机械系统动力学Rosenberg嵌入法的扩展与解耦：一阶约束与二阶约束的整合

doi:10.3901/JME.2023.09.101

机械工程学报 ›› 2023, Vol. 59 ›› Issue (9): 101-115.doi: 10.3901/JME.2023.09.101

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机械系统动力学Rosenberg嵌入法的扩展与解耦：一阶约束与二阶约束的整合

赵睿英¹, 余进¹, CHEN Y H², 冯艳丽³, 曹学鹏¹

1. 长安大学公路养护设备国家工程实验室西安 710065;
2. 佐治亚理工学院乔治·W·伍德拉夫机械工程学院亚特兰大 30332 美国;
3. 西安航天精密机电研究所西安 710199

收稿日期:2022-05-17 修回日期:2022-08-21 出版日期:2023-05-05 发布日期:2023-07-19
通讯作者: 曹学鹏(通信作者),男,1982年出生,博士,副教授,主要研究方向为机电系统的动力学建模与控制、环境感知及轨迹规划。E-mail:caoxp2011@163.com E-mail:caoxp2011@163.com
作者简介:赵睿英,女,1985年出生,博士,副教授。主要研究方向为机器人动力学建模与智能控制、工程机械集群控制。E-mail:ruiying.zhao@chd.edu.cn
基金资助:
中国博士后科学基金第14批特别资助(2021T140585)、陕西省重点研发计划(2021ZDLGY09-02)和陕西省自然科学基础研究计划面上(2020JM-240)资助项目。

Extending and Decoupling of the Rosenberg Embedding Method for Mechanical System Dynamics:The Integration of the First-order and the Second-order Constraints

ZHAO Ruiying¹, YU Jin¹, CHEN Y H², FENG Yanli³, CAO Xuepeng¹

1. National Engineering Laboratory for Highway Maintenance Equipment, Chang'an University, Xi'an, 710065;
2. The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta 30332, USA;
3. Xi'an Aerospace Precision Electromechanical Institute, Xi'an 710199

Received:2022-05-17 Revised:2022-08-21 Online:2023-05-05 Published:2023-07-19

摘要/Abstract

摘要： 如何正确又有效地将非完整约束引入一直是机械系统动力学建模的重点和难点。传统的机械系统动力学建模方法(如:拉格朗日乘子法、Gibbs-Appell方法和Kane方法)通过借助拉格朗日乘子、广义伪速度和伪加速度等辅助变量来建立系统的运动方程,这些方法正确,但步骤复杂、计算量大。相比而言,伯克利大学力学大师Reinhardt M. Rosenberg提出了一种完全不同的约束嵌入方法,该方法将约束嵌入到虚位移中,再借助基本方程来建立机械系统的动力学模型,可获得低维的机械系统动力学模型。与上述传统建模方法相比,Rosenberg嵌入法简单、直观且无需辅助变量,弥补了其他约束嵌入建模方法在机械系统动力学中可能存在不适用问题(特别是非完整约束)。然而,该方法借助"嵌入"约束对系统降维,存在降维后的动力学模型耦合性较强,且后续解耦困难等问题。因此,创新地提出将约束的一阶与二阶形式进行整合的思想,对Rosenberg方法进行扩展和补充,建立完全解耦的机械系统动力学模型,并利用经由完全不同理念所推导出的Udwadia-Kalaba方程对扩展后的Rosenberg嵌入法进行理论验证,确认扩展后的方法正确、有效。其次,基于扩展后的Rosenberg嵌入法,建立了太空悬浮机器人和三轮全向移动机器人的动力学解析模型,通过与Udwadia-Kalaba方程建立模型的数值仿真结果相对比,完全验证了所建动力学模型的正确性,为机械系统动力学建模提供了一个具有实际应用价值的新方法。

关键词: 机械系统, 约束, 动力学建模, Udwadia-Kalaba方程

Abstract: The proper approach to identify the constraint force generated by the constraints is paramount and the key for modeling of the mechanical systems. The traditional dynamics modeling methods for the nonholonomic mechanical systems (e.g., Lagrange multipliers, Gibbs-Appell equation, and Kane equation) have to utilize the auxiliary variables to establish the motion equations (like Lagrange multipliers, generalized pseudo-velocity, and pseudo-acceleration). While, by embedding the nonholonomic constraints into the virtual displacements, Rosenberg proposed a novel approach to formulate the motion equation without the auxiliary variables based on the fundamental equation. In contrast to the traditional methods, Rosenberg's approach is intuitive and the auxiliary variables free. It is applicable for both holonomic and nonholonomic mechanical systems and can be considered as a remediation for most of the existing constraint embedding methods. Due to "embedding" constraints into the modeling, the dimensions of the dynamics established by Rosenberg's approach are reduced, which may lead to the coupling issues for the following analysis and control. Hence, by virtue of the integration of the first order and the second order form of constraints, the paper creatively extends and supplements the Rosenberg embedding method to derive the uncoupled motion equation of the constrained mechanical systems. By the Udwadia-Kalaba equation, the extended Rosenberg's approach is theoretically verified. To demonstrate the application of the modeling procedures, the dynamics of a free-floating robot and a three-wheeled omnidirectional robot are constructed by the proposed approach. Through the numerical simulation of the illustrative examples, the established dynamics models are completely validated. The paper provides a solid theoretical basis for the future application of this extended Rosenberg embedding method.

Key words: mechanical system, constraint, dynamics modeling, Udwadia-Kalaba equation

中图分类号:

TP24

赵睿英, 余进, CHEN Y H, 冯艳丽, 曹学鹏. 机械系统动力学Rosenberg嵌入法的扩展与解耦：一阶约束与二阶约束的整合[J]. 机械工程学报, 2023, 59(9): 101-115.

ZHAO Ruiying, YU Jin, CHEN Y H, FENG Yanli, CAO Xuepeng. Extending and Decoupling of the Rosenberg Embedding Method for Mechanical System Dynamics:The Integration of the First-order and the Second-order Constraints[J]. Journal of Mechanical Engineering, 2023, 59(9): 101-115.

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机械系统动力学Rosenberg嵌入法的扩展与解耦：一阶约束与二阶约束的整合

Extending and Decoupling of the Rosenberg Embedding Method for Mechanical System Dynamics:The Integration of the First-order and the Second-order Constraints

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