基于移/转动Jacobian矩阵的少自由度并联机构性质研究
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摘要
和传统的6自由度并联机器人相比,少自由度并联机器人具有驱动件少、控制简单方便、制造容易、价格低廉等特点。近年来,少自由度并联机构成为国际机构学和机器人领域的研究热点,少自由度并联机构的性质研究包括运动学、奇异位形和工作空间等等。
运动学是机构分析的重要组成部分,Jacobian矩阵在并联机构的运动学分析当中占有十分重要的位置。传统意义的少自由度并联机构的Jacobian矩阵不是方阵,即机构的输入和输出之间不满足一一对应关系。这种现象出现的本质原因是动平台的各个输出速度分量并不是独立的。本文用移动Jacobian矩阵和转动Jacobian矩阵分别表示动平台的线速度和角速度与机构独立输出速度矢量之间的关系。通过移动和转动Jacobian矩阵,得到了n自由度并联机构(n<6)的n×n型Jacobian方阵并对其速度进行了分析;在移动和转动Jacobian矩阵的基础上,得到了表示动平台的线加速度和角加速度与机构的独立输出加速度矢量之间关系的移动和转动Hessian矩阵,进而对少自由度并联机构的加速度进行了分析。
少自由度并联机器人总体上具有比6自由度并联机器人更复杂的运动特性,特别是动平台的运动受到约束后会表现出很多特殊的性质,这使得少自由度并联机器人机型匮乏,与实际应用中的广泛要求相距甚远。本文综合出了几种3、4自由度并联机构,包括含约束分支的并联机构,驱动分支数小于自由度数的并联机构以及含非直线分支的并联机构等。以移动和转动Jacobian矩阵为基础,对所综合出来的机构进行了详细的运动学分析。
并联机构的奇异位形是并联机器人设计、应用和分析中的一个重要问题。Jacobian代数法是研究并联机构奇异位形最一般、最通用的方法。本文通过移动和转动Jacobian矩阵以及由这两个矩阵所得到的n×n型Jacobian方阵,对少自由度并联机构,特别是具有不同运动形式的3、4自由度并联机构的奇异位形进行了分析,包括三维移动并联机构,三维转动并联机构,具有混合运动的3自由度并联机构,三移一转4自由度并联机构,三转一移4自由度并联机构,以及移动和转动耦合的4自由度并联机构等等。
对于一种新型机构,能够准确、方便地分析其工作空间对于评定该机构的实际应用价值是十分有帮助的。本文介绍了求解少自由度并联机构工作空间的步骤,并以3、4自由度并联机构为重点详细阐述了其工作空间的构造过程。定义了描述工作空间的参数,讨论了运动副的布置对少自由度并联机构特别是3自由度并联机构工作空间的影响。
Comparing with the traditional 6-DOF (Degree of Freedom) parallel robot, a limited-DOF parallel robot has many advantages such as having less driving limbs, being simple in control, easy in manufacture, low in cost and so on. Recently, limited-DOF parallel manipulator has become a focus among the international mechanics and robotics fields. The main study of the limited-DOF parallel manipulator character includes the kinematics, singularity configurations, workspaces and so on.
Kinematics is an important part of mechanism analysis, and Jacobian matrix plays an important role in the kinematics analysis of parallel manipulators. The traditional Jacobian matrix of the limited-DOF parallel manipulator is not a square matrix, that is, the corresponding relationship between manipulator’s input and output is not satisfied. The essential reason for this phenomenon is that the elements of the moving platform output velocity are not independent. In this paper, the translational Jacobian matrix was used to denote the relationship between the linear velocity of the moving platform and the independent output velocity vector of the manipulator, while the rotational Jacobian matrix was used to denote the relationship between the angular velocity of the moving platform and the independent output velocity vector of the manipulator. Through the translational and rotational Jacobian matrices, an n×n square Jacobian matrix of n DOF (n<6) parallel manipulator was obtained, and the manipulator velocity was analyzed. On the basis of the translational and rotational Jacobian matrices, the translational and rotational Hessian matrices were obtained, which denoted the relationship between the linear and angular acceleration of the moving platform and the independent output acceleration vector of the manipulator, respectively. The acceleration of the limited-DOF parallel manipulator was further analyzed.
