## 1 Introduction
It can be noticed from the table that all the investigated PMs are symmetric, which means that all the legs are identical and configured in a symmetric manner and all the studies assumed that the moving platform is rigid. The flexible links under the studies are either RR links (which means that the links have revolute joints at both ends) or PR links (which means that the links have prismatic joint at one end and revolute joint at another end). While the flexible PR links were modeled by applying fixed-pinned and fixed-free boundary conditions, the flexible RR links were modeled by using various boundary conditions namely pinned-free, pinned-rolled, pinned-pinned, fixed-free and fixed-pinned boundary conditions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Reference**} & **PM** & \multirow{2}{*}{**Symmetry**} & **Moving** & **Boundary** & **Method** \\  & **Topology** & & **Platform** & **Condition** & **Condition** \\ \hline Kang and [PERSON] [10] & planar 3 PRR & Yes & rigid & Pinned-free RR & AMM \\ [PERSON] et al. [11] & planar 3 PRR & Yes & rigid & Pinned-free RR & AMM \\ [PERSON] et al. [12] & planar 3 PRR & Yes & rigid & Pinned-free RR & AMM \\ [PERSON] et al. [13] & planar 3 RRR & Yes & rigid & Pinned-free RR & AMM \\ [PERSON] et al. [14] & planar 3 PRR & Yes & rigid & Pinned-pinned RR & AMM \\ [PERSON] et al. [15] & planar 3 PRR & Yes & rigid & Pinned-pinned RR & AMM \\ [PERSON] et al. [16] & planar 3 PRR & Yes & rigid & Fixed-pinned PR & AMM \\ [PERSON] and & planar 3 PRR & Yes & rigid & Fixed-free PR & AMM \\ [PERSON] [17] & planar 3 RRR & Yes & rigid & Fixed-free RR & FEM \\ [PERSON] et al. [18] & planar 3 RRR & Yes & rigid & Pinned-rolled RR, & \\ [PERSON] and & & & & pinned-pinned RR, & FEM \\ [PERSON] [19] & planar 3 RRR & Yes & rigid & Fixed-pinned RR & \\ [PERSON] and [PERSON] [20] & planar 3 PRR & Yes & rigid & Fixed-free RR & FEM \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Previous works studying the flexible body dynamics of parallel manipulators (PMs) based on floating frame of reference formulation (FFRF). This work investigates the flexible multibody dynamics of a nonsymmetric planar 3 PRR PM by using FFRF [7; 8] with the following distinguished aspects:  
* While the previous works as summarized in Table 1 studied symmetric PMs, this work investigates a nonsymmetric planar 3 PRR manipulator previously discussed in References [24; 25]. In the previous works with symmetric topologies, only one branch, that is, one leg, can be used to evaluate the dynamics of the manipulator. As this work deals with a non-symmetric PM in which all the three legs are not identical and they are not configured in a symmetric manner, all the legs should be included in the dynamics analysis of the manipulator. * While the previous works as summarized in Table 1 assumed that the moving platform is rigid whereas another work such as [26] assumed that only the moving platform is flexible while the legs are rigid, this work assumes that the moving platform and all the legs are flexible. Hence, the links are called fully flexible. * While the previous works as summarized in Table 1 used only a single method, that is, either AMM or FEM, this work uses both the Rayleigh-Ritz and finite element methods in a comparative study. It should be noticed that the finite element model used here is based on the FFRF, not the incremental method. * While many existing works evaluated the elastic displacements of an intermediate or middle points of the links, this work evaluates the elastic displacement of the end-effector of the manipulator due to its significance in robotics application. In a nutshell, this work has two-fold goals. First, it aims at evaluating the pose of a link tip of a nonsymmetric planar 3 PRR PM, which represents the end-effector, due to the elasticity of the links. By knowing the deviation of the end-effector pose from its rigid body pose due to the link elasticity, a compensation can be performed to increase the accuracy of the manipulator motion.