Modeling of a One Flexible Link Manipulator

Dynamics and control of flexible robot manipulators have received wider attention during the last two decades. In manufacturing and space applications, the use of lightweight structures in robot manipulators is motivated by their capacity for high speed maneuvers, their high payload to arm weight ratio, higher mobility, reduced energy consumption, and lower inertia forces for accurate positioning. To insure satisfactory performances of such systems, their flexibility should be included in modeling and in control design. This flexibility becomes more significant in cases of larger structures and more stringent performance demands. In modeling flexible link manipulators, the most widely used methods to generate spatially discrete models are the Assumed-Mode Method (AMM), and the Finite Element Method (FEM). The accuracy of the dynamical model obtained from the analytical formulation is highly dependent on the adopted mode shapes of the link deflection and their number. In the AMM, the shape functions are typically eigenfunctions of a closely related simpler problem with standard boundary conditions (BCs). For example, the Euler-Bernoulli beam in one of the following configurations (Mirovitch, 1967): clamped-free, pinned-free, clampedmass, or pinned-mass. In the FEM, the shape functions, known as interpolation functions, are simple polynomials that verify the continuity conditions between two adjacent elements or nodes. Examples of interpolation functions are Hermite cubics (Chen & Menq, 1990), cubic splines (Cho et al., 1991; Saad et al., 2006) and cubic B-splines (Saad et al., 2006). In the literature, most of the comparison studies that have been done are for clamped versus pinned mode shapes (Barbieri & Ozguner, 1988; Cetinkunt & Yu, 1991; Hastings and Book, 1987). The general conclusion is that for a slewing beam, clampedmodes aremore appropriate than pinned modes. Meirovitch and Kwak (Mirovitch & Kwak, 1990) compared the convergence rate of a clamped-free assumed-mode model vs. a linear interpolation finite elements model in estimating the frequencies of a horizontal beam with longitudinal deformation. The convergence rate was slow in both cases. To accelerate the rate of convergence, assumedmodes that take into account the natural BCs and interpolation functions that have the ability to satisfy the differential equation of the system were introduced. Buffinton and Lam (Buffinton & Lam, 1992) compared a lumped-parameter model versus an AMM in modeling and control of a one-link flexible manipulator in the horizontal plane. They concluded that from a control viewpoint, the AMM based model yields better performances when compared to lumped-parameter model. In comparing two clamped-mass assumed-modes and two Hermite cubic finite elements, Theodore and Ghosal (Theodore & Ghosal, 1995) concluded that 4