Direct Linearization via the Gibbs Function

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Julie J. Parish ◽  
John E. Hurtado
Keyword(s):  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Linearization Method ◽  
Gibbs Function ◽  
Linearized Equations ◽  
Linearization Methods ◽  
Direct Linearization ◽  
Analysis Design ◽  
And Control

Linearized governing equations are often used in analysis, design, and control applications for dynamical systems. Linearized equations of motion can be formed in either an indirect or direct manner, that is, by first forming or bypassing the full nonlinear equations. Direct linearization is useful for easing the computation of linearized equations, particularly when the full nonlinear equations are not immediately desired. Currently, direct linearization methods derived from a Lagrangian perspective are available. In this paper, these methods are extended to reflect a Gibbs/Appell viewpoint. The resulting directly linearized equations take advantage of features of a Gibbs/Appellian formulation such as the ability to handle nonholonomic constraints and use of quasi-velocities. The Gibbs function and resulting equations are reviewed, the direct linearization method is explained, and a new method for directly linearizing equations via an augmented Gibbs function is presented with examples.

Direct Linearization of Continuous and Hybrid Dynamical Systems

2009 ◽  
Vol 4 (3) ◽  
Author(s):  
Julie J. Parish ◽  
John E. Hurtado ◽  
Andrew J. Sinclair
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Equations ◽  
Equilibrium Configuration ◽  
Equations Of Motion ◽  
Hybrid Dynamical Systems ◽  
Linearized Equations ◽  
Direct Linearization ◽  
Taylor Series Expansions ◽  
Nonlinear Equations Of Motion ◽  
And Control

Nonlinear equations of motion are often linearized, especially for stability analysis and control design applications. Traditionally, the full nonlinear equations are formed and then linearized about the desired equilibrium configuration using methods such as Taylor series expansions. However, it has been shown that the quadratic form of the Lagrangian function can be used to directly linearize the equations of motion for discrete dynamical systems. This procedure is extended to directly generate linearized equations of motion for both continuous and hybrid dynamical systems. The results presented require only velocity-level kinematics to form the Lagrangian and find equilibrium configuration(s) for the system. A set of selected partial derivatives of the Lagrangian are then computed and used to directly construct the linearized equations of motion about the equilibrium configuration of interest, without first generating the entire nonlinear equations of motion. Given an equilibrium configuration of interest, the directly constructed linearized equations of motion allow one to bypass first forming the full nonlinear governing equations for the system. Examples are presented to illustrate the method for both continuous and hybrid systems.

scholarly journals On modelling and control design for self-balanced two-wheel vehicle

2008 ◽  
Vol 30 (3) ◽  
Author(s):  
Nguyen Hoang Quang
Keyword(s):  
Numerical Simulation ◽  
Differential Equations ◽  
Mobile Robot ◽  
Equations Of Motion ◽  
Control Design ◽  
Modeling And Control ◽  
Linearized Equations ◽  
Stable Motion ◽  
Simulation Results ◽  
And Control

In this paper, the modeling and control design of a self-balancing mobile robot are presented. The method of sub-structures is employed to derive the differential equations of motion of the robot. Based on the linearized equations of motion, a controller is designed to maintain a stable motion of the robot. Some numerical simulation results are shown to clarify the designed controller.

scholarly journals Analysis of the brachistochronic motion of a variable mass nonholonomic mechanical system

2016 ◽  
Vol 43 (1) ◽  
pp. 19-32 ◽  
Author(s):  
Bojan Jeremic ◽  
Radoslav Radulovic ◽  
Aleksandar Obradovic
Keyword(s):  
Differential Equations ◽  
Mechanical System ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Variable Mass ◽  
Nonholonomic Constraints ◽  
Initial Position ◽  
Mass Particle ◽  
Control Forces ◽  
And Control

The paper considers the brachistochronic motion of a variable mass nonholonomic mechanical system [3] in a horizontal plane, between two specified positions. Variable mass particles are interconnected by a lightweight mechanism of the ?pitchfork? type. The law of the time-rate of mass variation of the particles, as well as relative velocities of the expelled particles, as a function of time, are known. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are created based on the general theorems of dynamics of a variable mass mechanical system [5]. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved, in this case, as the simplest task of optimal control by applying Pontryagin?s maximum principle [1]. A corresponding two-point boundary value problem (TPBVP) of the system of ordinary nonlinear differential equations is obtained, which, in a general case, has to be numerically solved [2]. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are determined. The analysis of the brachistochronic motion for different values of the initial position of a variable mass particle B is presented. Also, the interval of values of the initial position of a variable mass particle B, for which there are the TPBVP solutions, is determined.

A General Approach to the Dynamics of Nonholonomic Mobile Manipulator Systems

2002 ◽  
Vol 124 (4) ◽  
pp. 512-521 ◽  
Author(s):  
Qing Yu ◽  
I-Ming Chen
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Mobile Platform ◽  
Mobile Manipulator ◽  
Dynamic Interactions ◽  
Dynamics Of Multibody Systems ◽  
Minimum Number ◽  
Simulation And Control ◽  
And Control

This paper studies the dynamic modeling of a nonholonomic mobile manipulator that consists of a multi-degree of freedom serial manipulator and an autonomous wheeled mobile platform. The manipulator is rigidly mounted on the mobile platform, and the wheeled mobile platform moves on the ground subjected to nonholonomic constraints. Forward Recursive Formulation for the dynamics of multibody systems is employed to obtain the governing equation of the mobile manipulator system. The approach fully utilizes the existing equations of motion of the manipulator and that of the mobile platform. Furthermore, terms representing the dynamic interactions between the manipulator and the mobile platform can be observed. The resulting dynamic equation of the mobile manipulator has the minimum number of generalized coordinates and can be used for the purpose of dynamic simulation and control design, etc. The implementation issues of the model are discussed.

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

scholarly journals A SURVEY OF LAGRANGIAN MECHANICS AND CONTROL ON LIE ALGEBROIDS AND GROUPOIDS

2006 ◽  
Vol 03 (03) ◽  
pp. 509-558 ◽  
Author(s):  
JORGE CORTÉS ◽  
MANUEL DE LEÓN ◽  
JUAN C. MARRERO ◽  
D. MARTÍN DE DIEGO ◽  
EDUARDO MARTÍNEZ
Keyword(s):  
Control Systems ◽  
Classical Field Theory ◽  
Nonholonomic Constraints ◽  
Classical Field ◽  
Lie Algebroids ◽  
Hamiltonian Mechanics ◽  
Lagrangian Mechanics ◽  
Geometric Description ◽  
Lagrangian And Hamiltonian Mechanics ◽  
And Control

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.

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