Invariant Modeling of Mechanical Systems in Terms of Quasi-Velocities and Quasi-Forces

2009 ◽  
Author(s):  
Farhad Aghili
Keyword(s):  
Motion Control ◽  
Equations Of Motion ◽  
Control Action ◽  
Constraint Equations ◽  
Weighting Matrix ◽  
Rotational Components ◽  
Tracking Force ◽  
Invariant Formulation ◽  
Invariance Problem ◽  
Multi Body

A gauge-invariant formulation for deriving the dynamic equations of constrained multi-body systems (MBS) in terms of (reduced) quasi–velocities is presented. This formulation does not require any weighting matrix to deal with the gauge-invariance problem when both translational and rotational components are involved in the generalized coordinates or in the constraint equations. Moreover, in this formulation the equations of motion are decoupled from those of constrained force and each system has its own independent input. This allows the possibility to develop a simple force control action that is totally independent from the motion control action facilitating a hybrid force/motion control. Tracking force/motion control of constrained multi-body systems based on a combination of feedbacks on the vectors of the quasi–velocities and the configuration variables are presented.

A Gauge-Invariant Formulation for Constrained Mechanical Systems Using Square-Root Factorization and Unitary Transformation

2009 ◽  
Vol 4 (3) ◽  
Author(s):  
Farhad Aghili
Keyword(s):  
Motion Control ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Control Action ◽  
Gauge Invariant ◽  
Constrained Multibody Systems ◽  
Constrained Mechanical Systems ◽  
Weighting Matrix ◽  
Rotational Components ◽  
Invariant Formulation

A gauge-invariant formulation for deriving the dynamic equations of constrained multibody systems in terms of (reduced) quasivelocities is presented. This formulation does not require any weighting matrix to deal with the gauge-invariance problem when both translational and rotational components are involved in the generalized coordinates or in the constraint equations. Moreover, in this formulation the equations of motion are decoupled from those of constrained force, and each system has its own independent input. This allows the possibility to develop a simple force control action that is totally independent from the motion control action facilitating a hybrid force/motion control. Tracking force/motion control of constrained multibody systems based on a combination of feedbacks on the vectors of the quasivelocities and the configuration-variables are presented.

scholarly journals Motion Planning and Control of an Omnidirectional Mobile Robot in Dynamic Environments

Robotics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 48
Author(s):  
Mahmood Reza Azizi ◽  
Alireza Rastegarpanah ◽  
Rustam Stolkin
Keyword(s):  
Mobile Robot ◽  
Collision Avoidance ◽  
Motion Control ◽  
Equations Of Motion ◽  
Dynamic Environments ◽  
Discrete State ◽  
Omnidirectional Mobile Robot ◽  
Planning And Control ◽  
Avoidance Strategy ◽  
And Performance

Motion control in dynamic environments is one of the most important problems in using mobile robots in collaboration with humans and other robots. In this paper, the motion control of a four-Mecanum-wheeled omnidirectional mobile robot (OMR) in dynamic environments is studied. The robot’s differential equations of motion are extracted using Kane’s method and converted to discrete state space form. A nonlinear model predictive control (NMPC) strategy is designed based on the derived mathematical model to stabilize the robot in desired positions and orientations. As a main contribution of this work, the velocity obstacles (VO) approach is reformulated to be introduced in the NMPC system to avoid the robot from collision with moving and fixed obstacles online. Considering the robot’s physical restrictions, the parameters and functions used in the designed control system and collision avoidance strategy are determined through stability and performance analysis and some criteria are established for calculating the best values of these parameters. The effectiveness of the proposed controller and collision avoidance strategy is evaluated through a series of computer simulations. The simulation results show that the proposed strategy is efficient in stabilizing the robot in the desired configuration and in avoiding collision with obstacles, even in narrow spaces and with complicated arrangements of obstacles.