As a whole, a limited-DOF parallel robot has more complex kinematic characters than a 6-DOF parallel robot. Many special characters may show up because of the moving platform constraint, which leads to the lack of the limited-DOF parallel prototype and dissatisfaction of the practical wide requirement. Some 3-DOF and 4-DOF parallel manipulators were synthesized in this paper, including the parallel manipulator with a constrained limb, the limited-DOF parallel manipulator with less limbs and the parallel manipulator with a non-linear limb and so on. On the basis of the translational and rotational Jacobian matrices, the kinematics of these parallel manipulators was analyzed in detail.
Singularity configuration of the parallel manipulator is an important issue in the design, application and analysis of parallel robots. Jacobian algebra method is the most general and common method to study the parallel manipulator singularity configuration. According to the translational and rotational Jacobian matrices and the n×n square Jacobian matrix received from these two matrices, the limited-DOF parallel manipulators, especially the 3-DOF and 4-DOF parallel manipulators with different movement types were analyzed in this paper. The involved manipulators were the three-dimension translation parallel manipulator, the three-dimension rotation parallel manipulator, the 3-DOF parallel manipulator with mixed movement, the three-dimension translation and one-dimension rotation 4-DOF parallel manipulator, the three-rotation and one-dimension translation 4-DOF parallel manipulator, and the 4-DOF parallel manipulator with coupling translation and rotation.
Accurate and convenient workspace analysis is greatly helpful for the evaluation of a new manipulator. The general limited-DOF parallel manipulator workspace was solved; 3-DOF and 4-DOF parallel manipulators were used as examples to explain the detailed process. Parameters were defined to describe the workspace, the influence of joint distribution to limited-DOF parallel manipulators, especially to the 3-DOF parallel manipulator workspace was discussed.
运动学是机构分析的重要组成部分,Jacobian矩阵在并联机构的运动学分析当中占有十分重要的位置。传统意义的少自由度并联机构的Jacobian矩阵不是方阵,即机构的输入和输出之间不满足一一对应关系。这种现象出现的本质原因是动平台的各个输出速度分量并不是独立的。本文用移动Jacobian矩阵和转动Jacobian矩阵分别表示动平台的线速度和角速度与机构独立输出速度矢量之间的关系。通过移动和转动Jacobian矩阵,得到了n自由度并联机构(n<6)的n×n型Jacobian方阵并对其速度进行了分析;在移动和转动Jacobian矩阵的基础上,得到了表示动平台的线加速度和角加速度与机构的独立输出加速度矢量之间关系的移动和转动Hessian矩阵,进而对少自由度并联机构的加速度进行了分析。
少自由度并联机器人总体上具有比6自由度并联机器人更复杂的运动特性,特别是动平台的运动受到约束后会表现出很多特殊的性质,这使得少自由度并联机器人机型匮乏,与实际应用中的广泛要求相距甚远。本文综合出了几种3、4自由度并联机构,包括含约束分支的并联机构,驱动分支数小于自由度数的并联机构以及含非直线分支的并联机构等。以移动和转动Jacobian矩阵为基础,对所综合出来的机构进行了详细的运动学分析。
并联机构的奇异位形是并联机器人设计、应用和分析中的一个重要问题。Jacobian代数法是研究并联机构奇异位形最一般、最通用的方法。本文通过移动和转动Jacobian矩阵以及由这两个矩阵所得到的n×n型Jacobian方阵,对少自由度并联机构,特别是具有不同运动形式的3、4自由度并联机构的奇异位形进行了分析,包括三维移动并联机构,三维转动并联机构,具有混合运动的3自由度并联机构,三移一转4自由度并联机构,三转一移4自由度并联机构,以及移动和转动耦合的4自由度并联机构等等。
对于一种新型机构,能够准确、方便地分析其工作空间对于评定该机构的实际应用价值是十分有帮助的。本文介绍了求解少自由度并联机构工作空间的步骤,并以3、4自由度并联机构为重点详细阐述了其工作空间的构造过程。定义了描述工作空间的参数,讨论了运动副的布置对少自由度并联机构特别是3自由度并联机构工作空间的影响。
Comparing with the traditional 6-DOF (Degree of Freedom) parallel robot, a limited-DOF parallel robot has many advantages such as having less driving limbs, being simple in control, easy in manufacture, low in cost and so on. Recently, limited-DOF parallel manipulator has become a focus among the international mechanics and robotics fields. The main study of the limited-DOF parallel manipulator character includes the kinematics, singularity configurations, workspaces and so on.
