scholarly journals Exponential stabilization of flexural sway vibration of gantry crane via boundary control method

2019 ◽  
Vol 26 (1-2) ◽  
pp. 36-55
Author(s):  
Farshid Entessari ◽  
Ali Najafi Ardekany ◽  
Aria Alasty
Keyword(s):  
Differential Equation ◽  
Boundary Control ◽  
Control Method ◽  
Equations Of Motion ◽  
Large Angle ◽  
Uniform Boundedness ◽  
Gantry Crane ◽  
Partial Differential ◽  
Control Approach ◽  
Sway Motion

This paper aims to develop a boundary control solution for complicated gantry crane coupled motions. In addition to the large angle sway motion, the crane cable has a flexural transverse vibration. The Hamilton principle has been utilized to derive the governing partial differential equations of motion. The control objectives which are sought include: moving the payload to the desired position; reducing the payload swing with large sway angle; and finally suppressing the cable transverse vibrations in the presence of boundary disturbances simultaneously. These simultaneous boundary control objectives make the problem challenging. The proposed control approach is based on the original nonlinear hybrid partial differential equation–ordinary differential equation model without any simplifications of sway motion nonlinearities, coupling effects, and the effect of gravitational force. Using the Lyapunov method, a boundary control law has been designed which guarantees the exponential stability and uniform boundedness of the closed-loop system. In order to demonstrate the effectiveness of the proposed control method, numerical simulation results are provided by applying the finite difference method.

scholarly journals Vibration Suppression Control of a Flexible Gantry Crane System with Varying Rope Length

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Tung Lam Nguyen ◽  
Trong Hieu Do ◽  
Hong Quang Nguyen
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Vibration Suppression ◽  
Direct Method ◽  
Equations Of Motion ◽  
Gantry Crane ◽  
Partial Differential ◽  
Control Approach ◽  
Lyapunov’S Direct Method

The paper presents a control approach to a flexible gantry crane system. From Hamilton’s extended principle the equations of motion that characterized coupled transverse-transverse motions with varying rope length of the gantry is obtained. The equations of motion consist of a system of ordinary and partial differential equations. Lyapunov’s direct method is used to derive the control located at the trolley end that can precisely position the gantry payload and minimize vibrations. The designed control is verified through extensive numerical simulations.

scholarly journals The Boundary Proportion Differential Control Method of Micro-Deformable Manipulator with Compensator Based on Partial Differential Equation Dynamic Model

Micromachines ◽  
2021 ◽  
Vol 12 (7) ◽  
pp. 799
Author(s):  
Xiangli Pei ◽  
Ying Tian ◽  
Minglu Zhang ◽  
Ruizhuo Shi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dynamic Model ◽  
Control Method ◽  
System Modeling ◽  
Working Environment ◽  
Partial Differential ◽  
External Interference ◽  
Differential Control ◽  
Differential Control Method

It is challenging to accurately judge the actual end position of the manipulator—regarded as a rigid body—due to the influence of micro-deformation. Its precise and efficient control is a crucial problem. To solve the problem, the Hamilton principle was used to establish the partial differential equation (PDE) dynamic model of the manipulator system based on the infinite dimension of the working environment interference and the manipulator space. Hence, it resolves the common overflow instability problem in the micro-deformable manipulator system modeling. Furthermore, an infinite-dimensional radial basis function neural network compensator suitable for the dynamic model was proposed to compensate for boundary and uncertain external interference. Based on this compensation method, a distributed boundary proportional differential control method was designed to improve control accuracy and speed. The effectiveness of the proposed model and method was verified by theoretical analysis, numerical simulation, and experimental verification. The results show that the proposed method can effectively improve the response speed while ensuring accuracy.

Trajectory following control of lower limb exoskeleton robot based on Udwadia–Kalaba theory

2021 ◽  
pp. 107754632110317
Author(s):  
Jin Tian ◽  
Liang Yuan ◽  
Wendong Xiao ◽  
Teng Ran ◽  
Li He
Keyword(s):  
Lower Limb ◽  
Control Method ◽  
Uniform Boundedness ◽  
Adaptive Robust Control ◽  
Structural Vibration ◽  
Control Approach ◽  
Lower Limb Exoskeleton ◽  
Exoskeleton Robot ◽  
Constraint Problem ◽  
Trajectory Following

The main objective of this article is to solve the trajectory following problem for lower limb exoskeleton robot by using a novel adaptive robust control method. The uncertainties are considered in lower limb exoskeleton robot system which include initial condition offset, joint resistance, structural vibration, and environmental interferences. They are time-varying and have unknown boundaries. We express the trajectory following problem as a servo constraint problem. In contrast to conventional control methods, Udwadia–Kalaba theory does not make any linearization or approximations. Udwadia–Kalaba theory is adopted to derive the closed-form constrained equation of motion and design the proposed control. We also put forward an adaptive law as a performance index whose type is leakage. The proposed control approach ensures the uniform boundedness and uniform ultimate boundedness of the lower limb exoskeleton robot which are demonstrated via the Lyapunov method. Finally, simulation results have shown the tracking effect of the approach presented in this article.

scholarly journals Weighted multiple model adaptive boundary control for a flexible manipulator

