## 1 Introduction  
Commercial robotics composed of multiple links have been widely applied in several industrial domains. For serial robots with rigid links, they are usually invented to maintain their high repeated accuracy with heavy mechanical structures. This feature leads to a dynamic motion of the mechanical system in a relatively limited working speed because of the large mass-inertia of bulky structures. With the development of new composite materials and advanced manufacturing methods, the mechanical flexibility of robotic manipulators has attracted many researchers with the merits of high-speed motion, large-scale manipulation, and low manufacturing cost [1; 2; 3; 4; 5; 6]. One of the principal issues for using flexible-link manipulators with rotational actuators is to develop the accurate dynamic model of flexible systems, which had been studied by researchers and engineers for over 20 years [7; 8; 9; 10; 11; 12]. The ultimate performance of model-based controllers is mainly dependent on the accuracy of dynamic system models [13]. The particular feature of a flexible link is denoted as the capability of allowing a transversal deflection away from the predefined undeformed link. Regarding the flexible beam modeling methods, there are two well-known approaches, namely the Timoshenko beam theory [14] and the Euler-Bernoulli beam theory [7; 9; 12; 15; 16; 17; 18]. The Euler-Bernoulli beam theory is widely used with the assumption of neglecting both the shear deformation and rotary mass moment of inertia. A potential technique to reduce the influence of shear deformation during a large motion proposed in [17; 18] is to use two parallel cable units on both sides of the flexible beam. In addition, the Euler-Bernoulli beam theory is sufficient to predict the dynamic performance of slender composite beams with a slenderness ratio larger than 20 [19]. New materials for flexible systems can be applied in the design of soft exoskeletons, which have been reviewed in [20]. The governing equations of motion of a flexible link with tip payload can be modeled using six methods [21]. [PERSON]'s principle [7; 10; 12; 22], Lagrange equations [21], and Newton-Euler equations [23] are often applied for developing a dynamic system model of a mechanical system. A dynamic modeling of a single-link flexible robot with a tip payload using the Newton-Euler formulation was addressed by [PERSON] [23]. Natural frequencies and corresponding modal shapes were obtained for a study case in accordance with those in the literature. An accurate dynamic model of a flexible-link system can be expressed by Partial Differential Equations (PDEs) with a set of boundary conditions. Furthermore, a closed-form characteristic equation is yielded from the PDEs of flexible-link systems, which can be utilized to study the influence of structure design parameters on vibration features [9]. As for discrete flexible-link models, the Finite Element Method (FEM) is applied to transform PDEs into Ordinary Differential Equations (ODEs) for the purpose of investigating the vibration characteristics of a flexible-link system in the time and frequency domains [22]. Another method, the Assumed Mode Method (AMM) [24], is exploited by replacing the accurate shape models with some shape models of classical flexible beam models (e.g., clamped-free beams, clamped-clamped beams, clamped-roller beams, and roller-free beams). Generally, FEM with more degrees of freedom produces more precise results than when using AMM. However, FEM costs more time to calculate the results of flexible-link systems in comparison to AMM. Moreover, it is worth noting that the foreshortening (geometric stiffening or stress stiffness in [8], as shown in Figure 1) of the flexible link should be considered in the dynamic modeling for rotating structures [21; 25; 26; 27], while it is negligible for small angular velocity [7]. In [21], [PERSON] studied the dynamics models of a flexible beam using different methods based on the assumptions by neglecting the foreshortening effect.