空间刚体变换的倍矩阵描述方法

doi:10.3901/JME.2022.13.089

机械工程学报 ›› 2022, Vol. 58 ›› Issue (13): 89-100.doi: 10.3901/JME.2022.13.089

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空间刚体变换的倍矩阵描述方法

张英, 黄起能, 廖启征, 魏世民

北京邮电大学现代邮政学院(自动化学院) 北京 100876

收稿日期:2021-07-18 修回日期:2021-12-13 出版日期:2022-07-05 发布日期:2022-09-13
通讯作者: 张英(通信作者),女,1987年出生,博士,副教授。主要研究方向为机器人机构学。E-mail:graduate_yingzh@bupt.edu.cn
作者简介:黄起能,男,1996年出生,硕士。主要研究方向为机器人机构学。E-mail:hqn@bupt.edu.cn;廖启征,男,1947年出生,博士,教授。主要研究方向为机器人机构学。E-mail:qzliao@bupt.edu.cn;魏世民,男,1965年出生,博士,教授,博士研究生导师。主要研究方向为机器人机构学。E-mail:wsmly@bupt.edu.cn
基金资助:
国家自然科学基金资助项目（51605036）。

Double Matrix Representation Method for the Spatial Rigid Body Transformation

ZHANG Ying, HUANG Qineng, LIAO Qizheng, WEI Shimin

School of Modern Post (School of Automation), Beijing University of Posts and Telecommunications, Beijing 100876

Received:2021-07-18 Revised:2021-12-13 Online:2022-07-05 Published:2022-09-13

摘要/Abstract

摘要： 空间刚体变换的描述方法是机器人机构运动学分析的一个重要内容，描述方法对机构性能分析起着重要作用，不同的描述方法会导致机构运动学建模和求解的难易不同。基于对偶矩阵理论，类比倍四元数与对偶四元数的映射关系，提出并推导了空间刚体位移和变换的倍矩阵描述新方法。该方法对空间线变换进行描述，可以描述空间刚体的全位姿变换。详细推导并证明了四维旋转矩阵、倍四元数和倍矩阵这三种空间刚体变换的近似描述方法之间的相互转换关系。这三种描述方法将三维空间刚体的平移转换为4维空间的旋转，进而实现了刚体平移和旋转量纲的统一，为其比较大小提供了一种尺度。

关键词: 空间刚体变换, 对偶矩阵, 倍矩阵, 四维旋转矩阵, 倍四元数

Abstract: The representation of the spatial rigid body transformation for the robotic mechanisms is one of the important contents in kinematic analysis. The representation method plays a key role in the performance analysis of such mechanisms and whether or not the formulations and solutions to the kinematic analysis are readily obtained depending on the selective methods. In analogy with the mapping relationship between the double quaternion and the dual quaternion, the double matrix representation method for the spatial rigid body transformation is presented and derived based on the dual-number matrix. The double matrix method operates on lines in space and represents the pose transformation of the rigid body. The relationships among three representation methods such as 4D rotational matrices, double quaternions and double matrices for the spatial rigid body transformation as the limit of four-dimensional rotation transformation are derived in detail. The translation transformation of the rigid body is represented with the rotation transformation in the four-dimensional space by applying the three representation methods. Furthermore, the translational and rotational displacements of the rigid body are scaled and therefore they are compared in a physical scheme.

Key words: spatial rigid body transformation, dual matrices, double matrices, 4D rotational matrices, dual quaternions

中图分类号:

TH112

张英, 黄起能, 廖启征, 魏世民. 空间刚体变换的倍矩阵描述方法[J]. 机械工程学报, 2022, 58(13): 89-100.

ZHANG Ying, HUANG Qineng, LIAO Qizheng, WEI Shimin. Double Matrix Representation Method for the Spatial Rigid Body Transformation[J]. Journal of Mechanical Engineering, 2022, 58(13): 89-100.

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空间刚体变换的倍矩阵描述方法

Double Matrix Representation Method for the Spatial Rigid Body Transformation

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