拉格朗日与牛顿欧拉方法：差动机器人动力学建模详解
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本文主要探讨了动态建模在差动驱动移动机器人(Differential-drive Mobile Robots, DDR)领域的应用，由Rached Dhaouadi和Ahmad Abu Hatab两位作者在《先进机器人与自动化》(AdvRobotAutom)期刊的第二卷第二期发表。他们通过结合拉格朗日法(Lagrange)和牛顿欧拉法(Newton-Euler)对DDR的动力学模型进行了深入研究。
首先，背景部分指出近年来，随着移动机器人技术和教育技术的快速发展，对轮式移动机器人(如WDRI, Wheeled Differential Drive Robots)的控制及其应用的研究兴趣显著增长。然而，尽管关于轮式机器人运动学建模的文献丰富且易于理解，对于其动力学模型的探讨却相对不足。这是因为轮式机器人的动力学特性非常复杂，非holonomic约束使得动力学分析具有挑战性，对于刚接触这一领域的工程学生来说，这是一大难点。
作者强调了开发一个详尽而精确的动态模型的重要性，该模型能够为学生提供一个全面的框架，用于仿真分析和基于模型的设计。他们通过拉格朗日法构建了动力学模型，这是一种在机械系统动力学分析中常用的方法，它将系统的总动能和势能转化为一组称为拉格朗日函数的量，然后通过对拉格朗日方程求解来揭示系统的运动规律。牛顿欧拉法则是一种更为细致的局部分析方法，它将系统分解为各个质点或刚体，并考虑它们之间的相互作用力，以便于计算每个部分的运动。
在文中，Dhaouadi和Hatab可能详细阐述了如何处理DDR的特殊性，例如双轮布局带来的非对称性、驱动力矩的分配以及如何处理由于车轮滚动和地面摩擦等因素引起的复杂运动。他们可能还讨论了如何将这些模型应用于实际控制策略的设计，比如路径规划、稳定性和动态平衡问题，以及如何通过数值方法进行模拟，以便优化机器人的性能和避免潜在的不稳定行为。
这篇论文为理解和掌握差动驱动移动机器人动力学模型提供了宝贵的资源，为教育工作者和学生提供了一个清晰的动态模型基础，促进了机器人领域尤其是动态建模技术的发展和教学。
Volume 2 • Issue 2 • 1000107
Adv Robot Autom
ISSN: 2168-9695 ARA, an open access journal
Open Access
Research Article
Dhaouadi and Hatab, Adv Robot Autom 2013, 2:2
DOI: 10.4172/2168-9695.1000107
Keywords: Dierential-drive; Mobile robot; Dynamics; Modeling;
Lagrange; Newton-Euler
Introduction
In recent years, there has been a considerable interest in the area
of mobile robotics and educational technologies [1-7]. For control
engineers and researchers, there is a wealth of literature dealing
with wheeled mobile robots (WMR) control and their applications.
However, while the subject of kinematic modeling of WMR is
well documented and easily understood by students, the subject of
dynamic modeling of WMR has not been addressed adequately in the
literature. e dynamics of WMR are highly nonlinear and involve
non-holonomic constraints which makes dicult their modeling
and analysis especially for new engineering students starting their
research in this eld. erefore, a detailed and accurate dynamic model
describing the WMR motion need to be developed to oer students
a general framework for simulation analysis and model based control
system design.
In the case of a dierential drive mobile robot (DDMR), for
example, there is no textbook available that investigates thoroughly
the dynamic modeling approach taking into consideration the non-
holonomic constraints in a step by step procedure. e analysis is
available mainly in journals, conference papers, and technical reports
[8]. Moreover, the material presented diers from one paper to
another with dierent variables and reference frames used, and various
assumptions. In addition, some papers present dierent results for the
same DDMR used, which adds to the confusion of dynamic modeling.
For the case of DDMR, the methods used are either the Lagrangian
approach [9-15] or the Newton-Euler approach [16-19]. Other
formalisms such as the Kane’s method have been also suggested as
viable approaches of modeling DDMR [20]. erefore, it is not clear
for new engineering students and researchers which concept to use and
which method oers a better physical insight on the dynamic behavior
of the system and the eect of the non-holonomic constraints. Also, it
is not clear if both methods will lead to the same nal dynamic model.
In the Newton Euler method, one has to take into account two
kinds of forces applied to a system: the given forces and the constraint
forces. e given forces include the externally impressed forces by
the actuators while the constraint forces are the forces of interaction
between the robot platform and ground through the wheels. Moreover,
in a system with interconnected elements, the components may
interact with each other through gears, springs, and frictional elements.
erefore, we need to take into account all of these forces. It is clear that
the Newtonian approach includes a few practical diculties since in
most cases these forces are not easily quantiable.
e methodology developed by Lagrange overcomes these problems
by expressing the forces in terms of the energies in the system, i.e., the
kinetic energy and the potential energy, which are scalar quantities
easily expressible in terms of the system coordinates. e derivation of
the Lagrange equations requires also that the generalized coordinates
be independent.
e Lagrangian approach usually provides a powerful and versatile
method for the formulation of the equations of motion for holonomic
systems. However, for non-holonomic systems, the usual method is to
introduce the motion constraint equations into the dynamic equations
using the additional Lagrange multipliers. ese multipliers are not
constants and are usually functions of all the generalized coordinates
and oen of time as well. ey represent a set of unknowns whose
values should be obtained as a part of the solution. To solve this
computational complexity, additional methods have been suggested to
remove the presence of the multipliers from the dynamic equations of
the given system [21,22].
e focus of this paper is to derive simple and well-structured
dynamic equations of the DDMR taking into account the non-
holonomic constraints. First, the Lagrange formulation is presented.
Coordinates transformation is used to cancel the Lagrange multipliers
to obtain well-structured equations. Second, the Newton-Euler method
is used to derive the dynamic equations of the DDMR. Major diculties
experienced in using both methods are illustrated and procedures are
outlined to oer a systematic approach to the dynamic modeling of
*Corresponding author: Rached Dhaouadi, College of Engineering, American
University of Sharjah, P.O. Box 26666, Sharjah, UAE, E-mail: rdhaouadi@aus.edu
Received
June 19, 2013; Accepted September 16, 2013; Published September
19, 2013
Citation: Dhaouadi R, Hatab AA (2013) Dynamic Modelling of Differential-Drive
Mobile Robots using Lagrange and Newton-Euler Methodologies: A Unied
Framework. Adv Robot Autom 2: 107. doi: 10.4172/2168-9695.1000107
Copyright: © 2013 Dhaouadi R, et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.
Abstract
This paper presents a unied dynamic modeling framework for differential-drive mobile robots (DDMR). Two
formulations for mobile robot dynamics are developed; one is based on Lagrangian mechanics, and the other on
Newton-Euler mechanics. Major difculties experienced when modeling non-holonomic systems in both methods are
illustrated and design procedures are outlined. It is shown that the two formulations are mathematically equivalent
providing a check on their consistency. The presented work leads to an improved understanding of differential-
drive mobile robot dynamics, which will assist engineering students and researchers in the modeling and design of
suitable controllers for DDMR navigation and trajectory tracking.
Dynamic Modelling of Differential-Drive Mobile Robots using Lagrange and
Newton-Euler Methodologies: A Unified Framework
Rached Dhaouadi* and Ahmad Abu Hatab
College of Engineering, American University of Sharjah, Sharjah, UAE
Advances in Robotics
& Automation
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ISSN: 2168-9695
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