## 1 Introduction  
The increasing need for high inertia and ultra-precision in optical devices of observational instruments motivates researchers to develop new design of grating tiling device [1]. The research activities on grating device have newly gained greater attention for industrial applications as their advantages become better known and undoubtedly documented [2]. The kinematic and dynamic analysis of the grating device are the most paramount parameters used in optimizing the grating device design [3]. The stability of tiling grating device is one of the most important parameters which need to be achieved. The main factors affecting the stability of grating mechanism are environmental vibration and thermal loading. Grating device stability can be optimized by structural and control algorithm adjustments which are composed of capacitive sensors and piezoelectric ceramic (PZT) actuator [4].  
In previous works, modelling and design of the grating devices was carried out using finite element analysis (FEA) [5, 6]. However, FEA always comes at the expense of the increased computational effort and is limited by the computing capabilities. Moreover, the grating devices exhibit definite rigid body motion as well as elastic deformations that might be considered relatively small according to the device movement. Thus, it was pointed out in reference [7] that the multibody dynamics approach may actually be the most suitable for modeling the system.  
The goal of the DFD procedure is to obtain an efficient design cycle by integrating the dynamics of interconnected systems, including nonlinearities, vibration analysis and parameters estimation, with current design methodologies [8; 9]. In the last two decades, MBS became the most effective tool in modeling complex mechanical systems for simulation and system analysis. Many commercial and educational programs have adopted the MBS approach to develop systematized subroutines to build the dynamic structure of systems [10].  
In contrast to the finite element method, the MBS approach is the best-suited for modeling dynamic systems that exhibit rigid body motion with definite rotation as well as small and/or large deformation [11; 12]. Three formulations for modeling dynamic problems associated with flexible multibody system approach are found in literature, which include the floating frame of reference (FFR) [13], and the structure-preserving method [14] and the absolute nodal coordinate (ANC) method [15]. In view of the nature of the movement in the grating system application under study and its motion characteristic that is associated with limited deformation, the (FFR) formula was used in this work.  
In this paper, firstly, the grating system is described, the multibody model is introduced and the resulting terms of equations of motion are constructed using MATLAB Symbolic Math Toolbox. Second, the proposed DFD procedure is utilized to optimize the design of the grating device system. Finally, the designed grating system device is manufactured, assembled and exploited for experimental validation. The remainder of this study is organized as follows: Section 2 describes the grating device mechanism as a multibody system. In Section 3, the system equations of motion are derived and presented, and the mathematical model is constructed. Section 4 presents the numerical results and discussion that are followed by the experimental validation in Section 5, and the conclusion is listed in Section 6.