Kinematics is an important part of mechanism analysis, and Jacobian matrix plays an important role in the kinematics analysis of parallel manipulators. The traditional Jacobian matrix of the limited-DOF parallel manipulator is not a square matrix, that is, the corresponding relationship between manipulator’s input and output is not satisfied. The essential reason for this phenomenon is that the elements of the moving platform output velocity are not independent. In this paper, the translational Jacobian matrix was used to denote the relationship between the linear velocity of the moving platform and the independent output velocity vector of the manipulator, while the rotational Jacobian matrix was used to denote the relationship between the angular velocity of the moving platform and the independent output velocity vector of the manipulator. Through the translational and rotational Jacobian matrices, an n×n square Jacobian matrix of n DOF (n<6) parallel manipulator was obtained, and the manipulator velocity was analyzed. On the basis of the translational and rotational Jacobian matrices, the translational and rotational Hessian matrices were obtained, which denoted the relationship between the linear and angular acceleration of the moving platform and the independent output acceleration vector of the manipulator, respectively. The acceleration of the limited-DOF parallel manipulator was further analyzed.
As a whole, a limited-DOF parallel robot has more complex kinematic characters than a 6-DOF parallel robot. Many special characters may show up because of the moving platform constraint, which leads to the lack of the limited-DOF parallel prototype and dissatisfaction of the practical wide requirement. Some 3-DOF and 4-DOF parallel manipulators were synthesized in this paper, including the parallel manipulator with a constrained limb, the limited-DOF parallel manipulator with less limbs and the parallel manipulator with a non-linear limb and so on. On the basis of the translational and rotational Jacobian matrices, the kinematics of these parallel manipulators was analyzed in detail.
Singularity configuration of the parallel manipulator is an important issue in the design, application and analysis of parallel robots. Jacobian algebra method is the most general and common method to study the parallel manipulator singularity configuration. According to the translational and rotational Jacobian matrices and the n×n square Jacobian matrix received from these two matrices, the limited-DOF parallel manipulators, especially the 3-DOF and 4-DOF parallel manipulators with different movement types were analyzed in this paper. The involved manipulators were the three-dimension translation parallel manipulator, the three-dimension rotation parallel manipulator, the 3-DOF parallel manipulator with mixed movement, the three-dimension translation and one-dimension rotation 4-DOF parallel manipulator, the three-rotation and one-dimension translation 4-DOF parallel manipulator, and the 4-DOF parallel manipulator with coupling translation and rotation.
Accurate and convenient workspace analysis is greatly helpful for the evaluation of a new manipulator. The general limited-DOF parallel manipulator workspace was solved; 3-DOF and 4-DOF parallel manipulators were used as examples to explain the detailed process. Parameters were defined to describe the workspace, the influence of joint distribution to limited-DOF parallel manipulators, especially to the 3-DOF parallel manipulator workspace was discussed.
引文
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