; 7; 8]. In the former approach, the shape function can only describe a small (infinitesimal) rotation and a convected coordinate system is used in order to deal with the finite rigid body motion. Accordingly, this approach involves a linearization of the equations of motion. This approach cannot provide an exact modeling of the rigid body dynamics, that is, exact rigid body inertia, in the presence of rigid body rotation. Furthermore, this approach does not guarantee zero strains in an arbitrary rigid body motion [9]. These drawbacks are overcome by the latter approach in which the displacements are presented as reference (rigid) and elastic coordinates. In the latter approach, which is used in this work, the elastic displacements are defined in floating frames attached to the bodies. To avoid the linearization of the equations of motion, an intermediate element coordinate system is introduced in this approach. When the FFRF is used in a constrained flexible multibody dynamics, the constraints are defined for the rigid and elastic components. While the constraints for the rigid components are formulated in an identical manner to those in the rigid multibody dynamics, the constraints for the elastic components can be defined by using a certain boundary condition. Using the assumed modes model, the pinned-free [10; 11; 12; 13], pinned-pinned [14; 15], fixed-pinned [16] and fixed-free [17] boundary conditions have been used in symmetric planar PMs. Using the finite element model based on the FFRF or FEM-FFRF for short, [PERSON] et al. [18] imposed the pinned-free boundary condition to the flexible links of a symmetric planar 3 RRR PM while [PERSON] and [PERSON] [19] applied pinned-rolled, pinned-pinned and fixed-pinned boundary conditions to a similar PM. Using FEM-FFRF, [PERSON] and [PERSON] [20] applied both pinned-free and fixed-free boundary conditions to the flexible links of a symmetric 3 PRR PM. When the finite element model is used in an incremental approach, such boundary conditions are not used [21; 22; 23]. Table 1 summarizes the aforementioned use of various boundary conditions.