Systematic Construction of the Equations of Motion for Multibody Systems Containing Closed Kinematic Loops

1989 ◽  
Author(s):  
P. E. Nikravesh ◽  
G. Gim
Keyword(s):  
Equations Of Motion ◽  
Transformation Matrix ◽  
Velocity Transformation ◽  
Constraint Equations ◽  
Advantages And Disadvantages ◽  
Lagrange Multiplier Technique ◽  
Joint Coordinates ◽  
Minimum Number ◽  
Kinematic Loops ◽  
Equations Of Motions

Abstract This paper presents a systematic method for deriving the minimum number of equations of motion for multibody system containing closed kinematic loops. A set of joint or natural coordinates is used to describe the configuration of the system. The constraint equations associated with the closed kinematic loops are found systematically in terms of the joint coordinates. These constraints and their corresponding elements are constructed from known block matrices representing different kinematic joints. The Jacobian matrix associated with these constraints is further used to find a velocity transformation matrix. The equations of motions are initially written in terms of the dependent joint coordinates using the Lagrange multiplier technique. Then the velocity transformation matrix is used to derive a minimum number of equations of motion in terms of a set of independent joint coordinates. An illustrative example and numerical results are presented, and the advantages and disadvantages of the method are discussed.

Sliding Mode Control in Ziegler’s Pendulum With Tracking Force: Novel Modeling Considerations

2013 ◽  
Author(s):  
Ehsan Sarshari ◽  
Nastaran Vasegh ◽  
Mehran Khaghani ◽  
Saeid Dousti
Keyword(s):  
Stability Analysis ◽  
Sliding Mode Control ◽  
Dynamic System ◽  
Linear Stability Analysis ◽  
Sliding Mode ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Sliding Mode Controller ◽  
Gravity Effect ◽  
Tracking Force

Ziegler’s pendulum is an appropriate model of a non-conservative dynamic system. By considering gravity effect, new equations of motion are extracted from Newton’s motion laws. The instability of equilibriums is determined by linear stability analysis. Chaotic behavior of the model is shown by numerical simulations. Sliding mode controller is used for eliminating chaos and for stabilizing the equilibriums.

A Large Displacement Formulation Using Euler’s Angles

1999 ◽  
Author(s):  
G. Biakeu ◽  
F. Thouverez ◽  
J. P. Laine ◽  
L. Jezequel
Keyword(s):  
Kinetic Energy ◽  
Equations Of Motion ◽  
Static Solution ◽  
Element Formulation ◽  
First Order ◽  
Nonlinear Relation ◽  
Newton Raphson ◽  
Body Formulation ◽  
Multi Body ◽  
Euler’S Angles

Abstract The goal of this paper is to present a flexible multi-body formulation involving large displacements. This method is based on a separate discretisation of the kinetic and the internal energies. To introduce flexibility, we discretize the structure in elements (of two nodes): on each element of the beam discretisation, the local frame is defined using Euler’s angles. A finite element formulation is then applied to describe the evolution of these angles along the beam neutral fibre. For the kinetic energy, each element is cut into two rigid bars whose characteristics are given by a first order Taylor factorisation on the general kinetic energy expression. These bars are linked by a nonlinear relation. We obtain the equations of motion by applying the Lagrange’s equations to the system. These equations are solved using the Newmark method in dynamic and a Newton-Raphson technique while looking for a static solution. The method is then applied to very classic problems such as the curved beam problem proposed by authors such as Simo [6, 9], Lee [4] or the rotational rod presented by Avello [1] and Simo [7, 8] etc...

A Recursive Householder Transformation for Complex Dynamical Systems With Constraints

1988 ◽  
Vol 55 (3) ◽  
pp. 729-734 ◽  
Author(s):  
F. M. L. Amirouche ◽  
Tongyi Jia ◽  
Sitki K. Ider
Keyword(s):  
Dynamical Systems ◽  
General Solution ◽  
Equations Of Motion ◽  
New Method ◽  
Systems Dynamics ◽  
Complex Dynamical Systems ◽  
Governing Equations ◽  
Householder Transformation ◽  
Constraint Equations ◽  
Computer Automation

A new method is presented by which equations of motion of complex dynamical systems are reduced when subjected to some constraints. The method developed is used when the governing equations are derived using Kane’s equations with undetermined multipliers. The solution vectors of the constraint equations are determined utilizing the recursive Householder transformation to obtain a Pseudo-Uptriangular matrix. The most general solution in terms of new independent coordinates is then formulated. Methods previously used for handling such systems are discussed and the new method advantages are illustrated. The procedures developed are suitable for computer automation and especially in developing generic programs to study a large class of systems dynamics such as robotics, biosystems, and complex mechanisms.