2019 ◽  
Vol 103 (1) ◽  
pp. 003685041988646
Author(s):  
Weicun Zhang ◽  
Qing Li ◽  
Yuzhen Zhang ◽  
Ziyi Lu ◽  
Cheng Nian
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Control ◽  
Control Strategy ◽  
Flexible Manipulator ◽  
Multiple Model ◽  
Parameter Uncertainties ◽  
Partial Differential ◽  
Local Boundary ◽  
Adaptive Boundary Control

In this article, a weighted multiple model adaptive boundary control scheme is proposed for a flexible manipulator with unknown large parameter uncertainties. First, the uncertainties are approximatively covered by a finite number of constant models. Second, based on Euler–Bernoulli beam theory and Hamilton principle, the distributed parameter model of the flexible manipulator is constructed in terms of partial differential equation for each local constant model. Correspondingly, local boundary controllers are designed to control the manipulator movement and suppress its vibration for each partial differential equation model, which are based on Lyapunov stability theory. Then, a novel weighted multiple model adaptive control strategy is developed based on an improved weighting algorithm. The stability of the overall closed-loop system is ensured by virtual equivalent system theory. Finally, numerical simulations are provided to illustrate the feasibility and effectiveness of the proposed control strategy.

XIV.—On General Dynamics:—I. Hamilton's Partial Differential Equations and the Determination of their Complete Integrals

1913 ◽  
Vol 32 ◽  
pp. 164-174
Author(s):  
A. Gray
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Equations Of Motion ◽  
Partial Differential ◽  
General Dynamics ◽  
Elementary Manner ◽  
Derivatives Of

The present paper contains the first part of a series of notes on general dynamics which, if it is found worth while, may be continued. In § 1 I have shown how the first Hamiltonian differential equation is led up to in a natural and elementary manner from the canonical equations of motion for the most general case, that in which the time t appears explicitly in the function usually denoted by H. The condition of constancy of energy is therefore not assumed. In § 2 it is proved that the partial derivatives of the complete integral of Hamilton's equation with respect to the constants which enter into the specification of that integral do not vary with the time, so that these derivatives equated to constants are the integral equations of motion of the system.*

Free and Forced Localization in a Nonlinear Periodic Lattice

1993 ◽  
Author(s):  
Alexander F. Vakakis ◽  
Melvin E. King ◽  
Arne J. Pearlstein
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equation ◽  
Structural Disorder ◽  
Periodic Lattice ◽  
Periodic Motions ◽  
Partial Differential ◽  
Chaotic Motions ◽  
Solitary Solutions

Abstract Free and forced localized periodic motions in an infinite nonlinear periodic lattice are analytically investigated. The lattice consists of weakly coupled identical masses, each connected to the ground by a nonlinear stiffness. In order to study the localized motions of the discrete system a continuoum approximation is assumed, and the ordinary differential equations of motion are replaced by a single nonlinear partial differential equation. The time-periodic solutions of this equation are then obtained by an averaging method, and their stability is examined using an analytic linearized method. It is shown that localized periodic motions of the lattice correspond to standing solitary solutions of the partial differential equation of the continuous approximation. For the free lattice, localized free motions occur when the coupling stiffnesses forces are much smaller than the nonlinear effects of the grounding stiffnesses. Moreover, these free localized motions are detected in the perfectly periodic nonlinear lattice, i.e., even in the absence of structural disorder (a feature which is an essential prerequisite for linear mode localization). When harmonic forcing is applied to the chain, localized, non-localized, and chaotic motions occur, depending on the spatial distribution and the magnitude of the applied loads. A variety of spatially distributed harmonic loads and analytic expressions for the resulting localized motions of the chain are derived.

NEGATIVE IMAGINARY THEOREM WITH AN APPLICATION TO ROBUST CONTROL OF A CRANE SYSTEM

2016 ◽  
Vol 78 (6-11) ◽  
Author(s):  
Auwalu M. Abdullahi ◽  
Z. Mohamed ◽  
M. S. Zainal Abidin ◽  
R. Akmeliawati ◽  
A. R. Husain ◽  
...  
Keyword(s):  
Control Method ◽  
Sliding Mode ◽  
Sine Wave ◽  
Pole Placement ◽  
Gantry Crane ◽  
Linear Matrix ◽  
Control Scheme ◽  
Nonlinear Control Method ◽  
Pole Placement Controller ◽  
Sway Motion

This paper presents an integral sliding mode (ISM) control for a case of negative imaginary (NI) systems. A gantry crane system (GCS) is considered in this work. ISM is a nonlinear control method introducing significant properties of precision, robustness, stress-free tuning and implementation. The GCS model considered in this work is derived based on the x direction and sway motion of the payload. The GCS is a negative imaginary (NI) system with a single pole at the origin. ISM consist of two blocks; the inner block made up of a pole placement controller (NI controller),  designed using linear matrix inequality for robustness and outer block made up of sliding mode control to reject disturbances. The ISM is designed to control position tracking and anti-swing payload motion. The robustness of the control scheme is tested with an input disturbance of a sine wave signal. The simulation results show the effectiveness of the control scheme.

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