Vibration Control and Trajectory Tracking for General In-Plane Motion of an Euler–Bernoulli Beam Via Two-Time Scale and Boundary Control Methods

Abstract

In this paper, general in-plane trajectory tracking problem of a flexible beam is studied. To obtain the dynamic equations of motion of the beam, Hamiltonian dynamics is used and then Lagrange’s equations of beam dynamics and corresponding expressions for boundary conditions are derived. Resulting equations show that the coupled beam dynamics including beam vibration and its rigid in-plane motion take place in two different time domains. By using two-time scale (TTS) control theory, a control scheme is elaborated that makes the orientation and position of the mass center of the beam track a desired trajectory while suppressing its vibration. TTS composite controller has two parts: one is a tracking controller designed for the slow (rigid) subsystem, and the other one is a stabilizing controller for the fast (flexible) subsystem. For the fast subsystem, the proposed boundary control (BC) method does not require any information about vibration along the beam except at the end points, nor requires discretizing the partial differential equation of beam vibration to a set of ordinary differential equations. So, the method avoids the need for instruments to measure data from vibration of any point along the beam or designing an observer for estimating this information. Also, the proposed method prevents control spillover due to discretization. Simulation results show that fast BC is able to remove undesirable vibration of the flexible beam and the slow controller provides very good trajectory tracking with acceptable actuating forces/moments.

Robust Control of Collaborative Manipulators - Flexible Object System

“…Keeping that in mind, the term EI /  in (36), which has large magnitude compared to other coefficients [20], can be redefined as,…”

“… and also in R. These flexible parameters have to be uncoupled from the above matrices and vectors by using the singular perturbation technique. This technique also accounts for the neglected high frequency characteristics when the beam undergoes vibration [20]. Using a perturbation parameter, say 2 ,…”

“…As a matter of fact, there are some results available for single-link flexible manipulators where the PDE is adopted without the need of discretization or approximation, see, for example, [18] and [19]. In these schemes, the strain signal is used instead of the measurements of flexible coordinates and their time derivatives.…”

“…In these schemes, the strain signal is used instead of the measurements of flexible coordinates and their time derivatives. Quite recently, [20] and [21] also considered the PDE in their controller designs. In the present study the dynamics for the flexible object moved by two planar rigid manipulators will be described by the PDE, and the controller will be designed without using the assumed mode method.…”

“…The proposed robust controller has the following features: 1) it does not need the information of the modes; 2) exact knowledge of the manipulators parameters or the beam are not required; and 3) it satisfies Tikhnov's theorem. It should be noted that the singular perturbation technique has also been applied to the singlelink flexible beams, say, for example, in [20] and [21], where a slow system and fast subsystems were identified as well. Comparing with the case of the manipulators handling a flexible beam, the major difference lies in the so-called slow system, where it couples with manipulator dynamics.…”