Analysis of the Nonlinear Dynamics of a Railway Vehicle Wheelset

1973 ◽  
Vol 95 (1) ◽  
pp. 28-35 ◽  
Author(s):  
E. Harry Law ◽  
R. S. Brand
Keyword(s):  
Lateral Displacement ◽  
Equations Of Motion ◽  
Vertical Displacement ◽  
Linear Analysis ◽  
Design Parameters ◽  
Railway Vehicle ◽  
Nonlinear Variation ◽  
Constraint Equations ◽  
The Stability ◽  
Nonlinear Equations Of Motion

The nonlinear equations of motion for a railway vehicle wheelset having curved wheel profiles and wheel-flange/rail contact are presented. The dependence of axle roll and vertical displacement on lateral displacement and yaw is formulated by two holonomic constraint equations. The method of Krylov-Bogoliubov is used to derive expressions for the amplitudes of stationary oscillations. A perturbation analysis is then used to derive conditions for the stability characteristics of the stationary oscillations. The expressions for the amplitude and the stability conditions are shown to have a simple geometrical interpretation which facilitates the evaluation of the effects of design parameters on the motion. It is shown that flange clearance and the nonlinear variation of axle roll with lateral displacement significantly influence the motion of the wheelset. Stationary oscillations may occur at forward speeds both below and above the critical speed at which a linear analysis predicts the onset of instability.

Modeling and simulation of planar flexible link manipulators with rigid tip connections to revolute joints

Robotica ◽  
2004 ◽  
Vol 22 (3) ◽  
pp. 285-300 ◽  
Author(s):  
S. M. Megahed ◽  
K. T. Hamza
Keyword(s):  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Link ◽  
Link Length ◽  
Finite Segment ◽  
Revolute Joints ◽  
Body Dynamics ◽  
Lumped Masses ◽  
Multi Body ◽  
Rigid Zones

This paper presents the basis of a mathematical model for simulation of planar flexible-link manipulators, taking into consideration the effect of higher stiffness zones at the link tips. The proposed formulation is a variation of the finite segment multi-body dynamics approach. The formulation employs a consistent mass matrix in order to provide better approximation than the traditional lumped masses often encountered in the finite segment approach. The formulation is implemented into a computational code and tested through three examples; cantilever beam, rotating beam and three-link manipulator. In these examples, the length of the rigid tips at both sides of each link ranges from 0% to 6.25% of the whole link length. The zones of higher stiffness at the link tips are treated as short rigid zones. The effect of the rigid zones is averaged along with some portions of the flexible links, thereby allowing further simplification of the dynamic equations of motion. The simulation results demonstrate the effectiveness of the proposed modeling technique and show the importance of not ignoring the effect of the rigid tips.

Investigation of Dynamics and Vibration of a Three Unit PIG in Oil and Gas Pipelines

2008 ◽  
Author(s):  
Mohammad Durali ◽  
Amir Fazeli ◽  
Mohsen Azimi
Keyword(s):  
Oil And Gas ◽  
Equations Of Motion ◽  
Design Procedure ◽  
Pressure Waves ◽  
Gas Oil ◽  
Oil Pipeline ◽  
Under Construction ◽  
Multi Body ◽  
Dynamics And Vibration ◽  
Body Dynamic

In this paper, the transient motion of a three unit intelligent Pipe Inspection Gauge (PIG) while moving across anomalies and bends inside gas/oil pipeline has been investigated. The pipeline fluid has been considered as isothermal and compressible. In addition, the pipeline itself has also been considered to be flexible. The fluid continuity and momentum equations along with the 3D multi body dynamic equations of motion of the pig comprise a system of coupled dynamic differential equations which have been solved numerically. Pig’s position and velocity profiles as well as upstream and downstream fluid’s pressure waves are presented as simulation results which provide a better understanding of the complex behavior of pig motion through pipelines. This study has been conducted as a part of the design procedure for the Pig which is currently under construction.

Explicit Poincaré equations of motion for general constrained systems. Part I. Analytical results

2007 ◽  
Vol 463 (2082) ◽  
pp. 1421-1434 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Constrained Systems ◽  
Analytical Results ◽  
Multi Body ◽  
D'alembert's Principle

This paper gives the general constrained Poincaré equations of motion for mechanical systems subjected to holonomic and/or nonholonomic constraints that may or may not satisfy d'Alembert's principle at each instant of time. It also extends Gauss's principle of least constraint to include quasi-accelerations when the constraints are ideal, thereby expanding the compass of this principle considerably. The new equations provide deeper insights into the dynamics of multi-body systems and point to new ways for controlling them.

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