A Lagrangian Formulation of the Dynamic Model for Flexible Manipulator Systems

This paper presents a procedure for deriving dynamic equations for manipulators containing both rigid and flexible links. The equations are derived using Hamilton’s principle, and are nonlinear integro-differential equations. The formulation is based on expressing the kinetic and potential energies of the manipulator system in terms of generalized coordinates. In the case of flexible links, the mass distribution and flexibility are taken into account. The approach is a natural extension of the well-known Lagrangian method for rigid manipulators. Properties of the dynamic matrices, which lead to a less computation, are shown. Boundary-value problems of continuous systems are briefly described. A two-link manipulator with one rigid link and one flexible link is analyzed to illustrate the procedure.

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A Constrained Lagrange Formulation of Multilink Planar Flexible Manipulator

Journal of Vibration and Acoustics ◽

10.1115/1.2827455 ◽

2008 ◽

Vol 130 (3) ◽

Cited By ~ 10

Author(s):

M. Vakil ◽

R. Fotouhi ◽

P. N. Nikiforuk ◽

H. Salmasi

Keyword(s):

Closed Form ◽

Dynamic Model ◽

Vibration Suppression ◽

Equations Of Motion ◽

Flexible Manipulator ◽

Dynamic Equations ◽

Flexible Link ◽

Element Analysis ◽

Constraint Matrix ◽

Generalized Coordinates

In this article, the closed-form dynamic equations of planar flexible link manipulators (FLMs), with revolute joints and constant cross sections, are derived combining Lagrange’s equations and the assumed mode shape method. To overcome the lengthy and complicated derivative calculation of the Lagrangian function of a FLM, these computations are done only once for a single flexible link manipulator with a moving base (SFLMB). Employing the Lagrange multipliers and the dynamic equations of the SFLMB, the equations of motion of the FLM are derived in terms of the dependent generalized coordinates. To obtain the closed-form dynamic equations of the FLM in terms of the independent generalized coordinates, the natural orthogonal complement of the Jacobian constraint matrix, which is associated with the velocity constraints in the linear homogeneous form, is used. To verify the proposed closed-form dynamic model, the simulation results obtained from the model were compared with the results of the full nonlinear finite element analysis. These comparisons showed sound agreement. One of the main advantages of this approach is that the derived dynamic model can be used for the model based end-effector control and the vibration suppression of planar FLMs.

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Resonance Analysis of Manipulators with Flexible Link and Flexible Joint

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.383-390.2868 ◽

2011 ◽

Vol 383-390 ◽

pp. 2868-2874

Author(s):

Zhi Hui Gao ◽

Yu Shu Bian

Keyword(s):

Motion Planning ◽

Structure Design ◽

Flexible Manipulator ◽

Dynamic Equations ◽

Flexible Link ◽

Flexible Joint ◽

Joint Flexibility ◽

Resonance Analysis ◽

Link Flexibility ◽

Main Factors

Worse than common vibration, resonance is a form of severe vibration. It is very important and useful to know what factors and conditions can result in resonance of flexible manipulators, when both link flexibility and joint flexibility are taken into account. In this paper, resonance analysis of the flexible manipulator with both link flexibility and joint flexibility is studied. Based on the flexible dynamic equations, main factors resulting in resonance of the flexible manipulator are analyzed. Furthermore, several conditions exciting resonance are derived and verified with numerical simulations. These conclusions are helpful to predict resonance and useful to the structure design and motion planning for a flexible manipulator to evade resonance

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Singular perturbation adaptive control and vibration suppression of free-flying flexible space manipulators

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406214551777 ◽

2014 ◽

Vol 229 (11) ◽

pp. 1989-1997 ◽

Cited By ~ 16

Author(s):

Xiao Yan Yu ◽

Li Chen

Keyword(s):

Adaptive Control ◽

Singular Perturbation ◽

Lagrange Equation ◽

Momentum Conservation ◽

Flexible Manipulator ◽

Physical Parameters ◽

Flexible Link ◽

Linear Quadratic ◽

Flexible Links ◽

Space Manipulators

Singular perturbation adaptive control is designed for free-flying space manipulators with multiple flexible links and unknown physical parameters. The dynamical Lagrange equation was established based on assumed mode technique and linear momentum conservation theory. A singular perturbation model has been formulated and used for designing a reduced-order controller. This controller consisted of a slow control component and a fast control component. An adaptive control law was constructed for the slow counterpart of the flexible manipulator. The flexible-link fast subsystem controller would damp out the vibrations of flexible links by optimal linear quadratic regulator method. Numerical simulations by undertaking a computer simulation of a two-flexible-link space manipulator using the fourth-order Runge–Kutta integration method showed that the link vibrations had been stabilized effectively with good tracking performance.

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